Bootstrapping Universal Asymptotics of Conformal Field Theory via Thermal Effective Action Citation Lee, Jaeha (2026) Bootstrapping Universal Asymptotics of Conformal Field Theory via Thermal Effective Action. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/00ej-3z79. https://resolver.caltech.edu/CaltechTHESIS:10032025-195133177 Abstract This thesis explores the asymptotic behavior of Conformal Field Theory (CFT) data at high energies using thermal effective action methods. Well-established results from two dimensions like Cardy formula, OPE coefficient asymptotics, and spin-refined partition function are extended to higher-dimensional theories. In the first part (Chapter 2), we study the asymptotic density of states formula to CFTs with continuous symmetries. Building on recent work that established the formula for finite groups, we derive universal results for compact Lie groups G. Together with checking on various theories, the formula is explained with thermal effective action. In the second part (Chapter 3), we develop the systematic exploration of thermal effective action methods for Cardy formula for the general dimension. Additionally, by introducing the "hot spot hypothesis," shrinking circles in complex geometries act as local thermal circles, and we extends the applicability of thermal effective action from simple fibrated manifolds to diverse geometries with extreme focusing structures, opening new avenues for computing CFT observables. In the third part (Chapter 4), we uncover a fractal-like structure in spin-refined partition functions in higher dimension using a cutting and gluing technique, decomposing the geometry into successive quotients and identifying Kaluza-Klein vortex defects. This reveals how thermal effective action methods remain robust even for discrete geometries and rational rotations. Our methods are purely field-theoretic and apply to both holographic and non-holographic theories. The results have implications for understanding black hole microstates in AdS/CFT, the statistics of OPE coefficients, and potential extensions of bootstrap axioms beyond traditional crossing symmetry. The thermal effective action framework studied here provides a systematic approach to computing high-energy asymptotics in CFTs, opening new avenues for exploring the structure of conformal field theories in dimensions greater than two. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: High Energy Theory, Conformal Field Theory Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Ooguri, Hirosi Group: Walter Burke Institute for Theoretical Physics Thesis Committee: Simmons-Duffin, David (chair) Ooguri, Hirosi Yu, Tony Yue Cheung, Clifford W. Defense Date: 16 September 2025 Funders: Funding Agency Grant Number Department of Energy (DOE) DE-SC0011632 Record Number: CaltechTHESIS:10032025-195133177 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:10032025-195133177 DOI: 10.7907/00ej-3z79 Related URLs: URL URL Type Description https://doi.org/10.1103/PhysRevD.107.026021 DOI Article adapted for chapter 2 https://doi.org/10.1007/JHEP03(2024)115 DOI Article adapted for chapter 3 https://doi.org/10.1007/JHEP11(2024)134 DOI Article adapted for chapter 4 ORCID: Author ORCID Lee, Jaeha 0000-0001-9124-450X Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17713 Collection: CaltechTHESIS Deposited By: Jaeha Lee Deposited On: 07 Oct 2025 21:32 Last Modified: 14 Oct 2025 19:58 Thesis Files PDF - Final Version See Usage Policy. 2MB Repository Staff Only: item control page

Bootstrapping Universal Asymptotics of Conformal Field Theory via Thermal Effective Action Thesis by Jaeha Lee In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2026 Defended September 16, 2025 ii © 2026 Jaeha Lee ORCID: 0000-0001-9124-450X All rights reserved except where otherwise noted iii ACKNOWLEDGEMENTS My years in Pasadena left me with the most important stories of my life. This period has been about both realizing and shaping myself. It has been possible only because of the people and environment around me, and in this acknowledgement, I want to express my gratitude to them all. I want to begin these acknowledgments with the place where I belonged. Everything about studying abroad was exciting but new and challenging at the same time. I appreciate the PMA department and Caltech for offering me a great program including kind and helpful staff. I especially enjoyed the cozy fourth floor in the Downs-Lauritsen building. The well-designed common areas sparked many discussions I enjoyed throughout my PhD. Also, without our fourth floor’s longtime secretary Carol, many things would not have been as smooth as they were. I would next like to express my appreciation to my advisor, Hirosi, for his endless support. He has respected my freedom to explore many areas, but whenever I needed his help, he became supportive, showing me practical but also warm care. I feel fortunate to have been advised by him. It was enjoyable to talk to professors at Caltech. My thesis committee chair David has also been an amazing collaborator. Collaboration with him was a fun and exciting experience. Without him, it would have been impossible to have a series of thermal effective action research papers. I also appreciate the Simons Collaboration on the Nonperturbative Bootstrap for funding and hosting bootstrap meetings. Tony and Sergei’s interest in new technology for theoretical research aligned with my later PhD studies. It has been nice to talk with them and openly explore ideas about AI. I also want to show my gratitude to Cliff and Anton for serving as my committee members for my thesis and candidacy. It was my pleasure to have discussions and collaborate with postdocs at Caltech. Nathan has been giving me great guidance on my physics research from the early days of my studies. Ning has always had many ideas and a genuine excitement for them. His passion for pursuing exciting things has motivated me. I also value my collaboration with Sridip; we worked on calculations together. Lastly, I am grateful to Temple and Yuya for the discussions we had. My friends in the community will not be forgotten. There are many happy memories with my friends on the floor. Also, without them, I would have been more confused iv and struggled with my research direction and decision-making. Important decisions in my life were made during my PhD. If it were not for Aike and Jerry, I would have been unable to face and be serious about my decisions. I also want to thank Aike for putting up with me as my officemate and for sharing her thoughtful perspective on the world. I wish all my friends every happiness. Yoojung has been the one who has witnessed my struggles during my PhD closely. Even though she has had her own struggles during her PhD, she has been understanding. Also, it has been amazing to see how much she enjoys and loves doing research in astronomy. I am sure the purest devotion I have ever seen toward her research will take her much further, probably to points I cannot imagine. I appreciate my friends from undergrad for being my most comfortable friends even though we cannot see each other often. Because of them, from time to time I could go back to my undergrad self without many worries or burdens. Also, because of these friends I could go through my most difficult times during my PhD. My sister, my parents, and Yoojung’s parents have shown warm interest in my studies. Even though I was bad at sharing my life abroad, their constant support and check-ins meant more than I showed. To everyone mentioned here, thank you for making this journey possible. v ABSTRACT This thesis explores the asymptotic behavior of Conformal Field Theory (CFT) data at high energies using thermal effective action methods. Well-established results from two dimensions like Cardy formula, OPE coefficient asymptotics, and spinrefined partition function are extended to higher-dimensional theories. In the first part (Chapter 2), we study the asymptotic density of states formula to CFTs with continuous symmetries. Building on recent work that established the formula for finite groups, we derive universal results for compact Lie groups 𝐺. Together with checking on various theories, the formula is explained with thermal effective action. In the second part (Chapter 3), we develop the systematic exploration of thermal effective action methods for Cardy formula for the general dimension. Additionally, by introducing the "hot spot hypothesis," shrinking circles in complex geometries act as local thermal circles, and we extends the applicability of thermal effective action from simple fibrated manifolds to diverse geometries with extreme focusing structures, opening new avenues for computing CFT observables. In the third part (Chapter 4), we uncover a fractal-like structure in spin-refined partition functions in higher dimension using a cutting and gluing technique, decomposing the geometry into successive quotients and identifying Kaluza-Klein vortex defects. This reveals how thermal effective action methods remain robust even for discrete geometries and rational rotations. Our methods are purely field-theoretic and apply to both holographic and nonholographic theories. The results have implications for understanding black hole microstates in AdS/CFT, the statistics of OPE coefficients, and potential extensions of bootstrap axioms beyond traditional crossing symmetry. The thermal effective action framework studied here provides a systematic approach to computing highenergy asymptotics in CFTs, opening new avenues for exploring the structure of conformal field theories in dimensions greater than two. vi PUBLISHED CONTENT AND CONTRIBUTIONS [1] Nathan Benjamin et al. “Angular fractals in thermal QFT”. In: JHEP 11 (2024), p. 134. doi: 10.1007/JHEP11(2024)134. arXiv: 2405.17562 [hep-th]. Contributions • Derived the form of the Kaluza-Klein voltex defects terms. • Performed calculations for free theory examples. . [2] Nathan Benjamin et al. “Universal asymptotics for high energy CFT data”. In: JHEP 03 (2024), p. 115. doi: 10.1007/JHEP03(2024)115. arXiv: 2306.08031 [hep-th]. Contributions • Derived the exact form of the generalized Cardy formula. • Performed calculations for all example theories for the generalized Cardy formula. • Derived the regime of validity of the generalized Cardy formula. • Derived subleading terms for the universal asymptotic formula for OPE coefficients. . [3] Monica Jinwoo Kang, Jaeha Lee, and Hirosi Ooguri. “Universal formula for the density of states with continuous symmetry”. In: Phys. Rev. D 107.2 (2023), p. 026021. doi: 10.1103/PhysRevD.107.026021. arXiv: 2206. 14814 [hep-th]. Contributions • Derived the exact form of the universal formula for asymptotic density of states in theories with symmetry of compact groups. • Performed calculations for all example theories. . vii TABLE OF CONTENTS Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Published Content and Contributions . . . . . . . . . . . . . . . . . . . . . . vi Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter II: Universal Formula for the Density of States with Continuous Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Spurion analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Expansion in characters . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Examples 1: 𝑈 (1) symmetry . . . . . . . . . . . . . . . . . . . . . 9 2.5 Examples 2: non-abelian symmetry . . . . . . . . . . . . . . . . . . 16 2.6 Stability of black hole with non-abelian hair . . . . . . . . . . . . . 21 Chapter III: Universal Asymptotics for High Energy CFT Data . . . . . . . . 25 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 The thermal effective action . . . . . . . . . . . . . . . . . . . . . . 29 3.3 The density of high-dimension states . . . . . . . . . . . . . . . . . 36 3.4 Density of states: examples . . . . . . . . . . . . . . . . . . . . . . 51 3.5 A "genus-2" partition function . . . . . . . . . . . . . . . . . . . . . 56 3.6 Genus-2 global conformal blocks . . . . . . . . . . . . . . . . . . . 72 3.7 OPE coefficients of heavy operators . . . . . . . . . . . . . . . . . . 86 3.8 Asymptotics of thermal 1-point functions . . . . . . . . . . . . . . . 99 3.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Appendix A Thermal two-point function of momentum generators . . . 109 Appendix B Scheme independence . . . . . . . . . . . . . . . . . . . . 114 Appendix C More on free theories . . . . . . . . . . . . . . . . . . . . 115 Appendix D Wilson coefficients for the 3D Ising model . . . . . . . . . 125 Appendix E The shadow transform of a three-point function at large Δ . 126 Appendix F Gluing factors . . . . . . . . . . . . . . . . . . . . . . . . 128 Appendix G Counting quantum numbers in the genus-2 block . . . . . . 132 Chapter IV: Angular Fractals in Thermal QFT . . . . . . . . . . . . . . . . . 135 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.2 Folding and unfolding the partition function . . . . . . . . . . . . . 138 4.3 Kaluza-Klein vortex defects . . . . . . . . . . . . . . . . . . . . . . 150 4.4 Fermionic theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 4.5 Free theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 4.6 Topological terms: example in 2D CFT . . . . . . . . . . . . . . . . 174 4.7 Holographic theories . . . . . . . . . . . . . . . . . . . . . . . . . . 175 4.8 Journey to 𝛽 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 viii 4.9 Nonperturbative corrections . . . . . . . . . . . . . . . . . . . . . . 181 4.10 Discussion and future directions . . . . . . . . . . . . . . . . . . . . 183 Appendix A Qualitative picture of the 3D Ising partition function . . . . 186 Appendix B Review of plethystic sums . . . . . . . . . . . . . . . . . . 187 Appendix C More examples for 4D and 6D CFTs . . . . . . . . . . . . 193 Appendix D More on nonperturbative terms . . . . . . . . . . . . . . . 195 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 1 Chapter 1 INTRODUCTION Conformal Field Theory (CFT) is a special type of Quantum Field Theory that possesses strong symmetry, conformal symmetry. Because of this symmetry, we can write down all correlation functions once the "CFT data" are given: the scaling dimensions and charges of primary operators, and the OPE coefficients. Furthermore, checking consistency conditions of the theory imposes strong constraints on this data [194]. Crossing symmetry is a representative consistency condition concerning the associativity of the Operator Product Expansion (OPE). Although the concept is simple, the constraints it provides are deep and their constraining power has not been exhausted. As researchers overcome more technical difficulties in checking this consistency condition, it yields increasingly powerful constraints on CFT data, including that of the critical 3D Ising model [60, 87, 149]. Another famous consistency condition is modular invariance in 2D. The partition function of a CFT on the torus remains unchanged regardless of how we set the time direction. Modular invariance in 2D CFT fundamentally constrains local data. For example, it determines the spectrum of minimal models and provides bounds on physical quantities such as the Hellerman bound and HKS bound [146]. The S-transformation of modular invariance is particularly useful for obtaining high-energy asymptotic data like the Cardy formula and OPE coefficients in certain limits. This is because it can translate high-temperature conditions from a Laplacetransformed perspective to the domination of a single state or single Virasoro block. How to perform similar analyses for higher-dimensional (d>3) CFTs has not been clearly understood, and my collaboration during graduate study was a journey to answer this question. In Chapter 2, we further tested the high-energy asymptotic formula (2.1) conjectured in [102], which describes the density of states for given energy and charges under symmetry. We extended the conjecture to continuous symmetry as shown in equation (2.4), and we checked the formula on theories invariant under both 𝑈 (1) symmetry and non-abelian symmetry using free theory and holographic CFT. In this work, we first used the thermal effective action to explore asymptotic CFT data. 2 The concept of the thermal effective action was explained in [14, 127]. If a CFT does not have any protected gapless modes, we generically do not expect compactification of the CFT on a thermal circle to be gapless. The compactified theory typically has a nonzero mass gap 𝑚 gap ∝ 𝑇 and a finite correlation length 𝜉 = 1/𝑚 gap . On the other hand, when we couple a gapped quantum field theory to a background field, the partition function of the theory has an effective description using a local effective action of the background field. Through these two steps, we can approximate the partition function of a metric-coupled CFT in 𝑑 dimensions with a thermal circle using a local effective action in 𝑑 − 1 dimensions. We call this the thermal effective action. See also [192] and the introduction sections of Chapter 3 and Chapter 4 for details. In Chapter 3, we calculate the partition function of CFTs on various manifolds using the thermal effective action, which gave us universal formulas for the asymptotics. We establish that the density of states follows the equation (3.74), with systematic corrections from higher-derivative terms. For heavy-heavy-heavy OPE coefficients, we construct a higher-dimensional analog of the genus-2 Riemann surface by gluing two three-punctured spheres 𝑆 𝑑 with cylinders. Through analysis of "hot spots" where thermal circles shrink, we derive asymptotic formulas for squared OPE coefficients, finding the equation (3.243). We also determine universal formulas for thermal one-point functions of heavy operators. In Chapter 4, we develop a cutting and gluing technique to compute spin-refined partition functions, revealing an intricate fractal-like structure in the partition function 𝑍 (𝛽, 𝜃) as a function of angular fugacity. Near every rational angle 𝜃 = 2𝜋 𝑝/𝑞, the partition function exhibits universal asymptotic behavior with the free energy scaling as 𝑓 /𝑞 𝑑−1 . 3 Chapter 2 UNIVERSAL FORMULA FOR THE DENSITY OF STATES WITH CONTINUOUS SYMMETRY 2.1 Introduction In [102], a simple formula is derived for the density of black hole microstates in theory with finite group gauge symmetry 𝐺. The formula states that, if we pick a random state from a uniform distribution of all states of the black hole, the probability of it being in a unitary irreducible representation 𝑅 of 𝐺 is 𝑃𝑅 = (dim𝑅) 2 , |𝐺 | where |𝐺 | is the number of elements in 𝐺 so that ∑︁ 𝑃𝑅 = 1 . (2.1) (2.2) 𝑅 It was also conjectured in the paper that the formula applies to any conformal field theory (CFT) on a sphere with finite group global symmetry 𝐺. This generalizes the result of [174] from two dimensions to arbitrary dimensions. The conjecture is verified in the context of free field theories and weakly coupled theories in [42], and a general derivation is presented in [160] using the result of [54]. See also [66] for earlier results on black holes with discrete gauge charges in specific models. In this paper, we generalize this result to the case where 𝐺 is a compact Lie group. Since |𝐺 | is infinite and 𝐺 has infinitely many unitary irreducible representations, equation (2.1) needs modifications. We show that, at high temperature and on a compact Cauchy surface, the probability 𝑃 𝑅 for a random state to be in a representation 𝑅 of 𝐺 is given by  dim 𝐺/2    𝑐 2 (𝑅) 4𝜋 2 exp − 𝑑−1 + · · · , (2.3) 𝑃 𝑅 = (dim 𝑅) 𝑏𝑇 𝑑−1 𝑏𝑇 where 𝑇 is the temperature, 𝑑 is the dimensions of the spacetime of the CFT, 𝑐 2 (𝑅) is the second Casimir of 𝑅, and · · · represents terms subleading in 1/𝑇. An important point is that 𝑏 is a positive constant independent of 𝑅 and 𝑇. For small representations, where 𝑐 2 (𝑅) ≪ 𝑇 𝑑−1 , the 𝑅-dependence of 𝑃 𝑅 is captured by the (dim 𝑅) 2 factor as in the finite group case (2.1). For large representations where 𝑐 2 (𝑅) ≫ 𝑇 𝑑−1 , 𝑃 𝑅 decays exponentially. 4 We derive equation (2.3) by calculating the twisted partition function, h i b 𝑍 (𝑇, 𝑔) = Tr 𝑈 (𝑔) 𝑒 −𝛽 𝐻 , (2.4) where the trace is taken over the CFT Hilbert space, 𝑈 (𝑔) is the action of 𝑔 ∈ 𝐺 b is the Hamiltonian. When 𝑔 = 1, it is the on the Hilbert space, 𝛽 = 1/𝑇, and 𝐻 standard partition function with the universal large 𝑇 behavior,   𝑑−1 𝑍 (𝑇, 𝑔 = 1) = exp 𝑎 𝑇 +··· , (2.5) for some constant 𝑎. In two dimensions, it is related to the Cardy formula with 𝑎 = 𝜋 2 (𝑐 𝐿 + 𝑐 𝑅 )/6 , (2.6) where 𝑐 𝐿 and 𝑐 𝑅 are the central charges in the left and right movers. We employ the spurion analysis for the theory obtained by dimensional reduction of the CFT on the thermal circle and show that the 𝑔 dependence of 𝑍 (𝑇, 𝑔) is of the form   𝑏 𝑑−1 𝑖𝜙 𝑑−1 𝑍 (𝑇, 𝑔 = 𝑒 ) = exp 𝑎 𝑇 − 𝑇 ⟨𝜙, 𝜙⟩ + · · · , (2.7) 4 where the inner product ⟨𝜙, 𝜙⟩ is given by the Killing form. In this description, the constant 𝑏 is related to the tension of the domain wall which generates the 𝑔twisted sector and therefore is positive. We also verify this formula by calculating 𝑏 for free field theories and for holographic conformal field theories. Since the twisted partition function 𝑍 (𝑇, 𝑔) is a class function of 𝑔, i.e., invariant under the conjugation 𝑔 → ℎ𝑔ℎ−1 for any ℎ ∈ 𝐺, we can expand 𝑍 (𝑇, 𝑔) in characters 𝜒𝑅 (𝑔) of unitary irreducible representations of 𝐺. We calculate the coefficients for the expansion of equation (2.7) and obtain  4𝜋 𝑍 (𝑇, 𝑔)/𝑍 (𝑇, 1) = 𝑏𝑇 𝑑−1  dim 𝐺/2 ∑︁  𝑐 2 (𝑅) dim 𝑅 · 𝜒𝑅 (𝑔) exp − 𝑑−1 + · · · 𝑏𝑇  . 𝑅 (2.8) Our main result (2.3) then follows. For 𝑑 = 1, equation (2.3) is derived for BF gauge theory coupled to Jackiw– Teitelboim gravity [129]. For 𝑑 = 2, the formula for 𝐺 = 𝑈 (1) is derived using the modular invariance of 2D CFTs [174]. Our results generalize this to 𝑑 ≥ 3 and to non-abelian 𝐺. The exponential suppression factor in equation (2.7) is also mentioned for free field theories in a note added to [42]. We note that the right-hand side of equation (2.8) is in the same form as that of the partition function of the 5 two-dimensional Yang–Mills theory with gauge group 𝐺 and the coupling constant proportional to 1/𝑇 (𝑑−1)/2 [30, 90, 166, 184, 205]. There may also be a connection between our results and the recent study of the entanglement entropy in the presence of a global symmetry [53]. In the holographic derivation of equation (2.8), we use the Einstein gravity coupled to the Yang–Mills theory with gauge group 𝐺 and a finite number of matter fields in anti-de Sitter space (AdS). When 𝐺 is non-abelian, there are two types of relevant bulk geometries besides the thermal AdS: black holes with and without non-abelian hair. Both bulk geometries obey the same boundary condition at the infinity of AdS. However, the former has genuinely non-abelian configurations of the gauge field, while the gauge field in the latter is commutative. There is extensive literature on such solutions (see [202, 204] for some reviews). One of the outstanding questions in this area has been whether solutions with non-abelian hair are thermodynamically stable. As we will show in this paper, the two types of solutions, with and without non-abelian hair, converge in the high temperature limit 𝑇 → ∞. We compute the 1/𝑇 corrections to their thermodynamical quantities for purely electric solutions and show that the black holes with non-abelian hair have lower free energies. This determines that the black holes with non-abelian hair are thermodynamically more stable. The coefficients 𝑎 and 𝑏 computed for free field theories and holographic CFTs are summarized in Table 2.1 below. When we have 𝑁 free scalars or 𝑁 free fermions, both 𝑎 and 𝑏 are proportional to 𝑁. In holographic CFTs, both 𝑎 and 𝑏 are proportional to ℓ 𝑑−1 /𝐺 𝑁 assuming 𝐺 𝑁 ∼ 𝑒 2 , where 𝐺 𝑁 is the Newton’s constant, 𝑒 is the gauge coupling constant, and ℓ is the curvature radius of AdS. Thus, in both the free field theories and holographic CFTs, 𝑎 and 𝑏 are proportional to the number of degrees of freedom of the system. 6 𝑎 𝑏 A free scalar with 𝐺 = 𝑈 (1) 2𝜁 (𝑑) 4𝜁 (𝑑 − 2) A free scalar in a representation 𝜌 of 𝐺 2𝜁 (𝑑) dim 𝜌 A free spinor with 𝐺 = 𝑈 (1) Holographic CFT 4𝜁 (𝑑 − 2) 𝑐 2 (𝜌) 𝑑 = 2 : 𝜁 (2) = 𝜋 2 /6 𝑑=3: 3𝜁 (3)   𝑑−1 4𝜋 𝑤 𝑑−1 ℓ 𝑑−1 𝑑 4𝑑𝐺 𝑁 dim 𝜌 dim 𝐺 1 16 log 2  4𝜋 𝑑  𝑑−2 4(𝑑 − 2)𝑤 𝑑−1 ℓ 𝑑−1 𝑒2 Table 2.1: The coefficients 𝑎 and 𝑏 in equation (2.5) for a variety of CFTs. For the free scalar, the results are for 𝑑 > 3. 𝑤 𝑑−1 is the area of the unit (𝑑 − 1)-sphere. The organization of this paper is as follows. In Section 2.2, we give a general argument for the large 𝑇 behavior in equation (2.7) using the spurion analysis for the theory obtained by dimensional reduction of the CFT on the thermal circle. In Section 2.4, we derive the large 𝑇 behavior when 𝐺 = 𝑈 (1) for free field theories and holographic CFTs. In Section 2.5, we generalize these results to a non-abelian group 𝐺. The holographic dual in this case involves the Yang–Mills theory with gauge group 𝐺, and we need to consider two types of black hole solutions: those with and without non-abelian hair. We show that the two solutions converge at high temperature and reproduce the behavior in equation (2.8). In Section 2.6, we discuss the theormodynamical stability of the black hole with non-abelian hair. 2.2 Spurion analysis Consider a 𝑑-dimensional CFT on a (𝑑 − 1)-dimensional compact Cauchy surface Σ𝑑−1 times the thermal circle 𝑆 1𝛽 at temperature 𝑇 = 1/𝛽. We assume that the CFT is invariant under a compact Lie group 𝐺. To calculate the twisted partition function (2.4), we use the approach of [14, 75, 126] and couple the CFT to a background gauge field 𝐴 with gauge group 𝐺.1 Upon dimensional reduction on 𝑆 1𝛽 , dynamical degrees of freedom acquire thermal masses. The low energy theory on Σ𝑑−1 is then described by a gauge field 𝑎 = 𝐴 |Σ𝑑−1 coupled to a scalar field 𝜙 in the adjoint representation of 𝐺, which is related to the holonomy of the gauge field around the 1 We thank David Simmons-Duffin for discussion on this approach. 7 thermal circle as ∮ ! 𝐴 ≡ 𝑒𝑖𝜙 . 𝑔 = exp 𝑖 (2.9) 𝑆 1𝛽 The low energy effective Lagrangian in (𝑑 − 1) dimensions has the derivative expansion,   L = trAdj 𝑇 𝑑−1 𝑉 (𝑒𝑖𝜙 ) + 𝑐 𝑇 𝑑−3 (D𝜙) 2 + 𝑔𝑌−2𝑀 𝐹 2 + · · · , (2.10) where the trace is taken over the adjoint representation, the scalar potential Tr𝑉 (𝑔) is a class function of 𝑔 as required by gauge invariance in 𝑑 dimensions, D is the covariant derivative, 𝐹 = 𝑑𝑎 + 𝑎 2 , and · · · are terms suppressed by 1/𝑇. The twisted partition function 𝑍 (𝑇, 𝑔) is obtained by setting 𝑔 = 𝑒𝑖𝜙 to be constant and 𝑎 = 0. Therefore, its 𝑔-dependence is captured by the potential term Tr𝑉 (𝑔) in the effective Lagrangian as     (2.11) 𝑍 (𝑇, 𝑔)/𝑍 (𝑇, 1) = exp −trAdj 𝑇 𝑑−1 𝑉 (𝑔) vol(Σ𝑑−1 ) + · · · . Now, we relate the potential Tr𝑉 (𝑔) to the tension of the domain wall which generates the 𝑔-twisted sector in the CFT Hilbert space. To do so, we note that the Lagrangian density (2.10) is of the same form for any smooth compact manifold Σ𝑑−1 , provided we use the metric of Σ𝑑−1 to write L in a diffeomorphism invariant way. In particular, we can choose Σ𝑑−1 = 𝑆˜1 × Σ𝑑−2 , with 𝑆˜1 having unit circumference and the thermal boundary condition, and compute 𝑉 (𝑔) for this geometry. By exchanging the thermal circle 𝑆 1𝛽 with 𝑆˜1 , we can interpret the twisted partition function 𝑍 (𝑇, 𝑔) as the untwisted partition function in the 𝑔-twisted sector on 𝑆 1𝛽 × Σ𝑑−2 with the twist along the 𝑆 1𝛽 direction [189]. Since we are computing the partition function of the CFT, we can rescale the spacetime so that the thermal circle 𝑆 1𝛽 has unit circumference and the volume of 𝑆˜1 × Σ𝑑−2 is proportional to 𝑇 𝑑−1 . In the limit   of 𝑇 → ∞, the exponent Tr 𝑇 𝑑−1 𝑉 (𝑔) vol(Σ𝑑−1 ) + · · · of equation (2.11) can be interpreted as the ground state energy of the 𝑔-twisted sector on 𝑆 1𝛽 × Σ𝑑−2 times the circumference 𝑇 of the rescaled 𝑆˜1 . Since we expect that the ground state energy of the 𝑔-twisted sector with 𝑔 ≠ 1 is higher than that of the untwisted ground state, Tr𝑉 (𝑔) should have the global minimum at 𝑔 = 1. Therefore, in the high temperature limit, 𝑍 (𝑇, 𝑔)/𝑍 (𝑇, 1) → 𝐶 (𝑇) 𝛿(𝑔, 1), 𝑇 → ∞, (2.12) 8 for some 𝐶 (𝑇), where 𝛿(𝑔, 1) is the delta-function on the group manifold 𝐺 localized at 𝑔 = 1. Since the delta-function can be expanded in terms of characters as ∑︁ 𝛿(𝑔, 1) = dim 𝑅 · 𝜒𝑅 (𝑔), (2.13) 𝑅 where the sum is over unitary irreducible representations of 𝐺 and we normalized the volume of 𝐺 to be 1, we conclude that the probability 𝑃 𝑅 for a random state to be in the representation 𝑅 is proportional to (dim 𝑅) 2 , for fixed 𝑅 in the limit of 𝑇 → ∞. This explains the (dim 𝑅) 2 factor in equation (2.3).   To reproduce the exp −𝑐 2 (𝑅)/(𝑏 𝑇 𝑑−1 ) factor in equation (2.3), we expand the potential Tr𝑉 (𝑔) around 𝑔 = 1. Since it is a class function of 𝑔, the expansion should take the form,   𝑏 (2.14) trAdj 𝑇 𝑑−1 𝑉 (𝑔 = 𝑒𝑖𝜙 ) vol(Σ𝑑−1 ) = constant + 𝑇 𝑑−1 ⟨𝜙, 𝜙⟩ + · · · . 4 The coefficient 𝑏 must be non-negative since the minimum of Tr𝑉 (𝑔) is at 𝑔 = 1. This reproduces equation (2.7). As we will show in Section 4.4, this is equivalent to equation (2.8) and therefore to equation (2.3). 2.3 Expansion in characters We have shown that the twisted partition function has the universal high temperature behavior,   𝑏 𝑑−1 𝑖𝜙 (2.15) 𝑍 (𝑇, 𝑔 = 𝑒 )/𝑍 (𝑇, 1) = exp − 𝑇 ⟨𝜙, 𝜙⟩ + · · · . 4 Since it is a class function of 𝑔, we can expand it in characters 𝜒𝑅 (𝑔). The purpose of this section is to find the expansion coefficients and derive equation (2.8). To do so, we use the fact that the left-hand side of (2.15) approximately solves the heat equation for 𝑇 ≫ 1 as #   "  𝑑−1  dim 𝐺/2 𝑏𝑇 𝑑 𝜕 𝑏𝑇 𝑍 (𝑇, 𝑔) +Δ ≃ 0, (2.16) 𝑑 − 1 𝜕𝑇 4𝜋 𝑍 (𝑇, 1) and obeys the initial condition,  𝑑−1  dim 𝐺/2 𝑏𝑇 𝑍 (𝑇, 𝑔) 4𝜋 𝑍 (𝑇, 1) = 𝛿(𝑔, 1) . (2.17) 𝑇=∞ Here Δ is the Laplace operator on the group manifold 𝐺. Since each character is an eigenstate of the Laplace operator, Δ𝜒𝑅 (𝑔) = −𝑐 2 (𝑅) 𝜒𝑅 (𝑔), (2.18) 9 and since characters make an orthonormal basis of class functions, { 𝜒𝑅 (𝑔)𝑒 −𝑐2 (𝑅)/(𝑏𝑇 gives the complete set of solutions to the heat equation. Therefore, we can expand,  𝑑−1  dim 𝐺/2   𝑏𝑇 𝑍 (𝑇, 𝑔) ∑︁ 𝑐 2 (𝑅) ≃ (2.19) 𝑑 𝑅 𝜒𝑅 (𝑔) exp − 𝑑−1 . 4𝜋 𝑍 (𝑇, 1) 𝑏 𝑇 𝑅 𝑑−1 ) To determine the expansion coefficient 𝑑 𝑅 , we use the initial condition (2.17), which can be written as ∑︁ 𝑑 𝑅 𝜒𝑅 (𝑔) = 𝛿(𝑔, 1). (2.20) 𝑅 Since 𝛿(𝑔, 1) = Í 𝑅 dim 𝑅 · 𝜒𝑅 (𝑔), the expansion coefficients are determined as 𝑑 𝑅 = dim 𝑅 , (2.21) and we obtain  4𝜋 𝑍 (𝑇, 𝑔)/𝑍 (𝑇, 1) = 𝑏𝑇 𝑑−1  dim 𝐺/2 ∑︁ 𝑅 2.4   𝑐 2 (𝑅) dim 𝑅 · 𝜒𝑅 (𝑔) exp − 𝑑−1 + · · · . 𝑏𝑇 (2.22) Examples 1: 𝑈 (1) symmetry In the remainder of the paper, we will study free field theories and holographic CFTs on 𝑆 1𝛽 × 𝑆 𝑑−1 and calculate the coefficient 𝑏 explicitly. The circumference of the thermal circle 𝑆 1𝛽 is 𝛽, and the radius of the Cauchy surface 𝑆 𝑑−1 is normalized to be 1. We begin by studying CFTs with 𝐺 = 𝑈 (1). Each state in the Hilbert space can be labeled by a charge 𝑄, and the conjectured formula takes the form, √︂     4𝜋𝑏 𝑄2 1 𝑄2 𝑃𝑄 = , exp − 1+O . (2.23) 𝑇 𝑇 2𝑑−4 𝑇 𝑑−1 𝑏 𝑇 𝑑−1 We verify this by calculating the grand canonical partition function with an imaginary chemical potential 𝜇 = 𝑖𝑇 𝜃, h i b b 𝑍 (𝑇, 𝜇 = 𝑖𝑇 𝜃) = Tr 𝑒 −𝛽 𝐻+𝑖𝜃 𝑄 . (2.24) b is quantized in such a way that the field with the smallest non-zero We assume that 𝑄 𝑈 (1) charge has charge 1. In the limit of large 𝑇 and small 𝜇, we show        1 𝑏 𝑑−3 2 2 1 𝑑−1 𝑍 (𝑇, 𝜇) = exp 𝑎𝑇 1+O + 𝑇 𝜇 1+O 𝜇 , , (2.25) 𝑇 4 𝑇 for some constants 𝑎 and 𝑏. The Fourier transformation of this formula with respect to 𝜃 = −𝑖𝛽𝜇 gives the canonical partition function, which leads to equation (2.23). }𝑅 10 Free field theory Free scalar theories: Consider a massless complex free scalar field 𝜙 in 𝑑 spacetime dimensions.2 We b such that 𝜙 has charge 1. For such a theory on normalize the 𝑈 (1) generator 𝑄 R×𝑆 𝑑−1 , the grand canonical partition function with an imaginary chemical potential is given by [165], "∞ # ∑︁ 𝑒 −𝑛𝛽 𝑑−2 2 (1 − 𝑒 −2𝑛𝛽 ) 𝑍scalar (𝑇, 𝜇 = 𝑖𝑇 𝜃) = exp cos(𝑛𝜃) . (2.26) −𝑛𝛽 ) 𝑑 𝑛 (1 − 𝑒 𝑛=1 As we are interested in the high temperature limit, that is, when 𝜃 = −𝑖𝛽𝜇 is small, we first expand the exponent in powers of 𝜃 as "∞ # ∑︁ 𝑍scalar (𝑇, 𝜇) = exp 𝐶 𝑘 𝜃 2𝑘 , (2.27) 𝑘=0 where the coefficient 𝐶 𝑘 is given by ∞ (−1) 𝑘 ∑︁ 2𝑘−1 − (𝑑−2) 𝑛𝛽 (1 − 𝑒 −2𝑛𝛽 ) 𝑛 𝑒 2 𝐶𝑘 = . (2𝑘)! 𝑛=1 (1 − 𝑒 −𝑛𝛽 ) 𝑑 (2.28) At high temperature, one might think that the sum over 𝑛 in equation (2.28) can be approximated by an integral over 𝑥 = 𝑛𝛽 as ∫ −2𝑥 ) (−1) 𝑘 2𝑘 ∞ 𝑥 (1 − 𝑒 2𝑘−1 − 𝑑−2 2 . (2.29) 𝐶𝑘 ≈ 𝑇 𝑓 (𝑥)𝑑𝑥, 𝑓 (𝑥) = 𝑥 𝑒 (2𝑘)! (1 − 𝑒 −𝑥 ) 𝑑 0 However, we need to be careful when 2𝑘 ≤ 𝑑 − 1 as 𝑓 (𝑥) is singular at 𝑥 = 0 and the integral approximation will fail when 𝑥 is small. To take this into account, we introduce a cutoff at some small value 𝑥 0 and use the integral approximation only for 𝑥 > 𝑥 0 . The terms in the summation in equation (2.28) are not converted to integral form when 𝑛 is such that 𝑛𝛽 < 𝑥0 . Taking this singular behavior into account, the correct approximation is ! ∫ ∞ 𝑥 0𝑇 ∑︁ (−1) 𝑘 𝐶𝑘 ≈ 2𝑇 𝑑−1 𝑛2𝑘−𝑑 + 𝑇 2𝑘 𝑑𝑥 𝑓 (𝑥) . (2.30) (2𝑘)! 𝑥 0 𝑛=1 2 We generally assume that 𝑑 > 3 due to certain subtleties with massless scalar fields in two and three dimensions which we discuss later. 11 It is straightforward to show that equation (2.30) is independent of 𝑥 0 for large values of 𝑇. In this way, we find that the coefficients 𝐶 𝑘 can be approximated as  ∫ ∞  (1 − 𝑒 −2𝑥 ) (−1) 𝑘  2𝑘−1 − 𝑑−2  2 𝑥 𝑇 2𝑘 𝑑𝑥 𝑥 𝑒   −𝑥)𝑑  (2𝑘)! (1 − 𝑒 0       (−1) 𝑘   2 log 𝑇 + 2𝛾 + 2 log 𝑥0   (2𝑘)!     ∫ ∞  −2𝑥 )    𝑥 (1 − 𝑒 2𝑘−1 − 𝑑−2 2 𝑇 𝑑−1 + 𝑑𝑥 𝑥 𝑒 𝐶𝑘 ≈ −𝑥)𝑑 (1 − 𝑒  𝑥 0     (−1) 𝑘    2𝜁 (𝑑 − 2𝑘)𝑇 𝑑−1   (2𝑘)!   !  ∫ ∞  2𝑘−𝑑+1 𝑘 −2𝑥 )  2𝑥 𝑑−2 (−1) (1 − 𝑒  0   − 𝑇 2𝑘 + 𝑑𝑥 𝑥 2𝑘−1 𝑒 − 2 𝑥   (2𝑘)! 𝑑 − 2𝑘 − 1 (1 − 𝑒 − 𝑥 ) 𝑑 𝑥 0  2𝑘 ≥ 𝑑, (2.31a) 2𝑘 = 𝑑 − 1, (2.31b) 2𝑘 < 𝑑 − 1. (2.31c) The constant 𝛾 appearing in equation (2.31b) is the Euler–Mascheroni constant. At 𝜃 = 0, the partition function is 𝑍scalar (𝑇, 0) = 𝑒𝐶0 . Since equation (2.31c) gives 𝐶0 = 2𝜁 (𝑑) 𝑇 𝑑−1 , the coefficient 𝑎 in equation (2.5) is given by 𝑎 = 2𝜁 (𝑑) for the massless free scalar. For 𝑑 > 3, equation (2.31c) gives 𝐶1 ≈ 𝜁 (𝑑 − 2) 𝑇 𝑑−1 , (2.32) where we ignore the second term of equation (2.31c), since it is subleading in 1/𝑇. Thus we find that the grand canonical partition function is   𝑍scalar (𝑇, 𝜇) ≈ exp 𝜁 (𝑑 − 2) 𝑇 𝑑−3 𝜇2 (1 + O (𝜇2 , 1/𝑇)) 𝑍scalar (𝑇, 0) . (2.33) In summary, the grand canonical partition function of the massless free complex scalar field theory in 𝑑 > 3 demonstrates the universal behavior at high temperature as in equation (2.25), with constants 𝑎 = 2𝜁 (𝑑) , 𝑏 = 4𝜁 (𝑑 − 2) . (2.34) When 𝑑 = 3, we use equation (2.31b) to obtain   𝑍scalar (𝑇, 𝜇) ≈ exp (log 𝑇 + 2.96351...)𝜇2 (1 + O (𝜇2 , 1/𝑇)) 𝑍scalar (𝑇, 0) . (2.35) However, the massless scalar field at 𝑑 = 3 does not make sense at finite temperature since it has the same infrared issue as that of the massless scalar field at 𝑑 = 2. We believe that the appearance of the log 𝑇 singularity is a reflection of the infrared pathology in this case. 12 Free spinor theories: For the massless scalar field, we cannot consider theories in 𝑑 = 2, 3 due to the infrared problem. As it is good to also have an example in these dimensions, we consider the theory of a free spinor field. In two dimensions, the grand canonical partition function of a free complex Weyl spinor is given by 𝑍spinor (𝑇, 𝜇 = 𝑖𝑇 𝜃) = ∞ Ö 1 1 (1 + 𝑒 −𝛽(𝑛− 2 ) 𝑒𝑖𝜃 )(1 + 𝑒 −𝛽(𝑛− 2 ) 𝑒 −𝑖𝜃 ) . (2.36) 𝑛=1 We can transform this into the plethystic form as "∞ #  ∑︁  1 1 𝑍spinor (𝑇, 𝜇 = 𝑖𝑇 𝜃) = exp log(1 + 𝑒 −𝛽(𝑛− 2 ) 𝑒𝑖𝜃 ) + log(1 + 𝑒 −𝛽(𝑛− 2 ) 𝑒 −𝑖𝜃 ) # " 𝑛=1 ∞  ∑︁ (−1) 𝑚  1 1 𝑒 −𝛽𝑚(𝑛− 2 ) 𝑒𝑖𝑚𝜃 + 𝑒 −𝛽𝑚(𝑛− 2 ) 𝑒 −𝑖𝑚𝜃 = exp − 𝑚 𝑛,𝑚=1 " ∞ # ∑︁ (−1) 𝑚 cos(𝑚𝜃) = exp − . 𝑚𝛽 𝑚 sinh 𝑚=1 2 (2.37) As in the free scalar case, we expand the exponent of the partition function in 𝜃 as "∞ # ∑︁ 𝑍spinor (𝑇, 𝑖𝜇) = exp 𝐷 𝑘 𝜃 2𝑘 , (2.38) 𝑘=0 for some coefficients 𝐷 𝑘 . We find that ∞ ∞ 1 ∑︁ 𝑚 1 ∑︁ ©­ 2𝑛 2𝑛 − 1 ª® 𝑚   , − 𝐷1 = − (−1) = − (2.39) 2 𝑚=1 2 𝑛=1 ­ sinh(𝑛𝛽) sinh 2𝑛−1 𝛽 ® sinh 𝑚𝛽 2 2 « ¬ where we split the series into 𝑚 = 2𝑛 and 𝑚 = 2𝑛 − 1 terms and sum them as pairs, which is valid as 𝐷 1 converges due to the hyperbolic sine function in the denominator. At high temperature, we can approximate the summation as an integration over 𝑥 = 𝑛𝛽: ∫ ∫ T2 ∞ 𝑇 𝑥 T2 ∞ 𝑑𝑥( 𝑓 (𝑥 + 𝛽) − 𝑓 (𝑥)) ≈ − 𝑑𝑥 𝑓 ′ (𝑥) 𝛽 ≈ , 𝑓 (𝑥) = . 𝐷1 ≈ − 4 0 4 0 4 sinh 𝑥 (2.40) Similarly, ∞ ∑︁ (−1) 𝑚 𝐷0 = − . (2.41) 𝑚𝛽 𝑚 sinh 𝑚=1 2 13 In this case, we need the cutoff 𝑥0 to covert the sum into an integral as 𝐷 0 ≈ −2𝑇 𝑥 0𝑇 ∑︁ (−1) 𝑚 𝑚2 𝑚=1 𝑇 − 2 ∫ ∞ 𝑑𝑥𝑔′ (𝑥) ≈ 𝜁 (2)𝑇, 𝑔(𝑥) = 𝑥0 where we used the zeta function identity − Í∞ 𝑚=1 1 , 𝑥 sinh 2𝑥 (2.42) (−1) 𝑚 = 𝜁 (2) 2 . 𝑚2 Let us turn to 𝑑 = 3, where the grand canonical partition function of the free spinor theory is 𝑍spinor (𝑇, 𝜇 = 𝑖𝑇 𝜃) = ∞ Ö (1 + 𝑒 −𝛽(𝑛+1) 𝑒𝑖𝜃 ) 2𝑛+1 (1 + 𝑒 −𝛽(𝑛+1) 𝑒 −𝑖𝜃 ) 2𝑛+1 (2.43) 𝑛=0 " = exp − ∞ ∑︁ (−1) 𝑚 𝑒 − 𝑚𝛽 2 coth 𝑚𝛽 2 sinh 𝑚𝛽 2 𝑚 𝑚=1 # cos(𝑚𝜃) . (2.44) Expanding the exponent in powers of 𝜃, we find the coefficients to be ∫ ∞ 𝑥 1 − 𝑥 coth 2 4 𝑇 ′ 2 𝑒 𝑑𝑥 𝑓 (𝑥) ≈ 3𝜁 (3), 𝑓 (𝑥) = 𝐷0 ≈ − (−1) + 𝑥, 3 𝛽2 2 𝑥 sinh 𝑚 𝑥 0 2 𝑚=1 ∫ ∞ 𝑥 0𝑇 ∑︁ coth 2𝑥 𝑇 𝑚 4 ′ 2 − 2𝑥 𝐷1 ≈ − (−1) 𝑑𝑥𝑔 (𝑥) ≈ 4𝑇 log 2, 𝑔(𝑥) = 𝑥𝑒 + 𝑥 , 2 2 sinh 𝑚𝛽 𝑥 0 2 𝑚=1 𝑥 0𝑇 ∑︁ 𝑚 (2.45) where we used zeta function identities − 43 𝜁 (3). Í∞ 𝑚=1 (−1) 𝑚 𝑚 = log 2 and Combining these results, we find    1 2  2   exp  𝜇 (1 + 𝑂 (𝜇 , 1/𝑇)) 𝑍spinor (𝑇, 0) 4𝑇 𝑍spinor (𝑇, 𝜇) ≈      exp 4 log 2 𝜇2 (1 + 𝑂 (𝜇2 , 1/𝑇)) 𝑍spinor (𝑇, 0)  Í∞ 𝑚=1 (−1) 𝑚 𝑚3 = 𝑑 = 2 , (2.46a) 𝑑 = 3 , (2.46b) where the partition functions at 𝛽𝜇 = 0 for both dimensions are given by ( 𝑒 𝜁 (2)𝑇 𝑑 = 2, 𝑍spinor (𝑇) = 𝑍spinor (𝑇, 0) ≈ 2 𝑒 3𝜁 (3)𝑇 𝑑 = 3 . (2.46c) (2.46d) Equations (2.46c) and (2.46d) show that the free Weyl spinor theory also demonstrates the universal behavior in equation (2.25) at high tempearture with the coefficients 𝜋2 𝑎 = 𝜁 (2) = , 𝑏 = 1, (2.47) 6 14 for 𝑑 = 2, and 𝑎 = 3𝜁 (3) , 𝑏 = 16 log 2 , (2.48) for 𝑑 = 3. For 𝑑 = 2, the Cardy formula gives 𝑎 = 𝜋 2 (𝑐 𝐿 + 𝑐 𝑅 )/6, and the above value of 𝑎 is consistent with (𝑐 𝐿 , 𝑐 𝑅 ) = (1, 0) for the complex Weyl spinor. As expected, the result at 𝑑 = 3 is free from the log 𝑇 singularity we saw for the free scalar field in equation (2.35). Holographic CFT We now consider a holographic CFT, whose bulk theory is described at low energy in terms of the Einstein gravity coupled to the Maxwell field and a finite number of matter fields in AdS𝑑+1 . The action of the theory is given by     ∫ 𝑑 (𝑑 − 1) 1 1 2 𝑑+1 √ 𝑅+ (2.49) − 2𝐹 + ··· , 𝐼= 𝑑 𝑥 −𝑔 16𝜋𝐺 𝑁 ℓ2 4𝑒 where · · · represents matter field terms. The curvature radius ℓ is related to the cosmological constant as Λ = −𝑑 (𝑑 − 1)/2ℓ 2 . To calculate the grand canonical partition function, we impose the boundary condition that the boundary geometry is 𝑆 1𝛽 × 𝑆 𝑑−1 and the gauge field 𝐴 has the holonomy around the thermal circle 𝑆 1𝛽 at the boundary given by ! ∮ exp 𝑖 𝐴𝜏 = 𝑒 𝛽𝜇 , (2.50) 𝑆 1𝛽 where 𝜇 is identified with the chemical potential of the boundary CFT. We solve the Einstein and Maxwell equations assuming the spherical symmetry on 𝑆 𝑑−1 and setting all other matter fields to zero. There are two classical solutions under these conditions; one is the thermal AdS and the other is the AdS Reissner–Nordstrom (RN) black hole. At high temperature, the RN solution is dominant [56, 57]. The RN solution can be written in static coordinates as 𝑚 𝑣𝑞 2 𝑟2 𝑑𝑟 2 𝑑𝑠2 = 𝑉 (𝑟)𝑑𝜏 2 + + 𝑟 2 𝑑Ω2𝑑−1 , 𝑉 (𝑟) = 1 − 𝑑−2 + 2𝑑−4 + 2 , 𝑉 (𝑟) 𝑟 𝑟 ℓ √︄ ! 𝑑−1 𝑞 4𝜋𝐺 𝑁 𝑞 𝐴 = −𝑖 − 𝑑𝜏, 𝑣 = , 2(𝑑 − 2) 𝑟 𝐻𝑑−2 𝑟 𝑑−2 𝑒2 (2.51a) (2.51b) where 𝑚 and 𝑞 are related to the ADM mass and the charge of the black hole [56, 155, 183]. This solution has its ADM mass, charge, temperature, and entropy given 15 by ! 2 𝑟𝐻 (𝑑 − 1)𝑤 𝑑−1 𝑑−2 𝑣𝑞 2 𝑀= 𝑟 𝐻 1 + 2𝑑−4 + 2 , 16𝜋𝐺 𝑁 𝑙 𝑟𝐻   √︁ 𝑤 𝑑−1 𝑄 = 2(𝑑 − 1)(𝑑 − 2) 𝑣𝑞, 8𝜋𝐺 𝑁 ! 𝑑−2 𝑣𝑞 2 𝑟𝐻 𝑑 , 𝑇= 1 − 2𝑑−4 + 4𝜋𝑟 𝐻 4𝜋𝑙 2 𝑟𝐻 𝑤 𝑑−1 𝑑−1 𝑆= 𝑟 , 4𝐺 𝑁 𝐻 (2.52a) (2.52b) (2.52c) (2.52d) where 𝑤 𝑑−1 is the surface area of the unit (𝑑 − 1)-sphere, and the horizon radius 𝑟 𝐻 is the largest real positive root of 𝑉 (𝑟) [1, 11, 57, 106]. The chemical potential of the black hole system is related to the charge 𝑄 as √︄ 𝑑−1 𝑞 𝑄 𝑒2 𝜇= = . (2.53) 𝑑−2 𝑑−2 2(𝑑 − 2) 𝑟 𝐻 (𝑑 − 2)𝑤 𝑑−1 𝑟 𝐻 By the AdS/CFT correspondence, the grand canonical partition function of the CFT can be calculated using the Euclidean action for this solution. At high temperature, the horizon radius 𝑟 𝐻 of the stable black hole grows linearly in the temperature as 𝑇≈ 𝑟𝐻 𝑑 (1 − 𝑋) , 4𝜋ℓ 2 𝑋= 𝑑 − 2 𝑣𝑞 2 ℓ 2 , 2𝑑−2 𝑑 𝑟𝐻 (2.54) where we keep 𝑋 as small which is equivalent to small |𝜇| approximation in free field calculation. The grand potential Φ(𝑇, 𝜇) is related to the grant canonical partition function as 𝑍 𝐴𝑑𝑆 (𝑇, 𝜇) = 𝑒 −𝛽Φ(𝑇,𝜇) , (2.55) and is given by the Euclidean action of the RN solution,   𝑤 𝑑−1𝑟 𝐻𝑑 𝑑 1+ Φ(𝑇, 𝜇) = 𝑀 − 𝑇 𝑆 − 𝜇𝑄 ≈ − 𝑋 . 𝑑−2 16𝜋𝐺 𝑁 ℓ 2 (2.56) Using 4𝜋ℓ 2 𝑇 , 𝑟𝐻 = 𝑑 1−𝑋 𝑑 (𝑑 − 2) 2 𝑣𝜇2 𝑋= 2 2 + 𝑂 (𝑋 2 ), 2 8𝜋 ℓ (𝑑 − 1) 𝑇 (2.57) we find   𝑑−2 𝑤 𝑑−1 (4𝜋ℓ 2 /𝑑) 𝑑 𝑑−1 𝑤 𝑑−1 (𝑑 − 2) 4𝜋ℓ 2 −𝛽Φ(𝑇, 𝜇) ≈ 𝑇 + 𝜇2𝑇 𝑑−3 . 2 2 𝑑 16𝜋𝐺 𝑁 ℓ 𝑒 (2.58) 16 Rescaling the temperature as ℓ𝑇 → 𝑇, the grand canonical partition function of the dual CFT on the sphere with unit radius is given by "   𝑑−1  𝑑−1 # 𝑑−1 4𝜋 ℓ 𝑑 (𝑑 − 2)ℓ 𝑍𝐶𝐹𝑇 (𝑇, 𝜇) ≈ exp 𝑤 𝑑−1 𝑇 𝑑−1 + 𝜇2𝑇 𝑑−3 . (2.59) 𝑑 4𝑑𝐺 𝑁 4𝜋𝑒 2 This determines the coefficients 𝑎 and 𝑏 of equation (2.25) in this case as  4𝜋 𝑎= 𝑑 2.5  𝑑−1 𝑤 𝑑−1 ℓ 𝑑−1 , 4𝑑𝐺 𝑁  4𝜋 𝑏= 𝑑  𝑑−2 4(𝑑 − 2)𝑤 𝑑−1 ℓ 𝑑−1 . 𝑒2 (2.60) Examples 2: non-abelian symmetry When 𝐺 is non-abelian, we utlize the fact that the twisted partition function 𝑍 (𝑇, 𝑔) is a class function invariant under the conjugation 𝑔 → ℎ𝑔ℎ−1 for any ℎ. This allows us to restrict 𝑔 to the maximum torus of 𝐺 and simplify our calculation. In both free field theories and holographic CFTs, we find   𝑏 𝑑−1 𝑖𝜙 𝑍 (𝑇, 𝑔 = 𝑒 ) = exp − 𝑇 ⟨𝜙, 𝜙⟩ + · · · 𝑍 (𝑇, 𝑔 = 1), (2.61) 4 where 𝑔 = 𝑒 𝐻 and the constant 𝑏 depends on the theory but not on 𝑔 or 𝑇. In particular, the partition function 𝑍 (𝑇, 𝑔) is peaked at 𝑔 = 1. In fact, it is related to a solution to the heat equation on the group manifold 𝐺 with the diffusion time related to 1/𝑇 𝑑−1 . This will enable us to expand the partition function in characters of 𝐺 to obtain equation (2.8). Massless free scalar Suppose a compact Lie group 𝐺 has a faithful unitary representation 𝜌 with dim 𝜌 = 𝑛. Consider 𝑛 massless scalar fields in 𝑑 dimensions. Though the theory has a larger symmetry of 𝑂 (𝑛), we focus on its 𝐺 subgroup. We would like to calculate the finite temperature partition function of this theory with an insertion of 𝑔 ∈ 𝐺 as h i b 𝑍 (𝑇, 𝑔) = Tr 𝑈 (𝑔)𝑒 −𝛽 𝐻 . (2.62) Since 𝑍 (𝑇, 𝑔) is a class function of 𝑔, without loss of generality, 𝑔 can be restricted to the maxim torus of 𝐺. In this case, 𝑈 (𝑔) acts as a multiplication of a phase factor on each of the scalar fields. We can then apply equation (2.26) for 𝐺 = 𝑈 (1) to each scalar field and assemble the results to obtain "∞ # 𝑛 ∗ 𝑛 ∑︁ 𝑒 − 𝑑−2 2 𝑛𝛽 𝜒 𝜌 (𝑔 ) + 𝜒 𝜌 (𝑔 ) (1 − 𝑒 −2𝑛𝛽 ) 𝑍scalar (𝑇, 𝑔) = exp , (2.63) −𝑛𝛽 ) 𝑑 𝑛 2 (1 − 𝑒 𝑛=1 17 where 𝜒𝜌 is the character of the representation 𝜌 and 𝜒𝜌∗ is that for its conjugate. Writing 𝑔 = 𝑒𝑖𝜙 and expanding in powers of 𝜙,   𝜒𝜌 (𝑔 𝑛 ) + 𝜒𝜌∗ (𝑔 𝑛 ) = tr 𝜌 1 − 𝑛2 𝜙2 + · · · 2 dim 𝜌 = dim 𝜌 − 𝑐 2 (𝜌)⟨𝜙, 𝜙⟩𝑛2 + · · · , (2.64) dim 𝐺 where tr 𝜌 is the trace over the representation 𝜌 and ⟨𝜙, 𝜙⟩ = trAdj 𝜙2 . We can repeat the calculation of 𝐺 = 𝑈 (1) in Section 2.4 to obtain   𝑖𝜙 𝑑−1 dim 𝜌 𝑐 2 (𝜌)⟨𝜙, 𝜙⟩ + · · · 𝑍scalar (𝑇, 𝑔 = 1), 𝑍scalar (𝑇, 𝑒 ) ≈ exp −𝜁 (𝑑 − 2) 𝑇 dim 𝐺 (2.65) where we assumed 𝑑 > 3. Holographic CFT Consider a holographic CFT in 𝑑 dimensions, whose bulk theory is described in low energy in terms of the Einstein gravity coupled to the Yang–Mills field with gauge group 𝐺 and a finite number of matter fields in AdS𝑑+1 . The action of the theory is given by     ∫ 1 𝑑 (𝑑 − 1) 1 𝑑+1 √ 𝐼= 𝑑 𝑥 −𝑔 (2.66) 𝑅+ − 2 ⟨𝐹, 𝐹⟩ + · · · , 16𝜋𝐺 𝑁 𝑙2 4𝑒 where 𝐹 is in the Lie algebra of gauge group 𝐺 and · · · represents matter field terms. To calculate the grand canonical partition function, we impose the boundary condition that the boundary geometry is 𝑆 1𝛽 × 𝑆 𝑑−1 and the gauge field 𝐴 𝜇 has the holonomy around the thermal circle 𝑆 1𝛽 as ! ∮ 𝐴 = 𝑒 𝛽𝜇 = 𝑔. P exp 𝑖 (2.67) 𝑆 1𝛽 We assume that the solution is spherically symmetric on 𝑆 𝑑−1 , and all the other matter fields are set to zero. We calculate the field strength and the stress-energy tensor as     1 1 𝛼 𝛼𝛽 𝐹𝜇𝜈 = 𝜕𝜇 𝐴𝜈 − 𝜕𝜈 𝐴 𝜇 − 𝑖 𝐴 𝜇 , 𝐴𝜈 , 𝑇𝜇𝜈 = 2 ⟨𝐹𝜇𝛼 , 𝐹𝜈 ⟩ − 𝑔 𝜇𝜈 ⟨𝐹𝛼𝛽 , 𝐹 ⟩ . 4 𝑒 (2.68) There are three classical solutions under these conditions. The first is the thermal AdS,   Λ𝑟 2 𝑑𝑟 2 2 𝑑𝑠 = 1 − 𝑑𝜏 2 + + 𝑟 2 𝑑Ω2𝑑−1 , 𝐴 = −𝑖𝜇 𝑑𝜏 . (2.69) Λ𝑟 2 3 1− 3 18 The second makes use of the 𝑈 (1) RN solution (2.49), 𝑑𝑠𝑈2 (1) and 𝑎 𝜇 with the chemical potential 𝜇𝑈 (1) , by the substitution, 𝑑𝑠2 = 𝑑𝑠𝑈2 (1) , 𝐴𝜇 = 𝜙 𝑎𝜇 , ⟨𝜙, 𝜙⟩ 1/2 𝜇= 𝜙 𝜇𝑈 (1) . ⟨𝜙, 𝜙⟩ 1/2 (2.70) Since 𝐻 commutes with itself, the Yang–Mills equation for 𝐴 𝜇 reduces to the Maxwell equations for 𝑎 𝜇 . The rescaling by ⟨𝜙, 𝜙⟩ −1/2 is needed to match the stress energy tensors of both systems. The third is a genuinely non-abelian solution. A dyonic black hole solution with 𝑆𝑈 (𝑁) hair is known in 𝐴𝑑𝑆4 [191]. Here, we construct a purely electric black hole solution with 𝑆𝑈 (2) hair with the following ansatz [28] 𝑑𝑠2 = −𝜇(𝑟)𝜎(𝑟) 2 𝑑𝑡 2 + 𝑑𝑟 2 + 𝑟 2 𝑑𝜃 2 + 𝑟 2 sin2 𝜃𝑑𝜙2 , 𝜇(𝑟) 2𝑚(𝑟) Λ𝑟 2 − , 𝑟 3  𝜏3 𝜏1 𝜏3 𝜏2  𝐴 𝜇 = ℎ(𝑟) 𝑑𝑡 + 𝑤(𝑟) 𝑑𝜃 + cot𝜃 + 𝑤(𝑟) sin𝜃𝑑𝜙 , 2 2 2 2 𝜇(𝑟) = 1 − (2.71) where we use Pauli matrices 𝜏1,2,3 as generators of the Lie algebra of 𝑆𝑈 (2) and the inner product is defined as twice the trace of two elements’ multiplication. The 𝐴𝑑𝑆 boundary condition requires 𝜎(𝑟 → ∞) = 1. The functions, 𝜎(𝑟), 𝑚(𝑟), ℎ(𝑟), and 𝑤(𝑟), are determined by numerically solving the Einstein Yang–Mills equations, which take the form [28],  ′  2 2𝑤 2 ′′ ′ 𝜎 − ℎ =ℎ + 2 ℎ, (2.72a) 𝜎 𝑟 𝜇𝑟  ′  𝜇′ 𝑤ℎ2 𝑤(1 − 𝑤 2 ) ′′ ′ 𝜎 + =0 (2.72b) 𝑤 +𝑤 + 2 2+ 𝜎 𝜇 𝜎 𝜇 𝜇𝑟 2  2 ′2  𝑟 ℎ 𝑤 2 ℎ2 1 ′ ′2 2 2 𝑚 =𝑣 + 2 + 𝜇𝑤 + 2 (1 − 𝑤 ) , (2.72c) 2𝜎 2 𝜎 𝜇 2𝑟   2𝜎𝑤 ′2 2𝑤 2 ℎ2 ′ 𝜎 =𝑣 + , (2.72d) 𝑟 𝜎𝜇2𝑟 where the prime denotes differentiation with respect to 𝑟. The horizon radius 𝑟 𝐻 is defined as the largest solution to 𝜇(𝑟) = 0 and 𝑣 = 4𝜋𝐺 𝑁 /𝑒 2 . Since we have the three possible solutions, we should determine which one gives the dominant contribution to the partition function. Above the Hawking–Page temperature, we should consider either the second or the third solution. It turns out that the two solutions converge at high temperature. This is because, as the 19 temperature rises, the horizon grows and approaches the AdS boundary, where the interaction terms in the bulk equations of motion are suppressed. This expectation will be confirmed by the numerical computation below. In the asymptotically 𝐴𝑑𝑆 case, there are stable hairy black hole solutions, and those with 𝑆𝑈 (𝑁) hair have been extensively studied [28, 191, 201–203]. In particular, Bjoraker and Hosotani in [28] discussed the existence of a purely electric 𝑆𝑈 (2) charged black hole in 𝐴𝑑𝑆4 , which is of our interest, but it has not been constructed explicitly. Let us construct the genuinely non-abelian solution with 𝑆𝑈 (2) purely electric hair in 𝐴𝑑𝑆4 at high temperature. We determine 𝜎(𝑟), 𝑚(𝑟), ℎ(𝑟), and 𝑤(𝑟) by integrating equations (2.72) from the horizon to the infinity. Since a thermodynamically stable black hole has a large horizon at high temperature, we can expand them in the inverse powers of 𝑟 𝐻 as −1e −2 ), ℎ(𝑟) = 𝑟 𝐻 e ℎ0 (e 𝑟) + 𝑟𝐻 ℎ1 (e 𝑟 ) + 𝑂 (𝑟 𝐻 3 𝑚 e0 (e 𝑟) + 𝑟𝐻 𝑚 e1 (e 𝑟 ) + 𝑂 (1), 𝑚(𝑟) = 𝑟 𝐻 −3 −2 ), e 𝜎1 (e 𝑟 ) + 𝑂 (𝑟 𝐻 𝜎(𝑟) = 1 + 𝑟 𝐻 −3 −2 ), 𝑤 e1 (e 𝑟 ) + 𝑂 (𝑟 𝐻 𝑤(𝑟) = 𝑤 e0 (e 𝑟) + 𝑟𝐻 −1 2 ), 𝜇 e0 + 𝜇 e1 + 𝑂 (𝑟 𝐻 𝜇(𝑟) = 𝑟 𝐻 (2.73) where e 𝑟 = 𝑟/𝑟 𝐻 . Once we substitute this expansion into Einstein Yang–Mills equations (2.72), and solve the leading order equations, we get leading value of the functions: !   ′2 ′2 −Λ + 3𝑣ℎ 𝑣ℎ 1 𝐻 e ℎ0 (e 𝑟 ) = ℎ′𝐻 1 − − 𝐻 , 𝜎 , 𝑚 e0 (e 𝑟) = ˜ 0 (𝑟) ˜ = 1, (2.74) e 𝑟 6 2e 𝑟 and 𝑤˜ 0 is the solution of ¥0 + 0=𝑤 e 3𝑣ℎ′𝐻2 (e 𝑟 − 2) − Λe 𝑟 (1 + 2e 𝑟 3) ¤0 + 𝑤 e 9ℎ′𝐻2e 𝑟2 𝑤 e0 . (3𝑣ℎ′𝐻2 + Λ(e 𝑟 +e 𝑟2 + e 𝑟 3 )) 2 (2.75) 𝑑ℎ(𝑟) 𝑑e ℎ0 (e 𝑟) ′ Here, ℎ 𝐻 = 𝑑𝑟 |𝑟=𝑟 𝐻 ∼ 𝑑e𝑟 |e𝑟 =1 . Then, the leading thermodynamic quantities of e 𝑟 (e 𝑟 − 1)(−3𝑣ℎ′𝐻2 − Λ(e 𝑟 +e 𝑟2 + e 𝑟 3 )) 20 the black hole with non-abelian hair are given by [28], 4𝜋 ′ 4𝜋 2 ′ 𝜏3 𝑟 𝐻 →∞ 2 𝜏3 ℎ (𝑟)𝑟 𝑟 ℎ − − − − − → , 2 𝑟→∞ 𝑒2 𝑒2 𝐻 𝐻 2  4𝜋  4𝜋 𝑟 𝐻 →∞ 2 𝜏3 2 𝜏3 −−−−−→ 2 1 − 𝑤 e0 (e 𝑟) , 𝑄 𝑀 = 2 (1 − 𝑤(𝑟) ) 2 𝑟→∞ 2 𝑒 𝑒 −Λ + 3𝑣ℎ′𝐻2 3 𝑚(𝑟) 𝑟 𝐻 →∞ 𝑀= −−−−−→ 𝑟𝐻 , 𝐺 𝑁 𝑟→∞ 6𝐺 𝑁 1 𝑟𝐻 𝑟 𝐻 →∞ 𝑇= 𝜎(𝑟 𝐻 )𝜇′ (𝑟 𝐻 ) −−−−−→ (−Λ − 𝑣ℎ′𝐻2 ), 4𝜋 4𝜋 𝜋𝑟 2 𝑆 = 𝐻. 𝐺𝑁 𝑄𝐸 = (2.76a) (2.76b) (2.76c) (2.76d) (2.76e) The AdS boundary condition implies 𝑤 e0 (e 𝑟 → ∞) = 1. Since it is known that the black hole is unstable if 𝑤(𝑟) has a node (a nontrivial solution to 𝑤(𝑟) = 0) [29, 200], we require 𝑤 e0 (e 𝑟 ) be positive everywhere. Under these conditions, we find a unique solution for 𝑤 e0 when Λ, 𝑣 and ℎ′𝐻 are given. This establishes the existence of a stable (nodeless) solution in leading order for given values of 𝑟 𝐻 , Λ, 𝑣, and ℎ′𝐻 , provided ℎ′𝐻2 , 𝑣ℎ′𝐻2 < −Λ, which are always satisfied for large enough 𝑇. As expected, at high temperature, the thermodynamic quantities of the solution converge to those of the embedded 𝑈 (1) RN black hole as ! 2 Λ𝑟 𝐻 𝑒2 𝐺 𝑁 𝑄 2 𝜏3 𝑟𝐻 − + , 𝑄𝐸 = 𝑄 , 𝑀= 2 2𝐺 𝑁 3 2 4𝜋𝑟 𝐻 ! (2.77) 2 𝜋𝑟 𝐻 𝑒2 𝐺 𝑁 𝑄 2 1 2 , 𝑆= 𝑇= −Λ𝑟 𝐻 − . 2 4𝜋𝑟 𝐻 𝐺𝑁 4𝜋𝑟 𝐻 In Figure 2.1, we show the Helmholtz free energies of the two solutions as functions of 𝑇, with 𝑄, 𝑣 and Λ fixed as √ −Λ𝐺 𝑁 𝑄 = 100, 𝑣 = 1, Λ = −1. (2.78) We observe that Helmholtz free energies of the two solutions converge at high temperature. We also note that, if we look at smaller temperature, the free energy of the 𝑈 (1) RN black hole (in the orange curve) becomes larger than that of the genuinely non-abelian solution (in the dotted red curve). We will discuss more about it in Section 2.6. 21 (a) Helmholtz free energies with respect to the temperature. (b) The difference in Helmholtz free energies of the two solutions with respect to the temperature. √ Figure 2.1: They are plotted at a fixed value of Λ = −1, 𝑣 = 1 and −Λ 𝐺 𝑁 𝑄 = 100. Having our expectation confirmed, we can utilize the 𝑈 (1) RN solution to estimate the high temperature behavior of the holographic CFT with non-Abelian global symmetry 𝐺 in any dimensions. In particular,  
1 𝜙 𝜇 ∼ 𝑍 𝑇, 𝜇 , 𝑇 ≫ , (2.79) 𝑍𝐺 𝑇, 𝜇 = 𝑈 (1) 𝑈 (1) 𝑈 (1) ℓ ⟨𝜙, 𝜙⟩ 1/2 where 𝑍𝐺 denotes the grand canonical partition function for the Einstein Yang–Mills system in 𝐴𝑑𝑆 𝑑+1 with gauge group 𝐺 and 𝑍𝑈 (1) is that for the Einstein Maxwell system. By using equation (2.59), we obtain   𝑏 𝑑−3 𝑑−1 𝑍𝐺 (𝑇, 𝜇) = exp 𝑎 𝑇 + 𝑇 ⟨𝜇, 𝜇⟩ + · · · 4   𝑏 𝑑−1 𝑑−1 = exp 𝑎 𝑇 − 𝑇 ⟨𝜙, 𝜙⟩ + · · · , (2.80) 4 where 𝛽𝜇 = 𝑖𝜙. 2.6 Stability of black hole with non-abelian hair In the holographic CFT with non-abelian gauge symmetry, there are two types of black holes solutions, with and without non-abelian hair. In the previous section, we showed that the two solutions converge at high temperature. Since the two solutions differ at lower temperature, it is interesting to find out which solution is preferred theormodynamically. In this section, we calculate 1/𝑇 corrections to the Helmholtz free energies of the two solutions at the same temperature and charge. We find that the black hole with non-abelian hair has a lower free energy and is more stable. To be specific, we consider 𝐺 = 𝑆𝑈 (2) though we believe the results apply to any compact Lie group. 22 Since we know the exact solution without non-abelian hair, we focus on evaluating 1/𝑇 corrections to the solution with hair. We start with the equations, 2¤ ¥ e ℎ1 = − e ℎ1 + e 𝛿ℎ , e 𝑟   ¤ e ¤ ¤2 2e 2e ¤ e 𝑚 e1 = 𝑣 e 𝑟 ℎ0 ℎ1 − e 𝑟 ℎ 0e 𝜎1 + 𝛿𝑚 , ! 2e 2 ¤ 2 2e 𝑤 ℎ 2 𝑤 e 0 e 𝜎¤ 1 = 𝑣 + 20 0 , e 𝑟 𝜇 e0e 𝑟 2 2e 𝑤0 ¤ e e ℎ0 , 𝛿ℎ = e 𝜎¤ 1e ℎ0 + 𝜇 e0e 𝑟2 𝑤 e20e ℎ20 ¤ 2, e 𝛿𝑚 = +𝜇 e0 𝑤 e 0 𝜇 e0 (2.81) which are subleading order terms of equation (2.72) with respect to the expansion taken in equation (2.73). These differential equations depend on the zeroth order quantities; we note that e 𝜎0 , e ℎ0 and 𝑚 e0 are directly calculated to be equation (2.74), whereas 𝑤˜ 0 is solved numerically, when Λ and ℎ′𝐻 are given, using the differential equation in equation (2.75). Hence, we know all zeroth-order quantities, and we can ¤ decide e ℎ1 , 𝑚 e1 , 𝑤 e1 , and e 𝜎1 from equation (2.81). We first find e ℎ1 as 1 ¤ e ℎ1 (e 𝑟) = 2 e 𝑟 ∫ e𝑟 1 𝑑e 𝑟′e 𝑟 ′ 2e 𝛿 ℎ (e 𝑟 ′). (2.82) Since 𝜎(𝑟) goes to one when e 𝑟 → ∞, e 𝜎1 goes to zero as e 𝑟 → ∞, and e 𝜎1 (e 𝑟 ) = −Δe 𝜎+ ∫ e𝑟 1 ′ ¤ ′ 𝜎 1 (e 𝑟 ), 𝑑e 𝑟 e ∫ ∞ Δe 𝜎 := 1 𝑑e 𝑟e 𝜎¤ 1 (e 𝑟 ). (2.83) ¤ Then, by taking the quantities e ℎ1 and e 𝜎1 , given by equations (2.82) and (2.83), we can solve 𝑚 e1 in equation (2.81) as 𝑚 e1 (e 𝑟) 1 = + 𝑣 2𝑣 ∫ e𝑟 1 ∫ e𝑟 ′ ∫ e𝑟 ℎ′ ℎ′2 𝑑e 𝑟 ′ 𝐻′ 𝑑e 𝑟 ′′e 𝑟 ′′ 2 e 𝛿 ℎ (e 𝑟 ′′) + 𝑑e 𝑟 ′ 𝐻′ e 𝑟 1 e 𝑟 1 Δe 𝜎− ∫ e𝑟 ′ 1 ! ′′ ¤ ′′ 𝑑e 𝑟 e 𝜎 1 (e 𝑟 ) + ∫ e𝑟 1 𝑑e 𝑟 ′e 𝛿𝑚 (e 𝑟 ′). (2.84) We are interested in 𝑚 e1 (e 𝑟 → ∞) because it corresponds to the mass of the black hole. The subleading contribution to the mass of the non-abelian black hole is expressed as     ∫ ∞ 1 1 ¤ ′ e ′2 e 𝑚 e1 (e 𝑟 → ∞) = + 𝑑e 𝑟 𝑣 ℎ 𝐻e e 𝜎 1 (e 𝑟 ) + 𝛿𝑚 (e 𝑟 ) . (2.85) 𝑟 𝛿 ℎ (e 𝑟 ) + ℎ𝐻 1 − 2 e 𝑟 1 Now that we have computed e ℎ1 , 𝑚 e1 , and e 𝜎1 , we can estimate the thermodynamic 23 quantities of the black hole as   ∫ ∞ 𝑣 𝜏3 2 ′ 2e 𝑄≈ 𝑟 𝐻 ℎ𝐻 + 𝑑e 𝑟e 𝑟 𝛿 ℎ (e 𝑟) , 𝐺𝑁 2 1   𝑟𝐻 ¤ 1 ¤ −𝜇 ¤ Δe 𝑇≈ 𝜇 e0 + 𝜇 e e 𝜎 , 0 4𝜋𝐺 𝑁 4𝜋𝐺 𝑁 𝑟 𝐻 1 e 𝑟 =1 e 𝑟 =1 −Λ + 3𝑣ℎ′𝐻2 3 𝑚 e1 (e 𝑟 → ∞) 𝑀≈ 𝑟𝐻 + 𝑟𝐻 , 6𝐺 𝑁 𝐺𝑁 (2.86) where e 𝑟 = 𝑟/𝑟 𝐻 and the mass 𝑀 is evaluated in the infinite radius limit, provided from the value of 𝑚(𝑟) at 𝑟 → ∞. Finally, the Helmholtz free energy of black hole with non-abelian hair is given by, 𝐹 = 𝑀 − 𝑇𝑆 𝑟𝐻 3 e = 𝑟𝐻 𝐹0 + 𝐺𝑁  1 +𝑣 4 ∫ ∞ 1 (2.87)  1 ′2 𝜎 , 𝑑e 𝑟 Δ𝑚 e (𝑟) ˜ + (−Λ + 𝑣ℎ 𝐻 )Δe 4 where 𝑣Δ𝑚 e is the integrand of the equation (2.85) and   3 ′2 1 1 e Λ + 𝑣ℎ 𝐻 . 𝐹0 = 𝐺 𝑁 12 4 (2.88) Let us compare this with the free energy of the 𝑈 (1) RN black hole. For the solution to have the same temperature and charge, the radius of the horizon of the RN black hole must be ∫∞ −(−Λ + 𝑣ℎ′𝐻2 )Δe 𝜎 + 2𝑣ℎ′𝐻 1 𝑑e 𝑟e 𝑟 2e 𝛿 ℎ (e 𝑟) Δe 𝑟 𝑟 𝐻,𝑅𝑁 = 𝑟 𝐻 + , Δe 𝑟= . (2.89) 𝑟𝐻 −Λ + 3𝑣ℎ′𝐻2 The free energy is then 𝑟𝐻 3 e 𝐹𝑅𝑁 = 𝑟 𝐻 𝐹0 + 𝐺𝑁  1 + 𝑣ℎ′𝐻 4 ∫ ∞ 1 1 𝑑e 𝑟e 𝑟 2e 𝛿 ℎ (e 𝑟 ) + (−Λ + 𝑣ℎ′𝐻2 )Δ𝜎 ˜ 4  . (2.90) We remark again that the two free energies in equations (2.87) and (2.90) have same leading behavior. By taking the difference of the two free energies, we obtain  ∫ ∞  ∫ ∞ 𝑣 𝑟𝐻 ′ 2e 𝐹𝑅𝑁 − 𝐹 = ℎ𝐻 𝑑e 𝑟e 𝑟 𝛿 ℎ (e 𝑟) − 𝑑e 𝑟 Δ𝑚 e (e 𝑟) , (2.91) 𝐺𝑁 1 1 which comes from the 1/𝑇 correction. When we numerically calculate this difference, it has strictly positive value as shown in Figure 2.2. Therefore, the black hole with non-abelian hair has a smaller free energy and is thermodynamically preferred over the 𝑈 (1) RN black hole at finite temperature. 24 √︁ Figure 2.2: log [𝐺 𝑁 (𝐹𝑅𝑁 − 𝐹)/𝑟 𝐻 ] as a function of ℎ′𝐻 𝑣/−Λ when Λ = −1. 25 Chapter 3 UNIVERSAL ASYMPTOTICS FOR HIGH ENERGY CFT DATA 3.1 Introduction What is the behavior of conformal field theory (CFT) data at high energies? This question is well-studied in two dimensions. For instance, the density of states of any 2D CFT at high energies takes the following universal form, known as the Cardy formula [46]: √︂ √︂ √︂  𝑐 𝑐 𝑐 𝑑=2 𝜌 (Δ, 𝐽) ∼ exp 𝜋 Δ+𝐽− + Δ−𝐽− , Δ − |𝐽 | ≫ 𝑐. (3.1) 3 12 12 Here, 𝜌 𝑑=2 (Δ, 𝐽) is the density of local operators (equivalently states on 𝑆 1 ) with scaling dimension Δ and spin 𝐽. The entropy at high energies is controlled by a single theory-dependent number: the central charge 𝑐. The Cardy formula follows from modular invariance of the genus one partition function. Though (3.1) is valid for all 2D CFTs, it has a particularly nice interpretation for CFTs dual to quantum gravity in weakly-curved AdS3 . In such theories, the entropy log 𝜌 𝑑=2 (Δ, 𝐽) is interpreted as the area of a BTZ black hole with spin 𝐽 and mass 𝑀 given by [197] 𝑐  1  Δ− , (3.2) 𝑀= ℓAdS 12 in an AdS3 space with [38] 3ℓAdS 𝑐= . (3.3) 2𝐺 𝑁 The Cardy formula then becomes a statement of universality of black hole entropy, regardless of the microscopic details of the quantum gravity theory. OPE coefficients of heavy operators in 2D CFTs obey similar, though perhaps less well-known, universal formulas. In [44], a formula for average squared OPE coefficients of three heavy Virasoro primaries was derived using modular invariance of the genus two partition function. For example, when all three operators have roughly equal dimensions Δ𝑖 = Δ ≫ 𝑐, it takes the form 𝑑=2 2 (𝐶HHH ) ∼  27 16  3Δ 𝑒 −6𝜋 √︃ 𝑐−1 24 Δ 5𝑐−11 Δ 36 , Δ ≫ 𝑐. (3.4) 26 Similar formulas were derived for OPE coefficients with one or two heavy operator(s) (see e.g. [141]). These formulas were subsequently unified in [67], with interesting 𝑑=2 ) 2 connections to the DOZZ formula. In holographic theories, the formula for (𝐶HHH matches the contribution of a two-sided wormhole connecting a pair of boundary three-point functions [58]. In this paper, we explore whether similar universal formulas exist for higher dimensional CFTs. We will use purely field-theoretic methods, so our results will be applicable to both holographic and non-holographic theories. An immediate puzzle is that there is no simple analog of modular invariance in higher dimensional geometries like 𝑆 1 × 𝑆 𝑑−1 (𝑑 ≥ 3). (See [18, 21, 116, 156, 189, 190] for some discussion and progress on modular invariance in higher dimensions.) However, we can instead use a beautiful idea from [14, 27, 127], which was used to count the density of states in higher dimensional CFTs in [27, 188]. (Similar ideas were used for studying supersymmetric indices in [75].) The key point is that finding the leading asymptotics of CFT data doesn’t require full modular invariance — we just need a sufficiently powerful effective theory for a CFT dimensionally reduced on a circle. The dimensional reduction of a 𝑑-dimensional CFT is generically a gapped theory in 𝑑−1 dimensions. Fortunately for our purposes, the exponential decay of correlations in a gapped theory makes it very flexible: we can place it on many different geometries, and in this way extract myriad predictions for the 𝑑-dimensional CFT. A gapped theory can be described by a local action for background fields, obtained by integrating out the gapped degrees of freedom. In the context of a dimensionallyreduced CFT (with thermal boundary conditions), we call this local action the "thermal effective action." It describes hydrodynamic observables of the CFT in equilibrium. The derivative expansion of the thermal effective action is an expansion in the inverse temperature 𝛽 = 1/𝑇. This construction was explained in [14, 127], and has been explored extensively in the hydrodynamics literature; see e.g. [26, 71, 123, 124, 126]. We review it in Section 3.2, along the way discussing some subtleties related to the Weyl anomaly. Placing the thermal effective theory on 𝑆 1 ×𝑆 𝑑−1 leads to simple universal predictions for the density of CFT operators at large Δ. For example, in 3D CFTs, one obtains log 𝜌 𝑑=3 (Δ, 𝐽) = 3𝜋 1/3 𝑓 1/3 (Δ2 − 𝐽 2 ) 1/3 − 2 log(Δ2 − 𝐽 2 ) + O (Δ0 ). 3 (3.5) 27 Here, 𝑓 is a theory-dependent positive real number, equal to minus the free energy density of the CFT, as we review in Section 3.2. The leading term in the high temperature partition function for the canonical ensemble of a CFT was first written down using hydrodynamic techniques in [27]. It was subsequently transformed to the microcanonical ensemble in [188].1 The thermal effective theory approach in this work allows us to reproduce those results and systematically explore subleading corrections. The quantity 𝑓 controls the leading density of states in both 2D (where 𝑓 = 𝜋𝑐/6) and higher dimensions. However, unlike in 2D, where the Cardy formula is valid up to nonperturbative corrections in Δ, the entropy in higher dimensional CFTs receives perturbative corrections in 1/Δ, coming from higher-derivative terms in the thermal effective action. The derivation of (3.5) using the thermal effective action is given in Section 3.3. There, we also describe the leading higher-derivative corrections. (Furthermore in Section 3.3, we clarify some subtleties related to the Casimir energy on 𝑆 𝑑−1 in higher-dimensional CFTs.) We also briefly discuss nonperturbative corrections to the density of states in Section 3.3. Then, in Section 3.4, we compare these general formulas to free theories and holographic theories, determining Wilson coefficients in those cases by matching their partition functions to the effective theory. In addition to the density of states, we will also find universal formulas for OPE coefficients of three heavy operators in higher-𝑑 CFTs.2 Our strategy will be to put the theory on a higher-dimensional version of a genus-2 Riemann surface, obtained by gluing a pair of three-punctured 𝑆 𝑑 ’s with three cylinders 𝑆 𝑑−1 × 𝐼 (where 𝐼 is an interval). We describe this "genus-2" geometry in detail in Section 3.5.3 A glaring problem is that the "genus-2" geometry is not a circle fibration, so it is not immediately obvious how to apply the thermal effective action. However, in a "high temperature" limit where the cylinders get short, the geometry contains shrinking circles. We claim that these shrinking circles can be treated like thermal circles in local regions that we call "hot spots"; see Figure 3.1. We furthermore conjecture that the effective action of the hot spots gives the singular part of the partition function 1 The density of states was also studied [147, 199]. 3 2 Formulas for heavy OPE coefficients weighted by light OPE coefficients, e.g. 𝐶 HHH 𝐶HLL , were derived in [9] using crossing symmetry of six-point functions of local operators. By contrast, our focus will be on un-weighted heavy OPE coefficients 𝐶HHH , which are controlled by different physics. For example, the leading behavior of 𝐶HHH is determined by the free energy density 𝑓 , which does not (to our knowledge) appear in a simple way in a six-point function of light local operators. 3 A special case of this geometry with no angular fugacities was studied recently in [20]. 28 Figure 3.1: The "genus-2" geometry and its "hot spots." The top is a ball 𝐵 𝑑 with two balls removed. It is topologically equivalent to a three-punctured 𝑆 𝑑 . The bottom is the same. The top and bottom are glued together with three cylinders. In the limit that the cylinders get short, there are shrinking circles indicated in red that run down one cylinder and up another. The neighborhoods of each of these circles are "hot spots," where the thermal effective action receives a large contribution. on our "genus-2" geometry. (The remaining parts of the geometry are not described by thermal EFT, but contribute non-singular corrections to the partition function at high temperature.) With the "hot spot" conjecture, we can determine the partition function in the regime where it is dominated by heavy CFT data. To extract heavy-heavy-heavy OPE coefficients, we must furthermore understand the decomposition of the partition function into a higher-dimensional version of genus-2 (global) conformal blocks. These are interesting special functions that to our knowledge have not previously appeared in the CFT literature. We explore them in Section 3.6, determining their behavior at large Δ using the shadow formalism and saddle-point analysis. We then decompose the partition function into "genus2" blocks using an appropriate inverse Laplace transform on the moduli space of "genus-2" conformal structures in higher dimensions. In the end, we obtain a universal formula for average squared heavy-heavy-heavy OPE coefficients in a 𝑑dimensional CFT. For example, for three scalar operators with similar dimensions Δ in 𝑑 = 3, we find 𝜌 𝑑=3 3 𝑑=3 2 (Δ, 0) (𝐶HHH ) ∼   6Δ √ 3 𝑒 3 2𝜋 𝑓 Δ × . . . 2 (3.6) where ". . . " are subleading corrections in Δ. We give a formula for OPE coefficients of three operators with arbitrary Lorentz representations (with spin held constant as Δ → ∞) in arbitrary 𝑑 below in (3.242). In Section 3.8, we apply similar (but simpler) methods to compute asymptotic thermal 1-point functions of heavy operators. This can be viewed as a particularly simple limit of heavy-heavy-heavy OPE coefficients. 29 In Section 3.9, we discuss (3.6), its generalizations, and some implications and future directions. In holographic theories, we speculate that (3.6) describes a three-point function of three black holes surrounded by highly entangled matter. Three point functions of three "pure" black holes are likely atypical from the point of view of (3.6), but perhaps could be determined from an appropriate holographic calculation. In Appendix A, we discuss a simple warmup example of the thermal effective action for a two-point function of momentum generators. In Appendix C, we discuss some aspects of free theories, including novel formulas for nonperturbative corrections to density of states. Other appendices contain detailed calculations to supplement the main text. 3.2 The thermal effective action Consider a 𝑑-dimensional CFT at finite temperature 𝑇. Generically, thermal fluctuations cause equilibrium correlators to decay exponentially with distance: ⟨O ( 𝑥®1 )O (® 𝑥2 )⟩ 𝛽 ∼ 𝑒 −|𝑥®1 −®𝑥2 |/𝜉 . (3.7) By dimensional analysis, the correlation length 𝜉 must be inversely proportional to the temperature, 𝜉 ∝ 1/𝑇. Exponentially-decaying correlators can be expanded in a series in 𝛿-functions and their derivatives. (Equivalently, in momentum space, they can be expanded in a power series in momenta.) This expansion is summarized by a local effective action for background fields that we call the thermal effective action. It is useful to adopt the geometric perspective on the thermal effective action explained in [14]. Equilibrium thermal correlators are computed by compactifying the Euclidean theory on a thermal circle of length 𝛽 = 1/𝑇. Generically, when a 𝑑dimensional CFT is compactified on a circle, the result is a gapped theory in (𝑑−1) dimensions. A rough argument is that compactification of a CFT does not involve tuning any parameters, since all 𝛽 are equivalent by 𝑑-dimensional scale invariance. Thus, it would be non-generic for the resulting (𝑑−1)-dimensional theory to be at a critical point. Instead, it will typically have a nonzero mass gap 𝑚 gap ∝ 𝑇, and a finite correlation length 𝜉 = 1/𝑚 gap .4 We can think of this gapped theory as the modular transform of the 𝑑-dimensional CFT. This argument fails when a symmetry protects gapless modes in the compactified theory, such as in free theories or supersymmetric compactifications (where we twist 4 By contrast, when a theory with an intrinsic scale is compactified, one generally obtains different dynamics at different compactification radii. By tuning 𝛽 it may be possible to reach a critical point. An example is 4D SU(2) pure Yang-Mills theory, which is expected to possess a critical point in the Ising universality class at a particular temperature, see e.g. [93] for a review. 30 by (−1) 𝐹 around the circle). It could also fail in theories that spontaneously break a continuous symmetry at finite temperature, such as those recently constructed (in fractional spacetime dimensions) in [55]. It is currently unknown whether such theories exist in integer spacetime dimensions. See [55, 100] and references therein for more discussion. In this work, we will focus on theories that are gapped at finite temperature. An efficient way to capture correlators of the CFT is to couple to classical background fields. For example, stress tensor correlators are captured by coupling to a 𝑑dimensional background metric 𝐺 𝜇𝜈 . If the metric possesses a circle isometry, then by a suitable choice of coordinates, we can put it in Kaluza-Klein form 𝑥 )𝑑𝑥 𝑖 𝑑𝑥 𝑗 + 𝑒 2𝜙(𝑥®) (𝑑𝜏 + 𝐴𝑖 (® 𝑥 )) 2 , 𝐺 𝜇𝜈 𝑑𝑥 𝜇 𝑑𝑥 𝜈 = 𝑔𝑖 𝑗 (® 𝜏 ∈ [0, 1), (3.8) where the periodic direction is 𝑥 0 = 𝜏. The (𝑑−1)-dimensional fields are a metric 𝑔𝑖 𝑗 , a gauge field 𝐴𝑖 , and a dilaton 𝜙. We choose conventions so that 𝜏 has periodicity 1. Thus, for the thermal compactification 𝑆 1𝛽 ×R𝑑−1 with the flat metric, we have 𝑒 𝜙 = 𝛽. However, it will be interesting in what follows to allow the (𝑑−1)-dimensional fields 𝑔𝑖 𝑗 , 𝐴𝑖 , 𝜙 to be spatially varying. By our assumption above, the partition function of the 𝑑-dimensional CFT on the Kaluza-Klein geometry (3.8) becomes the partition function of a gapped (𝑑−1)dimensional theory coupled to (𝑑−1)-dimensional background fields: 𝑍CFT [𝐺] = 𝑍gapped [𝑔, 𝐴, 𝜙]. (3.9) The partition function of a trivially gapped QFT at long distances can be expanded in a sum of local counterterms in the background fields. In this case, we have5 𝑍CFT [𝐺] = 𝑍gapped [𝑔, 𝐴, 𝜙] ∼ 𝑒 −𝑆th [𝑔,𝐴,𝜙] . (3.10) The thermal effective action 𝑆th is a sum of local terms in 𝑔𝑖 𝑗 , 𝐴𝑖 , 𝜙 that captures Euclidean correlators at length scales that are large compared to the correlation length 𝜉 = 1/𝑚 gap (equivalently, at momenta small compared to 𝑚 gap ).6 In (3.10), 5 A theory that spontaneously breaks a discrete symmetry at finite temperature can display mild violations of (3.10); see [55]. In general, if the finite temperature theory is nontrivially gapped, then the thermal effective action must include a nontrivial TQFT. We focus on the trivially gapped case in this work, though most of our results are simple to adapt to a more general thermal TQFT. 6 Note that the thermal effective action does not in general capture long-distance real time observables, even at small nonzero frequencies 0 < 𝜔 ≪ 𝑚 gap , where dissipation is an important effect. 31 "∼" means agreement up to exponential corrections of the form 𝑒 −𝐿/𝜉 , where 𝐿 is a characteristic length scale. In QFT, we usually have the freedom to add arbitrary local counterterms in background fields. This is called a change of "scheme." In the thermal effective action (3.10), it is important that we are only allowed to add local 𝑑-dimensional counterterms to the CFT (which enter 𝑆th via dimensional reduction). We are not allowed to add arbitrary local 𝑑−1-dimensional counterterms. Thus, 𝑆th can contain physical, scheme-independent information. The thermal effective action is highly constrained by symmetries. Firstly, coordinateinvariance in 𝑑-dimensions implies that 𝑆th is invariant under (𝑑−1)-dimensional coordinate transformations, as well as gauge transformations of the KK gauge field 𝐴𝑖 . For simplicity, in this work we focus on CFT𝑑 ’s with vanishing gravitational anomaly. When the gravitational anomaly is non-vanishing (for example in a 2D CFT with 𝑐 𝐿 ≠ 𝑐 𝑅 ), the anomaly must be matched by gravitational Chern-Simons terms in the thermal effective action; see [59, 75, 98, 125, 126] for more details. Such terms could be easily incorporated into the analysis that follows.7 Secondly, 𝑆th is constrained by Weyl invariance of the 𝑑-dimensional theory. Under a Weyl transformation, the CFT partition function changes by 𝑍CFT [𝑒 2𝜎 𝐺] = 𝑍CFT [𝐺]𝑒 −𝑆anom [𝐺,𝜎] , (3.11) where 𝑆anom [𝐺, 𝜎] is the contribution from the Weyl anomaly. Because 𝜙 transforms with a shift 𝜙 → 𝜙 +𝜎 under 𝜏-independent Weyl transformations, we can use (3.11) to completely determine the 𝜙-dependence of 𝑍CFT [𝐺] [85]. Note that b b −𝑆anom [𝐺,𝜙] 𝑍CFT [𝐺] = 𝑍CFT [𝐺]𝑒 , (3.12) b ≡ 𝑒 −2𝜙 𝐺. Plugging in (3.10), this implies where 𝐺 b 𝜙] 𝑆th [𝑔, 𝐴, 𝜙] = 𝑆th [b 𝑔 , 𝐴, 0] + 𝑆anom [𝐺, b 𝜙]. ≡ 𝑆[b 𝑔, 𝐴] + 𝑆anom [𝐺, (3.13) b 𝜙] plays the role of matching the Weyl anomaly in the In equation (3.13), 𝑆anom [𝐺, thermal effective action. We discuss this contribution later in Section 3.2. The remaining term 𝑆[b 𝑔, 𝐴] is invariant under 𝜏-independent Weyl transformations. It depends only on 𝐴𝑖 , and the Weyl-invariant "effective metric" b 𝑔𝑖 𝑗 ≡ 𝑒 −2𝜙 𝑔𝑖 𝑗 . 7 We thank Yifan Wang for discussion on these points. (3.14) 32 Note that b 𝑔 transforms as a metric under coordinate-transformations. Thus, 𝑆[b 𝑔, 𝐴] can be organized in a derivative expansion in coordinate invariants built out of b 𝑔 and 𝐴. Classifying the terms in 𝑆[b 𝑔 , 𝐴] is similar to classifying local interactions in Einstein-Maxwell theory (without the freedom to perform field redefinitions). The first few terms are ∫  √︁  b + 𝑐2 𝐹 2 + . . . . 𝑆[b 𝑔, 𝐴] = 𝑑 𝑑−1 𝑥 b 𝑔 − 𝑓 + 𝑐1 𝑅 (3.15) b is the Riemann curvature built from the metric b Here, 𝑅 𝑔 , and 𝐹𝑖 𝑗 = 𝜕𝑖 𝐴 𝑗 − 𝜕 𝑗 𝐴𝑖 is the field strength of the KK gauge boson. Indices are everywhere contracted using b 𝑔 , for example 𝐹2 = b 𝑔𝑖𝑘 b 𝑔 𝑗𝑙 𝐹𝑖 𝑗 𝐹𝑘𝑙 . (3.16) This ensures that the derivative expansion for 𝑆th becomes an expansion in 𝑝/𝑇, ® since b 𝑔𝑖 𝑗 contains a factor of 𝑒 2𝜙 , where 𝑒 −𝜙 is a local temperature. Again, in (3.15), we have assumed gravitational anomalies are absent. The construction of the thermal effective action (3.15) closely mimics the construction of the dilaton effective action in theories where scale invariance is spontaneously broken to 𝑑-dimensional Poincare invariance [137]. The key difference is that here we have an effective action in (𝑑−1)-dimensions instead of 𝑑 dimensions. The cosmological constant term √︁ ∫ 𝑔𝑓 The leading term in the thermal effective action is a cosmological constant − 𝑑 𝑑 𝑥 b for the effective metric b 𝑔 . The coefficient 𝑓 has at least three important interpretations. 1. − 𝑓 is (the scheme-independent part of) the free energy density of the CFT at finite temperature in flat space.8 To see why, we place the theory on R𝑑−1 × 𝑆 1𝛽 , where the thermal circle has length 𝛽. In this geometry, the effective metric is b 𝑔𝑖 𝑗 = 𝛽−2 𝛿𝑖 𝑗 . Only the cosmological constant term in 𝑆th is nonzero, and it contributes ∫ 𝑑−1 1 𝐹/𝑇 = − log 𝑍CFT [R × 𝑆 𝛽 ] = 𝑆th = − 𝑑 𝑑 𝑥 𝑓 𝛽−(𝑑−1) = − 𝑓 𝑇 𝑑−1 vol R𝑑−1 . (3.17) 𝐹/𝑇 = − log 𝑍CFT [R 𝑑−1 × 𝑆 1𝛽 ] = 𝑆th = − ∫ 8 Our convention for 𝑓 is opposite to the one in [119], 𝑓 𝑑 𝑑 𝑥 𝑓 𝛽−(𝑑−1) = − 𝑓 𝑇 𝑑−1 vol R𝑑−1 . here = − 𝑓there . 33 Thus, the free energy density is − 𝑓 𝑇 𝑑 . 2. 𝑓 is proportional to the thermal one-point function of the stress tensor in flat space R𝑑−1 . Lorentz and scale invariance dictate that   1 𝜇𝜈 𝜇 𝜈 𝜇𝜈 𝑑 ⟨𝑇 (𝑡, 𝑥®)⟩ 𝛽 = 𝑏𝑇 𝑇 𝛿0 𝛿0 − 𝛿 , (3.18) 𝑑 for some dimensionless coefficient 𝑏𝑇 . Meanwhile, the energy density can be computed from the derivative of the partition function9 −⟨𝑇 00 (0, 𝑥®)⟩ 𝛽 = − 1 𝜕 log 𝑍CFT [R𝑑−1 × 𝑆 1𝛽 ] = (𝑑 − 1) 𝑓 𝑇 𝑑 . (3.19) 𝑑−1 𝜕𝛽 vol R Equating (3.18) and (3.19), we find 𝑏𝑇 = −𝑑𝑓 . In particular, positivity and extensivity of the energy density on R𝑑−1 implies that 𝑓 is positive: 𝑓 > 0. (3.20) 3. − 𝑓 is the Casimir energy density of the CFT compactified on a circle. To see why, we choose 𝑥 1 as a time direction. The Hamiltonian density for the compactified theory is then 𝐸 cas = −⟨𝑇 11 ⟩ 𝛽 = − 𝑓 𝑇 𝑑 . 1 𝑑−2 vol(𝑆 𝛽 × R ) (3.21) In particular, (3.20) gives a simple proof that the Casimir energy of a CFT compactified on a circle is always negative. Ultimately, this is a consequence of tracelessness of the stress tensor. The components 𝑇 00 compute the energy in the thermal ensemble (which is positive), while the components 𝑇 11 compute the energy in the compactified theory. The two have opposite signs by tracelessness of 𝑇 𝜇𝜈 . This proof applies, for example, to electromagnetism. Thus, the coefficient 𝑓 is a simple and important observable of the CFT. We will see that it controls a huge amount of the physics at finite temperature. As a simple warmup example, in Appendix A, we show how 𝑓 determines a (regularized) two-point function of momentum operators at finite temperature. This result can be understood both from "bootstrap" arguments using properties of stress tensor correlators, and from the thermal effective action. 9 Note that we are using the conventions of Euclidean field theory, where the generator of time ∫ translations includes a minus sign 𝐻 = − 𝑑 𝑥®𝑇 00 . The minus sign can be understood via Wick rotation from Lorentzian signature 𝑇𝐸00 = (𝑖) 2𝑇𝐿00 . 34 Weyl anomaly terms b 𝜙] in the thermal effective action were given in The Weyl anomaly terms 𝑆anom [𝐺, [85]. Let us write them down in detail. Such terms are of course absent when 𝑑 is odd, so we focus on even 𝑑 in this subsection. As a review, the infinitesimal form of the Weyl anomaly in 𝑑 dimensions is ∫ √ 𝛿𝜎 (− log 𝑍CFT [𝐺]) = 𝑑 𝑑 𝑥 𝐺𝜎A [𝐺], (3.22) where 𝛿𝜎 is defined by rescaling the metric 𝐺 → (1 + 2𝜎)𝐺 with 𝜎 infinitesimal. Here, A [𝐺] is a local functional of 𝐺 such that (3.22) solves the Wess-Zumino consistency condition [𝛿𝜎1 , 𝛿𝜎2 ] log 𝑍 = 0. The infinitesimal Weyl anomaly can be integrated by considering a family of metrics 𝑒 2𝑡𝜎 𝐺 where 𝑡 ∈ [0, 1], using (3.22) to write a differential equation in 𝑡, and solving the differential equation. The result is the finite Weyl transformation rule (3.11), where 𝑆anom is given by [187] ∫ 1 𝑆anom [𝐺, 𝜎] = ∫ 𝑑𝑡 √︁ 𝑑 𝑑 𝑥 det(𝑒 2𝑡𝜎 𝐺) 𝜎A [𝑒 2𝑡𝜎 𝐺]. (3.23) 0 The general solution of the Wess-Zumino consistency condition in 𝑑-dimensions is [33, 34]: ∫ ∫   √ √ Í 1 (𝑑) 𝑑 𝑑/2 𝑑 + 𝛿𝜎 𝑆ct . 𝑑 𝑥 𝐺𝜎A [𝐺] = 𝐺 𝜎 (−1) 𝑎 𝐸 − 𝑐 𝐼 𝑑 𝑥 𝑑 𝑑 𝑘 𝑑𝑘 𝑘 (4𝜋) 𝑑/2 (3.24) √ Here, 𝐸 𝑑 is the Euler density,10 and 𝐺 𝐼 𝑘(𝑑) are local Weyl-invariants of 𝐺. For example, in 4D there is one such Weyl-invariant, given by the square of the Weyl tensor: 𝐼1(4) = 𝐶 2 = 𝐶 𝜇𝜈𝜌𝜎 𝐶 𝜇𝜈𝜌𝜎 (𝑑 = 4). (3.25) (6) In 6D there are three such Weyl invariants 𝐼 𝑘=1,2,3 , and in general the number grows 𝑑/2 with 𝑑. The factor 1/(4𝜋) in (3.24) is a convention. The remaining terms 𝛿𝜎 𝑆ct in (3.24) are Weyl variations of local counterterms. For instance, in 4D we can have ∫ √ 2 𝑏 𝑑 𝑑 𝑥 𝐺𝑅 (𝑑 = 4), (3.26) 𝑆ct = − 12(4𝜋) 2 10 In 2D, we have 𝑎 2 = 𝑐/6, and in 4D we write 𝑎 4 = 𝑎. 35 which leads to a contribution 𝛿𝜎 𝑅 2 ∼ 𝑏□𝑅 in the Weyl anomaly. We sometimes refer to 𝛿𝜎 𝑆ct as "𝑏-type" terms. Such terms trivially obey the Wess-Zumino consistency condition. They are scheme-dependent because they can be shifted by adding local counterterms to the action of the CFT. We comment more on this below in Section 3.3. Plugging these results into (3.23), we find 𝑆anom for example in 𝑑 = 2 and 𝑑 = 4: ∫ √ 𝑐 𝑑 2 𝑥 𝐺 (𝜎𝑅 + (𝜕𝜎) 2 ) 24𝜋 ∫  √  𝑎 𝜇𝜈 4D 4 2 4 1 𝜇𝜈 𝐺 𝜎𝐸 − 4𝜕 𝜎𝜕 𝜎(𝑅 − 𝑆anom [𝐺, 𝜎] = 𝑑 𝑥 𝐺 𝑅) − 4(𝜕𝜎) □𝜎 − 2(𝜕𝜎) 4 𝜇 𝜈 2 (4𝜋) 2 ∫ √ 𝑐 𝑑 4 𝑥 𝐺𝜎𝐶 2 + 𝑆ct [𝑒 2𝜎 𝐺] − 𝑆ct [𝐺]. − (3.27) (4𝜋) 2 2D 𝑆anom [𝐺, 𝜎] = − Note that the scheme-dependent part of the Weyl anomaly 𝛿𝜎 𝑆ct integrates trivially to give 𝑆ct [𝑒 2𝜎 𝐺] − 𝑆ct [𝐺]. The Weyl-invariant terms are also simple to integrate because the integrand (3.23) is 𝑡-independent for those terms. Putting everything together, the Weyl-anomaly contribution to the thermal effective action is b 𝜙] = 𝑆Euler 𝑆anom [𝐺, (3.28) ∫ ∑︁ √︁ 1 b + DR[𝑆ct [𝐺]] − DR[𝑆ct [𝐺]], b 𝑔 𝜙 DR[𝐼 𝑘(𝑑) [𝐺]] 𝑐 𝑑𝑘 𝑑 𝑑−1 𝑥 b − 𝑑/2 (4𝜋) 𝑘 (3.29) where (−1) 𝑑/2 𝑎 𝑑 𝑆Euler = (4𝜋) 𝑑/2 ∫ 1 ∫ 𝑑𝑡 0 √︁ b 𝑔 𝜙 DR[𝐸 𝑑 [𝑒 2𝑡𝜙 𝐺]]. 𝑑 𝑑−1 𝑥 𝑒 𝑑𝑡𝜙 b (3.30) Here, the dimensional reduction operation DR[· · · ] means evaluating in the KK metric (3.8) and integrating over 𝜏. Note that the 𝐼 𝑘(𝑑) terms in (3.28) are linear in 𝜙, and thus can lead to temperature dependence of the form log(𝛽/𝛽0 ) in certain geometries. Here, we note that the coefficient of log 𝛽 is a genuine prediction of 𝑆th , but the scale 𝛽0 is schemedependent. The reason is that 𝛽0 can be shifted by adding local Weyl-invariant counterterms to the action of the CFT: ∫ √ ∑︁ 1 𝑑 𝑑 𝑥 𝐺 𝑟 𝑘 𝐼 𝑘(𝑑) . (3.31) 𝑆CFT → 𝑆CFT + 𝑑/2 (4𝜋) 𝑘 This ambiguity shifts the coefficients of DR[𝐼 𝑘(𝑑) ] in the thermal effective action: −𝑐 𝑑𝑘 𝜙 → 𝑟 𝑘 − 𝑐 𝑑𝑘 𝜙, (3.32) 36 and consequently shifts 𝛽0 . This ambiguity will not play a further role in this work, since we will always consider CFTs in conformally-flat11 geometries where 𝐼 𝑘(𝑑) vanishes. Two dimensions In two dimensions, a very nice thing happens. The cosmological constant term is the only local gauge-invariant combination of b 𝑔 and 𝐴𝑖 that we can write down. Furthermore, there are no nontrivial local Weyl invariants. Thus, the Weyl-invariant part of the thermal effective action truncates to a single term! ∫ √︁ 𝑆[b 𝑔 , 𝐴] = − 𝑑1𝑥 b 𝑔𝑓 (𝑑 = 2). (3.33) The action (3.33) describes equilibrium thermal physics in 2D to all perturbative orders in 1/𝑇. Using the connection between 𝑓 and the Casimir energy (3.21), we find 𝑓 = 2𝜋𝑐 , 12 (3.34) where 𝑐 is the central charge. This result is not a surprise. A 2D CFT at high temperature can be described by performing a modular transformation, reinterpreting the thermal circle as a spatial circle. The states propagating in the modular-transformed theory have energies 𝑐 𝐸𝑖 = 2𝜋 𝛽 (Δ𝑖 − 12 ), where Δ𝑖 are scaling dimensions in the CFT. The effective action (3.33) simply captures the contribution of the ground state in the modular 𝑐 transformed theory, with energy 𝐸 0 = − 2𝜋 𝛽 12 . The energy gap to the next state is the "mass gap" of the thermal theory 𝑚 gap = 𝐸 1 − 𝐸 0 = 2𝜋 Δ1 𝛽 (𝑑 = 2). (3.35) States with energies at or above the mass gap 𝐸𝑖 − 𝐸 0 ≥ 𝑚 gap contribute nonperturbative corrections in 𝛽 of the form 𝑒 −2𝜋Δ𝑖 /𝛽 , which are not captured by 𝑆th . 3.3 The density of high-dimension states The spectrum of a 𝑑-dimensional CFT is captured by the partition function on 𝑆 1𝛽 × 𝑆 𝑑−1 . In this section, we compute this partition function using the thermal 11 We call a manifold "conformally-flat" if in a neighborhood of each point, the metric is Weyl- equivalent to a flat metric. This is sometimes called "locally conformally-flat." A 3-manifold is conformally-flat if and only if the Cotton tensor vanishes, and a 𝑑-manifold with 𝑑 ≥ 4 is conformally-flat if and only if the Weyl tensor vanishes. 37 effective action, and decompose the result into conformal characters to extract the density of high dimension states. We will recover the leading-order formulas from [27, 188], and also discuss subleading corrections. The precise expression for the partition function involves the Casimir energy of the CFT on 𝑆 𝑑−1 . To start, we review the Casimir energy and discuss some details of its relation to the thermal effective action. The Casimir energy on 𝑆 𝑑−1 The partition function on 𝑆 1𝛽 × 𝑆 𝑑−1 is a sum over states on 𝑆 𝑑−1 weighted by Boltzmann factors 𝑒 −𝛽𝐸𝑖 . By the state-operator correspondence, states on 𝑆 𝑑−1 are in one-to-one correspondence with local CFT operators O𝑖 . In even dimensions the energy 𝐸𝑖 of the state |O𝑖 ⟩ is equal to the dimension Δ𝑖 plus a contribution from the Casimir energy on the sphere: 𝐸𝑖 = Δ𝑖 + 𝐸 0 , (3.36) where Δ𝑖 is the scaling dimension of O𝑖 . For example, in 2D, the Casimir energy is 𝑐 (in units where the 𝑆 𝑑−1 has radius 1). The Casimir energy 𝐸 0 will play 𝐸 0 = − 12 an important role in higher dimensions as well, so let us recall how to derive it. We follow the discussion of [13]. Let 𝑊 [𝐺] ≡ − log 𝑍CFT [𝐺], so that the Weyl anomaly is 𝑊 [𝑒 2𝜎 𝐺] − 𝑊 [𝐺] = 𝑆anom [𝐺, 𝜎]. To compute the stress tensor on the cylinder R × 𝑆 𝑑−1 , we consider the Weyl rescaling from the plane to the cylinder 𝑑𝑟 2 + 𝑟 2 𝑑Ω2𝑑−1 → 𝑑𝑟 2 + 𝑑Ω2𝑑−1 = 𝑒 −2 log 𝑟 𝛿 𝜇𝜈 𝑑𝑥 𝜇 𝑑𝑥 𝜈 , 𝑟2 (3.37) which corresponds to 𝜎 = − log 𝑟. Plugging this into 𝑆anom [𝐺, 𝜎], we obtain the partition function on the cylinder as a function of the partition function on the plane. Taking a derivative with respect to 𝐺 𝜇𝜈 and using that the one-point function ⟨𝑇 𝜇𝜈 ⟩ on the plane vanishes, we obtain ⟨𝑇 𝜇𝜈 ⟩ on the cylinder, from which we can read off the Casimir energy. There is a small shortcut that will be useful in what follows. We can simply compute 𝑊 [𝐺] on the infinite cylinder R × 𝑆 𝑑−1 using the Weyl anomaly. This has an infinite part of the form 𝐸 0 vol R, from which we can read off the Casimir energy 𝐸 0 . As an example, in 2D, we have ∫ ∫ 𝑐 1 𝑐 −2 log 𝑟 𝑊 [𝑒 𝛿] − 𝑊 [𝛿] = − 𝑟 𝑑𝑟 𝑑𝜃 2 = − 𝑑𝜏 (𝑑 = 2), (3.38) 24𝜋 12 𝑟 38 𝑐 where we defined 𝜏 = log 𝑟. This gives the expected result 𝐸 0 = − 12 . In 4D, we find   ∫ 𝑎 vol 𝑆 3 6 3 −2 log 𝑟 𝑟 𝑑𝑟 4 + 𝑆ct [𝑒 2𝜎 𝛿] 𝑊 [𝑒 𝛿] − 𝑊 [𝛿] = 2 (4𝜋) 𝑟  ∫ 3𝑎 3𝑏 = − 𝑑𝜏, (3.39) 4 8 where we used the form of 𝑆ct in (3.26), together with the fact that the curvature of 𝑆 3 is 𝑅 = 6. Thus, the Casimir energy in 4D is 𝐸0 = 3𝑎 3𝑏 − 4 8 (𝑑 = 4). (3.40) • Choice of scheme As noted in [13], the 4D Casimir energy (3.40) is scheme-dependent — it can be shifted by redefining the local counterterm coefficient 𝑏. Similar statements hold in any even 𝑑 ≥ 4. However, CFT data is scheme-independent. To study it, we are free to choose whatever scheme is most convenient. In what follows, we will choose a scheme where 𝑆ct = 0, so that 𝑏-type terms are absent from both the Casimir energy and the Weyl anomaly. To define such a scheme in practice, one must choose a regulator, compute the Weyl anomaly with that regulator, and then add appropriate local counterterms to cancel the 𝑏-type terms.12 In this 𝑆ct = 0 scheme, the 𝑏-type terms DR[𝑆ct ] are not present in the thermal effective action, and the partition function on 𝑆 1𝛽 × 𝑆 𝑑−1 is given by   Tr 𝑒 −𝛽(𝐷+𝜀0 ) ∼ 𝑒 −𝑆th = 𝑒 −𝑆[b𝑔,𝐴]−𝑆Euler , (3.41) where 𝜀0 is the 𝑎 𝑑 -type contribution to the Casimir energy alone. Here, "∼" means equality up to exponentially suppressed corrections in 1/𝛽. 𝑆Euler is given in (3.30). We have used that 𝑆 1𝛽 × 𝑆 𝑑−1 is conformally-flat to drop the Weyl-invariants 𝐼 𝑘(𝑑) . In Appendix B, we describe how (3.41) comes about in a general scheme. 12 This requires a sufficiently "flexible" regulator that we can compute the Weyl anomaly. For example it is not obvious how to do this with a lattice regulator. Furthermore, such a scheme choice might clash with other symmetries, e.g. SUSY [13]. We thank Zohar Komargodski for discussion on these points 39 The value of 𝜀0 was computed in general 𝑑 in [113]: 𝜀0 ≡ 𝑎 𝑑 -type contribution to Casimir energy on 𝑆 𝑑−1  √ (𝑑−1)!!   (−2)  𝑑 even, 𝜋 𝑑/2 𝑎 𝑑 , = 𝑎 = 𝑑  Γ( 1−𝑑 0 𝑑 odd, 2 )  (3.42) where (𝑑 − 1)!! = (𝑑 − 1) · · · 3 · 1 for even 𝑑. Note that in 2D, we have 𝑎 2 = 𝑐/6. The partition function from the thermal effective action Let us now study the density of CFT operators with various dimensions and spins. We can obtain this from the partition function of the CFT on 𝑆 1𝛽 × 𝑆 𝑑−1 with a spin fugacity: i h ® 𝑀 ® −𝛽(𝐷+𝜀 0 )+𝑖𝛽Ω· ® (3.43) ∼ 𝑒 −𝑆[b𝑔,𝐴]−𝑆Euler . 𝑍 (𝛽, Ω) = Tr 𝑒 ® are the 𝑛 := ⌊ 𝑑 ⌋ generators of the Cartan Here, 𝐷 is the dilatation operator, 𝑀 2 ® subalgebra of the rotation group SO(𝑑), and Ω are spin fugacities. Geometrically, (3.43) is computed by a path integral on 𝑆 1𝛽 × 𝑆 𝑑−1 , with a twist by ® as we move around the thermal circle. The metric is 𝛽Ω 2 2 𝑑𝑠cylinder = 𝛽2 𝑑𝜏 2 + 𝑑𝑠sphere , (3.44) 2 where 𝜏 ∈ [0, 1] is a coordinate on 𝑆 1 and 𝑑𝑠sphere is the metric on 𝑆 𝑑−1 . To write down the metric on the sphere, let us choose coordinates that make Cartan rotations manifest. The Cartan generators are rotations in 𝑛 orthogonal 2-planes. We use radius-angle coordinates {𝑟 𝑎 , 𝜃 𝑎 } for each plane (𝑎 = 1, . . . , 𝑛), so the Cartan generators are simply 𝑖𝜕𝜃 𝑎 . If 𝑑 is odd, we have an extra axis and we use the Í 2 coordinate 𝑟 𝑛+1 for it. The radii satisfy the constraint 𝑛+𝜖 𝑎=1 𝑟 𝑎 = 1, where 𝜖 = 0 for even 𝑑 and 𝜖 = 1 for odd 𝑑. In these coordinates, the metric of the sphere is 2 𝑑𝑠sphere = 𝑛+𝜖 ∑︁ 𝑑𝑟 𝑎2 + 𝑎=1 where we have the constraint 𝑛 ∑︁ 𝑟 𝑎2 𝑑𝜃 𝑎2 , (3.45) 𝑎=1 Í𝑛+𝜖 𝑎=1 𝑟 𝑎 𝑑𝑟 𝑎 = 0. ® we identify the points In the twisted geometry that computes 𝑍 (𝛽, Ω), (𝜏, 𝜃 𝑎 ) ∼ (𝜏 + 1, 𝜃 𝑎 − 𝛽Ω𝑎 ). (3.46) Because of this identification, shifts in 𝜏 with fixed 𝜃 𝑎 are not periodic isometries, and thus the metric (3.45) is not in Kaluza-Klein (KK) form. To place it in KK 40 form, we redefine 𝜃 𝑎 → 𝜃 𝑎 + 𝛽Ω𝑎 𝜏, which removes the twist (3.46), and produces the new metric 𝑑𝑠2 = 𝛽2 𝑑𝜏 2 + 𝑛+𝜖 ∑︁ 𝑑𝑟 𝑎2 + 𝑛 ∑︁ 𝑟 𝑎2 (𝑑𝜃 𝑎 + 𝛽Ω𝑎 𝑑𝜏) 2 𝑎=1 𝑎=1 !2 𝑛 2Ω ∑︁ 𝑟 1 𝑎 𝑎 𝑟 𝑎2 Ω2𝑎 𝑑𝜏 + = 𝛽2 1 + Í𝑛 2 2 𝑑𝜃 𝑎 𝛽 1 + 𝑏=1 𝑟 𝑏 Ω𝑏 𝑎=1 𝑎=1 ! 𝑛 𝑛+𝜖 2𝑟 2 Ω Ω ∑︁ ∑︁ 𝑟 𝑎 𝑏 𝑎 𝑏 𝑑𝑟 𝑎2 + 𝑟 𝑎 𝑟 𝑏 𝛿𝑎𝑏 − + Í𝑛 2 2 𝑑𝜃 𝑎 𝑑𝜃 𝑏 . 1 + 𝑐=1 𝑟 𝑐 Ω𝑐 𝑎,𝑏=1 𝑎=1 ! 𝑛 ∑︁ (3.47) Comparing (3.47) with (3.8), we identify a metric 𝑔𝑖 𝑗 , a KK gauge field 𝐴𝑖 , and a dilaton 𝜙 given by ! 𝑛 ∑︁ 𝑒 2𝜙 = 𝛽2 1 + 𝑟 𝑎2 Ω2𝑎 , 𝑎=1 𝑟 𝑎2 Ω𝑎 1 𝑑𝜃 𝑎 , 𝐴= Í 𝛽 𝑎=1 1 + 𝑛𝑏=1 𝑟 𝑏2 Ω2𝑏 𝑛 ∑︁ 𝑔= 𝑛+𝜖 ∑︁ 𝑎=1 𝑑𝑟 𝑎2 + ! 𝑟 𝑎2𝑟 𝑏2 Ω𝑎 Ω𝑏 𝑑𝜃 𝑎 𝑑𝜃 𝑏 . 𝑟 𝑎 𝑟 𝑏 𝛿𝑎𝑏 − Í 1 + 𝑛𝑐=1 𝑟 𝑐2 Ω2𝑐 𝑎,𝑏=1 𝑛 ∑︁ (3.48) The effective metric b 𝑔 = 𝑒 −2𝜙 𝑔, together with 𝐴, then appears in the thermal effective action 𝑆[b 𝑔, 𝐴]. Explicitly, the cosmological constant term in the effective Lagrangian is 𝑛 ∑︁ √︁ 𝑑−1 1+ 𝑟𝑖2 Ω𝑖2 b 𝑔=𝑇 𝑖=1 ! − 𝑑2 𝑛 Ö 𝑟𝑖 , (3.49) 𝑖=1 while the Maxwell and Einstein densities are ! Í𝑛 2 2 𝑛 2) ∑︁ 𝑟 Ω (1 + Ω 𝐹 2 = 8𝑇 −2 Ω𝑖2 − 𝑖=1 𝑖Í𝑛𝑖 2 2 𝑖 , 1 + 𝑖=1 𝑟𝑖 Ω𝑖 𝑖=1 !! Í𝑛 2 4 𝑛 ∑︁ 1 − 𝑟 Ω 𝑖=1 𝑖 𝑖 b = 𝑇 −2 𝑑 (𝑑 + 1) 𝑅 Ω𝑖2 . Í𝑛 2 2 − 2(2𝑑 − 1) 1 − 1 + 𝑖=1 𝑟𝑖 Ω𝑖 𝑖=1 (3.50) As expected, at high temperature, the cosmological constant term gives the leading contribution, while the Einstein and Maxwell terms are subleading by 1/𝑇 2 , since they are two-derivative terms. Finally, the thermal effective action on our geometry 41 is given by integrating over 𝑆 𝑑−1 : ∫  √︁  b + 𝑐2 𝐹 2 + . . . b 𝑔 − 𝑓 + 𝑐1 𝑅 𝑆 𝑑−1 " ! # 𝑛 ∑︁ vol 𝑆 𝑑−1 = Î𝑛 − 𝑓 𝑇 𝑑−1 + (𝑑 − 2) (𝑑 − 1)𝑐 1 + (2𝑐 1 + 𝑑8 𝑐 2 ) Ω2𝑖 𝑇 𝑑−3 + . . . , 2) (1 + Ω 𝑖=1 𝑖 𝑖=1 𝑆[b 𝑔 , 𝐴] = (3.51) 𝑑/2 2𝜋 where vol 𝑆 𝑑−1 = Γ(𝑑/2) is the volume of the 𝑑−1-sphere.13 Note that the cosmological constant predicts the entire leading term in (3.51) as a detailed function of the spin fugacities Ω𝑖 . This leading term was first written down in [27].14 Using the thermal effective action, it is straightforward to incorporate more terms in 𝑆th and characterize the form of subleading corrections. Overall, we obtain a formula for the thermal partition function of a CFT with a spin fugacity in a systematic expansion in 1/𝑇. The leading term in (3.51) has poles at Ω𝑖 = ±𝑖. These poles are related to the unitarity bound because states close to the unitarity bound are not penalized by Boltzmann factors 𝑒 −𝛽(Δ−𝑖Ω𝐽) in this regime of angular fugacities. In our calculation, the poles come from locations on the sphere where 𝑟𝑖 = 1 (and the remaining 𝑟 𝑗 vanish). Despite additional poles in the expressions (3.50), the higher-order b and 𝐹 2 do not lead to further enhanced poles in the partition function. corrections 𝑅 b and 𝐹 2 are actually finite at 𝑟𝑖 = 1 when Ω𝑖 = ±𝑖. We expect The reason is that 𝑅 that this remains true for all higher-order corrections in the thermal effective action, so that the pole structure of (3.51) holds to arbitrary (perturbative) order in 1/𝑇. 13 To evaluate (3.51), we use the following explicit coordinates on 𝑆 𝑑−1 . For even 𝑑 = 2𝑛, the integral is ∫ 1 ∫ 1 𝑑𝑟 1 . . . 0 ∫ 2𝜋 𝑑𝑟 𝑛 0 ∫ 2𝜋 𝑑𝜃 1 . . . 𝑑𝜃 𝑛 𝛿 0 0  √︁  √︃  b + 𝑐2 𝐹 2 + . . . , 𝑟 12 + · · · + 𝑟 𝑛2 − 1 b 𝑔 − 𝑓 + 𝑐1 𝑅 and for odd 𝑑 = 2𝑛 + 1, the integral is ∫ 1 ∫ 1 𝑑𝑟 1 . . . 0 ∫ 1 𝑑𝑟 𝑛 0 ∫ 2𝜋 𝑑𝑟 𝑛+1 −1 ∫ 2𝜋 𝑑𝜃 1 . . . 0 𝑑𝜃 𝑛 𝛿 0 √︃  √︁   2 −1 b + 𝑐2 𝐹 2 + . . . . 𝑟 12 + · · · + 𝑟 𝑛+1 b 𝑔 − 𝑓 + 𝑐1 𝑅 To compute either case, we used the Feynman parametrization identity: 1 Γ(𝛼1 + · · · + 𝛼 𝑘 ) 𝛼𝑘 = 𝛼1 Γ(𝛼1 ) . . . Γ(𝛼 𝑘 ) 𝐴1 . . . 𝐴 𝑘 ∫ 1 ∫ 1 𝑑𝑢 1 · · · 0 𝑑𝑢 𝑘 0 𝛿(𝑢 1 + · · · + 𝑢 𝑘 − 1)𝑢 1𝛼1 −1 . . . 𝑢 𝑘𝛼𝑘 −1 (𝑢 1 𝐴1 + · · · + 𝑢 𝑘 𝐴 𝑘 ) 𝛼1 +···+𝛼𝑘 . (3.52) In order for (3.51) to be valid, we require that the convex hull of 1 + Ω2𝑖 not contain 0. Otherwise, the integral diverges (as expected from the unitarity bound of the CFT). 14 Our Ω ® is related to the one in [27] by 𝑖 Ω ® here = Ω ® there . 42 Specifically, we expect that the coefficient of 𝑇 𝑑−2𝑘−1 is a degree-2𝑘 polynomial in Î𝑛 the Ω𝑖 , times an overall factor 1/ 𝑖=1 (1 + Ω𝑖2 ). Finally, let us compute 𝑆Euler by plugging (3.48) into (3.30). In 𝑑 = 2, 4, 6, we find 𝑑=2 𝑆Euler = 0, 𝑎 4 𝛽 (Ω21 − Ω22 ) 2 , 12 (1 + Ω21 )(1 + Ω22 ) 1 3𝑎 6 𝛽 𝑑=6 𝑆Euler =− 2 80 (1 + Ω1 )(1 + Ω22 )(1 + Ω23 ) h × Ω61 + Ω62 + Ω63 − 4(Ω41 (Ω22 + Ω23 ) + Ω42 (Ω23 + Ω21 ) + Ω43 (Ω21 + Ω22 )) i + 21Ω22 Ω23 Ω21 + 5(Ω21 Ω22 + Ω22 Ω23 + Ω23 Ω21 ) − 5(Ω41 + Ω42 + Ω43 ) . (3.53) 𝑑=4 𝑆Euler =− In all these cases, 𝑆Euler has the same functional form as expected from the 𝑂 (𝑇 −1 ) terms in the Weyl-invariant part of the thermal effective action 𝑆[b 𝑔 , 𝐴] — namely a Î𝑛 2 polynomial of degree 𝑑 in the Ω𝑖 ’s times 𝛽/ 𝑖=1 (1 + Ω𝑖 ). Thus, the effects of 𝑆Euler cannot be distinguished from 𝑆[b 𝑔, 𝐴] in the CFT partition function on 𝑆 1 × 𝑆 𝑑−1 . It would be interesting to try to distinguish these terms in some example theories, perhaps by studying stress-tensor correlators in thermal flat space. Leading asymptotic formula From (3.51) we can extract the high energy density of states for any CFT15 . Let us first consider the leading term of the high-temperature partition function: vol 𝑆 𝑑−1 𝑓 𝑇 𝑑−1 + 𝑂 (𝑇 𝑑−3 ), log 𝑍 (𝑇, Ω𝑖 ) = Î𝑛 2) (1 + Ω 𝑖=1 𝑖 (3.54) where 𝑛 = ⌊𝑑/2⌋. To extract the density of states, we perform an inverse Laplace transform on the partition function, which can be done by saddle point approximation. Before we do the general 𝑑 case however, let us first do 𝑑 = 2 and 𝑑 = 3 explicitly. We pause to note that we can compute either the asymptotic density of all operators of the CFT, including both primary and descendent operators, or the asymptotic density of only the conformal primary operators. We compute the latter by decomposing the partition function into the conformal characters. In 𝑑 dimensions the characters 15 A previous version of this paper had minor typos in the density of states that we have corrected. We thank Sasha Diatlyk and Yifan Wang for pointing them out to us. 43 are given by16 𝑒 −𝛽Δ 𝜒𝐽 (𝛽Ω𝑖 )    Î𝑛 (1−𝑒 −𝛽 (1+𝑖Ω𝑖𝑖) ) (1−𝑒 −𝛽 (1−𝑖Ω𝑖 ) ) ,  𝑖=1 𝜒Δ,𝐽𝑖 (𝛽, Ω𝑖 ) = 𝑒 −𝛽Δ 𝜒𝐽𝑖 (𝛽Ω𝑖 )   Î , 𝑛  (1−𝑒 −𝛽 ) 𝑖=1 (1−𝑒 −𝛽 (1+𝑖Ω𝑖 ) ) (1−𝑒 −𝛽 (1−𝑖Ω𝑖 ) ) 𝑑 even, (3.55) 𝑑 odd, where 𝜒𝐽𝑖 (𝜃 𝑖 ) is the character of the SO(𝑑) representation 𝜆 = (𝐽1 , . . . , 𝐽𝑛 ). The partition function is a sum over characters, with an additional inclusion of the Casimir energy 𝜀0 defined in (3.42): ∑︁ 𝑍 (𝑇, Ω𝑖 ) = 𝑒 −𝛽𝜀0 𝜒Δ𝑖 ,𝐽𝑖 (𝛽, Ω𝑖 ). (3.56) Δ𝑖 ,𝐽𝑖 • 𝑑=2 From (3.54), our high-temperature expression for the partition function in 𝑑 = 2 is   2𝜋 𝑓 𝑇 𝑍 (𝑇, Ω) 𝑑=2 ≈ exp 1 + Ω2   4𝜋 2 𝑐𝑇 = exp , (3.57) 12(1 + Ω2 ) where we used the relation between 𝑓 and 𝑐 for 2D CFTs (3.34). We would like to take the inverse Laplace transform to extract the high-energy density of states. This calculation is precisely Cardy’s calculation for the high-energy density of states [46], but we include it for completeness. It is convenient to first change variables: 𝛽 𝐿 := 1 𝑖Ω + , 𝑇 𝑇 𝛽 𝑅 := 1 𝑖Ω − , 𝑇 𝑇 (3.58) which gives  𝑍 (𝛽 𝐿 , 𝛽 𝑅 ) 𝑑=2 := Tr 𝑒 𝑐 𝑐 −𝛽 𝐿 ( Δ−2 𝐽 − 24 ) −𝛽 𝑅 ( Δ+𝐽 2 − 24 )  𝑒 ≈𝑒 2 𝜋2 𝑐 12  1 1 𝛽 𝐿 + 𝛽𝑅  , (3.59) where we include the Casimir shift described in Sec 3.3. Taking the inverse Laplace transform then gives the following integral: 1 𝜌 states 𝑑=2 (Δ, 𝐽) ∼  1 2 2𝜋𝑖 ∫ 𝛾+𝑖∞ 𝑑𝛽 𝐿 𝑒 𝛾−𝑖∞ 2 𝜋2 𝑐 12𝛽 𝐿 +𝛽 𝐿 ( Δ−2 𝐽 − 24𝑐 )  1 2𝜋𝑖 ∫ 𝛾+𝑖∞ 𝑑𝛽 𝑅 𝑒 2 𝜋2 𝑐 12𝛽𝑅 +𝛽𝑅  𝑐 ( Δ+𝐽 2 − 24 ) . 𝛾−𝑖∞ (3.60) 16 The characters (3.55) are for long representations of the conformal group. For special values of Δ, 𝐽𝑖 (e.g. states at the unitarity bound), the representation may be shortened and the expression for the character will be modified. However, since we are interested in reading off the density of primary operators at large dimension, short representations will not play a role. 44 Although we can do this integral by saddle, it actually can be done exactly. The tree level piece is "√︂ !# √︂ √︂ 2𝑐 Δ + 𝐽 Δ − 𝐽 𝑐 𝑐 𝜌 states 𝜋 − + − . (3.61) 𝑑=2 (Δ, 𝐽) ∼ exp 3 2 24 2 24 We see this is none other than the Cardy formula [46]. If we do the integrals in (3.60) exactly17 , we get 𝜌 states 𝑑=2 (Δ, 𝐽) ! ! √︂ √︂ Δ+𝐽 𝑐 2𝑐 Δ−𝐽 𝑐 𝐼1 − 𝐼1 𝜋 − ∼ √︁ 𝑐 𝑐 2 24 3 2 24 ) (Δ − 𝐽 − 12 ) 6 (Δ + 𝐽 − 12 " !# √︂ √︂ √︂ √ 𝑐 𝑐 𝑐 2𝑐 Δ+𝐽 Δ−𝐽 =√ exp 𝜋 − + − 𝑐 3/4 𝑐 3/4 3 2 24 2 24 48(Δ + 𝐽 − 12 ) (Δ − 𝐽 − 12 )   × 1 + 𝑂 (Δ−1/2 ) , (3.63) √︂ 𝜋2 𝑐 2𝑐 𝜋 3 √︂ where 𝐼1 is a modified Bessel function of the first kind. The expression (3.63) indeed gives the known logarithmic corrections to Cardy’s formula [47]. So far, (3.63) is counting the density of all states rather than the density of global or Virasoro primaries. A more natural object from the CFT perspective may be to count the density of primary operators. In order to generalize more easily to higher dimensions, we will now compute the asymptotic density of global (not Virasoro) primary operators. The calculation is almost identical, except now instead of taking the inverse Laplace transform of (3.59), we include the characters (3.55):   ∫ 1 2 𝜋2 𝑐 1 𝑐 𝑐 primaries ) −𝛽 𝑅 ( Δ+𝐽 − 24 ) −𝛽 𝐿 ( Δ−2 𝐽 − 24 12 𝛽 𝐿 + 𝛽𝑅 2 (1−𝑒 −𝛽 𝐿 )(1−𝑒 −𝛽𝑅 ). 𝑒 ≈𝑒 𝑑Δ𝑑𝐽 𝜌 𝑑=2 (Δ, 𝐽)𝑒 (3.64) 17 Strictly speaking the integral in (3.60) diverges. The precise statement is 1 2𝜋𝑖 ∫ 𝛾+𝑖∞ 𝑑𝛽𝑒 𝛾−𝑖∞ 𝛽Δ  𝜋𝑓  √︂ 𝜋 𝑓 √︁  𝑒 𝛽 −1 = 𝐼1 4𝜋 𝑓 Δ . Δ (3.62) This leads to an additional factor of 𝛿(Δ) in the inverse Laplace transform. However, since we are using this method to read off the large energy density of states, it does not affect our final expression (3.63). 45 Taking the inverse Laplace transform we then get primaries 𝜌 𝑑=2 • 𝑐3/2 𝜋 2 (Δ, 𝐽) = √ 𝑐 5/4 𝑐 5/4 432(Δ + 𝐽 − 12 ) (Δ − 𝐽 − 12 ) "√︂ !# √︂ √︂ 2𝑐 Δ+𝐽 𝑐 Δ−𝐽 𝑐 × exp 𝜋 − + − 3 2 24 2 24   × 1 + 𝑂 (Δ−1/2 ) . (3.65) 𝑑=3 Now let us redo this analysis for 𝑑 = 3. The logic is the same except now the form of the partition function is different:   4𝜋 𝑓 𝑇 2 𝑍 (𝑇, Ω) 𝑑=3 ≈ exp . (3.66) 1 + Ω2 Using the same change variables (3.58) we get  𝑍 (𝛽 𝐿 , 𝛽 𝑅 ) 𝑑=3 := Tr 𝑒 Δ+𝐽 −𝛽 𝐿 ( Δ−𝐽 2 ) −𝛽 𝑅 ( 2 ) 𝑒    4𝜋 𝑓 ≈ exp . 𝛽𝐿 𝛽𝑅 (3.67) To extract the density of states we again use an inverse Laplace transform:  2 ∫ 𝛾+𝑖∞     Δ−𝐽 4𝜋 𝑓 Δ+𝐽 + 𝛽𝐿 𝑑𝛽 𝐿 𝑑𝛽 𝑅 exp + 𝛽𝑅 . 𝛽𝐿 𝛽𝑅 2 2 𝛾−𝑖∞ (3.68) The integral in (3.68) is more complicated so now we do it via saddle point analysis. The saddles in 𝛽 𝐿 , 𝛽 𝑅 are located at  1 𝜌 states 𝑑=3 (Δ, 𝐽) ≈ 1 2 2𝜋𝑖 𝛽∗𝐿 =  8𝜋 𝑓 (Δ + 𝐽) (Δ − 𝐽) 2   1/3 , 𝛽∗𝑅 =  8𝜋 𝑓 (Δ − 𝐽) (Δ + 𝐽) 2  1/3 . (3.69) This gives a remarkably simple expression for the tree-level density of states: h i 1/3 1/3 states 1/3 1/3 (Δ + 𝐽) (Δ − 𝐽) 𝜌 𝑑=3 (Δ, 𝐽) ∼ exp 3𝜋 𝑓 . (3.70) Keeping the one-loop terms we get 𝜌 states 𝑑=3 (Δ, 𝐽) ∼ √ 𝑓 1/3 3𝜋 2/3 (Δ + 𝐽) 2/3 (Δ − 𝐽) 2/3 h exp 3𝜋 1/3 1/3 𝑓 (Δ + 𝐽) 1/3 (Δ − 𝐽) 1/3 i . (3.71) The expression (3.71) again is counting the asymptotic density of states rather than conformal primaries. We can read off the density of primaries from the character 46 formula (3.55). We get primaries 𝜌 𝑑=3 (Δ, 𝐽) ∼ √ " 8𝜋 2/3 𝑓 5/3 (2𝐽 + 1)Δ 3(Δ + 𝐽 + 12 ) 7/3 (Δ − 𝐽 − 12 ) 7/3 exp 3𝜋 1/3 1/3  𝑓 1 Δ+𝐽+ 2  1/3  1 Δ−𝐽− 2  1/3 # . (3.72) Note that in (3.72), 2𝐽 + 1 is the dimension of the spin 𝐽 representation of 𝑆𝑂 (3). This is reminiscent of the formulas found in [102, 128], where the density of a global symmetry representation 𝜌 is proportional to its dimension dim 𝜌. This comes about because the high temperature partition function can be approximated by a delta-function on a group (the rotation group in this case) centered at the identity, whose harmonic transform is the Plancherel measure dim 𝜌/vol 𝐺. • General 𝑑 For general 𝑑 > 3, there are several chemical potentials to turn on, so the final leading formula for the density of states is a little more cumbersome. For simplicity, we will simply compute the tree-level asymptotic density of states (i.e. the value at the saddle-point, not including the Gaussian determinant). Note that because in this section we are computing the tree-level contribution, the formula is identical for states and for primaries. First, let us consider the case where we only turn on information about one spin, for simplicity. The saddles in temperature and chemical potential are located at  1/𝑑  (Δ + 𝜀0 − 𝑖𝐽Ω∗ )(1 + Ω2∗ ) , 𝑇∗ = (𝑑 − 1) 𝑓 vol 𝑆 𝑑−1 √︁ (Δ + 𝜀0 ) 2 + (𝑑 − 3)(𝑑 − 1)𝐽 2 − (Δ + 𝜀 0 ) , (3.73) Ω∗ = −𝑖 𝐽 (𝑑 − 3) which lead to a high energy density of states of log 𝜌 𝑑 (Δ, 𝐽) ∼ 𝑑−1 © 𝑑 (Δ + 𝜀 0 ) 𝑑 ­ 𝑑−1 « vol 𝑆 𝑑−1 𝑓 (𝑑 − 1) 1 ©𝑑 − 2 ×­ − 𝑑−3 𝑑−3 « 2 √︄ 1+ √︄ © ­1 + « 1+ (𝑑 − 3) (𝑑 − 1)𝐽 2 (Δ + 𝜀 0 ) 2 (𝑑 − 3) (𝑑 − 1)𝐽 2 (Δ + 𝜀 0 ) 2 ªª ®® ¬¬ . (3.74) 1− 𝑑2 ª ® ¬ 1 𝑑 This leading order formula matches the result in Equation (49) of [188]. The expression (3.74) reproduces (3.61) and (3.70) for 𝑑 = 2 and 𝑑 = 3 respectively. Note that to reproduce (3.70) we set the Casimir energy 𝜀0 = 0 and use the fact that √︁ (𝑑 − 2)Δ − Δ2 + (𝑑 − 3)(𝑑 − 1)𝐽 2 Δ2 − 𝐽 2 lim = . (3.75) 𝑑→3 𝑑−3 Δ 47 Now let us consider all the chemical potentials Ω𝑖 turned on. The leading term for the partition function is given in (3.54), which means ! ∫ 𝑛 Δ + 𝜀0 vol 𝑆 𝑑−1 𝑓 𝑇 𝑑−1 ∑︁ 𝑖Ω𝑖 𝐽𝑖 𝜌 𝑑 (Δ, 𝐽𝑖 ) ∼ 𝑑𝑇 𝑑Ω𝑖 exp + Î𝑛 , (3.76) − 2) 𝑇 𝑇 (1 + Ω 𝑖=1 𝑖 𝑖=1 where for even dimensions, 𝜀0 is the Casimir energy as defined in (3.42). The saddle for the 𝑇 integral is easy to compute, and is located at ! 1/𝑑 Î Í (Δ + 𝜀0 − 𝑖 𝑖 𝐽𝑖 Ω𝑖 ) 𝑖 (1 + Ω𝑖2 ) . (3.77) 𝑇∗ = vol 𝑆 𝑑−1 𝑓 (𝑑 − 1) Plugging this back into (3.76), we get 𝜌 𝑑 (Δ, 𝐽𝑖 ) ∫ ∼ ! 𝑑−1  𝑑 Ö   − 𝑑1  ∑︁  1 𝑑−1 − 𝑑 𝑑−1 2  . 𝑑 𝑑Ω𝑖 exp 𝑑 (𝑑 − 1) 𝑓 ) Δ + 𝜀0 − 𝑖 (vol 𝑆 𝐽𝑖 Ω𝑖 1 + Ω𝑖    𝑖 𝑖   (3.78) We now want to find the saddles in the chemical potentials. Taking a derivative with respect to each Ω 𝑗 gives us the following equations to solve for the saddle Ω∗, 𝑗 : − ∑︁ Ω∗, 𝑗 𝑖(𝑑 − 1)𝐽 𝑗 . = (Δ + 𝜀0 − 𝑖 𝐽𝑖 Ω∗,𝑖 ) 2 1 + Ω2∗, 𝑗 𝑖 (3.79) If we solve (3.79) for Ω∗, 𝑗 for 𝑗 = 1, 2, . . . , ⌊ 𝑑2 ⌋, and plug into (3.78), we get the tree-level density of states (or primaries) in 𝑑 dimensions. To describe the solution for (3.79), it is useful to define the quantity 𝑎 as the following: Í (Δ + 𝜀0 ) − 𝑖 𝑖 𝐽𝑖 Ω∗,𝑖 𝑎 := . (3.80) 𝑑−1 𝑎 is a function of Δ + 𝜀0 and the spins 𝐽𝑖 , and it is a symmetric function of 𝐽𝑖 . Each saddle point of Ω satisfies the following equations: √︃ −𝑎 + 𝑎 2 + 𝐽 2𝑗 2𝑖𝑎Ω2∗, 𝑗 , Ω∗, 𝑗 = −𝑖 . (3.81) 𝐽𝑗 = 𝐽𝑗 1 + Ω2∗, 𝑗 When we substitute this relation into (3.79), we get the density of states at leading order to be 1 √︃ 2 𝑑 𝐽 𝑖 𝑛 ©1 + 1 + Ö ª 1 𝑑−1 𝑎2 ® ­ log 𝜌 𝑑 (Δ, 𝐽1 , . . . , 𝐽𝑛 ) ∼ 𝑑 (vol 𝑆 𝑑−1 𝑓 ) 𝑑 𝑎 𝑑 (3.82) ® , ­ 2 𝑖=1 « ¬ 48 where 𝑎 is the positive real solution of Δ + 𝜀0 = ⌊ 𝑑−1 2 ⌋𝑎 + ∑︁ √︃ 𝑎 2 + 𝐽𝑖2 . (3.83) 𝑖 Because the right-hand side of (3.83) is a monotonically increasing function of 𝑎, there is only one positive real 𝑎 satisfying (3.83) for a given energy Δ + 𝜀0 and spins 𝐽𝑖 . • Perturbative corrections to the density of states Besides corrections coming from higher-loop terms in the saddle point analysis, the first nontrivial correction for 𝑑 > 2 comes from higher derivative terms in the effective action of 𝑆th in 3.51. Note that in 𝑑 = 2 these corrections are absent (see 3.33). This is another way of understanding that the corrections to the first line of 3.63 are non-perturbatively suppressed in Δ (and come from the modular 𝑆 transformation of the lightest non-vacuum operator). For 𝑑 > 2, the first correction comes from the Maxwell and Einstein terms in the effective action. This will turn out to induce a correction to the entropy of the form 𝑑−3 Δ 𝑑 . To see this, let us look at our expression for the thermal effective action (3.51). In 𝑑 = 3 we have   4𝜋 8 2 2 −2 log 𝑍 𝑑=3 (𝑇, Ω) = 𝑓 𝑇 − (2𝑐 1 + (2𝑐 1 + 𝑐 2 )Ω ) + 𝑂 (𝑇 ) . 3 1 + Ω2 (3.84) From this we can extract a correction to the density of states from our saddle (3.69). We get 2 1 1/3 1/3 (Δ + 𝐽) 1/3 (Δ − 𝐽) 1/3 − log(Δ2 − 𝐽 2 ) + log 𝑓 log 𝜌 states 𝑑=3 (Δ, 𝐽) = 3𝜋 3 3  𝑓  √ 3 3𝜋 2 32𝑐 2 𝐽 2 𝜋 + 𝑂 (Δ−1/3 ), 3(Δ2 − 𝐽 2 ) !   1/3   1/3 1 1 Δ(2𝐽 + 1) primaries log 𝜌 𝑑=3 (Δ, 𝐽) = 3𝜋 1/3 𝑓 1/3 Δ + 𝐽 + Δ−𝐽− + log 2 2 (Δ2 − (𝐽 + 12 ) 2 ) 7/3  2/3 5/3  32𝑐 2 (𝐽 + 12 ) 2 𝜋 8𝜋 𝑓 + log − 8𝜋𝑐 1 + + 𝑂 (Δ−1/3 ). (3.85) √ 3(Δ2 − (𝐽 + 12 ) 2 ) 3 − 8𝜋𝑐 1 + Note that the size of the Maxwell term (𝑐 2 ) compared to higher-derivative terms depends on the order of limits in Δ, 𝐽. If we take Δ ≫ 𝐽 ≫ 1, then the Maxwell term scales as Δ−2 (instead of Δ0 ), can be neglected at this order in the derivative expansion. However, if we instead take a limit where Δ/𝐽 is fixed and then take Δ to infinity, then it is important. 49 For 𝑑 > 3, we have the following correction to the density of states: 1 √︃ 𝐽𝑖2 𝑑 𝑛 Ö 1 + 1 + © ª 1 𝑑−1 𝑎2 ® ­ log 𝜌 𝑑 (Δ, 𝐽1 , . . . , 𝐽𝑛 ) ∼ 𝑑 (vol 𝑆 𝑑−1 𝑓 ) 𝑑 𝑎 𝑑 ­ ® 2 𝑖=1 « ¬ 3 √︃ √︃ 𝐽𝑖2 𝑑   ∑︁ 1 + 𝐽 2 /𝑎 2 − 1 Ö 𝑛 ª © 1 + 1 + 𝑎2 ª 𝑖 3 𝑑−3 © 8 𝑑−2 ® ® ­ − (vol 𝑆 𝑑−1 𝑓 ) 𝑑 𝑎 𝑑 ­­ (𝑑 − 1)𝑐 1 − 2𝑐 1 + 𝑐 2 √︃ ® ® ­ 𝑓 𝑑 2 2 2 𝑖 1 + 𝐽𝑖 /𝑎 + 1 𝑖=1 « ¬ ¬ «   𝑑−5 +𝑂 𝑎 𝑑 (3.86) , where 𝑎 is defined in (3.83). In order to trust the high-temperature expansion, we demand that the temperature at the saddle point be large. The saddle temperature 1 (3.77) is proportional to 𝑎 𝑑 , with the remaining factors being O (1). Therefore, we can expand our formula in 𝑎. We will discuss the regime of validity of our formulas in more detail in Sec 3.3. If we keep the information from only one spin 𝐽, then we get a correction of the form 1   1− 𝑑2   𝑑1  𝑑−1 𝑑 (𝑑 − 3)𝛼𝐽 𝑑 𝛼𝐽 𝑑−1 𝑑 log 𝜌 𝑑 (Δ, 𝐽) ∼ 1+ (Δ + 𝜀 0 − 𝛼𝐽) (𝑑 − 1)vol 𝑆 𝑓 1− 𝑑−1 Δ + 𝜀0 Δ + 𝜀0   𝑑−1 3/𝑑 𝑑−3 (𝑑 − 2) (vol 𝑆 𝑓) 8𝑐 2 + 2𝑐 1 𝑑 2 2 −3/𝑑 𝑑 (Δ + 𝜀 0 − 𝛼𝐽) (1 − 𝛼 ) − 𝛼 (𝑑 − 1)𝑐 1 − 𝑑−3 𝑑 (𝑑 − 1) 𝑑 𝑓   𝑑−5 (3.87) + 𝑂 (Δ + 𝜀 0 ) 𝑑 , 𝐽 where 𝛼 := Δ+𝜀 0 √ 𝑑−1 . 1+ 1+(𝑑−1) (𝑑−3)𝐽 2 /(Δ+𝜀 0 ) 2 Because Δ + 𝜀0 is larger than 𝐽, |𝛼| is always in between 0 and 1. The corrections from the Maxwell and Einstein terms are in fact more important than the Gaussian fluctuations about the saddle point. Again, if Δ + 𝜀0 ≫ |𝐽 |, then |𝛼| ≪ 1, so the Einstein (𝑐 1 ) term dominates in 3.87 and the Maxwell (𝑐 2 ) term is subleading. However, if (Δ + 𝜀0 )/𝐽 is fixed as Δ → ∞, then 𝛼 ∼ 𝑂 (1), and the two terms are comparable. • Regime of validity Because we are doing a high-temperature expansion, in order for our formulas to be valid, we need the saddle-point value of temperature, 𝑇∗ , to be large. From (3.77), we see that we need to take Δ ≫ 𝑓 . However, large Δ is not a sufficient condition — it is possible that the saddles in Ω are sufficiently close to 𝑖 to make the saddle in Í 𝑇 no longer large. This puts a condition on the twist Δ − 𝑖 |𝐽𝑖 | of the operators. In Í particular, if 𝑚 of the ⌊ 𝑑2 ⌋ spins are large, meaning |𝐽1 |, . . . |𝐽𝑚 | ≫ Δ − 𝑖 |𝐽𝑖 |, then 50 our universal entropy formula is only valid when Δ− ∑︁ 𝑖 |𝐽𝑖 | ≫ 𝑓 𝑚 Ö 1 ! 𝑚+1 |𝐽𝑖 | . (3.88) 𝑖=1 For example, in CFT3 , this is equivalent to the entropy formula only being valid in the regime √︁ Δ − |𝐽 | ≫ 𝑓 Δ. (3.89) (This can be seen easily by demanding the saddles in (3.69) are very small.) Operators outside of this window (for example operators along a Regge trajectory with √ Δ large but Δ − |𝐽 | growing slower than 𝐽) will not obey our universal entropy formula. In 𝑑 = 2, the regime of validity is larger. In order to trust the saddle point analysis, it is only necessary to take Δ → ∞ with Δ − |𝐽 | ≫ 𝑐 (3.90) (or equivalently take ℎ, ℎ ≫ 𝑐). • Non-perturbative corrections to the density of states So far, we have discussed an infinite set of perturbative corrections in 1/𝑇 to log 𝑍 (𝑇, Ω𝑖 ), parametrized by an infinite set of terms in the thermal effective action. In this section we briefly consider nonperturbative corrections to log 𝑍 (𝑇, Ω𝑖 ), namely corrections that scale as 𝑒 −#𝑇 . In general we expect the first nonperturbative correction to be proportional to 𝑒 −2𝜋𝑚 , (3.91) where 𝑚 is the mass of the lightest massive state in the dimensionally reduced theory. By dimensional analysis, 𝑚 ∝ 𝑇, so (3.91) is indeed a nonperturbative correction. The reason for (3.91) is the following. Consider the CFT on 𝑆 1𝛽 × 𝑆 𝑑−1 as a gapped theory on 𝑆 𝑑−1 . Corrections of the form (3.91) will be generated by world-line instantons associated with a massive particle moving along a great circle of 𝑆 𝑑−1 of length 2𝜋. Similar world line instantons were studied in [39, 79, 99, 109, 110] in the context of the large-charge expansion.18 18 We thank Yifan Wang for pointing out this interpretation and associated references. 51 We can also understand (3.91) from a Hamiltonian perspective. We can compute the partition function of the gapped theory on 𝑆 𝑑−1 by slicing the path integral on 𝑆 𝑑−2 spatial slices at various polar angles 𝜃 ∈ [0, 𝜋]. These slices have varying radius sin 𝜃, and therefore varying Hamiltonian 𝐻 (𝜃). Overall, we time-evolve by Euclidean time 𝜋 as we move from the south pole to the north pole of the 𝑆 𝑑−1 . This gives an expression for the partition function of the form   ∫ 𝜋 𝑍 = ⟨𝜓0 |T exp − 𝑑𝜃 𝐻 (𝜃) |𝜓0 ⟩, (3.92) 0 where |𝜓0 ⟩ is the state in the 𝑆 𝑑−2 Hilbert space created by the path integral near the south pole, and T denotes Euclidean time ordering. In general, the spectrum of 𝐻 (𝜏) could be quite complicated. However, when 𝛽 is small, most spatial slices are large compared to the mass gap, and we expect the low-lying spectrum of 𝐻 (𝜏) to be close to the gapped spectrum in flat space R𝑑−2 . In particular, there is a contribution from a particle-antiparticle pair nucleated at the south pole, which propagate for Euclidean time 𝜋 before annihilating at the north pole. This leads to 3.91. In general, we expect similar corrections of the form 𝑒 −2𝜋𝑚𝑖 for each massive state in the gapped theory on R𝑑−2 . There will also be Lüscher corrections [159] that it would be interesting to study in more detail. We can check the prediction (3.91) explicitly in free theories. In Appendix C, we write down the high-temperature expansions of the partition function for a 𝑑dimensional free boson and free Dirac fermion respectively. In even dimensions, we write down the exact expression, including all non-perturbative corrections; in odd dimensions, the perturbative expansion is asymptotic rather than convergent, but we are still able to write down the first non-perturbative correction. In all cases, we show that the first non-perturbative correction at high temperature to the partition function take the form as predicted by (3.91). 3.4 Density of states: examples In this section, we study partition functions of various CFTs to illustrate the general results of the previous sections. The examples we consider are the free scalar, the free scalar with a Z2 twist, the free fermion, and holographic CFTs where the entropy is well-approximated by that of a Kerr-AdS black hole. In these examples, we check the partition function against our general formula (3.51) and determine the unknown coefficients 𝑓 , 𝑐 1 , and 𝑐 2 when the thermal effective action applies. Furthermore, in Appendix 3.9, we compare the predictions of the thermal effective 52 action to numerical bootstrap data for the 3D Ising model, obtaining an estimate for 𝑓. Free scalar Our first example is the free scalar in 𝑑 dimensions. When compactified, this theory contains a gapless sector corresponding to a free scalar in one lower dimension. Therefore, it violates the central assumption of the thermal effective action, and the predictions of the thermal EFT should be violated in some way. The partition function of this theory can be computed exactly. For a review, see Appendix C. Expanding the result at high temperature, we find for example Í ⌊ 𝑑2 ⌋ 2 𝑑 − 4 − 2 © ª 𝑖=1 Ω𝑖 𝜁 (𝑑 − 2)𝑇 𝑑−3 + 𝑂 (𝑇 𝑑−5 ) ® . log 𝑍 (𝑇, Ω𝑖 ) = ­2𝜁 (𝑑)𝑇 𝑑−1 − Î ⌊ 𝑑2 ⌋ 12 2 ¬ 𝑖=1 (1 + Ω𝑖 ) « (3.93) 1 Importantly, in even 𝑑, we find that the high temperature expansion contains a term proportional to 𝑇 0 , while in odd dimensions there is a term proportional to log 𝑇. (When 𝑑 = 3, the logarithm is visible in (3.93) via the pole in the 𝜁-function at 1.) Such terms are inconsistent with the derivative expansion of the thermal effective action (which contains powers of the form 𝑇 𝑑−2𝑘−1 for integer 𝑘). They represent contributions from the gapless sector. We discuss these terms more explicitly in Appendix C. Free scalar with a Z2 twist To remove the gapless sector in the compactified free scalar, we can insert a Z2 twist on the 𝑆 1 , where we identify 𝜙(𝜏 = 1) = −𝜙(𝜏 = 0) as we go around the thermal circle. Computing the partition function with this twist inserted (this can be treated using methods in e.g. [80]), we find 1 × 2) (1 + Ω 𝑖 𝑖=1 Í ⌊ 𝑑2 ⌋ 2     (𝑑 − 4) − 2 1 1 © ª 𝑖=1 Ω𝑖 𝜁 (𝑑 − 2)𝑇 𝑑−3 + 𝑂 (𝑇 𝑑−5 ) ® . ­−2 1 − 𝑑−1 𝜁 (𝑑)𝑇 𝑑−1 + 1 − 𝑑−3 12 2 2 « ¬ (3.94) log 𝑍 (𝑇, Ω𝑖 ) = Î ⌊ 𝑑2 ⌋ This result is now consistent with the thermal effective action (even though the compactification is no longer "thermal"). For example, when 𝑑 = 3, the extra factor (1 − 1/2𝑑−3 ) cancels the pole in 𝜁 (𝑑 − 2), so there is no log 𝑇 term. Matching with 53 (3.51), we get   2𝜁 (𝑑) 1 𝑓 =− 1 − 𝑑−1 , vol 𝑆 𝑑−1 2   1 (𝑑 − 4)𝜁 (𝑑 − 2) 1 − 𝑑−3 , 𝑐1 = − 12(𝑑 − 1)(𝑑 − 2)vol 𝑆 𝑑−1 2   𝑑 (2𝑑 − 5)𝜁 (𝑑 − 2) 1 𝑐2 = 1 − 𝑑−3 , 48(𝑑 − 1)(𝑑 − 2)vol 𝑆 𝑑−1 2 (3.95) for a free scalar with a Z2 twist. Note that 𝑓 < 0 in (3.95). This is not a contradiction because the partition function computed in (3.94) is not a positive-definite sum of states (due to the insertion of the Z2 twist). In fact 𝑓 < 0 implies a strong cancellation between Z2 -even and odd operators in the theory. Free fermion Next, let us consider a free Dirac fermion in 𝑑 dimensions. We compactify the theory with thermal boundary conditions, where we do not insert (−1) 𝐹 . This leads to a massive (𝑑 − 1)-dimensional theory. (With the (−1) 𝐹 operator inserted, there would be a gapless sector.) We compute the partition function explicitly in Appendix C. The leading two terms in the free energy are given by 𝑑 2 ⌊ 2 ⌋+1 × 2) (1 + Ω 𝑖 𝑖=1 Í ⌊ 𝑑2 ⌋ 2        (𝑑 − 1) + 1 1 𝑖=1 Ω𝑖 𝑑−1 𝑑−3 𝑑−5   1− 𝜁 (𝑑)𝑇 − 1 − 𝑑−3 𝜁 (𝑑 − 2)𝑇 + 𝑂 (𝑇 ) .  𝑑−1 24 2 2     (3.96) log 𝑍 (𝑇,Ω𝑖 ) = Î ⌊ 𝑑2 ⌋ Unlike in the free scalar case, the expression (3.96) has no log 𝑇 terms or 𝑇 0 terms in even 𝑑, consistent with the thermal compactification being gapped. Matching with (3.51), we find   𝑑 1 2 ⌊ 2 ⌋+1 1 − 2𝑑−1 𝜁 (𝑑) 𝑓 = , 𝑑−1 vol 𝑆   𝑑 1 2 ⌊ 2 ⌋+1 1 − 2𝑑−3 𝜁 (𝑑 − 2) 𝑐1 = , 24(𝑑 − 2)vol 𝑆 𝑑−1   𝑑 1 2 ⌊ 2 ⌋+1 1 − 2𝑑−3 𝜁 (𝑑 − 2) 𝑐2 = − , (3.97) 96(𝑑 − 2)vol 𝑆 𝑑−1 54 for the free fermion. Holographic theories Finally, let us consider CFTs dual to semiclassical Einstein gravity via AdS/CFT. We can estimate the partition functions of such theories by studying the thermodynamics of Kerr-AdS black holes (see e.g. [95]). By the holographic principle, a holographic CFT has the same partition function as its dual in AdS space. We get the following high-temperature partition function for a holographic CFT𝑑 , 𝑑 ≥ 3:  Í ⌊𝑑/2⌋ 2  2   𝑑−1 𝑑−1 𝑑 (𝑑 − 1) + 𝑖=1 Ω𝑖 © ℓAdS 𝑇 vol 𝑆 𝑑−1 (4𝜋) 𝑑−1 1 ª® ­ log 𝑍 = 1 − + O ,
Î ⌊𝑑/2⌋ 2 ­ 4𝑑 𝑑 𝐺 𝑁 16𝜋 2𝑇 2 𝑇4 ® 1 + Ω 𝑖 𝑖=1 « ¬ (3.98) where ℓAdS is the characteristic length of the dual asymptotic AdS𝑑+1 spacetime and 𝐺 𝑁 is the 𝑑 + 1-dimensional Newton constant. This is indeed consistent with the result (3.51) from the thermal effective action. Matching coefficients, we find19 𝑓 = 𝑐1 = 𝑑−1 (4𝜋) 𝑑−1 ℓAdS , 4𝑑 𝑑 𝐺 𝑁 𝑑−1 (4𝜋) 𝑑−3 ℓAdS , 4(𝑑 − 2)𝑑 𝑑−2 𝐺 𝑁 𝑑−1 (4𝜋) 𝑑−3 ℓAdS 𝑐2 = − , 32(𝑑 − 2)𝑑 𝑑−3 𝐺 𝑁 (3.99) for the thermal Wilson coefficients of a holographic CFT. • Extended regime of validity for holographic theories For holographic theories, the entropy of local operators with certain dimensions and spins can be approximated by the entropy of a black hole with the same quantum numbers, as long as the black hole is stable and has large area (in Planck units). In this section we examine where the entropy of Kerr black holes is trustworthy, and compare it to the range of validity of the EFT expansion in Section 3.3. We will find that for holographic theories, the universal formula for entropy has an extended regime of validity, compared to general CFTs. This is reminiscent of what happens 19 The coefficients in (3.99) were also independently computed by Edgar Shaghoulian. We thank him for discussions related to these coefficients. 55 in two-dimensional CFTs, where the Cardy formula has an extended regime of validity for holographic theories [104]. When Kerr black holes in AdS spin too quickly, they suffer from a phenomenon called superradiant instability [43]. In the case of the Kerr-AdS black hole, it has −1 . In an instability when any of the angular velocities Ω𝑖 become larger than ℓAdS Í −1 . particular, a stable Kerr-AdS black hole has a bound on quantity 𝐸 − 𝑖 |𝐽𝑖 |ℓAdS For instance, in AdS4 , the condition for stability is (see e.g. [133]) √︄ 𝐸ℓAdS −1 𝐸 − |𝐽 |ℓAdS > , (3.100) 2𝐺 𝑁 when the black hole has large mass and spin.20 Translated to CFT data, the entropy for holographic theories is trustworthy when the twist obeys Δ − |𝐽 | ≳ √︁ 𝑓 Δ, (3.101) where by ≳ we allow for an 𝑂 (1) constant on the RHS that we do not compute.21 A similar calculation shows that, for a holographic theory where 𝑚 of the spins are Í ⌊ 𝑑2 ⌋ |𝐽𝑖 |), then taken to be large compared to the twist (i.e. Δ, |𝐽1 |, . . . |𝐽𝑚 | ≫ Δ − 𝑖=1 the entropy is trustworthy when the twist obeys Δ− ∑︁ 𝑖 |𝐽𝑖 | ≳ 𝑓 𝑚 Ö 1 ! 𝑚+1 |𝐽𝑖 | . (3.102) 𝑖=1 We see that the functional form of the stability bound for Kerr-AdS black holes is very similar to the regime of validity (3.89) and (3.88) for the general entropy formula. This is reminiscent of the extended regime of the Cardy formula for the case of CFT2 : for theories holographically dual to large-radius gravity in AdS3 , it was shown that there is a further extension of the validity of the Cardy formula in [104]. For holographic CFTs, the Cardy formula matches the Bekenstein-Hawking entropy of BTZ black holes, which only requires Δ − |𝐽 | ≳ 𝑐, (3.103) −1 −2 𝑁 ℓAdS , |𝐽 |𝐺 𝑁 ℓAdS ≫ 1. 21 The reason we allow this freedom is the possibility of the black holes being stable, but not 20 Meaning 𝐸𝐺 yet dominating the canonical ensemble. This can occur for sufficiently light black holes, sometimes called "enigmatic black holes" [32, 104]. 56 with 𝑐 → ∞ (rather than the usual Δ − |𝐽 | ≫ 𝑐 condition). In particular, it was shown that theories with a sufficiently sparse light spectrum (which is a necessary, but not sufficient condition for the theory to have a semiclassical Einstein gravity dual) have an extended Cardy regime of entropy. It would be interesting if one could prove a similar statement for higher dimensional CFTs.22 3.5 A "genus-2" partition function While the density of states of a CFT is encoded in the partition function on 𝑆 1𝛽 ×𝑆 𝑑−1 , OPE coefficients are encoded in partition functions on other manifolds. In this work, we will be interested in "heavy-heavy-heavy" OPE coefficients 𝑐𝑖 𝑗 𝑘 between operators with parametrically large scaling dimension. To study them, we can consider the partition function of the CFT on a manifold constructed by gluing a pair of three-punctured spheres along their punctures. In two dimensions, this produces a genus-2 Riemann surface. However, a similar construction works in higher dimensions. We will continue to refer to such a manifold as "genus-2" in higher dimensions, by analogy with the 2D case. We pause to note that our final expression for "heavy-heavy-heavy" OPE coefficients is given at the end of Sec. 3.7, in Eqn (3.242). Readers only interested in the final result can skip to this part. Conformal structures of a genus-2 manifold In higher dimensions we can build a genus-2 manifold 𝑀2 by taking two copies of the plane R𝑑 (more precisely its conformal compactification 𝑆 𝑑 ), removing three balls from each plane, and gluing the boundaries of the balls with cylinders. In this construction, we can choose the positions and radii of the balls, as well as the lengths of the cylinders. We can additionally add angular twists by elements of SO(𝑑) as we move along each cylinder. This is a large number of parameters, but many of them are related by conformal symmetry. In addition, if the CFT has a global symmetry Γ, we can introduce topological defects on 𝑀2 , or equivalently a flat Γ-bundle over 𝑀2 . Such flat bundles are parametrized by homomorphisms of the fundamental group of 𝑀2 to Γ. For 𝑑 > 2, R𝑑 with balls removed is simply connected, and the only homotopically non-trivial cycles on 𝑀2 are those going through the cylinders between the two copies of R𝑑 . Therefore, the 22 Some works studying sparseness in higher-𝑑 CFT include [21, 116, 164]. In particular it would be interesting if the precise sparseness conditions in [164] implied the extended entropy formulas described in this section. 57 fundamental group is a free group with two generators, and flat Γ-bundles can be parametrized by decorating the cylinders by inserting topological defects transverse to these cycles. The situation is different for 𝑑 = 2 since R2 with balls removed is already not simply connected, and there are additional generators of the fundamental group which go around the boundaries of the balls. Inserting topological defects transverse to these additional cycles on 𝑀2 is equivalent to considering twisted sectors along the cylinders. To understand the implications of conformal symmetry, it is helpful to ignore the Weyl anomaly and focus on the conformal structure of the manifold 𝑀2 — i.e. properties of 𝑀2 that are independent of the Weyl class of the metric. First, we can associate an (orientation-reversing) conformal group element to each cylinder as follows. Let 𝑥, 𝑥 ′ be flat coordinates on the two copies of R𝑑 . A cylinder 𝐶 that connects the two planes is Weyl-equivalent to an annulus in each plane. Using this Weyl-equivalence, each coordinate 𝑥 and 𝑥 ′ can be extended to cover 𝐶. Inside 𝐶, the coordinates 𝑥 and 𝑥 ′ are identified by an orientation-reversing conformal group element: 𝑥 = 𝑔𝑥 ′, 𝑔 ∈ 𝐺− (inside 𝐶). (3.104) Here, we denote the conformal group as 𝐺 = SO(𝑑 + 1, 1), and we write the orientation-reversing component of 𝑂 (𝑑 + 1, 1) as 𝐺 − . For example, if a cylinder of length 𝛽 connects the unit spheres in each copy of the plane, then we have 𝑔 = 𝑒 −𝛽𝐷 𝐼, (3.105) where 𝐼 (𝑥) = 𝑥𝑥2 is an inversion. More generally, suppose the cylinder is centered at 𝑥 = 𝑎, has radius 𝑟 and length 𝛽𝑟, and includes an angular twist by ℎ ∈ SO(𝑑). Then23 𝑔 = 𝑒 𝑎·𝑃 𝑒 −(𝛽−2 log 𝑟)𝐷 ℎ𝐼𝑒 −𝑎·𝑃 . (3.106) In the case of interest, we have three cylinders connecting two copies of the plane. This gives three group elements (𝑔1 , 𝑔2 , 𝑔3 ) ∈ (𝐺 − ) 3 . However, the conformal structure of the resulting manifold is unchanged if we perform a conformal transformation 𝑥 ↦→ 𝑔𝑥 on the first plane or 𝑥 ′ ↦→ 𝑔′𝑥 ′ on the second. These conformal 23 The appearance of the quantity 𝛽 − 2 log 𝑟 reflects the fact that two planes glued by a cylinder with radius 𝑟 and length 𝛽𝑟 is Weyl-equivalent to two planes glued by a cylinder with radius 1 and length 𝛽 − 2 log 𝑟. The Weyl transformation breaks the cylinder into three pieces of lengths 𝑟 log 𝑟, 𝑟 (𝛽 − 2 log 𝑟), 𝑟 log 𝑟, and flattens out the first and third piece into annuli. 58 transformations become gauge redundancies acting on the 𝑔𝑖 : (𝑔1 , 𝑔2 , 𝑔3 ) ∼ (𝑔𝑔1 𝑔′−1 , 𝑔𝑔2 𝑔′−1 , 𝑔𝑔3 𝑔′−1 ). (3.107) Modding out by this gauge-redundancy, we obtain the moduli space of conformal structures as a double-quotient M = 𝐺\(𝐺 − ) 3 /𝐺, (3.108) where the left and right factors of 𝐺’s act on (𝐺 − ) 3 via (3.107). The action of the 𝐺 × 𝐺 gauge redundancies on (𝐺 − ) 3 is almost free, so the dimension of M is dim M = 3 dim 𝐺 − − 2 dim 𝐺 = dim 𝐺 = (𝑑 + 1)(𝑑 + 2) . 2 (3.109) For orientation, in 2D the parametrization of a genus-2 Riemann surface in terms of (𝑔1 , 𝑔2 , 𝑔3 ) is called a Whittaker parametrization. (The closely-related Schottky parametrization can be obtained by forming the combinations 𝛾𝑖 𝑗 = 𝑔𝑖 𝑔 −1 𝑗 , which satisfy 𝛾12 𝛾23 𝛾31 = 1.) In 2D, the true moduli space of genus-2 surfaces is a quotient of M by the mapping class group. The action of the mapping class group is unfortunately somewhat complicated in the Whittaker/Schottky parameterizations. In higher dimensions, M is again a covering space of the moduli space of conformal structures on 𝑀2 . Topologically, 𝑀2 is equivalent to a connected sum of two copies of 𝑆 1 × 𝑆 𝑑−1 :24 𝑀2  (𝑆 1 × 𝑆 𝑑−1 ) # (𝑆 1 × 𝑆 𝑑−1 ). (3.110) This can be seen by decomposing 𝑀2 in the "dumbbell" channel where we slice Figure 3.3 down the middle into a left half and a right half. Each half is topologically a copy of 𝑆 1 × 𝑆 𝑑−1 with a ball removed, and the two halves are glued along an 𝑆 𝑑−1 . The mapping class group of this space was computed for 𝑑 = 3 in [37]. This mapping class group will not play a further role in the present work. It will be interesting to explore its implications and other global aspects of higher-dimensional "highergenus" surfaces in future work. An important set of functions on M are eigenvalues of the group elements 𝑔𝑖−1 𝑔 𝑗 ∈ SO(𝑑 + 1, 1):    (𝑒 ±𝛽𝑖 𝑗 , 𝑒 ±𝑖 𝜃®𝑖 𝑗 )  (even 𝑑),  (𝑒 ±𝛽𝑖 𝑗 , 𝑒 ±𝑖 𝜃®𝑖 𝑗 , 1)  (odd 𝑑). eigenvalues (𝑑+2)×(𝑑+2) (𝑔𝑖−1 𝑔 𝑗 ) =  24 We thank Yifan Wang for pointing this out and directing us to reference [37]. (3.111) 59 𝜕𝐵3 𝜕𝐵1 𝜕𝐵2 𝑌 Figure 3.2: The space 𝑌 = 𝐵3 \(𝐵1 ∪ 𝐵2 ). The boundary of 𝑌 has three 𝑆 𝑑−1 components given by 𝜕𝐵1 , 𝜕𝐵2 , 𝜕𝐵3 with radii 1, 1, 2, respectively. Note that in 𝑑 ≥ 3, 𝑌 has an SO(𝑑 − 1) rotational symmetry around the horizontal axis, and here we are depicting only a 2-dimensional slice. These are indeed invariant under gauge-redundancies (3.107). We refer to the 𝛽𝑖 𝑗 and 𝜃®𝑖 𝑗 as "relative" inverse temperatures and angles, for reasons that will become clear shortly. Note that there are 3⌊ 𝑑+2 2 ⌋ relative temperatures/angles, which is not enough to parametrize the full moduli space when 𝑑 ≥ 3. Choice of geometry The partition function of a CFT on 𝑀2 factors into a theory-independent part that depends on the precise metric and is determined by the Weyl anomaly, times a theory-dependent part that depends only on the conformal structure of 𝑀2 . To get nontrivial information about the theory, it suffices to study only a single representative geometry for each conformal structure. We will choose our geometry as follows. Let 𝐵1 be the unit ball centered at (−1, 0, . . . , 0), let 𝐵2 be the unit ball centered at (1, 0, . . . , 0), and let 𝐵3 be the ball of radius 2 centered at the origin. All three balls are mutually tangent. From 𝐵3 , we remove 𝐵1 and 𝐵2 . The resulting space 𝑌 = 𝐵3 \(𝐵1 ∪ 𝐵2 ) has three 𝑆 𝑑−1 boundaries given by 𝜕𝐵1 , 𝜕𝐵2 , 𝜕𝐵3 , see Figure 3.2. We now take a second copy of 𝑌 , which we call 𝑌 ′, containing boundaries 𝜕𝐵′1 , 𝜕𝐵′2 , 𝜕𝐵′3 . Finally, we glue each 𝜕𝐵𝑖 to 𝜕𝐵𝑖′ with cylinders 𝐶𝑖 whose ratios of length/radius are 𝛽𝑖 , and we include angular twists ℎ𝑖 ∈ SO(𝑑) along each cylinder. See Figure 3.3 for an illustration. This construction gives a particular choice of metric that is flat on 𝑌 , 𝑌 ′ and is the usual cylinder metric on the 𝐶𝑖 . Note that the metric has curvature localized at the junctions between 𝑌 , 𝑌 ′ and the cylinders. However, it is everywhere conformallyflat because the plane, the cylinder, and a plane-cylinder junction are all conformally- 60 𝑌 𝐶1 𝐶2 𝐶3 𝑌′ Figure 3.3: The manifold 𝑀2 is obtained by taking two copies of 𝑌 and gluing corresponding boundary components with cylinders 𝐶1 , 𝐶2 , 𝐶3 of inverse temperatures 𝛽1 , 𝛽2 , 𝛽3 and angular twists ℎ1 , ℎ2 , ℎ3 , colored red, blue, and green, respectively. In the figure, we slightly bent the edges of the cylinders to help visualize them, but in the actual geometry the cylinders have constant radii. The figure also naively suggests that the lengths of the cylinders must be equal, but in reality they need not be related to each other. flat. In terms of conformal structures, our geometry corresponds to 1 1 𝑔1 = 𝑒 −𝑃 𝑒 −𝛽1 𝐷 ℎ1 𝐼𝑒 𝑃 , 1 1 𝑔2 = 𝑒 𝑃 𝑒 −𝛽2 𝐷 ℎ2 𝐼𝑒 −𝑃 , 𝑔3 = 𝑒 𝐷 (2 log 2+𝛽3 ) ℎ3 𝐼, (3.112) which gives a point in M, parametrized by 𝛽1 , 𝛽2 , 𝛽3 , ℎ1 , ℎ2 , ℎ3 . A slight disadvantage of the parametrization (3.112) is that it is not permutationsymmetric among the three cylinders — 𝐶3 is treated differently. However, an advantage is that it makes manifest an important SO(𝑑 − 1) symmetry that rotates all three balls around the 𝑥 1 axis, preserving their points of tangency.25 We can act with an SO(𝑑 − 1) rotation on either 𝑌 or 𝑌 ′, which means that the angular twists (ℎ1 , ℎ2 , ℎ3 ) are subject to a residual gauge redundancy (ℎ1 , ℎ2 , ℎ3 ) ∼ (𝑘 ℎ1 𝑘 ′−1 , 𝑘 ℎ2 𝑘 ′−1 , 𝑘 ℎ3 𝑘 ′−1 ), 𝑘, 𝑘 ′ ∈ SO(𝑑 − 1). (3.113) Thus, overall, we can think of the parametrization (3.112) as a map SO(𝑑 − 1)\(SO(1, 1) × SO(𝑑)) 3 /SO(𝑑 − 1) → M, (3.114) 25 By contrast, we could restore manifest permutation symmetry by taking the balls to all have the same radius and be mutually tangent, but then the SO(𝑑 − 1) would act via a nontrivial conformal transformation. 61 where 𝛽𝑖 parametrize the SO(1, 1)’s, the ℎ𝑖 parametrize the SO(𝑑)’s, and the SO(𝑑 − 1)’s act on the SO(𝑑)’s via (3.113). We claim that (3.112) is injective and covers an open subset of M — in particular an open subset that contains the physical loci that will be important in what follows. These loci include the low temperature regime of large 𝛽𝑖 , and a high temperature limit that will control the asymptotics of OPE coefficients. A first important check is that the two spaces in (3.114) have the same dimension. Indeed, they do, since   𝑑 (𝑑 − 1) (𝑑 − 1)(𝑑 − 2) 3(1 + dim SO(𝑑)) − 2 dim SO(𝑑 − 1) = 3 1 + −2 2 2 (𝑑 + 1)(𝑑 + 2) = . (3.115) 2 In Section 3.7, we will show that the natural measure on M is nonzero in the coordinates (3.112) at both low and high temperatures. This establishes that 𝛽1 , 𝛽2 , 𝛽3 , ℎ1 , ℎ2 , ℎ3 (modulo the gauge redundancy (3.113)) furnish good coordinates on M for our purposes. Consequently, it suffices to consider geometries of the form described above. These geometries contain all possible theory-dependent information about OPE coefficients of the CFT. In Appendix G, we show that there is a matching between quantum numbers specifying OPE coefficients and the dimension of the genus-2 moduli space dim M. The partition function as a sum over states The partition function on the above geometry is a weighted sum of squares of OPE coefficients. In this section, we derive this fact in detail, taking care with some of the subtleties of cutting and gluing in higher-dimensional CFTs. Consider first the space 𝑌 = 𝐵3 \(𝐵1 ∪ 𝐵2 ). This space has boundaries given by 𝜕𝑌 = 𝜕𝐵3 ⊔ −𝜕𝐵1 ⊔ −𝜕𝐵2 . Thus, the partition function 𝑍 (𝑌 ) is an element of H2 ⊗ H1∗ ⊗ H1∗ , or equivalently a map H1 ⊗ H1 → H2 , where H𝑟 is the Hilbert space on a sphere of radius 𝑟. A basis of states |O (𝑥)⟩𝑟 in H𝑟 is given by the insertion of an operator O (𝑥) inside a ball of radius 𝑟. The defining property of 𝑍 (𝑌 ) is that its pairing with three basis elements is a conformal three-point function. Let us state this more precisely. The Hermitian conjugate state to |O (𝑥)⟩𝑟 ∈ H𝑟 can be obtained by inserting the following conjugate operator in a flat geometry outside 62 the ball of radius 𝑟: 𝑎 [O (𝑥)] †𝑟 𝐷 ≡ 𝑟 [𝑟 −𝐷  𝑟 O (𝑥)] = |𝑥| † 𝑎  2Δ 𝐼 −1 (b 𝑥 )𝑎 O𝑎† 𝑎  2  𝑟 𝑥 . 𝑥2 (3.116) Note that [· · · ] † = [· · · ] †1 is the usual BPZ conjugation. The inversion tensor 𝐼𝑎 𝑎 is the solution to the conformal Ward identities for a two-point function of O 𝑎 and O𝑎† : ⟨O𝑎† (𝑥)O 𝑎 (0)⟩ = 𝐼𝑎 𝑎 (b 𝑥) , 2Δ 𝑥 (3.117) normalized so that 𝐼 𝐼 † = 1. With this notation, a projector onto the conformal multiplet of O inside H𝑟 can be written ′ ′ |O|𝑟 = |O 𝑎 (0)⟩𝑟 ⟨O 𝑎 (0) [O 𝑎 (0)] †𝑟 ⟩ −1 𝑟 ⟨O 𝑎 (0)| + descendants, (3.118) ′ where the inverse two-point function ⟨O 𝑎 (0) [O 𝑎 (0)] †𝑟 ⟩ −1 should be understood as a matrix with indices 𝑎, 𝑎′. The indices 𝑎, 𝑎′ are implicitly summed over in (3.118). The form of the sum over descendants is determined by the conformal algebra. A Í resolution of the identity on H𝑟 is given by summing over projectors 1 = O |O|𝑟 . By composing 𝑍 (𝑌 ) with resolutions of the identity on each of its three boundaries, we find an expression in terms of three-point functions: ∑︁ 𝑍 (𝑌 ) = |O3† | 2 𝑍 (𝑌 )(|O1 | 1 ⊗ |O2 | 1 ) O1 ,O2 ,O3 = ∑︁ ⟨O1 (−𝑒)O2 (𝑒)O3 (∞𝑒)⟩⟨O1 (0) [O1 (0)] † ⟩ −1 ⟨O2 (0) [O2 (0)] † ⟩ −1 O1 ,O2 ,O3 × ⟨O3 (∞) [O3 (∞)] †2 ⟩ −1 |[O3 (∞)] †2 ⟩2 1 ⟨O1 (0)| ⊗ 1 ⟨O2 (0)|. (3.119) For simplicity, we have omitted spin indices. This is a sum over states with coefficients given by a three-point function, where 𝑒 = (1, 0, . . . , 0) is a unit vector along the 𝑥 1 direction. Note that we define a primary operator at infinity without an inversion tensor: O3𝑐 (∞𝑒) ≡ lim 𝐿 2Δ3 O3𝑐 (𝐿𝑒). (3.120) 𝐿→∞ To help restore symmetry among the three operators, we have chosen to insert the projector |O3† |, as opposed to |O3 | — this ensures that the three-point function contains O3 (∞) and not O3† (∞). 63 C𝑟,𝛽 𝑟 𝑆𝑑 Weyl 𝑟𝛽 Figure 3.4: The sphere 𝑆 𝑑 is Weyl-equivalent to a "capped cylinder" C𝑟,𝛽 with radius 𝑟 and length 𝑟 𝛽. Each end cap is a ball (the interior of an 𝑆 𝑑−1 ) of radius 𝑟. The "closed junctions" where the cylinder meets the end caps are highlighted in red. Figure 3.5: An "open junction," where a cylinder meets the complement of a ball in a flat plane. The junction is highlighted in blue. The partition function of each cylinder 𝐶𝑖 is simply 𝑒 −𝛽𝑖 (𝐷+𝜀0 ) ℎ𝑖 , where 𝜀0 is the Casimir energy (3.42), and ℎ𝑖 ∈ SO(𝑑) is the angular twist along the cylinder. One subtle ingredient is that we must also associate nontrivial "gluing" factors to junctions between cylinders and flat planes. To derive these gluing factors, let us start with the partition function on 𝑆 𝑑 (with unit radius). This is Weyl-equivalent to a cylinder of radius 𝑟 and length 𝑟 𝛽, capped off by flat balls; see Figure 3.4. Let the capped cylinder be C𝑟,𝛽 , and denote the Weyl factor going from 𝑆 𝑑 to C𝑟,𝛽 by 𝑒 2𝜔𝑟 ,𝛽 . The Weyl anomaly implies 𝑍 (C𝑟,𝛽 ) = 𝑒 −𝑆anom [𝑔,𝜔𝑟 ,𝛽 ] 𝑍 (𝑆 𝑑 ). (3.121) At the same time, 𝑍 (C𝑟,𝛽 ) can be computed by cutting and gluing. The end caps are simply identity operators in radial quantization, |1⟩𝑟 . The cylinder contributes 𝑒 −𝜀0 𝛽 , where 𝜀 0 is the Casimir energy on 𝑆 𝑑−1 . Let us define 𝑍glue (𝑟) as the factor associated to a junction between a cylinder and a flat end-cap, which we call a "closed junction". We find 𝑍 (C𝑟,𝛽 ) = |𝑍glue (𝑟)| 2 𝑒 −𝜀0 𝛽 =⇒ 1 1 1 |𝑍glue (𝑟)| = 𝑒 2 𝜀0 𝛽 𝑒 − 2 𝑆anom [𝑔,𝜔𝑟 ,𝛽 ] 𝑍 (𝑆 𝑑 ) 2 . (3.122) 64 We calculate these gluing factors in various dimensions in Appendix F. For example, we find26    1       𝑒 𝑐/12 (𝑟/2) 𝑐/6  1 |𝑍glue (𝑟)| = 𝑍 (𝑆 𝑑 ) 2 ×   𝑒 −7𝑎/6 (𝑟/2) −2𝑎       𝑒 37𝑎6 /10 (𝑟/2) 6𝑎6  𝑑 odd, 𝑑 = 2, (3.123) 𝑑 = 4, 𝑑 = 6. Our geometry also contains four "open junctions" with the opposite curvature, where a cylinder joins a flat region that locally looks like the complement of a ball; see Figure 3.5. The gluing factor associated to an open junction is the inverse 𝑍glue (𝑟) −1 of the one associated to a closed junction. The reason is that we can perform an infinitesimal Weyl transformation on a plane to create a closed junction infinitesimally-close to an open junction. The Weyl anomaly is infinitesimal, so the gluing factors must multiply to 1. Putting everything together, the partition function on our genus-2 manifold is |𝑍glue (2)| 2 −𝛽1 (𝐷+𝜀 0 ) 𝑍 (𝑀2 ) = Tr(𝑍 (𝑌 ) † 𝑒 −𝛽3 (𝐷+𝜀0 ) ℎ−1 ℎ1 ⊗ 𝑒 −𝛽2 (𝐷+𝜀0 ) ℎ2 )). 3 𝑍 (𝑌 )(𝑒 |𝑍glue (1)| 4 (3.124) Each group element 𝑒 −𝛽𝑖 𝐷 ℎ𝑖 acts on the Hilbert space corresponding to the boundary component 𝜕𝐵𝑖 . Inserting our expression (3.119) for 𝑍 (𝑌 ), we obtain the partition function as a sum over a triplet of primary operators 𝑍 (𝑀2 ) = |𝑍glue (2)| 2 −𝜀0 (𝛽1 +𝛽2 +𝛽3 ) 𝑒 |𝑍glue (1)| 4 ∑︁  × 𝑒 −𝛽1 Δ1 −𝛽2 Δ2 −𝛽3 Δ3 O1 ,O2 ,O3 ′ ′ ′ ⟨O1𝑎 (−𝑒)O2𝑏 (𝑒)O3𝑐 (∞𝑒)⟩ ∗ ⟨ℎ1 · O1𝑎 (−𝑒) ℎ2 · O2𝑏 (𝑒) ℎ3 · O3𝑐 (∞𝑒)⟩ ′ ′ ′ × ⟨O1𝑎 (0) [O1𝑎 (0)] † ⟩ −1 ⟨O2𝑏 (0) [O2𝑏 (0)] † ⟩ −1 ⟨O3𝑐 (∞) [O3𝑐 (∞)] †2 ⟩ −1  + descendants . (3.125) Here, ℎ · O denotes the action of a rotation ℎ ∈ SO(𝑑) on a local operator: ℎ · O 𝑎 = ℎO 𝑎 ℎ−1 = 𝜆(ℎ−1 ) 𝑎 𝑏 O 𝑏 , where 𝜆 is the SO(𝑑) representation of O. 26 In 𝑑 = 4, we write 𝑎 4 as 𝑎. (3.126) 65 Let us choose the O𝑖 to be an orthonormal basis of primaries with respect to the BPZ inner product. Using (3.116) and (3.120), we find ′ ⟨O3𝑐 (∞) [O3𝑐 (∞)] †2 ⟩ = 22Δ3 𝛿 𝑐 𝑐′ , (3.127) while the other two-point functions in (3.125), which involve standard BPZ conjugates [· · · ] † , are identity matrices. Let us furthermore expand the three-point functions in a basis of conformallyinvariant three-point structures: ⟨O1𝑎 (−𝑒)O2𝑏 (𝑒)O3𝑐 (∞𝑒)⟩ = = 1 2Δ1 +Δ2 −Δ3 ⟨O1𝑎 (0)O2𝑏 (𝑒)O3𝑐 (∞𝑒)⟩ 1 𝑐 𝑠 𝑉 𝑠;𝑎𝑏𝑐 (0, 𝑒, ∞). Δ +Δ 2 1 2 −Δ3 123 (3.128) Here, 𝑠 is a structure label, which runs over a finite-dimensional space of solutions 𝑉 𝑠 to the 3-point conformal Ward identities. Meanwhile, 𝑎, 𝑏, 𝑐 are spin indices in the SO(𝑑) representations associated to the three operators. Each three-point structure 𝑠 , and a sum over 𝑠 is implicit. We comes with an associated OPE coefficient 𝑐 123 will discuss the space of three-point structures in more detail in Section 3.6. Plugging everything in, we find an expression for the "genus-2" partition function as a sum over conformal blocks |𝑍glue (2)| 2 −𝜀0 (𝛽1 +𝛽2 +𝛽3 ) ∑︁ 𝑠′ ∗ 𝑠 𝑠′ 𝑠 (𝑐 123 ) 𝑐 123 𝐵123 , 𝑍 (𝑀2 ) = 𝑒 4 |𝑍glue (1)| O O O 1 2 (3.129) 3 ′ 𝑠𝑠 where we have introduced the "genus-2" block 𝐵123 ′ ′ 𝑠𝑠 𝐵123 (𝛽𝑖 , ℎ𝑖 ) = 2−2Δ1 −2Δ2 𝑒 −𝛽1 Δ1 −𝛽2 Δ2 −𝛽3 Δ3 (𝑉 𝑠 ;𝑎𝑏𝑐 (0, 𝑒, ∞)) ∗ (ℎ1 ℎ2 ℎ3 · 𝑉 𝑠 ) 𝑎𝑏𝑐 (0, 𝑒, ∞) + descendants. (3.130) The first term in (3.130) comes from primary states, and dominates in the "low temperature" limit 𝛽1 , 𝛽2 , 𝛽3 ≫ 1. The descendent terms involve three-point functions of descendant operators, contracted using the inverse of the Gram matrix. Such terms are determined by the conformal algebra. ′ 𝑠 𝑠 is naturally a function on the moduli space M of conformal structures, The block 𝐵123 and doesn’t depend on the Weyl class of the metric. This fact is already hinted at in (3.130). Note that the factor 2−2Δ1 −2Δ2 seems to violate permutation symmetry among the three operators. However, this is an artifact of our asymmetric conformal frame. We can restore manifest permutation symmetry by rewriting the block in 66 𝐶1 𝐶2 𝐶3 Figure 3.6: The thermal circle near the hot spot at the origin, highlighted in red. Starting from the top, we move along cylinder 𝐶2 to the top, and then back along cylinder 𝐶1 to the top. terms of the "relative" temperatures 𝛽𝑖 𝑗 , defined in (3.111), which are permutationsymmetric functions on M. At low temperatures, we find 2−2Δ1 −2Δ2 𝑒 −𝛽1 Δ1 −𝛽2 Δ2 −𝛽3 Δ3 = 𝑒 − 𝛽 +𝛽 −𝛽31 𝛽 +𝛽 −𝛽12 𝛽12 +𝛽31 −𝛽23 Δ1 − 12 23 Δ2 − 31 23 Δ3 2 2 2 +..., (3.131) which is manifestly symmetric under permuting 1, 2, 3. This is a nontrivial check on 3.130. In Appendix G, we point out that the number of unbounded quantum numbers 𝑠′ 𝑠 matches the dimension of the moduli space M. needed to specify the block 𝐵123 This is analogous to the fact that the number of unbounded quantum numbers needed to specify a four-point conformal block (two: Δ and 𝐽) matches the number of cross-ratios for a four-point function (two: 𝑧 and 𝑧). Hot spots and the thermodynamic limit Because the geometry described in Section 3.5 is not a circle fibration, it is not immediately obvious how to compute the partition function using the thermal effective action. To make progress, we adopt the following assumption Assumption. The thermal effective action describes the contribution to the partition function from any region where the geometry locally looks like a circle fibration with a large local temperature. We call such a region a "hot spot." For example, consider the origin in one of the copies of R𝑑 , where the balls 𝐵1 and 𝐵2 are tangent. Starting at the origin, there is a circular path of length 𝛽1 + 𝛽2 that runs along one cylinder 𝐶2 , and then back along the other 𝐶1 ; see Figure 3.6. In the limit where 𝛽1 , 𝛽2 are both small, this circular path shrinks and we have a hot spot. 67 𝜕𝐵1 𝜕𝐵2 𝜕𝐵′1 𝜕𝐵′2 𝜃 Figure 3.7: An approximation to the thermal circle, at an angle 𝜃 away from the origin. Starting at the top left, we move horizontally from 𝜕𝐵1 to 𝜕𝐵2 . Then we move down along 𝐶2 to 𝜕𝐵′2 . Then horizontally to the left to 𝜕𝐵′1 , then up along 𝐶1 to the starting point. The path has approximate length 𝛽1 + 𝛽2 + 2𝜃 2 . To build some intuition, let us compute the leading contribution to the thermal effective action near this hot spot. The local temperature is highest at the origin, and decays away from it. To determine the local temperature precisely, we should find a (locally defined) conformal Killing vector field that moves around the hot spot’s thermal circle. At leading order near the origin, we can guess what it looks like without too much calculation. Consider a path that starts at the point (−1 + cos 𝜃, sin 𝜃, 0, . . . ) on 𝜕𝐵1 . Move horizontally to the point (1−cos 𝜃, sin 𝜃, 0, . . . , 0) on 𝜕𝐵2 , through cylinder 2 to 𝜕𝐵′2 , horizontally from 𝜕𝐵′2 to 𝜕𝐵′1 , and back through cylinder 1 to the initial point on 𝜕𝐵1 , see Figure 3.7. This path has length 𝛽1 + 𝛽2 + 4(1 − cos 𝜃) ≈ 𝛽1 + 𝛽2 + 2𝜃 2 (𝜃 ≪ 1). (3.132) When 𝜃 is small, we expect this path to be close to the orbit of a local conformal Killing vector. Thus, the local temperature is approximately 𝛽(𝜃) ≈ 𝛽1 + 𝛽2 + 2𝜃 2 . (3.133) The leading contribution to the thermal effective action near this hot spot is thus ∫ −𝑆hot (𝛽1 , 𝛽2 ) ∼ 𝑓 vol 𝑆 𝑑−2 𝑑𝜃 sin𝑑−2 𝜃 . (𝛽1 + 𝛽2 + 2𝜃 2 ) 𝑑−1 (3.134) Here, vol 𝑆 𝑑−2 sin𝑑−2 𝜃 comes from an integral over azimuthal angles. When 𝛽1 + 𝛽2 √ is small, the integral will be dominated by small 𝜃 ∼ 𝛽1 + 𝛽2 . To compute it, we can approximate sin𝑑−2 𝜃 ≈ 𝜃 𝑑−2 and extend the 𝜃-integral from 0 to ∞: ∫ ∞ 𝑑𝜃 𝜃 𝑑−2 𝑓 vol 𝑆 𝑑−1 𝑑−2 −𝑆hot (𝛽1 , 𝛽2 ) ∼ 𝑓 vol 𝑆 = . (3.135) (𝛽1 + 𝛽2 + 2𝜃 2 ) 𝑑−1 (8(𝛽1 + 𝛽2 )) 𝑑−1 2 0 68 Note that the integral is dominated near the hot spot, i.e. in the neighborhood √ 𝜃 ∼ 𝛽1 + 𝛽2 . This justifies our use of the thermal effective action everywhere inside the integrand. Furthermore, when 𝛽1 + 𝛽2 is small, we find a large negative action from the hot spot, which translates into a large multiplicative contribution to the partition function 𝑍 ∼ 𝑒 −𝑆 . A more precise formula for the action of a hot spot The fact that the integral (3.135) is dominated near the hot spot suggests a more precise and illuminating way to derive it. Let us assume that the action of a hot spot √ doesn’t depend on the geometry far outside the neighborhood 𝜃 ∼ 𝛽1 + 𝛽2 . Thus, to compute it, it suffices to consider a "genus-1" version of the geometry discussed in Section 3.5, where we have only two balls 𝐵1 and 𝐵2 . We claim that this "genus-1" geometry is Weyl equivalent to 𝑆 1𝛽12 ×𝑆 𝑑−1 with a special inverse temperature 𝛽12 that depends on 𝛽1 and 𝛽2 . The Weyl transformation that implements this equivalence essentially spreads out the hot-spot over the entire 𝑆 𝑑−1 , resulting in a uniform inverse temperature 𝛽12 . The result is −𝑆hot (𝛽1 , 𝛽2 ) ∼ log 𝑍 𝑆1 ×𝑆 𝑑−1 (𝛽12 ) + Weyl terms, (3.136) where "Weyl terms" are possible contributions from the Weyl anomaly, and "∼" indicates that both sides have the same singular parts as 𝛽1 , 𝛽2 → 0. Now, 𝛽12 is determined by the conformal structure of our "genus-1" manifold, so we can read it off from the gluing group elements 𝑔1 and 𝑔2 associated to the two cylinders, given in (3.112). Gluing two copies of the plane with 𝑔1 and 𝑔2 is equivalent to gluing a single copy of the plane to itself with 𝑔1−1 𝑔2 . To read off 𝛽12 , we must simply diagonalize 𝑔1−1 𝑔2 : ® ® 𝑔1−1 𝑔2 = 𝑈𝑒 −𝛽12 𝐷+𝑖 𝜃 12 · 𝑀 𝑈 −1 , 𝑈 ∈ SO(𝑑 + 1, 1). (3.137) In other words, 𝛽12 is precisely the "relative" inverse temperature defined in (3.111). There is a particularly nice expression for 𝛽12 when the angular fugacities are turned off. In this case, the group elements 𝑔1 , 𝑔2 are built from conformal generators 𝑃1 , 𝐷, 𝐾 1 , that generate a PSL(2, R) subgroup of the conformal group. Thus, we can obtain 𝛽12 by computing 𝑔1−1 𝑔2 inside SL(2, R) and comparing the trace of both sides of (3.137) as 2 × 2 matrices. We should compare them up to a sign, since the 69 1d conformal group PSL(2, R) is a quotient of SL(2, R) modulo ±1. This gives ! 𝛽1 −𝛽2 𝛽1 −𝛽2 𝛽2 −𝛽1 𝛽1 +𝛽2 ! 𝑒 −𝛽12 /2 0 𝑒 2 (1 − 2𝑒 𝛽2 ) 𝑒 2 + 𝑒 2 − 2𝑒 2 ±Tr = Tr . 𝛽2 −𝛽1 𝛽1 +𝛽2 0 𝑒 𝛽12 /2 𝑒 2 (1 − 2𝑒 𝛽1 ) −2𝑒 2 (3.138) To find a solution, we must choose the − sign, which gives    𝛽1 +𝛽2 𝛽1 − 𝛽2 −1 𝛽12 = 2 cosh 2𝑒 2 − cosh . 2 (3.139) This is the inverse temperature at which we should evaluate (3.136). In the limit where 𝛽1 , 𝛽2 become small, the relative inverse temperature 𝛽12 has the expansion 𝛽12 ∼ √︁ 8(𝛽1 + 𝛽2 ) − 𝛽12 − 10𝛽1 𝛽2 + 𝛽22 + 𝑂 (𝛽𝑖5/2 ) √ √ 12 2 𝛽1 + 𝛽2 (𝛽1 , 𝛽2 ≪ 1). (3.140) Consequently, the leading contribution to the thermal effective action (3.54) is − log 𝑍 𝑆1 ×𝑆 𝑑−1 (𝛽12 ) ∼ 𝑓 vol 𝑆 𝑑−1 − 𝑑−3 2 + 𝑂 (𝛽𝑖 𝑑−1 (8(𝛽1 + 𝛽2 )) 2 ), (𝛽1 , 𝛽2 ≪ 1), (3.141) in perfect agreement with (3.135)! We have recovered our earlier result for the leading action of a hot spot. However, an advantage of this more abstract derivation is that we expect (3.136) and (3.139) to encompass all singular terms in the small 𝛽1 , 𝛽2 limit. This derivation is also straightforward to generalize to the case with angular fugacities. Let us think of the 𝑆 1 × 𝑆 𝑑−1 partition function as a class function 𝑍 𝑆1 ×𝑆 𝑑−1 (𝑔) ® is obtained by of a conformal group element 𝑔. The old notation 𝑍 𝑆1 ×𝑆 𝑑−1 (𝛽, 𝜃) ® ® setting 𝑔 = 𝑒 −𝛽𝐷+𝜃· 𝑀 . Then the above argument implies that the singular part of the action of a hot spot associated to two group elements 𝑔1 , 𝑔2 is −𝑆hot,12 ∼ log 𝑍 𝑆1 ×𝑆 𝑑−1 (𝑔1−1 𝑔2 ) + Weyl terms. (3.142) Let us comment on the Weyl anomaly terms in (3.142). In the genus-1 case, one can check that contributions from the Weyl anomaly to (3.142) vanish in the limit 𝛽1 , 𝛽2 → 0. In particular, they do not contribute to the singular part of the partition function in the high temperature limit. In what follows, we will assume that the same is true at higher genus, so that analogous Weyl terms can be ignored for our purposes. It would be nice to make these contributions more precise in an example theory. 70 The hot spot action for the genus-2 case Let us finally apply this result to our "genus-2" partition function. We conjecture that the singular part of the log of the partition function as 𝛽1 , 𝛽2 , 𝛽3 → 0 is given by a sum of hot spot actions for each pair of tangent balls. Combined with (3.142), this implies log 𝑍 (𝑀2 ) ∼ −𝑆hot,12 − 𝑆hot,23 − 𝑆hot,31 ∼ log 𝑍 𝑆1 ×𝑆 𝑑−1 (𝑔1−1 𝑔2 ) + log 𝑍 𝑆1 ×𝑆 𝑑−1 (𝑔2−1 𝑔3 ) + log 𝑍 𝑆1 ×𝑆 𝑑−1 (𝑔3−1 𝑔1 ). (3.143) Another way to state the conjecture is as follows. Consider the ratio 𝑅(𝛽𝑖 ) = 𝑍 (𝑀2 ) . 𝑍 𝑆1 ×𝑆 𝑑−1 (𝑔1−1 𝑔2 )𝑍 𝑆1 ×𝑆 𝑑−1 (𝑔2−1 𝑔3 )𝑍 𝑆1 ×𝑆 𝑑−1 (𝑔3−1 𝑔1 ) (3.144) We conjecture that 𝑅(𝛽𝑖 ) has a finite limit as 𝛽𝑖 → 0: 𝑅 = lim 𝑅(𝛽𝑖 ) < ∞. 𝛽𝑖 →0 (3.145) Intuitively, we imagine that dividing by the hot-spot partition function 𝑍 𝑆1 ×𝑆 𝑑−1 (𝑔𝑖−1 𝑔 𝑗 ) allows us to define a kind of renormalized "hot-spot operator" in the limit 𝛽𝑖 → 0 — a CFT operator that lives at a location where a circle shrinks to zero size. The quantity 𝑅 is then a correlator of three such hot-spot operators. It would be very interesting to make this statement more precise and compute 𝑅 in some example theories. In this work, we will mostly be concerned with the leading singularity of the partition function that follows from (3.143). Using (3.54), this is ! 𝑓 vol 𝑆 𝑑−1 𝑓 vol 𝑆 𝑑−1 𝑓 vol 𝑆 𝑑−1 + 𝑑−1 Î + 𝑑−1 Î , 𝑍 (𝑀2 ) ∼ exp 𝑑−1 Î 2 2 2 𝛽12 𝛽23 𝛽31 𝑎 (1 + Ω12,𝑎 ) 𝑎 (1 + Ω23,𝑎 ) 𝑎 (1 + Ω31,𝑎 ) (3.146) where the relative angular velocities are given by Ω𝑖 𝑗,𝑎 = 𝛽𝑖 𝑗 𝜃 𝑖 𝑗,𝑎 . We define the ®𝑖𝑗 "high temperature" regime of the genus-2 partition function as 𝛽𝑖 𝑗 → 0, with Ω held fixed. This is the physical regime where the thermal effective action can be applied to each hot spot. Here, "∼" means that the logs of both sides agree, up to subleading terms as 𝛽𝑖 𝑗 → 0. If the CFT has a global symmetry Γ, we can decorate each cylinder by a topological defect associated to a group element 𝛾𝑖 (𝑖 = 1, 2, 3). If we do so, the coefficient 71 𝑓 in (3.146) for the (𝑖, 𝑗) hot spot becomes a function of the conjugacy class of 𝛾𝑖 𝛾 −1 𝑗 ∈ Γ. Thus, in general, we can have a different 𝑓 for each hot spot. Note that the angular parameters 𝜃®𝑖 𝑗 scale to zero at high temperature. In this work, ® 𝑖 𝑗 will scale to we will be particularly interested in a limit of low spin, where the Ω zero as well at an appropriate saddle point (so that the 𝜃®𝑖 𝑗 are parametrically smaller than the 𝛽𝑖 𝑗 ). Let us further expand the partition function in this regime. It will be convenient to parametrize the rotations ℎ𝑖 as a product of a rotation away from the 𝑥 1 axis, times an SO(𝑑 − 1) rotation: ! ! 𝑑 ∑︁ ∑︁ ℎ𝑖 = exp 𝑖𝛼𝑖,𝑏 𝑀1𝑏 exp 𝑖Φ𝑖,𝑎𝑏 𝑀𝑎𝑏 , (3.147) 𝑏=2 2≤𝑎<𝑏≤𝑑 Here, 𝛼 ® 𝑖 = (𝛼𝑖,2 , . . . , 𝛼𝑖,𝑑 ) transforms like a vector under SO(𝑑 − 1), while the ® 𝑖 = (Φ𝑖,23 , . . . , Φ𝑖,𝑑−1,𝑑 ) transform like an adjoint under SO(𝑑 − 1) parameters Φ SO(𝑑 − 1). Recall that the ℎ𝑖 are subject to the gauge-redundancy (3.113). The right action ® 𝑖 (to leading order). Thus, the partition of SO(𝑑 − 1) simultaneously shifts the Φ ® 𝑖 . Under the left action by SO(𝑑 − 1), function must be translation-invariant in the Φ ® 𝑖 transform linearly as SO(𝑑 − 1) vectors and adjoints, respectively. the 𝛼 ® 𝑖 and Φ So the partition function must also be invariant under SO(𝑑 − 1) rotations of these variables. Indeed, expanding (3.146) in small angles, we find ! ®1 − Φ ® 2)2 (Φ (𝛼 ®1 + 𝛼 ®2)2 1 1 − 8(𝑑 + 1) +... , = 𝑑−1 1 − 𝑑−1 Î𝑛 (1 + Ω2 ) 2 4 𝛽12,0 𝛽12,0 𝛽12 𝛽12,0 𝑎=1 12,𝑎 ! ®2 − Φ ® 3)2 ( 14 𝛼 ® 2 − 12 𝛼 ®3)2 1 (Φ 1 − 8(𝑑 + 1) +... , = 𝑑−1 1 − 𝑑−1 Î𝑛 (1 + Ω2 ) 2 4 𝛽23,0 𝛽23,0 𝛽23 𝛽23,0 𝑎=1 23,𝑎 ! ®3 − Φ ® 1)2 ( 14 𝛼 ® 1 − 12 𝛼 ®3)2 1 1 (Φ = 𝑑−1 1 − − 8(𝑑 + 1) +... . 𝑑−1 Î𝑛 (1 + Ω2 ) 2 4 𝛽31,0 𝛽31,0 𝛽31 𝛽31,0 𝑎=1 31,𝑎 (3.148) ® 1/2 ≪ 1, and 𝛽 ≪ 1. Here, 𝛽𝑖 𝑗,0 This formula is valid when 𝛼 ® /𝛽 ≪ 1, and Φ/𝛽 denote the relative temperatures when the angles are set to zero. They are given by √︁ 𝛽12,0 = 8(𝛽1 + 𝛽2 ) + . . . , √︃ 𝛽23,0 = 8( 14 𝛽2 + 12 𝛽3 ) + . . . , √︃ 𝛽31,0 = 8( 14 𝛽1 + 12 𝛽3 ) + . . . . (3.149) 72 𝛼 ®1 𝛼 ®2 𝛼 ®3 Figure 3.8: The partition function (3.148) penalizes rotations 𝛼 ® 𝑖 in such a way that the spheres behave like three interlocked gears. No matter what signs we choose for the 𝛼 ® 𝑖 , there is no way to rotate the gears, since two of them will always be counter-rotating at their points of contact. Let us understand some physical implications of (3.148). Terms like ( 𝛼 ®1 + 𝛼 ®2)2 come from rotating the spheres so that they rub against each other (Figure 3.8). The thermal effective action penalizes such rotations — the spheres behave like gears that are interlocked. It follows that there is no zero mode associated with moving the 𝛼 ® 𝑖 ’s: three mutually interlocked circular gears cannot be rotated. Meanwhile, ® 𝑖 −Φ ® 𝑗 ) 2 represent the effect of twisting the spheres by different amounts terms like ( Φ around their point of tangency. Such twists are also penalized by the effective action, but by a smaller power of 𝛽𝑖2𝑗,0 . To summarize, the only zero mode in the angular parameters is associated to the gauge symmetry of right multiplication by SO(𝑑 −1). 3.6 Genus-2 global conformal blocks To determine the asymptotics of CFT OPE coefficients, we must invert the conformal block expansion (3.129) of the genus-2 partition function 𝑍 (𝑀2 ). In particular, 𝑠′ 𝑠 in the highwe will need the large-Δ limit of the genus-2 conformal blocks 𝐵123 temperature regime discussed in Section 3.5. Our strategy will be to write an integral representation for the block using the "shadow formalism." In the large-Δ regime, the integral can be evaluated by saddle point, yielding simple closed-form expressions in the regimes of interest. Review: shadow integrals for four-point blocks Let us first review this strategy in the more familiar case of conformal blocks for four-point functions of local operators on R𝑑 [78, 89, 130, 193]. We will follow the notation and conventions of [130]. 73 The central objects in the shadow formalism are principal series representations and their matrix elements. Let 𝜋 = (Δ, 𝜆) denote a conformal representation, where 𝜆 is a representation of SO(𝑑). The principal series corresponds to (unphysical) complex dimensions of the form Δ = 𝑑2 +𝑖𝑠, where 𝑠 ∈ R. States in a principal series representation are given by functions 𝑓 𝑎 (𝑥) that transform like conformal primaries with dimension Δ and rotation representation 𝜆. Here, 𝑎 is an index for 𝜆. Such states admit a Hermitian inner product ∫ (𝑔| 𝑓 ) ≡ 𝑑 𝑑 𝑥(𝑔 𝑎 (𝑥)) ∗ 𝑓 𝑎 (𝑥), (3.150) 𝑎 ∗ where the index  𝑎 is summed over. Note that (𝑔 (𝑥)) has scaling dimension 𝑑 𝑑 𝑑 2 − 𝑖𝑠 = 𝑑 − 2 + 𝑖𝑠 , so the integrand (including the measure 𝑑 𝑥) has scaling dimension 0. Furthermore, it transforms in the dual rotation representation 𝜆∗ , so the integrand is rotation-invariant. It follows that the pairing (𝑔| 𝑓 ) is conformallyinvariant. The principal series representation 𝜋 = ( 𝑑2 +𝑖𝑠, 𝜆) is isomorphic to the "shadow" representation e 𝜋 = ( 𝑑2 −𝑖𝑠, 𝜆 𝑅 ), where 𝜆 𝑅 denotes the reflection of 𝜆. This isomorphism is implemented by the shadow transform: ∫ e(𝑥) O e† (𝑦)⟩ 𝑓 (𝑦), S[ 𝑓 ] (𝑥) = 𝑑 𝑑 𝑦⟨O (3.151) e(𝑥) O e† (𝑦)⟩ denotes the unique (up to scale) conformal two-point structure where ⟨O between operators in the representations e 𝜋 and e 𝜋 † ≡ ( 𝑑2 − 𝑖𝑠, 𝜆∗ ). Conformal three-point functions can be thought of as Clebsch-Gordon coefficients for a tensor product of principal series representations. Such three-point functions carry a structure label 𝑠 that corresponds to different solutions of the conformal Ward identities: 𝑉 𝑠;𝑎𝑏𝑐 (𝑥 1 , 𝑥2 , 𝑥3 ) = ⟨O1𝑎 (𝑥 1 )O2𝑏 (𝑥2 )O3𝑐 (𝑥 3 )⟩ (𝑠) . (3.152) Here, ⟨· · ·⟩ (𝑠) denotes a solution to the conformal Ward identities — not a physical three-point function. (In particular, it does not include an OPE coefficient.) The space of three-point structures is given by (𝜆 1 ⊗ 𝜆 2 ⊗ 𝜆 3 ) SO(𝑑−1) , where SO(𝑑 − 1) indicates the SO(𝑑 − 1)-invariant subspace [142]. We sometimes write 𝑉 𝑠 for a three-point structure, and we sometimes use the notation on the right-hand side of 3.152. With these ingredients, we are ready to build conformal blocks. Four-point blocks are eigenfunctions of the conformal Casimir acting simultaneously on points 1 and 74 2, obeying certain boundary conditions. Using the inner product on principal series representations, we can instead easily build an eigenfunction called a "conformal partial wave" from two three-point structures: ∫ 𝑠′ 𝑠 e† (𝑥)⟩ (𝑠′ ) ⟨O1 (𝑥 1 )O2 (𝑥 2 )O (𝑥)⟩ (𝑠) . Ψ𝜋 (𝑥 1 , · · · , 𝑥4 ) = 𝑑 𝑑 𝑥⟨O3 (𝑥 3 )O4 (𝑥4 ) O (3.153) e† has representation e Here, O 𝜋 † = ( 𝑑2 − 𝑖𝑠, 𝜆∗ ), so that it can be paired with O (𝑥) inside the integral. We omit spin indices for brevity. ′ The partial wave Ψ𝜋𝑠𝑠 satisfies the same Casimir differential equations as a conformal block, but obeys different boundary conditions. However, it gets us "most of the way" to a block, and the block can be extracted from it with a small amount of extra work. The key point is that the space of solutions of the Casimir equations is two-dimensional. It is spanned by the conformal block, and a so-called "shadow" block for the representation e 𝜋 . It follows that the partial wave can be written as a linear combination of the block and its shadow: ′ ′ ′ ′ Ψ𝜋𝑠 𝑠 = 𝑆(𝜋3 𝜋4 [e 𝜋 † ]) 𝑠 𝑡 ′ 𝐺 𝑡𝜋 𝑠 + 𝑆(𝜋1 𝜋2 [𝜋]) 𝑠 𝑡 𝐺 e𝜋𝑠 𝑡 . (3.154) Here, the "shadow" coefficients 𝑆(𝜋1 𝜋2 [𝜋]) 𝑠 𝑡 are obtained by applying shadow transformations to a three-point structure. For example, S3𝑉 𝑠 (𝑥 1 , 𝑥2 , 𝑥3 ) = 𝑆(𝜋1 𝜋2 [𝜋]) 𝑠 𝑡 𝑉 𝑡 (𝑥 1 , 𝑥2 , 𝑥3 ), (3.155) where S3 denotes the shadow transform (3.151) acting at 𝑥 3 . The reason these coefficients appear in (3.154) is explained in [130]. Starting from (3.154), we can ′ isolate the block 𝐺 𝜋𝑠 𝑠 using a "monodromy projection" [193], as we explain in more detail later. To summarize, the shadow formalism gives a convenient integral representation of a partial wave, satisfying the same differential equations as a conformal block, and from which the block can be extracted. This approach will work for the genus-2 𝑠′ 𝑠 as well. Integral representations are particularly useful for studying blocks 𝐵123 large quantum number asymptotics, since we can use saddle point methods. There exist alternative constructions of four-point conformal blocks via shadow-like integrals in Lorentzian signature [179]. These have the advantage of giving the block "on the nose," eschewing the need for a monodromy projection. Finding a similar Lorentzian shadow representation for the genus-2 block is an interesting problem for the future. 75 𝑔1 ′ 𝑉𝑠 ∗ 𝑉𝑠 𝑔2 𝑔3 Figure 3.9: A "tensor diagram" for the genus-2 partial wave (3.158). The three′ point structures 𝑉 𝑠 and 𝑉 𝑠 ∗ are invariant tensors for a tensor product of three principal series representations, so they are represented as trivalent nodes. Each group element acts as a linear operator on a representation, so is a bivalent node. Lines connect together using the inner product (3.150). We act on each of the legs ′ of 𝑉 𝑠 with group elements 𝑔1 , 𝑔2 , 𝑔3 before contracting with 𝑉 𝑠 ∗ . A genus-2 partial wave ′ 𝑠 𝑠 is a simultaneous eigenfunction of the conformal The genus-2 conformal block 𝐵123 Casimir operators acting on each of the group elements 𝑔1 , 𝑔2 , 𝑔3 . In more detail, let 𝐿 𝐴 ( 𝐴 = 1, . . . , dim 𝐺) be the generators of the Lie algebra of 𝐺, realized as left-invariant vector fields on 𝐺. Then D = 𝐿 𝐴 𝐿 𝐴 is a differential operator on 𝐺 such that for any irrep 𝜋, we have D𝜋(𝑔) = 𝜋(𝑔)𝐶2 (𝜋), (3.156) where 𝐶2 (𝜋) is the Casimir eigenvalue for 𝜋. Any matrix element of 𝑔 in the representation 𝜋 is thus an eigenfunction of D. Viewing the conformal block as a matrix element of three group elements 𝑔1 , 𝑔2 , 𝑔3 in the representations 𝜋𝑖 , it follows that it must be a simultaneous eigenfunction of D, acting on each of the 𝑔𝑖 : ′ ′ 𝑠𝑠 𝑠𝑠 D𝑖 𝐵123 = 𝐶2 (𝜋𝑖 )𝐵123 𝑖 = 1, 2, 3, (3.157) where D𝑖 indicates the action of D on 𝑔𝑖 . The block also diagonalizes the higher Casimirs of the conformal group, acting on each 𝑔𝑖 . By analogy with the four-point case, we can define a genus-2 partial wave as a principal series matrix element of 𝑔𝑖 ’s between a pair of three-point structures: ′ ′ 𝑠𝑠 Ψ123 ≡ (𝑉 𝑠 |𝑔1 ⊗ 𝑔2 ⊗ 𝑔3 |𝑉 𝑠 ) ∫ ′ = 𝑑 𝑑 𝑥1 𝑑 𝑑 𝑥 2 𝑑 𝑑 𝑥 3𝑉 𝑠 ∗ (𝑥 1 , 𝑥2 , 𝑥3 )𝑔1 𝑔2 𝑔3 · 𝑉 𝑠 (𝑥1 , 𝑥2 , 𝑥3 ). (3.158) This is illustrated diagrammatically in Figure 3.9. The action of a conformal group element on an operator is 𝑔 · O 𝑎 (𝑥) = Ω(𝑥 ′) Δ𝜆 𝑎 𝑏 (𝑅 −1 (𝑥 ′))O 𝑏 (𝑥 ′). (3.159) 76 The notation 𝑔1 𝑔2 𝑔3 · 𝑉 𝑠 (𝑥1 , 𝑥2 , 𝑥3 ) indicates the simultaneous actions of 𝑔1 , 𝑔2 , 𝑔3 on the three-point structure 𝑉 𝑠 , using the formula (3.159) at each point. The spin indices of the operators are implicitly contracted in (3.158). By construction, the partial wave also solves the Casimir equations (3.157). Though we have not proven it, we expect that the space of solutions of the Casimir 𝑠′ 𝑠 and seven equations (3.157) is eight-dimensional, and is spanned by the block 𝐵123 "shadow" blocks obtained by replacing 𝜋𝑖 → e 𝜋𝑖 in various combinations: ′ ′ ′ 123 123 123 {𝐵e𝑠 𝑠 , 𝐵 𝑠e𝑠 , · · · , 𝐵e𝑠e𝑠e}. The genus-2 partial wave is a linear combination of these 8 solutions. Applying similar logic to the derivation of (3.154), we expect it to have the form ′ ′ ′ 𝑠𝑠 𝑡 𝑠 Ψ123 = (𝐼 −3 𝑆e3†e†e† ) 𝑠 𝑡 ′ 𝐵123 + (7 shadow blocks). (3.160) 1 2 3 Here (𝑆e3†e†e† ) 𝑠 𝑡 denotes the product of three shadow coefficients coming from 1 2 3 e†𝑠′ on each of its external legs: performing the shadow transform of 𝑉 𝑠′ ′ 𝑣 ′ e†𝑠′ = (𝑆 3 S1 S2 S3𝑉 e†e†e† ) 𝑣 𝑉 1 2 3 ′ ′ ′ ′ = 𝑆([e 𝜋1 ] † e 𝜋2† e 𝜋3† ) 𝑠 𝑡 ′ 𝑆(𝜋1† [e 𝜋2† ]e 𝜋3† ) 𝑡 𝑢′ 𝑆(𝜋1† 𝜋2† [e 𝜋3† ]) 𝑢 𝑣 ′ 𝑉 𝑣 . (3.161) (other expressions are possible, coming from doing the shadow transforms in other orders). Meanwhile, 𝐼 −3 indicates the action of inverse inversion tensors (𝐼 (𝑒) −1 )𝑎 𝑎 on each operator. These are needed in order for the resulting three-point structure to transform in the dual representations 𝜆∗1 , 𝜆∗2 , 𝜆∗3 , so that it can be paired with 𝑉 𝑠 . The seven shadow blocks in (3.160) will have similar coefficients, though we have not written them explicitly for brevity. We have not attempted to give a rigorous proof of (3.160), but instead have motivated it by analogy with the four-point case. We will verify (3.160) explicitly in the large-Δ limit where it is needed in Section 3.6. Symmetries of the saddle point equations We will be interested in the blocks and partial waves in the large-Δ limit. In this limit, the shadow integral (3.158) for the genus-2 partial wave can be evaluated by saddle point. The structure of the saddle point equations is complicated, and we will not attempt to solve them exactly for arbitrary 𝑔𝑖 . However, there are some simple operations that permute the saddle points that will be helpful in exploring their structure. We derive them in this section. 77 Let us begin with the integral for a partial wave. For simplicity, we will work in 𝑑 = 1; the results of this section will generalize straightforwardly to any 𝑑. The integral takes the form ∫ ΨΔ1 ,Δ2 ,Δ3 = 𝑑𝑧 1 𝑑𝑧2 𝑑𝑧3𝑉eΔ1 ,eΔ2 ,eΔ3 (𝑧 1 , 𝑧2 , 𝑧3 )𝑔1 𝑔2 𝑔3 · 𝑉Δ1 ,Δ2 ,Δ3 (𝑧1 , 𝑧2 , 𝑧3 ), (3.162) where 𝑉Δ1 ,Δ2 ,Δ3 (𝑧 1 , 𝑧2 , 𝑧3 ) = 1 |𝑧 12 | Δ1 +Δ2 −Δ3 |𝑧 23 | Δ2 +Δ3 −Δ1 |𝑧 Δ +Δ −Δ 31 | 3 1 2 (3.163) is a three-point structure for primaries with dimensions Δ1 , Δ2 , Δ3 in 1𝑑. Thinking of each 𝑔𝑖 as an SL(2, R) element, the action of 𝑔𝑖 on each operator is given by   𝑎 𝑖 𝑧𝑖 + 𝑏 𝑖 −2Δ𝑖 𝑔𝑖 · O (𝑧𝑖 ) = (𝑐𝑖 𝑧𝑖 + 𝑑𝑖 ) O . (3.164) 𝑐 𝑖 𝑧 𝑖 + 𝑑𝑖 In the limit of large Δ𝑖 , the integral is dominated by saddle points. Let us split the integrand into a rapidly-varying part that depends exponentially on Δ𝑖 , and a part that is slowly-varying at large Δ𝑖 . We define the saddle point equations as stationarity equations for the rapidly-varying part of the integrand. Concretely they are   𝜕𝑧𝑖 log 𝑉−Δ1 ,−Δ2 ,−Δ3 (𝑧 1 , 𝑧2 , 𝑧3 )𝑔1 𝑔2 𝑔3 · 𝑉Δ1 ,Δ2 ,Δ3 (𝑧 1 , 𝑧2 , 𝑧3 ) = 0 (𝑖 = 1, 2, 3). (3.165) This is a system of three coupled polynomial equations in the 𝑧𝑖 , with coefficients that depend on the 𝑔𝑖 . Note that the saddle-point equations are homogeneous in the Δ𝑖 in our conventions. We denote the coordinates of the three points collectively as 𝑝® = (𝑧 1 , 𝑧2 , 𝑧3 ). Suppose that we can find a saddle point 𝑝®∗ = (𝑧 1∗ , 𝑧2∗ , 𝑧3∗ ), i.e. a solution of (3.165), as a function of the Δ𝑖 . A simple operation that relates different saddle points is 𝜏 𝑝®∗ ≡ (𝑔1−1 𝑧 1 , 𝑔2−1 𝑧2 , 𝑔3−1 𝑧3 ) Δ𝑖 →eΔ𝑖 , (3.166) where we can approximate e Δ𝑖 ≈ −Δ𝑖 at large Δ𝑖 . The fact that 𝜏 𝑝®∗ is a saddle point of (3.162) follows from symmetry of the integrand under the change of variables 𝑧𝑖 → 𝑔𝑖−1 𝑧𝑖 and Δ𝑖 → e Δ𝑖 . However there is another less-obvious operation that relates saddle points to each other, coming from the Z32 shadow symmetry of the partial wave. We start by 78 rewriting (3.162) by introducing a shadow transformation on 𝑧1 : ΨΔ1 ,Δ2 ,Δ3 1 = 𝑆([Δ1 ] e Δ2 e Δ3 ) ∫ 1 𝑑𝑧 1 𝑑𝑧′1 𝑑𝑧 2 𝑑𝑧3 2Δ 𝑉Δ1 ,eΔ2 ,eΔ3 (𝑧′1 , 𝑧2 , 𝑧3 )𝑔1 𝑔2 𝑔3 · 𝑉Δ1 ,Δ2 ,Δ3 (𝑧 1 , 𝑧2 , 𝑧3 ) 𝑧 11′1 𝑆([Δ1 ]Δ2 Δ3 ) = 𝑆([Δ1 ] e Δ2 e Δ3 ) ∫ 𝑑𝑧′1 𝑑𝑧 2 𝑑𝑧 3𝑉Δ1 ,eΔ2 ,eΔ3 (𝑧′1 , 𝑧2 , 𝑧3 )𝑔1 𝑔2 𝑔3 · 𝑉eΔ1 ,Δ2 ,Δ3 (𝑧′1 , 𝑧2 , 𝑧3 ). (3.167) In the second line, we performed the integral over 𝑧1 and used that the two-point −2Δ1 function 𝑧 11 is 𝐺-invariant. ′ The resulting integral (3.167) has the same form as (3.162), except that Δ1 and e Δ1 have been swapped, and 𝑧1 has been swapped with 𝑧′1 . Thus, a saddle point of (3.167) is given by (𝑧′1 , 𝑧2 , 𝑧3 ) = 𝑝®∗ | Δ1 →eΔ1 . (3.168) So far, we have managed to find a saddle point for a different integral — not the original integral we started with. However, in the large Δ𝑖 limit, 𝑧′1 can be related to 𝑧1 using the integral on the first line of (3.167). The integral over 𝑧1 takes the form of a shadow transform, and the shadow transform is dominated by its own saddle point at large Δ𝑖 , as we explain in Appendix 3.9. Let us denote the saddle point obtained by shadow-transforming 𝑉Δ1 ,Δ2 ,Δ3 (𝑧 1 , 𝑧2 , 𝑧3 ) at site 𝑧1 by 𝑠 [Δ1 ]Δ2 Δ3 ( 𝑝) ® ≡− 2Δ1 𝑧2 𝑧3 − (Δ1 + Δ2 − Δ3 )𝑧 1 𝑧3 − (Δ1 − Δ2 + Δ3 )𝑧1 𝑧2 2Δ1 𝑧 1 − (Δ1 + Δ2 − Δ3 )𝑧2 − (Δ1 − Δ2 + Δ3 )𝑧3 (Δ𝑖 ≫ 1). (3.169) Note that 𝑠 [Δ1 ]Δ2 Δ3 satisfies the identity 𝑧1 = 𝑠 [eΔ1 ]Δ2 Δ3 (𝑠 [Δ1 ]Δ2 Δ3 (𝑧1 , 𝑧2 , 𝑧3 ), 𝑧2 , 𝑧3 ), (3.170) which is related to the fact that the square of the shadow transform is proportional to the identity. In our case, we have 𝑧′1 = 𝑠 [Δ1 ] eΔ2 eΔ3 (𝑧 1 , 𝑧2 , 𝑧3 ). (3.171) Using (3.170), we can solve this as 𝑧1 = 𝑠 [eΔ1 ] eΔ2 eΔ3 (𝑧′1 , 𝑧2 , 𝑧3 ). (3.172) 79 Putting everything together, we find a saddle point of the original integrand (3.162) 𝜎1 𝑝®∗ ≡ (𝑠 [eΔ1 ] eΔ2 eΔ3 ( 𝑝®∗ | Δ1 →eΔ1 ), 𝑧 2∗ | Δ1 →eΔ1 , 𝑧 3∗ | Δ1 →eΔ1 ) = (𝑠 [Δ1 ] eΔ2 eΔ3 ( 𝑝®∗ ), 𝑧2∗ , 𝑧3∗ )| Δ1 →eΔ1 . (3.173) Note that 𝜎1 𝑝®∗ may or may not coincide with 𝑝®∗ . We can similarly define operations 𝜎2 and 𝜎3 by cyclic permutations of (3.173). Note that 𝜎𝑖2 = 1 and the 𝜎𝑖 are mutually commuting. One can also show that 𝜏 = 𝜎1 𝜎2 𝜎3 . The 𝜎𝑖 operations give a homomorphism from Z32 into the group of permutations of the saddle solutions. The behavior of this homomorphism can jump when any of the Δ𝑖 crosses 0, since the saddle point analysis of the shadow integral for 𝑧1 , 𝑧′1 becomes invalid if Δ1 ’s is small. Indeed, we will see that such jumps happen in practice. Low temperature saddles We are now ready to explore the saddle points of the partial wave integral in different regimes. Let us begin by exploring low temperature 𝛽𝑖 → ∞, where it will be easy to distinguish the block from shadow blocks. We study high-temperature saddles (which will be our main interest) in the next section. As a reminder, we will use the parametrization of M given in (3.112). For simplicity, let us first turn off the angular fugacities by setting ℎ𝑖 = 1. The shadow integral (3.158) then has an SO(𝑑 − 1) symmetry, so we can locate its saddle points by specializing the points to the 𝑥 1 axis: 𝑥𝑖 = (𝑧𝑖 , 0, . . . , 0). The SO(𝑑 − 1)-symmetry also means that the local rotations 𝑅 𝜇 𝜈 (𝑥 ′) associated to each group element 𝑔𝑖 are trivial, since the centralizer of SO(𝑑 − 1) ⊂ SO(𝑑) is trivial. Thus, each group element acts in a simple way on the 𝑥-axis: 𝑔𝑖 · O 𝑎 (𝑥) = Ω𝑖 (𝑥 ′) Δ O 𝑎 (𝑔𝑖 𝑥). (3.174) The tensor structures in the numerators of the three-point structures are identical to what they would be in a standard configuration (𝑥 1 , 𝑥2 , 𝑥3 ) = (0, 𝑒, ∞). The remaining factors are the same as in the SL(2, R) transformation of conformal three-point structures in 1D. Thus, the integrand restricted to the 𝑥 1 -axis becomes ′ 𝐼 (𝑥𝑖 ) = 𝑉 𝑠 ∗ (𝑥 1 , 𝑥2 , 𝑥3 )𝑔1 𝑔2 𝑔3 · 𝑉 𝑠 (𝑥1 , 𝑥2 , 𝑥3 ) ′ = 𝑉 𝑠 ∗ (0, 𝑒, ∞)𝑉 𝑠 (0, 𝑒, ∞)𝑉eΔ1 ,eΔ2 ,eΔ3 (𝑧1 , 𝑧2 , 𝑧3 )𝑔1 𝑔2 𝑔3 · 𝑉Δ1 ,Δ2 ,Δ3 (𝑧1 , 𝑧2 , 𝑧3 ), (3.175) 80 where 𝑉Δ1 ,Δ2 ,Δ3 (𝑧1 , 𝑧2 , 𝑧3 ) are 1d conformal three-point functions, and the 𝑔𝑖 act via (3.164). In the small temperature limit 𝛽𝑖 → ∞, it is straightforward to find at least one solution to the saddle point equations (3.165). We naively expand the equations in the 𝛽𝑖 → ∞ limit to obtain (Δ1 − Δ2 + Δ3 )𝑧2 + (Δ1 + Δ2 − Δ3 )𝑧3 − 2Δ1 𝑧1 2Δ1 − + 𝑂 (𝑒 −𝛽𝑖 ), 𝑧12 𝑧 31 𝑧1 + 1 (Δ2 + Δ3 − Δ1 )𝑧1 + (Δ1 + Δ2 − Δ3 )𝑧3 − 2Δ2 𝑧2 2Δ2 0= − + 𝑂 (𝑒 −𝛽𝑖 ), 𝑧12 𝑧 23 𝑧2 − 1 (Δ2 + Δ3 − Δ1 )𝑧1 + (Δ1 + Δ3 − Δ2 )𝑧2 − 2Δ3 𝑧3 0= + 𝑂 (𝑒 −𝛽𝑖 ). (3.176) 𝑧31 𝑧 23 0= These have the solution   3Δ1 − Δ2 + Δ3 Δ1 − 3Δ2 − Δ3 Δ2 − Δ1 𝑝®0,0,0 ≡ , , + 𝑂 (𝑒 −𝛽𝑖 ). Δ1 + Δ2 − Δ3 Δ1 + Δ2 − Δ3 Δ3 (3.177) Then, using the operations defined in Section 3.6, we can generate seven additional saddles: 𝑝®1,0,0 ≡ 𝜎1 𝑝®0,0,0 , 𝑝®0,1,0 ≡ 𝜎2 𝑝®0,0,0 , 𝑝®0,0,1 ≡ 𝜎3 𝑝®0,0,0 , 𝑝®1,1,0 ≡ 𝜎1 𝜎2 𝑝®0,0,0 = 𝜏 𝑝®0,0,1 , 𝑝®1,0,1 ≡ 𝜎1 𝜎3 𝑝®0,0,0 = 𝜏 𝑝®0,1,0 , 𝑝®1,0,1 ≡ 𝜎1 𝜎3 𝑝®0,0,0 = 𝜏 𝑝®1,0,0 , 𝑝®1,1,1 ≡ 𝜎1 𝜎2 𝜎3 𝑝®0,0,0 = 𝜏 𝑝®0,0,0 . (3.178) As an aside, these additional saddles are more subtle to see directly from the saddle point equations because they involve points scaling towards singularities. For example, in the solution   Δ1 − Δ2 + Δ3 −𝛽1 Δ1 + 3Δ2 + Δ3 Δ1 + Δ2 𝑝®1,0,0 = −1 + 𝑒 +..., +... , +..., 4Δ1 Δ1 − Δ2 + Δ3 Δ3 (3.179) the point 𝑧 1 approaches the center of the ball 𝐵1 at 𝑧1 = −1, which is a singularity of the saddle point equations at low temperatures. To find the solution 𝑝®1,0,0 directly, we cannot use (3.176). Instead, we must re-expand the equations near the singularity and re-solve them in a small-temperature expansion, resulting in (3.179). 81 Let us denote the saddle point integral along a steepest descent contour through 𝑝®𝑎,𝑏,𝑐 by 𝐼𝑎,𝑏,𝑐 . Plugging in the different solutions, we find that the 𝐼𝑎,𝑏,𝑐 have the following behavior in the small temperature regime (as a function of the 𝛽𝑖 ): 𝐼0,0,0 ∼ 𝑒 −𝛽1 Δ1 −𝛽2 Δ2 −𝛽3 Δ3 , 𝐼1,0,0 ∼ 𝑒 −𝛽1 Δ1 −𝛽2 Δ2 −𝛽3 Δ3 , e 𝐼0,1,0 ∼ 𝑒 −𝛽1 Δ1 −𝛽2 Δ2 −𝛽3 Δ3 , e ... 𝐼1,1,1 ∼ 𝑒 −𝛽1 Δ1 −𝛽2 Δ2 −𝛽3 Δ3 . e e e (3.180) More formally, if we consider monodromies 𝑀𝑖 : 𝛽𝑖 → 𝛽𝑖 + 2𝜋𝑖, then each saddle point integral is an eigenfunction of the monodromies 𝑀1,2,3 with different eigenvalues. The block is the solution to the Casimir equations with monodromies 𝑠′ 𝑠 = 𝑒 −2𝜋𝑖Δ𝑖 𝐵 𝑠′ 𝑠 . It follows that 𝐼 𝑀𝑖 𝐵123 0,0,0 is the block at low temperatures, while 123 the other saddle point contours give shadow blocks. Let us finally turn back on the angular fugacities ℎ1 , ℎ2 , ℎ3 . In the low-temperature limit, they do not move the saddle point 𝑝®0,0,0 . Performing the gaussian integral around 𝑝®0,0,0 , and multiplying by the inverse of the triple shadow coefficient computed in (3.346), we find a nontrivial cancellation of Δ-dependent factors, resulting in ′ ′ 𝑡 𝑠 ((𝐼 −3 𝑆e3†e†e† ) −1 ) 𝑠 𝑡 ′ Ψ123 1 2 3 𝑝®0,0,0 = 2−2Δ1 −2Δ2 𝑒 −Δ1 𝛽1 −Δ2 𝛽2 −Δ3 𝛽3 ′ × 𝑉 𝑠 ∗ (0, 𝑒, ∞)(ℎ1 ℎ2 ℎ3 · 𝑉 𝑠 )(0, 𝑒, ∞) × (1 + 𝑂 (Δ𝑖−1 , 𝑒 −𝛽𝑖 )), (3.181) where the 𝑝®0,0,0 subscript means we evaluate the saddle point integral around 𝑝®0,0,0 . This result agrees precisely with the formula for the block from summing over states (3.130). This is a check on our assertion that 𝐼0,0,0 computes the block, and also on our ansatz (3.160) for the partial wave as a sum of blocks. High temperature saddles ® 𝑖 𝑗 = 𝛽−1 𝜃®𝑖 𝑗 fixed. In terms We define the high temperature regime as 𝛽𝑖 𝑗 → 0 with Ω 𝑖𝑗 of the coordinates 𝛽𝑖 , ℎ𝑖 , this means that ℎ𝑖 → 1 at high temperatures. Our strategy will be to start with the infinite temperature case 𝛽𝑖 = 0 and ℎ𝑖 = 1, and then work in perturbation theory at small 𝛽𝑖 . At infinite temperature, we restore SO(𝑑 − 1) symmetry, so we can again look for solutions along the 𝑥 1 axis. 82 It is not obvious a-priori that perturbation theory around infinite temperature makes sense — what if the block had a singularity at infinite temperature? However, we find in practice that the block is nonsingular at infinite temperature, and this strategy works. Relatedly, we do not find any evidence of a nontrivial difference in the order of limits 𝛽𝑖 → 0 and Δ𝑖 → ∞. This situation is somewhat different from the "𝑡-channel" 𝑧, 𝑧 → 1 limit of four-point conformal blocks, where the blocks have nontrivial log or power-law singularities, and one must be careful about orders of limits [176, 185]. It would be nice to understand these differences in more detail. In the infinite temperature limit 𝛽𝑖 = 0, one saddle point is relatively easy to find. We naively expand the saddle point equations and solve them to give   2(Δ2 −Δ1 ) 2Δ3 3 𝑞®0 ≡ 2Δ2Δ , − , + 𝑂 (𝛽𝑖 ). (3.182) 2Δ1 −Δ3 Δ1 +Δ2 2 −Δ3 Interestingly, it turns out that 𝜎1 𝑞®0 = 𝜎2 𝑞®0 = 𝜎3 𝑞®0 = 𝜏 𝑞®0 , so the operations defined in Section 3.6 generate only one additional high temperature saddle, namely 𝜏 𝑞®0 . However, it turns out that there are three additional high temperature saddles where the points 𝑥 1 , 𝑥2 , 𝑥3 scale towards each other in the high temperature limit. For example, we find a solution 𝑞®12 given by  2 2 2 2 2 2  𝛽2 Δ1 +𝛽1 (Δ1 +Δ3 ) 𝛽1 Δ2 +𝛽2 (Δ2 +Δ3 ) 𝛽1 ( Δ1 −Δ2 −Δ3 ) +𝛽2 ( Δ1 −Δ2 +Δ3 ) , , + 𝑂 (𝛽𝑖2 ). 𝑞®12 ≡ − 2Δ3 2Δ3 4Δ2 3 (3.183) Here, all three points scale toward 𝑥 = 0 (the point where balls 𝐵1 and 𝐵2 are tangent) as 𝛽𝑖 → 0. Similarly, we find a solution 𝑞®23 where the 𝑥𝑖 scale toward the point where balls 𝐵2 and 𝐵3 are tangent, and a solution 𝑞®31 where the 𝑥𝑖 scale toward the point where balls 𝐵3 and 𝐵1 are tangent. The action of 𝜎𝑖 and 𝜏 on these solutions is given by 𝜎2 𝑞®12 = 𝜎1 𝑞®12 = 𝑞®12 , 𝜎3 𝑞®12 = 𝜏 𝑞®12 , (3.184) and cyclic permutations of these relations. The saddle points 𝜏 𝑞®𝑖 𝑗 are new — they involve two points scaling towards a singularity, while one remains at a finite position. Thus, overall, we have eight high-temperature saddles given by 𝑞®0 , 𝑞®12 , 𝑞®23 , 𝑞®31 , and their images under 𝜏. These saddle points yield eight solutions of the conformal Casimir equations in the high temperature regime. But which one(s) corresponds to the block? To answer this, let us start at low temperature, where we know that the block corresponds to 83 𝑝®0,0,0 . As we dial from low to high temperature, we find that each low temperature saddle point transitions smoothly to a high temperature saddle. Although we have not proved analytically which saddle becomes which, we can track them numerically; see for example Figure 3.10. Interestingly, the matching between low- and high-temperature saddles depends on the signs of the Δ𝑖 . The saddle point equations depend projectively on the Δ𝑖 , so more precisely only the signs of their ratios matter. We will be most interested in the case where all Δ𝑖 /Δ 𝑗 are positive, where we find low temperature −→ 𝑝®0,0,0 𝑝®1,1,0 𝑝®0,1,1 𝑝®1,0,1 −→ −→ −→ −→ high temperature 𝑞®0 𝑞®12 𝑞®23 𝑞®31 (Δ𝑖 /Δ 𝑗 > 0). (3.185) The remaining mappings from low to high temperature are obtained by acting with 𝜏, for example 𝜏 𝑝®0,0,0 → 𝜏 𝑞®0 . p0,0,0 q0 p1,1,0 q12 ∞ ∞ 2 1 2 1 x2 0 -1 x1 0 x3 -1 -2 ∞ -2 ∞ p0,1,1 q23 p1,0,1 q31 ∞ ∞ 2 1 2 1 0 -1 0 -1 -2 ∞ -2 ∞ Figure 3.10: Evolution of saddle points from low to high temperature when Δ𝑖 /Δ 𝑗 > 0 for some representative values of Δ𝑖 . The blue curve is 𝑥 1 , the orange curve is 𝑥 2 , and the green curve is 𝑥 3 . Here, we set all three temperatures equal 𝛽𝑖 = 𝛽. The horizontal axis is 𝑡 = 𝑒 −𝛽 , with low temperatures near 𝑡 = 0 and high temperatures near 𝑡 = 1. How is this compatible with the fact that the 𝜎𝑖 operations act differently on the high temperature and low temperature saddle points? The key is that the map from 84 low to high temperature depends on the signs of the Δ𝑖 ’s, and the 𝜎𝑖 operations can flip these signs. We can capture these rules as follows. Let us define maps 𝐻±,±,± that take low temperature saddle points to high temperature saddle points by continuation in 𝛽𝑖 , with the signs of (Δ1 , Δ2 , Δ3 ) corresponding to the signs in the subscript of 𝐻±,±,± . For example, when all of the Δ𝑖 are positive, the map 𝐻+++ is given by 3.185. We claim that the 𝜎𝑖 interchange the maps 𝐻±±± in the following way: 𝐻𝑠1 𝑠2 𝑠3 ( 𝑝) ® = 𝜎1 𝐻 (−𝑠1 )𝑠2 𝑠3 (𝜎1 𝑝), ® 𝐻𝑠1 𝑠2 𝑠3 ( 𝑝) ® = 𝜎2 𝐻𝑠1 (−𝑠2 )𝑠3 (𝜎2 𝑝), ® 𝐻𝑠1 𝑠2 𝑠3 ( 𝑝) ® = 𝜎3 𝐻𝑠1 𝑠2 (−𝑠3 ) (𝜎3 𝑝). ® (3.186) In other words, conjugating by 𝜎𝑖 flips the sign of the 𝑖-th subscript in 𝐻±±± . Intuitively, this is because 𝜎𝑖 flips the sign of Δ𝑖 in 3.173. With the rules (3.185) defining 𝐻+++ , the relations (3.186), and the action of 𝜎𝑖 , 𝜏 on the low and high temperature saddles, we can predict how any low temperature saddle continues to a high temperature saddle, for various signs of the Δ𝑖 . We have verified these predictions numerically in examples. It would be nice to prove them analytically. To summarize, as we continue from low to high temperature, with Δ𝑖 /Δ 𝑗 > 0, the saddle point 𝑝®0,0,0 continues to 𝑞®0 . We will assume that no Stokes phenomena occur during this continuation, so that the saddle point integral through 𝑝®0,0,0 continues to the saddle point integral through 𝑞®0 . Another way to think about this is that the saddle point integral automatically solves the conformal Casimir equation, in perturbation theory in 1/Δ. The assumption of no Stokes phenomena is the same as the assumption that perturbation theory in 1/Δ can be used to solve the Casimir equations at large Δ for all temperatures. Thus, let us focus on the saddle point 𝑞®0 . Away from infinite temperature, the positions of the points in the 𝑞®0 saddle get corrected, and we can compute these ® 𝑖 defined in corrections in a systematic expansion in 𝛽𝑖 and the angles 𝛼 ® 𝑖 and Φ 85 (3.147) (still in the large-Δ limit). For example, 𝑥 1 shifts by 𝛿𝑥11 = Δ2 (4Δ21 − Δ23 )(Δ21 + Δ22 − Δ23 ) (Δ21 + (Δ2 − Δ3 ) 2 )(2Δ2 + Δ3 ) 𝛽1 + 𝛽2 4Δ2 (2Δ2 − Δ3 )Δ3 4Δ21 Δ3 (2Δ2 − Δ3 ) 2 − 𝛿® 𝑥1 = (Δ21 − Δ22 )Δ3 (Δ21 − Δ22 + Δ23 ) 2Δ21 Δ2 (2Δ2 − Δ3 ) 2 𝛽3 + 𝑂 (𝛽2 , 𝛼 ® 2 ), Δ2 (4Δ21 − Δ23 )(Δ21 + Δ22 − Δ23 ) (Δ21 + (Δ2 − Δ3 ) 2 )(2Δ2 + Δ3 ) 𝛼 ®1 + 𝛼 ®2 4Δ2 Δ3 (2Δ2 − Δ3 ) 4Δ21 Δ3 (2Δ2 − Δ3 ) 2 + (Δ21 − Δ22 )Δ3 (Δ21 − Δ22 + Δ23 ) 2Δ21 Δ2 (2Δ2 − Δ3 ) 2 ® 𝛼). 𝛼 ® 3 + 𝑂 (𝛽𝛼 ® , Φ® (3.187) Here, 𝛿® 𝑥 1 indicates the components of 𝑥1 perpendicular to the 𝑥 11 axis, and 𝑂 (𝛽2 , 𝛼 ® 2) ® 𝛼) stand for quadratic corrections in the 𝛽𝑖 , 𝛼 ® 𝑖 of the indicated form. and 𝑂 (𝛽𝛼 ® , Φ® ®𝑖 , Φ The 𝛼 ® 𝑖 appear at second order in 𝛿𝑥11 , as required by SO(𝑑 − 1) invariance. Plugging this corrected 𝑞®0 into the saddle point integral, and taking into account the 1-loop determinant, we finally find the high temperature behavior of the block at large Δ: ′ (2Δ1 ) 2Δ1 −𝑑 (2Δ2 ) 2Δ2 −𝑑 (2Δ3 ) 2Δ3 −𝑑 𝑠′ ∗ 𝑉 (0, 𝑒, ∞)ℎ1 ℎ2 ℎ3𝑉 𝑠 (0, 𝑒, ∞) 23𝑑/2 (Δ1 + Δ2 + Δ3 ) 2Δ1 +2Δ2 +2Δ3 −3𝑑       Δ1 Δ2 Δ2 Δ3 𝛽2 𝛽3 Δ1 Δ3 𝛽1 𝛽3 2 2 × exp − (𝛽1 + 𝛽2 ) − + − + + 𝑂 (Δ𝛽 , Δ® 𝛼 ) Δ3 Δ1 4 2 Δ2 4 2 ® 𝑖 | ≪ 1, Δ𝑖 ≫ 1). (𝛽𝑖 , | 𝛼 ® 𝑖 |, | Φ (3.188) 𝑠𝑠 𝐵123 = Recall that this formula only holds in the chamber Δ𝑖 /Δ 𝑗 > 0. When the signs of ratios Δ𝑖 /Δ 𝑗 are different, the high temperature behavior of the block is in general controlled by a different saddle. Our derivation so far has been for principal series representations Δ𝑖 ∈ 𝑑2 + 𝑖R≥0 . However, we can now analytically continue the result to real Δ𝑖 by simultaneously rotating the Δ𝑖 clockwise in the complex plane. In (3.188), we only kept linear order terms in 𝛽𝑖 in the exponent. Later, we will argue ® do not contribute to the leading asymptotics that the higher order terms in 𝛽, 𝛼 ®, Φ of OPE coefficients, at large Δ and finite 𝐽. Such terms can potentially become important at large-𝐽, but we leave the analysis of this case to future work. Note also that in (3.188), we have ℎ𝑖 = 1 + 𝑂 (𝛼𝑖 , Φ𝑖 ). We study one consequence of the Φ𝑖2 terms in ℎ𝑖 later in Section 3.7. As was the case at low temperatures, the apparent breaking of 1-2-3 permutation symmetry in (3.188) is due to using non-permutation symmetric coordinates on the 86 moduli space M. Switching to the relative temperatures 𝛽𝑖 𝑗 , the exponent in 3.188 becomes the beautifully permutation-symmetric   Δ1 Δ2 2 Δ2 Δ3 2 Δ3 Δ1 2 2 2 exp − 𝛽 − 𝛽 − 𝛽 + 𝑂 (Δ𝛽 , Δ® 𝛼 ) . (3.189) 8Δ3 12 8Δ1 23 8Δ2 31 Let us make a few additional observations about the result (3.188). We define a "scaling block" as the primary term in 3.130. Because the full block is a sum of scaling blocks with nonnegative coefficients, we should have the inequality 𝐵123 ≥ 2−2Δ1 −2Δ2 𝑒 −𝛽1 Δ1 −𝛽2 Δ2 −𝛽3 Δ3 (ℎ𝑖 = 1). (3.190) Let us check this at high temperatures. Our calculation of the block is valid in the regime Δ ≫ 1 and 𝛽 ≪ 1. Thus, we can ignore 𝛽Δ compared to Δ and just compare the Δ-dependent terms out front: (2Δ1 ) 2Δ1 −𝑑 (2Δ2 ) 2Δ2 −𝑑 (2Δ3 ) 2Δ3 −𝑑 ≥ 2−2Δ1 −2Δ2 . 3𝑑/2 2Δ +2Δ +2Δ −3𝑑 1 2 3 2 (Δ1 + Δ2 + Δ3 ) (3.191) Indeed, we find that numerically, the above inequality holds when Δ𝑖 > 0. It is saturated when Δ1 = Δ2 = Δ3 /2, and in this case the high temperature block and the scaling block are exactly the same (up to the order we’ve computed them)! One speculative interpretation is that the full genus-2 block at large Δ may be a scaling block in an appropriate Weyl frame that depends on the Δ𝑖 . Our choice of Weyl frame in Section 3.5 happens to be the appropriate frame for Δ1 = Δ2 = Δ3 /2. Other Weyl frames would be best suited to other Δ𝑖 . The large-Δ limit of Virasoro blocks also simplifies in an appropriate Weyl frame [161, 206]. 3.7 OPE coefficients of heavy operators "Heavy-heavy-heavy" OPE coefficients are encoded in the partition function of the CFT on the genus-2 manifold 𝑀2 via (3.129). In Section 3.5, using the thermal effective action and the "hot spot" hypothesis, we calculated the leading expression (3.146) for the partition function in the high-temperature regime discussed in Section 3.5. In Section 3.6, we obtained an expression (3.188) for a conformal block in the same regime. Finally, in this section, we will combine these ingredients to obtain an asymptotic formula for "heavy-heavy-heavy" OPE coefficients by inverting the conformal block decomposition of the partition function. Review: inverting a genus-1 partition function Before discussing how to invert a genus-2 partition function, let us revisit the genus1 case, phrasing it in language that will generalize to genus-2. Conformal blocks for 87 the genus-1 partition function on 𝑆 1 ×𝑆 𝑑−1 are just conformal characters 𝜒Δ,𝐽 (𝛽, Ω𝑖 ). For simplicity, let us work in 𝑑 = 1, where the characters have the simple form 𝜒Δ (𝛽) = 𝑒 −Δ𝛽 1 − 𝑒 −𝛽 (𝑑 = 1), (3.192) and the partition function has the decomposition ∫ 𝑍 (𝛽) = 𝑑Δ 𝑝(Δ) 𝜒Δ (𝛽), (3.193) which is essentially a Laplace transform of the density of states 𝑝(Δ). It is straightforward to decompose 𝑍 (𝛽) into characters via an inverse Laplace transform, as we did in Section 3.3. However, let us pause to understand this transform in grouptheoretic language. The conformal characters can be viewed as functions on the group SL(2, R) that are invariant under conjugation, i.e. class functions. They are naturally eigenfunctions of the Casimir differential operator D defined in Section 3.6. In terms of 𝛽, this leads to the eigenvalue equation D 𝜒Δ (𝛽) = 1 + 𝑒 −𝛽 ′ 𝜒Δ (𝛽) + 𝜒Δ′′ (𝛽) = Δ(Δ − 1) 𝜒Δ (𝛽). −𝛽 1−𝑒 (3.194) (Here, we abuse notation and write D both for the differential operator 𝐿 𝐴 𝐿 𝐴 on the group SL(2, R), and for the differential operator (3.194) acting on 𝛽.) Because the Casimir eigenvalue Δ(Δ − 1) is the same for Δ and e Δ = 1 − Δ, the shadow character 𝜒eΔ (𝛽) satisfies the same differential equation as 𝜒Δ (𝛽). Because of its group-theoretic origin, D is naturally self-adjoint in the Haar measure on SL(2, R). When acting on class functions, this implies that D defined in (3.194) is self-adjoint with respect to the quotient measure on the space of conjugacy classes of SL(2, R). The quotient measure is given by the famous Weyl integration formula (and can be computed using the Faddeev-Popov procedure): 𝑑𝜇 = 𝑑𝛽(𝑒 𝛽/2 − 𝑒 −𝛽/2 ) 2 . Self-adjointness of D immediately implies an orthogonality relation ∫ 𝑑𝜇 𝜒Δ (𝛽) 𝜒eΔ′ (𝛽) = 0 unless Δ = Δ′ or Δ = e Δ′, (3.195) (3.196) where the integral can follow any contour such that the boundary terms from integrating D by parts vanish. For our applications, we can integrate over an infinite 88 contour parallel to the imaginary axis 𝛽 = 𝛽0 + 𝑖𝑡, which gives ∮ ∮ ′ 𝑑𝜇 𝜒Δ (𝛽) 𝜒eΔ′ (𝛽) = 𝑑𝛽 𝑒 (Δ −Δ) 𝛽 = 2𝜋𝑖𝛿(Δ − Δ′). (3.197) This allows us to invert the partition function by integrating against a shadow block: ∮ ∮ 1 1 𝑝(Δ) = 𝑑𝜇 𝜒eΔ (𝛽)𝑍 (𝛽) = 𝑑𝛽𝑒 Δ𝛽 (1 − 𝑒 −𝛽 )𝑍 (𝛽), (3.198) 2𝜋𝑖 2𝜋𝑖 which is the usual inverse Laplace transform. Before proceeding, let us make a comment about the choice of contour. By analogy with Euclidean inversion formulae for local correlation functions, we could have instead tried to decompose 𝑍 (𝛽) in characters for principal series representations, which take the form 𝜒𝑠′ (𝛽) = 𝜒 1 +𝑖𝑠 (𝛽) + 𝜒 1 −𝑖𝑠 (𝛽) = 2 2 𝑒𝑖𝑠𝛽 + 𝑒 −𝑖𝑠𝛽 . 𝑒 𝛽/2 − 𝑒 −𝛽/2 (3.199) Principal series characters are naturally orthonormal with respect to 𝑑𝜇, when integrated along a real contour 𝛽 ∈ R. However, this kind of orthogonality is unsuitable for decomposing a physical partition function. The reason is that 𝑍 (𝛽) typically possesses a high-temperature singularity on the real axis of the form 𝑎 𝑍 (𝛽) ∼ 𝑒 1/𝛽 for some positive power 𝑎. This high-temperature singularity cannot be integrated against 𝜒𝑠′ (𝛽) along a real contour in a simple way.27 Using a complex contour as in 3.198 bypasses this issue by avoiding the singularity. An inverse Laplace transform for genus-2 blocks Let us now assemble analogous ingredients in the genus-2 case. Let 𝑑𝜇 be the natural quotient measure on the moduli space M = 𝐺\(𝐺 − ) 3 /𝐺, descending from the product of Haar measures on (𝐺 − ) 3 . Consider a contour integral of a block against a shadow block ∮ ′ 𝑠′ 𝑠 𝑑𝜇 𝐵123 𝐵e𝑡 ′𝑡†e′ †e′ † . (3.200) 1 2 3 (We do not specify the precise contour for now.) The block and shadow block are both simultaneous eigenfunction of the Casimir operators D𝑖 acting on each group element 𝑔𝑖 . Because of their group-theoretic origin, these operators are naturally 27 Interestingly, there is usually no problem with decomposing correlators of local operators in principal series representations. Doing so leads to Euclidean inversion formulas, which typically involve integrable power-like singularities. It would be nice to better understand the distinction between these cases. 89 self-adjoint in the measure 𝑑𝜇. If the contour is such that there are no boundary terms from integrating the D𝑖 by parts, then the above integral must be proportional to 𝛿-functions restricting 𝜋𝑖 and 𝜋𝑖′ to be the same ∮ ′ 𝑠′ 𝑠 (3.201) 𝑑𝜇 𝐵123 𝐵e𝑡 ′𝑡†e′ †e′ † ∝ 𝛿 𝜋1 𝜋1′ 𝛿 𝜋2 𝜋2′ 𝛿 𝜋3 𝜋3′ , 1 2 3 where 𝛿 𝜋𝜋′ = 𝛿(Δ − Δ′)𝛿𝜆𝜆′ . (3.202) To find the constant of proportionality in (3.201), we will assume that the contour is a complex contour in 𝛽𝑖 that can be deformed to low temperature and evaluated in that regime. This is analogous to the inverse Laplace transform (3.198), where we can choose any contour of the form 𝛽 = 𝛽0 + 𝑖𝑡. Moving the contour to low temperatures, we can evaluate the orthogonality relation (3.201) using the lowtemperature expansion of the blocks (3.130). When inverting a partition function, we can instead deform the contour into the high temperature regime and look for a saddle point. Computing the measure The first step is to compute the quotient measure 𝑑𝜇 via the Faddeev-Popov procedure. There are a few wrinkles in doing so, so let us work through the computation in full. Recall that the quotient space M can be redundantly parametrized by (𝑔1 , 𝑔2 , 𝑔3 ) ∈ 𝐺 − × 𝐺 − × 𝐺 − . We would like to fix a gauge by writing 𝑔𝑖 in terms of 𝛽𝑖 , ℎ𝑖 as in (3.112). For the moment, let us also imagine that we have chosen a non-redundant parametrization of the ℎ𝑖 in terms of angles, so that overall the 𝑔𝑖 are specified in terms of 𝑛 = dim 𝐺 parameters which we call 𝑦. Our gauge-fixing condition is 𝑔𝑖 = 𝑔𝑖 (𝑦). An appropriate gauge-fixing function is ∫ 𝑄(𝑔1 , 𝑔2 , 𝑔3 ) = 𝑑𝑦 𝛿(𝑔1 , 𝑔 1 )𝛿(𝑔2 , 𝑔 2 )𝛿(𝑔3 , 𝑔 3 ), (3.203) where 𝑑𝑦 is any measure on the coordinates 𝑦, and 𝛿(𝑔, 𝑔′) is a unit-normalized 𝛿-function (in the Haar measure) on the group supported at 𝑔 = 𝑔′. Consider now a gauge-invariant function 𝑓 (𝑔1 , 𝑔2 , 𝑔3 ). We formally define its integral over the moduli space as ∫ ∫ 𝑑𝑔1 𝑑𝑔2 𝑑𝑔3 𝑑𝜇 𝑓 = 𝑓 (𝑔1 , 𝑔2 , 𝑔3 ), (3.204) (vol𝐺) 2 M 90 where 𝑑𝑔𝑖 are Haar measures. Following the usual FP procedure, we insert 1 in the form of an integral over gauge orbits of 𝑄, divided by its average over gauge orbits ∫ ∫ ′ −1 ′ ′ 𝑑𝑔1 𝑑𝑔2 𝑑𝑔3 , 𝑔𝑔2 𝑔 −1 , 𝑔𝑔3 𝑔 −1 ) ′ 𝑄(𝑔𝑔1 𝑔 𝑑𝜇 𝑓 = 𝑑𝑔𝑑𝑔 𝑓 (𝑔1 , 𝑔2 , 𝑔3 ), b 1 , 𝑔2 , 𝑔3 ) (vol𝐺) 2 M 𝑄(𝑔 (3.205) where ∫ b 1 , 𝑔2 , 𝑔3 ) ≡ 𝑄(𝑔 ′ ′ ′ 𝑑𝑔 𝑑𝑔′𝑄(𝑔𝑔1 𝑔 −1 , 𝑔𝑔2 𝑔 −1 , 𝑔𝑔3 𝑔 −1 ). (3.206) b to Now we change variables 𝑔𝑖 → 𝑔 −1 𝑔𝑖 𝑔′ and use gauge-invariance of 𝑓 and 𝑄 obtain ∫ ∫ ∫ 𝑓 (𝑔 1 , 𝑔 2 , 𝑔 3 ) 𝑓 (𝑔1 , 𝑔2 , 𝑔3 ) . 𝑑𝜇 𝑓 = 𝑑𝑔1 𝑑𝑔2 𝑑𝑔3 𝑄(𝑔1 , 𝑔2 , 𝑔3 ) = 𝑑𝑦 b 1 , 𝑔2 , 𝑔3 ) b 1 , 𝑔2 , 𝑔3 ) M 𝑄(𝑔 𝑄(𝑔 (3.207) b is the FP determinant, which we now This is our gauge-fixed integral and 1/𝑄 compute. We have ∫ b 1 , 𝑔2 , 𝑔3 ) = 𝑄(𝑔 ′ 𝑑𝑔 𝑑𝑔 𝑑𝑦 ′ 3 Ö 𝛿(𝑔𝑔𝑖 (𝑦)𝑔′−1 , 𝑔𝑖 (𝑦′)). (3.208) 𝑖=1 The 𝛿-functions are supported for 𝑔, 𝑔′ near the identity and 𝑦′ near 𝑦. Thus, we can write 𝑔 = 1 + 𝜉, 𝑔′ = 1 + 𝜉 ′, (3.209) where 𝜉, 𝜉 ′ are elements of the Lie algebra of 𝐺, and we can furthermore Taylor expand the 𝑔𝑖 in 𝑦′ = 𝑦 + 𝑑𝑦. We have 𝛿(𝑔𝑔𝑖 (𝑦)𝑔′−1 , 𝑔𝑖 (𝑦′)) = 𝛿(𝑔𝑖 (𝑦) −1 𝑔𝑔𝑖 (𝑦)𝑔′−1 , 𝑔𝑖 (𝑦) −1 𝑔𝑖 (𝑦′)) = 𝛿(1 + 𝑔𝑖 (𝑦) −1 𝜉𝑔𝑖 (𝑦) − 𝜉 ′, 1 + 𝑑𝑦 · 𝑔𝑖 (𝑦) −1 𝜕𝑦 𝑔𝑖 (𝑦)) = 𝛿(Ad𝑔 −1 𝜉 − 𝜉 ′ − 𝑑𝑦 · 𝑔𝑖−1 𝜕𝑦 𝑔𝑖 ), (3.210) 𝑖 where in the last line, we have a 𝛿 function on the Lie algebra 𝔤, and Ad𝑔 denotes the adjoint action of 𝑔. The gauge-fixed measure is thus −1 Ad𝑔 −1 −1 −𝑔 −1 1 𝜕𝑦 1 𝑔 1 · · · −𝑔 1 𝜕𝑦 𝑛 𝑔 1 ª © 1 𝑑𝑦 ­ ® 𝜕𝑦1 𝑔 2 · · · −𝑔 −1 𝜕𝑦 𝑛 𝑔 2 ® 𝑑𝑦 1 · · · 𝑑𝑦 𝑛 𝑑𝜇 = = det ­Ad𝑔 −1 −1 −𝑔 −1 2 2 2 ­ ® b 𝑄 −1 −1 𝑛𝑔 Ad −1 −𝑔 𝜕 𝑔 · · · −𝑔 𝜕 −1 1 𝑦 3¬ 3 𝑦 3 3 « 𝑔3 (3.211) 91 The object inside the determinant is a 3𝑛 × 3𝑛 matrix. Choosing an orthonormal basis of 𝔤, Ad𝑔 −1 becomes an 𝑛 × 𝑛 block, and each 1 becomes an 𝑛 × 𝑛 identity 𝑖 matrix. Finally, −𝑔𝑖−1 𝜕𝑦 𝑗 𝑔𝑖 is an element of 𝔤, which we can think of as a column vector of height 𝑛. • Partial gauge fixing In our parametrization of M in terms of temperatures 𝛽1 , 𝛽2 , 𝛽3 and rotations ℎ1 , ℎ2 , ℎ3 ∈ SO(𝑑), we write the moduli space as M = 𝐺\(𝐺 − ) 3 /𝐺  SO(𝑑 − 1)\(SO(1, 1) × 𝑆𝑂 (𝑑)) 3 /SO(𝑑 − 1), (3.212) where the two copies of SO(𝑑 − 1) act by left and right multiplication on the ℎ𝑖 . Above, we obtained the measure from fully gauge-fixing both the left and right action of 𝐺. However, it will be more convenient to only partially fix the gauge, leaving the SO(𝑑 − 1) × SO(𝑑 − 1) gauge redundancy un-fixed. Let us determine how the above computation should be modified in this case. Let 𝑦 = (𝛽1 , 𝛽2 , 𝛽3 , ℎ1 , ℎ2 , ℎ3 ) now be coordinates on (SO(1, 1) × 𝑆𝑂 (𝑑)) 3 , and let us write 𝐾 = SO(𝑑 − 1), with Lie algebra 𝔨. The 𝑦 have an action of 𝐾 × 𝐾 given by 𝑦 ↦→ 𝑘 𝑦𝑘 ′−1 and an invariant measure 𝑑𝑦. Once again, we should consider the average over gauge orbits of the gauge-fixing function ∫ Ö b 𝑄(𝑔1 , 𝑔2 , 𝑔3 ) ≡ 𝑑𝑔𝑑𝑔′ 𝑑𝑦 𝛿(𝑔𝑔1 𝑔′−1 , 𝑔𝑖 (𝑦)). (3.213) 𝑖 We can factorize 𝑔 into 𝑔 = 𝑘 𝛾, (3.214) where 𝑘 ∈ 𝐾, and 𝛾 is a representative of the quotient 𝐾\𝐺. The measure on 𝐺 similarly splits as 𝑑𝑔 = 𝑑𝑘 𝑑𝛾, (3.215) where 𝑑𝑘 is the measure on 𝐾, and 𝑑𝛾 is a right-𝐺-invariant measure on 𝐾\𝐺. To be more precise, let 𝑇 𝑎 be an orthonormal basis of generators of 𝔤, and let the generators of 𝔨 ⊂ 𝔤 be the 𝑇 𝑎 with 𝑎 = 1, . . . , dim 𝐾. Then we can take the 𝛾 to be the image under the exponential map of the remaining generators 𝑇 𝑎 with 𝑎 = dim 𝐾 + 1, . . . , dim 𝐺. 92 Inserting this decomposition into (3.213), we find ∫ Ö b 1 , 𝑔2 , 𝑔3 ) ≡ 𝑄(𝑔 𝑑𝑘 𝑑𝛾 𝑑𝑘 ′ 𝑑𝛾 ′ 𝑑𝑦 𝛿(𝑘𝛾𝑔1 𝛾 ′−1 𝑘 ′−1 , 𝑔𝑖 (𝑦)) 𝑖 ∫ = 𝑑𝑘 𝑑𝛾 𝑑𝑘 ′ 𝑑𝛾 ′ 𝑑𝑦 Ö 𝛿(𝛾𝑔1 𝛾 ′−1 , 𝑔𝑖 (𝑘 −1 𝑦𝑘 ′)) 𝑖 = (vol𝐾) 2 ∫ 𝑑𝛾 𝑑𝛾 ′ 𝑑𝑦 Ö 𝑑𝜉 𝑑𝜉 ′ 𝑑𝑦 Ö 𝛿(𝛾𝑔1 𝛾 ′−1 , 𝑔𝑖 (𝑦)) 𝑖 = (vol𝐾) 2 ∫ 𝛿(Ad𝑔 −1 𝜉 − 𝜉 ′ − 𝑑𝑦 · 𝑔𝑖−1 𝜕𝑦 𝑔), (3.216) 𝑖 𝑖 where we have written 𝛾 = exp(𝜉), 𝛾 ′ = exp(𝜉 ′). (3.217) The only differences from before are that now 𝜉 and 𝜉 ′ are restricted to generators of 𝛾 — i.e. 𝑇 𝑎 with 𝑎 = dim 𝐾 + 1, . . . , dim 𝐺, and furthermore we must divide by (vol𝐾) 2 . The partially gauge-fixed measure is thus −1 Ad𝑔 −1 Π𝛾† −Π𝛾† −𝑔 −1 1 𝜕𝑦 1 𝑔 1 · · · −𝑔 1 𝜕𝑦 𝑚 𝑔 1 ª © 1 1 ­ ® 1 −1 𝑚 det ­Ad𝑔 −1 Π𝛾† −Π𝛾† −𝑔 −1 𝑑𝜇 = 2 𝜕𝑦 1 𝑔 2 · · · −𝑔 2 𝜕𝑦 𝑚 𝑔 2 ® 𝑑𝑦 · · · 𝑑𝑦 , 2 ­ ® (vol𝐾) 2 † † −1 −1 𝑚 «Ad𝑔3−1 Π𝛾 −Π𝛾 −𝑔 3 𝜕𝑦1 𝑔 3 · · · −𝑔 3 𝜕𝑦 𝑔 3 ¬ (3.218) where Π𝛾 is an (𝑛 − dim 𝐾) × 𝑛 matrix implementing the orthogonal projection onto the generators of 𝛾, and Π𝛾† is its adjoint. Finally, 𝑚 = 𝑛 + 2 dim 𝐾. Overall, this again gives a 3𝑛 × 3𝑛 matrix. Let us finally plug in (3.112) to write the measure (3.218) in terms of the parameters 𝛽𝑖 , ℎ𝑖 . We found it difficult to compute the measure exactly for generic parameters.28 However, we can compute it in various limits. At low temperatures, we find 𝑑𝜇 = 3 Ö 25𝑑 𝑒 𝑑 𝛽𝑖 𝑑𝛽𝑖 𝑑ℎ𝑖 , (vol SO(𝑑 − 1)) 2 𝑖=1 (low temperature), (3.219) where 𝑑ℎ𝑖 denotes Haar measures on SO(𝑑). At high temperature, we find 3 Ö 24𝑑 ® 𝑖, 𝑑𝜇 = 𝑑𝛽𝑖 𝑑 𝛼 ®𝑖 𝑑 Φ (vol SO(𝑑 − 1)) 2 𝑖=1 (high temperature), (3.220) 28 The reason is that (3.218) is the determinant of a large symbolic matrix, which is extremely difficult for Mathematica to handle. 93 ® 𝑖 are the angles defined in (3.147).29 where 𝛼 ® 𝑖 and Φ • Orthogonality relation for genus-2 blocks With the measure in hand, let us determine the correct orthogonality relation for genus-2 blocks. We will take the contour to be a real contour for the group elements ℎ𝑖 ∈ SO(𝑑), and a complex contour running parallel to the imaginary axis for the 𝛽𝑖 . The key idea will be to deform the 𝛽𝑖 contour into the low-temperature region, where the blocks are given by the simple formula ′ ′ 𝑠𝑠 𝐵123 = 2−2Δ1 −2Δ2 𝑒 −Δ1 𝛽1 −Δ2 𝛽2 −Δ3 𝛽3 𝑉 𝑠 (0, 𝑒, ∞) ∗ ℎ1 ℎ2 ℎ3 𝑉 𝑠 (0, 𝑒, ∞) × (1 + 𝑂 (𝑒 −𝛽𝑖 )). (3.222) Consider a shadow block with complex-conjugated three-point structures ′ ′∗ ∗ ′ ′ ′ ′ 𝐵e𝑡 ′ †𝑡e′ †e′ † = 2−2(𝑑−Δ1 )−2(𝑑−Δ2 ) 𝑒 −(𝑑−Δ1 ) 𝛽1 −(𝑑−Δ2 ) 𝛽2 −(𝑑−Δ3 ) 𝛽3 1 2 3 ′ × 𝑉 𝑡 (0, 𝑒, ∞)ℎ1 ℎ2 ℎ3 𝑉 𝑡 (0, 𝑒, ∞) ∗ (1 + 𝑂 (𝑒 −𝛽𝑖 )). (3.223) Integrating the block and the shadow block against each other using the lowtemperature measure (3.219), we find ∮ 𝑠′ 𝑠 𝑡 ′∗ 𝑡 ∗ 𝑑𝜇𝐵123 𝐵e′ †e′ †e′ † 1 2 3 ! ∫ Ö ′ ′ ′ 25𝑑 = 𝑑𝛽𝑖 𝑑ℎ𝑖 2−4𝑑 𝑒 −𝛽1 (Δ1 −Δ1 )−𝛽2 (Δ2 −Δ2 )−𝛽3 (Δ3 −Δ3 ) 2 (vol SO(𝑑 − 1)) 𝑖 ′ ′ × 𝑉 𝑠 (0, 𝑒, ∞) ∗ ℎ1 ℎ2 ℎ3 𝑉 𝑠 (0, 𝑒, ∞) × 𝑉 𝑡 (0, 𝑒, ∞)ℎ1 ℎ2 ℎ3 𝑉 𝑡 (0, 𝑒, ∞) ∗ . (3.224) To perform the integral over the ℎ𝑖 , we can use the Schur orthogonality formula, which states that for any compact group 𝐺 with unitary representations 𝜆, 𝜆′ ∫ vol𝐺 𝑑𝑔⟨𝑎|𝜆(𝑔)|𝑏⟩⟨𝑐|𝜆′ (𝑔 −1 )|𝑑⟩ = ⟨𝑎|𝑑⟩⟨𝑐|𝑏⟩ 𝛿𝜆𝜆′ . (3.225) dim 𝜆 29 We can find an interpolating result between high and low temperatures by setting the angles to ® 𝑖 = 0. In this case, Mathematica is able to compute the determinant for general 𝛽𝑖 , zero, 𝛼 ® 𝑖 = 0, Φ giving  𝑑 1 𝛽1 −𝛽2 −𝛽3 1 −𝛽1 +𝛽2 −𝛽3 𝛽1 𝛽2 𝛽1 +𝛽2 −𝛽3 −𝛽3 𝛽1 +𝛽2 +𝛽3 𝑑𝜇 = −8𝑒 − 8𝑒 + 𝑒 + 𝑒 − 2𝑒 +𝑒 + 32𝑒 2 2 3 Ö 1 ®𝑖 ® 𝑖 = 0). × 𝑑𝛽𝑖 𝑑 𝛼 ®𝑖 𝑑Φ (𝛼 ® 𝑖 = 0, Φ (3.221) (vol SO(𝑑 − 1)) 2 𝑖=1 This indeed agrees with both 3.219 and 3.220 in the appropriate limits. 94 The integral over 𝛽𝑖 gives 𝛿-functions of the form 𝛿(Δ𝑖 − Δ′𝑖 ), as in the inverse Laplace transform (3.198). Overall, we find the orthogonality relation ′ ′ 3 𝑑 𝑉 𝑡 (0, 𝑒, ∞) ∗𝑉 𝑠 (0, 𝑒, ∞) 𝑉 𝑠 (0, 𝑒, ∞) ∗𝑉 𝑡 (0, 𝑒, ∞) Ö ′ 2 vol SO(𝑑) = 𝛿𝜆𝑖 𝜆′𝑖 2𝜋𝛿(Δ − Δ ) 𝑖 𝑖 dim 𝜆 2𝑑 vol SO(𝑑 − 1) 2𝑑 vol SO(𝑑 − 1) 𝑖 𝑖=1 ′ ′ = 𝑇 𝑡𝑠𝑇 𝑠 𝑡 3 Ö 2𝜋𝛿(Δ𝑖 − Δ′𝑖 ) 𝑖=1 2𝑑 vol SO(𝑑) 𝛿𝜆𝑖 𝜆′𝑖 , dim 𝜆𝑖 (3.226) where we have introduced the three-point pairing matrix 𝑇 𝑡𝑠 = 𝑉 𝑡 (0, 𝑒, ∞) ∗𝑉 𝑠 (0, 𝑒, ∞) . 2𝑑 vol SO(𝑑 − 1) (3.227) Recall that the structure 𝑉 𝑠 is a tensor with an index for each of the representations 𝜆 1 , 𝜆2 , 𝜆3 , and 𝑉 𝑡∗ carries indices for the dual representations 𝜆∗1 , 𝜆∗2 , 𝜆∗3 . The indices of 𝑉 𝑠 and 𝑉 𝑡∗ are implicitly contracted in the three point pairing (3.227). The pairing matrix 𝑇 𝑡𝑠 shows up in other contexts related to harmonic analysis on the conformal group, and is discussed more extensively in [130, 151]. To summarize, if the partition function has an expansion in conformal blocks ∑︁ ∫ 𝑠𝑠′ 𝑠′ 𝑠 𝑍= 𝑑Δ1 𝑑Δ2 𝑑Δ3 𝑃123 𝐵123 , (3.228) 𝜆 1 ,𝜆2 𝜆 3 ′ 𝑠 𝑠 are given by the inversion formula then the conformal block coefficients 𝑃123 ′ ′ 𝑠𝑠′ 𝑃123 = (𝑇 −1 ) 𝑠𝑡 (𝑇 −1 ) 𝑡 𝑠 ∮ 3  ′∗ ∗ dim 𝜆𝑖 1 Ö 𝑑𝜇 𝑍 𝐵e𝑡 †e𝑡 †e† . 3 𝑑 1 2 3 (2𝜋) 𝑖=1 2 vol SO(𝑑) (3.229) In both (3.228) and (3.229), a sum over repeated three point structure indices 𝑠, 𝑡, 𝑠′, 𝑡 ′ is implicit. Putting everything together We are finally ready to put everything together and perform the genus-2 Laplace transform at high temperature. Let us recall the important formulas. The shadow block at high temperature is given by  𝑑−2Δ𝑖 ! Δ Δ 3  Ö 2 1 2 𝛽2 + Δ2 Δ3 𝛽2 + Δ1 Δ3 𝛽2 +𝑂 (Δ𝛽2 ,Δ® ′∗ ∗ 1 2Δ 𝑖 𝑖 𝛼 ) 𝐵𝑡e𝜋1 𝑡e𝜋2 e𝜋3 = 3𝑑/2 𝑒 8Δ3 12,0 8Δ1 23,0 8Δ2 31,0 Δ1 + Δ2 + Δ3 2 𝑖=1 𝑡′ × 𝑉 (0, 𝑒, ∞) 3 Ö 𝑖=1 𝑒𝑖𝛼𝑖 ·𝑀𝑖, 𝛼 𝑒𝑖Φ𝑖 ·𝑀𝑖,Φ 𝑉 𝑡 (0, 𝑒, ∞) ∗ , (3.230) 95 where we use the shorthand notation 𝛼𝑖 · 𝑀𝑖,𝛼 = 𝑑 ∑︁ 𝛼𝑖,𝑏 𝑀𝑖,1𝑏 , 𝑏=2 Φ𝑖 · 𝑀𝑖,Φ = ∑︁ Φ𝑖,𝑎𝑏 𝑀𝑖,𝑎𝑏 , (3.231) 2≤𝑎<𝑏≤𝑑 where 𝑀𝑖,𝑎𝑏 denotes a rotation generator with indices 𝑎𝑏 acting on the 𝑖-th point in the representation 𝜆𝑖 . The partition function at high temperature is given by ! ®1 − Φ ® 2)2 𝑓12 vol 𝑆 𝑑−1 (𝛼 ®1 + 𝛼 ®2)2 (Φ 𝑍 = exp 1− − 8(𝑑 + 1) +... 𝑑−1 2 4 𝛽12,0 𝛽12,0 𝛽12,0 ! ®2 − Φ ® 3)2 ® 2 − 12 𝛼 ®3)2 ( 14 𝛼 𝑓23 vol 𝑆 𝑑−1 (Φ + 1− − 8(𝑑 + 1) +... 𝑑−1 2 4 𝛽23,0 𝛽23,0 𝛽23,0 !! ®3 − Φ ® 1)2 ® 1 − 12 𝛼 ®3)2 ( 14 𝛼 (Φ 𝑓31 vol 𝑆 𝑑−1 1− + − 8(𝑑 + 1) +... . 𝑑−1 2 4 𝛽31,0 𝛽31,0 𝛽31,0 (3.232) Here, we have allowed for different free energy densities 𝑓𝑖 𝑗 at each hot spot. This would arise if we inserted topological defects into the partition function, for example symmetry operators, as discussed in Section 3.5. Our main case of interest is where 𝑓𝑖 𝑗 = 𝑓 (the thermal free energy density), but it is just as straightforward to do the computation for general 𝑓𝑖 𝑗 . Finally, the measure at high temperature is 3 Ö (𝑑−1) (𝑑−2) 24𝑑 ® 𝑖. 2 𝑑𝛽𝑖 𝑑 𝑑−1 𝛼 ®𝑖 𝑑 Φ 𝑑𝜇 = 2 (vol SO(𝑑 − 1)) 𝑖=1 (3.233) We would now like to integrate (3.229) to extract the density of OPE coefficients. We deform the contour so that it passes through the regime of high temperature. We will organize the calculation as follows. We split the integrand into the form (quickly varying) × (slowly varying). (3.234) Here "quickly varying" includes terms that are exponential in large parameters like Δ or 1/𝛽# , while "slowly varying" includes everything else. We look for a saddle point of the "quickly varying" terms, writing them as a gaussian centered at this saddle point, times perturbative corrections. Meanwhile, we expand the (slowly varying) part perturbatively around the saddle point. One simplification of this way of organizing the calculation is that, because we are working in the regime 𝐽 ≪ Δ, terms in the conformal block of the form 𝑉 ∗ ℎ1 ℎ2 ℎ3𝑉 96 will be included among the slowly-varying terms, and will not affect the location of ® 𝑖 = 0. the saddle point. By symmetry, the saddle point will be located at 𝛼 ®𝑖 = Φ Let us analyze the size of fluctuations around the saddle point. In particular, we would like to determine which terms must be kept in our approximation to the conformal block and the partition function. The quickly-varying (i.e. exponential in Δ) part of the block has the schematic form 2 2 𝐵 ∼ 𝑒 −Δ𝛽−Δ𝛽 −Δ®𝛼 +... , (3.235) where ". . . " are higher-order corrections in 𝛽, 𝛼, Φ. Meanwhile, the quickly-varying part of the partition function has the form !! ®2 𝛼 ®2 1 Φ 𝑍 ∼ exp 𝑑−1 1 − − 2 . (3.236) 𝛽 𝛽 𝛽 2 Here, 𝛽 schematically denotes the individual 𝛽𝑖 , not the relative 𝛽𝑖 𝑗,0 . The saddle point equation for 𝛽 will set Δ ∼ 𝛽−(𝑑+1)/2 . Plugging this in, we find ! ®2 Φ 1 𝛼 ®2 𝛼 ®2 𝑍 𝐵 ∼ exp 𝑑−1 − 𝑑+1 − 𝑑+3 − 𝑑+1 . (3.237) 𝛽 2 𝛽 2 𝛽 2 𝛽 2 There are two 𝛼 ® 2 terms: one coming from the block (3.235) and one coming from the partition function (3.236). We see that the 𝛼 ® 2 term coming from the partition function is more important — it is enhanced by an additional power of 1/𝛽. Thus, we can ignore the quadratic 𝛼 ® -dependence of the conformal block. In other words, the terms written explicitly in (3.230) are sufficient for our purposes. Overall, the characteristic size of fluctuations in the angular variables coming from (3.148) is 1 𝑑+3 1 𝛼 ® ∼ 𝛽 4 ∼ Δ− 2 − 𝑑+1 , 1 ® ∼ 𝛽 𝑑+1 4 ∼ Δ− 2 . Φ (3.238) ® 𝑖 = 0, and The saddle point for the quickly-varying terms is located at 𝛼 ® 𝑖 = 0, Φ  4(𝑑 − 1) 𝑓12 vol 𝑆 𝑑−1 Δ3 𝛽12,0 = Δ1 Δ2 1  𝑑+1 , (3.239) together with cyclic permutations of (3.239). The Hessian matrix at the saddle point splits into three separate blocks: a block for the 𝛽𝑖 , a block for the 𝛼 ® 𝑖 , and a block for ® the Φ𝑖 . Thus, we can calculate the 1-loop determinant separately in each of these 97 sets of variables. The determinant for the 𝛽𝑖 and 𝛼 ® 𝑖 variables is straightforward: 1-loop 𝛽𝑖 factor = (𝛽12,0 𝛽23,0 𝛽31,0 4 𝑓12 𝑓23 𝑓31  ) 𝑑+3  3 ! 12 𝜋 , 2(𝑑 2 − 1)vol 𝑆 𝑑−1   3 ! 𝑑−1 2 (𝛽12,0 𝛽23,0 𝛽31,0 ) 𝑑+3 𝜋 1-loop 𝛼 ® 𝑖 factor = . 4 𝑓12 𝑓23 𝑓31 2(𝑑 + 1)vol 𝑆 𝑑−1 (3.240) ® 𝑖 determinant, we must fix the SO(𝑑 − 1) gauge redundancy that To compute the Φ ® 𝑖 . For example, we can set Φ ® 3 = 0 and compute the simultaneously shifts the Φ ® 1, Φ ® 2 , multiplying the result by vol SO(𝑑 − 1) to account for 1-loop determinant in Φ the volume of the SO(𝑑 − 1) orbit. This gives ® 𝑖 factor 1-loop Φ  = vol SO(𝑑 − 1) 2 vol 𝑆 𝑑−1 𝜋 𝑓12 𝑓23 𝛽12,0 𝛽23,0
𝑑+1 + 𝑓12 𝑓31 𝛽12,0 𝛽31,0
𝑑+1 + ! ! − (𝑑−2)4(𝑑−1) 𝑓23 𝑓31 𝛽23,0 𝛽31,0 .
𝑑+1 (3.241) Putting everything together, plugging in the saddle point values (3.239), we find the asymptotic conformal block coefficients ′ 𝑠𝑠 ′ 𝑃123 ∼ (𝑇 −1 ) 𝑠𝑠 ! (𝑑+2) (𝑑−2) 3 5 𝑑2  𝑑 2 𝜋 dim 𝜆𝑖 (4(𝑑 − 1)) 2 − 𝑑+1 + 2  𝑑−1 3 𝑑+1 ) 8 𝑓 𝑓 𝑓 (vol 𝑆 12 23 31 3𝑑 𝑑 vol SO(𝑑) 2 2 (𝑑 + 1) 2 𝑖=1 3 Ö (Δ1 + Δ2 + Δ3 ) 2(Δ1 +Δ2 +Δ3 ) −3𝑑 (𝑑−1) (𝑑−2) (𝑑−1) Î 2Δ𝑖 − 𝑑2(𝑑+1) 3 4 (Δ21 + Δ22 + Δ23 ) 𝑖=1 (2Δ𝑖 ) " !#   2   𝑑−1 𝑑+1 2 vol 𝑆 𝑑−1 𝑑+1 Δ1 Δ2 𝑑+1 × exp (𝑑 + 1) 𝑓12 + cycl. , 2 8(𝑑 − 1)Δ3 × (3.242) where "+cycl." denotes a sum over cyclic permutations of 123. This result is valid for large Δ𝑖 with the spin-representations 𝜆𝑖 fixed, up to subleading corrections at large Δ𝑖 . Our approximation for the asymptotic squared OPE coefficients is then ′ 𝑠′ 𝑠 (𝑐 123 ) ∗ 𝑐 123 ∼ 𝑠𝑠 𝑃123 , (3.243) 𝜌1 𝜌2 𝜌3 where 𝜌𝑖 are the densities of states of the CFT computed in Section 3.3 for the representations 𝜋𝑖 = (Δ𝑖 , 𝜆𝑖 ). (In the case, where we refine the partition function with topological defects, the density 𝜌𝑖 should be the appropriate density of states with that defect inserted.) 98 ′ 𝑠𝑠 for identical-dimension scalars in various dimenAs an example, let us study 𝑃123 sions. In 2D, the rotation representations are one dimensional, there is a unique conformal three-point structure, and the corresponding 𝑇 matrix is simply 𝑇 = 1/2. Plugging this in, we find the high energy density of (global primary) OPE coefficients in 2D:   6Δ−9 2 9 (𝜋2 𝑓 2 Δ) 1/3 3 𝑓 𝑒2 2𝐷 : 𝑃ΔΔΔ ∼ . (3.244) 2 𝜋Δ5 In 3D, the 𝑇 matrix is diagonalized in the 𝑞-basis of [142]. Specifically, it is given by [130]  −1 3  1 Ö 2𝐽𝑖 [𝑞 1 𝑞 2 𝑞 3 ],[𝑞 1′ 𝑞 2′ 𝑞 3′ ] 𝑇 = 3 𝛿 𝑞𝑖 𝑞𝑖′ . (3.245) 2 2𝜋 𝑖=1 𝐽𝑖 + 𝑞𝑖 Plugging this into (3.242), we find the high energy density of OPE coefficients in 3D:   6Δ 49 9 3√2𝜋 𝑓 Δ Ö   3 3 2 4 𝑓 4𝑒 2𝐽𝑖 [𝑞 1 𝑞 2 𝑞 3 ] [𝑞 1′ 𝑞 2′ 𝑞 3′ ] (2𝐽𝑖 + 1) 3𝐷 : 𝑃 (Δ,𝐽 ) (Δ,𝐽 ) (Δ,𝐽 ) ∼ 𝛿 𝑞𝑖 𝑞𝑖′ . (3.246) 31 19 1 1 2 3 2 𝐽 + 𝑞 𝑖 𝑖 3 2 𝜋4Δ 4 𝑖=1 • A subleading correction It is straightforward to take into account perturbative corrections around the saddle point. For example, let us highlight the leading correction that is not proportional ′ to the 𝑇 𝑠𝑠 three-point structure matrix. It comes from the Φ2 term in the expansion of Ö ′ 𝑉 𝑡 (0, 𝑒, ∞) 𝑒𝑖Φ𝑖 ·𝑀𝑖,Φ 𝑉 𝑡 (0, 𝑒, ∞) ∗ (3.247) 𝑖 in the genus-2 block. Performing the Gaussian integral and using the fact that 𝑀1,Φ + 𝑀2,Φ + 𝑀3,Φ = 0 (because the three-point structures are SO(𝑑 − 1)-invariant), we find a multiplicative correction of the form ! 2 + Δ2 Δ2 𝑀 2 + Δ2 Δ2 𝑀 2 Δ21 Δ22 𝑀3,Φ 2 3 1,Φ 3 1 2,Φ 2 ® 𝑖 -correction = 1 − (𝑑 − 1) . (3.248) Φ Δ1 Δ2 Δ3 (Δ21 + Δ22 + Δ23 ) 2 are the Casimirs of the SO(𝑑 − 1) subgroup of SO(𝑑). As discussed Here, 𝑀𝑖,Φ ′ in [130], the three-point pairing matrix 𝑇 𝑠 𝑠 can be simultaneously diagonalized 2 . For example, in three dimensions, we have 𝑀 2 = 𝑞 2 in the together with the 𝑀𝑖,Φ 𝑖 𝑖,Φ 𝑞-basis. It will be interesting in the future to compute the full spin-dependence of the asymptotic OPE coefficients by computing the genus-2 blocks in the regime of finite 𝐽/Δ. 99 𝛽0 = 2 𝛽 𝛽0 = 2 Figure 3.11: A geometry that encodes a sum of squares of thermal one-point functions. The top surface is a copy of thermal flat space 𝑆 1𝛽0 =2 × R𝑑−1 , with a unit ball removed. The ball is tangent to itself because it wraps completely around the thermal circle of length 𝛽0 = 2. The bottom is the same as the top. The top and bottom are connected by a cylinder of length 𝛽 and angular twist ℎ ∈ SO(𝑑). The periodicity of each copy of thermal flat space is illustrated via arrow marks, indicating loci that should be identified. The hot spot thermal circle (red) runs down the cylinder in the front of the figure, and back up the cylinder in the back of the figure. 3.8 Asymptotics of thermal 1-point functions We can also use the techniques developed in this work to determine asymptotics of thermal 1-point functions. (In fact, one can think of high-energy thermal 1-point functions as a particular limit of heavy-heavy-heavy OPE coefficients.) One nice thing about this exercise is that, because the blocks are so simple, we can easily invert the partition function for arbitrary 𝐽/Δ. For brevity, we will only determine the leading exponential form of the thermal 1-point coefficients, leaving 1-loop determinants and subleading corrections for later work. Recall that the 1-point function of a primary operator O at inverse temperature 𝛽0 is fixed by symmetries to be [119] ⟨O 𝜇1 ···𝜇 𝐽 ⟩ 𝛽0 = 𝑏 O 𝜇1 (𝑒 · · · 𝑒 𝜇 𝐽 − traces), 𝛽0Δ (3.249) where 𝑒 = (1, 0, . . . , 0) is a unit vector in the Euclidean time direction, and 𝑏 O is an operator-dependent thermal 1-point coefficient. Only even-spin traceless symmetric tensors have nonvanishing thermal 1-point functions. We can build a geometry that measures squares of the 1-point coefficients 𝑏 O as follows. We start with two copies of thermal flat space 𝑆21 × R𝑑−1 at inverse temperature 𝛽0 = 2. From each copy, we drill out unit balls, so that the balls wrap around the thermal circle and are self-tangent. We then glue the boundaries of the two balls together with a cylinder of length 𝛽 (not equal to 𝛽0 !) and angular twist ℎ ∈ SO(𝑑); see Figure 3.11. By the cutting and gluing arguments of Section 3.5, 100 this geometry computes 𝑍= ∑︁ 1 ⟨O⟩ 𝛽0 =2 ⟨ℎ · O⟩ 𝛽0 =2 𝑒 −𝛽(Δ+𝜀0 ) 2 |𝑍glue (1)| O = ∑︁ 𝑏 1 O 𝑞 𝐽 P𝐽 (cos 𝜃)𝑒 −𝛽(Δ+𝜀0 ) . 2 2Δ |𝑍glue (1)| O 2 2 (3.250) Here, we used the fact that thermal 1-point functions are SO(𝑑 − 1)-invariant to write ℎ as a rotation away from the 𝑒-axis by an angle 𝜃. The function P𝐽 (cos 𝜃) is a Gegenbauer polynomial, given by 1−𝑥 P𝐽 (𝑥) = 2 𝐹1 (−𝐽, 𝐽 + 𝑑 − 2, 𝑑−1 2 , 2 ), (3.251) and 𝑞 𝐽 is given by 𝑞𝐽 = Γ( 𝑑−2 2 )Γ(𝐽 + 𝑑 − 2) . 2𝐽 Γ(𝑑 − 2)Γ(𝐽 + 𝑑−2 2 ) (3.252) In the limit 𝛽 → 0, this geometry develops a hot spot. The emergent thermal circle goes down the cylinder starting at a point of tangency, and then back up the diametrically opposite side of the cylinder to the starting point. To determine the hot-spot partition function, we should find the conformal group element that glues the plane to itself near this hot spot. On the each copy of the plane, we have a thermal periodicity 1 𝑥 ∼ 𝑒 2𝑃 𝑥, 1 𝑥 ′ ∼ 𝑒 2𝑃 𝑥 ′, (3.253) where 𝑃1 generates translations in the Euclidean time direction. Meanwhile, the cylinder induces an identification between the two coordinates 𝑥 = 𝑒 −𝛽𝐷 ℎ𝐼𝑥 ′ . (3.254) The hot-spot thermal circle thus corresponds to the group element 1 1 𝑔hot = 𝑒 −2𝑃 (𝑒 −𝛽𝐷 ℎ𝐼) −1 𝑒 2𝑃 (𝑒 −𝛽𝐷 ℎ𝐼). (3.255) We can define a relative temperature and angle from the eigenvalues of 𝑔hot : (𝑒 ±𝛽rel , 𝑒 ±𝑖𝜃 rel , 1, . . . , 1) = eigenvalues(𝑔hot ). (3.256) 101 Note that ℎ can be brought to the form of a rotation between the 1 and 2 axes. Then SO(𝑑 − 2) symmetry guarantees that the eigenvalues of 𝑔hot take the above form. The leading contribution to the partition function from the hot spot is !! ! 𝑓 vol𝑆 𝑑−1 𝜃2 𝑓 vol𝑆 𝑑−1 𝑍 ∼ exp 𝑑−1 1 − 32(𝑑 + 1) 4 + . . . , = exp 𝑑−1 𝛽rel,0 𝛽rel (1 + Ω2rel ) 𝛽rel,0 (3.257) where   √︁ 𝛽rel,0 = cosh−1 1 − 8𝑒 𝛽 + 8𝑒 2𝛽 = 4 𝛽 + . . . (3.258) is the relative inverse temperature when 𝜃 = 0, and this formula is valid for 𝜃 2 /𝛽2 ≪ 1 and 𝛽 ≪ 1. Formula (3.257) is the analog of (3.148) from the genus-2 calculation. All that remains is to invert (3.257) to determine the asymptotics of the coefficients 𝑏 O . In doing so, we can use the orthogonality relation for Gegenbauer polynomials ∫ 𝜋 𝑑𝜃 sin𝑑−2 𝜃 P𝐽 (cos 𝜃)P𝐽 ′ (cos 𝜃) = 𝑛 𝐽 𝛿 𝐽𝐽 ′ , (3.259) 0 where 𝑛𝐽 = 𝜋Γ(𝐽 + 1) . 𝑑−2 2 Γ( 𝑑−2 2 ) (𝐽 + 2 )Γ(𝐽 + 𝑑 − 2) (3.260) When we integrate P𝐽 (cos 𝜃) in 𝜃 against the Gaussian in (3.257), the integral will be dominated by small 𝜃 with fixed 𝜃𝐽. In this regime, we can use an approximation for the Gegenbauer function in terms of a Bessel function (which plays an important role in dispersive bounds on scattering amplitudes [52]): lim 𝐽→∞, 𝜃𝐽 fixed P𝐽 (cos 𝜃) = Γ( 𝑑−1 2 ) (𝜃𝐽/2) 𝑑−3 2 ∫ 𝐽 𝑑−3 (𝜃𝐽) = 2 𝑑 𝑛® ® 𝑒𝑖𝐽 𝑛®·𝜃 . 𝑑−2 vol 𝑆 (3.261) Here, 𝐽𝛼 (𝑥) is a Bessel function. In the right-hand formula, 𝑛® is a point on the unit ® = 𝜃. The idea is that 𝑆 𝑑−2 , and we think of 𝜃® as a vector in R𝑑−1 with norm | 𝜃| P𝐽 (cos 𝜃) satisfies a wave equation on 𝑆 𝑑−1 . In the limit of large 𝐽 with small 𝜃, we can zoom in near the locus 𝜃 = 0, where the 𝑆 𝑑−1 becomes flat space R𝑑−1 . We are left with a linear combination of solutions to the wave equation in flat space, i.e. plane waves. In this limit, the measure 𝑑𝜃 sin𝑑−2 𝜃 becomes equivalent to the usual measure 𝑑 𝑑−1 𝜃® on R𝑑−1 . 102 Thus, overall, the angular integral from inverting the partition function takes the form ! ∫ ∫ 𝑑−1 32(𝑑 + 1) 𝑓 vol𝑆 𝑑 𝑛® ® 𝑑 𝑑−1 𝜃® 𝑒𝑖𝐽 𝑛®·𝜃 exp − 𝜃®2 𝑑+3 vol 𝑆 𝑑−2 𝛽rel,0 ! 𝑑+3 𝛽rel,0 = exp − 𝐽 2 × 1-loop determinant. (3.262) 𝑑−1 128(𝑑 + 1) 𝑓 vol 𝑆 At the same time, we must perform an inverse Laplace transform in 𝛽. This integral can be done by saddle point, with the overall result  2 !   𝑏 2O 𝜌(Δ, 𝐽) 1 𝑑 + 1 2 𝑑 − 1 2 (𝑑 − 1) 𝑓 vol 𝑆 𝑑−1 𝑑+1 𝑞 𝐽 𝑛 𝐽 ∼ exp , Δ − 𝐽 Δ 𝑑−1 𝑑+1 22𝑑−1 Δ 2Δ (3.263) where 𝜌(Δ, 𝐽) is the density of states for traceless symmetric tensors. 3.9 Discussion In this paper, we studied the asymptotic behavior of CFT data at large energy. Using the thermal effective action, we looked at both the density of states and the threepoint-functions of heavy operators as a function of Δ, 𝐽. There are a number of interesting future directions to study. Density of states The formula (3.74) for the density of states is valid in a specific region of Δ, 𝐽. For example, in CFT3 , it is valid when Δ − |𝐽 | ≫ √︁ 𝑓 Δ. (3.264) This notably has no overlap with the regime of large spin with fixed twist described by the lightcone bootstrap [92, 138]. Naively, the lightcone bootstrap suggests that the spectrum of interacting CFT should look like Mean Field Theory in this regime [6, 91, 195]. It would be interesting if one could prove this statement using some kind of effective action, perhaps by compactifying the CFT on a null circle. It would be also interesting to study how the spectrum of operators can behave between these regions. For instance, for interacting 3D CFTs, is the density of states at large spin with twist obeying √︁ Δ0 ≪ Δ − |𝐽 | ≪ 𝑓 Δ (3.265) universal or theory-dependent? 103 One could also ask to further refine our general entropy formulas. In [128], a universal formula for CFT𝑑 with global symmetry was found. It would be nice to combine them and obtain universal formulas as a function of energy, spin, and global charges. In Section 3.4, we compared the predictions of the thermal effective action to exact results in free theories and Einstein gravity, finding excellent agreement. In Appendix 3.9, we give a preliminary comparison between the thermal effective action for the 𝑆 1 × 𝑆 2 partition function and numerical bootstrap data for the 3D Ising CFT. One could also consider other theories where a large number of operators are known from numerics, e.g. the 𝑂 (2) model [63, 152]. Obtaining accurate information about large-twist operators is a challenge for the numerical bootstrap, which seems to be most sensitive to the lowest-twist Regge trajectories [195]. (Computing a large number of heavy-heavy-heavy OPE coefficients with the numerical bootstrap is likely even more challenging.) In 2D CFTs, it is possible to make very precise statements about the spectrum of high energy states using more sophisticated tools than Laplace transforms and saddle point approximations; see e.g. [24, 73, 94, 168, 170, 174, 175]. Such techniques typically rely on nonperturbative input coming from modular invariance. Is it possible to derive similarly precise statements in higher dimensions? What additional information about the partition function is needed? Effective actions We parametrized our ignorance of the 𝑑−1-dimensional gapped theory upon compactifying on a thermal circle via an effective action, with an infinite set of Wilson coefficients. Can we place bounds on these Wilson coefficients? For instance, as discussed in Section 3.2, we know that 𝑓 > 0. Are there similar bounds (in either direction) on 𝑐 1 , 𝑐 2 , or other higher-derivative Wilson coefficients? One possible approach is to consider Weinberg-like sum rules relating two-point functions in the IR (described by the thermal effective action) and the UV (described by the CFT), as was recently done in [70]. Another approach is to consider the compactified theory in (𝑑 − 2, 1) (Lorentzian) signature and study dispersive bounds on scattering, following e.g. [2, 50, 137]. It may also be interesting to study perturbative examples. For example, the value of 𝑓 in the 3D 𝑂 (𝑁) models at large 𝑁 was computed long ago by Sachdev [186]. To our knowledge, higher Wilson coefficients in the thermal effective action, like the 104 b have not yet been computed for the 𝑂 (𝑁) models. coefficients of 𝐹 2 and 𝑅, In this work, we obtained all of our results purely using equilibrium hydrodynamical information. Recently, there has been a surge of progress in non-equilibrium hydrodynamics for CFTs; see [35, 150] for reviews. What additional CFT data can be predicted using this more sophisticated machinery? See [74, 131] for recent work in this direction. Can one study non-equilibrium dynamics at higher genus? There has also been tremendous recent progress applying other effective actions to characterize asymptotic CFT data, for example the effective theory of large charge [111, 121, 167]. One way of summarizing our "hot spot" analysis of the genus-2 partition function is the idea of using an EFT in the part of a geometry where it is valid (the hot spots), and factoring out the part where the EFT is not valid (the region away from the hot spots). Can this "hot spot" idea be useful in other contexts like large charge? Three-point functions and genus-2 blocks So far, we calculated asymptotic OPE coefficients to leading order at large Δ, with fixed spin. It would interesting to allow the spin to grow large with Δ, as we did for the density of states. In particular, this would require a more general expression for the genus-2 conformal block at large quantum numbers. Genus-2 blocks are interesting objects in their own right, and it would be interesting to study their properties more systematically, both at large and non-large quantum numbers. For example, can we find recursion relations for genus-2 blocks similar to those in [86, 139, 140, 177, 206]? Can we explore genus-2 blocks from the perspective of integrability [120]? Is there a clearer understanding of the interesting saddle-point dynamics uncovered in Section 3.6? Do there exist Lorentzian shadow representations [179] or holographic representations [114] for higher-genus blocks, and do they admit any interesting kinematic limits? The literature on global conformal blocks for correlation functions of local operators is vast, but global genus-2 blocks are essentially unexplored. We would also like to understand how to systematically improve our three-point function result. For the density of states, we understand how to systematically improve the result by keeping further terms in the thermal effective action. However, for the three-point function, corrections come in two types: higher derivative terms in the effective action (which are easy to include), and corrections to the hot spot assumption, as discussed in Sec. 3.5. In order to understand how to systematically 105 improve the estimate for the HHH three-point functions, we need to understand contributions to the partition function outside of the hot spot regions, namely understanding the quantity 𝑅 defined in (3.145). Explicitly computing examples of "genus-two" partition functions, either for free or holographic theories, could be instructive. It is also worthwhile to compare our result to known results CFT2 . In [67], it was shown that HHH, HHL, and HLL asymptotic density of states for Virasoro primary operators are all related to analytic continuations of the DOZZ formula — the structure constants of Liouville theory. In higher dimensions, asymptotic formulas for HLL OPE coefficients have been studied using Tauberian techniques and inversion formulae [169, 176, 185]. Furthermore, it is well-known that HHL OPE coefficients are related to thermal one-point functions. These computations, together with our genus-2 computation seem to involve different physics. It would be extremely interesting if there were a unifying perspective or formula similarly to 2D. Bootstrap axioms and crossing equations To what extent are our results for the density of states and OPE coefficients encoded in the usual bootstrap conditions — namely unitarity and crossing symmetry of local correlation functions? In 2D, modular invariance is known to be independent from crossing symmetry of local correlators. By analogy, this suggests that perhaps the formulas we derived from the thermal effective action are independent from the usual bootstrap axioms.30 If so, should we enlarge the axioms to include them? What is the minimal set of extra axioms that we need? In 2D, modular invariance can be interpreted as crossing symmetry of twist operators. Can our results in higher dimensions be interpreted in terms of traditional bootstrap axioms applied to appropriate twist operators? As we mentioned briefly in Section 3.5, there exists another decomposition of the genus-2 partition function 𝑍 (𝑀2 ) into a sum over states: the "dummbell" channel, which expresses 𝑍 (𝑀2 ) as a sum of squares of 1-point functions on 𝑆 1 × 𝑆 𝑑−1 . The dumbell channel has its own conformal blocks, which as far as we know have not been studied in detail. (The blocks discussed in Section 3.8 can be thought of as a limiting case of these dumbbell blocks.) Furthermore, one can formulate a crossing equation relating the dumbbell channel to the channel considered in this work. As 30 We thank Dalimil Mazáč for pointing this out. 106 pointed out in [64] for 𝑑 = 2, this crossing equation enjoys manifest positivity properties needed for numerical bootstrap applications. It would be very interesting to explore it in both two-dimensional and higher-dimensional theories. Ensembles and holographic theories There is an important difference between our higher-dimensional result for asymptotic OPE coefficients and the 2D results of [44, 67]. The results of [44, 67] were for OPE coefficients of Virasoro primaries, while our results are for OPE coefficients of global primaries. In the case of the density of states, there isn’t a huge difference between Virasoro and global primaries. But the story is different for OPE coefficients, where descendant states play an important role. We can see this by comparing the leading exponential behavior of Virasoro and global OPE coefficients in 2D: global 𝑃ΔΔΔ ∼   6Δ 3 2 ≫ Virasoro 𝑃ΔΔΔ ∼  27 16  3Δ . (3.266) We see that typical global primaries have much larger OPE coefficients than typical Virasoro primaries.31 In other words, the statistics of CFT data in a theory with global Virasoro symmetry has more structure than is captured by 𝑃ΔΔΔ . These statements are interesting to consider in a holographic CFT. In a holographic 2D CFT, a high energy Virasoro primary is interpreted as a black hole microstate, while a Virasoro descendant is a black hole orbited by boundary gravitons. We have found that states with boundary gravitons typically have much larger OPE coefficients than pure black hole microstates. While we don’t have an analog of Virasoro symmetry in higher dimensions, we can conjecture an analogous statement for higher-dimensional CFTs: we expect that typical states of black holes with orbiting matter have much larger OPE coefficients than pure black hole microstates. It would be very interesting to make this more precise, for example by performing a holographic computation of OPE coefficients of pure black hole microstates via an appropriate wormhole geometry. Virasoro to define an interesting "ensemble" of CFT data. The authors of [58] used 𝑃ΔΔΔ In their ensemble, OPE coefficients are (almost) gaussian random variables whose Virasoro /𝜌(Δ) 3 . Remarkably, the predictions of this ensemble turn variance is set by 𝑃ΔΔΔ out to agree with bulk 3D gravity. The result (3.266) indicates that an analogous global ensemble based on 𝑃ΔΔΔ would not have refined-enough information to recover 31 Similarly, by comparing scaling blocks and full genus-2 blocks in the high temperature regime, we conclude that typical states have much larger OPE coefficients than typical global primaries. 107 bulk gravity in 2D (presumably also in higher 𝑑). However, it is interesting to ask global whether any interesting physics would be captured by an ensemble built from 𝑃ΔΔΔ . In the spirit of [17, 20, 58, 122], we can imagine starting with a completely general ensemble of CFT data. We can refine this ensemble with knowledge of the partition functions on 𝑆 1𝛽 × 𝑆 𝑑−1 and the "genus-2" geometry 𝑀2 . We could additionally refine the ensemble with other information like local correlation functions and thermal one-point functions. At what point does the refined ensemble begin to make nontrivial predictions that can be tested in additional observables, and what are those predictions? Bulk locality for thermal observables HPPS famously conjectured that any unitary CFT with large 𝑐𝑇 and a large gap Δgap in the spectrum of higher-spin single-trace operators should agree with a local gravitational EFT in AdS [107]. Recently, there has been significant progress proving this statement for correlators of local operators in the CFT vacuum [3, 19, 51, 103, 135]. However, holography implies analogous statements in nontrivial backgrounds as well — in particular a thermal background. For example, the Wilson coefficients 𝑓 , 𝑐 1 , 𝑐 2 , etc. in the thermal effective action of a theory satisfying HPPS conditions should agree with those of Einstein gravity, up to small corrections suppressed by 1/Δgap . How can we prove the emergence of black hole physics using field-theoretic methods? Can we formulate dispersion relations in a black hole background? For recent work in these direction, see [49]. Completing the square in the thermal bootstrap An interesting feature of our formulas (3.250) and (3.257) is that they provide a kind of sum rule for squares of thermal 1-point coefficients 𝑏 2O . Such a sum rule could in principle be used to "complete the square" in the bootstrap equations studied in [118, 119]. The works [118, 119] studied crossing symmetry of thermal two-point functions, which have an expansion in products 𝑐 𝜙𝜙O 𝑏 O , where 𝑐 𝜙𝜙O are bulk OPE coefficients. Unfortunately, we do not know the sign of 𝑐 𝜙𝜙O 𝑏 O , and this prevents one from applying traditional numerical bootstrap techniques [181]. (The same issue appears in the study of boundary and defect two-point functions [148].) Ideally, one would like to complete the square by finding other crossing equations in which 𝑐 𝜙𝜙O and 𝑏 O appear quadratically. Then, one can treat 𝑐 𝜙𝜙O 𝑏 O as an off-diagonal element in a positive-definite 2 × 2 matrix and apply numerical bootstrap techniques for mixed 108 correlators [139]. A crossing equation where 𝑐 𝜙𝜙O appears quadratically is easy to find: it is just the usual crossing equation for vacuum four-point functions! Tantalizingly, the formulas (3.250) and (3.257) have 𝑏 O appearing quadratically, but unfortunately they are not as precise as the usual four-point crossing equation, due to our use of the hot-spot ansatz. It would be interesting to go beyond the hot-spot ansatz and find a sum rule precise enough to be used in the numerical bootstrap. "Sphere packing" and other hot-spot geometries In addition to the "genus-2" manifold 𝑀2 studied in this work, there are many additional geometries that encode statistics of CFT data and can be studied using the hot-spot idea. Partition functions on these geometries are examples of "generalized spectral form factors" [20]. As a simple example, consider a higher-genus generalization of 𝑀2 , where we take two copies of R𝑑 , drill out 𝑛 > 3 balls from each copy, and connect the boundaries of the balls with cylinders. The partition function on this geometry encodes a sum of squared 𝑛-point correlation functions of the CFT, schematically ∑︁ Í |⟨O1 · · · O𝑛 ⟩| 2 𝑒 − 𝑖 𝛽𝑖 Δ𝑖 . (3.267) O1 ,...,O𝑛 If two balls are tangent in (both copies of) R𝑑 , we obtain a hot spot when the corresponding cylinders shrink to zero length. The hot spot ansatz is most useful when there are a maximal number of hot spots, and all other moduli of the geometry are frozen. Thus, we should consider configurations where most of the balls are mutually tangent — i.e. sphere packings!32 For example, the packing shown in Figure 3.12 encodes interesting asymptotics of CFT four-point functions. We can construct an even more general "higher genus" manifold as follows. We take 𝑚 copies of R𝑑 and drill out various numbers of balls from each copy, such that there are an even number of balls in total. We then connect pairs of balls with cylinders. This computes a sum of products of 𝑚 correlation functions ∑︁ Í ⟨· · ·⟩ · · · ⟨· · ·⟩ 𝑒 − 𝑖 𝛽𝑖 Δ𝑖 . (3.268) | {z } O ,··· ,O 1 𝑛 𝑚 32 See [76, 105] for other connections between sphere packing and conformal field theory. 109 Figure 3.12: A ball with three mutually-tangent balls removed. If we take two copies of this space and glue the boundaries of the balls together with cylinders, analogously to Figure 3.3, we obtain a geometry that computes a sum of squares of CFT four-point functions. When the cylinders shrink, this "sphere packing" geometry contains hot spots at each of the six points of tangency. Again, hot spots can emerge when cylinders shrink to zero. Of course, CFT 𝑛-point functions for 𝑛 > 3 are determined in terms of 2- and 3-point functions. In this work we have determined asymptotics of 2- and 3-point functions, and it is interesting to ask whether our results can be used to predict partition functions on higher genus geometries, and whether the results agree with the hotspot ansatz at higher genus. In two dimensions, it is known that crossing symmetry of local four-point functions and torus one-point functions implies crossing symmetry on arbitrary Riemann surfaces. However, we expect that in higher dimensions, hot spot results for higher genus manifolds provide a nontrivial refinement of the statistics computed in this work. Appendix A Thermal two-point function of momentum generators In Section 3.3, we study the response of a CFT on 𝑆 1𝛽 × 𝑆 𝑑−1 when we twist by a rotation of 𝑆 𝑑−1 . In this appendix, as a warmup, we study the leading term in the twisting parameter in the high temperature limit. This computation reduces to a two-point function of momenta in thermal flat space. We determine this two-point function directly from Ward identities, and then show how the same result can be understood using the thermal effective action. At high temperature, the radius of curvature of the sphere becomes unimportant, and we can approximate 𝑆 1𝛽 × 𝑆 𝑑−1 as thermal flat space 𝑆 1𝛽 × R𝑑−1 . A rotation generator on the sphere locally looks like a translation on R𝑑−1 . Thus, it suffices to study the 110 thermal expectation value of a translation group element 1 ® 𝑃® ⟨𝑒 𝑎· ⟩ 𝛽 = ⟨1⟩ 𝛽 + 𝑎𝑖 𝑎 𝑗 ⟨𝑃𝑖 𝑃 𝑗 ⟩ 𝛽 + . . . . 2 (3.269) The momentum 𝑃𝑖 is the integral of 𝑇 0𝑖 on a spatial slice at fixed Euclidean time 𝜏 (which can take any value, by conservation): ∫ 𝑖 𝑃 =− 𝑑 𝑥® 𝑇 0𝑖 (𝜏, 𝑥®). (3.270) The 𝑂 (𝑎) term in (3.269) vanishes by rotation-invariance. For simplicity, in this section we set 𝛽 = 1, restoring it when needed by dimensional analysis. Using Ward identities Let us focus on the quadratic term in (3.269), given by an integrated two-point function of stress tensors: ∫ 1 1 1 𝑖 𝑗 𝑑 𝑥®1 ⟨𝑇 0𝑖 (0, 𝑥®1 )𝑇 0 𝑗 (𝑥 2 )⟩ 𝛽 . (3.271) 𝑎𝑖 𝑎 𝑗 ⟨𝑃 𝑃 ⟩ 𝛽 = 𝑎𝑖 𝑎 𝑗 𝑑−1 2 volR 2 Here, we separated the momentum generators in Euclidean time, placing the first at time 𝜏1 = 0 and the second at time 𝜏2 . We furthermore divided by vol R𝑑−1 to obtain a finite result, and used translation-invariance in R𝑑−1 to fix the second stress tensor at 𝑥 2 = (𝜏2 , 𝑥®2 ). We claim that the integrated two-point correlator (3.271) is determined by the onepoint function of the stress tensor at finite temperature. To understand why, we must express it in terms of operators in the dimensionally-reduced 𝑑−1-dimensional theory. The first step is to average over Euclidean times 𝜏1 and 𝜏2 . However, this averaging is subtle because 𝑇 0𝑖 (𝑥 1 ) and 𝑇 0 𝑗 (𝑥2 ) become coincident, and contact terms can contribute. Such contact terms are actually crucial to the calculation, so let us take a moment to define them carefully. We define (un-normalized) one- and two-point functions of the stress tensor by √︁ 𝛿𝑍 𝐺 (𝑥)⟨𝑇 𝜇𝜈 (𝑥)⟩ = 2 , (3.272) 𝛿𝐺 𝜇𝜈 (𝑥) √︁ √︁ 𝛿2 𝑍 𝐺 (𝑥) 𝐺 (𝑦)⟨𝑇 𝜇𝜈 (𝑥)𝑇 𝜌𝜎 (𝑦)⟩ = 4 , (3.273) 𝛿𝐺 𝜇𝜈 (𝑥)𝛿𝐺 𝜌𝜎 (𝑦) where 𝑍 [𝐺] is the partition function. Our definition of the two-point function (3.273) applies at both coincident and non-coincident points, and thus suffices to specify all contact terms. Diffeomorphism invariance of 𝑍 [𝐺] implies that ∫ 𝛿𝑍 0= 𝑑 𝑑 𝑥L𝜉 𝐺 𝜇𝜈 , (3.274) 𝛿𝐺 𝜇𝜈 111 where L𝜉 denotes the Lie derivative with respect to a vector field 𝜉 𝜇 . Taking an additional derivative with respect to 𝐺 𝜌𝜎 (𝑦), and evaluating in a locally flat metric 𝐺 𝜇𝜈 = 𝛿 𝜇𝜈 , we obtain the following Ward identity for conservation of the stress tensor inside a two-point correlator ∫ 𝑑 𝑑 𝑥 (𝜕𝜇 𝜉 𝜈 (𝑥))⟨𝑇 𝜇𝜈 (𝑥)𝑇 𝜌𝜎 (𝑦)⟩ = (𝜉 · 𝜕)⟨𝑇 𝜌𝜎 (𝑦)⟩ + (𝜕 · 𝜉)⟨𝑇 𝜌𝜎 (𝑦)⟩ − 𝜕𝜇 𝜉 𝜌 ⟨𝑇 𝜇𝜎 (𝑦)⟩ − 𝜕𝜇 𝜉 𝜎 ⟨𝑇 𝜇𝜌 (𝑦)⟩. (3.275) We can use this identity to average the correlator (3.271) over Euclidean time. Consider the vector field 𝜉 𝑖 (𝜏, 𝑥®) = −𝑎𝑖 𝜏, 𝜉 0 (𝜏, 𝑥®) = 0, (3.276) where 𝜏 ∈ [0, 1]. Since 𝜏 is periodic, 𝜉 𝑖 has the shape of a "sawtooth" function, with a discontinuity at 𝜏 = 0. In particular, we have 𝜕𝜏 𝜉 𝑖 = 𝑎𝑖 (𝛿(𝜏) − 1). (3.277) Applying (3.275), we find ∫ ∫ 0𝑖 0𝑗 𝑑 𝑥®1 𝑎𝑖 𝑎 𝑗 ⟨𝑇 (0, 𝑥®1 )𝑇 (𝑦)⟩ 𝛽 = 𝑑𝜏1 𝑑 𝑥®1 𝑎𝑖 𝑎 𝑗 ⟨𝑇 0𝑖 (𝜏1 , 𝑥®1 )𝑇 0 𝑗 (𝑦)⟩ 𝛽 + 𝑎 2 ⟨𝑇 00 (𝑦)⟩ 𝛽 , (3.278) where we used 𝜕 · 𝜉 = 0 and translation invariance 𝜉 · 𝜕⟨𝑇 𝜌𝜎 ⟩ 𝛽 = 0. The righthand side of (3.278) is the two-point correlator averaged over Euclidean time, plus a nontrivial contact term 𝑎 2 ⟨𝑇 00 (𝑦)⟩ that is a consequence of diffeomorphism invariance. It is natural to define the 𝑑−1 dimensional stress tensor 𝑡 𝑖 𝑗 (® 𝑥 ) and KK current 𝑗 𝑖 (® 𝑥) via derivatives with respect to 𝑔𝑖 𝑗 and 𝐴𝑖 in the Kaluza-Klein parametrization (3.8). For example, we have 𝛿𝑍 , 𝛿𝑔𝑖 𝑗 (® 𝑥) 𝛿2 𝑍 ⟨ 𝑗 𝑖 (® 𝑥 ) 𝑗 𝑗 ( 𝑦®)⟩ ≡ . 𝛿 𝐴𝑖 (® 𝑥 )𝛿 𝐴 𝑗 ( 𝑦®) √︁ 𝑔(® 𝑥 )⟨𝑡 𝑖 𝑗 (® 𝑥 )⟩ ≡ (3.279) A key property of the KK parametrization (3.8) is that gauge transformations 𝐴𝑖 ( 𝑥®) → 𝐴𝑖 (® 𝑥 ) + 𝜕𝑖 𝜆(® 𝑥 ) are diffeomorphisms of the 𝑑-dimensional metric. Consequently, diff-invariance implies that ⟨ 𝑗 𝑖 (® 𝑥 ) 𝑗 𝑗 ( 𝑦®)⟩ is exactly conserved, even at coincident points. This will be important in a moment. 112 At separated points 𝑡 𝑖 𝑗 (® 𝑥 ) and 𝑗 𝑖 (® 𝑥 ) are equivalent to Euclidean time averages of 𝑇 𝑖 𝑗 (𝜏, 𝑥®) and 𝑇 0𝑖 (® 𝑥 ), respectively. However, at coincident points, they differ from naïve averages by contact terms. In particular, the definitions (3.279) and (3.273) imply (on a flat geometry) ∫ ∫ 𝑖 𝑗 0𝑖 0𝑗 ⟨ 𝑗 (® 𝑥 1 ) 𝑗 (® 𝑥 2 )⟩ = 𝑑𝜏1 𝑑𝜏2 ⟨𝑇 (𝑥 1 )𝑇 (𝑥 2 )⟩ + 𝛿(® 𝑥1 − 𝑥®2 ) 𝑑𝜏1 ⟨𝑇 𝑖 𝑗 (𝑥1 )⟩. (3.280) The contact term on the right-hand side arises from the quadratic term in 𝐴𝑖 in the KK metric: 𝐺 𝑖 𝑗 = 𝑔𝑖 𝑗 + 𝑒 2𝜙 𝐴𝑖 𝐴 𝑗 . Integrating (3.280) over 𝑥®1 , and combining it with the average of (3.278) over 𝜏2 , we find ∫ ∫ 0𝑖 0𝑗 𝑑 𝑥®1 ⟨ 𝑗 𝑖 (® 𝑥1 ) 𝑗 𝑗 (® 𝑥 2 )⟩ − 𝑎𝑖 𝑎 𝑗 ⟨𝑇 𝑖 𝑗 ⟩ 𝛽 + 𝑎®2 ⟨𝑇 00 ⟩ 𝛽 . 𝑑 𝑥®1 𝑎𝑖 𝑎 𝑗 ⟨𝑇 (0, 𝑥®1 )𝑇 (𝑦)⟩ 𝛽 = 𝑎𝑖 𝑎 𝑗 (3.281) ∫ Finally, we will argue that the integrated correlator 𝑑 𝑥®1 ⟨ 𝑗 𝑖 (® 𝑥1 ) 𝑗 𝑗 (® 𝑥 2 )⟩ vanishes. We can think of it as the momentum-space two-point function ⟨ 𝑗 𝑖 ( 𝑝) ® 𝑗 𝑗 (− 𝑝)⟩, ® evaluated at zero momentum. Rotation-invariance and conservation imply that the momentum-space two-point function takes the form ⟨ 𝑗 𝑖 ( 𝑝) ® 𝑗 𝑗 (− 𝑝)⟩ ® = ( 𝑝®𝑖 𝑝® 𝑗 − 𝛿𝑖 𝑗 𝑝®2 )𝐺 (| 𝑝|). ® (3.282) If the finite-temperature theory has a nonzero mass gap, then 𝐺 (| 𝑝|) ® must be regular near zero momentum (otherwise its Fourier transform would have support at long distances.) Thus, at low momenta, we have ⟨ 𝑗 𝑖 ( 𝑝) ® 𝑗 𝑗 (− 𝑝)⟩ ® = 𝑐( 𝑝®𝑖 𝑝® 𝑗 − 𝛿𝑖 𝑗 𝑝®2 ) + 𝑂 ( 𝑝®4 ). (3.283) In particular, ⟨ 𝑗 𝑖 ( 𝑝) ® 𝑗 𝑗 (− 𝑝)⟩ ® vanishes at 𝑝® = 0. Note that conservation of 𝑗 𝑖 (® 𝑥 ) at coincident points is crucial here. Without it, the momentum-space correlator could have an 𝑂 ( 𝑝®0 ) contact term of the form 𝛿𝑖 𝑗 . It is instructive to understand this vanishing result in position space as well. In ∫ the position-space integral 𝑑 𝑥®1 𝑎𝑖 𝑎 𝑗 ⟨ 𝑗 𝑖 (® 𝑥 1 ) 𝑗 𝑗 (® 𝑥 2 )⟩, we can write 𝑎𝑖 = 𝜕𝑖 ( 𝑎® · 𝑥®1 ). Integrating by parts and using conservation, we obtain a boundary term at infinity: ∫ ∫ 𝑖 𝑗 𝑎𝑖 𝑎 𝑗 𝑑 𝑥®1 ⟨ 𝑗 (® 𝑥 1 ) 𝑗 (® 𝑥2 )⟩ = lim 𝑎 𝑗 𝑑𝑆𝑖 ( 𝑎® · 𝑥®1 )⟨ 𝑗 𝑖 (® 𝑥1 ) 𝑗 𝑗 (® 𝑥 2 )⟩, (3.284) 𝑅→∞ | 𝑥®|=𝑅 113 where 𝑑𝑆𝑖 is the surface normal for the sphere |® 𝑥 | = 𝑅. This boundary term (3.284) vanishes provided the two-point function decays sufficiently quickly at large |® 𝑥 |. In other words, ∫ 𝑎𝑖 𝑎 𝑗 𝑑 𝑥®1 ⟨ 𝑗 𝑖 (® 𝑥 1 ) 𝑗 𝑗 (® 𝑥2 )⟩ = 0 if lim |® 𝑥 | 𝑑−1 ⟨ 𝑗 𝑖 (® 𝑥 ) 𝑗 𝑗 (0)⟩ = 0. (3.285) | 𝑥®|→∞ This condition certainly holds when the finite-temperature theory has a mass gap (since the correlator decays exponentially). However, it also holds more generally. For example, if the finite-temperature theory possesses a massless sector, we expect the current two-point function to decay no slower than a correlator of conserved currents in a 𝑑−1 dimensional CFT: ⟨ 𝑗 (® 𝑥 ) 𝑗 (0)⟩ ∼ |® 𝑥 | −2(𝑑−2) . In that case, (3.285) will be satisfied as long as 𝑑 > 3. Finally, using (3.285) in (3.281), we find ∫ 𝑑 𝑥®1 𝑎𝑖 𝑎 𝑗 ⟨𝑇 0𝑖 (0, 𝑥®1 )𝑇 0 𝑗 (𝑦)⟩ 𝛽 = −𝑎𝑖 𝑎 𝑗 ⟨𝑇 𝑖 𝑗 ⟩ 𝛽 + 𝑎®2 ⟨𝑇 00 ⟩ 𝛽 = −( 𝑓 𝑑) 𝑎®2 , (3.286) where we used (3.18) for the one-point functions ⟨𝑇 𝜇𝜈 ⟩ 𝛽 . We conclude 𝑓 𝑑 𝑑+1 2 𝑇 𝑎® vol R𝑑−1 + . . . , 2 ® ®𝑃 ⟨𝑒 𝑎· ⟩𝛽 = 1 − (3.287) where we have restored factors of 𝑇 by dimensional analysis. To apply this result to 𝑆 1𝛽 × 𝑆 𝑑−1 with a twist by a rotation of 𝑆 𝑑−1 , we can make the replacement ∫ 2 𝑑−1 𝑎® vol R → 𝑑Ω𝑑−1 |𝑣| 2 , (3.288) 𝑆 𝑑−1 where 𝑣 is the Killing vector on 𝑆 𝑑−1 implementing the rotation. Using the thermal effective action The thermal effective action gives an efficient way to package the Ward identity calculations above and extend them to arbitrary nonlinear order in 𝑎. ® Let us see ® 𝑎· ® 𝑃 how it recovers the result (3.269). The correlator ⟨𝑒 ⟩ is captured by the geometry 𝑆 1𝛽 × R𝑑−1 with a twist of 𝑎® around the thermal circle, i.e. an identification (𝜏, 𝑥®) ∼ (𝜏 + 1, 𝑥 − 𝑎). ® (3.289) To use the thermal effective action, we must put the metric into Kaluza Klein form. We undo the twist with a coordinate transformation 𝑥®′ = 𝑥® − 𝜏 𝑎. ® (3.290) 114 This essentially implements averaging over Euclidean time (3.278) by spreading out the twist over the thermal circle. The metric changes to 𝑑 𝑥®2 + 𝑑𝜏 2 = (𝑑 𝑥®′ + 𝑎𝑑𝜏) ® 2 + 𝑑𝜏 2  2 ( 𝑎® · 𝑑 𝑥®′) 2 𝑎® · 𝑑 𝑥®′ 2 = 𝑑 𝑥® − + (1 + 𝑎® ) 𝑑𝜏 + . 1 + 𝑎®2 1 + 𝑎®2 ′2 (3.291) The effective metric is thus 𝑎𝑖 𝑎 𝑗  1  𝛿𝑖 𝑗 − , b 𝑔𝑖 𝑗 = 1 + 𝑎®2 1 + 𝑎®2 and the thermal effective action is √︁ 𝑔 = − 𝑓 vol R𝑑−1 (1 + 𝑎®2 ) −𝑑/2 . 𝑆[b 𝑔 , 𝐴] = − 𝑓 vol R𝑑−1 b (3.292) (3.293) Finally, the partition function is 𝑒 −𝑆[b 𝑔 ,𝐴] =𝑒 𝑓 vol R𝑑−1 (1+𝑎®2 ) −𝑑/2 =𝑒 𝑓 vol R𝑑−1   𝑓𝑑 2 𝑑−1 1− 𝑎® vol R +... , 2 (3.294) in agreement with (3.269). These manipulations are clearly easier and more efficient than those in the previous section. However, detailed manipulations of correlators are instructive as well. For instance, they tell us that (3.269) holds even when the thermal theory is not gapped, as long as 𝑑 > 3. It would be interesting to determine which other results from the thermal effective action continue to hold in non-gapped thermal theories. We leave this problem for future work. Appendix B Scheme independence In Section 3.3, we derived (3.41) by working in a scheme where 𝑏-type terms 𝑆ct are absent from the Weyl anomaly. In this appendix, we describe how (3.41) comes about in a general scheme. The point is that the scheme dependence of the Casimir energy and the thermal effective action cancel each other. For concreteness, let us work in 4D. The partition function on 𝑆 1𝛽 × 𝑆 3 is h i 3𝑎 3𝑏 b Tr 𝑒 −𝛽(𝐷+ 4 − 8 ) ∼ 𝑒 −𝑆th = 𝑒 −𝑆[b𝑔,𝐴]−𝑆Euler −DR[𝑆ct [𝐺]]+DR[𝑆ct [𝐺]] , (3.295) where we have used that 𝑆 1𝛽 × 𝑆 𝑑−1 is conformally-flat to drop Weyl-invariants. Meanwhile, we have   ∫ ∫ 1 𝑏 3𝑏 3 √ 2 b DR[𝑆ct [𝐺]] = DR[𝑆ct [𝐺]] = 𝑑 𝑥 𝑔 𝛽𝑑𝜏 − 𝑅 = − 𝛽, 2 8 12(4𝜋) 𝑆3 0 (3.296) 115 where we used that 𝑅 = 6 on 𝑆 3 . Thus, we can cancel the 𝑏-dependence on the left-hand side with DR[𝑆ct [𝐺]] on the right-hand side, leaving h i 3𝑎 b Tr 𝑒 −𝛽(𝐷+ 4 ) ∼ 𝑒 −𝑆[b𝑔,𝐴]+DR[𝑆ct [𝐺]]−𝑆Euler . (3.297) b is then scheme-independent, and equal to The combination −𝑆[b 𝑔 , 𝐴] + DR[𝑆ct [𝐺]] 𝑆[b 𝑔, 𝐴] in the scheme of Section 3.3. Of course, this cancellation was not an accident. The 𝑏-term contribution to the Casimir energy is 1 𝑆ct | R×𝑆 𝑑−1 . vol R (3.298) 𝛽𝐸 0 | 𝑏-type = DR[𝑆ct ] | 𝑆1 ×𝑆 𝑑−1 . (3.299) 𝐸 0 | 𝑏-type = Because 𝑆ct is local, it follows that 𝛽 Said another way, the 𝑏-term in the Casimir energy is precisely what matches the 𝑏-term contribution to the Weyl anomaly on the cylinder. Since the 𝑏-term in the thermal effective action was determined by Weyl anomaly matching, it must cancel with the Casimir energy. This argument generalizes to arbitrary 𝑑. Appendix C More on free theories Partition function of free scalar theories In this section we review the partition function of a free scalar theory on R × 𝑆 𝑑−1 𝑅 . 𝑑 This space is conformally equivalent to the Euclidean space R . The energy 𝐸 of the state on R × 𝑆 𝑑−1 𝑅 is related to the scaling dimension Δ of the corresponding field 𝑑 on R via: 𝐸 = Δ/𝑅. (3.300) The equation of motion of the free scalar field on R × 𝑆 𝑑−1 𝑅 is   𝜕2 2 − 2 + ∇𝑆 𝑑−1 − 𝜉R 𝜙 = 0, 𝑅 𝜕𝑡 (3.301) 𝑑−2 where 𝜉 is the conformal coupling in 𝑑 dimensions, 𝜉 = 4(𝑑−1) , and R is the Ricci (𝑑−1) (𝑑−2) scalar of 𝑆 𝑑−1 . The spherical harmonics in 𝑑 dimension, 𝑌𝑙(𝑑) , are 𝑅 , R = 𝑅2 eigenfunctions of ∇2 𝑑−1 , with eigenvalue −𝑙 (𝑙 + 𝑑 − 2), where 𝑙 is a non-negative 𝑆𝑅 integer. We can then construct an orthonormal set of solutions as 𝜙𝑙 ∝ 𝑒 −𝑖𝐸𝑡 𝑌𝑙(𝑑) , 𝑙 + 𝑑−2 2 𝐸= , 𝑅 (3.302) 116 whose elements become each mode after quantization. Note that 𝑌𝑙(𝑑) is the representation of 𝑆 𝑝𝑖𝑛(𝑑) with the highest weight (𝑙, 0, . . . , 0) in the standard Cartan-Weyl labeling scheme. We will write down the case of even dimension and odd dimension separately because the group structure of 𝑆𝑂 (𝑑) is slightly different in these two cases. Even dimensions The spherical harmonics 𝑌𝑙(𝑑) are also eigenfunctions of 𝜕𝜃 𝑎 in the coordinate system (3.45). The eigenvalues 𝑚 𝑎 (𝑎 = 1, . . . , 𝑛) of 𝜕𝜃 𝑎 are integers, and they obey the following relation: 𝑙 = 2𝑚 0 + |𝑚 1 | + · · · + |𝑚 𝑛 |, (3.303) where 𝑚 0 is a non-negative integer. The multiplicity of this specific eigenvalue is 𝑛+𝑚 0 −2
. Therefore, in even 𝑑 (𝑑 > 2), the partition function of a free scalar field is 𝑚0 ∞ ∞ ∞ Ö Ö 1 © Ö ª 𝑍 (𝑇, Ω𝑖 ) = ··· ­ ® Í Í 1 − (2𝑚 0 + 𝑖 |𝑚 𝑖 |+𝑑/2−1+𝑖 𝑖 𝑚 𝑖 Ω𝑖 ) 𝑚 𝑑/2 =−∞ 1 − 𝑒 𝑇 𝑚 0 =0 «𝑚 1 =−∞ ¬ 0 −2) ( 𝑑/2+𝑚 𝑚0 , (3.304) where the sums over 𝑖 in (3.304) run from 𝑖 = 1, · · · , 𝑑2 . From this we can read off log 𝑍. The first two terms in the high-temperature expansion are as in (3.93). The higher order terms in log 𝑍 come with a factor proportional to ∼ 𝜁 (𝑑 − 2𝑘)𝑇 𝑑−2𝑘−1 for integer 𝑘. Because the zeta function vanishes at negative even integers, this means the high-temperature perturbative expansion for log 𝑍 of a free scalar field in even dimensions truncates after the 𝑂 (1/𝑇) term [147]. The further corrections after the perturbative expansion in 1/𝑇 are non-perturbatively suppressed in 𝑇. In fact, using techniques similar to [45], we can get an exact expression for log 𝑍 for free scalar field theories in even 𝑑. For completeness, we write these expressions in the following section. Odd dimensions In odd 𝑑, the relation between the eigenvalues 𝑙 and 𝑚 𝑎 is 𝑙 = 𝑚′0 + |𝑚 1 | + · · · + |𝑚 𝑑−1 |, 2 (3.305) where 𝑚′0 is a non-negative integer. The multiplicity of this specific eigenvalue state
𝑑−1 𝑚′ 0 −1 is 2 +𝑚 , where 𝑚 0 = ⌊ 20 ⌋. Therefore, the partition function of a free scalar 𝑚0 117 field is 𝑑−3 𝑍 (𝑇, Ω𝑖 ) = ∞ ∞ ∞ © Ö Ö Ö ª 1 ­ ® · · · Í Í 1 ­ − 𝑇 (2𝑚 0 + 𝑖 |𝑚 𝑖 |+𝑑/2−1+𝑖 𝑖 𝑚 𝑖 Ω𝑖 ) ® 1 − 𝑒 𝑚 𝑑−1 =−∞ 𝑚 0 =0 𝑚 1 =−∞ 2 « ¬ 0) ( 2𝑚+𝑚 0 ∞ © Ö ∞ ∞ Ö Ö ª 1 ­ ® × · · · Í Í 1 ­ ® − (2𝑚 + |𝑚 |+𝑑/2+𝑖 𝑚 Ω ) 𝑖 𝑖 𝑖 0 𝑖 𝑖 𝑇 𝑚 𝑑−1 =−∞ 1 − 𝑒 𝑚 0 =0 𝑚 1 =−∞ 2 « ¬ 𝑑−3 0) ( 2𝑚+𝑚 0 , (3.306) where the sums over 𝑖 in (3.306) run from 𝑖 = 1, · · · , 𝑑−1 2 . When we take the log and expand at high temperature, we arrive at the same result as in (3.93). In even 𝑑, the expansion in inverse powers of 𝑇 truncates after the 𝑂 (𝑇 −1 ) term. In odd 𝑑, however, the expansion never truncates. This is because the higher order terms have a factor proportional to ∼ 𝜁 (𝑑 − 2𝑘)𝑇 𝑑−2𝑘−1 . For odd 𝑑, this never vanishes. Moreover, due to the factorial growth of the zeta function at large, negative, odd values of the argument, the expansion in inverse powers of 𝑇 is in fact asymptotic rather than convergent. Finally (due to a pole of the zeta function at argument 1), there is a log 𝑇 term in the high-temperature expansion as well (see (4.13) of [128] for this log 𝑇 term in the case of 𝑑 = 3.) We write an explicit expression for all perturbative terms in odd dimensions in the previous section. Gapless sector in the free scalar As noted in Sec 3.4, the free scalar in 𝑑 dimensions is a somewhat pathological example for our purposes, due to the presence of a gapless sector upon compactification on 𝑆 1 , namely the 𝑑−1-dimensional free scalar CFT. As a result, the partition function at high temperature contains terms proportional to 𝑂 (𝑇 0 ) and 𝑂 (log 𝑇) in even-𝑑 and odd-𝑑 respectively. These terms cannot be produced by the thermal effective action, and must come from the gapless sector. In this appendix we understand them explicitly (see also [61] for earlier discussion of such terms). An important subtlety is that the 𝑑−1 dimensional gapless sector is not conformally coupled to curvature in 𝑑−1 dimensions. To see why, we start with a conformallycoupled scalar in 𝑑-dimensions. This contains a term in the Lagrangian 21 𝜉 𝑑 R𝜙2 with coefficient 𝜉𝑑 = 𝑑−2 4(𝑑 − 1) (conformal coupling in 𝑑-dimensions). (3.307) 118 When we dimensionally reduce on 𝑆 1 , we do not obtain a conformally-coupled scalar in 𝑑−1 dimensions, because 𝜉 𝑑 ≠ 𝜉 𝑑−1 . Instead, the 𝑑−1-dimensional scalar has a particular mass deformation turned on. To be more precise, it satisfies the equation of motion   D𝜙 = −∇2𝑑−1 + 𝜉 𝑑 R 𝜙 = 0, (3.308) where ∇2𝑑−1 is the 𝑑−1 dimensional Laplacian. By contrast, a conformally-coupled scalar would satisfy   2 e D𝜙 = −∇𝑑−1 + 𝜉 𝑑−1 R 𝜙 = 0 (𝑑−1-dimensional conformal coupling). (3.309) The partition function of the gapless sector in our case is (det D) −1/2 . By contrast, reference [134] computed the sphere partition function of a conformally-coupled e −1/2 . For our purposes, we can follow the methods of [134], free scalar, i.e. (det D) but we will obtain different results because we have a different equation of motion (3.308).33 Following [134], the contribution to − log 𝑍 𝑆1 ×𝑆 𝑑−1 from the gapless sector is    ∞ ∑︁ 𝑑−2 , (3.310) 𝐹= 𝑚 𝑛 − log(𝜇𝑅) + log 𝑛 + 2 𝑛=0 where (2𝑛 + 𝑑 − 2)(𝑛 + 𝑑 − 3)! (3.311) (𝑑 − 2)!𝑛! is the dimension of the 𝑛-th traceless symmetric tensor representation of SO(𝑑). Here, 𝑅 is the radius of the 𝑆 𝑑−1 , and 𝜇 is a mass scale coming from the regulator. In our case, the temperature sets the regulator scale, so we have 𝜇 = 𝑇. 𝑚 𝑛 := The sum (3.310) diverges, but we can use 𝜁-function regularization to make it finite. In even 𝑑, the 𝑅-dependence of (3.310) formally drops out. However, in odd 𝑑 it remains, giving a nontrivial log 𝑇 term in log 𝑍. The 𝜁-function regulated sum is ∞ ∑︁ 𝑚𝑛 𝐹𝑠 := (3.312)  𝑠 . 𝑑−2 𝑛 + 𝑛=0 2 The log 𝑇 term is given by 𝐹𝑠=0 and the 𝑇 0 term is given by 𝜕𝑠 𝐹𝑠 | 𝑠=0 . These values for the first few 𝑑 are given in Table 3.1. They indeed agree with the corresponding terms in the high-temperature expansion of the partition function of the free scalar on 𝑆 1 × 𝑆 𝑑−1 , as we explicitly write in Appendix C. 33 We are grateful to Yifan Wang for discussions related to this point. 119 𝑑 3 4 5 𝐹𝑠=0 1 12 0 17 − 2880 6 7 0 367 483840 8 0 𝜕𝑠 𝐹𝑠 | 𝑠=0 log 2 − 12 − 𝜁 ′ (−1) − 𝜁4𝜋(3)2 11 log 2 𝜁 ′ (−1) 7𝜁 ′ (−3) 2880 + 24 − 24 𝜋 2 𝜁 (3)+3𝜁 (5) 48𝜋 4 ′ ′ (−5) 211 log 2 3𝜁 ′ (−1) (−3) − 483840 − 640 + 7𝜁192 − 31𝜁1920 4 2 𝜁 (5)+45𝜁 (7) − 8𝜋 𝜁 (3)+30𝜋 2880𝜋 6 Table 3.1: Values for 𝐹𝑠=0 and 𝜕𝑠 𝐹𝑠 | 𝑠=0 for various dimensions, with 𝐹𝑠 defined in (3.312). These provide the coefficients of the 𝑂 (log 𝑇) and 𝑂 (𝑇 0 ) terms respectively in the free energy of a free scalar field in 𝑑 dimensions. The 𝑎-anomaly of the free scalar As an aside, we can use similar techniques to compute the 𝑎-anomaly of a free scalar theory in 𝑑 dimensions. The value of the 𝑎 anomaly is well-known in 𝑑 = 2, 4, 6 [82]; see e.g. [15] for a calculation in 6D. In general 𝑑 it was computed in [96, 97], which we review here. Here we study the free scalar field in 𝑑 dimensions conformally coupled to 𝑆 𝑑 , as was precisely done in [134]. As discussed in Appendix F, the 𝑎-anomaly is related to the sphere partition function by34 log 𝑍 (𝑆𝑟𝑑 ) = − (−1) 𝑑/2 𝑎 𝑑 𝑑! vol 𝑆 𝑑 log(𝜇𝑟), 𝑑/2 (4𝜋) (3.313) where 𝜇 is a regulator scale. Thus, we can read off 𝑎 𝑑 from the log 𝑟 term in the sphere free energy. We find the general answer (−1) 2 +1 ∫ 1 2Γ( 𝑑+2 2 )Γ(𝑑 + 2) 0 𝑑 𝑎𝑑 =  𝑑 𝑑𝑡 (𝑑 + 4𝑡 ) 𝑡 − + 1 2 𝑑−1 2  (𝑑 even, 𝑑 > 2), (3.314) where (𝑥)𝑛 is the Pochhammer symbol (𝑥)𝑛 := Γ(𝑥+𝑛) Γ(𝑥) . The integrand is of course a polynomial in 𝑡 for positive integer 𝑑, so the integral is trivial in practice. This formula requires 𝑑 > 2 because the sphere partition function of the (noncompact) free boson in 𝑑 = 2 is ill-defined. The first few values of 𝑎 𝑑 are listed in Table 3.2. 𝑑/2 here 𝑑 34 References [96, 97] use a different convention for the 𝑎 anomaly where (−1) 𝑎𝑑 𝑑!vol 𝑆 (4 𝜋 ) 𝑑/2 𝑎 there 𝑑 . = 120 𝑑 𝑎𝑑 4 6 8 10 12 1 360 1 9072 23 5443200 263 1796256000 133787 29422673280000 Table 3.2: Values for the conformal 𝑎-anomaly of a free scalar field in 𝑑 dimensions, with 𝑑 even. For general 𝑑, see (3.314). Non-perturbative corrections for free scalars From the second section of this appendix, we have the perturbative corrections in 1/𝑇 for the free scalar field in 𝑑 dimensions. For even 𝑑, they truncate after the 𝑂 (1/𝑇) term (see e.g. [41, 81]). From the techniques in [45] we can in fact compute the exact high-temperature partition function. It is given by the following. Define the auxiliary function for even 𝑑: 𝑓 (𝑑, 𝑇) := 𝜁 (𝑑)𝑇 𝑑−1 " # 2 ∞ 𝑑−2 (−1) 𝑑/2 𝜁 (𝑑 − 1) (𝑑 − 1)𝜁 (𝑑) ∑︁ 𝑒 −4𝜋 𝑇 𝑛 𝜎𝑑−1 (𝑛) ∑︁ (4𝜋 2𝑇 𝑛) 𝑖 − + , − 𝑑−1 2 Γ(𝑖 + 1) (2𝜋) 𝑑−2 4𝜋 2𝑇 𝑛 𝑛=1 𝑖=0 (3.315) where 𝜎 is the divisor sigma function: 𝜎𝑑−1 (𝑛) := 𝑑 free scalar is log 𝑍 𝑑 (𝑇) = 𝑑/2−2 ∑︁ Í ℓ|𝑛 ℓ 𝑑−1 . Then the general even 𝑐 2𝑖−(𝑑−1) (𝑑) 𝑓 (𝑑 − 2𝑖, 𝑇), (3.316) 𝑖=0 where 𝑐 2𝑖 (𝑑) is the coefficient of the 𝛽2𝑖 term in the expansion of 2𝑑−1sinh(𝛽) about sinh𝑑 (𝛽/2) 𝛽 = 0.35 For example at 𝑑 = 4 and 𝑑 = 6, (3.316) reduces to the following two equations36 :   2 ∞ ∑︁ 𝜋 4 3 𝜁 (3) 1 4𝜎3 (𝑛)𝑒 −4𝜋 𝑇 𝑛 2 2 𝑇 1 + − + log 𝑍 𝑑=4 (𝑇) = 𝑇 − 𝜋 𝑇 + , 45 𝑛 2𝑛 8𝜋 2 𝑛2 4𝜋 2 240𝑇 𝑛=1 (3.319) 35 This function comes about from writing the logarithm of the free boson partition function as log 𝑍 (𝑇) = −   ∞ ∑︁ 𝑗 + 𝑑/2 − 1 𝑗 + 𝑑 − 3 𝑗=0 𝑑/2 − 1 𝑑−3 1 𝑑−2 log(1 − 𝑒 − 𝑇 ( 𝑗+ 2 ) ), (3.317) doing the Taylor expansion of the logarithm, and finally resumming over 𝑗, giving (see e.g. [61]) log 𝑍 (𝑇) = ∞ ∑︁ 𝑛=1 𝑛2 sinh( 𝑇𝑛 ) . 𝑑−1 sinh 𝑑 ( 𝑛 ) 2𝑇 36 These perturbative terms here reproduce, e.g. [165]. (3.318) 121 log 𝑍 𝑑=6 (𝑇) = 2𝜋 6 5 𝜋 4 3 𝜋 2 𝜁 (3) + 3𝜁 (5) 31 − 𝑇 − 𝑇 + 4 945 540" 60480𝑇 48𝜋   ∞ 2 3 ∑︁ 𝜋 𝑇 3𝑇 2 3𝑇 3 4 4 −4𝜋 2 𝑛𝑇 4𝜎5 (𝑛) 𝜋 𝑇 + + 2 + 3 2+ + 𝑒 3𝑛 𝑛 4𝑛 8𝑛 𝜋 32𝑛4 𝜋 4 𝑛=1 #  𝜎3 (𝑛) 2 2 𝑇 1 + 𝜋 𝑇 + + . (3.320) 3𝑛 2𝑛 8𝑛2 𝜋 2 For a free scalar field in odd dimensions, the perturbative expansion in 1/𝑇 no longer truncates. The perturbative expansion is 𝑑−1 log 𝑍 𝑑 (𝑇) ∼ 2 ∑︁ 𝑐 −2𝑛 (𝑑)𝜁 (2𝑛 + 1)𝑇 2𝑛 𝑛=1 + 𝐹𝑠=0 log 𝑇 + 𝜕𝑠 𝐹𝑠 | 𝑠=0 + ∞ ∑︁ 𝑐 2𝑛 (𝑑)𝜁 (−2𝑛 + 1)𝑇 −2𝑛 , (3.321) 𝑛=1 where in (3.321), 𝐹𝑠=0 and 𝜕𝑠 𝐹𝑠 | 𝑠=0 are defined in (3.312), and 𝑐 2𝑖 (𝑑) is defined about 𝛽 = 0. again as the coefficient of the 𝛽2𝑖 term in the expansion of 2𝑑−1sinh(𝛽) sinh𝑑 (𝛽/2) (2𝑛)! At large 𝑛, |𝑐 2𝑛 (𝑑)| ∼ (2𝜋) −2𝑛 . On the other hand, |𝜁 (−2𝑛 + 1)| ∼ (2𝜋) 2𝑛 . Thus the expression (3.321) is an asymptotic series with divergent piece growing like log 𝑍 𝑑 (𝑇) ∼ ∑︁ 𝑛 (2𝑛)! . (4𝜋 2𝑇) 2𝑛 (3.322) From techniques in resurgence, this implies the first non-perturbative correction to 2 (3.321) scales as 𝑒 −4𝜋 𝑇 , just as in even dimensions. These results are consistent with the worldline instanton corrections discussed in Section 3.3 (even though in this example there is also a gapless sector upon compactification). When the free boson is compactified on a circle with thermal boundary conditions, the mass of the lightest KK mode is 𝑚 𝐾𝐾 = 2𝜋𝑇. Therefore, (3.91) 2 predicts a correction to log 𝑍 of the form 𝑒 −4𝜋 𝑇 , which is precisely consistent with what we found in both even and odd 𝑑. We can also study the free scalar with a Z2 twist around the thermal circle. For example, in 𝑑 = 4, we find Z2 -twisted log 𝑍 𝑑=4 (𝑇) = − 2 7𝜋 4 3 1 𝑇 + + 𝑂 (𝑒 −2𝜋 𝑇 ). 360 240𝑇 (3.323) In this case, the lightest KK mode has mass 𝑚 𝐾𝐾 = 𝜋𝑇, and the nonperturbative 2 corrections are indeed of the form 𝑒 −2𝜋𝑚 𝐾 𝐾 = 𝑒 −2𝜋 𝑇 . 122 Partition function of free fermion theories In this section, we review the partition function of a free massless Dirac fermion in 𝑑 dimensions on R × 𝑆 𝑑−1 𝑅 . We can construct the partition function from the solution of the Dirac equation:   0 𝜕 𝑖 Γ + Γ ∇𝑖 𝜓 = 0, (3.324) 𝜕𝑡 where Γ 𝜇 (𝜇 = 0, 1, . . . , 𝑑 − 1) are gamma matrices, and ∇𝑖 (𝑖 = 1, . . . , 𝑑 − 1) is the covariant derivative on the sphere. The spectrum of the Dirac operator on 𝑆 𝑑−1 has been considered in e.g. [40]. The Dirac operator on 𝑆 𝑑−1 , ∇ ≡ Γ𝑖 ∇𝑖 , has the following eigenvalues:   𝑑−1 𝜓±𝜌 , 𝜌 = 0, 1, 2, . . . . (3.325) ∇𝜓±𝜌 = ±𝑖 𝜌 + 2 2 Because Γ0 = −1 and {Γ0 , ∇} = 0, we find the solution of the Dirac equation as   𝑑−1 𝜓 = 𝑒𝑖𝐸𝑡 𝜓±𝜌 ± Γ0 𝜓±𝜌 , 𝐸 = 𝜌 + . (3.326) 2 From [40] we see  the solutions are representations of Spin(𝑑) with highest weight  1 1 1 𝜌 + 2 , 2 , . . . , 2 for odd 𝑑 and 𝜌 + 21 , 21 , . . . , 12 , ± 12 for even 𝑑 where 𝜌 is the eigenvalue of the Dirac operator in (3.325). In odd dimensions, we have a complete set of solution of the Dirac equation with eigenvalues as follows: 𝑑−1 , 2 𝜌 = 𝑚 0 + 𝑚′1 + · · · + 𝑚′𝑑−1 ,    2  1 𝑑−1 ′ 𝑚𝑎 = ± 𝑚𝑎 + , 𝑎 = 1, . . . , , 2 2 𝐸 =𝜌+ (3.327) where 𝑚 0 , 𝑚′1 , . . . , 𝑚′𝑛 are non-negative integers. Here 𝐸 is the energy of the state and 𝑚 𝑎 is the eigenvalue of the rotation generator 𝜕𝜃 𝑎 . The multiplicity of states
𝑑−3 0 with eigenvalues (𝐸, 𝑚 1 , . . . , 𝑚 𝑑−1 ) is 2 𝑚+𝑚 . Finally we get an additional tower 0 2 of states from quantizing the field 𝜓. Therefore, the partition function of a free fermion in odd dimensions is 𝑍 (𝑇, Ω𝑖 ) = ∞ Ö Ö 𝑚 0 =0 𝑚 1 ∈Z+ 1 2 ··· Ö  1+𝑒  2( Í Í − 𝑇1 (𝑚 0 + 𝑖 |𝑚 𝑖 |+ 𝑑−1 4 +𝑖 𝑖 𝑚 𝑖 Ω𝑖 ) 𝑑−3 +𝑚 0 2 𝑚0 ) , 𝑚 𝑑−1 ∈Z+ 12 2 (3.328) 123 where the sums over 𝑖 run from 1, . . . , 𝑑−1 2 . Taking a log and expanding at high temperature recovers (3.96). In even dimensions, we can do a very similar calculation. We have a complete set of solutions to the Dirac equation with eigenvalues as follows: 𝑑 𝐸 =𝜌+ , 2 𝜌 = 𝑚 0 + 𝑚′1 + · · · + 𝑚′𝑑 ,    2  1 𝑑 ′ 𝑚𝑎 = ± 𝑚𝑎 + , 𝑎 = 1, . . . , , 2 2 (3.329) where 𝑚 0 , 𝑚′1 , . . . , 𝑚′𝑛 are nonnegative integers. The multiplicity of the states with 
𝑑 +𝑚 −2 eigenvalues 𝐸, 𝑚 1 , . . . , 𝑚 𝑑 is 2 𝑚00 . Therefore, the partition function of a free 2 fermion in even dimensions is ® = 𝑍 (𝑇, Ω) ∞ Ö Ö ··· 𝑚 0 =0 𝑚 1 ∈Z+ 1 2  Ö 1+𝑒 − 𝑇1 (𝑚 0 + Í Í 𝑑−2 𝑖 |𝑚 𝑖 |+ 4 +𝑖 𝑖 𝑚 𝑖 Ω𝑖 )  2 ( 𝑑2 +𝑚0 −2) 𝑚0 , 𝑚 𝑑 ∈Z+ 12 2 (3.330) where the sums over 𝑖 in (3.330) run from 1, . . . , 𝑑2 . This gives the same answer as (3.96). Non-perturbative corrections for free fermions We can repeat the same analysis in previous section to find the non-perturbative corrections for the free energy of a free Dirac fermion in 𝑑 dimensions. When we turn off the spin fugacities, we can rewrite (3.328), (3.330), as 𝑓 log 𝑍 𝑑 (𝑇) = (𝑑−1)𝑛 𝑑 ∞ ∑︁ 2 ⌊ 2 ⌋+1 (−1) 𝑛+1 𝑒 − 2𝑇 𝑛 𝑛=1 𝑛(1 − 𝑒 − 𝑇 ) 𝑑−1 . (3.331) In even dimensions, this admits the following (exact) high-temperature expansion. First, we define the auxiliary function for even 𝑑 as   (−1) 𝑑/2 (𝑑 − 1)(2𝑑 − 2)𝜁 (𝑑) 1 𝑑/2 𝑔(𝑑, 𝑇) := 2 1 − 𝑑−1 𝜁 (𝑑)𝑇 𝑑−1 + 2 𝜋 𝑑 23𝑑/2𝑇 ∞ 𝑑−2 −2𝜋 2𝑇 𝑛 (−1) 𝑛 𝜎 odd (𝑛) ∑︁ (2𝜋 2𝑇 𝑛) 𝑖 (−1) 𝑑/2 ∑︁ 𝑒 𝑑−1 + 𝑑 , (3.332) Γ(𝑖 + 1) 𝑛 𝑑−1 2 2 −2 𝜋 𝑑−2 𝑛=1 𝑖=0 where odd 𝜎𝑑−1 (𝑛) := ∑︁ ℓ|𝑛, ℓ odd ℓ 𝑑−1 . (3.333) 124 𝑑 − 𝛽 (𝑑−1) 2 𝑒 We also define the function 𝑐˜𝑖 (𝑑) as the 𝛽𝑖 term in the expansion of 𝛽(1−𝑒 −𝛽 ) 𝑑−1 about 𝛽 = 0.37 Then the partition function of a free Dirac fermion in even 𝑑 dimensions at temperature 𝑇 is 𝑓 log 𝑍 𝑑 (𝑇) = 𝑑−1 ∑︁ 𝑖+1 2 2 𝑐˜𝑖 (𝑑)𝑔(𝑑 + 1 − 𝑖, 𝑇). (3.334) 𝑖=1 For instance, (3.334) for 𝑑 = 4, 6 reduces to 𝑓 log 𝑍 𝑑=4 (𝑇) = 7𝜋 4 3 𝜋 2 17 𝑇 − 𝑇+ 90 12 480𝑇 ! ∞ 2 ∑︁ 𝜎1odd (𝑛) 2𝜎3odd (𝑛) 4 odd 𝑛 −2 𝜋 2 𝑇 𝑛 4𝜋 odd 2 + (−1) 𝑒 𝜎 (𝑛)𝑇 + 2 𝜎3 (𝑛)𝑇 + + , 𝑛 3 𝑛 𝑛 𝑛3 𝜋 2 𝑛=1 (3.335) and 367 31𝜋 6 5 7𝜋 4 3 𝜋 2 𝑇 − 𝑇 + 𝑇− 1890 " 216 32 24192𝑇 ! ∞ 4 ∑︁ 2𝜎5odd (𝑛) 5𝜋 2 𝜎3odd 2 4𝜋 2 odd 𝑛+1 −2 𝜋 2 𝑇 𝑛 2𝜋 odd 4 3 + (−1) 𝑒 + 𝜎5 (𝑛)𝑇 + 2 𝜎5 (𝑛)𝑇 + 𝑇 3 3𝑛 3𝑛 3𝑛 𝑛 𝑛=1 ! !# 2𝜎5odd (𝑛) 5𝜎3odd (𝑛) 𝜎5odd (𝑛) 5𝜎3odd (𝑛) 3𝜎1odd (𝑛) + + 𝑇+ + . + 8𝑛 𝜋 2 𝑛4 3𝑛2 6𝜋 2 𝑛3 𝜋 4 𝑛5 𝑓 log 𝑍 𝑑=6 (𝑇) = (3.336) In odd dimensions, the perturbation theory in 1/𝑇 no longer truncates. Rather, it looks like ∞   ∑︁ 𝑑+1 𝑓 (3.337) log 𝑍 𝑑 (𝑇) ∼ 2 2 1 − 2𝑛−𝑑 𝜁 (𝑑 + 1 − 𝑛) 𝑐˜𝑛 (𝑑)𝑇 𝑑−𝑛 , 𝑛=1 𝑑+1 where the 𝑛 = 𝑑 term in (3.337) is 𝑐˜𝑑 (𝑑)2 2 log 2. The sum in (3.337) is an 𝑛! asymptotic expansion. At large odd 𝑛, | 𝑐˜𝑛 (𝑑)| ∼ (2𝜋) −𝑛 and |𝜁 (𝑑 + 1 − 𝑛)| ∼ (2𝜋) 𝑛. The sum then diverges, growing as ∑︁ 𝑛! 𝑓 log 𝑍 𝑑 (𝑇) ∼ . (3.338) (2𝜋 2𝑇) 𝑛 𝑛 From the techniques in resurgence, this implies the first non-perturbative correction 2 of (3.337) scales as 𝑒 −2𝜋 𝑇 , just as in even dimensions. In a free fermion theory compactified on a circle with thermal boundary conditions, we have 𝑚 𝐾𝐾 = 𝜋𝑇, so that (3.91) predicts a non-perturbative correction of the form 2 𝑒 −2𝜋 𝑇 , consistent with what we found in both even and odd dimensions. 37 Like in the free scalar case, up to an overall constant, this comes from (3.331) upon setting 𝑛 = 1. 125 Appendix D Wilson coefficients for the 3D Ising model In this appendix, we discuss estimates of the high-temperature partition function of the 3D Ising model. In Appendix A of [119], the coefficient 𝑓 for the 3D Ising model was estimated by constructing the partition function with Ω = 0 as a sum over the spectrum of known operators, which has numerically been computed up to about dimension 8 [195]. In this appendix, we perform a similar analysis, but including spin-dependence. Note that there is a balance in choosing the temperature — if the temperature is too low, then truncating the thermal effective action becomes a poor approximation; if the temperature is too high, then truncating the partition function to a finite sum of characters becomes inaccurate.38 0.7 2 09 0.73 1. 1 0.4 1 0.3 1. 03 0.82 1 0.9 1. 1.0 9 0.7 06 1. 0.5 1 0.85 0.2 0.7 0.97 1.15 0.85 15 0.5 0.97 1 0.76 1.15 1.3 1 0.94 1.2 1.5 0.6 0.73 1 1.06 1.1 8 0.79 1.12 1.3 4 1.2 0.85 1 1 2 0.79 0.76 0 0.1 1.03 0.7 4 6 0 0.7 2 4 6 Loading [MathJax]/extensions/MathMenu.js Figure 3.13: Contour plots of the ratio of the estimated partition function to the leading term in the  thermal effective action with 𝑓 = 0.153, i.e. 0.153 3D Ising 𝑍 (𝛽, Ω)/exp 4𝜋 𝛽2 (1+Ω2 ) , as a function of 𝛽, Ω for real Ω (left) and imaginary Ω (right). The ratio is very close to 1 for intermediate temperatures 1 ≲ 𝛽 ≲ 5 and small angles |Ω| ≲ 0.5. From Monte-Carlo techniques, it has been estimated that 𝑓3D Ising ∼ 0.153 [143, 145, 198]. In Figure 3.13, we plot the ratio of the computed partiton function to the estimate from the first term in the thermal effective action (3.54) with 𝑓 = 0.153. For the values of 𝛽 and Ω shown in the figure, the ratio is quite close to 1. We can also independently fit 𝑓 from the partition function. By studying temperatures with 1.5 < 𝛽 < 3 and chemical potentials |Ω| < 0.6, we estimate that 𝑓 ∼ 0.15, consistent with [143, 145, 198]. 38 Much like the porridge in Goldilocks and the Three Bears [196], it is important to pick a temperature that is just right. 126 We can also try to estimate higher-derivative Wilson coefficients in the effective action of the 3D Ising model, such as 𝑐 1 , 𝑐 2 . Unfortunately, we do not have a clear enough picture of the high-dimension spectrum to estimate these coefficients with reliable precision. However, our best fits suggest that 𝑐 2 < 0 (we are not yet able to reliably estimate the sign of 𝑐 1 ). In general, accessing large twist operators is challenging for the numerical bootstrap, which seems most sensitive to low-twist operators—particularly "double twist" operators [195]. Furthermore, the numerical bootstrap studies done so far are blind to certain parts of the operator spectrum of the 3D Ising model, such as odd-spin Z2 even operators, or parity-odd operators.39 Such sectors would need to be included to reliably compute the partition function at higher temperatures. See [117, 207] for recent work accessing these sectors with other methods. Appendix E The shadow transform of a three-point function at large Δ The formula for OPE coefficients depends on the triple shadow coefficient 𝑆e3†e†e† 1 2 3 given in (3.161). In this appendix, we compute this coefficient at large Δ𝑖 . First consider a single shadow transform applied to a three-point function with large Δ’s. The shadow transform is ∫ 𝑐 (𝑠) 𝑎 𝑏 e𝑐 (𝑥 3 ) O e† (𝑥 0 )⟩⟨O 𝑎 (𝑥 1 )O 𝑏 (𝑥 2 )O 𝑐 (𝑥 0 )⟩ (𝑠) . 𝑑 𝑑 𝑥0 ⟨O ⟨O1 (𝑥 1 )O2 (𝑥 2 )S[O3 ] (𝑥 3 )⟩ ≡ 𝑐 1 2 3 (3.339) e† has Here, 𝑎, 𝑏, 𝑐 are spin indices for the representations 𝜆 1 , 𝜆2 , 𝜆3 . The operator O Lorentz representation 𝜆∗3 , so we write its index as a lowered 𝑐 index. The operator e has Lorentz representation 𝜆 𝑅 (the reflected representation), and we indicate this O 3 with a barred index. The three-point function is ⟨O1𝑎 (𝑥 1 )O2𝑏 (𝑥 2 )O3𝑐 (𝑥 3 )⟩ (𝑠) = 𝑉 𝑠;𝑎𝑏𝑐 (𝑥 1 , 𝑥2 , 𝑥3 ). (3.340) We will be interested in restricting this three-point function to a single axis 𝑥𝑖 = 𝑧𝑖 𝑒 with unit vector 𝑒 ∈ 𝑆 𝑑−1 ⊂ R𝑑 . We get two different answers, depending on the cyclic ordering of the points  𝑉 𝑠;𝑎𝑏𝑐 (0,𝑒,∞)  (𝑧1 < 𝑧2 < 𝑧3 , or cycl.),   Δ +Δ −Δ Δ +Δ −Δ Δ +Δ −Δ 23 | 2 3 1 |𝑧 31 | 3 1 2 𝑉 𝑠;𝑎𝑏𝑐 (𝑧1 𝑒, 𝑧2 𝑒, 𝑧3 𝑒) = |𝑧12 | 1 2 3 |𝑧𝑠;𝑎𝑏𝑐 𝑉 (𝑒,0,∞)   (𝑧2 < 𝑧1 < 𝑧3 , or cycl.).  |𝑧12 |Δ1 +Δ2 −Δ3 |𝑧23 |Δ2 +Δ3 −Δ1 |𝑧31 |Δ3 +Δ1 −Δ2 (3.341) 39 Though there are preliminary results for some of these sectors from the stress tensor bootstrap [84]. 127 The tensors 𝑉 𝑠;𝑎𝑏𝑐 (0, 𝑒, ∞) and 𝑉 𝑠;𝑎𝑏𝑐 (𝑒, 0, ∞) are related to each other by a rotation by 𝜋 in the 1-2 direction applied simultaneously to all three indices. The operator at ∞ is defined by (3.120). To compute the shadow transform, we can use conformal symmetry to choose a simple configuration of the points. We pick (𝑥 1 , 𝑥2 , 𝑥3 ) = (0, 𝑒, ∞), where 𝑒 is a unit vector in the 𝑥 1 direction. The two-point function becomes a tensor depending on the unit vector 𝑒 that maps 𝜆 3 → 𝜆 3𝑅 : e𝑐 (∞𝑒) O e† (0)⟩ = 𝐼 𝑐 𝑐 (𝑒). ⟨O 𝑐 (3.342) For example, in the case of a spinor representation in 4D, we have 𝐼 𝛼¤ 𝛼 (𝑒) ∝ (𝑒·𝜎) 𝛼¤ 𝛼 . The shadow transform will be dominated by a saddle point on the 𝑥 1 -axis, by SO(𝑑 − 1) invariance. Its location depends only on the 𝑧0 -dependent factors in the three-point function 𝑉 𝑠;𝑎𝑏𝑐 (0, 𝑒, 𝑧 0 𝑒) ∝ |𝑧0 | Δ1 −Δ2 −Δ3 |1 − 𝑧 0 | Δ2 −Δ1 −Δ3 . (3.343) This has the saddle solution 𝑧0∗ = Δ2 + Δ3 − Δ1 . 2Δ3 (3.344) The tensor structure that multiplies the answer depends on the location of the saddle. Taking into account the gaussian fluctuations around the saddle, we find ⟨O1𝑎 (0)O2𝑏 (𝑒)S[O3 ] 𝑐 (∞)⟩ (𝑠)   𝑑/2  Δ1 + Δ3 − Δ2 2Δ3    𝑉 𝑠;𝑎𝑏𝑐 (0, 𝑒, ∞)  𝑐 × 𝐼 𝑐 (𝑒) ×   𝑉 𝑠;𝑎𝑏𝑐 (𝑒, 0, ∞)  =𝑖 𝜋 Δ3 Δ −Δ1 −Δ3 + 𝑑 2 2 2! 2  Δ2 + Δ3 − Δ1 2Δ3 0 < 𝑧0∗ < 1, Δ −Δ2 −Δ3 + 𝑑 2 2 2! 1 (3.345) otherwise. If we perform the shadow transform three times, we find that 𝑧0∗ ∈ (0, 1) twice, and once it lies outside of this range. Thus, the cyclic ordering of the arguments to 𝑉 𝑠 get swapped twice, resulting in the same ordering after three transforms. The overall effect is to multiply by Δ-dependent factors and act on each index with 𝐼 𝑐 𝑐 (𝑒). 128 Overall, we find e† ] 𝑎 (0)S[ O e† ] (𝑒)S[ O e† ] 𝑐 (∞)⟩ (𝑠′ ∗) (𝐼 (𝑒) −1 )𝑎 𝑎 (𝐼 (𝑒) −1 ) 𝑏 𝑏 (𝐼 (𝑒) −1 )𝑐 𝑐 ⟨S[ O 1 2 𝑏 3   𝑑/2   𝑑/2   𝑑/2 𝑖 𝜋 (𝑑−2) 𝜋𝑖 𝜋𝑖 𝜋𝑖 𝑠′ ∗ = 𝑉𝑎𝑏𝑐 (0, 𝑒, ∞) × 𝑒 4 Δ1 Δ2 Δ3 𝑑   Δ +Δ −Δ −   Δ +Δ −Δ − 𝑑 (Δ1 +Δ2 −Δ3 ) (Δ1 +Δ2 +Δ3 ) 1 2 3 2 (Δ3 +Δ1 −Δ2 ) (Δ1 +Δ2 +Δ3 ) 3 1 2 2 × 4Δ1 Δ2 4Δ3 Δ1   Δ2 +Δ3 −Δ1 − 𝑑2 1 +Δ2 +Δ3 ) × (Δ2 +Δ3 −Δ4Δ1 )2(Δ . (3.346) Δ3 ′ ′ Here, 𝑉 𝑠 ∗ indicates the complex conjugate of the three-point structure 𝑉 𝑠 . For the purposes of this calculation, we should think of it simply as a three-point structure for operators in the representations e 𝜋𝑖† . We have written (3.346) so that its phase is manifest when Δ𝑖 is on the principal series Δ𝑖 ∈ 𝑑2 + 𝑖R≥0 (with positive imaginary part). Finally, we included inverse two-point structures 𝐼 (𝑒) −1 , since they are needed in (3.181). Appendix F Gluing factors In this appendix, we determine the gluing factor |𝑍glue (𝑟)| coming from a junction between a 𝑑-dimensional cylinder of radius 𝑟 and a flat end-cap given by a 𝑑dimensional ball. Our strategy is to start with the partition function on 𝑆 𝑑 (with radius 1) and perform a Weyl transformation to a cylinder C𝑟,𝛽 of radius 𝑟 and length 𝛽𝑟 with two flat end-caps. We will integrate the Weyl anomaly to compute 𝑆anom and deduce |𝑍glue (𝑟)| via (3.122). Recall that on a conformally-flat geometry, with the scheme 𝑆ct = 0 discussed in Section 3.3, the finite form of the Weyl anomaly is log 𝑍 [𝑒 2𝜔 𝑔] − log 𝑍 [𝑔] = −𝑆anom [𝑔, 𝜔] ∫ 1 ∫ (−1) 𝑑/2 𝑎 𝑑 √ 𝑑𝑡 𝑑 𝑑 𝑥𝜔 𝑔𝑒 𝑑𝑡𝜔 𝐸 𝑑 [𝑒 2𝑡𝜔 𝑔]. =− 𝑑/2 (4𝜋) 0 (3.347) To compute the Weyl anomaly between the sphere and the capped cylinder, we first need the Riemann tensor for a Weyl rescaling of 𝑆 𝑑 . Let us write the metric on 𝑆 𝑑 as 𝑑𝑠2𝑆 𝑑 = 𝑑𝜙2 + sin2 𝜙 𝑑𝑠2𝑆 𝑑−1 . (3.348) 129 We will be interested in Weyl rescalings e 𝑔 = 𝑒 2𝜔 𝑔𝑆 𝑑 , where 𝜔 is a function of 𝜙 alone. In this case, the Riemann tensor simplifies: h e𝜇𝜈 𝜌𝜎 = 𝑒 −2𝜔 (1 − 2𝜔′ cot 𝜙 − 𝜔′2 )(𝛿 𝜇𝜌 𝛿 𝜈𝜎 − 𝛿 𝜎𝜇 𝛿 𝜈𝜌 ) 𝑅 i 𝜌 𝜌 𝜌 𝜌 + (𝜔′ cot 𝜙 + 𝜔′2 − 𝜔′′)(𝛿1𝜇 𝛿1 𝛿 𝜈𝜎 − 𝛿1𝜈 𝛿1 𝛿 𝜎𝜇 − 𝛿1𝜇 𝛿1𝜎 𝛿 𝜈 + 𝛿1𝜈 𝛿1𝜎 𝛿 𝜇 ) , (3.349) where the index 1 represents the 𝜙 coordinate, and 𝜔′, 𝜔′′ denote derivatives of 𝜔 with respect to 𝜙. The Euler density is e𝑑 = 1 𝜖 𝜇1 ···𝜇 𝑑 𝜖 𝜈1 ···𝜈𝑑 𝑅 e𝜇1 𝜇2 𝜈1 𝜈2 · · · 𝑅 e𝜇 𝑑−1 𝜇 𝑑 𝜈𝑑−1 𝜈𝑑 𝐸 2𝑑/2 𝑑−2 = 𝑑!𝑒 −𝑑𝜔 (1 − 2𝜔′ cot 𝜙 − 𝜔′2 ) 2 (1 − 𝜔′ cot 𝜙 − 𝜔′′). (3.350) Plugging this result into (3.347), we find log 𝑍 [𝑒 2𝜔 𝑔] − log 𝑍 [𝑔] ∫ 1 ∫ 𝜋 (−1) 𝑑/2 𝑑!𝑎 𝑑 𝑑−1 ′ 2 ′2 𝑑−2 =− 𝑑𝑡 𝑑𝜙 𝜔(sin 𝜙) (1 − 2𝑡𝜔 cot 𝜙 − 𝑡 𝜔 ) 2 (1 − 𝑡𝜔′ cot 𝜙 − 𝑡𝜔′′) . 𝑑 𝑑−1 2 Γ( 2 ) 0 0 (3.351) Now let us examine the Weyl factor that relates the sphere to C𝑟,𝛽 . We will impose a symmetry under 𝜙 → 𝜋 − 𝜙, so that it suffices to consider the range 0 ≤ 𝜙 ≤ 𝜋2 . The Weyl factor is 𝛽/2 tan(𝜙/2)    log 𝑒 𝑟sin  𝜙 𝜔(𝜙) =   log sin𝑟 𝜙  0 ≤ 𝜙 ≤ 𝜙0 (3.352) 𝜙0 ≤ 𝜙 ≤ 𝜋/2, where 𝜙0 = 2 tan−1 (𝑒 −𝛽/2 ). As a check, consider first the range 0 ≤ 𝜙 < 𝜙0 . There, we have  2  2𝜔 2 2 2 𝛽 2 tan (𝜙/2) 2 2 2 𝑒 (𝑑𝜙 + sin 𝜙 𝑑𝑠 𝑆 𝑑−1 ) = 𝑒 𝑟 𝑑𝜙 + tan (𝜙/2)𝑑𝑠 𝑆 𝑑−1 sin2 𝜙 = 𝑑𝜌 2 + 𝜌 2 𝑑𝑠2𝑆 𝑑−1 , (3.353) where 𝜌 = 𝑒 𝛽/2𝑟 tan(𝜙/2). This is the metric on the flat ball, i.e. the first end cap. Similarly, for 𝜙0 ≤ 𝜙 ≤ 𝜋/2, we have 𝑒 2𝜔 2 (𝑑𝜙 + sin 2 𝜙 𝑑𝑠2𝑆 𝑑−1 ) =  𝑟𝑑𝜙 sin 𝜙 2 + 𝑟 2 𝑑𝑠2𝑆 𝑑−1 = 𝑑𝜏 2 + 𝑟 2 𝑑𝑠2𝑆 𝑑−1 , (3.354) 130 where 𝜏 = 𝑟 log tan(𝜙/2), which is the metric on the cylinder. Thus, (3.352) describes how the first hemisphere 0 ≤ 𝜙 ≤ 𝜋/2 maps to half of the capped cylinder. The remaining hemisphere should be treated symmetrically under 𝜙 → 𝜋 − 𝜙. In practice, this means integrating the anomaly over 𝜙 ∈ [0, 𝜋/2] and including a factor of 2. Plugging the Weyl factor (3.352) into (3.351) is subtle because 𝜔′′ has a 𝛿-function singularity at 𝜙 = 𝜙0 . In (3.351), this 𝛿-function gets multiplied by a function of 𝜔′, which is discontinuous at the support of the 𝛿-function! To get the correct result, we must regularize 𝜔 by smoothing out the discontinuities in its derivatives. For example, we can model 𝜔′ near 𝜙0 as  1 ′ 0 𝜔+ + 𝜔′− + (𝜔′+ − 𝜔′− )erf( 𝜙−𝜙 ) , (3.355) 𝜔′ (𝜙) = 𝜖 2 where 𝜖 is a small regulator, and 𝜔′± are the values of 𝜔′ to the left and the right of the discontinuity. Plugging this into (3.351), expanding to leading order in 𝜖, and writing 𝜙 = 𝜙0 + 𝜖𝑥, we obtain integrals of the form √ ∫ 1 + (−1) 𝑛 𝜋 −𝑥 2 𝑛 𝑑𝑥 𝑒 erf (𝑥) = , (3.356) 2 𝑛+1 which give finite, calculable contributions. Applying this procedure, we can obtain the contribution to (3.351) from an infinitesimal neighborhood of 𝜙0 :  𝛽 𝑎2    2 log(𝑟 cosh 2 )    𝛽−4 cosh 𝛽−6 (contribution near 𝜙0 ) = 𝑎44 log(𝑟 cosh 2𝛽 ) sinh4(cosh 𝛽+1)       𝑑 = 2, 𝑑 = 4, (3.357) ··· The detailed expressions here will not matter for our purposes. The important observation is that the contributions (3.357) all vanish when 𝑟 = 1 and 𝛽 = 0. We will take advantage of this fact shortly. Before computing the full result from plugging (3.352) into (3.351), let us use a shortcut to determine its 𝑟-dependence. From cutting and gluing, we expect the capped cylinder partition function to take the form log 𝑍 (C𝑟,𝛽 ) = log |𝑍glue (𝑟)| 2 − 𝜀0 𝛽, (3.358) where 𝜀0 is the Casimir energy on a unit 𝑆 𝑑−1 , given in (3.42). We can determine the 𝑟-dependence of the right-hand side by starting with a cylinder C1,𝛽 of radius 1 and performing a Weyl rescaling 𝑔 → 𝑟 2 𝑔 to get C𝑟,𝛽 . Because the integral of the 131 𝑑 𝑓 (𝑑) 2 1 4 6 8 10 12 14 7 3 37 5 1066 35 3254 21 72428 77 949484 143 Table 3.3: Values of 𝑓 (𝑑) for the first few even 𝑑. Euler density is topological, a constant Weyl rescaling gives the same anomaly on the capped cylinder as on the sphere. In other words, we have log 𝑍 (C𝑟,𝛽 ) − log 𝑍 (C1,𝛽 ) = log 𝑍 (𝑆𝑟𝑑 ) − log 𝑍 (𝑆 𝑑 ), (3.359) where 𝑆𝑟𝑑 is the sphere with radius 𝑟. On the sphere, we can easily integrate the Weyl anomaly using 𝐸 𝑑 [𝑔𝑆 𝑑 ] = 𝑑! to give log 𝑍 (𝑆𝑟𝑑 ) − log 𝑍 (𝑆 𝑑 ) = − (−1) 𝑑/2 𝑎 𝑑 𝑑! vol 𝑆 𝑑 log 𝑟 = −2(−1) 𝑑/2 ( 𝑑2 )!𝑎 𝑑 log 𝑟. (4𝜋) 𝑑/2 (3.360) Combining (3.358), (3.359), and (3.360), we conclude log |𝑍glue (𝑟)| 2 = log |𝑍glue (1)| 2 − 2(−1) 𝑑/2 ( 𝑑2 )!𝑎 𝑑 log 𝑟. (3.361) Thus, we have completely fixed the 𝛽 and 𝑟 dependence of log 𝑍 (C𝑟,𝛽 ), and the only remaining unknown is log 𝑍 (C1,0 ) = log |𝑍glue (1)| 2 . As noted above, when 𝑟 = 1 and 𝛽 = 0, the contribution to the Weyl anomaly near 𝜙0 vanishes. Furthermore, the contribution from the cylinder region vanishes as well since 𝜙0 = 𝜋2 . We are left with an integral over the end cap 𝜙 ∈ [0, 𝜋/2] alone: log |𝑍glue (1)| 2 − log 𝑍 (𝑆 𝑑 ) ∫ ∫ 1 𝑑−2 (−1) 𝑑/2 𝑎 𝑑 𝑑! 1 𝑑𝑡 𝑑𝑥 [(1 − 𝑡)(1 − 𝑥)(1 + 𝑡 + 𝑥 − 𝑥𝑡)] 2 (1 − 𝑡) log(𝑥 + 1), = 𝑑−2 2 Γ(𝑑/2) 0 0 (3.362) where we made the change of variables 𝑥 = cos 𝜙. We have not found a simple closed-form formula for log |𝑍glue (1)| 2 in general 𝑑. However, it is straightforward to plug different values of 𝑑 into (3.362) and perform the resulting elementary integrals. We find that 𝑑 ) − (−1) 𝑑/2 𝑓 (𝑑)𝑎 𝑑 , log |𝑍glue (𝑟)| 2 = log 𝑍 (𝑆𝑟/2 (3.363) 𝑑 ) is the partition function on a sphere of radius 𝑟/2 (determined by where 𝑍 (𝑆𝑟/2 (3.360)), and 𝑓 (𝑑) takes rational values for even 𝑑; see table 3.3. 132 Appendix G Counting quantum numbers in the genus-2 block A four point function of local operators depends on two independent cross ratios 𝑧 and 𝑧. These cross ratios are roughly conjugate to the quantum numbers Δ and 𝐽 labeling the internal representation. By varying 𝑧, 𝑧 independently, we can extract information about Δ and 𝐽 independently (for example, using Caron-Huot’s Lorentzian inversion formula [48]). Note that the internal operator in a four-point function may transform in a complicated SO(𝑑) representation whose Young diagram has multiple rows with lengths (𝑚 1 , 𝑚 2 , . . . , 𝑚 𝑛 ), where 𝑛 = ⌊ 𝑑2 ⌋ and 𝑚 1 = 𝐽. However, in a fixed four-point function, only 𝑚 1 = 𝐽 is unbounded. The remaining 𝑚𝑖 are bounded. In this appendix, we point out a similar match between unbounded quantum num𝑠′ 𝑠 and the dimension of the moduli space of genus-2 bers in the genus-2 block 𝐵123 conformal structures dim M = dim SO(𝑑 + 1, 1). Before explaining the general case, let us describe the matching in 𝑑 = 2 and 𝑑 = 3. In 𝑑 = 2, there is a unique three-point structure, so the labels 𝑠, 𝑠′ take only one value. The only unbounded quantum numbers in the block are the dimensions and spins of the three exchanged operators. This gives six quantum numbers, which matches dim M = 6 in 𝑑 = 2. In 𝑑 = 3, we again have six unbounded quantum numbers from the dimensions and spins of the exchanged operators. However, we must also take into account the 3point structure labels 𝑠, 𝑠′. One way to count them is to work in the embedding space. (The counting is even easier in the 𝑞-basis, but our embedding-space discussion will be useful later.) In the embedding space formalism, a spin-𝐽 operator becomes a homogeneous function O (𝑋, 𝑍) of an embedding coordinate 𝑋 ∈ R𝑑+1,1 and a polarization vector 𝑍 ∈ C𝑑+1,1 , subject to orthogonality conditions 𝑋 2 = 𝑍 2 = 𝑋 · 𝑍 = 0, and a gauge redundancy 𝑍 ∼ 𝑍 + 𝜆𝑋. The operator O (𝑋, 𝑍) has homogeneity −Δ in 𝑋 and 𝐽 in 𝑍. A general three-point structure for such operators is given by ⟨O1 (𝑋1 , 𝑍1 )O2 (𝑋2 , 𝑍2 )O3 (𝑋3 , 𝑍3 )⟩ ∋ ℓ1 ℓ2 ℓ3 𝑉1𝐽1 −ℓ2 −ℓ3 𝑉2𝐽2 −ℓ3 −ℓ1 𝑉3𝐽3 −ℓ1 −ℓ2 𝐻23 𝐻31 𝐻12 Δ1 +Δ2 −Δ3 Δ2 +Δ3 −Δ2 Δ3 +Δ1 −Δ2 𝑋12 𝑋23 𝑋31 (ℓ2 + ℓ3 ≤ 𝐽1 , ℓ3 + ℓ1 ≤ 𝐽2 , ℓ1 + ℓ2 ≤ 𝐽3 ), (3.364) where 𝐻𝑖 𝑗 and 𝑉𝑖 are standard polynomials in the polarization vectors [69]. The three point structure is labeled by integers ℓ1 , ℓ2 , ℓ3 , which are constrained by the 133 requirement that the correlator should be a polynomial in the 𝑍𝑖 . The ℓ𝑖 are unbounded in the limit of large spin 𝐽𝑖 . However, there is a relation between the 𝐻𝑖 𝑗 and 𝑉𝑖 : (𝑉1 𝐻23 + 𝑉2 𝐻13 + 𝑉3 𝐻12 + 2𝑉1𝑉2𝑉3 ) 2 = 2𝐻12 𝐻13 𝐻23 . (3.365) Using this relation, we can always reduce one of the 𝜆𝑖 to zero, so the number of unbounded quantum numbers labeling the three-point structures in 3D is 3 − 1 = 2. The conformal block is labeled by two three-point structures, so this gives 2 × 2 = 4 additional unbounded quantum numbers for the block. Overall, we have 6 + 4 = 10 = dim M in 3𝐷. We are now ready to tackle the 𝑑-dimensional case. In the 𝑑-dimensional embedding formalism, an operator O becomes a homogeneous function of an embedding coordinate 𝑋 ∈ R𝑑+1,1 and polarization vectors 𝑊1 , . . . , 𝑊𝑛 ∈ C𝑑+1,1 , where 𝑛 = ⌊ 𝑑2 ⌋. (We conventionally write 𝑊1 = 𝑍.) More formally, O is a locally holomorphic section of a line bundle over the flag variety V𝑑+1,1 of SO(𝑑 + 1, 1), which has (𝑋, 𝑊1 , · · · , 𝑊𝑛 ) as projective coordinates. The number of these coordinates, subject to orthogonality relations and modulo gauge redundancies, is the same as the complex dimension of the flag variety, which is    1 (𝑑 + 1)(𝑑 + 2) 𝑑+2 1 − dimC V𝑑+1,1 = (dim SO(𝑑 + 1, 1) − dim 𝑇) = , 2 2 2 2 (3.366) where 𝑇 is the maximal torus of SO(𝑑 + 1, 1). See [68, 136] for more discussion on the embedding formalism for general tensors. For example, in 𝑑 = 3, this gives dimC V4,1 = 4, which is the correct number of degrees of freedom in the vectors 𝑋, 𝑍 ∈ R4,1 . We can see this explicitly by restricting to the Poincare section 𝑋 = (1, 𝑥 2 , 𝑥) and 𝑍 = (1, 2𝑥 · 𝑧, 𝑧). Here 𝑥 ∈ R3 is unconstrained, and 𝑧 is a null vector in 3D, modulo rescaling, which corresponds to a single angle on the celestial circle. A three-point function ⟨O1 O2 O3 ⟩ is a section of a line bundle over three copies of V𝑑+1,1 , satisfying invariance under SO(𝑑 + 1, 1). In the large quantum number limit, the number of quantum numbers labeling such sections is #(Z-valued 3-pt structure labels) = 3 dimC V𝑑+1,1 − dim SO(𝑑 + 1, 1). (3.367) 134 Finally, a genus-2 block has two three-point structure labels 𝑠, 𝑠′, together with dim 𝑇 = ⌊ 𝑑+2 2 ⌋ quantum numbers for each of the three internal operators. Overall, the number of unbounded quantum numbers is
2 3 dim V𝑑+1,1 − dim SO(𝑑 + 1, 1) + 3 dim 𝑇 = dim SO(𝑑 + 1, 1) = dim M. (3.368) 135 Chapter 4 ANGULAR FRACTALS IN THERMAL QFT 4.1 Introduction Many aspects of conformal field theories (CFTs) are universal at high energies. A famous example is Cardy’s formula, which states that the entropy of local operators at sufficiently high energies takes a universal form in all unitary, compact 2D CFTs [46] (see [168, 170, 175] for a precise formulation). Equivalently, the partition function of a 2D CFT   Tr 𝑒 −𝛽𝐻+𝑖𝜃𝐽 (4.1) is universal in the high temperature regime 𝛽 → 0 with 𝜃 ∼ 𝑂 (𝛽). The derivation of Cardy’s formula uses invariance of the torus partition function under the modular transformation 𝑆 : 𝜏 ↦→ −1/𝜏. By instead using the full modular group PSL(2, Z), one finds similar universal behavior as 𝛽 → 0, near any rational angle 𝜃 = 2𝜋𝑞 𝑝 , see e.g. [24]. This leads to universal "spin-refined" versions of the density of states. For example, in the case 𝑞𝑝 = 12 , the modular transformation −𝜏 gives the universal behavior of 𝜏 ↦→ 2𝜏−1     Tr 𝑒 −𝛽(𝐻−𝑖Ω𝐽) (−1) 𝐽 = Tr 𝑒 −𝛽𝐻+𝑖𝜃𝐽 𝜃=𝜋+𝛽Ω , (4.2) in the regime 𝛽 → 0 with Ω ∼ 𝑂 (1). For any given 2D CFT, the logarithm of (4.2) is 1/4 the logarithm of (4.1) at high temperature, leading to a universal result for the difference between densities of even- and odd-spin operators in 2D CFTs.1 While modular invariance is not available on 𝑆 𝑑−1 × 𝑆 1 in higher dimensions, higher dimensional CFTs still display forms of universality at high energies, both in their density of states [7, 25, 27, 147, 188, 189, 199], and OPE coefficients [25, 74]. A central insight from [14, 27, 127] is that the high temperature behavior of a CFT can be captured by a "thermal Effective Field Theory (EFT)" that efficiently encodes the constraints of conformal symmetry and locality. In [7, 25, 188, 189], thermal EFT plays the role of a surrogate for the modular 𝑆-transformation (as well as modular transformations on genus-2 surfaces). 1 Modular invariance on higher genus surfaces also leads to universal results for OPE coefficients in 2D CFTs, as derived in [36, 44, 72, 115, 141], and unified in [67]. 136 Figure 4.1: A qualitative picture of log(log(𝑍)) in the 3D Ising CFT, where 𝑍 = Tr(𝑒 −𝛽𝐻+𝑖𝜃𝐽 ) is the 𝑆 2 × 𝑆 1 partition function. To construct this picture, we took the leading terms in the EFT description around each rational angle (up to denominator 15), and combined them with a root-mean-square. We give more detail in Appendix 4.10. In this work, we will be interested in "spin-refined" information about the CFT density of states in general dimensions. In particular, we will study the partition ® (In higher function (4.1) with high temperature (𝛽 ≪ 1) and finite angles 𝜃. dimensions we promote 𝜃® and 𝐽® to vectors with ⌊𝑑/2⌋ components coming from ® with fixed Ω ® is captured by thermal EFT the rank of 𝑆𝑂 (𝑑).) The regime 𝜃® = 𝛽Ω as discussed in [14, 25, 27, 127, 188]. However, when 𝜃® does not scale to zero as 𝛽 → 0, the naïve EFT description breaks down. A simple example of a partition function with finite 𝜃® is (4.2): the relative density of even-spin and odd-spin operators with respect to some particular Cartan generator 𝐽 of the rotation group. This observable is naïvely outside the regime of validity of the thermal EFT, since 𝜃 remains finite as 𝛽 → 0. More generally, we can consider a partition function that includes a rotation by finite rational angles in each of the Cartan directions:2   h i 𝑝 ® 𝐽) 2𝜋𝑖 𝑞1 𝐽1 +···+ 𝑞𝑝𝑛𝑛 𝐽𝑛 ® −𝛽(𝐻−𝑖 Ω· 1 . (4.3) Tr 𝑒 𝑅 , where 𝑅 = 𝑒 Using a trick that was applied in [10] to study superconformal indices near roots of unity, we will find a different EFT description for this partition function, in terms of the thermal EFT on a background geometry with inverse temperature 𝑞𝛽 and spatial manifold 𝑆 𝑑−1 /Z𝑞 , where 𝑞 = lcm(𝑞 1 , . . . , 𝑞 𝑛 ). This determines the small-𝛽 2 In parity-invariant theories, we can also include reflections. 137 expansion of (4.3) in terms of the usual Wilson coefficients of thermal EFT, up to new subleading contributions from "Kaluza-Klein vortices" that we classify. For example, the effective free energy density of (4.3), coming from the leading term in the thermal effective action, is smaller than the usual free energy density by a factor of 1/𝑞 𝑑 . In particular, the effective free energy density of even-spin minus odd-spin operators described by (4.2) is smaller by 1/2𝑑 . (This generalizes the factor of 1/4 in 2D.)3 The EFT descriptions around each rational angle patch together to create fractal-like behavior in the high-temperature partition function; see Figure 4.1 for an illustration in the 3D Ising CFT. It is remarkable that effective field theory constrains the asymptotics of the partition function in such an intricate way, even in higher dimensions. Kaluza-Klein vortices appear whenever the rational rotation 𝑅 does not act freely on the sphere 𝑆 𝑑−1 . Each vortex creates a defect in the thermal EFT, whose action can be written systematically in a derivative expansion in background fields. By contrast, when 𝑅 generates a group that acts freely, no vortex defects are present, and the complete perturbative expansion of (4.3) in 𝛽 is determined in terms of thermal EFT Wilson coefficients, with no new undetermined parameters. While most of our discussion and examples are focused on CFTs, our formalism also applies to general QFTs. In particular, using thermal effective field theory, we derive a relation between the partition function at temperature 𝑇 with a discrete isometry of order 𝑞 inserted, to the partition function with no insertion at temperature 𝑇/𝑞, in the thermodynamic limit.4 For example, we have     1 − log TrH (M 𝐿 ) 𝑒 −𝛽𝐻 𝑅 ∼ − log TrH (M 𝐿 ) 𝑒 −𝑞𝛽𝐻 + topological + KK defects 𝑞 (as 𝐿 → ∞). (4.4) Here, M 𝐿 is a spatial manifold of characteristic size 𝐿, with associated Hilbert space H (M 𝐿 ), 𝑅 is a discrete isometry of order 𝑞, and "∼" denotes agreement to all perturbative orders in 1/𝐿. The relation (4.4) holds whenever the theory is gapped at inverse temperature 𝑞𝛽. We write the most general relation in (4.32), which we check 3 Note that simply taking the density of states computed in [25] and inserting the phase 𝑅 into the trace will not give the correct answer to the partition function. For a demonstration of this in 2D, see Appendix B of [24]. 4 We are extremely grateful to Luca Delacretaz for emphasizing the general QFT case to us. 138 in both massive and massless examples. An interesting consequence of this simple formula is that twists by discrete isometries can be sensitive to lower-temperature phases of the theory. For example, the partition function of QCD at temperature 𝑇 > ΛQCD , twisted by a discrete isometry with order 𝑞, becomes sensitive to physics below the confinement scale when 𝑇/𝑞 < ΛQCD . This universality of partition functions with spacetime symmetry insertions is in contrast to the case for global symmetry insertions. The insertion of a global symmetry generator operator is equivalent to turning on a new background field in the thermal EFT. The dependence of the effective action on this background field introduces new Wilson coefficients that are not necessarily related in a simple way to the Wilson coefficients without the global symmetry background; see e.g. [102, 128, 174]. The paper is organized as follows. In Section 4.2, we present a derivation of our main result: a systematic study of the high temperature expansion of the partition function of any quantum field theory with the insertion of a discrete isometry. In Section 4.3, we look in more detail at the Kaluza-Klein vortices that appear on 𝑆 𝑑−1 when the discrete isometry (which is a rational rotation in this case) has fixed points. In Section 4.4, we discuss subtleties that appear for fermionic theories. In Section 4.5, we give several examples in free theories that illustrate our general results. In Section 4.6, we consider thermal effective actions with topological terms. In Section 4.7, we apply our results to holographic CFTs. In Section 4.8, we look at irrational 𝜃. In Section 4.9, we discuss non-perturbative corrections in temperature. Finally in Section 4.10, we conclude and discuss future directions. 4.2 Folding and unfolding the partition function Thermal effective action and finite velocities Equilibrium correlators of generic interacting QFTs at finite temperature are expected to have a finite correlation length. Equivalently, the dimensional reduction of a generic interacting QFT on a Euclidean circle is expected to be gapped. When this is the case, long-distance finite-temperature observables of the QFT can be captured by a local "thermal effective action" of background fields [14, 27, 127]. For example, consider the partition function of a QFT𝑑 on M 𝐿 × 𝑆 1𝛽 , where the spatial 139 𝑑−1-manifold M 𝐿 has size 𝐿. In the thermodynamic limit of large 𝐿, we have TrH (M 𝐿 ) [𝑒 −𝛽𝐻 𝐿 ] = 𝑍QFT [M 𝐿 × 𝑆 1𝛽 ] = 𝑍gapped [M 𝐿 ] ∼ 𝑒 −𝑆th [𝑔,𝐴,𝜙] + nonperturbative in 1/𝐿 (𝐿 → ∞), (4.5) where H (M 𝐿 ) is the Hilbert space of states on M 𝐿 , and 𝐻 𝐿 is the Hamiltonian. Here, the thermal effective action 𝑆th depends on a 𝑑−1-dimensional metric 𝑔𝑖 𝑗 , a Kaluza-Klein gauge field 𝐴𝑖 , and a dilaton 𝜙, which can be obtained by placing the 𝑑-dimensional metric in Kaluza-Klein (KK) form 𝐺 𝜇𝜈 𝑑𝑥 𝜇 𝑑𝑥 𝜈 = 𝑔𝑖 𝑗 (® 𝑥 )𝑑𝑥 𝑖 𝑑𝑥 𝑗 + 𝑒 2𝜙(𝑥®) (𝑑𝜏 + 𝐴𝑖 (® 𝑥 )) 2 , (4.6) where 𝜏 ∼ 𝜏 + 𝛽 is a periodic coordinate along the thermal circle. The derivative expansion for 𝑆th becomes an expansion in inverse powers of the length 𝐿. If the spatial manifold M 𝐿 possesses a continuous isometry 𝜉, then we can additionally twist the partition function by the corresponding charge 𝑄 𝜉 :   TrH (M 𝐿 ) 𝑒 −𝛽(𝐻 𝐿 −𝑖𝛼𝑄 𝜉 ) . (4.7) Geometrically, this twist corresponds to a deformation of the background fields 𝑔, 𝐴, 𝜙 that depends on 𝛼𝜉. In the thermodynamic limit, we can describe (4.7) using the thermal effective action, provided that the background fields 𝑔, 𝐴, 𝜙 remain finite as 𝐿 → ∞. In particular, the combination 𝛼𝜉 must remain finite as 𝐿 → ∞. The physical reason is that 𝑖𝛼 represents the velocity of the system in the direction of 𝜉 in the canonical ensemble. This velocity must remain finite in order to have a good thermodynamic limit. By contrast, suppose that M 𝐿 possesses a nontrivial discrete isometry 𝑅 with finite order 𝑅 𝑞 = 1. If we twist the partition function by 𝑅,   TrH (M 𝐿 ) 𝑒 −𝛽𝐻 𝑅 , (4.8) then physically this corresponds to a system whose "velocity" is of order 𝐿. The background fields 𝑔, 𝐴, 𝜙 naïvely do not have a good thermodynamic limit, and we cannot apply the thermal effective action in an obvious way. • Example: CFT partition function An important example for us is the partition function of a CFT𝑑 on 𝑆 𝑑−1 × 𝑆 1𝛽 . Conformal invariance dictates that h i h i ® 𝐽) ® 𝐽®𝐿 ) ® −𝛽(𝐻−𝑖 Ω· −𝐿 𝛽(𝐻 𝐿 −𝑖 Ω· Tr 𝑒 = TrH (𝑆 𝑑−1 ) 𝑒 . (4.9) 𝐿 140 On the left-hand side, we have the usual partition function of the CFT on a sphere of radius 1. On the right-hand side, 𝐻 𝐿 denotes the Hamiltonian on a sphere 𝑆 𝑑−1 𝐿 of radius 𝐿, and 𝐽®𝐿 are generators of isometries of the sphere, normalized so that the corresponding Killing vectors are finite in the flat-space limit 𝐿 → ∞. (For example, for a rotation of the sphere by an angle 𝜙, a Killing vector with a finite 𝜕 .) flat-space limit is 𝐿1 𝜕𝜙 When 𝛽 is small, we can set 𝐿 = 𝑂 (1/𝛽) on the right-hand side and try to apply the thermal effective action (4.5). We find that in order to have a good thermodynamic ® must remain finite. Phrased in terms of the limit as 𝛽 → 0, the angular potentials Ω ® we find that 𝜃® must scale to zero as 𝛽 → 0. Provided this rotation angle 𝜃® = 𝛽Ω, is the case, the 1/𝐿 expansion of the thermal effective action gives an expansion in small 𝛽 for the CFT partition function. We can also understand condition 𝜃® → 0 more explicitly from a direct computation using the thermal effective action. In a CFT, the thermal effective action is constrained by 𝑑-dimensional Weyl invariance. The most general coordinate- and Weyl-invariant action takes the form5,6 ∫ 𝑑−1 √︁   𝑑 𝑥® 2b 2 2 b 𝑔 − 𝑓 + 𝑐 1 𝛽 𝑅 + 𝑐 2 𝛽 𝐹 + . . . + 𝑆anom . (4.10) 𝑆th = 𝛽 𝑑−1 b is the Ricci scalar built from b Here b 𝑔 = 𝑒 −2𝜙 𝑔, 𝑅 𝑔 , 𝐹 2 is a Maxwell term, etc. The term 𝑆anom accounts for Weyl anomalies (which are not important for the present discussion). On the geometry 𝑆 𝑑−1 × 𝑆 1𝛽 , we can easily determine 𝑔, 𝐴, 𝜙 and evaluate 𝑆th [25]: " ! #   ∑︁ 𝑛 vol 𝑆 𝑑−1 8 𝑑−1 2 𝑑−3 𝑆th = Î𝑛 −𝑓𝑇 + (𝑑 − 2) (𝑑 − 1)𝑐 1 + 2𝑐 1 + 𝑐 2 +... . Ω𝑖 𝑇 2 𝑑 𝑖=1 (1 + Ω𝑖 ) 𝑖=1 (4.11) We see that terms of order 𝑇 𝑑−1−𝑘 = 𝛽 𝑘−𝑑+1 in the high-temperature expansion of 𝑆th are multiplied by a polynomial in the angular potentials Ω𝑖 of degree 𝑘 (see e.g. examples in [31]). Consequently, 𝜃 𝑖 → 0 as 𝛽 → 0 is necessary for the high-temperature expansion to be well-behaved. 5 For simplicity, here we assume that the theory is free of gravitational anomalies. 6 Note that [25] worked in conventions where 𝜏 has periodicity 1, and 𝛽 is absorbed into the field 𝜙. In this paper, we instead use conventions where 𝜏 has dimensionful periodicity 𝛽 (later we will also have other periodicities) so that explicit powers of 𝛽 appear in the action (4.10), as required by dimensional analysis. To convert from the conventions of [25] to the conventions in this work, one shifts the dilaton by 𝜙 → 𝜙 + log 𝛽. 141 To summarize, the thermal effective action can describe "small" angles 𝜃 ∼ 𝛽Ω, where the angular velocity remains finite in the thermodynamic limit. However, results from the thermal effective action like (4.11) break down outside this regime. How can we access more general angles? Spin-refined partition functions: warm-up in 2D CFT As a warm-up, in 2D CFT, we can compute partition functions at more general angles using modular invariance. Let us review how this works and derive some example results. For convenience, we write the partition function as h i 𝑐 𝑐 𝑍 (𝜏, 𝜏) = Tr 𝑒 2𝜋𝑖𝜏 ( 𝐿 0 − 24 ) −2𝜋𝑖𝜏 ( 𝐿 0 − 24 ) , (4.12) 𝑖𝛽 𝜃 + 2𝜋 and 𝜏 = 𝜏 ∗ .7 The high temperature behavior of 𝑍 (𝜏, 𝜏) at where 𝜏 = 2𝜋 small angles can be obtained by performing the modular transformation 𝜏 → −1/𝜏 (similarly for 𝜏) and approximating by the contribution of the vacuum state. The result agrees with the thermal effective action:      −𝛽(𝐻−𝑖Ω𝐽)  vol 𝑆 1 𝑓 𝑐 4𝜋 2 −𝑆th = exp Tr 𝑒 ∼𝑒 = exp (CFT2 ), (1 + Ω2 ) 𝛽 𝛽(1 + Ω2 ) 12 (4.13) where 𝑓 = 2𝜋𝑐 12 . Here, we assume 𝑐 𝐿 = 𝑐 𝑅 for simplicity. In this case, only the cosmological constant term appears in the thermal effective action in 2D. 𝜃 is close to a nonzero rational angle 𝑞𝑝 , so that Now let us instead assume that 2𝜋 𝜏, 𝜏 are very close to 𝑞𝑝 . Following [24], we can perform a different modular transformation to map (𝜏, 𝜏) close to ±𝑖∞ and approximate the partition function by the vacuum state in the new channel. For example, let us study the partition function with an insertion of (−1) 𝐽 given in (4.2). In this case, we have 1 𝛽Ω 𝑖𝛽 + + 2 2𝜋 2𝜋 1 𝛽Ω 𝑖𝛽 𝜏= + − , 2 2𝜋 2𝜋 𝜏= 𝛽 ≪ 1, Ω ∼ 𝑂 (1). (4.14) Modular invariance is the statement 𝑍 (𝛾 ◦ 𝜏, 𝛾 ◦ 𝜏) = 𝑍 (𝜏, 𝜏) , 𝛾 ∈ PSL(2, Z). (4.15) 7 Note that in this section, 𝜏 denotes the modular parameter of the torus, while in other sections 𝜏 denotes Euclidean time. We hope this will not cause confusion. 142 An appropriate transformation in this case is ! −1 0 𝛾=± ∈ PSL(2, Z), 2 −1 (4.16) which leads to   h i 2  −𝛽(𝐻−𝑖Ω𝐽)  𝑐 𝑐 4𝜋 𝑐 1 e 𝐽 2𝜋𝑖e 𝜏 ( 𝐿 0 − 24 −2𝜋𝑖 𝜏 𝐿 − ) ( 0 24 ) ∼ exp Tr 𝑒 (−1) = Tr 𝑒 , 4 𝛽(1 + Ω2 ) 12 1 𝜋𝑖 where e 𝜏=− + , e 𝜏=e 𝜏∗ . (4.17) 2 2𝛽(1 − 𝑖Ω) On the right-hand side, we approximated the trace by the contribution of the vacuum state in the 𝛽 → 0 limit. We find that the partition function weighted by (−1) 𝐽 grows exponentially in 1/𝛽, with an exponent that is 1/4 of the un-weighted case (4.13). 𝜃 close to 𝑞𝑝 , we repeat the same logic above but with a more For a general angle 2𝜋 complicated modular transformation, namely ! −( 𝑝 −1 ) 𝑞 𝑏 𝛾=± ∈ PSL(2, Z), (4.18) 𝑞 −𝑝 where ( 𝑝 −1 ) 𝑞 is the inverse of 𝑝 modulo 𝑞, and 𝑏 is chosen so the matrix has determinant 1. We get   h 𝑝 i 1 𝑐 4𝜋 2 −𝛽(𝐻−𝑖Ω𝐽) 2𝜋𝑖 𝑞 𝐽 ∼ exp 2 Tr 𝑒 𝑒 . (4.19) 𝑞 𝛽(1 + Ω2 ) 12 𝑝 In general, we find that the partition function of a 2D CFT weighted by 𝑒 2𝜋𝑖 𝑞 𝐽 grows exponentially in 1/𝛽, with an exponent that is 1/𝑞 2 of the un-weighted case (4.13). Because modular invariance is not available in higher dimensions, it will be useful to rederive (4.19) in a different way. We now describe two (related) approaches that can generalize to higher dimensions. Folding and unfolding Thermal EFT naively breaks down in spin-refined partition functions like (4.2) because the large spacetime symmetry (−1) 𝐽 moves us outside the thermodynamic limit. One way to recover an EFT description is to perform a change of coordinates that makes (−1) 𝐽 look more like a global symmetry. For example, consider a spin-refined partition function of a 2D QFT (not necessarily conformal) on 𝑆 1𝐿 × 𝑆 1𝛽 ,   Tr 𝑒 −𝛽𝐻 (−1) 𝐽 , (4.20) 143 where (−1) 𝐽 denotes a rotation of the spatial circle 𝑆 1𝐿 by 𝜋. We can reinterpret one copy of the QFT on 𝑆 1𝐿 × 𝑆 1𝛽 as two copies of the QFT on (𝑆 1𝐿 /Z2 ) × 𝑆 1𝛽 , with topological defects that glue the two copies to each other, see the middle of Figure 4.2. In this picture, the operator (−1) 𝐽 becomes a topological defect that simply permutes the two copies of the QFT as we move along the time direction. If we begin in one copy of the QFT and move by 𝛽 in Euclidean time, we pass once through the (−1) 𝐽 defect and go to the other copy. Moving by 𝛽 again, we pass through the (−1) 𝐽 defect again and end up in the first copy. Thus, inserting (−1) 𝐽 into the partition function creates a new effective thermal circle of length 2𝛽. =⇒ 𝛽 =⇒ 2𝛽 𝐿 𝐿/2 Figure 4.2: Left: The torus partition function with spatial cycle of length 𝐿, inverse temperature 𝛽, and an insertion of (−1) 𝐽 . The (−1) 𝐽 insertion means we must glue the top and bottom of the figure with a half shift around the spatial circle. We split the figure into a left and right half using the trivial defect (vertical dashed line), and for convenience we color the right half grey. Middle: Placing the black and grey rectangles on top of each other, we can interpret this same observable as the partition function of two copies of the QFT (black and grey) on an (𝐿/2) × 𝛽 rectangle, with boundary conditions inherited from the left figure. Right: Finally, we can re-stack the two copies of the QFT, resulting in a single copy of the QFT with a new spatial circle of length 𝐿/2 and an effective thermal circle of length 2𝛽. Note that the effective thermal circle is nontrivially fibered over the new spatial circle. This reinterpretation of the path integral with a (−1) 𝐽 insertion is illustrated in 1 Figure 4.2. One wrinkle (that is clear in the figure) is that the effective thermal 𝑆2𝛽 is nontrivially fibered over the spatial circle 𝑆 1𝐿 /Z2 : when we go once around the 1 shifts by 𝛽. new spatial circle, the 𝑆2𝛽 So far, we have considered a rotation angle of 𝜋. However, it is straightforward to study nearby rotation angles of the form 𝜃 = 𝜋 + 𝛽Ω. On the left-hand side of Figure 4.2, we simply insert an additional topological operator along the spatial cycle that implements the small rotation 𝑒𝑖𝛽Ω𝐽 . Following the manipulations in the 144 figure, we end up with a product of two such operators on the new spatial cycle 𝑆 1𝐿 /Z2 , which together implement a rotation of 2𝛽Ω. The advantage of this rewriting of the path integral is that we can now smoothly take the thermodynamic limit 𝐿 → ∞ and use the thermal effective action. The effective inverse temperature is 2𝛽, the rotation angle is 2𝛽Ω, and the effective spatial cycle is 𝑆 1𝐿 /Z2 . In fact, the above construction is straightforward to generalize to twists by any rational angle: h 𝑝 i Tr 𝑒 −𝛽(𝐻−𝑖Ω𝐽)+2𝜋𝑖 𝑞 𝐽 . (4.21) We interpret (4.21) as the partition function of 𝑞 copies of the QFT on the space 𝑆 1𝐿 /Z𝑞 , with appropriate topological defects that glue the copies together. The 𝑝 operator 𝑒 2𝜋𝑖 𝑞 𝐽 becomes a topological defect that permutes the copies of the QFT as we move around the Euclidean time circle. This creates an effective thermal circle 𝑆 1𝑞𝛽 , which is fibered over 𝑆 1𝐿 /Z𝑞 . We can now apply thermal EFT on 𝑆 1𝐿 /Z𝑞 . • Example: 2D CFT As an example application, we can recover our previous answer for the spin-refined partition function of a 2D CFT. For a twist by (−1) 𝐽 , we find      −𝛽 (𝐻 −𝑖Ω𝐽 )  1 ] 2𝜋𝑐 vol(𝑆 1 /Z2 ) 1 4𝜋 2 𝑐 −𝑆th [𝑆 1 /Z2 ×𝑆2𝛽 𝐽 (−1) ∼ 𝑒 . Tr 𝑒 = exp = exp 12 2𝛽(1 + Ω2 ) 4 𝛽(1 + Ω2 ) 12 (4.22) In the action, we obtain one factor of 21 from the smaller spatial cycle 𝑆 1 /Z2 , and another factor of 12 from the larger thermal circle, resulting in an overalll factor of 41 that agrees with the result from modular invariance (4.17).8 More generally, for a twist by 2𝜋𝑞 𝑝 , the thermal effective action gives h Tr 𝑒 −𝛽 (𝐻 −𝑖Ω𝐽 ) 𝑒 2 𝜋𝑖 𝑝 𝑞 𝐽 i ∼𝑒 1 ] −𝑆th [𝑆 1 /Z𝑞 ×𝑆𝑞𝛽 " #   2𝜋𝑐 vol(𝑆 1 /Z𝑞 ) 1 4𝜋 2 𝑐 . = exp = exp 2 12 𝑞𝛽(1 + Ω2 ) 𝑞 𝛽(1 + Ω2 ) 12 (4.23) 8 The fact that the thermal circle is nontrivially fibered plays no role here because the thermal effective action is the integral of a local coordinate-invariant quantity that does not detect global features of the bundle. In a theory with a gravitational anomaly, the thermal effective action would contain an additional 1-dimensional Chern-Simons for the Kaluza-Klein gauge field, which can detect the nontrivial topological structure of the thermal circle bundle; see Section 4.6. The nontrivial topology also enters into nonperturbative corrections; see Section 4.9. 145 We find that the effective free energy at high temperature for the spin-refined partition function (4.21) is down by a factor of 𝑞 2 , in agreement with (4.19). Note 𝑝 that the precise permutation of the copies of the CFT implemented by 𝑒 2𝜋𝑖 𝑞 𝐽 depends on 𝑝, but the length of the resulting thermal circle does not. Consequently the partition function is independent of 𝑝, up to nonperturbative corrections as 𝛽 → 0.9 • Higher dimensions The above construction works for 𝑑 > 2 as well, and on more general geometries. Consider a QFT𝑑 on any (𝑑−1)-dimensional spatial manifold M 𝐿 with a discrete isometry 𝑅 of finite order 𝑅 𝑞 = 1. Again, we can reinterpret one copy of the QFT on M 𝐿 × 𝑆 1𝛽 as 𝑞 copies of the QFT on (M 𝐿 /Z𝑞 ) × 𝑆 1𝛽 , with topological defects that glue the copies to each other. In this picture, 𝑅 is represented as a topological defect that simply permutes the 𝑞 copies of the QFT as we move along the time direction, creating an effective inverse temperature 𝑞𝛽. The EFT bundle Before exploring further consequences of this idea, it will be helpful to adopt a more abstract, geometrical perspective on this construction. Consider again a 𝑑dimensional QFT with spatial manifold M 𝐿 . Given an isometry 𝑈 ∈ Iso(M 𝐿 ), the partition function twisted by 𝑈 10   TrH (M 𝐿 ) 𝑒 −𝛽𝐻 𝑈 (4.24) is computed by the path integral of the CFT on the mapping torus 𝑀 𝛽,𝑈 ≡ (M 𝐿 × R)/Z, (4.25) where Z = ⟨ℎ⟩ is generated by ℎ : M 𝐿 × R → M 𝐿 × R, ℎ : (® 𝑥 , 𝜏) ↦→ (𝑈 𝑥®, 𝜏 + 𝛽), (4.26) where 𝑥® is a coordinate on M 𝐿 . Now let us specialize to 𝑈 = 𝑅, where 𝑅 has order 𝑞. In this case, the 𝑞-th power of ℎ acts very simply: it leaves M 𝐿 invariant, and shifts 𝜏 by 𝑞𝛽: ℎ 𝑞 : (® 𝑥 , 𝜏) ↦→ (® 𝑥 , 𝜏 + 𝑞𝛽). (4.27) 9 There is 𝑝-dependence if the theory is fermionic (see Section 4.4) or has a gravitational anomaly (see Section 4.6). 10 Here, we abuse notation and write 𝑈 for both the isometry and the operator implementing its action on the Hilbert space H (M 𝐿 ). 146 Consequently, it is useful to decompose Z  𝑞Z × Z𝑞 = ⟨ℎ 𝑞 ⟩ × (⟨ℎ⟩/⟨ℎ 𝑞 ⟩), and obtain the mapping torus 𝑀 𝛽,𝑅 via two successive quotients. We first quotient by 𝑞Z  ⟨ℎ 𝑞 ⟩ (which turns R into 𝑆 1𝑞𝛽 ), and then quotient by Z𝑞 = ⟨ℎ⟩/⟨ℎ 𝑞 ⟩: 𝑀 𝛽,𝑅 = ((M 𝐿 × R)/𝑞Z)/Z𝑞 = (M 𝐿 × 𝑆 1𝑞𝛽 )/Z𝑞 . (4.28) The quotient (M 𝐿 × 𝑆 1𝑞𝛽 )/Z𝑞 on the right-hand side of (4.28) can be viewed as a bundle in two different ways. Firstly, it is a M 𝐿 -bundle over 𝑆 1𝑞𝛽 /Z𝑞  𝑆 1𝛽 . This is the usual point of view of the trace as a spatial manifold evolving over Euclidean time 𝛽. However, we can alternatively view (M 𝐿 × 𝑆 1𝑞𝛽 )/Z𝑞 as an 𝑆 1𝑞𝛽 bundle over M 𝐿 /Z𝑞 . We call this latter description the "EFT bundle." In Section 4.2, the EFT bundle was a nontrivial 𝑆 1𝑞𝛽 bundle over 𝑆 1𝐿 /Z𝑞 . As we saw, the virtue of the EFT bundle is that the thermodynamic limit 𝐿 → ∞ is straightforward: we can dimensionally reduce along the effective thermal circle 𝑆 1𝑞𝛽 without leaving the thermodynamic limit. The theory is then described by thermal EFT with effective inverse temperature 𝑞𝛽 and spatial cycle M 𝐿 /Z𝑞 . Suppose for the moment that the action of 𝑅 on M 𝐿 is free, so that M 𝐿 /Z𝑞 is smooth. (This is the case, for example, for a rational rotation of the spatial circle in 2D.) For any term in the thermal effective action that is the integral of a local density, the effect of the quotient by Z𝑞 is simply to multiply its contribution by 1/𝑞. Thus, we conclude     1 (if the 𝑅 action is free). − log Tr 𝑒 −𝛽𝐻 𝑅 ∼ − log Tr 𝑒 −𝑞𝛽𝐻 + topological 𝑞 (4.29) Here, "∼" denotes agreement to all perturbative orders in the 1/𝐿 expansion. The term "topological" indicates potential contributions from a finite number of terms capable of detecting the topology of the EFT bundle, which cannot be written as the integral of a local gauge/coordinate-invariant density. We discuss such terms in Section 4.6. Let us pause to note that the result (4.29) really only requires that the theory be gapped at inverse temperature 𝑞𝛽 (not necessarily at inverse temperature 𝛽), since we only use locality of the thermal effective action on the right-hand side. • Adding "small" isometries Just as before, we can also consider inserting into the trace an additional "small" isometry 𝑈 = 𝑒𝑖𝛽(𝛼𝑄 𝜉 ) , where 𝜉 is a Killing vector on M 𝐿 , 𝑄 𝜉 is its corresponding 147 charge, and 𝛼 is the corresponding thermodynamic potential. We will be mainly interested in the case where 𝑈 commutes with the discrete isometry 𝑅, so we assume this henceforth. The insertion of 𝑈 can be thought of as a topological defect that wraps M 𝐿 . Consequently, the defect wraps 𝑞 times around the base of the EFT bundle M 𝐿 /Z𝑞 , resulting in an effective rotation 𝑈 𝑞 . We conclude that 1 − log Tr [𝑔𝑅] ∼ − log Tr [𝑔 𝑞 ] + topological 𝑞 (if the 𝑅 action is free), (4.30) where 𝑔 = 𝑒 −𝛽𝐻 𝑈. In fact, this argument applies to any global symmetry element 𝑉 as well, so (4.30) holds when 𝑔 is multiplied by a global symmetry group element: 𝑔 = 𝑒 −𝛽𝐻 𝑈𝑉. We can think of 𝑉 as implementing a nontrivial flat connection for a background gauge field coupled to the global symmetry. In this case, the "topological" terms in (4.30) could include contributions from nontrivial topology of this connection. We can also understand the insertion of "small" isometries geometrically. Again, the idea is to view the mapping torus 𝑀 𝛽,𝑈 𝑅 as the result of two successive quotients 𝑀 𝛽,𝑈 𝑅 = ((M 𝐿 × R)/⟨ℎ 𝑞 ⟩)/Z𝑞 = 𝑀𝑞𝛽,𝑈 𝑞 /Z𝑞 , where ℎ : (® 𝑥 , 𝜏) ↦→ (𝑈 𝑅 𝑥®, 𝜏 + 𝛽), (4.31) where Z𝑞 = ⟨ℎ⟩/⟨ℎ 𝑞 ⟩, and we have used (𝑈 𝑅) 𝑞 = 𝑈 𝑞 . On the right-hand side, we have the mapping torus 𝑀𝑞𝛽,𝑈 𝑞 which is described by the thermal effective action at inverse temperature 𝑞𝛽, with small isometries 𝑈 𝑞 turned on. The effect of the Z𝑞 quotient is to multiply the contribution of any integral of a local density by 1/𝑞. This again leads to (4.30).11 The work [10] uses similar ideas to characterize superconformal indices of 4D CFTs near roots of unity. Our novel contribution is to apply these ideas in not-necessarilysupersymmetric, not-necessarily-conformal theories, on general spatial geometries, and also to describe the effects of Kaluza-Klein vortices (see below), which do not appear in superconformal indices. • Non-free actions and Kaluza-Klein vortex defects What happens if the action of 𝑅 is not free? For example, in a 3D QFT on 𝑆 2 × 𝑆 1𝛽 , the action of (−1) 𝐽 (where 𝐽 is the Cartan generator of the rotation group) has fixed 11 When 𝑈 and 𝑅 don’t commute, the same logic works but we have 𝑀 𝑞𝛽, (𝑈𝑅) 𝑞 /Z𝑞 on the 𝑞 right-hand side of (4.31). We can still use thermal EFT, since (𝑈 𝑅) is 𝑂 (𝛽) close to the identity. 148 points at the north and south poles of 𝑆 2 . In this case, the EFT bundle degenerates at the fixed loci of nontrivial elements of Z𝑞 , namely 𝑅, . . . , 𝑅 𝑞−1 ∈ Z𝑞 . After dimensional reduction, these degeneration loci becomes defects 𝔇𝑖 (with 𝑖 labelling the set of defects) in the 𝑑 − 1 dimensional thermal effective theory. We call them "Kaluza-Klein vortex defects" because the KK gauge field 𝐴 has nontrivial holonomy around them, as we explain in Section 4.3. Each defect 𝔇𝑖 contributes to the partition function a coordinate-invariant effective action 𝑆𝔇𝑖 of the background fields 𝑔, 𝐴, 𝜙 in the infinitesimal neighborhood of 𝔇𝑖 . We then have the more general result ∑︁ 1 − log Tr [𝑔𝑅] ∼ − log Tr [𝑔 𝑞 ] + topological + 𝑆𝔇𝑖 . 𝑞 𝔇 (4.32) 𝑖 We conjecture that for generic interacting QFTs, the KK vortex defects will be gapped. (In fact, in this work, we will study several examples of free theories where the appropriate defects are still gapped.) In this case, each 𝑆𝔇𝑖 will be a local functional of 𝑔, 𝐴, 𝜙. In CFTs, the defect actions 𝑆𝔇𝑖 are additionally constrained by Weyl-invariance, just like the bulk terms in the thermal effective action. We will determine the explicit Í form of 𝑆𝔇 := 𝔇𝑖 𝑆𝔇𝑖 in CFTs later in Section 4.3. For now, we simply note that the leading term in the derivative expansion of 𝑆𝔇𝑖 in a CFT is a cosmological constant localized on 𝔇𝑖 :12 ∫ √︃ 𝑑 𝑛𝑖 𝑦 b 𝑔 | 𝔇𝑖 + higher derivatives. (4.33) 𝑆𝔇𝑖 = 𝑎 𝔇𝑖 𝑛𝑖 𝔇𝑖 (𝑞𝛽) Here, we assume that 𝔇𝑖 is 𝑛𝑖 -dimensional, 𝑦 are coordinates on the defect, and b 𝑔 | 𝔇𝑖 denotes the pullback of b 𝑔 = 𝑒 −2𝜙 𝑔 to 𝔇𝑖 . This term behaves like 𝛽−𝑛𝑖 as 𝛽 → 0. In the case 𝑛𝑖 = 0, i.e. when 𝔇𝑖 is point-like (for example the north/south poles of 𝑆 2 ), the "cosmological constant" becomes simply a constant. • Example: CFT in general 𝑑 As an example application, consider a 𝑑-dimensional CFT on 𝑆 𝑑−1 × 𝑆 1𝛽 . Although our discussion so far has been somewhat abstract, and we have used only basic geometry and principles of EFT, our conclusion (4.32) makes powerful predictions 12 𝑆 𝔇𝑖 itself can also have topological terms; if the topological term has no derivatives (i.e. the Wilson line of the KK photon), it will contribute at the same order in 𝛽 as the defect cosmological constant. 149 about CFT spectra. For example, to leading order as 𝛽 → 0, the defect term 𝑆𝔇 does not contribute, so very generally we obtain a higher dimensional generalization of (4.23), ® ® log Tr[𝑒 −𝛽(𝐻−𝑖Ω·𝐽) 𝑅] ∼ vol𝑆 𝑑−1 1 𝑓 +..., Î 𝑛 𝑑 𝑑−1 2 𝑞 𝛽 𝑖=1 (1 + Ω𝑖 ) (4.34) valid for any element 𝑅 of the Cartan subgroup of SO(𝑑) with order 𝑞. For example, the relative density of even- and odd-spin operators (with respect to any Cartan generator) grows exponentially at a rate precisely 1/2𝑑 times the rate for the un-weighted density of states. Unlike in 𝑑 = 2, the thermal effective action in 𝑑 > 2 can have more than just a cosmological constant term. Consequently, the ". . ." in (4.34) includes higherderivative corrections (in addition to possible vortex defect contributions). However, these higher-derivative corrections can be predicted in the same way: they differ from the un-spin-refined case by replacing 𝛽 → 𝑞𝛽 and multiplying by 1/𝑞 to account for the smaller spatial manifold. The results (4.34) and (4.32) display an important difference between partition functions weighted by spacetime symmetries and partition functions weighted by global symmetries. If we replace 𝑅 with a global symmetry element, this corresponds to turning on new background gauge fields in the thermal effective action, whose contributions are captured by Wilson coefficients that are not active when the global symmetry generators are turned off. For example, the density of states weighted by a global symmetry generator 𝑈 is controlled by a 𝑈-dependent free energy density 𝑓𝑈 with no (obvious) relation to 𝑓 when 𝑈 ≠ 1 (see e.g. [102, 128]). By contrast, the density of states weighted by different discrete spacetime symmetries are all controlled by the same 𝑓 (and the same higher Wilson coefficients like 𝑐 1 , 𝑐 2 , . . . ), in a predictable way. Finally, let us describe the possible discrete rotations 𝑅 for which (4.34) applies. ®® Let us write 𝑅 = 𝑒𝑖 𝜃·𝐽 . In order for 𝑅 to have finite order 𝑞, we must have 𝜃® = 2𝜋( 𝑞𝑝11 , . . . , 𝑞𝑝𝑛𝑛 ), where the 𝑝𝑖 /𝑞𝑖 are rational numbers (which we assume are in reduced form, so that 𝑝𝑖 and 𝑞𝑖 are relatively prime). The order of 𝑅 is 𝑞 = lcm(𝑞 1 , . . . , 𝑞 𝑛 ). When 𝑑 is even, the action of Z𝑞 is free if all 𝑞𝑖 = 𝑞. In this case, the quotient 𝑆 𝑑−1 /Z𝑞 is a lens space 𝐿(𝑞; 𝑝 1 , . . . , 𝑝 𝑛 ), and there are no vortex defects 𝔇. If instead there exists at least one 𝑞𝑖 ≠ 𝑞, then the group element 𝑅 𝑞𝑖 will have a 150 fixed locus 𝑆 2𝑘−1 , where 𝑘 is the number of 𝑞 𝑗 ’s such that 𝑞 𝑗 |𝑞𝑖 , and there will be a corresponding defect 𝔇 at this location (or rather its image after quotienting by Z𝑞 ). Note that it is possible for fixed loci to intersect, creating higher codimension defects. For example, if 𝜃® = 2𝜋(1, 12 , 31 ), the element 𝑅 2 has a fixed 𝑆 3 , the element 𝑅 3 has its own fixed 𝑆 3 , and the two 𝑆 3 ’s intersect along an 𝑆 1 . Quotienting by Z6 , we obtain a defect localized on 𝑆 3 /Z3 ⊂ 𝑆 5 /Z6 , a defect localized on 𝑆 3 /Z2 ⊂ 𝑆 5 /Z6 , and they intersect along an 𝑆 1 ∈ 𝑆 5 /Z6 . In this case, the thermal effective action will include terms localized on the defects and their intersection. When 𝑑 is odd, any element of SO(𝑑) necessarily has a nontrivial fixed locus, since there is a direction left invariant by the Cartan generators. If the theory has a reflection symmetry, then we can more generally consider 𝑅 ∈ 𝑂 (𝑑). The above arguments continue to hold, essentially unmodified. When 𝑅 includes a reflection, the base of the EFT bundle 𝑆 𝑑−1 /Z𝑞 can be non-orientable. For example, if we take 𝑅 to be the parity operator 𝑅 : 𝑛® → −® 𝑛, then 𝑆 𝑑−1 /Z2 = RP𝑑−1 , which is non-orientable in odd 𝑑. Note that the parity operator acts freely, so in this case we can apply (4.30). 4.3 Kaluza-Klein vortex defects In this section, we explore the form of the defect action 𝑆𝔇 that contributes whenever the group generated by the discrete rotation 𝑅 does not act freely. For simplicity, we will restrict our attention to CFT’s in 𝑑-dimensions on a spatial sphere 𝑆 𝑑−1 . Background fields and EFT gauge As before, we wish to compute the partition function of a CFT on the geometry 𝑀𝑞𝛽,𝑈 𝑞 /Z𝑞 , where the mapping torus in the numerator is 𝑀𝑞𝛽,𝑈 𝑞 = (𝑆 𝑑−1 × R)/⟨ℎ 𝑞 ⟩, the group in the denominator is Z𝑞 = ⟨ℎ⟩/⟨ℎ 𝑞 ⟩, and the action of ℎ is given by 2 𝜋 𝑝𝑎 ℎ : ( 𝑛®, 𝜏) ↦→ (𝑈 𝑅 𝑛®, 𝜏 + 𝛽) = (𝑒𝑖( 𝑞𝑎 +𝛽Ω𝑎 )𝐽 𝑛®, 𝜏 + 𝛽). 𝑎 (4.35) First, let us be more precise about the form of the background fields in this geometry. Following [25], we use radius-angle coordinates on the sphere 𝑆 𝑑−1 . These are given by a pair of radius and angle {𝑟 𝑎 , 𝜃 𝑎 } for each orthogonal 2-plane (𝑎 = 1, . . . , 𝑛 = ⌊ 𝑑2 ⌋). If 𝑑 is odd, we have an additional radial coordinate 𝑟 𝑛+1 . Together, the radii Í 2 satisfy the constraint 𝑛+𝜖 𝑎=1 𝑟 𝑎 = 1, where 𝜖 = 0 in even 𝑑 and 𝜖 = 1 in odd 𝑑. To write the metric on 𝑀𝑞𝛽,𝑈 𝑞 in Kaluza-Klein form, we switch to co-rotating 151 coordinates 𝜑𝑎 ≡ 𝜃 𝑎 − Ω𝑎 𝜏, (4.36) where 𝜏 is the coordinate on R. In co-rotating coordinates, the action of ℎ simplifies to ℎ : (𝑟 𝑎 , 𝜑𝑎 , 𝜏) ↦→ (𝑟 𝑎 , 𝜑𝑎 + 2𝜋𝑞 𝑎𝑝 𝑎 , 𝜏 + 𝛽). (4.37) In particular ℎ 𝑞 , becomes simply a shift ℎ 𝑞 : 𝜏 ↦→ 𝜏 + 𝑞𝛽. Thus, quotienting by ⟨ℎ 𝑞 ⟩ to obtain 𝑀𝑞𝛽,𝑈 𝑞 makes 𝜏 periodic with period 𝑞𝛽. The metric of 𝑀𝑞𝛽,𝑈 𝑞 in co-rotating coordinates takes the Kaluza-Klein form 𝑑𝑠2 = 𝑔 + 𝑒 2𝜙 (𝑑𝜏 + 𝐴) 2 , (4.38) where the fields 𝑔, 𝐴, 𝜙 are given by [25] 𝑒 2𝜙 = 1 + 𝑛 ∑︁ 𝑟 𝑎2 Ω2𝑎 , (4.39) 𝑟 𝑎2 Ω𝑎 Í 2 2 𝑑𝜑𝑎 , 1 + 𝑏 𝑟 𝑏 Ω𝑏 𝑎=1 (4.40) 𝑎=1 𝐴= 𝑛 ∑︁ ! 2𝑟 2 Ω Ω 𝑟 𝑎 𝑏 𝑎 𝑏 𝑔= 𝑑𝑟 𝑎2 + 𝑟 𝑎 𝑟 𝑏 𝛿𝑎𝑏 − Í𝑛 2 2 𝑑𝜑𝑎 𝑑𝜑 𝑏 . 1 + 𝑐=1 𝑟 𝑐 Ω𝑐 𝑎=1 𝑎,𝑏=1 𝑛+𝜖 ∑︁ 𝑛 ∑︁ (4.41) The metric of the EFT bundle 𝑀𝑞𝛽,𝑈 𝑞 /Z𝑞 is locally the same as (4.38). Consequently, we can choose a local trivialization of the EFT bundle such that the fields 𝑔, 𝐴, 𝜙 are identical to (4.39) in each patch. However, such a local trivialization will have nontrivial transition functions between patches that contribute to holonomies of the Kaluza-Klein connection along various cycles (including around the defect locus). If we like, we can perform a gauge transformation that makes the transition functions trivial, at the cost of introducing new contributions to 𝐴. We refer to such a gauge as "EFT gauge" because it will be convenient for discussing the EFT limit of the CFT on this geometry. In EFT gauge, the curvature 𝐹 = 𝑑𝐴 has 𝛿-function type singularities at the fixed-loci of 𝑅 whose coefficients reflect the topology of the EFT bundle. 152 • Example: 2D CFT Let us illustrate these ideas with an example. Consider a 2D CFT, where the action of ℎ is given by ℎ : (𝜑, 𝜏) ↦→ (𝜑 + 2𝜋𝑞 𝑝 , 𝜏 + 𝛽). The metric on 𝑀𝑞𝛽,𝑈 𝑞 is 𝑑𝑠2 = 𝑑𝜏 2 + 𝑑𝜃 2 = 𝑑𝜏 2 + (𝑑𝜑 + Ω𝑑𝜏) 2 2 2 2 2 Ω Ω = (1 − 1+Ω 2 )𝑑𝜑 + (1 + Ω ) (𝑑𝜏 + 1+Ω2 𝑑𝜑 ) . | {z } | {z } | {z } 𝑒 2𝜙 𝑔 (4.42) 𝐴 To choose a local trivialization of the EFT bundle, we first specify two intervals in the 𝜑 coordinate: 𝐼1 = {𝜑 : 0 < 𝜑 < 2𝜋 𝑞 }, 𝐼2 = {𝜑 : −𝜖 < 𝜑 < 𝜖 }, (4.43) with 0 < 𝜖 < 𝜋𝑞 . We denote their images in 𝑆 1 /Z𝑞 by 𝑈1 and 𝑈2 , respectively. Together 𝑈1 and 𝑈2 cover the quotient space 𝑆 1 /Z𝑞 ; see Figure 4.3. 𝜑 ∈ 𝑆1 𝐼2 ( ( ) −𝜖 0 𝜖 𝐼1 ) 2𝜋 𝑞 𝑈1 [𝜑] ∈𝑆 1 /Z𝑞 ( ) ( ) 𝑈2 Figure 4.3: The open intervals 𝐼1 and 𝐼2 are subsets of 𝑆 1 . Their images under the quotient map 𝑆 1 → 𝑆 1 /Z𝑞 are 𝑈1 and 𝑈2 , respectively, which together cover 𝑆 1 /Z𝑞 . We can choose a gauge where the KK fields 𝑔, 𝐴, 𝜙 are given by (4.42) in each of 𝑈1 and 𝑈2 . However, in this gauge, there will be a nontrivial transition function between 𝑈1 and 𝑈2 . The bundle projection 𝜋 : 𝑀𝑞𝛽,𝑈 𝑞 /⟨ℎ⟩ → 𝑆 1 /Z𝑞 acts by 𝜋 : [(𝜑, 𝜏)] ↦→ [𝜑], where [(𝜑, 𝜏)] denotes an equivalence class modulo the action of ℎ, and [𝜑] denotes an equivalence class modulo 2𝜋𝑞 𝑝 . Over each open set 𝑈1 , 𝑈2 , we must define trivialization maps 𝜙𝑈𝑖 : 𝜋 −1 (𝑈𝑖 ) → 𝑈𝑖 × 𝑆 1𝑞𝛽 . (4.44) We choose them as follows. Given 𝑝 ∈ 𝜋 −1 (𝑈𝑖 ), thought of as an equivalence class modulo ⟨ℎ⟩, let (𝜑, 𝜏) be a representative of the equivalence class such that 𝜑 is contained in 𝐼𝑖 . Then we define 𝜙𝑈𝑖 ( 𝑝) = ([𝜑], 𝜏). (4.45) 153 Note that 𝜏 is well-defined modulo 𝑞𝛽 because the only elements of ⟨ℎ⟩ that map the 𝐼𝑖 to themselves are powers of ℎ 𝑞 . With this local trivialization, the fields 𝑔, 𝐴, 𝜙 are Ω given by (4.42) in each patch. In particular, we have 𝐴 = 1+Ω 2 𝑑𝜑 in both patches. However, the data of the Kaluza-Klein connection includes both the value of 𝐴 in each patch, as well as the transition functions between patches. We must also determine these transition functions. There are two overlap regions to consider. The first is (0, 𝜖). In this region, the transition function is trivial. The second overlap region is the image of (−𝜖, 0) ⊂ 𝐼2 2𝜋 1 in 𝑆 1 /Z𝑞 , which coincides with the image of ( 2𝜋 𝑞 − 𝜖, 𝑞 ) ⊂ 𝐼1 in 𝑆 /Z𝑞 . Note that −1 (𝜑, 𝜏) for 𝜑 ∈ (−𝜖, 0) is equivalent modulo ⟨ℎ⟩ to (𝜑 + 2𝜋 𝑞 , 𝜏 + ( 𝑝 ) 𝑞 𝛽), where 2𝜋 2𝜋 −1 𝜑 + 2𝜋 𝑞 ∈ ( 𝑞 − 𝜖, 𝑞 ). Here, ( 𝑝 ) 𝑞 denotes the inverse of 𝑝 mod 𝑞, i.e. it satisfies 𝑝( 𝑝 −1 ) 𝑞 = 𝑞𝑛 + 1 for some integer 𝑛. Thus, the transition function in this second overlap region is −1 𝜙𝑈2 ◦ 𝜙𝑈 : ([𝜑], 𝜏) ↦→ ([𝜑], 𝜏 − ( 𝑝 −1 ) 𝑞 𝛽). 1 (4.46) The holonomy of the connection gets a nontrivial contribution from the transition functions:13 ∮ 2𝜋 Ω − 𝐴=− − ( 𝑝 −1 ) 𝑞 𝛽. (4.47) 𝑞 1 + Ω2 (The holonomy is an example of the "topological" terms discussed in Section 4.6 which can contribute in the thermal effective action, but are not the integral of a local gauge/coordinate invariant density.) To go to EFT gauge, we perform a gauge transformation (i.e. a 𝜑-dependent redefinition of 𝜏) that trivializes the transition functions. One possible choice is k j 𝜑 . (4.48) 𝜏′ = 𝜏 − ( 𝑝 −1 ) 𝑞 𝛽 2𝜋/𝑞 j k 𝜑 Note that the function 2𝜋/𝑞 is multi-valued on the entire circle 𝑆 1 , but there is no problem defining it inside the intervals 𝐼1 , 𝐼2 where we perform the gauge transformation. In terms of 𝜏′, the transition functions are now trivial in both overlap regions. (A quick way to see why is to note that 𝜏′ is invariant under the ℎ-action (4.37).) The gauge field becomes Ω 𝐴′ = 𝐴 + 𝑑𝜏 − 𝑑𝜏′ = 𝑑𝜑 + ( 𝑝 −1 ) 𝑞 𝛽 𝛿(𝜑)𝑑𝜑 (EFT gauge). (4.49) 1 + Ω2 ∮ The holonomy − 𝐴′ is still given by (4.47), but that is now manifest in the local expression for the gauge field (4.49). 13 The parallel transport equation is 𝑑𝜏 + 𝐴 = 0, so the holonomy of 𝜏 is computed by − ∮ 𝐴. 154 • Example: 3D CFT Now consider the same setup in a 3D CFT. The metric on 𝑆 2 is 𝑑𝑠2𝑆2 = 𝑑𝑟 12 + 𝑑𝑟 22 + 𝑟 12 𝑑𝜑2 (𝑟 12 + 𝑟 22 = 1). (4.50) Essentially all of the above discussion goes through un-modified, with the radii 𝑟 1 , 𝑟 2 coming along for the ride. We can again go to EFT gauge, and the gauge field (4.49) now gets interpreted as a gauge field on 𝑆 2 . This time, 𝐴 has 𝛿-function-localized curvature at the north and south poles: 𝑑𝐴 = ∓( 𝑝 −1 ) 𝑞 𝛽𝛿( 𝑛®, 𝑛®± )𝑑 2 𝑛® + nonsingular, (4.51) where 𝛿( 𝑛®, 𝑛®′) represents a 𝛿-function on 𝑆 2 , and 𝑛®± are the north/south poles. • EFT gauge in general More generally, we go to EFT gauge as follows. First choose a fundamental domain 𝐹 for the quotient map 𝑆 𝑑−1 → 𝑆 𝑑−1 /Z𝑞 . Define an integer valued function 𝑘 ( 𝑛®) by 𝑘 ( 𝑛®) = 0 if 𝑛® ∈ 𝐹, 𝑘 (𝑅 𝑛®) = 1 + 𝑘 ( 𝑛®). (4.52) In words, 𝑘 ( 𝑛®) counts the power of 𝑅 needed to move from somewhere in 𝐹 to 𝑛®. Again, 𝑘 ( 𝑛®) is multi-valued if we try to define it on the entire sphere, but we only need to define it inside a collection of open sets that cover 𝑆 𝑑−1 /Z𝑞 . For example, 𝜑 ⌋. in the 2D case considered above, we had 𝑘 (𝜑) = ( 𝑝 −1 ) 𝑞 ⌊ 2𝜋/𝑞 Finally, we define 𝜏′ ≡ 𝜏 − 𝛽𝑘 ( 𝑛® (𝑟 𝑎 , 𝜑𝑎 )). (4.53) In the coordinates (𝑟 𝑎 , 𝜑𝑎 , 𝜏′), ℎ acts simply by shifting angles 𝜑𝑎 : ℎ : (𝑟 𝑎 , 𝜑𝑎 , 𝜏′) ↦→ (𝑟 𝑎 , 𝜑𝑎 + 2𝜋𝑞 𝑎𝑝 𝑎 , 𝜏′). (4.54) Consequently, a local trivialization of the EFT bundle defined using the 𝜏′ coordinate has trivial transition functions. The gauge field is given by ′ ′ 𝐴 = 𝐴 + 𝑑𝜏 − 𝑑𝜏 = 𝑟 𝑎2 Ω𝑎 𝑑𝜑𝑎 + 𝛽𝑑𝑘 ( 𝑛® (𝑟 𝑎 , 𝜑𝑎 )). Í 1 + 𝑏 𝑟 𝑏2 Ω2𝑏 𝑎=1 𝑛 ∑︁ (4.55) The curvature 𝑑𝐴′ has 𝛿-function contributions 𝛽𝑑 2 𝑘 ( 𝑛®) at the fixed loci of powers of 𝑅. 155 Effective action Following the logic of the thermal effective action, let us now equip the EFT bundle with a more general metric 𝐺 and try to write down a local action of 𝐺. We will demand that 𝐺 satisfy the following conditions: • It possesses a circle isometry, so that it can be written in Kaluza-Klein form (4.38). • In EFT gauge, the curvature 𝑑𝐴 is a sum of 𝛿-function singularities of the form 𝛽𝑑 2 𝑘 ( 𝑛®), plus something smooth on 𝑆 𝑑−1 /Z𝑞 . (This ensures that the Kaluza-Klein bundle has the same topology as 𝑀𝑞𝛽,𝑉 𝑞 /Z𝑞 .) • 𝑔 and 𝑒 2𝜙 should be smooth on 𝑆 𝑑−1 /Z𝑞 . Here, a field is "smooth on 𝑆 𝑑−1 /Z𝑞 " if it lifts to a smooth Z𝑞 -invariant field on 𝑆 𝑑−1 . In the limit 𝛽 → 0, we can separate each of the background fields into a longwavelength part, with wavelengths much longer than 𝛽, and a short-wavelength part, with wavelengths comparable to (or smaller than) 𝛽. The long-wavelength parts become background fields for the thermal EFT. Meanwhile, the short-wavelength parts become operator insertions in that EFT. In our case, the 𝛿-function curvature singularities 𝑑𝐴 ∼ 𝛽𝑑 2 𝑘 ( 𝑛®) are short-wavelength. They determine the insertion of an operator in the thermal EFT, which is described by the defect action 𝑆𝔇 . This action is a functional of the long-wavelength parts of 𝑔, 𝐴, 𝜙. As mentioned in Section 4.2, we will assume that the defect is gapped, so that the action functional is local and can be organized in a derivative expansion. To construct it, we should compute curvatures and other invariants of 𝑔, 𝐴, 𝜙, and throw away 𝛿-function singularities. Since 𝑔 and 𝜙 are smooth, this effectively amounts to the replacement e ≡ 𝑑𝐴 − 𝛽𝑑 2 𝑘 ( 𝑛®). 𝑑𝐴 → 𝑑 𝐴 (4.56) Henceforth, we leave this replacement implicit. In other words, when we write 𝑑𝐴 e with 𝛿-functions thrown in the defect action, we mean its long-wavelength part 𝑑 𝐴, away. The defects live at singularities in the quotient space 𝑆 𝑑−1 /Z𝑞 . How should we write an action for long wavelength fields near these singularities? Recall that the long-wavelength parts of 𝑔, 𝐴, 𝜙 lift to smooth Z𝑞 -invariant fields on 𝑆 𝑑−1 . We will 156 write 𝑆𝔇 as a functional of these Z𝑞 -invariant lifts, integrated over the preimage of e We also conventionally divide the defect locus modulo Z𝑞 , which we denote by 𝔇. by 𝑞, which ensures that Wilson coefficients of defects living at singularities with the same local structure (but possibly different global structure) are the same. The action should be invariant under gauge/coordinate transformations that preserve the defect locus. For now, we ignore the possibility of nontrivial Weyl anomalies on the defect 𝔇, and we impose that 𝑆𝔇 be Weyl-invariant as well. Consequently, it will be a functional of 𝐴 and the Weyl-invariant combination b 𝑔 = 𝑒 −2𝜙 𝑔. e is the fixed locus of an Consider an 𝑛-dimensional defect 𝔇 whose preimage 𝔇 e we can choose a vielbein element 𝑅 𝑙 ∈ ⟨𝑅⟩ with order 𝑚. Given a point 𝑝 on 𝔇, b 𝑒𝑖𝑎 at 𝑝 satisfying 𝛿𝑎𝑏 b 𝑒𝑖𝑎 b 𝑒 𝑏𝑗 = b 𝑔𝑖 𝑗 , where 𝑎, 𝑏 are indices for the local rotation group 𝑙 SO(𝑑 − 1). The group ⟨𝑅 ⟩  Z𝑚 acts as a subgroup of the local rotation group SO(𝑑 − 1), so the b 𝑒𝑖𝑎 can be classified into representations of this Z𝑚 . Singlets under Z𝑚 represent directions parallel to the defect. They are acted upon by an SO(𝑛) ⊂ SO(𝑑 − 1) that commutes with Z𝑚 . Hence, altogether the b 𝑒𝑖𝑎 can be classified into representations of Z𝑚 × SO(𝑛). To build the defect action, we enumerate curvature tensors built from b 𝑔 and 𝐴, in a 𝑎 derivative expansion, and contract them with b 𝑒𝑖 to build Z𝑚 × SO(𝑛) invariants 𝐼𝑖 with 𝑑𝑖 derivatives. The defect action is then ! ∫ √︃ ∑︁ 1 𝑆𝔇 = 𝑔 |𝔇 𝑎𝑖 (𝑞𝛽) 𝑑𝑖 −𝑛 𝐼𝑖 , (4.57) 𝑑𝑛 𝑦 b e 𝑞 𝔇 e 𝑖 e and 𝑦 are coordinates on the defect. The where b 𝑔 |𝔇 𝑔 to 𝔇, e denotes the pullback of b factors of 𝑞𝛽 are supplied using dimensional analysis. Finally, to evaluate the defect action on 𝑀𝑞𝛽,𝑉 𝑞 /Z𝑞 , we simply plug in the expressions (4.39), (4.40), (4.41), which are precisely the Z𝑞 -lifts to 𝑆 𝑑−1 of the long-wavelength parts of 𝑔, 𝐴, 𝜙. In what follows, we will sometimes use the notation 𝔇 to refer to both a defect on e of the defect locus to 𝑆 𝑑−1 . We hope this will not cause 𝑆 𝑑−1 /Z𝑞 and the lift 𝔇 confusion. Example: point-like vortex defects in 3D CFTs As an example, consider a 3D CFT, where 𝑅 acts by the discrete rotation 𝜑 → 𝜑+ 2𝜋𝑞 𝑝 . The action of 𝑅 fixes the north and south poles of 𝑆 2 . Consequently, there are two 157 point-like vortex defects: 𝔇 𝑝/𝑞 located at the north pole, and its orientation reversal 𝔇−𝑝/𝑞 located at the south pole. Let us focus on 𝔇 𝑝/𝑞 . Classifying the vielbein at the north pole into representations of Z𝑞 , we have basis elements b 𝑒𝑖+ , b 𝑒𝑖− with charges +𝑝 and −𝑝, respectively. We normalize them so that b 𝑒 ± · (b 𝑒 ± ) ∗ = 1. To build basic Z𝑞 -invariant curvatures, we begin with tensors b b and b b is the curvature scalar built from ∇𝑖 · · · b ∇𝑗 𝑅 ∇𝑖 · · · b ∇ 𝑗 𝐹𝑘𝑙 , where 𝐹 = 𝑑𝐴, 𝑅 b 𝑔 , and b ∇ denotes a covariant derivative with respect to b 𝑔 . We then contract their 𝑖 indices with b 𝑒 ± in such a way that the total Z𝑞 charge vanishes. The action at each order in a derivative expansion is a polynomial in these basic Z𝑞 -invariants. Note that we cannot build valid terms in the action by multiplying two Z𝑞 -charged b 𝑒𝑖− b b is not admissible. The objects to obtain a Z𝑞 -singlet. For example, (b 𝑒𝑖+ b ∇𝑖 𝑅)(b ∇𝑖 𝑅) b individually vanishes, due to Z𝑞 -invariance. reason is that b 𝑒𝑖+ b ∇𝑖 𝑅 Proceeding in this way, the leading invariants in a derivative expansion are b (★ b𝐹, 𝑅, b𝐹) 2 , . . . , 𝑆𝔇 𝑝/𝑞 ∋ 1, ★ (4.58) b𝐹 = 𝑖b where ★ 𝑒 +𝑘 b 𝑒 −𝑙 𝐹𝑘𝑙 is the Hodge star of 𝐹 in the metric b 𝑔 . Concretely, the action is     1 2 2 b + 𝑎 3,𝑝/𝑞 (★ b𝐹 + (𝑞𝛽) 𝑎 2,𝑝/𝑞 𝑅 b𝐹) + . . . 𝑎 0,𝑝/𝑞 + (𝑞𝛽)𝑎 1,𝑝/𝑞 ★ , 𝑆𝔇 𝑝/𝑞 = 𝔇 𝑝/𝑞 𝑞 (4.59) where (· · · )| 𝔇 𝑝/𝑞 denotes evaluation at the preimage of the defect on 𝑆 2 — in this case the north pole. We have written the Wilson coefficients as 𝑎𝑖,𝑝/𝑞 to emphasize that they depend on the rotation fraction 𝑝/𝑞. In bosonic theories, the 𝑎𝑖,𝑥 are periodic in 𝑥 with period 1, while in fermionic theories, they are periodic in 𝑥 with period 2. At higher orders in derivatives, we can also include laplacians b ∇2 , as well as 𝑞-th powers of charged derivatives (b 𝑒 ± · ∇) 𝑞 . However, note that the background fields on 𝑀𝑞𝛽,𝑈 𝑞 given in (4.39), (4.40), and (4.41) are invariant not only under Z𝑞 , but under the full maximal torus SO(2). Consequently, terms involving charged derivatives (b 𝑒 ± · ∇) 𝑞 will actually vanish on 𝑀𝑞𝛽,𝑈 𝑞 /Z𝑞 , and in practice we only need to keep b b𝐹 and b polynomials in b ∇2𝑘 ★ ∇2𝑘 𝑅. Plugging in the fields on 𝑀𝑞𝛽,𝑈 𝑞 , we find b𝐹 | ± = ±2Ω, ★ b ± = 2 + 10Ω2 , 𝑅| (4.60) 158 where (· · · )| ± denotes the north/south poles of 𝑆 2 . Thus, summing up the contributions from the north and south poles, the total defect contribution to Tr[𝑒 −𝛽𝐻 𝑈 𝑅] is 𝑆𝔇 = 𝑎 0,𝑝/𝑞 + 𝑎 0,−𝑝/𝑞 + 2𝛽Ω(𝑎 1,𝑝/𝑞 − 𝑎 1,−𝑝/𝑞 ) 𝑞   (𝑞𝛽) 2 + ((2 + 10Ω2 ) 𝑎 2,𝑝/𝑞 + 𝑎 2,−𝑝/𝑞 ) + 4Ω2 (𝑎 3,𝑝/𝑞 + 𝑎 3,−𝑝/𝑞 ) + . . . . 𝑞 (4.61) In general, the point-like defect action at each order 𝛽 𝑘 is a polynomial in Ω that is even if 𝑘 is even and odd if 𝑘 is odd. We will verify this structure in several examples below. There is an important distinction between the terms (4.61) arising in the defect action 𝑆𝔇 and the "bulk" terms (4.11). Note that the bulk terms contain poles at Ω𝑎 = ±𝑖. Physically, such poles arise because a great circle 𝑟 𝑎 = 1 of the spinning √︁ 𝑆 𝑑−1 approaches the speed of light as Ω𝑎 → ±𝑖. The measure b 𝑔 becomes singular at the great circle, and the integral over 𝑟 𝑎 cannot be deformed away from the singularity because it is at an endpoint of the integration contour. By contrast, the defects 𝔇±𝑝/𝑞 are located at the north and south poles of 𝑆 2 , where this phenomenon does not occur, and thus their contributions do not have poles at Ω = ±𝑖. In general, the action of a defect 𝔇 on 𝑀𝑞𝛽,𝑉 𝑞 /Z𝑞 can have poles at Ω𝑎 = ±𝑖 if and only if the support of 𝔇 intersects the great circle 𝑟 𝑎 = 1. We will see an example in the next subsection. Example: vortex defects in 4D CFTs Consider now a 4D CFT, where 𝑅 acts by discrete rotations on each of the angles 𝜑1 → 𝜑1 + 2𝜋𝑞1𝑝1 and 𝜑2 → 𝜑2 + 2𝜋𝑞2𝑝2 . If 𝑞 1 ≠ 𝑞 2 , we have two 1-dimensional vortex defects 𝔇 (1) and 𝔇 (2) . The first defect 𝔇 (1) is located at the fixed locus of 𝑅 𝑞1 , which := lcm(𝑞 1 , 𝑞 2 ). The second is given by (𝑟 1 , 𝑟 2 ) = (1, 0) with 𝜑1 ∈ [0, 2𝜋 𝑞 ), where 𝑞 defect 𝔇 (2) is located at the fixed locus of 𝑅 𝑞2 , which is given by (𝑟 1 , 𝑟 2 ) = (0, 1) with 𝜑2 ∈ [0, 2𝜋 𝑞 ). Let us focus on 𝔇 (1) for now. On 𝔇 (1) , the leading term in the effective action is ∫ √︁ a cosmological constant 𝑑𝜙1 b 𝑔 | 𝔇 (1) , as usual. At the first subleading order in a ∫ √︁ derivative expansion, we have the term 𝑑𝜑1 b 𝑔 | 𝔇 (1) 𝑖b 𝑒 +𝑘 b 𝑒 −𝑙 𝐹𝑘𝑙 , which can be written ∫ b𝐹. more simply as ★ The Wilson coefficients of a defect depend only on the geometry of the singularity 159 where the defect lives. To describe this geometry, it is helpful to introduce the co-prime integers 𝑝1 𝑞2 , (𝑞 1 , 𝑞 2 ) 𝑝2 𝑞1 𝑃2 := , (𝑞 1 , 𝑞 2 ) 𝑃1 := 𝑞1 , (𝑞 1 , 𝑞 2 ) 𝑞2 𝑄 2 := , (𝑞 1 , 𝑞 2 ) 𝑄 1 := (4.62) where (𝑞 1 , 𝑞 2 ) is the greatest common divisor of 𝑞 1 , 𝑞 2 . The structure of the singularity at 𝔇 (1) is determined by the action of 𝑅 𝑞1 , which is 𝑅 𝑞1 : 𝜑2 ↦→ 𝜑2 + 2𝜋𝑃2 . 𝑄2 (4.63) Thus, the Wilson coefficients of 𝔇 (1) should depend only on 𝑃2 /𝑄 2 . However, there is a subtlety in fermionic theories: Note that 𝑅 𝑞1 implements a rotation by 2𝜋 𝑝 1 , which is (−1) 𝑝1 𝐹 in fermionic theories. Thus, the Wilson coefficients of 𝔇 (1) can additionally depend on (−1) 𝑝1 in that case. Consequently, we will write the Wilson coefficients of 𝔇 (1) as 𝑎𝑖,𝑃2 /𝑄 2 ,(−1) 𝑝1 to emphasize the data they depend on. (We will see subtleties of a similar flavor in Section 4.4.) Putting everything together, the action 𝑆𝔇 (1) takes the form   ∫ ∫ √︁ 1 𝑎 0,𝑃2 /𝑄 2 ,(−1) 𝑝1 b𝐹 + . . . 𝑔 | 𝔇 + 𝑎 1,𝑃2 /𝑄 2 ,(−1) 𝑝1 ★ 𝑑𝜑1 b 𝑆𝔇 (1) = 𝑞 𝑞𝛽   𝑎 0,𝑃2 /𝑄 2 ,(−1) 𝑝1 2𝜋 = + 2𝑎 1,𝑃2 /𝑄 2 ,(−1) 𝑝1 Ω2 + . . . , (4.64) 𝑞𝛽 𝑞(1 + Ω21 ) where in the second line, we evaluated the action in the background 𝑀𝑞𝛽,𝑈 𝑞 /Z𝑞 . Note that because 𝔇 (1) lives on the great circle 𝑟 1 = 1, its action has poles at Ω1 = ±𝑖. Adding similar terms for 𝔇 (2) , the total defect contribution to Tr[𝑒 −𝛽𝐻 𝑈 𝑅] is ! 2𝜋 𝑎 0,𝑃2 /𝑄 2 ,(−1) 𝑝1 𝑎 0,𝑃1 /𝑄 1 ,(−1) 𝑝2 𝑆𝔇 = 2 + 𝑞 𝛽 1 + Ω21 1 + Ω22 ! (4.65) 4𝜋 Ω2 Ω1 + 𝑎 1,𝑃2 /𝑄 2 ,(−1) 𝑝1 + 𝑎 1,𝑃1 /𝑄 1 ,(−1) 𝑝2 +..., 𝑞 1 + Ω21 1 + Ω22 where ". . . " represents higher-order terms in 𝛽 coming from higher dimension operators in the defect action. 4.4 Fermionic theories In this section, we describe some subtleties associated with partition functions of fermionic theories. Again, for simplicity we mostly restrict our discussion to CFT𝑑 160 on a spatial sphere 𝑆 𝑑−1 , though the final conclusion (4.84) holds in a general QFT. In short, the results (4.30) and (4.32) work in fermionic CFTs as well, but we must take care to keep track of the spin structure of the manifold (in particular whether we have periodic or antiperiodic boundary conditions for fermions around 𝑆 1𝛽 and 𝑆 1𝑞𝛽 ), and we must consider the rotation 𝑅 as an element of Spin(𝑑). Review of 2D Let us first review fermionic CFTs in 2D. In 2D, we need to specify the boundary conditions of the fermions around both the space and time circles. This defines four different fermion partition functions:   𝑐 𝑐 2𝜋𝑖𝜏(𝐿 0 − 24 ) −2𝜋𝑖𝜏(𝐿 0 − 24 ) 𝑍R,+ (𝜏, 𝜏) := TrR 𝑒 𝑒   𝑐 𝑐 𝑍R,− (𝜏, 𝜏) := TrR (−1) 𝐹 𝑒 2𝜋𝑖𝜏(𝐿 0 − 24 ) 𝑒 −2𝜋𝑖𝜏(𝐿 0 − 24 )   𝑐 𝑐 𝑍NS,+ (𝜏, 𝜏) := TrNS 𝑒 2𝜋𝑖𝜏(𝐿 0 − 24 ) 𝑒 −2𝜋𝑖𝜏(𝐿 0 − 24 )   𝑐 𝑐 (4.66) 𝑍NS,− (𝜏, 𝜏) := TrNS (−1) 𝐹 𝑒 2𝜋𝑖𝜏(𝐿 0 − 24 ) 𝑒 −2𝜋𝑖𝜏(𝐿 0 − 24 ) . The partition functions in (4.66) are not independent. The partition functions 𝑍R,+ , 𝑍NS,+ , and 𝑍NS,− are invariant under different subgroups of 𝑆𝐿(2, Z) and can transform into each other. More precisely, 𝑍R,− is invariant under all of 𝑆𝐿(2, Z); and 𝑍R,+ , 𝑍NS,+ , and 𝑍NS,− are invariant under the congruence subgroups Γ0 (2), Γ𝜃 , and Γ0 (2) respectively, which are defined as ( ! ) 𝑎 𝑏 Γ0 (2) = ∈ 𝑆𝐿(2, Z), 𝑐 even 𝑐 𝑑 ( ! ) 𝑎 𝑏 Γ𝜃 = ∈ 𝑆𝐿(2, Z), 𝑎 + 𝑏 odd , 𝑐 + 𝑑 odd 𝑐 𝑑 ( ! ) 𝑎 𝑏 Γ0 (2) = ∈ 𝑆𝐿(2, Z), 𝑏 even . (4.67) 𝑐 𝑑 Finally they transformation into each other as 𝑍R,+ (−1/𝜏, −1/𝜏) = 𝑍NS,− (𝜏, 𝜏), 𝑍NS,+ (𝜏 + 1, 𝜏 + 1) = 𝑍NS,− (𝜏, 𝜏). (4.68) The NS sector partition function (with or without a (−1) 𝐹 insertion) at low temperature is well-approximated by the vacuum state (which is a bosonic state), with 𝑐 Casimir energy − 12 : 𝑐 𝛽𝑐 TrNS (𝑒 −𝛽(Δ− 12 ) ) ∼ 𝑒 12 , 𝛽 ≫ 1. (4.69) 161 𝑐 The Ramond sector ground state, in contrast, has a Casimir energy of 𝐸 𝑔𝑠 − 12 where 𝐸 𝑔𝑠 is the Ramond ground-state energy, a non-negative number that is theorydependent.14 Finally, the Ramond ground-state may not necessarily be unique, so we call the degeneracy 𝑁𝑔𝑠 ∈ N. 𝑐 𝑐 TrR (𝑒 −𝛽(Δ− 12 ) ) ∼ 𝑁𝑔𝑠 𝑒 −𝛽(𝐸 𝑔𝑠 − 12 ) , 𝛽 ≫ 1. (4.70) To study the high temperature behavior of the NS-sector partition function with an arbitrary phase 𝑒 2𝜋𝑖 𝑝/𝑞𝐽 inserted (with 0 ≤ 𝑝/𝑞 < 2 and 𝑝, 𝑞 coprime) TrNS (𝑒 −𝛽 [ (Δ− 12 )−𝑖Ω𝐽 ] 𝑒 2𝜋𝑖𝐽 𝑝/𝑞 ), 𝑐 𝛽≪1 (4.71) we can use an 𝑆𝐿 (2, Z) transform. In particular we would like to apply a modular transformation of the form ! 𝑎 𝑏 (4.72) 𝑞 −𝑝 to (4.71). The result crucially depends on the parity of 𝑝, 𝑞. If 𝑝 + 𝑞 is odd, then we can choose (4.72) to be in Γ𝜃 and map the partition function to the NS sector at low temperature. However, if 𝑝 + 𝑞 is even, we map the partition function to the R sector at low temperature instead. We therefore get, for 𝛽 ≪ 1 and 0 ≤ 𝑝/𝑞 < 2: TrNS (𝑒 𝑐 )−𝑖Ω𝐽 ] 2𝜋𝑖𝐽 𝑝/𝑞 −𝛽 [ (Δ− 12 𝑒 )∼𝑒 𝑐 4 𝜋2 𝑞 2 𝛽 (1+Ω2 ) 12 𝑝 + 𝑞 odd, 𝛽 ≪ 1, 4 𝜋2 𝑐 ( 𝑐 −𝐸 𝑔𝑠 ) TrNS (𝑒 −𝛽 [ (Δ− 12 )−𝑖Ω𝐽 ] 𝑒 2𝜋𝑖𝐽 𝑝/𝑞 ) ∼ 𝑁𝑔𝑠 𝑒 𝑞2 𝛽 (1+Ω2 ) 12 𝑝 + 𝑞 even, 𝛽 ≪ 1. (4.73) Equivalently we can always take 0 ≤ 𝑞𝑝 < 1 with the insertion of a (−1) 𝐹 : TrNS (𝑒 𝑐 −𝛽 [ (Δ− 12 )−𝑖Ω𝐽 ] 2𝜋𝑖𝐽 𝑝/𝑞 𝑒 )∼𝑒 4 𝜋2 𝑐 𝑞 2 𝛽 (1+Ω2 ) 12 𝑝 + 𝑞 odd, 𝛽 ≪ 1, 4 𝜋2 𝑐 ( 𝑐 −𝐸 𝑔𝑠 ) TrNS (𝑒 −𝛽 [ (Δ− 12 )−𝑖Ω𝐽 ] 𝑒 2𝜋𝑖𝐽 𝑝/𝑞 ) ∼ 𝑁𝑔𝑠 𝑒 𝑞2 𝛽 (1+Ω2 ) 12 4 𝜋2 𝑐 TrNS ((−1) 𝐹 𝑒 −𝛽 [ (Δ− 12 )−𝑖Ω𝐽 ] 𝑒 2𝜋𝑖𝐽 𝑝/𝑞 ) ∼ 𝑒 𝑞2 𝛽 (1+Ω2 ) 12 𝑐 𝑝 + 𝑞 even, 𝛽 ≪ 1, 𝑝 odd, 𝛽 ≪ 1, 2 4𝜋 𝑐 ( 𝑐 −𝐸 𝑔𝑠 ) TrNS ((−1) 𝐹 𝑒 −𝛽 [ (Δ− 12 )−𝑖Ω𝐽 ] 𝑒 2𝜋𝑖𝐽 𝑝/𝑞 ) ∼ 𝑁𝑔𝑠 𝑒 𝑞2 𝛽 (1+Ω2 ) 12 𝑝 even, 𝛽 ≪ 1. (4.74) We see that in (4.74), there are two real numbers that can determine the behavior of fermionic partition functions: the central charge 𝑐 and the Ramond ground state 14 For supersymmetric theories, 𝐸 𝑐 or below or equal to 12 . 𝑐 𝑔𝑠 = 12 , but for generic fermionic theories, 𝐸 𝑔𝑠 can be above 162 𝑐 energy 𝐸 𝑔𝑠 . Moreover, which of the two high temperature behaviors we get ( 12 or 𝑐 12 − 𝐸 𝑔𝑠 multiplying temperature in the free energy) depends on the parity of 𝑝 and 𝑞. We will see this exact same behavior repeat itself for fermionic theories in higher dimensions in Section 4.4. In 2D, we can also analyze the behavior of the partition function in the Ramond sector. This does not have a direct analog as far as we are aware in higher dimension, but we include it for completeness. By using the same modular transformation properties discussed earlier, we find 4 𝜋2 𝑐 TrR (𝑒 −𝛽 [ (Δ− 12 )−𝑖Ω𝐽 ] 𝑒 2𝜋𝑖𝐽 𝑝/𝑞 ) ∼ 𝑒 𝑞2 𝛽 (1+Ω2 ) 12 𝑐 𝑞 even, 𝛽 ≪ 1, 2 4𝜋 𝑐 ( 𝑐 −𝐸 𝑔𝑠 ) TrR (𝑒 −𝛽 [ (Δ− 12 )−𝑖Ω𝐽 ] 𝑒 2𝜋𝑖𝐽 𝑝/𝑞 ) ∼ 𝑁𝑔𝑠 𝑒 𝑞2 𝛽 (1+Ω2 ) 12 𝑞 odd, 𝛽 ≪ 1, 4 𝜋2 𝑐 ( 𝑐 −𝐸 𝑔𝑠 ) TrR ((−1) 𝐹 𝑒 −𝛽 [ (Δ− 12 )−𝑖Ω𝐽 ] 𝑒 2𝜋𝑖𝐽 𝑝/𝑞 ) ≲ 𝑁𝑔𝑠 𝑒 𝑞2 𝛽 (1+Ω2 ) 12 𝛽 ≪ 1. (4.75) Because the final spin structure (where the fermion is periodic in both space and time directions) is invariant under the full modular group, it has the same universal behavior at high temperature regardless of the phase. In general this spin structure cannot be directly derived from the other three (although there are potentially powerful constraints coming from unitarity, and knowledge of 𝑍R,+ [23]). We write ≲ rather than ∼ in the last line of (4.75) due to possible cancellations between fermionic and bosonic Ramond ground states, which would effectively reduce 𝑁𝑔𝑠 (by an even number) for the final spin structure. Higher 𝑑 Let us now consider fermionic CFTs in 𝑑 > 2. For simplicity let us first consider turning on only a single spin and consider: 𝑝 Tr[𝑒 −𝛽(𝐻−𝑖Ω𝐽) 𝑒 2𝜋𝑖 𝑞 𝐽 ], (4.76) with 𝛽 ≪ 1, Ω ∼ 𝑂 (1), 0 ≤ 𝑝/𝑞 < 2. As discussed before, this can be computed from a path integral of 𝑆 𝑑−1 × 𝑆 1𝛽 , where we insert a defect that rotates the sphere by 2𝜋 𝑝/𝑞. Moreover, due to the fermions in the theory, we need to specify a spin structure on this geometry. In (4.76) we compute the path integral with anti-periodic boundary conditions for the fermion around the 𝑆 1𝛽 . We now imagine the setup as in Figure 4.2 to reduce the setup into a geometry we can obtain a thermal EFT. In particular we need to stack 𝑞 copies of the CFT, and perform a 2𝜋 𝑝 rotation in the spatial direction. The final geometry we get — in addition to the spatial sphere being modded by Z𝑞 and the thermal circle increasing by a factor of 𝑞 — has different 163 periodicity for the fermions about the time circle depending on the parity of 𝑝 + 𝑞. If 𝑝 + 𝑞 is odd, the fermions remain antiperiodic, and we can use the original EFT description for fermionic CFTs on a long, thin cylinder. If 𝑝 + 𝑞 is even, however, we need a new EFT, for fermionic CFTs with periodic boundary conditions on a long, thin cylinder. We now see there are two different thermal EFTs we consider, depending on the periodicity of the fermions around 𝑆 1𝛽 . Each EFT comes with its own set of Wilson coefficients. We write them as 𝑍CFT [𝑆 𝑑−1 × 𝑆 1𝛽, anti-periodic ] = 𝑍gapped [𝑆 𝑑−1 ] ∼ 𝑒 −𝑆th [𝑔𝑖 𝑗 ,𝐴𝑖 ,𝜙] , ˜ 𝑍CFT [𝑆 𝑑−1 × 𝑆 1𝛽, periodic ] = 𝑍˜ gapped [𝑆 𝑑−1 ] ∼ 𝑒 −𝑆th [𝑔𝑖 𝑗 ,𝐴𝑖 ,𝜙] . (4.77) ˜ The two expressions 𝑒 −𝑆th and 𝑒 −𝑆th respectively compute the partition function with and without the insertion of (−1) 𝐹 :   Tr 𝑒 −𝛽(𝐻−𝑖Ω𝐽) ∼ 𝑒 −𝑆th ,   ˜ (4.78) Tr (−1) 𝐹 𝑒 −𝛽(𝐻−𝑖Ω𝐽) ∼ 𝑒 −𝑆th . Let us write the two thermal actions as ∫ 𝑑−1 √︁   𝑑 𝑥® 2b 2 2 𝑆th = b 𝑔 − 𝑓 + 𝑐 1 𝛽 𝑅 + 𝑐 2 𝛽 𝐹 + . . . + 𝑆anom , 𝛽 𝑑−1 ∫ 𝑑−1 √︁   𝑑 𝑥® ˜ ˜ + 𝑐˜1 𝛽2 𝑅 b + 𝑐˜2 𝛽2 𝐹 2 + . . . + 𝑆anom . 𝑆th = b 𝑔 − 𝑓 𝛽 𝑑−1 (4.79) The most general leading term behavior of (4.76) then goes as h log Tr(𝑒 𝑝 −𝛽(𝐻−𝑖Ω𝐽) 2𝜋𝑖 𝑞 𝐽 𝑒  vol 𝑆 𝑑−1   𝑞1𝑑 𝛽𝑓𝑑−1  +..., (1+Ω2 ) ) ∼ 𝑑−1 ˜ vol 𝑆   1𝑑 𝑓𝑑−1 +...,  𝑞 𝛽 (1+Ω2 ) i 𝑝 + 𝑞 odd, (4.80) 𝑝 + 𝑞 even. Note that if we specialize (4.80) to 𝑑 = 2, this is precisely the behavior we get for (4.73), if we identify 2𝜋𝑐 , 12  𝑐 𝑓˜ = 2𝜋 − 𝐸 𝑔𝑠 . 12 𝑓 = (4.81) We can think of the difference between the two sets of Wilson coefficients in (4.79) as the higher-dimensional analog of the "Ramond ground state energy". We can generalize (4.76) to turning on 𝑛 spins and consider i h 𝑝 𝑝𝑛 2𝜋𝑖 1 𝐽 Tr 𝑒 −𝛽(𝐻−𝑖(Ω1 𝐽1 +···+Ω𝑛 𝐽𝑛 )) 𝑒 𝑞1 1 · · · 𝑒 2𝜋𝑖 𝑞𝑛 𝐽𝑛 (4.82) 164 for 𝛽 ≪ 1, Ω𝑖 ∼ 𝑂 (1), 0 ≤ 𝑞𝑝𝑖𝑖 < 2. If we define 𝑞 := lcm(𝑞 1 , . . . , 𝑞 𝑛 ), (4.83) then in our EFT, we will go around the time circle 𝑞 times and go around the spatial Í Í sphere 𝑖 𝑝𝑞𝑖𝑖𝑞 times. Thus depending on if 𝑞 + 𝑖 𝑝𝑞𝑖𝑖𝑞 is odd or even, we get the thermal EFT described by 𝑆th or 𝑆˜th . Finally, all of these results are consistent with (4.32) if we simply interpret 𝑒𝑖𝛽Ω·𝐽 𝑅 as an element of Spin(𝑑) and interpret Tr as imposing periodic boundary conditions for both bosonic and fermionic variable. With this understanding, (4.32) is the general recipe. If we instead interpret Tr as imposing periodic boundary conditions for bosons and antiperiodic boundary conditions for fermions, (4.32) should be modified to ∑︁   1 − log Tr [𝑔𝑅] ∼ − log Tr (−1) (𝑞−1)𝐹 𝑔 𝑞 + topological + 𝑆𝔇𝑖 . (4.84) 𝑞 𝔇 𝑖 We will see explicit examples of this in Section 4.5. 4.5 Free theories We now present several examples involving free fields to show explicitly that (4.30) and (4.32) hold (along with their appropriate generalizations to fermionic theories). We begin in Section 4.5 by checking a massive quantum field theory, namely a massive free boson in 2D. We then consider free CFTs in 3D and 4D. Our main tool for computing partition functions in free CFTs is the plethystic exponential, which we review in Appendix 4.10 (see e.g. [88, 112, 165]). In Section 4.5, we consider various examples involving free scalar fields and free fermions in 3 dimensions. In Section 4.5, we explore few more examples in 4 dimension. We present additional 4D and 6D examples in Appendix 4.10. Lastly, in Section 4.6, we consider 2D CFTs with a local gravitational anomaly. Such theories can include Chern-Simons term in their thermal effective action, which are not integrals of local gauge/coordinate invariant densities. These furnish examples of the "topological" terms in (4.30) and (4.32). Massive free boson in 2D As a first check on our formalism (and to illustrate that the basic ideas do not require conformal symmetry), let us study the partition function of a free scalar with mass 𝑚 165 in 2D. We begin by computing the partition function on a rectangular torus 𝑆 1𝐿 × 𝑆 1𝛽 : 1 log 𝑍 = − Tr log(𝑚 2 + Δ) − 𝑆ct 2  2  2! 1 ∑︁ 2𝜋𝑟 2𝜋𝑠 =− log 𝑚 2 + + − 𝑆ct 2 𝐿 𝛽 2 (𝑟,𝑠)∈Z √︃ 2𝜋𝑟 ∑︁ © 𝛽 𝑚2 + ( 𝐿 )2 ª ® − 𝑆ct , =− log 2 sinh ­­ ® 2 𝑟∈Z ¬ « (4.85) where 𝑆ct is a cosmological constant counterterm, and in the third line we performed Í the sum over 𝑟 and threw away a constant using the fact that 𝑠∈Z 1 = 0 in 𝜁-function regularization. Because the summand is slowly-varying in 𝑟, we can take the thermodynamic limit 𝐿 → ∞ by replacing the sum over 𝑟 with an integral over momentum 𝑘 = 2𝜋𝑟 𝐿 : ! ∫   𝛽 √𝑚 2 + 𝑘 2 √ 𝐿 ∞ 2 2 log 𝑍 ∼ − 𝑑𝑘 log 1 − 𝑒 −𝛽 𝑚 +𝑘 − − 𝑆ct (𝐿 → ∞). 𝜋 0 2 (4.86) The second term in the integrand is UV-divergent, but proportional to 𝐿 𝛽. Hence, it takes the form of a cosmological constant and can be removed with an appropriate choice of 𝑆ct . We will choose 𝑆ct to simply subtract this contribution, which is equivalent to setting the free energy density to zero in flat R2 . We find log 𝑍 ∼ 𝐿 𝑓 (𝛽𝑚) 𝛽 (𝐿 → ∞), (4.87) where minus the effective free energy density is 𝑦 𝑓 (𝑦) = − 𝜋 ∫ ∞  𝑑𝑥 log 1 − 𝑒 0  √ −𝑦 1+𝑥 2    𝜋6  ∼ √︃ 𝑦   2𝜋 𝑒 −𝑦  𝑦 ≪ 1, (4.88) 𝑦 ≫ 1. Note that the limit of 𝑓 (𝑦) as 𝑦 → 0 is consistent with 𝑓 = 2𝜋𝑐 12 with 𝑐 = 1 for the massless free boson in 2D. Before continuing to the twisted partition function, let us make two observations. Firstly, the partition function 𝑍 obeys a form of modular invariance. To formulate it, let us write T2 = R2 /Λ, where the lattice Λ is spanned by basis vectors 𝑒®1 , 𝑒®2 . We can arrange the basis vectors into a matrix 𝐸 ∈ R2×2 whose columns are 𝑒®1 and 𝑒®2 , and consider the partition function as a function of 𝐸. (Above, we studied 166   the case where 𝑒®1 = (𝐿, 0) and 𝑒®2 = (0, 𝛽), and thus 𝐸 = 𝐿0 0𝛽 .) Rotational invariance implies that 𝑍 (𝐸) is invariant under left-multiplication by an orthogonal group element 𝑔 ∈ SO(2). Modular invariance is the statement that 𝑍 (𝐸) is also invariant under an integer change of basis of the lattice Λ, which is equivalent to right-multiplication by 𝛾 ∈ SL(2, Z): 𝑍 (𝐸) = 𝑍 (𝑔𝐸𝛾). (4.89) In other words, 𝑍 is a function on the moduli space SO(2)\R2,2 /SL(2, Z). The second key observation is that in the thermodynamic limit, 𝑍 is unchanged under a small shift of the basis vector that grows as 𝐿 → ∞: 𝑒®1 → 𝑒®1 + 𝛼 𝑒®2 . To see why, note that the corresponding dual basis shifts by b 𝑒2 → b 𝑒 2 − 𝛼b 𝑒 1 (with b 𝑒1 unchanged). Thus, the sum (4.85) changes by shifting 𝑟 → 𝑟 − 𝑠𝛼:  2  2!   ∑︁ 2𝜋(𝑟 − 𝑠𝛼) 2𝜋𝑠 1 𝐿 0 log 𝑚 2 + + − 𝑆ct =− log 𝑍 𝛼𝛽 𝛽 2 𝐿 𝛽 2 (𝑟,𝑠)∈Z 𝐿 ∼ 𝑓 (𝛽𝑚) 𝛽 (𝐿 → ∞). (4.90) In the continuum limit, we can identify the momentum 𝑘 = 2𝜋(𝑟−𝑠𝛼) and rewrite 𝐿 the sum as an integral over 𝑘. The shift 𝑟 → 𝑟 − 𝑠𝛼 becomes immaterial in this approximation. Now let us finally consider the partition function with the insertion of a discrete rotation of the spatial circle 𝑆 1𝐿 by an angle 2𝜋 𝑝/𝑞. This corresponds to the matrix ! ! ! 𝑝𝐿 𝐿 0 𝐿 𝑞 𝑝 𝑞 𝑞 = 𝛾 ′, where 𝛾 ′ = ∈ SL(2, Z). 𝐸′ = −( 𝑝 −1 ) 𝑞 −𝑏 0 𝛽 ( 𝑝 −1 ) 𝑞 𝛽 𝑞𝛽 (4.91) As before ( 𝑝 −1 ) 𝑞 denotes the inverse of 𝑝 modulo 𝑞, and 𝑏 is chosen so that 𝛾 ′ has determinant 1. Applying modular invariance (4.89) and the result (4.90), we find   𝐿 𝐿 0 ′ 𝑞 log 𝑍 (𝐸 ) = log 𝑍 ∼ 2 𝑓 (𝑞𝛽𝑚) (𝐿 → ∞), (4.92) ( 𝑝 −1 )𝑞 𝛽 𝑞𝛽 𝑞 𝛽 consistent with (4.32). Using slightly fancier technology (see e.g. [79]), we can be more precise about what information is thrown away in the "∼" in equation (4.92). We rewrite the partition function in terms of a spectral zeta function log 𝑍 (𝐸) = 1 ′ 𝜁 (0) − 𝑆ct , 2 𝐸 (4.93) 167 =⇒ 2𝛽 𝛽 𝐿 Figure 4.4: Left: the lattice Λ for a rectangular torus 𝑆 1𝐿 × 𝑆 1𝛽 . In the thermodynamic limit 𝐿 → ∞, the sum (4.95) is dominated by 𝜆 ∈ Λshort (depicted in red), which are spaced apart by 𝛽. Right: the lattice after twisting by a spatial rotation by 𝜋. In the limit 𝐿 → ∞, the partition function is dominated by 𝜆 ∈ Λ′short , which are now spaced apart by 2𝛽. where ∫ ∞ 𝑑𝑡 𝑠 h −𝑡 (𝑚 2 +Δ) i 1 𝑡 Tr 𝑒 𝜁 𝐸 (𝑠) = Tr (𝑚 + Δ) = Γ(𝑠) 0 𝑡 ∫ ∞ 1 𝑑𝑡 𝑠 −𝑡𝑚 2 ∑︁ −𝑡 (2𝜋𝐸 −𝑇 𝑟®) 2 = 𝑡 𝑒 𝑒 Γ(𝑠) 0 𝑡 2 𝑟®∈Z ∫ ∞ 1 𝑑𝑡 𝑠 −𝑡𝑚 2 det 𝐸 ∑︁ − 𝜆2 𝑡 𝑒 𝑒 4𝑡 . = Γ(𝑠) 0 𝑡 4𝜋𝑡 𝜆∈Λ  −𝑠 2  (4.94) In the last line, we performed Poisson resummation to obtain a sum over lattice points 𝜆 ∈ Λ. (Recall that Λ is the lattice spanned by the columns of 𝐸.) We can now perform the integral over 𝑡 and plug the result into (4.93). The term 𝜆 = (0, 0) precisely cancels against 𝑆ct , and we find log 𝑍 (𝐸) = 𝑚 det 𝐸 2𝜋 ∑︁ 𝐾1 (𝑚|𝜆|) , |𝜆| (4.95) 𝜆∈Λ−(0,0) where 𝐾1 (𝑥) is a modified Bessel function. Now we can see clearly that in the thermodynamic limit, the lattice vectors 𝜆 ∈ Λ that become longer give an exponentially suppressed contribution to log 𝑍 (since 𝐾1 (𝑟) ∼ 𝑒 −𝑟 for large 𝑟). Indeed, we have log 𝑍 (𝐸) = 𝑚 det 𝐸 2𝜋 ∑︁ 𝜆∈Λshort −(0,0) 𝐾1 (𝑚|𝜆|) + 𝑂 (𝑒 −𝐿𝑚 ) |𝜆| (𝐿 → ∞), (4.96) where Λshort are the lattice vectors that do not get longer in the thermodynamic limit 𝐿 → ∞; see Figure 4.4. Applying the result (4.96) to the rectangular torus 𝑆 1𝐿 × 𝑆 1𝛽 , we find another expression for the effective free energy density: ∞ 𝑦 ∑︁ 1 𝑓 (𝑦) = 𝐾1 (𝑦ℓ), 𝜋 ℓ=1 ℓ (4.97) 168 √ √ Í 2 2 which agrees with (4.88), as we can see by expanding log(1−𝑒 −𝑦 1+𝑥 ) = − ℓ 1ℓ 𝑒 −ℓ𝑦 1+𝑥 and integrating term-by term. The result (4.96) also immediately implies (4.90). Combining this with modular invariance, we find that (4.92) holds up to exponential corrections of the form 𝑒 −𝑚𝐿 . Note that this conclusion relies on finiteness of 𝑚 in the thermodynamic limit. If instead we take 𝑚 → 0, then the thermal theory has a gapless sector, and we do not expect (4.32) to hold. We can also understand (4.92) more directly from (4.96) as follows. When the partition function has a twist by a rational angle 2𝜋𝑞 𝑝 , then a new Λ′short emerges, as depicted in Figure 4.4. The emergent Λ′short looks like Λshort in the un-twisted case, but with the replacement 𝛽 → 𝑞𝛽. It is also straightforward to generalize this analysis to a massive free scalar in 𝑑 dimensions. The torus partition function is15 log 𝑍 (𝐸) = det 𝐸  𝑚  𝑑/2 ∑︁ 𝐾 2𝜋 𝜆∈Λ 𝑑/2 (𝑚|𝜆|) . |𝜆| 𝑑/2 (4.98) This is again consistent with (4.32) (with vanishing topological and defect contributions) through the mechanism depicted in Figure 4.4. 3D CFTs • Free scalar We now turn our attention to partition functions of higher dimensional CFTs on a spatial 𝑆 𝑑−1 . Let us begin by studying the partition function of a free scalar in 3D, with various discrete rotations inserted. The usual KK reduction of a free scalar on a circle possesses a zero mode. However, in order to apply thermal EFT, the KK reduced theory must be gapped. Thus, in order to avoid zero modes, we will study a complex scalar charged under a Z 𝑘 flavor symmetry. We will turn on the Z 𝑘 fugacity as appropriate to eliminate zero modes, and study the thermal EFT description of the resulting twisted partition functions. Note that in a generic interacting CFT (as opposed to a free theory) we would not expect to have zero modes, and it would not be necessary to consider flavor symmetry. Let 𝑅(𝜃) ∈ SO(3) denote a rotation around the 𝑧-axis, and let Ψ denote a reflection in the 𝑧 direction. We will consider insertions of 𝑅(𝜃, 𝑠) = 𝑅(𝜃) ◦ Ψ 𝑠 ∈ 𝑂 (3), where 𝜃 = 2𝜋𝑞 𝑝 . Note that the reflection potential 𝑠 takes values in {0, 1}, since Ψ2 = 1. 15 This is an example of an Epstein 𝜁-function; see e.g. [83]. 169 The partition function is given by h i −𝛽(𝐻−𝑖Ω𝐽) 2 𝜋𝑖ℓ𝑄 𝑘 log Tr 𝑒 𝑒 𝑅(𝜃, 𝑠)  ∑︁ 1  2 𝜋𝑖ℓ𝑛 2 𝜋𝑖ℓ𝑛 𝑒 −𝑛𝛽/2 (1 − 𝑒 −2𝑛𝛽 ) = 𝑒 𝑘 + 𝑒− 𝑘 , 𝑛𝑠 𝑒 −𝑛𝛽 )(1 − 𝑒 −𝑛𝛽 𝑥 𝑛 )(1 − 𝑒 −𝑛𝛽 𝑥 −𝑛 ) 𝑛 (1 − (−1) 𝑛 (4.99) where 𝑥 = 𝑒 2𝜋𝑖 𝑝/𝑞+𝑖𝛽Ω , and 𝑄 is the charge of the scalar under 𝑈 (1) (of which Z 𝑘 is a subgroup). When we turn on a big rotation 𝑅(𝜃, 𝑠) with order 𝑞 and include the global symmetry 2 𝜋𝑖ℓ𝑄 operator 𝑉 = 𝑒 𝑘 , a zero mode will be present on the EFT bundle whenever 𝑉 𝑞 = 1, since 𝑉 wraps 𝑞 times around the base of the EFT bundle — see the discussion around (4.30). Let us consider the case 𝑝/𝑞 = 1/2, corresponding to a rotation by an angle 𝜋, which fixes the north and south poles of 𝑆 2 . If the flavor group were Z2 , then we would necessarily have a zero mode on the EFT bundle. Hence we instead consider Z3 flavor symmetry. The thermal partition function of a free scalar field with non-zero Z3 flavor and small rotation-fugacity has the following high temperature expansion: i h 4 𝜋𝑖𝑄 log Tr 𝑒 −𝛽(𝐻−𝑖Ω𝐽) 𝑒 3 =− 16𝜁 (3) (1 + 2Ω2 ) log 3 (21 + 4Ω2 − 24Ω4 ) 𝛽2 − + + 𝑂 (𝛽4 ). 2 2 2 2 9(1 + Ω ) 𝛽 12(1 + Ω ) 4320(1 + Ω ) (4.100) Now we would like to turn on the fugacity for the big rotation with or without a reflection, leading to absence or presence of non trivial defect action, respectively. Free action: without defect Let us first consider an insertion of 𝑅(𝜋, 1) = 𝑅(𝜋) ◦ Ψ. This is a parity transformation 𝑛® ↦→ −® 𝑛 on 𝑆 2 , and it does not have any fixed points. Thus, the thermal effective action will be free of defects. Concretely, we find h i −𝛽(𝐻−𝑖Ω𝐽) 2 𝜋𝑖𝑄 3 log Tr 𝑒 𝑒 𝑅(𝜋, 1) (4.101) 2𝜁 (3) (1 + 2Ω2 ) log 3 (21 + 4Ω2 − 24Ω4 ) 𝛽2 4 − + + 𝑂 (𝛽 ). =− 9(1 + Ω2 ) 𝛽2 24(1 + Ω2 ) 2160(1 + Ω2 ) Comparing (4.100) with (4.101) we find h i 1 i h 2 𝜋𝑖𝑄 4 𝜋𝑖𝑄 log Tr 𝑒 −𝛽(𝐻−𝑖Ω𝐽) 𝑒 3 𝑅(𝜋, 1) ∼ log Tr 𝑒 −2𝛽(𝐻−𝑖Ω𝐽) 𝑒 3 , 2 so (4.30) holds. (4.102) 170 Non-free action: defect Without the reflection fugacity, the action of 𝑅 has fixed points and thermal EFT predicts a non-trivial 𝑆𝔇 as in (4.32). We find h i 4 𝜋𝑖𝑄 log Tr 𝑒 −𝛽(𝐻−𝑖Ω𝐽) 𝑒 3 𝑅(𝜋, 0) 2𝜁 (3) (7 + 8Ω2 ) log 3 (−69 + 94Ω2 + 156Ω4 ) 𝛽2 =− − + + 𝑂 (𝛽4 ) . 2 2 2 2 9(1 + Ω ) 𝛽 24(1 + Ω ) 2160(1 + Ω ) (4.103) Comparing with (4.100), we obtain i h i h 4 𝜋𝑖𝑄 2 𝜋𝑖𝑄 1 − log Tr 𝑒 −𝛽(𝐻−𝑖Ω𝐽) 𝑒 3 𝑅(𝜋, 0) ∼ − log Tr 𝑒 −2𝛽(𝐻−𝑖Ω𝐽) 𝑒 3 + 𝑆𝔇 , 2 (4.104) where the total defect action 𝑆𝔇 (𝛽, Ω) is given by 𝑆𝔇 (𝛽, Ω) = − log 3 2Ω2 − 1 2 + 𝛽 + 𝑂 (𝛽4 ) . 4 24 (4.105) Note that 𝑆𝔇 has precisely the form predicted in (4.61), with 𝑎 0,1/2 = − log 3 , 4 𝑎 2,1/2 = − 1 , 192 𝑎 3,1/2 = 7 . 384 (4.106) Furthermore, the linear term in 𝛽 vanishes because 𝑎 1,1/2 = 𝑎 1,−1/2 in bosonic theories. More generally, we can consider a rotation 𝑅( 2𝜋𝑞 𝑝 , 0), where 𝑝 and 𝑞 are coprime and 𝑞 is not divisible by 3 (so that there is no zero mode upon dimensional reduction). The leading defect Wilson coefficients in this case satisfy: 𝑞−1 h       1 ∑︁ 1 2𝜋(𝑘+𝑞) 𝑘 1 𝑘 𝑎 0,𝑝/𝑞 + 𝑎 0,−𝑝/𝑞 = − cos 𝜓 + − 𝜓 3 3𝑞 3 3𝑞 3 𝑘=1 sin( 𝜋𝑘 𝑝 ) 2 3      i 𝑘 2 𝑘 + cos 2𝜋(𝑘+2𝑞) 𝜓 + − 𝜓 , 3 3𝑞 3 3𝑞 (4.107) 𝜋𝑘 𝑝 𝑞−1    i 1 ∑︁ cos( 3 ) h 2𝜋(𝑘+𝑞) 2𝜋(𝑘+2𝑞) 𝑎 1,𝑝/𝑞 − 𝑎 1,−𝑝/𝑞 = cos + 2 cos , 3 3 6 𝑘=1 sin( 𝜋𝑘 𝑝 ) 3 3 where 𝜓(𝑧) is the digamma function. (4.108) 171 • Free fermion We can perform a similar exercise with free Dirac fermions. The partition function is given by h i −𝛽𝐻 2 𝜋𝑖ℓ𝑄 𝐹𝑤 𝑘 log Tr 𝑒 𝑒 𝑅(𝜃, 0)(−1) h √   i −𝑛𝛽 𝑛 + √1 −𝑛𝛽 ( √𝑥) 𝑛 + √1   𝑒 ( 𝑥) − 𝑒 𝑛+1 ∑︁ 2 𝜋𝑖ℓ𝑛 2 𝜋𝑖ℓ𝑛 (−1) ( 𝑥) 𝑛 ( 𝑥) 𝑛 𝑒 𝑘 + 𝑒− 𝑘 . = (−1) 𝑛𝑤 √ √ −𝑛𝛽 )(1 − 𝑒 −𝑛𝛽 ( 𝑥) 2𝑛 )(1 − 𝑒 −𝑛𝛽 ( 𝑥) −2𝑛 ) 𝑛 (1 − 𝑒 𝑛 (4.109) Fermions are antiperiodic in the time direction at finite temperature. Hence, when no fugacities corresponding to a big rotation or fermion number are turned on, there is no zero mode contribution to the thermal EFT description of the partition function. In this case, we have the following high temperature behavior:   log Tr 𝑒 −𝛽(𝐻−𝑖Ω𝐽) = 3𝜁 (3) (2 + Ω2 ) log 2 (24 − 4Ω2 − 21Ω4 ) 𝛽2 − + + 𝑂 (𝛽4 ). 2 2 2 2 (1 + Ω ) 𝛽 6(1 + Ω ) 5760(1 + Ω ) (4.110) However, when (−1) 𝐹 and big rotations are turned on, we must be careful about zero modes. If the big rotation is given by 𝑒 2𝜋𝑖 𝑝/𝑞 , then we have various cases depending on the parity of 𝑝, 𝑞 and presence or absence of (−1) 𝐹 . We consider all possible cases in the Table4.1. Let us consider the case when 𝑞 is even. We take 𝑞 = 2 for simplicity. According to table4.1, we do not need a flavor twist to remove zero modes. We compute   log Tr 𝑒 −𝛽(𝐻−𝑖Ω𝐽) 𝑅(±𝜋, 0) 3𝜁 (3) (2 + Ω2 ) log 2 Ω𝛽 (24 − 4Ω2 − 21Ω4 ) 𝛽2 − ∓ + + 𝑂 (𝛽3 ). 4 8(1 + Ω2 ) 𝛽2 12(1 + Ω2 ) 2880(1 + Ω2 ) (4.111)  −𝛽(𝐻−𝑖Ω𝐽)   −𝛽(𝐻−𝑖Ω𝐽)  𝐹 Note that log Tr 𝑒 (−1) 𝑅(𝜋, 0) = log Tr 𝑒 𝑅(−𝜋, 0) . = Comparing (4.110) and (4.111), we verify the appropriate generalizations of (4.32) to fermionic CFT. In particular, see (4.84) and note that as an element of spin group, we have 𝑅 2 (±𝜋, 0) = (−1) 𝐹 . We find     1 − log Tr 𝑒 −𝛽(𝐻−𝑖Ω𝐽) 𝑅(±𝜋, 0) ∼ − log Tr 𝑒 −2𝛽(𝐻−𝑖Ω𝐽) + 𝑆𝔇± , 2 where 𝑆𝔇± is given by 𝑆𝔇± = ∓ Ω𝛽 + 𝑂 (𝛽3 ) . 4 (4.112) (4.113) 172 (−1) 𝐹 ? Rotation Boundary Condition Flavor? Equation 𝑝/𝑞 (−1) 𝑞(𝑤+1) (−1) 𝑝 No (𝑤 = 0) 1/2 (−1) 2 (−1) 1 = −1 No (4.111), (4.112) Yes (𝑤 = 1) 1/2 (−1) 4 (−1) 1 = −1 No (4.111), (4.112) No (𝑤 = 0) 1/3 (−1) 3 (−1) 1 = +1 Z2 (4.115), (4.116) Yes (𝑤 = 1) 1/3 (−1) 6 (−1) 1 = −1 No (4.115), (4.116) No (𝑤 = 0) 2/3 (−1) 3 (−1) 2 = −1 No (4.118), (4.119) Yes (𝑤 = 1) 2/3 (−1) 6 (−1) 2 = +1 Z2 (4.118), (4.119) Table 4.1: Turning on fugacities corresponding to flavor, rotation and fermionic number. 𝑤 = 0, 1 refers to turning off and on the fugacity corresponding to (−1) 𝐹 respectively. The fugacity corresponding to big rotation is 𝑒 2𝜋𝑖 𝑝/𝑞 . The third column lists the effective spin structure on the EFT bundle, ±1 in this column refers to periodic and antiperiodic boundary condition respectively. The fourth column lists whether we need a flavor twist to make sure that there is no zero mode. Note that antiperiodic boundary condition rules out the presence of zero modes. The final column refers to relevant equations in the main text. This has precisely the form predicted in (4.61), with 𝑎 1,1/2 − 𝑎 1,−1/2 = 1 , 8 𝑎 0,1/2 + 𝑎 0,−1/2 = 0 . (4.114) Next we consider the case when 𝑞 is odd. Now 𝑝 can be either even or odd. For example, let us consider 𝑞 = 3 and 𝑝 = 1 or 𝑝 = 2. We introduce a flavor twist according to Table. 4.1. Note that, operationally the insertion of (−1) 𝐹 is same as introducing a Z2 flavor twist. The insertion of both amounts to inserting nothing. In what follows, we choose to insert (−1) 𝐹 to eliminate the zero mode. For 𝑝/𝑞 = 1/3, we find that   log Tr 𝑒 −𝛽(𝐻−𝑖Ω𝐽) (−1) 𝐹 𝑅(2𝜋/3, 0) (10 + 9Ω2 ) log 2 5Ω𝛽 (−568 + 1028Ω2 + 1617Ω4 ) 𝛽2 𝜁 (3) − + √ + + 𝑂 (𝛽3 ). 2 2 2 2 9(1 + Ω ) 𝛽 18(1 + Ω ) 5760(1 + Ω ) 3 3 (4.115) Comparing (4.110) and (4.115) we find that ∼   1   log Tr 𝑒 −𝛽(𝐻−𝑖Ω𝐽) (−1) 𝐹 𝑅(2𝜋/3, 0) ∼ log Tr 𝑒 −3𝛽(𝐻−𝑖Ω𝐽) + 𝑆𝔇 (𝛽, Ω) , 3 (4.116) 173 where the total defect action 𝑆𝔇 (𝛽, Ω) takes the form predicted in (4.61): 𝑆𝔇 (𝛽, Ω) ∼ − 4 log 2 5Ω𝛽 (8 − 21Ω2 ) 𝛽2 + √ − + 𝑂 (𝛽3 ) . 9 72 3 3 (4.117) The 𝑝/𝑞 = 2/3 case can be easily be done using the following identity     log Tr 𝑒 −𝛽(𝐻−𝑖Ω𝐽) 𝑅(4𝜋/3, 0) = log Tr 𝑒 −𝛽(𝐻+𝑖Ω𝐽) (−1) 𝐹 𝑅(2𝜋/3, 0) , (4.118) from which it follows that   1   log Tr 𝑒 −𝛽(𝐻−𝑖Ω𝐽) 𝑅(4𝜋/3, 0) ∼ log Tr 𝑒 −3𝛽(𝐻−𝑖Ω𝐽) + 𝑆𝔇 (𝛽, −Ω) , (4.119) 3 where 𝑆𝔇 (𝛽, Ω) is given by (4.117). More examples with a defect action: 4d CFTs In 4D, the rotation group has two Cartan generators, which we call 𝐽1 and 𝐽2 . Let us consider inserting 𝑅 = exp(2𝜋𝑖 𝑞𝑝11 𝐽1 + 2𝜋𝑖 𝑞𝑝22 𝐽2 ) into the trace. When 𝑞 1 = 𝑞 2 , the action of 𝑅 is free and there will be no vortex defects. However, when 𝑞 1 ≠ 𝑞 2 , we will have two 1-dimensional vortex defects 𝔇 (1) and 𝔇 (2) , whose combined action is given by (4.65). As a concrete example, consider the free Dirac fermion in 4D. Using plethystic exponentials, we find h i i h 𝑝 𝑝 1 ® ® ® ® 2𝜋𝑖 1 𝐽 +2𝜋𝑖 𝑞2 𝐽2 2 − log Tr 𝑒 −𝛽(𝐻−𝑖Ω·𝐽) 𝑒 𝑞1 1 ∼ − log Tr 𝑒 −𝑞𝛽(𝐻−𝑖Ω·𝐽) + 𝑆𝔇 . (4.120) 𝑞   Here we impose that 𝑞 1 + 𝑞𝑝11 + 𝑞𝑝22 is odd. This ensures absence of the (−1) 𝐹 insertion in the right hand side above. Thus the zero mode is absent and the thermal EFT applies. Furthermore, 𝑆𝔇 has precisely the form predicted by (4.65) with the leading Wilson coefficients given by the following function of (−1) 𝑝 and coprime integers 𝑃, 𝑄:    𝑄−1 ∑︁ (−1) 𝑝𝑘 cos(𝜋𝑘 𝑄𝑃 )   1 𝑘 𝑘 ′ ′ (4.121) 𝜓 𝑎 0,𝑃/𝑄,(−1) 𝑝 = + −𝜓 , 2 2𝑄 2𝑄 8𝜋 sin2 (𝜋𝑘 𝑄𝑃 ) 𝑘=1 where 𝜓 ′ (𝑧) is the derivative of the digamma function. This is a highly nontrivial check of (4.65). Matching (4.65) requires not just reproducing the correct function of 𝛽, Ω1 , Ω2 , but also the fact that the Wilson coefficient of 𝔇 (1) depends only on 𝑃2 /𝑄 2 and (−1) 𝑝1 (and analogously for 𝔇 (2) ). Note that when 𝑞 1 = 𝑞 2 , we have 𝑄 1 = 𝑄 2 = 1, hence the sum is non-existent, leading to 𝑎 0,𝑃1 /𝑄 1 ,(−1) 𝑝2 = 𝑎 0,𝑃2 /𝑄 2 ,(−1) 𝑝1 = 0, which is consistent with the fact that the action of 𝑅 becomes free and 𝑆𝔇 should vanish. 174 4.6 Topological terms: example in 2D CFT So far, we have focused on cases where the thermal effective action can be written as the integral of a local gauge- and coordinate-invariant density. As discussed in Section 4.3, we can choose a gauge ("EFT gauge") where the background fields 𝑔, 𝐴, 𝜙 on the EFT bundle look locally the same as on 𝑆 𝑑−1 × 𝑆 1𝛽 (possibly with "small" angular twists turned on). Curvature invariants built out of these fields are then also the same as on 𝑆 𝑑−1 × 𝑆 1𝛽 , and their integral is not sensitive to global properties of the EFT bundle. By contrast, when 𝑑 is even, the thermal effective action can also include ChernSimons-type terms that are not integrals of local curvature invariants, and hence can be sensitive to global properties of the EFT bundle. The contributions of such terms were called "topological" in (4.30) and (4.32). The coefficients of such ChernSimons terms can be determined systematically from the anomaly polynomial of the CFT [124–126, 153, 154, 171]. The simplest case is when 𝑑 = 2, where the thermal effective action contains a 1d Chern Simons term whose coefficient is proportional to the local gravitational anomaly 𝑐 𝐿 − 𝑐 𝑅 . In more detail, consider such a 2D CFT with a local gravitational anomaly, 𝑐 𝐿 ≠ 𝑐 𝑅 . We assume that 𝑐 𝐿 − 𝑐 𝑅 = 24𝑘 with 𝑘 ∈ Z. From modular invariance we have a high-temperature expansion of the partition function as 16  h log Tr 𝑒 𝑝 −𝛽(𝐻−𝑖Ω𝐽) 2𝜋𝑖 𝑞 𝐽 𝑒 i 4𝜋 2 (𝑐 𝐿 + 𝑐 𝑅 − 24𝑖𝑘Ω) 2𝜋𝑖𝑘 ( 𝑝 −1 ) 𝑞 ∼ − , 𝑞 24𝑞 2 (1 + Ω2 ) 𝛽 (4.122) where (4.122) is accurate to all orders in perturbation theory in 𝛽. This generalizes (4.19) to theories with 𝑐 𝐿 ≠ 𝑐 𝑅 . In the thermal effective action, we can reproduce the terms in (4.122) with a ChernSimons term from the KK gauge field in the action. In particular, we add a term of the form ∮ 2𝜋𝑖𝑘 − 𝐴 (4.123) 𝑞𝛽 to the thermal effective action. From (4.47), we see that (4.123) precisely reproduces the additional terms in (4.122). Note that (4.123) is properly quantized precisely when 𝑘 is an integer, since 𝐴 is a connection on a circle bundle where the fiber has circumference 𝑞𝛽. 16 To derive (4.122), we apply the modular transform (4.18) to the vacuum state 𝑒 − 𝑝 with 𝜏 = 2𝑖𝛽𝜋 + 𝛽Ω 2 𝜋 + 𝑞 and expand in small 𝛽. 2 𝜋𝑖 𝜏𝑐 𝐿 −2 𝜋𝑖 𝜏𝑐 𝑅 24 175 4.7 Holographic theories In this section we consider CFTs dual to semiclassical Einstein gravity in AdS. ®® The high temperature behavior of the thermal partition function, Tr[𝑒 −𝛽𝐻+𝑖 𝜃·𝐽 ] around 𝜃 𝑖 = 0, of such holographic CFTs in 𝑑 dimensions are captured by the thermodynamics of black hole solutions in AdS𝑑+1 . We would like to understand ®® the bulk solution that captures the high temperature behavior of Tr[𝑒 −𝛽𝐻+𝑖 𝜃·𝐽 ] near 𝜃 𝑖 = 2𝜋 𝑝𝑖 /𝑞𝑖 , where at least one of the 𝑝𝑖 ≠ 0 and 𝑞𝑖 ⩾ 2. First, let us consider a 𝑑 = 2 dimensional CFT in the context of AdS3 /CFT2 duality. The relevant bulk solution is given by the rotating BTZ black hole with the metric  (𝑟 2 − 𝑟 +2 )(𝑟 2 − 𝑟 −2 ) 𝑑𝑟 2 𝑟 +𝑟 −  2 𝑑𝑠2 = − 𝑓 (𝑟)𝑑𝑡 2 + + 𝑟 2 𝑑𝜙 − 2 𝑑𝑡 , 𝑓 (𝑟) := , 𝑓 (𝑟) 𝑟 𝑟2 (4.124) where 𝑟 ± are radius of outer and inner horizon respectively. The BH temperature 𝛽−1 and angular potential Ω = 𝜃/𝛽 are given by 𝑟− Ω= , 𝑟+ 𝛽 −1 𝑟 +2 − 𝑟 −2 = . 2𝜋𝑟 + (4.125) Asymptotically, the metric is Weyl equivalent to −𝑑𝑡 2 + 𝑑𝜙2 . In Euclidean signature, 𝑡 𝐸 = 𝑖𝑡 and we have (𝜙, 𝑡 𝐸 ) ∼ (𝜙 − 𝛽Ω, 𝑡 𝐸 + 𝛽). The Euclidean action evaluated on this bulk saddle reproduces the high temperature behavior of Tr[𝑒 −𝛽(𝐻−𝑖Ω𝐽) ]. Now let us compute Tr[𝑒 −𝛽(𝐻−𝑖Ω𝐽)+2𝜋𝑖 𝑝/𝑞𝐽 ] for a holographic 2D CFT. As explained in Section 4.2, this can be computed by doing a path integral over 𝑀𝑞𝛽,Ω /Z𝑞 ,17 where the action of Z𝑞 = ⟨ℎ⟩ is given by ℎ : (𝜙, 𝑡 𝐸 ) ↦→ (𝜙 + 2𝜋 𝑝/𝑞 − 𝛽Ω, 𝑡 𝐸 + 𝛽) , (4.126) and 𝑀𝑞𝛽,Ω is obtained from 𝑆 1 × R by quotienting by ⟨ℎ 𝑞 ⟩. Note that the action of Z𝑞 has a natural extension into the AdS bulk, where the radial direction goes along for the ride ℎ : (𝑟, 𝜙, 𝑡 𝐸 ) ↦→ (𝑟, 𝜙 + 2𝜋 𝑝/𝑞 − 𝛽Ω, 𝑡 𝐸 + 𝛽) . (4.127) We can use this natural extension to build a bulk dual solution. We start with a BTZ black hole Σ𝑞𝛽,Ω with parameters 𝑟˜± = 𝑟 ± /𝑞, such that the black hole is at inverse temperature 𝛽˜ = 𝑞𝛽. The Euclidean metric is given by  2 (𝑟 2 − 𝑟˜+2 )(𝑟 2 − 𝑟˜−2 ) 𝑑𝑟 2 𝑟˜+𝑟˜− 2 2 2 ˜ + 𝑟 𝑑𝜙 + 𝑖 2 𝑑𝑡 𝐸 , 𝑓˜(𝑟) := . 𝑑𝑠 = 𝑓 (𝑟)𝑑𝑡 𝐸 + 𝑟 𝑟2 𝑓˜(𝑟) (4.128) 17 In this section, we use the compressed notation 𝑀 𝛽,𝑒𝑖 𝐽Ω → 𝑀 𝛽,Ω . 176 Note that (𝜙, 𝑡 𝐸 ) ∼ (𝜙 − 𝑞𝛽Ω, 𝑡 𝐸 + 𝑞𝛽). We can then quotient Σ𝑞𝛽,Ω by the action of Z𝑞 = ⟨ℎ⟩, given by (4.127). The manifold Σ𝑞𝛽,Ω /Z𝑞 is smooth because the action of ℎ in the bulk (4.127) is free. Recall that on the boundary, the quotient 𝑀𝑞𝛽,Ω /Z𝑞 is a nontrivial bundle over 𝑆 1 /Z𝑞 . The discussion in 4.3 applies here, with the radial direction of AdS going along for the ride. In short, we consider open sets in the total space and consider the bulk region that asymptotes to this set. Locally in this bulk region, the metric is precisely given by (4.128). However, there are non-trivial transition functions when we go from one open patch to a different one. By construction, the quotient Σ𝑞𝛽,Ω /Z𝑞 solves Einstein’s equations with the appropriate asymptotic geometry. Compared to a rotating BTZ black hole at temperature 𝑞𝛽, the above quotient geometry has its angular variable restricted to 0 < 𝜙 < 2𝜋/𝑞. (Note that this almost covers the full manifold except the locus 𝜙 = 0. This locus is covered by another open patch and we have nontrivial transition function between these two patches.) Thus the evaluation of the bulk action amounts to an integral over the angular variable in the range 0 < 𝜙 < 2𝜋/𝑞 producing a factor of 1/𝑞, and a replacement 𝛽 → 𝑞𝛽 which produces another factor of 1/𝑞 compared to the evaluation of Euclidean action on BTZ black hole at temperature 𝛽. Thus, the Euclidean action evaluated on the saddle Σ𝑞𝛽,Ω /Z𝑞 reproduces   log Tr[𝑒 −𝛽(𝐻−𝑖Ω𝐽)+2𝜋𝑖 𝑝/𝑞𝐽 ] ∼ log 𝑍grav Σ𝑞𝛽,Ω /Z𝑞   1 1 = log 𝑍grav Σ𝑞𝛽,Ω ∼ log Tr[𝑒 −𝑞𝛽(𝐻−𝑖Ω𝐽) ], 𝑞 𝑞 (4.129) where 𝑍grav [Σ] is the gravitational path integral evaluated on the saddle Σ. Overall, we have a quotient of a BTZ black hole whose boundary has a temporal cycle of length 𝛽 𝐿 = 𝑞𝛽(1 − 𝑖Ω) and a spatial cycle of length 𝐿 = 2𝜋 𝑞 . This naively leads to the modular parameter e 𝜏 "=" 𝑖𝐿/𝛽 𝐿 = 2𝜋𝑖 𝑞 2 𝛽(1 − 𝑖Ω) . (4.130) The above is almost correct, but we must amend it by recalling the presence of nontrivial transition functions. This leads to the following identification: ( 𝑝 −1 ) 𝑞 2𝜋𝑖 − e 𝜏= 2 , 𝑞 𝑞 𝛽(1 − 𝑖Ω) ( 𝑝 −1 ) 𝑞 2𝜋𝑖 e 𝜏=− 2 − , 𝑞 𝑞 𝛽(1 + 𝑖Ω) (4.131) 177 which precisely matches the modular parameter e 𝜏 obtained after applying a modular transformation, as explained around (4.17). Note that we have ( 𝑝 −1 ) 𝑞 1  e 2𝜋Ω e 𝜏+𝜏 =− − 2 , 2 𝑞 𝑞 𝛽(1 + Ω2 ) (4.132) which is consistent with the holonomy of 𝐴 derived in (4.47). We conclude that Σ𝑞𝛽,Ω /Z𝑞 is simply a modular transformation of the BTZ black hole geometry. In dimensions greater than two, we follow the same prescription. Let Σ 𝛽,Ω® be a black hole solution that captures the high temperature behavior of the thermal partition function with small angular fugacity, i.e. the AdS-Kerr black hole with ® We claim that the insertion of a big inverse temperature 𝛽 and angular velocities 𝑖 Ω. angular fugacity 𝑅 with 𝑅 𝑞 = 1 is captured by the bulk manifold Σ𝑞𝛽,Ω® /Z𝑞 , where Z𝑞 is the natural extension of the boundary Z𝑞 into the bulk. To be precise, the AdS-Kerr solution possesses a time-translation isometry, which we parametrize by 𝑡 𝐸 . It furthermore possesses isometries under the Cartan subgroup of the rotation group, which we parametrize by 𝜙𝑎 . Then the Z𝑞 = ⟨ℎ⟩/⟨ℎ 𝑞 ⟩ group is generated by ℎ : (𝑟, 𝑟 𝑎 , 𝜙𝑎 , 𝑡 𝐸 ) ↦→ (𝑟, 𝑟 𝑎 , 𝜙𝑎 + 2𝜋𝑞 𝑎𝑝 𝑎 , 𝑡 𝐸 + 𝛽), (4.133) where 𝑟 𝑎 and 𝑟 are the remaining bulk coordinates. When the bulk action of ℎ is free, Σ𝑞𝛽,Ω® /Z𝑞 is a smooth solution to Einstein’s equations with the correct asymptotic geometry to describe the partition function with an insertion of 𝑅. At very high temperatures, the gravitational path integral on this geometry matches the field theory prediction (4.30) because the quotient by Z𝑞 just divides the semiclassical gravitational action by 𝑞. Thus we expect Σ𝑞𝛽,Ω® /Z𝑞 is the dominant solution at high temperatures. As we move away from high temperatures, we conjecture that Σ𝑞𝛽,Ω® /Z𝑞 remains the dominant solution down to a finite temperature. Unlike in 2d, in higher dimensions it is possible for the bulk Z𝑞 action (4.133) to have a fixed locus. Such a fixed locus must occur at the horizon of the black hole (where the Euclidean time circle degenerates), and at a location on the sphere 𝑆 𝑑−1 where the boundary action of ℎ has a fixed point as well. For example, consider a 3D CFT, with a four-dimensional bulk dual, and consider the case 𝑝 1 /𝑞 1 = 1/2. The bulk action of ℎ rotates the 𝜙 circle by 𝜋, and also shifts 𝑡 𝐸 → 𝑡 𝐸 + 𝛽 (which is equivalent to rotating the thermal circle by 𝜋). Fixed points of ℎ occur at the north and south poles of the horizon 𝑆 2 . For example, near the north pole, we can choose 178 coordinates so that the metric locally takes the form 𝑑𝑟 12 + 𝑟 12 𝑑𝜙2 + 𝑑𝑦 2 + 𝑦 2 𝑑𝜓 2 , (4.134) √ 2𝜋 where 𝜓 = 2𝛽 𝑡 𝐸 , and 𝑦 ∝ 𝑟 − 𝑟 𝐻 (with 𝑟 𝐻 the location of the horizon). In these coordinates, the Z2 action rotates both angles 𝜙, 𝜓 by 𝜋. Quotienting by this Z2 , we obtain an orbifold singularity of the form C2 /Z2 . When Σ𝑞𝛽,Ω® /Z𝑞 contains an orbifold singularity, it no longer furnishes a smooth solution to Einstein’s equations. To understand the correct bulk dual of the 𝑅twisted partition function, we must understand the fate of the orbifold singularity in the bulk theory. The physics of the singularity (or its resolution) essentially determines the defect action 𝑆𝔇 in holographic CFTs. Note that in a spacetime with vanishing cosmological constant, a C2 /Z2 singularity can be resolved by the "gravitational instanton" described by the Eguchi-Hansen metric. Perhaps orbifold singularities occurring in the Σ𝑞𝛽,Ω® /Z𝑞 can be resolved similarly. Or perhaps the orbifold singularity is resolved by stringy effects. We leave these questions to future work. For fermionic theories dual to Einstein gravity (which all known examples are), there is another set of Wilson coefficients 𝑓˜, 𝑐˜1 , . . . that can be defined (see (4.79)) coming from periodic boundary conditions for the fermion. The behavior of the partition function with (−1) 𝐹 inserted was explored in [62], which found a black hole solution that lead to an exponentially subleading (in temperature) contribution in the large 𝑁 limit. Since generically we expect 𝑓˜ to be nonvanishing, this means the black hole solution found in [62] should not be the dominant contribution in the 𝑇 ≫ 𝑁 ≫ 1 limit. It would be interesting to explore further if there are universal results or constraints on the Wilson coefficients 𝑓˜, 𝑐˜1 , . . . in (fermionic) holographic CFTs (for instance by looking at the theories studied in [31]). Finally, let us note that our construction works also in the case when the boundary theory possesses a reflection symmetry and the group element 𝑅 includes a reflection. In this case, the bulk dual solution Σ𝑞𝛽,Ω® /Z𝑞 is non-orientable. This is allowed because the boundary reflection symmetry must be gauged in the bulk, which means we must include contributions from non-orientable geometries. It so happens that a non-orientable geometry dominates in this case.18 18 See [101] for a recent discussion of gauging spacetime symmetries in the bulk. 179 4.8 Journey to 𝛽 = 0 ® at high So far we have described the structure of CFT partition functions 𝑍 (𝛽, 𝜃) ® 𝜃 temperature near rational angles 2𝜋 ∈ Q𝑛 . In this section we will use these results to sketch the behavior of the partition function at high temperature around any (possibly irrational) angle. For simplicity let us consider turning on only one angle 𝜃, and study the partition function   𝑍 (𝛽, 𝜃) = Tr 𝑒 −𝛽𝐻+𝑖𝐽𝜃 (4.135) at small 𝛽 with 𝜃 fixed. Suppose the angle has the following continued fraction expansion: 𝜃 1 = 𝑎0 + , (4.136) 1 2𝜋 𝑎1 + 1 𝑎 2 + 𝑎 +... 3 𝑝𝑖 𝑞𝑖 where 𝑎𝑖 ∈ N. We also define the fractions as truncations of the continued fraction, i.e. 𝑝𝑖 1 := 𝑎 0 + . (4.137) 𝑞𝑖 𝑎 1 + . . . 𝑎1𝑖 For a generic angle, the probability of 𝑎𝑖 = 𝑛 scales as 𝑛12 for large 𝑛, which means that the distribution of 𝑎𝑖 ’s for a generic angle has infinite mean.19 Suppose we have an angle 𝜃 where 𝑎𝑖 ≫ 1 for some 𝑖. How does the partition function behave at high temperature? Since we assume 𝑎𝑖 ≫ 1, we have 𝑞𝑖 ≈ 𝑎𝑖 𝑞𝑖−1 ≫ 𝑞𝑖−1 . So for 𝑞𝑖−1 ≪ 𝑇 ≪ 𝑞𝑖 , (4.138) we expect −𝑑 log 𝑍 (𝑇) ∼ vol𝑆 𝑑−1 𝑓 𝑇 𝑑−1 𝑞𝑖−1 . (4.139) 𝑇 ∼ 𝑞𝑖 (4.140) However, when the constant or proportionality in (4.139) suddenly shrinks. In Figure 4.5, we illustrate this explicitly for two example CFTs. We plot both the Klein invariant 𝑗 function (which is the partition function of some 2D CFTs at central charge 24) and the 3D free boson partition function as a function of temperature, 𝜃 3−𝜋 with chemical potential 2𝜋 = 7𝜋−22 . This has the continued fraction expansion: 𝜃 3−𝜋 1 = = 15 + . 1 2𝜋 7𝜋 − 22 292 + 1+... (4.141) 19 One way to see this scaling is, for a random real number between 0 and 1, the probability that 1 𝑎 1 = 𝑛 is roughly 𝑛1 − 𝑛+1 ∼ 𝑛12 . 180 8 10 θ 3-π = 2 π 7 π - 22 6 8 6 loglogZ2 d (T, θ) 4 θ 3-π = 2 π 7 π - 22 loglogZ3 d (T, θ) 4 2 2 0 0 2 4 6 log(T) 8 10 0 0 12 1 2 3 4 5 6 7 log(T) (a) 24 free bosons in 2D. (b) A free boson in 3D. Figure 4.5: In blue, the log of the free energy of two CFTs evaluated at chemical 𝜃 3−𝜋 potential 2𝜋 = 7𝜋−22 vs. log 𝑇. In red, a line with slope 𝑑 − 1. (a): 𝑑 = 2. (b): 𝑑 = 3. We see that at some temperature, there is a region where the slope matches 𝑑 − 1, meaning the effective field theory is a good description. If we continue this plot to higher and higher temperatures, there will be infinitely many times the slope matches 𝑑 − 1. 8 2.4 θ 2.2 2π = 1+ 6 5 θ 2 2π loglogZ loglogZ2 d (T, θ) 2.0 1.8 3d = 1+ 5 2 (T, θ) 4 2 1.6 7.0 7.5 8.0 8.5 log(T) 9.0 (a) 24 free bosons in 2D. 9.5 10.0 0 0 1 2 3 4 5 6 7 log(T) (b) A free boson in 3D. Figure 4.6: In blue, the log of the free energy of two CFTs evaluated at chemical √ 1+ 5 𝜃 potential 2𝜋 = 2 vs. log 𝑇. In red, a line with slope 𝑑 − 1. (a): 𝑑 = 2. (b): 𝑑 = 3. We see that at no temperature is there a region where the slope matches 𝑑 − 1, meaning the effective field theory is never a good description. Since we have fine-tuned the chemical potential to be a real number whose continued fraction has no large numbers in it, we do not guarantee any region in temperature where we have a good EFT description of our theory. (Although any generic real number would serve to illustrate the partition function’s behavior, we choose (4.141) for convenience because of the large 292 showing up immediately.) We see that there indeed is a region of 𝑇 where the free energies scale as we predict from the effective field theory; but when we increase 𝑇 further, the scaling breaks down. At large 𝑇 for generic chemical potential, there will be an infinite number of times the plot in Figure 4.5 has slope 𝑑 − 1, which is when the continued fraction approximation is well-approximated by a rational number (i.e. whenever 𝑎𝑖 ≫ 𝑎𝑖−1 in the continued fraction). 181 Finally we note that if we fine-tune the angle 𝜃 so that none of the 𝑎𝑖 ’s ever become large, we can make there be no regime where the partition function obviously grows as 𝑇 𝑑−1 . For example, choosing the angle to be the golden ratio √ 𝜃 1+ 5 1 (4.142) = =1+ 1 2𝜋 2 1 + 1+... gives an angle where there’s never a large enough regime in 𝑇 trust our effective field theory. In Figure 4.6, we again plot the Klein invariant 𝑗 function and the free boson partition functions as a√ function of temperature, but this time with the 𝜃 chemical potential set to 2𝜋 = 1+2 5 . We see that the slope of log log |𝑍 | against log 𝑇 never matches 𝑑 − 1 in a large region, so there is no good EFT description of our system. 4.9 Nonperturbative corrections In this section, we consider nonperturbative corrections to the thermodynamic limit 𝐿 → ∞ at finite 𝛽. For concreteness, we focus on a CFT𝑑 and its dimensional reduction on 𝑆 1𝛽 to a 𝑑−1-dimensional gapped theory. By conformal symmetry, the thermodynamic limit is equivalent to 𝛽 → 0 (with a fixed-size spatial manifold). ® though it would be For simplicity, we will not turn on "small" angular twists 𝛽Ω, straightforward to incorporate them. The thermal effective action essentially captures the dynamics of the ground state of the 𝑑−1 dimensional gapped theory, while nonperturbative corrections come from particle excitations. On the geometry R𝑑−1 × 𝑆 1𝛽 , the excitations can be classified into irreps of the 𝑑−1-dimensional Poincaré group and the Kaluza-Klein 𝑈 (1) that rotates the 𝑆 1𝛽 . Irreps of the Poincaré group are labeled by a mass and a little group representation — for simplicity, we will focus on scalars. Thus, each excitation of interest is labeled by a mass 𝑚𝑖 and a KK charge 𝔮𝑖 ∈ Z. The lightest mass 𝑚 𝔮 for each KK charge 𝔮 is sometimes called the "𝔮-th screening mass", while the lightest nonzero mass overall is the "thermal mass" of the theory. Note that when 𝑑 = 2, the spectrum of masses (𝑚𝑖 , 𝔮𝑖 ) are simply ( 2𝜋 𝛽 Δ𝑖 , ℓ𝑖 ), where (Δ𝑖 , ℓ𝑖 ) are scaling dimensions and spins of local operators. However, in higher dimensions, the masses 𝑚𝑖 are not related in an obvious way to the local operator spectrum. In the partition function 𝑍CFT [M 𝑑−1 × 𝑆 1𝛽 ], the leading nonperturbative effects at small 𝛽 are expected to come from "worldline instantons" associated with particles of mass 𝑚𝑖 propagating along geodesics of M 𝑑−1 , see e.g. [39, 79, 99, 109, 110]. 182 Such contributions can be computed from the worldline path integral  ∫ ∮  ∑︁ ∫ √︁ 2𝜋𝑖𝔮𝑖 𝜇 𝜇 log 𝑍 = 𝐷𝑥 (𝜏) exp −𝑚𝑖 𝑑𝜏 𝑥¤ 𝜇 𝑥¤ + 𝐴 . single-particle 𝛽 (𝑚 𝑖 ,𝔮𝑖 ) (4.143) ∫ Here, for each particle, we have included a length term −𝑚𝑖 𝑑𝑠 proportional to the ∮ 𝑖 mass, along with a coupling 2𝜋𝑖𝔮 𝐴 to the background KK gauge field. (Note that 𝛽 2𝜋 we include a factor of 𝛽 because in our conventions, 𝐴 is a connection on a circle bundle where the fiber has circumference 𝛽, instead of the usual 2𝜋.) In Appendix 4.10, we compute the wordline path integral (4.143) on some geometries of interest. For example, by computing (4.143) on 𝑆 𝑑−1 , we find that the leading nonperturbative terms in the partition function 𝑍CFT [𝑆 𝑑−1 × 𝑆 1𝛽 ] have the form log(TrH𝑆 𝑑−1 [𝑒 −𝛽𝐻 ]) ⊃ ∑︁ 𝑚𝑖 𝑒 −2𝜋𝑚 𝑖    (±𝑖) 𝑑−2 𝑚𝑖𝑑−2 1 1+𝑂 . Γ(𝑑 − 1) 𝑚𝑖 (4.144) Note that the effect of each particle is exponential in the mass 𝑒 −2𝜋𝑚𝑖 , where 2𝜋 is the length of a great circle on 𝑆 𝑑−1 . By dimensional analysis, the masses are proportional to 1/𝛽, and hence these are indeed nonperturbative corrections in 𝛽. In addition to the exponential dependence, the worldline path integral makes an unambiguous prediction for the leading coefficient of the exponential, coming from a gaussian determinant. We immediately see that the interpretation of (4.144) is subtle because the coefficient becomes imaginary when 𝑑 is odd (even when the partition function must be real). We discuss this phenomenon and its interpretation in Appendix 4.10. Specializing to 4D, we can similarly compute leading nonperturbative corrections to a partition function on a lens space 𝐿 (𝑞; 1), coming from a "short" geodesic of length 2𝜋/𝑞:    2 𝜋𝑚  Í − 𝑞 𝑖  𝑚𝑖  1 + 𝑂 𝑚1𝑖  𝑚𝑖 𝑒  2 sin( 2𝑞𝜋 ) −𝛽𝐻    log(TrH𝐿 (𝑞;1) [𝑒 ]) ⊃ Í 𝑚2    𝑚𝑖 𝑒 −𝜋𝑚𝑖 2𝑖 1 + 𝑂 𝑚1𝑖  (𝑞 ≠ 2), (4.145) (𝑞 = 2). By our discussion of the EFT bundle, this result is closely related to the partition function on 𝑆 3 , with a twist by a rational angle of order 𝑞. To obtain the latter, we must replace 𝛽 → 𝑞𝛽, and account for the presence of a nontrivial holonomy for the 183 KK gauge field (since 𝑆 1𝑞𝛽 is nontrivially fibered over 𝐿 (𝑞; 1)). We find    2 𝜋𝑚 2 𝜋𝑖𝔮  Í − 2𝑖+ 𝑞 𝑖  𝑚𝑖 1  𝑞 𝑒 1 + 𝑂 (𝑞 ≠ 2),   (𝑚 𝑖 ,𝔮𝔦 ) 𝑚𝑖 −𝛽𝐻− 2𝑞𝜋𝑖 𝐽12 − 2𝑞𝜋𝑖 𝐽34 2𝑞 sin( 2𝑞𝜋 ) log(TrH𝑆3 [𝑒 ]) ⊃ Í    𝜋𝑚𝑖 𝑚2   (𝑞 = 2).  (𝑚𝑖 ,𝔮𝑖 ) 𝑒 − 2 +𝜋𝑖𝔮𝑖 8𝑖 1 + 𝑂 𝑚1𝑖  (4.146) In particular, nonperturbative corrections to the twisted partition function go like 2 𝑒 −2𝜋𝑚𝑖 /𝑞 , as opposed to 𝑒 −2𝜋𝑚𝑖 in the un-twisted case. More generally, in any dimension 𝑑, when the quotient by Z𝑞 creates a short geodesic of length ℓshort = 2𝜋/𝑞, the leading nonperturbative corrections will 2 behave like 𝑒 −(𝑚𝑖 /𝑞)ℓshort = 𝑒 −2𝜋𝑚𝑖 /𝑞 , where 𝑚𝑖 /𝑞 comes from the replacement 𝛽 → 𝑞𝛽. Note that this matches the result from modular invariance in 2D. In 2D, after happlying a modular transformation, the twisted partition function bei 𝑐 𝑐 e −2𝜋𝑖 𝜏 𝐿 − 2𝜋𝑖e 𝜏 ( 𝐿 0 − 24 ( 0 24 ) , where e ) Thus, the leading 𝜏 = const + 𝑞2𝜋𝑖 comes Tr 𝑒 2𝛽 . 𝛽-dependence of the contribution of excited states to the twisted partition function 2 2 2 𝑖 is 𝑒 −4𝜋 Δ𝑖 /(𝑞 𝛽) = 𝑒 −2𝜋𝑚𝑖 /𝑞 , where 𝑚𝑖 = 2𝜋Δ 𝛽 . Note that the action (4.143) is only the leading approximation to the effective action of a worldline instanton in the small 𝛽 limit. In particular, there can be power-law corrections in 𝛽 coming from higher curvature terms. Thus, while the tree-level and 1-loop terms (4.144), (4.145), and (4.146) in the worldline path integral are universal, the subleading corrections in 1/𝑚𝑖 are not necessarily universal, since they get contributions both from (computable) loops and from higher curvature terms.20 In Appendix 4.10, we derive (4.144), (4.145), and (4.146) by performing the worldline path integral explicitly. We also verify the universal leading terms in several examples from free field theory. 4.10 Discussion and future directions In this work, we found that the high-temperature partition function, twisted by a finite-order discrete rotation 𝑅, is captured by the same thermal EFT as the untwisted partition function. One consequence is that "spin-refined" densities of states (like the difference between the density of even-spin and odd-spin operators) are determined by the same Wilson coefficients as the usual density of states, up to 20 Furthermore, our choice of 𝜁-function regularization does not in general respect coordinate- invariance of the worldline path integral. To restore coordinate invariance one must add non𝛼 Γ 𝛽 with the appropriate coefficients. See coordinate-invariant counterterms like 𝑔 𝜇𝜈 𝑔 𝜌𝜏 𝑔 𝛼𝛽 Γ𝜇𝜌 𝜈𝜏 [16] for discussion. 184 subleading contributions from Kaluza-Klein vortices. Furthermore, the partition ® itself has an intricate fractal-like structure as a function of angle at function 𝑍 (𝛽, 𝜃) small 𝛽, with the same universal asymptotics controlling the neighborhood of every rational angle. These results follow entirely from effective field theory, together with the assumption that generic CFTs develop a mass gap when dimensionally-reduced. It would be interesting to revisit this assumption and understand how our results are modified in the presence of potential gapless modes. It would also be interesting to investigate whether Kaluza-Klein vortex defects can support gapless excitations, contribute to Weyl anomalies, and/or have nontrivial topological terms in their effective action21 . Our central construction is simple: it is essentially the observation that it is useful to construct a mapping torus 𝑀 𝛽,𝑅 = (M 𝐿 × R)/Z from two successive quotients: first quotienting by 𝑞Z, and then by Z𝑞 = Z/𝑞Z. This idea is applicable on other geometries besides 𝑆 𝑑−1 ×R, and it would be interesting to explore its implications for other types of partition functions. For example, one could explore "spin-refined" statistics of OPE coefficients by studying the behavior of discrete spacetime symmetries on the "genus-2" geometry of [25], or spin-refined lens-space partition functions [182], or the interaction of discrete spacetime symmetries with other forms of higherdimensional "modular invariance" [7, 156, 189, 190]. Supersymmetric partition functions have been studied on a wide variety of geometries; see e.g. [178]. It is an enduring challenge to understand observables of non-supersymmetric (potentially nonperturbative) theories on these geometries. One can also consider applying thermal EFT to BCFTs to study the asymptotic spectra of boundary operators. In two dimensions, this boils down to studying the partition function on a finite cylinder in the 𝛽 → 0 limit and writing down an EFT on a finite interval with two end points. The end points will become point-like defects in the thermal effective action. In higher dimensions, by introducing defects one may break the 𝑆𝑂 (𝑑) invariance down to some subgroup 𝐻. We can imagine turning on a rotation belonging to 𝐻 and applying thermal EFT ideas in this context. So far, our main tool for computing partition functions has been thermal EFT, which is organized in an expansion in small 𝛽. This expansion is likely asymptotic in general. In fact we can see its asymptotic nature explicitly in odd-dimensional free theories. It is an important question whether one can obtain more precise results 21 See e.g. [22, 65] for a study of similar vortex defects in the context of supersymmetric quantum field theories. 185 about high temperature partition functions, potentially including convergent expansions and/or numerical bounds (as is possible in 2D using the modular bootstrap [108]). One possible approach is through a better understanding of resurgence in the small𝛽 expansion. In particular, it would be nice to better understand the structure of nonperturbative terms beyond the worldline instantons discussed in Section 4.9. We expect that in an interacting theory, there should also be contributions from an infinite sum of "instanton graphs," representing massive particles propagating and interacting. An old example of instanton graphs are Lüscher corrections in field theories on torii [157, 158]. However, to our knowledge, the rules for general instanton graphs on general geometries in general massive QFTs have not been spelled out. Relatedly, modular invariance in 2D CFTs constrains some of the nonperturbative behavior of the partition function. Given some input light spectra, modular invariance highly constrains the resulting spectra, which in effect forces the partition functions with any phase inserted to behave in a certain way. Techniques such as Poincare series and Rademacher series have been used to complete the light spectrum of a 2D CFT (see e.g. [5, 77, 132, 162]), which roughly take the form of a (convergent) sum over rational angles. It would be interesting if there were related techniques in higher dimensions, resumming all rational angle insertions in the partition function trace. Another potential avenue to making the high-temperature expansion precise is using Tauberian techniques, which have been applied successfully to correlation functions [73, 173, 176, 180] and torus partition functions in 2D [168, 174, 175]. An essential ingredient in Tauberian methods is positivity,22 which has not yet played an important role in applications of thermal EFT. 22 See [163] where, even if OPE coefficent can become negative, Tauberian theorems were used along with some boundedness conditions from below, to predict asymptotics of OPE coefficent averaged over a large microcanonical window. However, it is not clear how to extend the result rigorously for an order one window. The same theorem is used in [172] as well. 186 Appendix A Qualitative picture of the 3D Ising partition function In this appendix, we explain our procedure for producing Figure 4.1. Let 𝑆 𝑝/𝑞 denote the leading contribution to the thermal effective action around the angle 2𝜋𝑞 𝑝 in 3D: −𝑆 𝑝/𝑞 = 1 𝑓 vol 𝑆 2 . 𝑞 3 𝛽2 + (𝜃 − 2𝜋 𝑝 ) 2 (4.147) 𝑞 We expect −𝑆 𝑝/𝑞 to be a good estimate for the action when it is large. Thus, to patch together different EFT descriptions, we roughly want to choose the action −𝑆 𝑝/𝑞 that is largest for each (𝛽, 𝜃). This would lead to the approximation −𝑆 ≈ max 𝑝,𝑞 −𝑆 𝑝/𝑞 , where 𝑝, 𝑞 runs over co-prime pairs of integers. However, such an estimate would not be smooth, so we instead combine the different actions with a root-mean-square: ! 1/2 ∑︁ 2 −𝑆 ≈ (−𝑆 𝑝/𝑞 ) . (4.148) 𝑝,𝑞 In Figure 4.1, we plot log log 𝑍 = log(−𝑆), for 𝑆 given in (4.148), with coprime pairs of integers up to denominator 15. We use the value of minus the free energy density 𝑓 ≈ 0.153 determined from Monte-Carlo simulations [144, 145, 198]. Note that the approximation (4.148) has some unrealistic features. Firstly, its expansion around each rational angle contains subleading corrections in 𝛽 that do not conform with the expectation from thermal EFT (4.32). Secondly, it does not incorporate the nonperturbative effects discussed in Section 4.9. Our goal with this approximation is simply to give a qualitative picture of the partition function. It is interesting to ask whether there is a natural basis of functions for 𝑍 that naturally incorporates these constraints. In principle, the qualitative features of Figure 4.1 can be checked. For instance, one can explicitly build the partition function of the 3D Ising model with a phase (e.g. (−1) 𝐽 ) inserted, by simply using the operator scaling dimensions that have been estimated from the conformal bootstrap (or other methods). One can then plot it as a function of 𝛽, Ω, and check that, for instance, the leading Wilson coefficient is approximately 0.153 8 (similar to what was done in Appendix A of [119] and Appendix D of [25]). An important technical obstacle we run into when attempting this is: when computing the partition function with the phase (−1) 𝐽 , the EFT is valid when 2𝛽 ≪ 1 (rather than 𝛽 ≪ 1), so in effect, one needs to keep more operators in the partition function to get a trustworthy estimate. It would be interesting if other techniques to estimate the scaling dimensions of the 3D Ising model know about enough high energy operators to see explicitly the qualitative features of Figure 4.1. 187 Appendix B Review of plethystic sums In this appendix, we review some facts about the high temperature expansion of plethystic sums. Plethysic sums can be written as derivatives of various spectral zeta functions and to compute the high temperature expansion, one "resums" the zeta functions following a procedure very similar to (4.94). The same resummation technique is also used to compute the high temperature expansion of massive free field free energy, as we discussed in the Section 4.10. Plethystic sums and spectral zeta functions We start with some general Í discussion. Let 𝑓 (𝛽) = 𝑘 𝑑 𝑘 𝑒 −𝛽𝜆 𝑘 be a generating function. We define a few related quantities: • Plethystic sum: log(PE[ 𝑓 ]) := ∑︁ 1 𝑓 (𝑛𝛽). (4.149) 𝑛 𝑛=1 • Spectral zeta function: 𝜁 (𝑠; 𝑓 ) := ∑︁ ∑︁ " 2 # −𝑠 𝑑𝑘 2𝜋𝑛 𝛽 ∑︁ 𝑑 𝑘 𝑒 −𝑡𝜆 𝑘 . + 𝜆2𝑘 . (4.150) 𝑛∈Z 𝑘 • Heat trace: 𝐻 (𝑡; 𝑓 ) := 2 (4.151) 𝑘 The plethystic sum is related to the spectral zeta function in a simple way: 𝛽 ∑︁ 1 𝑑 log(PE[ 𝑓 ]) − 𝑑𝑘 𝜆𝑘 = 𝜁 (𝑠; 𝑓 ), (4.152) 2 𝑘 2 𝑑𝑠 𝑠=0 where the second term on the left hand side is the zeta function regularized Casimir energy.23 To derive (4.152), we start by using the Schwinger parametrization to obtain: " # −𝑠 2 ∫ ∞ ∑︁ ∑︁ 2 2 2𝜋𝑛 1 ∑︁ ∑︁ 𝑑𝑡 𝑠 −𝑡 ( 2 𝜋𝑛 2 𝛽 ) −𝑡𝜆 𝑘 𝜁 (𝑠; 𝑓 ) = 𝑑𝑘 + 𝜆𝑘 = 𝑑𝑘 𝑡 𝑒 𝛽 Γ(𝑠) 𝑛∈Z 𝑘 𝑡 0 𝑛∈Z 𝑘 ∫ ∞ 𝑑𝑡 𝑠− 1 𝑚2 𝛽2 −𝑡𝜆2 1 𝛽 ∑︁ ∑︁ 𝑘, 𝑑𝑘 = 𝑡 2 𝑒 4𝑡 √ Γ(𝑠) 2 𝜋 𝑚∈Z 𝑘 𝑡 0 (4.153) 23 Note that the Casimir energy computed using zeta function regularization is NOT the same as the Casimir energy included in Appendix 4.10. See Section 3.1 in [25] for a discussion of schemes and [113] for the difference with zeta function regularization. 188 where in the last line we’ve Poisson resummed with respect to 𝑛. Now we split the summation over 𝑚 into an "excited part" with 𝑚 ≠ 0 and a "vacuum part" where 𝑚 = 0: ∫ ∞ ∞ 𝑑𝑡 𝑠− 1 𝑚2 𝛽2 −𝑡𝜆2 1 𝛽 ∑︁ ∑︁ 𝑘, 𝑑𝑘 𝜁excited (𝑠; 𝑓 ) = 𝑡 2 𝑒 4𝑡 √ Γ(𝑠) 𝜋 𝑚=1 𝑘 𝑡 0 ∫ ∞ 1 𝛽 ∑︁ 𝑑𝑡 𝑠− 1 −𝑡𝜆2 𝜁vacuum (𝑠; 𝑓 ) = 𝑡 2 𝑒 𝑘. (4.154) 𝑑𝑘 √ Γ(𝑠) 𝜋 𝑘 𝑡 0 Performing the integral over 𝑡, we get ∫ ∞ ∞ 𝛽 ∑︁ ∑︁ 𝑑 𝑑𝑡 −1/2 − 𝑚2 𝛽2 −𝑡𝜆2 𝑘 𝜁excited (𝑠; 𝑓 ) = √ 𝑡 𝑒 4𝑡 𝑑𝑘 𝑑𝑠 𝑠=0 𝑡 𝜋 𝑚=1 𝑘 0 =2 ∞ ∑︁ 1 ∑︁ 𝑚 𝑚=1 𝑑𝑘 𝑒 −𝛽𝑚𝜆 𝑘 (4.155) = 2 log(PE[ 𝑓 ]). 𝑘 For 𝜁vacuum (𝑠; 𝑓 ), we invert the Schwinger parametrization: ∑︁ ∑︁ 𝑑 𝛽 𝜁vacuum (𝑠; 𝑓 ) = √ Γ(−1/2) 𝑑 𝑘 𝜆 𝑘 = −𝛽 𝑑𝑘 𝜆𝑘 . 𝑑𝑠 𝑠=0 2 𝜋 𝑘 𝑘 (4.156) Putting everything together we arrive at (4.152). To compute partition functions with flavor twists and rational angular fugacities we need to consider plethystic sums taking the following form: ∞ ∑︁ 𝑒 −𝑖𝑛𝜃 𝑛  𝐹 (𝑛𝛽, 𝜔𝑛𝑞 ),  2𝜋𝑖 𝜔 𝑞 = exp . 𝑞 (4.157) 𝑛=1 Some formal manipulation shows that when the "twisted generating function" 𝐹 (𝛽, 𝜔 𝑞 ) satisfies 𝐹 (𝛽, 𝜔 𝑞 ) = 𝐹 (𝛽, 𝜔−1 𝑞 ), the following relation holds: ∞ ∑︁ 𝑒 −𝑖𝑛 𝜃 𝑛 𝑛=1 ∞ ∑︁ 𝑒 𝑖𝑛 𝜃 1 𝑑 𝛽 ∑︁ 𝜁 [𝑞𝜃, 𝑠; 𝐹 (𝑞𝛽, 1)] + 𝑑𝑘𝜆𝑘 𝑛 𝑞 𝑑𝑠 𝑠=0 𝑞 𝑘 𝑛=1      𝑞−1 1 ∑︁ 𝑑 ℓ ℓ ℓ ℓ + 𝐿 𝑞𝜃, , 𝑠; 𝐹 (𝑞𝛽, 𝜔𝑞 ) + 𝐿 −𝑞𝜃, , 𝑠; 𝐹 (𝑞𝛽, 𝜔𝑞 ) , 2𝑞 ℓ=1 𝑑𝑠 𝑠=0 𝑞 𝑞 𝐹 (𝑛𝛽, 𝜔𝑞𝑛 ) + 𝐹 (𝑛𝛽, 𝜔𝑞𝑛 ) = (4.158) where • In the second sum on the right hand side, 𝑑 𝑘 and 𝜆 𝑘 are supposed to be read Í from 𝐹 (𝑞𝛽, 1), that is, 𝐹 (𝑞𝛽, 1) = 𝑘=0 𝑑 𝑘 𝑒 −𝛽𝜆 𝑘 189 • 𝜁 (𝜃, 𝑠; 𝑓 ) = ∑︁ ∑︁ " 𝑑𝑘 2𝜋𝑛 𝜃 + 𝛽 𝛽 2 # −𝑠 + 𝜆2𝑘 , with 𝑓 (𝛽) = 𝑛∈Z 𝑘 ∑︁ 𝑑 𝑘 𝑒 −𝛽𝜆 𝑘 . 𝑘=0 (4.159) • 𝐿 (𝜃, 𝑎, 𝑠; 𝑓 ) = ∑︁ ∑︁ 𝑑𝑘   𝑛∈Z 𝑘 𝑒 2𝜋𝑖𝑛𝑎 𝑠 , 2 𝜃 2𝜋𝑛 + 𝜆2𝑘 𝛽 + 𝛽 with 𝑓 (𝛽) = ∑︁ 𝑑 𝑘 𝑒 −𝛽𝜆 𝑘 . 𝑘=0 (4.160) Now that we’ve related different plethystic sums to various spectral zeta functions,the high temperature expansion can be computed by "resumming" all these zeta functions. For concreteness, we look at some examples: Example: free scalar on 𝑆 1 × 𝑆 3 𝑓 (𝛽) = The relevant generating function is24 𝑒 −𝛽 (1 − 𝑒 −2𝛽 ) ∑︁ 2 −𝛽𝑘 = 𝑘 𝑒 . (1 − 𝑒 −𝛽 ) 4 𝑘=1 (4.161) The spectral zeta function is 1 ∑︁ 𝜁 (𝑠; 𝑓 ) = Γ(𝑠) 𝑛∈Z ∫ ∞ 0 2 𝑑𝑡 𝑠 −𝑡 ( 2 𝜋𝑛 𝛽 ) 𝐻 (𝑡; 𝑓 ) 𝑡 𝑒 𝑡 (4.162) with 𝐻 (𝑡; 𝑓 ) = ∞ ∑︁ √ 2 −𝑡𝑘 2 𝑘 𝑒 𝑘=1   ∞ 𝜋 −3/2 √ ∑︁ −ℓ2 𝜋2 /𝑡 1 −3/2 2 2 −5/2 = 𝑡 + 𝜋 𝑒 𝑡 −ℓ 𝜋 𝑡 . 4 2 𝑙=1 Here we performed a Poisson resummation with respect to 𝑘 and separated the piece with ℓ = 0 from ℓ ≠ 0. • The term with 𝑛 = 0 in the sum (4.162) is the contribution of zero mode: 𝜁 (𝑠; 𝑓 ) = ∞ ∑︁ 𝑑 𝑘 𝜆−𝑠 𝑘 = ∞ ∑︁ 𝑘 2−2𝑠 = 𝜁 𝑅 (2𝑠 − 2). (4.163) 𝑛=0 𝑘=1 𝑘=1 24 This is the spectral generating function of the operator Δ + (𝑑−2) 𝑅 with Δ being the Laplacian 4(𝑑−1) on 𝑆 3 and 𝑑 = 4. 190 • Terms with 𝑛 ≠ 0 and ℓ = 0 generate the perturbative series: √ ∑︁ ∫ ∞ 𝑑𝑡 2 𝜋𝑛 2 𝜋 𝜁 (𝑠; 𝑓 ) = 𝑡 𝑠 𝑒 −𝑡 ( 𝛽 ) 𝑡 −3/2 𝑛≠0,ℓ=0 4Γ(𝑠) 𝑛∈Z,𝑛≠0 0 𝑡   √ Γ(𝑠 − 23 ) 2𝜋 3−2𝑠 = 𝜋 𝜁 𝑅 (2𝑠 − 3). 2Γ(𝑠) 𝛽 (4.164) • Terms with 𝑛 ≠ 0 and ℓ ≠ 0 generate the non-perturbative corrections: √ "   3 −𝑠  2    52 −𝑠  2 # ∞ ∞ 9 𝜋 ∑︁ ∑︁ 5 −𝑠 𝑛 2 4𝜋 ℓ𝑛 𝑛 4𝜋 ℓ𝑛 −𝑠 2 2 = 𝜁 (𝑠; 𝑓 ) 22 − 22 𝜋 ℓ . 𝐾 − 3 +𝑠 𝐾 − 5 +𝑠 2 2 𝑛≠0,ℓ≠0 Γ(𝑠) 𝑛=1 ℓ=1 ℓ𝛽 𝛽 ℓ𝛽 𝛽 (4.165) Following (4.152), we find ∞ ∞ 𝛽 𝜋4 𝜁 (3) 1 ∑︁ ∑︁ − 4 𝜋𝛽2 𝑛ℓ 1 4𝑛𝜋 2 8𝑛2 𝜋 4 log(PE[ 𝑓 ]) − = − − 2 𝑒 ( 3+ 2 + ). 240 45𝛽3 4𝜋 2 2𝜋 𝑛=1 ℓ=1 ℓ ℓ 𝛽 ℓ𝛽2 (4.166) With the same 𝑑 𝑘 and 𝜆 𝑘 , we can construct a slightly different zeta function: 𝜁𝑚 (𝑠) = ∞ ∑︁ 𝑑𝑘  𝑘=0 𝜆2𝑘 + 𝑚 2 𝑠 (4.167) this spectral zeta function satisfies 1 𝑑 𝜁𝑚 (𝑠) = log(𝑍massive [𝑆 3 ]), 2 𝑑𝑠 𝑠=0 (4.168) where 𝑍massive [𝑆 3 ]) is given in eqn(4.216). Applying the same resummation procedure, we get (4.217). The lens space partition functions 𝑍massive [𝐿(𝑞; 1)] can be computed in an almost identical way. One simply replace 𝑑 𝑘 = 𝑘 2 by [12]: h i h i  𝑘 𝑘  𝑘 , 𝑘 − 𝑞   𝑞 𝑞 ∈ 2Z, h i  h i 𝑑𝑘 = (𝑞 ∈ 2Z + 1),   𝑘 𝑞𝑘 + 1 , 𝑘 − 𝑞 𝑞𝑘 ∈ 2Z + 1,     0,  𝑘 ∈ 2Z,  h i  𝑑𝑘 = (𝑞 ∈ 2Z). (4.169)   𝑘 2 𝑞𝑘 + 1 , 𝑘 ∈ 2Z + 1,  (𝑑−2) which are the degeneracy for Δ + 4(𝑑−1) 𝑅 with Δ being the Laplacian in 𝐿(𝑞; 1) and 𝑑 = 4. The same calculations will yield (4.221) and (4.222). 191 Example: free scalar on 𝑆 1 × 𝑆 5 The relevant generating function is 𝑓 (𝛽) = ∞ 𝑞 2 (1 − 𝑞 2 ) ∑︁ 𝑘 2 (𝑘 2 − 1) −𝛽𝑘 = 𝑒 12 (1 − 𝑞) 6 𝑘=1 and the heat trace is: 2 1 ∑︁ 2 2 𝐻 (𝑡; 𝑓 ) = 𝑘 (𝑘 − 1)𝑒 −𝑡𝑘 24 𝑘 ∈Z       44 ∞ 2 2 √ √ ∑︁ 1 𝜋2𝑙 2 1 𝜋2𝑙 2 1 1 1 𝜋 𝑙 − 𝑙 𝑡𝜋 𝑒 = 𝜋 − + 𝜋 − + + − . 16 32𝑡 5/2 48𝑡 3/2 12𝑡 9/2 4𝑡 7/2 𝑡 5/2 12 24𝑡 3/2 𝑙=1 (4.170) Following the same procedure as in the example above, we obtain 31 𝜁 (5) 𝜁 (3) 2𝜋 6 𝜋4 log(PE[ 𝑓 ] − + + (4.171) 𝛽= − 60480 945𝛽5 540𝛽3 16𝜋 4 48𝜋 2  4 4    ∞ ∑︁ ∞ ∑︁ 2 4𝜋 2 𝑛3 1 𝑛2 𝜋 2 𝑛2 𝑛  1 1 𝑛 1 − 4 𝜋𝛽 𝑛ℓ 4𝜋 𝑛 + 𝑒 + + 2 3+ + 2 + 4 5+ . + 4ℓ 2 2ℓ4 2ℓ3 3ℓ 𝛽 3𝛽 3𝛽ℓ 𝛽 ℓ 2𝜋 6ℓ 24𝜋 8𝜋 ℓ 𝑛=1 ℓ=1 (4.172) If we replace 𝜁 (𝑠; 𝑓 ) by 𝜁𝑚 (𝑠), we reproduce (4.218). Example: free scalar on 𝑆 1 × 𝑆 3 with Z2 flavor twist The plethystic sum in interest is ∞ ∑︁ (−1) 𝑛 𝑓 (𝑛𝛽), 𝑓 (𝛽) = 𝑛 𝑛=1 𝑒 −𝛽 (1 − 𝑒 −2𝛽 ) ∑︁ 2 −𝛽𝑘 = 𝑘 𝑒 . (1 − 𝑒 −𝛽 ) 4 𝑘=1 (4.173) Setting 𝑞 = 1 and 𝜃 = 𝜋 in (4.158), we find that ∞ ∑︁ (−1) 𝑛 𝑛 𝑛=1 𝑓 (𝑛𝛽) = 1 𝑑 𝛽 ∑︁ 𝜁 (𝜋, 𝑠; 𝑓 ) + 𝑑𝑘 𝜆𝑘 . 2 𝑑𝑠 𝑠=0 2 𝑘 (4.174) where 𝜁 (𝜋, 𝑠; 𝑓 ) is defined in (4.159). After resumming the zeta function, we find ∞ ∑︁ (−1) 𝑛   ∞ ∞ 2 𝜋2 𝜋2 𝛽 7𝜋 4 ∑︁ ∑︁ − (4𝑛−2)ℓ (2𝑛 − 1) 2𝑛 − 1 1 𝛽 𝑓 (𝑛𝛽)− =− − 𝑒 + 2 + 3 2 . 3 2 𝑛 240 360𝛽 ℓ𝛽 ℓ 𝛽 ℓ 𝜋 𝑛=1 𝑛=1 ℓ=1 (4.175) Finally, we discuss twisted partition function with rational angular fugacity. These are the main objects of interest in this paper. Even though the techniques outlined 192 in this appendix can be very easily generalized to higher dimensions and to more complicated combinations of angles, we will focus on 4D scalar partition functions 2 𝜋𝑖 2 𝜋𝑖 twisted by 𝑒 − 𝑞 𝐽12 − 𝑞 𝐽34 for simplicity. As explained in Section 4.10, the subtle difference between the lens space partition function and the twisted partition function is related to the non-trivial topology of the EFT bundle. Example: free scalar in 𝑆 1 × 𝑆 3 with Z𝑞 rotation, 𝑞 odd   −𝛽𝐻− 2𝑞𝜋𝑖 𝐽12 − 2𝑞𝜋𝑖 𝐽34 ] and restrict We focus on the twisted partition function log Tr[𝑒 to the case where 𝑞 is odd. The plethystic sum in interest is ∞ ∑︁ 1 𝑛 𝐹 (𝑛𝛽, 𝜔𝑛𝑞 ), 𝐹 (𝛽, 𝜔 𝑞 ) = 𝑛=1 𝑒 −𝛽 (1 − 𝑒 −2𝛽 ) . −𝛽 2 (1 − 𝜔 𝑞 𝑒 −𝛽 ) 2 (1 − 𝜔−1 𝑞 𝑒 ) (4.176) It is easy to verify  Í∞ 2 −𝑞𝛽𝑘   , ℓ=0  𝑘=1 𝑘 𝑒     𝐹 (𝑞𝛽, 𝜔ℓ𝑞 ) =   Í∞ 𝑘 sin( 2 𝜋𝑞𝑘ℓ ) −𝑞𝛽𝑘   , ℓ≠0   𝑘=1 sin( 2 𝑞𝜋ℓ ) 𝑒  (4.177) Setting 𝜃 = 0 in (4.158), we find ∞ ∑︁ 1 𝑛 1 𝑑 𝛽 ∑︁ 𝑑𝑘 𝜆𝑘 𝜁 [0, 𝑠; 𝐹 (𝑞𝛽, 1)] + 2𝑞 𝑑𝑠 𝑠=0 𝑞 𝑘   𝑞−1 1 ∑︁ 𝑑 ℓ ℓ + 𝐿 𝑞𝜃, , 𝑠; 𝐹 (𝑞𝛽, 𝜔 𝑞 ) . 2𝑞 ℓ=1 𝑑𝑠 𝑠=0 𝑞 𝐹 (𝑛𝛽, 𝜔𝑛𝑞 ) = 𝑛=1 (4.178) where 𝑑 𝑘 = 𝑘 2 , 𝜆 𝑘 = 𝑞𝑘 for the second sum on the right hand side. After resumming all the zeta functions, we find ∞ ∑︁ 1 𝑒 −𝑛𝛽 (1 − 𝑒 −2𝑛𝛽 ) 𝜋4 𝛽 𝜁 (3) = + − 2 𝑛 −𝑛 −𝑛𝛽 2 −𝑛𝛽 2 3 4 𝑛 (1 − 𝜔 𝑞 𝑒 ) (1 − 𝜔 𝑞 𝑒 ) 240 4𝜋 𝑞 45𝛽 𝑞 𝑛=1   ∞ ∞ 1 ∑︁ ∑︁ − 4 𝜋𝛽𝑞2 ℓ𝑛 1 2𝑛 4𝜋 2 𝑛2 − 𝑒 + + 𝑞 𝑛=1 ℓ=1 2𝜋 2 ℓ 3 𝛽ℓ 2 𝑞 𝛽2 ℓ𝑞 2   ∞ ∑︁ ∞ cos( 2𝜋𝑛ℓ ) 2 ∑︁ 𝑞 − 4 𝜋2 𝑛ℓ 2𝜋𝑛 𝑞 𝑞 𝛽 + 𝑒 + 2𝜋ℓ ℓ𝑞𝛽 2ℓ 2 𝜋 𝑛=1 ℓ=1 sin( 𝑞 ) ℓ∉𝑞Z + ∞ ∑︁ 1 1 . 2 4𝜋ℓ sin( 2𝜋ℓ ℓ=1 𝑞 ) ℓ∉𝑞Z (4.179) 193 Example: free scalar in 𝑆 1 × 𝑆 3 with Z𝑞 rotation and Z2 flavor twist, 𝑞=odd The relevant plethystic sum is ∞ ∑︁ (−1) 𝑛 𝑛 𝐹 (𝑛𝛽, 𝜔𝑛𝑞 ), 𝑛=1 𝐹 (𝛽, 𝜔 𝑞 ) = 𝑒 −𝛽 (1 − 𝑒 −2𝛽 ) . −𝛽 2 (1 − 𝜔 𝑞 𝑒 −𝛽 ) 2 (1 − 𝜔−1 𝑞 𝑒 ) (4.180) Setting 𝜃 = 𝜋 in (4.158) and resum all the zeta functions, we get ∞ ∑︁ (−1) 𝑛 𝑒 −𝑛𝛽 (1 − 𝑒 −2𝑛𝛽 ) 7𝜋 4 𝛽 = − + 𝑛 −𝑛 𝑛 (1 − 𝜔 𝑞 𝑒 −𝑛𝛽 ) 2 (1 − 𝜔 𝑞 𝑒 −𝑛𝛽 ) 2 360𝛽3 𝑞 4 240 𝑛=1   ∑︁ ∑︁ 2ℓ (2𝑛−1) 𝜋 2 (2𝑛 − 1) 2 𝜋 2 2𝑛 − 1 1 − 𝑞𝛽 − 𝑒 + 2 2 + 3 2 ℓ𝑞 3 𝛽2 𝛽ℓ 𝑞 2ℓ 𝑞𝜋 𝑛=1 ℓ=1   ∞ ∑︁ ∞ cos( 2𝜋ℓ ( 𝑞−1 − 𝑛)) ∑︁ 𝜋2 (2𝑛 + 1)𝜋 𝑞 − 2ℓ (2𝑛+1) 𝑞 2 2 𝑞 𝛽 + 𝑒 + 2 . 2𝜋ℓ ℓ𝑞𝛽 2ℓ 𝜋 sin( ) ℓ=1 𝑛=0 𝑞 (4.181) ℓ∉𝑞Z Appendix C More examples for 4D and 6D CFTs In this appendix, we compute a few more examples of the high temperature expansion of CFTs with a large rotation inserted. We will focus on insertions 𝑅 without fixed points (so that 𝑆 D vanishes). In odd 𝑑, 𝑆𝑂 (𝑑) necessarily has fixed locus,   a nontrivial  𝑝1 so we focus on even 𝑑. In even 𝑑, if we insert 𝑅 = exp 2𝜋𝑖 𝑞 𝐽1 + · · · + 𝑝𝑞𝑛 𝐽𝑛 (namely all the denominators are equal, with gcd( 𝑝𝑖 , 𝑞) = 1), then 𝑅 has no fixed points and there is no defect action. Previously in Section 4.5, we described 4d examples with a defect coming from 𝑞 1 ≠ 𝑞 2 ; in this example we will consider the (simpler) case with no defect action. We note that the formulas in this section are accurate to all orders in perturbation theory in 𝛽; this is because the free energy of free theories in even dimensions truncates at 𝑂 (𝛽). This is an accident of free theories in even dimensions and does not generalize. Finally, we note that in even dimensions, the Hamiltonian 𝐻 = 𝐷 + 𝜀0 includes a contribution from the Casimir energy, which is not accounted for in the sum over characters using plethystic exponentials (see Section 3.1 of [25] for details on this scheme, [113] for a calculation of the Casimir energy, and e.g. [96] for values of the 𝑎 anomaly). We have included this contribution to the final expression (although terms linear in 𝛽 are invariant when acting on with (4.32)). 194 Example: 4D Z2 -twisted complex scalar Here we consider a 4D free complex scalar. We insert a Z2 twist (which we denote as 𝑒 𝜋𝑖𝑄 ) where we identify the field 𝜙 with −𝜙 as we go around the thermal circle, in order to remove the gapless sector. Without any insertion of a rotation 𝑅, we get the following free energy:   log Tr 𝑒 −𝛽(𝐻−𝑖Ω1 𝐽1 −𝑖Ω2 𝐽2 ) 𝑒 𝜋𝑖𝑄 ! Í2 Í2 Í2 Ω𝑖4 ) 𝛽 Ω𝑖2 − 10Ω21 Ω22 + 6 𝑖=1 𝜋 2 𝑖=1 Ω𝑖2 (3 + 6 𝑖=1 7𝜋 4 ∼ Î2 − + − . 2 36𝛽 720 180𝛽3 𝑖=1 (1 + Ω𝑖 ) (4.182) 1 Now let us consider a rotation by 2𝜋/3 in each Cartan direction. As expected, when we insert the large rotation, we find    𝐽1 𝐽2   1 −𝛽(𝐻−𝑖Ω1 𝐽1 −𝑖Ω2 𝐽2 ) 𝜋𝑖𝑄 2𝜋𝑖 3 + 3 log Tr 𝑒 𝑒 𝑒 ∼ log Tr 𝑒 −3𝛽(𝐻−𝑖Ω1 𝐽1 −𝑖Ω2 𝐽2 ) 𝑒 3𝜋𝑖𝑄 . 3 (4.183) Example: 4D Dirac fermion As discussed in Section 4.4, for fermionic theories we need to compute the partition function without and with a (−1) 𝐹 insertion:   log Tr 𝑒 −𝛽 (𝐻 −𝑖Ω1 𝐽1 −𝑖Ω2 𝐽2 ) ! Í Í Í 𝜋 2 (3 + 2𝑖=1 Ω2𝑖 ) (18 + 6 2𝑖=1 Ω2𝑖 + 10Ω21 Ω22 − 21 2𝑖=1 Ω4𝑖 ) 𝛽 7𝜋 4 − + ∼ Î2 2 36𝛽 1440 90𝛽3 𝑖=1 (1 + Ω𝑖 ) (4.184) 1   log Tr (−1) 𝐹 𝑒 −𝛽 (𝐻 −𝑖Ω1 𝐽1 −𝑖Ω2 𝐽2 ) 𝑒 4 𝜋𝑖𝑄/3 ! Í Í Í 𝜋 2 (3 + 2𝑖=1 Ω2𝑖 ) (18 + 6 2𝑖=1 Ω2𝑖 + 10Ω21 Ω22 − 21 2𝑖=1 Ω4𝑖 ) 𝛽 1 52𝜋 4 ∼ Î2 − + . 2 54𝛽 1440 1215𝛽3 𝑖=1 (1 + Ω𝑖 ) (4.185) Fermions carry 𝑈 (1) charge. We use a convention where the particle carries charge +1 and anti-particle carries charge −1. To make sure that the zero mode does not contribute in the thermal effective field theory description, we turned on the fugacity corresponding to 𝑈 (1) and set it to 𝑒 4𝜋𝑖/3 , along with a (−1) 𝐹 insertion. This amounted to turning on Z3 ⊂ 𝑈 (1) flavor symmetry for the partition function with a (−1) 𝐹 insertion. After turning on a large rotation 𝑒 2𝜋𝑖𝐽1 𝑝1 /𝑞 𝑒 2𝜋𝑖𝐽2 𝑝2 /𝑞 , we get 195 the following results (exactly as predicted in Section 4.4):    𝐽1 𝐽2   1 −𝛽(𝐻−𝑖Ω1 𝐽1 −𝑖Ω2 𝐽2 ) 2𝜋𝑖 3 + 3 log Tr 𝑒 𝑒 (4.186) ∼ log Tr 𝑒 −3𝛽(𝐻−𝑖Ω1 𝐽1 −𝑖Ω2 𝐽2 ) 3    𝐽1 𝐽2   1 −𝛽(𝐻−𝑖Ω1 𝐽1 −𝑖Ω2 𝐽2 ) 2𝜋𝑖𝑄/3 2𝜋𝑖 2 + 2 log Tr 𝑒 𝑒 𝑒 ∼ log Tr (−1) 𝐹 𝑒 −2𝛽(𝐻−𝑖Ω1 𝐽1 −𝑖Ω2 𝐽2 ) 𝑒 4𝜋𝑖𝑄/3 . 2 (4.187) Example: 6D twisted complex scalar Here we consider a 6D free scalar with a Z2 twist: h i Í3 log Tr 𝑒 −𝛽 (𝐻 −𝑖 𝑘=1 Ω𝑘 𝐽𝑘 ) 𝑒 𝜋𝑖𝑄 Í Í Í Í 7𝜋 4 (1 − 3𝑖=1 Ω2𝑖 ) 𝜋 2 (8 3𝑖=1 Ω2𝑖 − 5 𝑐𝑦𝑐 Ω21 Ω22 − 3 3𝑖=1 Ω4𝑖 ) 31𝜋 6 − + + ∼ Î3 2 2160𝛽 2160𝛽3 7560𝛽5 𝑖=1 (1 + Ω𝑖 ) ! Í Í Í3 Í Í3 Í 2 2 2 4 2 2 2 (37 + 62 𝑖=1 Ω𝑖 − 154 𝑐𝑦𝑐 Ω1 Ω2 − 104 𝑖=1 Ω𝑖 + 70Ω1 Ω2 Ω3 + 42 3𝑖=1 Ω2𝑖 3𝑖=1 Ω4𝑖 − 22 3𝑖=1 Ω6𝑖 ) 𝛽 . − 60480 (4.188) 1 With the insertion of a large rotation, we get    h i 𝐽1 𝐽2 𝐽3 Í3 Í 1 −𝛽(𝐻−𝑖 3𝑘=1 Ω 𝑘 𝐽 𝑘 ) 𝜋𝑖𝑄 2𝜋𝑖 3 + 3 + 3 log Tr 𝑒 𝑒 𝑒 ∼ log Tr 𝑒 −3𝛽(𝐻−𝑖 𝑘=1 Ω𝑘 𝐽𝑘 ) 𝑒 3𝜋𝑖𝑄 3 (4.189) Example: 6D Dirac fermion Finally we consider a 6D free Dirac fermion. We get h i Í3 log Tr 𝑒 −𝛽 (𝐻 −𝑖 𝑘=1 Ω𝑘 𝐽𝑘 ) Í Í Í Í 7𝜋 4 (5 + 3𝑖=1 Ω2𝑖 ) 𝜋 2 (135 + 26 3𝑖=1 Ω2𝑖 + 10 𝑐𝑦𝑐 Ω21 Ω22 − 21 3𝑖=1 Ω4𝑖 ) 31𝜋 6 ∼ Î3 − + 2 4320𝛽 1080𝛽3 1890𝛽5 𝑖=1 (1 + Ω𝑖 ) ! Í3 Í Í Í Í Í (−880 + 55 𝑖=1 Ω2𝑖 − 14 𝑐𝑦𝑐 Ω21 Ω22 − 743 3𝑖=1 Ω4𝑖 − 70Ω21 Ω22 Ω23 + 147 3𝑖=1 Ω2𝑖 3𝑖=1 Ω4𝑖 − 302 3𝑖=1 Ω6𝑖 ) 𝛽 + . 120960 (4.190) 1 With an insertion of a rotation 𝑅, we get the following:    h i 𝐽1 𝐽2 𝐽3 Í Í3 1 −𝛽(𝐻−𝑖 3𝑘=1 Ω 𝑘 𝐽 𝑘 ) 2𝜋𝑖 2 + 2 + 2 log Tr 𝑒 ∼ log Tr 𝑒 −2𝛽(𝐻−𝑖 𝑘=1 Ω𝑘 𝐽𝑘 ) (4.191) 𝑒 2 Appendix D More on nonperturbative terms In this appendix, we discuss more details on the nonperturbative parts of the partition function outlined in Section 4.9. In Appendix 4.10, we derive equations (4.144), (4.145) and (4.146) by performing the worldline path integral. In appendices 4.10, 196 𝑦 𝑎 (𝜏) 𝜃 (𝜏) Figure 4.7: Coordinates on 𝑆 𝑑−1 . The classical path is 𝜃 (𝜏) = 2𝜋𝜏 and perpendicular fluctuations 𝑦 𝑎 (𝜏) are integrated over. 4.10, and 4.10, we verify the universal leading terms in several examples from free field theory. Worldline path integral • Worldline path integral on the sphere Let us begin by computing the worldline path integral on 𝑆 𝑑−1 . Let 𝜃 denote a coordinate along a great circle and let 𝑦𝑖 (𝑖 = 1, . . . , 𝑑 − 2) parametrize the perpendicular directions, see Figure 4.7. The metric on 𝑆 𝑑−1 takes the form: 𝑑𝑠2 = 𝑑𝑦 2 + cos2 |𝑦| 𝑑𝜃 2 . (4.192) In the absence of a background gauge field, a single worldline instanton contribution is given by √︄    2   2 ∫ ∫ ∫ 1 ∫  𝑑𝜃 𝑑𝑦  𝜇 −𝑚 𝑖 𝑑𝑠 𝑖 2  + 𝐷𝑥 (𝜏) 𝑒 = 𝐷𝑦 (𝜏) exp −𝑚𝑖 𝑑𝜏 cos |𝑦| 𝑑𝜏 𝑑𝜏  0      2  ∫ ∫ 1 𝑦 ¤ 𝐷𝑦𝑖 (𝜏) exp −𝑚𝑖 𝑑𝜏 ∼ 𝑒 −2𝜋𝑚𝑖 − 𝜋𝑦 2 , (4.193) 4𝜋 0 where in the last line we Taylor-expanded the square root in fluctuations around the great circle.25 We also used reparameterization invariance of the Nambu-Goto action to set 𝜃 (𝜏) = 2𝜋𝜏. In the path integral, we sum over periodic paths 𝑦𝑖 (𝜏) satisfying 𝑦𝑖 (0) = 𝑦𝑖 (1), with the following mode expansion: 𝑖 𝑦 (𝜏) = ∞ ∑︁ 𝑛=1 𝑎𝑖𝑛 sin(2𝜋𝑛𝜏) + ∞ ∑︁ 𝑏𝑖0 𝑖 𝑏 𝑛 cos(2𝜋𝑛𝜏) + √ . 𝑛=1 25 This approximation is allowed as long as the thermal wavelength the size of the sphere. 2 (4.194) √︁ 2𝜋/𝑚 is much smaller than 197 In writing this mode expansion, we have defined the mode numbers and coefficients Î 𝑎𝑖𝑛 , 𝑏𝑖𝑛 so that the measure on the moduli space of geodesics is locally 𝑖 𝑑𝑎𝑖1 𝑑𝑏𝑖1 . We furthermore imposed that each mode should be unit-normalized under the inner ∫ product ⟨𝑦, 𝑦⟩ = 2 𝑑𝜏|𝑦| 2 . The path integral can then be reduced to a product of ordinary integrals over 𝑎𝑖𝑛 , 𝑏𝑖𝑛 :26   2  ∫ ∫ 1 𝑦¤ 𝑖 2 𝐷𝑦 (𝜏) exp −𝑚𝑖 𝑑𝜏 − 𝜋𝑦 4𝜋 0 ) (∞ ∫ 𝑑−2 Ö h 𝑚𝜋  i ∫ h𝑚 𝜋 i Ö 𝑖 𝑖 𝑑𝑎𝑖𝑛 𝑑𝑏𝑖𝑛 exp − 𝑛2 − 1 (𝑎𝑖𝑛 ) 2 + (𝑏𝑖𝑛 ) 2 × 𝑑𝑏𝑖0 exp (𝑏𝑖0 ) 2 . = 2 2 𝑖=1 𝑛=1 (4.195) Modes with 𝑛 > 1 are massive and correspond to oscillations around the geodesic. The modes with 𝑛 = 0 are tachyonic. After adding these tachyonic fluctuations, the trajectory of the particle will deviate from the great circle and shrink towards a point, reducing its length (see e.g. Figure 4 and the surrounding discussions in [79]). Consequently, the classical trajectory we are expanding around is an unstable saddle point of the Nambu-Goto action. To define the integral over the tachyonic directions, ∫∞ 2 2 we need to make sense of integral of the form −∞ 𝑑𝑥 𝑒 𝛼 𝑥 where Re(𝛼2 ) > 0. We do so by analytically continuing from the region where Re(𝛼2 ) < 0 and defining: √ ∫ ∞ 𝜋 𝛼2 𝑥 2 , Re 𝛼2 > 0. (4.196) 𝑑𝑥 𝑒 := (±𝑖) |𝛼| −∞ Furthermore, we make the same choice of sign in (4.196) for integrals over all such tachyonic modes. This prescription, however, does not remove all ambiguities. Since there are 𝑑 − 2 perpendicular directions and each of them contributes a single tachyonic mode, the overall phase factor is (±𝑖) (𝑑−2) . When 𝑑 is even, both choices of sign give the same result. But when 𝑑 is odd, the ambiguity remains. As we will see in Appendix 4.10, this remaining two-fold ambiguity is related to a contour ambiguity in performing a Borel resummation. The zero modes with 𝑛 = 1 represent rotations of the great circle. The corresponding mode coefficients 𝑎𝑖1 , 𝑏𝑖1 are local coordinates on the moduli space of geodesics. 26 The path integral measure 𝐷 𝑦(𝜏) is induced from a choice of inner product. This inner product must be local — i.e. an integral over 𝜏 — in order for the path integral to satisfy proper cutting and gluing rules. Î However, the overall coefficient of the inner product does not matter because det(const) = 𝑛∈Z const = 1 in 𝜁-function regularization. As long as the modes are all normalized the same way with respect to the inner product, the measure factorizes into an integral over each Î Î mode coefficient 𝐷 𝑦 = 𝑖 𝑑𝑏 𝑖0 𝑛,𝑖 𝑑𝑎 𝑖𝑛 𝑑𝑏 𝑖𝑛 . 198 Every geodesic on 𝑆 𝑑−1 is an intersection of 𝑆 𝑑−1 with a 2-plane in R𝑑 containing the origin, so this moduli space is the same as the Grassmannian Gr(2, R𝑑 ). Hence, the zero mode integral is the volume of this Grassmannian: 𝑑−2 ∫ Ö vol(𝑂 (𝑑)) vol(𝑆 𝑑−1 ) × vol(𝑆 𝑑−2 ) 𝑑𝑎𝑖1 𝑑𝑏𝑖1 = vol(Gr(2, R𝑑 )) = = . 1) vol(𝑂 (2)) × vol(𝑂 (𝑑 − 2)) 2vol(𝑆 𝑖=1 (4.197) We can finally explicitly calculate the path-integral over 𝑦𝑖 (𝜏):   2  ∫ ∫ 1 𝑦¤ 𝑖 2 𝐷𝑦 (𝜏) exp −𝑚𝑖 𝑑𝜏 − 𝜋𝑦 4𝜋 0 # " √︂ ∞ ′ ∞ ′ 𝑑−2 Ö Ö Ö 2 1 2 × = vol(Gr(2, R𝑑 )) (±𝑖) 𝑚 𝑚𝑖 𝑛=1 𝑛2 − 1 𝑖 𝑛=1 𝑖=1 (±𝑖) 𝑑−2 𝑚𝑖𝑑−2 = , Γ(𝑑 − 1) (4.198) Î where 𝑛 ′ means we skip the mode 𝑛 = 1. We computed the infinite product above using zeta function regularization [8]. In particular, the following formulas are useful for such computations: ∞ ∞ Ö Ö √ 𝑎 = 𝑎 −1/2 (𝑎 > 0). (4.199) 𝑛 = 2𝜋, and 𝑛=1 𝑛=1 Putting everything together, we obtain (4.144). • Worldline path integral on lens space Three dimensional homogeneous lens space 𝐿(𝑞; 1) can be defined as the quo tient of the three sphere 𝑆 3 = (𝑧1 , 𝑧2 ) ∈ C2 |𝑧1 | 2 + |𝑧2 | 2 = 1 by the equivalence relation (𝑧1 , 𝑧2 ) ∼ (𝑒 2𝜋𝑖/𝑞 𝑧1 , 𝑒 2𝜋𝑖/𝑞 𝑧2 ). Geodesics in 𝐿(𝑞; 1) comes in two types: the contractable "long geodesics" with length 2𝜋, and the non-contractable "short geodesics" with length 2𝜋/𝑞. If 𝑞 ≠ 2, each short geodesic is an intersection of a complex line 𝛼𝑧1 + 𝛽𝑧2 = 0 with 𝑆 3 , so the corresponding moduli space is Gr(1, C2 ). When 𝑞 = 2, every real two-plane 𝛼𝑧1 + 𝛽𝑧 2 + 𝛾𝑧 1 + 𝛿𝑧 2 in R4 intersects 𝑆 3 at a short geodesic, so the corresponding moduli space is Gr(2, R4 ). To compute worldline instanton contributions to the lens space partition function, we choose the classical trajectory 𝜃 (𝜏) to wind around a short geodesic and integrate over perpendicular fluctuations obeying a slightly different boundary condition: 2𝜋 © 𝑦 1 (1) ª ©cos( 2𝜋 © 𝑦 1 (0) ª 𝑞 ) − sin( 𝑞 ) ª ­ ®=­ ®­ ®. ­ ® ­ ®­ ® 2𝜋 2𝜋 2 (1) 2 (0) 𝑦 sin( ) cos( ) 𝑦 𝑞 𝑞 ¬« « ¬ « ¬ (4.200) 199 Introducing the complex variable 𝑧(𝜏) = 𝑦 1 (𝜏) +𝑖𝑦 2 (𝜏), we have the following mode expansion: ∑︁ 𝑧(𝜏) = 𝑐 𝑛 𝑒 2𝜋𝑖(𝑛+1/𝑞)𝜏 . (4.201) 𝑛∈Z Plugging in 𝜃 (𝜏) = 2𝜋𝜏 𝑞 , we find the worldline path integral √︄   2 ∫ 1 2   𝑑𝜃 𝑑𝑧  𝐷𝑧(𝜏) exp −𝑚𝑖 𝑑𝜏 (1 − |𝑧| 2 ) + 𝑑𝜏 𝑑𝜏  0   ∫ h   i 2𝑚𝑖 𝜋 Ö = 𝑒− 𝑞 𝑑𝑐 𝑛 𝑑𝑐 𝑛 exp −𝑚𝑖 𝜋 𝑛2 𝑞 + 2𝑛 |𝑐 𝑛 | 2 . ∫ (4.202) 𝑛∈Z When 𝑞 ≠ 2, the only zero mode is 𝑐 0 𝑒 (2𝜋𝑖/𝑞)𝜏 . In this case, following the treatment of zero modes outlined above, we find h   i Ö∫ Ö 1 1 𝑚𝑖 , 𝑑𝑐 𝑛 𝑑𝑐 𝑛 exp −𝑚𝑖 𝜋 𝑛2 𝑞 + 2𝑛 |𝑐 𝑛 | 2 = vol(Gr(1, C2 )) = 2𝜋 (2 𝑚 𝑛 + 𝑛𝑞) 2 sin( ) 𝑖 𝑛∈Z 𝑛∈Z 𝑞 ≠0 (4.203) where we’ve used vol(𝑈 (𝑑)) 𝜋 𝑘 (𝑑−𝑘) 𝐺 (𝑘 + 1)𝐺 (𝑑 − 𝑘 + 1) = , vol(𝑈 (𝑘)) × vol(𝑈 (𝑑 − 𝑘)) 𝐺 (𝑑 + 1) (4.204) where 𝐺 (𝑧) is the Barnes G-function. vol(Gr(𝑘, C𝑑 )) = When 𝑞 = 2, there are two zero modes 𝑐 0 𝑒 𝜋𝑖𝜏 and 𝑐 −1 𝑒 −𝜋𝑖𝜏 . Thus, h   i ∫ Ö∫ Ö 2 2 𝑑𝑐 𝑛 𝑑𝑐 𝑛 exp −𝑚𝑖 𝜋 2𝑛 + 2𝑛 |𝑐 𝑛 | = 𝑑𝑐 0 𝑑𝑐0 𝑑𝑐 −1 𝑑𝑐−1 𝑛∈Z 𝑛∈Z≠0,−1 1 1 . 2𝑚𝑖 𝑛 (1 + 𝑛) (4.205) The integral over zero mode coefficients 𝑐 0 and 𝑐 −1 is not exactly the same as the volume of Gr(2, R4 ): the correct local coordinates on Gr(2, R4 ) should be the coefficient in front of cos(𝜋𝜏), 𝑖 cos(𝜋𝜏) sin(𝜋𝜏) and 𝑖 sin(𝜋𝜏), therefore 𝑑Vol(Gr(2, R4 )) = 4 𝑑𝑐 0 𝑑𝑐0 𝑑𝑐 −1 𝑑𝑐−1 . Combing zero modes with massive modes, we get h   i 𝑚2 Ö∫ 2 2 𝑑𝑐 𝑛 𝑑𝑐 𝑛 exp −𝑚𝑖 𝜋 2𝑛 + 2𝑛 |𝑐 𝑛 | = 𝑖 . 2 𝑛∈Z (4.206) (4.207) Notice that tachyonic modes never appeared in the computations above. This is consistent with the fact that short geodesics are non-contractable! 200 • World line contribution to spin-refined partition function in 4D We would like to compute single-particle non-perturbative corrections for a spinrefined sphere partition function in 4D. As explained in Section 4.2, the corresponding EFT bundle is non-trivial. By winding around the fundamental cycle in 𝐿(𝑞; 1), one also advances by 𝛽 along the thermal circle whose total length is 𝑞𝛽. Thus, after Kaluza-Klein reduction, we have a flat connection on an 𝑆 1𝑞𝛽 bundle over 𝐿(𝑞; 1) ∮ whose holonomy along a short geodesic is given by 𝐴 = 𝛽. The corresponding 2𝜋𝑖𝔮𝑖 𝑖 phase in the worldline action is thus exp( 2𝜋𝑖𝔮 𝑞𝛽 𝛽) = exp( 𝑞 ). Since the connection is flat, the classical equations of motion will not be affected, and we can compute the "Nambu-Goto part" of the worldline path integral in exactly the same way as for the lens space partition function. Putting everything together, we arrive at (4.146). A closer look at free field theory In this subsection, we discuss non-perturbative corrections in free field partition functions. In particular, we will show that they indeed have the structure predicted in Section 4.9. Actually, thanks to the Fock space structure of the Hilbert space, we can easily predict the leading terms in all non-perturbative corrections to free theory partition functions using worldline instantons. The "free-theory upgraded predictions" in (4.211), (4.213), and (4.214) can also be checked explicitly, as we will do in Section 4.10. In the language of Section 4.10, there is a worldline instanton correction for each Poincaré irrep in the 𝑑−1 dimensional massive theory on R𝑑−1 . In a free theory, the Hilbert space furthermore has the structure of a Fock space, so that for any symmetry generator 𝑔, we have ! ∞ ∞ ∞ ∑︁ ∑︁ ∑︁ 1 TrH1 [𝑔ℓ ] , (4.208) TrH [𝑔] = TrH𝑛 [𝑔] = TrSym𝑛 (H1 ) [𝑔] = exp ℓ 𝑛=1 𝑛=1 ℓ=1 where H1 is the single-particle Hilbert space. We can apply this result to a free massive theory on 𝑆 𝑑−1 as follows. Consider the sphere 𝑆 𝑑−1 in "angular" quantization, where we choose an angle 𝜙 ∈ [0, 2𝜋) and slice the path integral on slices of constant 𝜙. Each spatial slice is a 𝑑−2dimensional hemisphere, with some boundary conditions at the equator of the hemisphere. (Our arguments will be heuristic because we will be vague about the nature of these boundary conditions, see [4] for a recent discussion.) Let the 201 corresponding Hamiltonian be 𝐻. The sphere partition function is then ! ∞ ∑︁ 1 −2𝜋𝐻 −2𝜋ℓ𝐻 TrH (𝑒 ) = exp TrH1 [𝑒 ] . ℓ ℓ=1 (4.209) Each single-particle worldline instanton discussed in Section 4.10 computes the leading contribution to TrH1 [𝑒 −2𝜋𝐻 ]. To instead compute TrH1 [𝑒 −2𝜋ℓ𝐻 ], we should expand around a classical trajectory 𝜃 (𝜏) = 2𝜋ℓ𝜏 that winds ℓ times around the great circle. Expanding the worldline path integral around this trajectory, we find:    (𝑑−2) (2ℓ−1) 𝑚 𝑑−2 ∑︁ 1 −2𝜋ℓ𝐻 −2𝜋ℓ𝑚 (±𝑖) 1+𝑂 . (4.210) TrH1 [𝑒 ]= 𝑒 Γ(𝑑 − 1) 𝑚 𝑚 For a free scalar CFT on 𝑆 𝑑−1 × 𝑆 1𝛽 , the single-particle states are KK modes, so that we expect log(𝑍free [𝑆 𝑑−1 × 𝑆 1𝛽 ]) ∼ ∞ ∑︁ ∑︁ 1 ℓ 𝑚 ℓ=1 TrH1 𝑒 −2𝜋ℓ𝐻 , (4.211) where each TrH1 𝑒 −2𝜋ℓ𝐻 is given by (4.210), and 𝑚 runs over the spectrum of KK masses. Here, "∼" means that the right-hand side displays the nonperturbative terms in the left-hand side. We can derive similar results for a free massive theory on the lens space 𝐿 (𝑞; 1). If we consider an instanton on the locus |𝑧 1 | = 1, wrapping ℓ times around the "short 2 𝜋ℓ𝑖 2 𝜋ℓ geodesic," then the lens space worldline path integral evaluates TrH1 [𝑒 − 𝑞 𝐻− 𝑞 𝐽 ], where 𝐽 generates a rotation in the 𝑧2 plane. Hence for the free scalar on 𝐿 (𝑞; 1) ×𝑆 1𝛽 , when 𝑞 is odd: log(𝑍free [𝐿 (𝑞; 1) × 𝑆 1𝛽 ]) = = ∞ ∑︁ ∑︁ 1 ℓ 2 𝜋ℓ 2 𝜋ℓ𝑖 TrH1 [𝑒 − 𝑞 𝐻− 𝑞 𝐽 ] 𝑚 ℓ=1 ∞ ∑︁ ∑︁ ∑︁ ∑︁ 1 2 𝜋ℓ 2 𝜋𝑖ℓ 1 TrH1 [𝑒 −2𝜋ℓ𝐻 ] + TrH1 [𝑒 − 𝑞 𝐻− 𝑞 𝐽 ]. 𝑞ℓ ℓ 𝑚 ℓ=1 𝑚 ℓ∉𝑞Z (4.212) In the last line, we singled out terms where ℓ ∈ 𝑞Z. Up to a 1/𝑞 factor, the first sum is exactly the same as the nonperturbative terms in log(𝑍free [𝑆 𝑑−1 ×𝑆 1𝛽 ]). Indeed, since the short geodesic generates 𝜋1 (𝐿 (𝑞; 1)) ≃ Z𝑞 , after winding around 𝑞 times, the loop becomes contractable. Notice that each contractable long geodesic in 𝐿(𝑞; 1) will be lifted to 𝑞 different geodesics in 𝑆 3 , so the moduli space of a long geodesic 202 in 𝐿(𝑞; 1) should have 1/𝑞-th the volume of Gr(2, R4 ) and this is the origin of the 1/𝑞 factor. Combining everything, the leading order terms are log(𝑍free [𝐿 (𝑞; 1) × 𝑆 1𝛽 ]) ∼ ∑︁ 𝑚 ∞ ∞ 1 ∑︁ −2𝜋ℓ𝑚 𝑚 2 ∑︁ − 2 𝜋ℓ𝑚 𝑚 𝑒 𝑒 𝑞 + − 𝑞 ℓ=1 2ℓ ℓ=1 2ℓ sin( 2𝜋ℓ 𝑞 ) ! (odd 𝑞), ℓ∉𝑞Z (4.213) where "∼" means we show nonperturbative corrections. Similarly when 𝑞 is even, we find log(𝑍free [𝐿(𝑞; 1) × 𝑆 1𝛽 ]) ∼ ∑︁ 𝑚 ∞ ∞ ∞ ∑︁ 2 𝜋ℓ𝑚 1 ∑︁ −𝜋(2ℓ−1)𝑚 𝑚 2 𝑚 1 ∑︁ −2𝜋ℓ𝑚 𝑚 2 𝑒− 𝑞 + 𝑒 + 𝑒 − 2𝜋ℓ 𝑞 ℓ=1 2ℓ 𝑞 2ℓ − 1 ) 2ℓ sin( ℓ=1 ℓ=1 𝑞 ! ℓ∉(𝑞/2)Z (4.214) Partition function from functional determinants In this section, we use the methods of [12] to compute the partition function of free scalar theories by explicitly summing over the Laplacian spectrum. We will find agreement with (4.211), (4.213), and (4.214). In particular, this will verify the general predictions of Section 4.10 for free theories. Consider a massive scalar field on 𝑆 𝑑−1 with the action   ∫ 1 1 (𝑑 − 2) 2 𝑑−1 √ 𝜇 𝜈 2 2 𝑑 𝑥 𝑔 𝑔 𝜇𝜈 𝜕 𝜙𝜕 𝜙 + 𝑚 𝜙 + 𝑅𝜙 , 𝑆free = 2 4 (𝑑 − 1) (4.215) with 𝑅 = (𝑑 − 1)(𝑑 − 2). Here, we have chosen the curvature coupling 12 𝜉 𝑅𝜙2 to have the form appropriate for a conformally-coupled scalar in 𝑑-dimensions, dimensionally reduced to 𝑑−1 dimensions. However, this coupling will not affect the leading form of nonperturbative corrections that are our main focus. The partition function is the functional determinant   −1/2 (𝑑 − 2) 2 𝑑−1 2 𝑍massive [𝑆 ] = det Δ + 𝑚 + , (4.216) 4 where Δ is the Laplace operator on 𝑆 𝑑−1 , which has eigenvalues 𝜆 𝑘 = 𝑘 (𝑘 + 𝑑 − 2) with degneracies 𝑑 𝑘 = (𝑑−3+𝑘)!(𝑑−2+2𝑘) . Let us focus on 𝑑 = 4 and 𝑑 = 6, i.e the 𝑘!(𝑑−2)! sphere partition function of a massive theory on 𝑆 3 and 𝑆 5 . (even 𝑞). 203 Using zeta function regularization, we find   ∞ 𝜋𝑚 3 ∑︁ 𝑒 −2𝜋ℓ𝑚 𝑚 1 2 log(𝑍massive [𝑆 ]) = − 𝑚 + 2+ 2 3 , 6 2ℓ 𝜋ℓ 2𝜋 ℓ ℓ 3 (4.217)   ∞ 𝜋 5 𝜋 3 ∑︁ 𝑒 −2𝜋ℓ𝑚 4 2𝑚 3 3 2 log(𝑍massive [𝑆 ]) = − 𝑚 − 𝑚 + +𝑚 1+ 2 2 𝑚 + 120 72 24ℓ 𝜋ℓ 𝜋 ℓ ℓ=1  ! 3 3 1 1 +𝑚 + 2 2+ 4 4+ 2 2 . (4.218) 𝜋ℓ 𝜋 ℓ 2𝜋 ℓ 2𝜋 ℓ 5 We now need to sum over the KK masses to obtain 𝑍free [𝑆 𝑑−1 × 𝑆 1𝛽 ]. Let us consider Z2 -twisted free scalar fields as an example, where the Z2 twist is introduced to remove a zero mode upon dimensional reduction. The KK mass spectrum is 𝑚 = |(2𝑛−1)/𝛽| for 𝑛 ∈ Z. Thus, the free energy in 𝑑 = 4 and 𝑑 = 6 is   ∞ 2 𝜋2 2 ∑︁ ∑︁ 2𝑛 − 1 1 1 (2𝑛 − 1) − 2 𝜋 (2𝑛−1)ℓ 3 1 𝛽 + + 2 3 , log(𝑍free [𝑆 × 𝑆 𝛽 ]) ∼ − 𝑒 2 2 ℓ 𝛽 𝛽ℓ 2𝜋 ℓ 𝑛=1 ℓ=1 (4.219) log(𝑍free [𝑆 5 × 𝑆 1𝛽 ]) 2 ∞ ∑︁ ∞ ∑︁ 𝑒 −2𝜋 (2𝑛−1)ℓ/𝛽 𝜋 4 (2𝑛 − 1) 4 2𝜋 2 (2𝑛 − 1) 3 ∼ + 4 12ℓ 𝛽 ℓ𝛽3 𝑛=1 ℓ=1    ! 𝜋 2 (2𝑛 − 1) 2 3 3 𝜋(2𝑛 − 1) 1 3 1 + + 1+ 2 2 + + + , 𝛽 𝜋ℓ 𝜋 2 ℓ 2 2𝜋 4 ℓ 4 2𝜋 2 ℓ 2 𝛽2 𝜋 ℓ (4.220) where "∼" means we only show non-perturbative corrections. These results are in agreement with the general form (4.211) predicted from worldline instantons, together with the Fock-space structure of the free theory. The same approach can be generalized to compute a lens space partition function in 4D. We display the result here for a more direct comparison with the twisted partition function:   ∞ 𝑚 𝜋𝑚 3 1 ∑︁ 𝑒 −2𝜋ℓ𝑚 1 2 log(𝑍massive [𝐿(𝑞; 1)]) = − 𝑚 + + 6𝑞 𝑞 ℓ 2ℓ 𝜋ℓ 2𝜋 2 ℓ 2 2 𝜋ℓ𝑚 ∞ 𝑚 ∑︁ 𝑒− 𝑞 𝑞    + + 2ℓ 4𝜋ℓ 2 2𝜋ℓ ℓ=1 sin 𝑞 ℓ∉𝑞Z (𝑞 ∈ 2Z + 1), (4.221) 204   ∞ 𝜋𝑚 3 1 ∑︁ 𝑒 −2𝜋ℓ𝑚 𝑚 1 log(𝑍massive [𝐿 (𝑞; 1)]) = − 𝑚2 + + 2 2 6𝑞 𝑞 ℓ 2ℓ 𝜋ℓ 2𝜋 ℓ   ∞ 1 ∑︁ 𝑒 −𝜋(2ℓ−1)𝑚 2𝑚 2 2 + 𝑚 + + 𝑞 ℓ 2ℓ − 1 𝜋(2ℓ − 1) 𝜋 2 (2ℓ − 1) 2 2 𝜋ℓ𝑚 ∞ 𝑚 ∑︁ 𝑞  𝑒− 𝑞   + + 2ℓ 4𝜋ℓ 2 2𝜋ℓ ℓ=1 sin 𝑞 𝑞 (𝑞 ∈ 2Z). (4.222) ℓ∉ 2 Z In Appendix 4.10, we computed the high temperature expansion of a free scalar 2 𝜋𝑖 2 𝜋𝑖 twisted by 𝑒 − 𝑞 𝐽12 − 𝑞 𝐽34 . Restricting to odd 𝑞 and inserting a Z2 twist to remove the zero mode, we find  h i −𝛽𝐻− 2𝑞𝜋𝑖 𝐽12 − 2𝑞𝜋𝑖 𝐽34 𝑁 log Tr 𝑒 (−1)   ∑︁ ∑︁ 2ℓ (2𝑛−1) 𝜋 2 (2𝑛 − 1) 2 𝜋 2 2𝑛 − 1 1 − 𝑞𝛽 + 2 2 + 3 2 ∼− 𝑒 3 𝛽2 ℓ𝑞 𝛽ℓ 𝑞 2ℓ 𝑞𝜋 𝑛=1 ℓ=1   ∞ ∑︁ ∞ cos( 2𝜋ℓ ( 𝑞−1 − 𝑛)) ∑︁ 𝜋2 (2𝑛 + 1)𝜋 𝑞 − 2ℓ (2𝑛+1) 𝑞 2 2𝛽 𝑞 + 2 , (4.223) + 𝑒 ℓ𝑞𝛽 2ℓ 𝜋 sin( 2𝜋ℓ ℓ=1 𝑛=0 𝑞 ) ℓ∉𝑞Z where "∼" means we only show non-perturbative corrections and 𝑁 is the 𝜙-number operator, so that (−1) 𝑁 implements the Z2 twist. Note that this is not quite identical to the lens space partition function (4.221) summed over the mass spectrum 𝑚 = |2𝑛+1|𝜋 𝑞𝛽 . To obtain the twisted partition function (4.223), we must additionally modify the lens space result to account for the nontrivial background gauge field. To do so, we 𝑖 multiply each term in the summation over short geodesics by a phase exp( 2𝜋𝑖ℓ𝔮 𝑞 ). For a Z2 twisted free scalar field, the KK charge spectrum is 𝔮 = 𝑛 + 12 + 𝑞2 . After putting in all the phases, we recover (4.223). Finally, note that this result is consistent with the general prediction (4.146) from worldline instantons. Free theories in odd dimension The examples presented in Section 4.10 were all in even 𝑑. In odd 𝑑, we face the puzzle that the contribution of a worldline instanton (4.144) is imaginary (even when the partition function should be real), and furthermore its phase depends on how we choose the integration contours for tachyonic modes. The proper interpretation of these kinds of contributions is explained for example in [79]. They can be understood as characterizing singularities in the Borel plane when computing the partition function via Borel resummation. 205 In this appendix, we provide a quick summary of the discussion from [79] in the example of a massive free scalar on 𝑆 2 . (We can think of this theory as the contribution of a single KK mode to the partition function of the 3D free scalar on 𝑆 2 × 𝑆 1𝛽 .) We can compute the partition function in terms of the heat trace, following Appendix 4.10. The heat trace on 𝑆 2 is h Tr 𝑒 2 −𝑡 (Δ+ ( 𝑑−2 2 ) ) i = ∞ ∑︁ 1 2 (2𝑘 + 1)𝑒 −𝑡 (𝑘+ 2 ) = 𝑘=0 = 1 2 1 |𝑘 + |𝑒 −𝑡 (𝑘+ 2 ) 2 𝑘∈Z ∑︁   2𝜋ℓ 𝜋ℓ − 3/2 𝐹 √ , 𝑡 𝑡 𝑡 ∑︁  1 (4.224) ℓ∈Z where 𝐹 (𝑧) is Dawson’s function which admits the following asymptotic expansion near 𝑧 = ∞:   2𝑛+1 ∞ ∑︁ (2𝑛 − 1)!! 1 𝐹 (𝑧) ∼ . (4.225) 𝑛+1 𝑧 2 𝑛=0 The heat trace therefore has the following expansion near 𝑡 = 0: h Tr 𝑒 2 −𝑡 (Δ+ ( 𝑑−2 2 ) ) i ∼ ∞ ∑︁ 𝑎𝑛𝑡 𝑛−1 (−1) 𝑛+1 (1 − 21−2𝑛 ) , 𝑎𝑛 = 𝐵2𝑛 , 𝑛! (4.226) 𝑛=0 where 𝐵2𝑛 are the Bernoulli numbers. Í 𝑛 Let us now perform a Borel resummation of the series Φ ≡ ∞ 𝑛=0 𝑎 𝑛 𝑡 : ∫ ∞ 2 2 S [Φ] (𝑡) = √ 𝑑𝜁 𝑒 −𝜁 /𝑡 BΦ(𝜁), 𝑡 0 (4.227) where the Borel transformed series BΦ(𝜁) is BΦ(𝜁) = ∞ ∑︁ 𝑎𝑛 1 𝑛=0 Γ(𝑛 + 2 ) 1 𝜁 𝜁 2𝑛 = √ . 𝜋 sin(𝜁) (4.228) With this expression for BΦ, the integral in (4.227) is apparently ill-defined since there are poles located at 𝜁 = ℓ𝜋, ℓ ∈ Z on the positive real axis. A contour prescription is needed to define the integral. One natural option is to deform the integration contour to pass over (we will refer to the corresponding contour as C+ ) or under (we will refer to the corresponding contour as C− ) the poles. However, these two choices are not equivalent. Their difference is the sum of residues at the poles: ∫  𝜋  3/2 ∑︁ 2 2 2 2 𝑑𝜁 𝑒 −𝜁 /𝑡 BΦ(𝜁) = 2𝑖𝑡 (−1) ℓ+1 |ℓ|𝑒 −ℓ 𝜋 /𝑡 . (4.229) √ 𝑡 𝑡 C+ −C− ℓ≠0 206 The ambiguity in integration contour leads to an ambiguity in the Borel-resummed heat trace: ∫ 2 h  𝜋  3/2 ∑︁ 2 i ℓ2 𝜋2 𝜁 𝑒 −𝜁 /𝑡 2 −𝑡 (Δ+ ( 𝑑−2 ) ) 2 𝑑𝜁 Tr 𝑒 𝜎ℓ± (−1) ℓ+1 |ℓ|𝑒 − 𝑡 , = √ 3/2 + 2𝑖 sin 𝜁 𝑡 𝜋𝑡 C± ℓ≠0 (4.230) where 𝜎ℓ± are arbitrary coefficients which jump as we switch from C+ to C− . If we require the heat trace to be real when 𝑡 ∈ R+ , then the 𝜎ℓ± are fixed to be ± 21 . This is equivalent to a principal value prescription for the Borel integral: " # ∫ ∞ h i −𝜁 2 /𝑡 𝑑−2 2 𝜁 𝑒 2 Tr 𝑒 −𝑡 (Δ+ ( 2 ) ) = √ 3/2 𝑑𝜁 P . (4.231) sin 𝜁 𝜋𝑡 0 ∫ 2 To compute log 𝑍, we must supply a factor of 𝑒 −𝑡𝑚 and integrate 𝑑𝑡/𝑡. If we start with (4.231), we get a valid integral representation for the partition function. However, if we start with (4.230) and perform the 𝑡-integral term by term, we obtain the series of nonperturbative corrections 2 log(𝑍free [𝑆 ]) = ±𝑖  ∞ ∑︁ (−1) ℓ 𝑒 −2𝜋𝑚ℓ ℓ non-perturbative ℓ=1  1 𝑚+ , 2𝜋ℓ (4.232) whose leading terms in 𝑚 agree precisely with the worldline instanton predictions (4.210) and (4.211) when 𝑑 = 3. To summarize, worldline instantons encode residues of certain singularities in the Borel plane. We conjecture that this remains true in interacting theories. In general, the thermal effective action gives an asymptotic expansion in 𝛽. When we Borel-resum this expansion, we encounter singularities in the Borel plane coming from worldline instantons (together with other nonperturbative effects like instanton graphs). 207 BIBLIOGRAPHY [1] Larry F Abbott and Stanley Deser. “Stability of gravity with a cosmological constant”. In: Nuclear Physics B 195.1 (1982), pp. 76–96. [2] Allan Adams et al. “Causality, analyticity and an IR obstruction to UV completion”. In: JHEP 0610 (2006), p. 014. doi: 10.1088/1126-6708/ 2006/10/014. arXiv: hep-th/0602178 [hep-th]. [3] Nima Afkhami-Jeddi et al. “Einstein gravity 3-point functions from conformal field theory”. In: JHEP 12 (2017), p. 049. doi: 10.1007/JHEP12(2017) 049. arXiv: 1610.09378 [hep-th]. [4] Nicholas Agia and Daniel L. Jafferis. “Angular Quantization in CFT”. In: (Apr. 2022). arXiv: 2204.11872 [hep-th]. [5] Luis F. Alday and Jin-Beom Bae. “Rademacher Expansions and the Spectrum of 2d CFT”. In: JHEP 11 (2020), p. 134. doi: 10.1007/JHEP11(2020) 134. arXiv: 2001.00022 [hep-th]. [6] Luis F. Alday and Juan Martin Maldacena. “Comments on operators with large spin”. In: JHEP 11 (2007), p. 019. doi: 10.1088/1126-6708/2007/ 11/019. arXiv: 0708.0672 [hep-th]. [7] Kuroush Allameh and Edgar Shaghoulian. “Modular invariance and thermal effective field theory in CFT”. In: (Feb. 2024). arXiv: 2402.13337 [hep-th]. [8] Jean-Paul Allouche. “Zeta-regularization of arithmetic sequences”. In: European Physical Journal Web of Conferences. Vol. 244. European Physical Journal Web of Conferences. Jan. 2020, 01008, p. 01008. doi: 10.1051/ epjconf/202024401008. [9] Tarek Anous et al. “OPE statistics from higher-point crossing”. In: JHEP 06 (2022), p. 102. doi: 10.1007/JHEP06(2022)102. arXiv: 2112.09143 [hep-th]. [10] Arash Arabi Ardehali and Sameer Murthy. “The 4d superconformal index near roots of unity and 3d Chern-Simons theory”. In: JHEP 10 (2021), p. 207. doi: 10.1007/JHEP10(2021)207. arXiv: 2104.02051 [hep-th]. [11] Richard Arnowitt, Stanley Deser, and Charles W Misner. “Dynamical structure and definition of energy in general relativity”. In: Physical Review 116.5 (1959), p. 1322. [12] M. Asorey et al. “Thermodynamics of conformal fields in topologically non-trivial space-time backgrounds”. In: JHEP 04 (2013), p. 068. doi: 10.1007/JHEP04(2013)068. arXiv: 1212.6220 [hep-th]. 208 [13] Benjamin Assel et al. “The Casimir Energy in Curved Space and its Supersymmetric Counterpart”. In: JHEP 07 (2015), p. 043. doi: 10.1007/ JHEP07(2015)043. arXiv: 1503.05537 [hep-th]. [14] Nabamita Banerjee et al. “Constraints on Fluid Dynamics from Equilibrium Partition Functions”. In: JHEP 09 (2012), p. 046. doi: 10 . 1007 / JHEP09(2012)046. arXiv: 1203.3544 [hep-th]. [15] F. Bastianelli, S. Frolov, and Arkady A. Tseytlin. “Conformal anomaly of (2,0) tensor multiplet in six-dimensions and AdS / CFT correspondence”. In: JHEP 02 (2000), p. 013. doi: 10.1088/1126-6708/2000/02/013. arXiv: hep-th/0001041. [16] Fiorenzo Bastianelli and Peter van Nieuwenhuizen. “Path integrals for quantum mechanics in curved space”. In: Path Integrals and Anomalies in Curved Space. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2006, pp. 1–2. [17] Alexandre Belin, Jan de Boer, and Diego Liska. “Non-Gaussianities in the statistical distribution of heavy OPE coefficients and wormholes”. In: JHEP 06 (2022), p. 116. doi: 10.1007/JHEP06(2022)116. arXiv: 2110.14649 [hep-th]. [18] Alexandre Belin, Jan De Boer, and Jorrit Kruthoff. “Comments on a stateoperator correspondence for the torus”. In: SciPost Phys. 5.6 (2018), p. 060. doi: 10.21468/SciPostPhys.5.6.060. arXiv: 1802.00006 [hep-th]. [19] Alexandre Belin, Diego M. Hofman, and Gregoire Mathys. “Einstein gravity from ANEC correlators”. In: (2019). arXiv: 1904.05892 [hep-th]. [20] Alexandre Belin et al. “Generalized spectral form factors and the statistics of heavy operators”. In: JHEP 11 (2022), p. 145. doi: 10 . 1007 / JHEP11(2022)145. arXiv: 2111.06373 [hep-th]. [21] Alexandre Belin et al. “Universality of sparse 𝑑 > 2 conformal field theory at large 𝑁”. In: JHEP 03 (2017), p. 067. doi: 10.1007/JHEP03(2017)067. arXiv: 1610.06186 [hep-th]. [22] Francesco Benini and Stefano Cremonesi. “Partition Functions of N = (2, 2) Gauge Theories on S2 and Vortices”. In: Commun. Math. Phys. 334.3 (2015), pp. 1483–1527. doi: 10.1007/s00220-014-2112-z. arXiv: 1206.2356 [hep-th]. [23] Nathan Benjamin and Ying-Hsuan Lin. “Lessons from the Ramond sector”. In: SciPost Phys. 9.5 (2020), p. 065. doi: 10.21468/SciPostPhys.9.5. 065. arXiv: 2005.02394 [hep-th]. [24] Nathan Benjamin et al. “Light-cone modular bootstrap and pure gravity”. In: Phys. Rev. D 100.6 (2019), p. 066029. doi: 10.1103/PhysRevD.100. 066029. arXiv: 1906.04184 [hep-th]. 209 [25] Nathan Benjamin et al. “Universal asymptotics for high energy CFT data”. In: JHEP 03 (2024), p. 115. doi: 10.1007/JHEP03(2024)115. arXiv: 2306.08031 [hep-th]. [26] Sayantani Bhattacharyya. “Entropy current and equilibrium partition function in fluid dynamics”. In: JHEP 08 (2014), p. 165. doi: 10 . 1007 / JHEP08(2014)165. arXiv: 1312.0220 [hep-th]. [27] Sayantani Bhattacharyya et al. “Large rotating AdS black holes from fluid mechanics”. In: JHEP 09 (2008), p. 054. doi: 10 . 1088 / 1126 - 6708 / 2008/09/054. arXiv: 0708.1770 [hep-th]. [28] Jeff Bjoraker and Yutaka Hosotani. “Monopoles, dyons and black holes in the four-dimensional Einstein-Yang-Mills theory”. In: Phys. Rev. D 62 (2000), p. 043513. doi: 10.1103/PhysRevD.62.043513. arXiv: hepth/0002098. [29] Jefferson Bjoraker and Yutaka Hosotani. “Stable monopole and dyon solutions in the Einstein-Yang-Mills theory in asymptotically Anti-de Sitter space”. In: Phys. Rev. Lett. 84 (2000), p. 1853. doi: 10.1103/PhysRevLett. 84.1853. arXiv: gr-qc/9906091. [30] Matthias Blau and George Thompson. “Quantum Yang-Mills theory on arbitrary surfaces”. In: Int. J. Mod. Phys. A 7 (1992), pp. 3781–3806. doi: 10.1142/S0217751X9200168X. [31] Nikolay Bobev, Junho Hong, and Valentin Reys. “Holographic thermal observables and M2-branes”. In: JHEP 12 (2023), p. 054. doi: 10 .1007 / JHEP12(2023)054. arXiv: 2309.06469 [hep-th]. [32] Jan de Boer et al. “Black hole bound states in AdS(3) x S**2”. In: JHEP 11 (2008), p. 050. doi: 10 . 1088 / 1126 - 6708 / 2008 / 11 / 050. arXiv: 0802.2257 [hep-th]. [33] L Bonora, P Pasti, and M Bregola. “Weyl cocycles”. In: Classical and Quantum Gravity 3.4 (1986), p. 635. doi: 10.1088/0264-9381/3/4/018. url: https://dx.doi.org/10.1088/0264-9381/3/4/018. [34] Nicolas Boulanger. “General solutions of the Wess-Zumino consistency condition for the Weyl anomalies”. In: JHEP 07 (2007), p. 069. doi: 10. 1088/1126-6708/2007/07/069. arXiv: 0704.2472 [hep-th]. [35] Tomas Brauner et al. “Snowmass White Paper: Effective Field Theories for Condensed Matter Systems”. In: Snowmass 2021. Mar. 2022. arXiv: 2203.10110 [hep-th]. [36] Enrico M. Brehm, Diptarka Das, and Shouvik Datta. “Probing thermality beyond the diagonal”. In: Phys. Rev. D 98.12 (2018), p. 126015. doi: 10. 1103/PhysRevD.98.126015. arXiv: 1804.07924 [hep-th]. 210 [37] Tara Brendle, Nathan Broaddus, and Andrew Putman. “The mapping class group of connect sums of 𝑆 2 × 𝑆 1 ”. In: Trans. Amer. Math. Soc. 376 (2023), pp. 2557–2572. doi: 10.1090/tran/8758. arXiv: 2012.01529 [math.GT]. [38] J. David Brown and M. Henneaux. “Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity”. In: Commun. Math. Phys. 104 (1986), pp. 207–226. doi: 10 . 1007/BF01211590. [39] Joao Caetano, Shota Komatsu, and Yifan Wang. “Large charge ’t Hooft limit of N = 4 super-Yang-Mills”. In: JHEP 02 (2024), p. 047. doi: 10.1007/ JHEP02(2024)047. arXiv: 2306.00929 [hep-th]. [40] Roberto Camporesi and Atsushi Higuchi. “On the eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces”. In: J. Geom. Phys. 20 (1996), pp. 1–18. doi: 10 . 1016 / 0393 - 0440(95 ) 00042 - 9. arXiv: gr-qc/9505009. [41] P. Candelas and J. S. Dowker. “Field theories on conformally related spacetimes: Some global considerations”. In: Phys. Rev. D 19 (1979), p. 2902. doi: 10.1103/PhysRevD.19.2902. [42] Weiguang Cao, Tom Melia, and Sridip Pal. “Universal fine grained asymptotics of weakly coupled Quantum Field Theory”. In: (Nov. 2021). arXiv: 2111.04725 [hep-th]. [43] Vitor Cardoso and Oscar J. C. Dias. “Small Kerr-anti-de Sitter black holes are unstable”. In: Phys. Rev. D 70 (2004), p. 084011. doi: 10 . 1103 / PhysRevD.70.084011. arXiv: hep-th/0405006. [44] John Cardy, Alexander Maloney, and Henry Maxfield. “A new handle on three-point coefficients: OPE asymptotics from genus two modular invariance”. In: JHEP 10 (2017), p. 136. doi: 10.1007/JHEP10(2017)136. arXiv: 1705.05855 [hep-th]. [45] John L. Cardy. “Operator content and modular properties of higher dimensional conformal field theories”. In: Nucl. Phys. B 366 (1991), pp. 403–419. doi: 10.1016/0550-3213(91)90024-R. [46] John L. Cardy. “Operator Content of Two-Dimensional Conformally Invariant Theories”. In: Nucl. Phys. B 270 (1986), pp. 186–204. doi: 10.1016/ 0550-3213(86)90552-3. [47] Steven Carlip. “Logarithmic corrections to black hole entropy from the Cardy formula”. In: Class. Quant. Grav. 17 (2000), pp. 4175–4186. doi: 10.1088/0264-9381/17/20/302. arXiv: gr-qc/0005017. [48] Simon Caron-Huot. “Analyticity in Spin in Conformal Theories”. In: JHEP 09 (2017), p. 078. doi: 10.1007/JHEP09(2017)078. arXiv: 1703.00278 [hep-th]. 211 [49] Simon Caron-Huot. “Holographic cameras: an eye for the bulk”. In: JHEP 03 (2023), p. 047. doi: 10.1007/JHEP03(2023)047. arXiv: 2211.11791 [hep-th]. [50] Simon Caron-Huot and Vincent Van Duong. “Extremal Effective Field Theories”. In: JHEP 05 (2021), p. 280. doi: 10.1007/JHEP05(2021)280. arXiv: 2011.02957 [hep-th]. [51] Simon Caron-Huot et al. “AdS bulk locality from sharp CFT bounds”. In: JHEP 11 (2021), p. 164. doi: 10.1007/JHEP11(2021)164. arXiv: 2106.10274 [hep-th]. [52] Simon Caron-Huot et al. “Sharp boundaries for the swampland”. In: JHEP 07 (2021), p. 110. doi: 10.1007/JHEP07(2021)110. arXiv: 2102.08951 [hep-th]. [53] Horacio Casini et al. “Entanglement entropy and superselection sectors. Part I. Global symmetries”. In: JHEP 02 (2020), p. 014. arXiv: 1905.10487 [hep-th]. [54] Horacio Casini et al. “Entropic order parameters for the phases of QFT”. In: JHEP 04 (2021), p. 277. doi: 10.1007/JHEP04(2021)277. arXiv: 2008.11748 [hep-th]. [55] Noam Chai et al. “Thermal Order in Conformal Theories”. In: Phys. Rev. D 102.6 (2020), p. 065014. doi: 10.1103/PhysRevD.102.065014. arXiv: 2005.03676 [hep-th]. [56] Andrew Chamblin et al. “Charged AdS black holes and catastrophic holography”. In: Phys. Rev. D 60 (1999), p. 064018. doi: 10.1103/PhysRevD. 60.064018. arXiv: hep-th/9902170. [57] Andrew Chamblin et al. “Holography, thermodynamics and fluctuations of charged AdS black holes”. In: Phys. Rev. D 60 (1999), p. 104026. doi: 10.1103/PhysRevD.60.104026. arXiv: hep-th/9904197. [58] Jeevan Chandra et al. “Semiclassical 3D gravity as an average of large-c CFTs”. In: JHEP 12 (2022), p. 069. doi: 10.1007/JHEP12(2022)069. arXiv: 2203.06511 [hep-th]. [59] Chi-Ming Chang et al. “Proving the 6d Cardy Formula and Matching Global Gravitational Anomalies”. In: SciPost Phys. 11.2 (2021), p. 036. doi: 10. 21468/SciPostPhys.11.2.036. arXiv: 1910.10151 [hep-th]. [60] Cyuan-Han Chang et al. “Bootstrapping the 3d Ising stress tensor”. In: JHEP 03 (2025), p. 136. doi: 10.1007/JHEP03(2025)136. arXiv: 2411.15300 [hep-th]. [61] Peter Chang and J. S. Dowker. “Vacuum energy on orbifold factors of spheres”. In: Nucl. Phys. B 395 (1993), pp. 407–432. doi: 10.1016/05503213(93)90223-C. arXiv: hep-th/9210013. 212 [62] Yiming Chen and Gustavo J. Turiaci. “Spin-statistics for black hole microstates”. In: JHEP 04 (2024), p. 135. doi: 10.1007/JHEP04(2024)135. arXiv: 2309.03478 [hep-th]. [63] Shai M. Chester et al. “Carving out OPE space and precise 𝑂 (2) model critical exponents”. In: JHEP 06 (2020), p. 142. doi: 10.1007/JHEP06(2020) 142. arXiv: 1912.03324 [hep-th]. [64] Minjae Cho, Scott Collier, and Xi Yin. “Genus Two Modular Bootstrap”. In: JHEP 04 (2019), p. 022. doi: 10.1007/JHEP04(2019)022. arXiv: 1705.05865 [hep-th]. [65] Cyril Closset, Heeyeon Kim, and Brian Willett. “Seifert fibering operators in 3d N = 2 theories”. In: JHEP 11 (2018), p. 004. doi: 10 . 1007 / JHEP11(2018)004. arXiv: 1807.02328 [hep-th]. [66] Sidney R. Coleman, John Preskill, and Frank Wilczek. “Quantum hair on black holes”. In: Nucl. Phys. B 378 (1992), pp. 175–246. doi: 10.1016/ 0550-3213(92)90008-Y. arXiv: hep-th/9201059. [67] Scott Collier et al. “Universal dynamics of heavy operators in CFT2 ”. In: JHEP 07 (2020), p. 074. doi: 10.1007/JHEP07(2020)074. arXiv: 1912. 00222 [hep-th]. [68] Miguel S. Costa and Tobias Hansen. “Conformal correlators of mixedsymmetry tensors”. In: JHEP 02 (2015), p. 151. doi: 10.1007/JHEP02(2015) 151. arXiv: 1411.7351 [hep-th]. [69] Miguel S. Costa et al. “Spinning Conformal Correlators”. In: JHEP 11 (2011), p. 071. doi: 10 . 1007 / JHEP11(2011 ) 071. arXiv: 1107 . 3554 [hep-th]. [70] Paolo Creminelli, Oliver Janssen, and Leonardo Senatore. “Positivity bounds on effective field theories with spontaneously broken Lorentz invariance”. In: JHEP 09 (2022), p. 201. doi: 10.1007/JHEP09(2022)201. arXiv: 2207.14224 [hep-th]. [71] Michael Crossley, Paolo Glorioso, and Hong Liu. “Effective field theory of dissipative fluids”. In: JHEP 09 (2017), p. 095. doi: 10 . 1007 / JHEP09(2017)095. arXiv: 1511.03646 [hep-th]. [72] Diptarka Das, Shouvik Datta, and Sridip Pal. “Universal asymptotics of three-point coefficients from elliptic representation of Virasoro blocks”. In: Phys. Rev. D 98.10 (2018), p. 101901. doi: 10 . 1103 / PhysRevD . 98 . 101901. arXiv: 1712.01842 [hep-th]. [73] Diptarka Das, Yuya Kusuki, and Sridip Pal. “Universality in asymptotic bounds and its saturation in 2D CFT”. In: JHEP 04 (2021), p. 288. doi: 10.1007/JHEP04(2021)288. arXiv: 2011.02482 [hep-th]. 213 [74] Luca V. Delacretaz. “Heavy Operators and Hydrodynamic Tails”. In: SciPost Phys. 9.3 (2020), p. 034. doi: 10.21468/SciPostPhys.9.3.034. arXiv: 2006.01139 [hep-th]. [75] Lorenzo Di Pietro and Zohar Komargodski. “Cardy formulae for SUSY theories in 𝑑 = 4 and 𝑑 = 6”. In: JHEP 12 (2014), p. 031. doi: 10.1007/ JHEP12(2014)031. arXiv: 1407.6061 [hep-th]. [76] Óscar J. C. Dias, Jorge E. Santos, and Benson Way. “Lattice Black Branes: Sphere Packing in General Relativity”. In: JHEP 05 (2018), p. 111. doi: 10.1007/JHEP05(2018)111. arXiv: 1712.07663 [hep-th]. [77] Robbert Dijkgraaf et al. “A Black hole Farey tail”. In: (May 2000). arXiv: hep-th/0005003. [78] F. A. Dolan and H. Osborn. “Conformal four point functions and the operator product expansion”. In: Nucl. Phys. B599 (2001), pp. 459–496. doi: 10. 1016/S0550-3213(01)00013-X. arXiv: hep-th/0011040 [hep-th]. [79] Nicola Dondi et al. “Resurgence of the large-charge expansion”. In: JHEP 05 (2021), p. 035. doi: 10.1007/JHEP05(2021)035. arXiv: 2102.12488 [hep-th]. [80] J. S. Dowker. “Remarks on spherical monodromy defects for free scalar fields”. In: (Apr. 2021). arXiv: 2104.09419 [hep-th]. [81] J. S. Dowker and Gerard Kennedy. “Finite Temperature and Boundary Effects in Static Space-Times”. In: J. Phys. A 11 (1978), p. 895. doi: 10 . 1088/0305-4470/11/5/020. [82] M. J. Duff. “Twenty years of the Weyl anomaly”. In: Class. Quant. Grav. 11 (1994), pp. 1387–1404. doi: 10.1088/0264-9381/11/6/004. arXiv: hep-th/9308075. [83] Gerald V. Dunne. “Functional determinants in quantum field theory”. In: J. Phys. A 41 (2008). Ed. by Manuel Gadella et al., p. 304006. doi: 10.1088/ 1751-8113/41/30/304006. arXiv: 0711.1178 [hep-th]. [84] Anatoly Dymarsky et al. “The 3d Stress-Tensor Bootstrap”. In: JHEP 02 (2018). [,343(2017)], p. 164. doi: 10.1007/JHEP02(2018)164. arXiv: 1708.05718 [hep-th]. [85] Christopher Eling et al. “Conformal Anomalies in Hydrodynamics”. In: JHEP 05 (2013), p. 037. doi: 10.1007/JHEP05(2013)037. arXiv: 1301. 3170 [hep-th]. [86] Rajeev S. Erramilli, Luca V. Iliesiu, and Petr Kravchuk. “Recursion relation for general 3d blocks”. In: Journal of High Energy Physics 2019.12 (2019), p. 116. doi: 10.1007/JHEP12(2019)116. url: https://doi.org/10. 1007/JHEP12(2019)116. 214 [87] Rajeev S. Erramilli et al. “The Gross-Neveu-Yukawa archipelago”. In: JHEP 02 (2023), p. 036. doi: 10.1007/JHEP02(2023)036. arXiv: 2210.02492 [hep-th]. [88] Bo Feng, Amihay Hanany, and Yang-Hui He. “Counting gauge invariants: The Plethystic program”. In: JHEP 03 (2007), p. 090. doi: 10.1088/11266708/2007/03/090. arXiv: hep-th/0701063. [89] S. Ferrara et al. “The shadow operator formalism for conformal algebra. Vacuum expectation values and operator products”. In: Lettere al Nuovo Cimento (1971-1985) 4.4 (May 1972), pp. 115–120. issn: 1827-613X. doi: 10.1007/BF02907130. url: https://doi.org/10.1007/BF02907130. [90] D. S. Fine. “Quantum Yang-Mills on the two-sphere”. In: Commun. Math. Phys. 134 (1990), pp. 273–292. doi: 10.1007/BF02097703. [91] A. Liam Fitzpatrick et al. “Eikonalization of Conformal Blocks”. In: (2015). arXiv: 1504.01737 [hep-th]. [92] A. Liam Fitzpatrick et al. “The Analytic Bootstrap and AdS Superhorizon Locality”. In: JHEP 1312 (2013), p. 004. doi: 10.1007/JHEP12(2013) 004. arXiv: 1212.3616 [hep-th]. [93] Kenji Fukushima and Tetsuo Hatsuda. “The phase diagram of dense QCD”. In: Rept. Prog. Phys. 74 (2011), p. 014001. doi: 10.1088/0034- 4885/ 74/1/014001. arXiv: 1005.4814 [hep-ph]. [94] Shouvik Ganguly and Sridip Pal. “Bounds on the density of states and the spectral gap in CFT2 ”. In: Phys. Rev. D 101.10 (2020), p. 106022. doi: 10.1103/PhysRevD.101.106022. arXiv: 1905.12636 [hep-th]. [95] G. W. Gibbons, M. J. Perry, and C. N. Pope. “The first law of thermodynamics for Kerr-anti-de Sitter black holes”. In: Class. Quant. Grav. 22 (2005), pp. 1503–1526. doi: 10.1088/0264- 9381/22/9/002. arXiv: hep-th/0408217. [96] Simone Giombi and Igor R. Klebanov. “Interpolating between 𝑎 and 𝐹”. In: JHEP 03 (2015), p. 117. doi: 10.1007/JHEP03(2015)117. arXiv: 1409.1937 [hep-th]. [97] Simone Giombi et al. “AdS Description of Induced Higher-Spin Gauge Theory”. In: JHEP 10 (2013), p. 016. doi: 10.1007/JHEP10(2013)016. arXiv: 1306.5242 [hep-th]. [98] Siavash Golkar and Savdeep Sethi. “Global Anomalies and Effective Field Theory”. In: JHEP 05 (2016), p. 105. doi: 10.1007/JHEP05(2016)105. arXiv: 1512.02607 [hep-th]. [99] Alba Grassi, Zohar Komargodski, and Luigi Tizzano. “Extremal correlators and random matrix theory”. In: JHEP 04 (2021), p. 214. doi: 10.1007/ JHEP04(2021)214. arXiv: 1908.10306 [hep-th]. 215 [100] Bertrand I. Halperin. “On the Hohenberg–Mermin–Wagner Theorem and Its Limitations”. In: Journal of Statistical Physics 175.3-4 (2018), pp. 521–529. doi: 10.1007/s10955-018-2202-y. arXiv: 1812.00220 [cond-mat.stat-mech]. [101] Daniel Harlow and Tokiro Numasawa. “Gauging spacetime inversions in quantum gravity”. In: (Nov. 2023). arXiv: 2311.09978 [hep-th]. [102] Daniel Harlow and Hirosi Ooguri. “A universal formula for the density of states in theories with finite-group symmetry”. In: Class. Quant. Grav. 39.13 (2022), p. 134003. doi: 10 . 1088 / 1361 - 6382 / ac5db2. arXiv: 2109.03838 [hep-th]. [103] Thomas Hartman, Sachin Jain, and Sandipan Kundu. “Causality Constraints in Conformal Field Theory”. In: JHEP 05 (2016), p. 099. doi: 10.1007/ JHEP05(2016)099. arXiv: 1509.00014 [hep-th]. [104] Thomas Hartman, Christoph A. Keller, and Bogdan Stoica. “Universal Spectrum of 2d Conformal Field Theory in the Large c Limit”. In: JHEP 09 (2014), p. 118. doi: 10 . 1007 / JHEP09(2014 ) 118. arXiv: 1405 . 5137 [hep-th]. [105] Thomas Hartman, Dalimil Mazáč, and Leonardo Rastelli. “Sphere Packing and Quantum Gravity”. In: JHEP 12 (2019), p. 048. doi: 10.1007/ JHEP12(2019)048. arXiv: 1905.01319 [hep-th]. [106] Stephen W Hawking and Gary T Horowitz. “The gravitational Hamiltonian, action, entropy and surface terms”. In: Classical and Quantum Gravity 13.6 (1996), p. 1487. [107] Idse Heemskerk et al. “Holography from Conformal Field Theory”. In: JHEP 0910 (2009), p. 079. doi: 10.1088/1126- 6708/2009/10/079. arXiv: 0907.0151 [hep-th]. [108] Simeon Hellerman. “A Universal Inequality for CFT and Quantum Gravity”. In: JHEP 08 (2011), p. 130. doi: 10.1007/JHEP08(2011)130. arXiv: 0902.2790 [hep-th]. [109] Simeon Hellerman. “On the exponentially small corrections to N = 2 superconformal correlators at large R-charge”. In: (Mar. 2021). arXiv: 2103. 09312 [hep-th]. [110] Simeon Hellerman and Domenico Orlando. “Large R-charge EFT correlators in N=2 SQCD”. In: (Mar. 2021). arXiv: 2103.05642 [hep-th]. [111] Simeon Hellerman et al. “On the CFT Operator Spectrum at Large Global Charge”. In: JHEP 12 (2015), p. 071. doi: 10.1007/JHEP12(2015)071. arXiv: 1505.01537 [hep-th]. [112] Brian Henning et al. “Operator bases, 𝑆-matrices, and their partition functions”. In: JHEP 10 (2017), p. 199. doi: 10.1007/JHEP10(2017)199. arXiv: 1706.08520 [hep-th]. 216 [113] Christopher P. Herzog and Kuo-Wei Huang. “Stress Tensors from Trace Anomalies in Conformal Field Theories”. In: Phys. Rev. D 87 (2013), p. 081901. doi: 10 . 1103 / PhysRevD . 87 . 081901. arXiv: 1301 . 5002 [hep-th]. [114] Eliot Hijano et al. “Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks”. In: JHEP 01 (2016), p. 146. doi: 10.1007/JHEP01(2016) 146. arXiv: 1508.00501 [hep-th]. [115] Yasuaki Hikida, Yuya Kusuki, and Tadashi Takayanagi. “Eigenstate thermalization hypothesis and modular invariance of two-dimensional conformal field theories”. In: Phys. Rev. D 98.2 (2018), p. 026003. doi: 10.1103/ PhysRevD.98.026003. arXiv: 1804.09658 [hep-th]. [116] Gary T. Horowitz and Edgar Shaghoulian. “Detachable circles and temperatureinversion dualities for CFT𝑑 ”. In: JHEP 01 (2018), p. 135. doi: 10.1007/ JHEP01(2018)135. arXiv: 1709.06084 [hep-th]. [117] Liangdong Hu, Yin-Chen He, and W. Zhu. “Operator Product Expansion Coefficients of the 3D Ising Criticality via Quantum Fuzzy Sphere”. In: (Mar. 2023). arXiv: 2303.08844 [cond-mat.stat-mech]. [118] Luca Iliesiu, Murat Koloğlu, and David Simmons-Duffin. “Bootstrapping the 3d Ising model at finite temperature”. In: JHEP 12 (2019), p. 072. doi: 10.1007/JHEP12(2019)072. arXiv: 1811.05451 [hep-th]. [119] Luca Iliesiu et al. “The Conformal Bootstrap at Finite Temperature”. In: JHEP 10 (2018), p. 070. doi: 10.1007/JHEP10(2018)070. arXiv: 1802. 10266 [hep-th]. [120] Mikhail Isachenkov and Volker Schomerus. “Integrability of conformal blocks. Part I. Calogero-Sutherland scattering theory”. In: JHEP 07 (2018), p. 180. doi: 10.1007/JHEP07(2018)180. arXiv: 1711.06609 [hep-th]. [121] Daniel Jafferis, Baur Mukhametzhanov, and Alexander Zhiboedov. “Conformal Bootstrap At Large Charge”. In: JHEP 05 (2018), p. 043. doi: 10.1007/JHEP05(2018)043. arXiv: 1710.11161 [hep-th]. [122] Daniel Louis Jafferis et al. “Matrix models for eigenstate thermalization”. In: (Sept. 2022). arXiv: 2209.02130 [hep-th]. [123] Kristan Jensen. “Triangle Anomalies, Thermodynamics, and Hydrodynamics”. In: Phys. Rev. D 85 (2012), p. 125017. doi: 10.1103/PhysRevD.85. 125017. arXiv: 1203.3599 [hep-th]. [124] Kristan Jensen, R. Loganayagam, and Amos Yarom. “Anomaly inflow and thermal equilibrium”. In: JHEP 05 (2014), p. 134. doi: 10 . 1007 / JHEP05(2014)134. arXiv: 1310.7024 [hep-th]. [125] Kristan Jensen, R. Loganayagam, and Amos Yarom. “Chern-Simons terms from thermal circles and anomalies”. In: JHEP 05 (2014), p. 110. doi: 10.1007/JHEP05(2014)110. arXiv: 1311.2935 [hep-th]. 217 [126] Kristan Jensen, R. Loganayagam, and Amos Yarom. “Thermodynamics, gravitational anomalies and cones”. In: JHEP 02 (2013), p. 088. doi: 10. 1007/JHEP02(2013)088. arXiv: 1207.5824 [hep-th]. [127] Kristan Jensen et al. “Towards hydrodynamics without an entropy current”. In: Phys. Rev. Lett. 109 (2012), p. 101601. doi: 10.1103/PhysRevLett. 109.101601. arXiv: 1203.3556 [hep-th]. [128] Monica Jinwoo Kang, Jaeha Lee, and Hirosi Ooguri. “Universal formula for the density of states with continuous symmetry”. In: Phys. Rev. D 107.2 (2023), p. 026021. doi: 10.1103/PhysRevD.107.026021. arXiv: 2206. 14814 [hep-th]. [129] Daniel Kapec, Raghu Mahajan, and Douglas Stanford. “Matrix ensembles with global symmetries and ’t Hooft anomalies from 2d gauge theory”. In: JHEP 04 (2020), p. 186. arXiv: 1912.12285 [hep-th]. [130] Denis Karateev, Petr Kravchuk, and David Simmons-Duffin. “Harmonic Analysis and Mean Field Theory”. In: JHEP 10 (2019), p. 217. doi: 10. 1007/JHEP10(2019)217. arXiv: 1809.05111 [hep-th]. [131] Robin Karlsson et al. “Thermal stress tensor correlators, OPE and holography”. In: JHEP 09 (2022), p. 234. doi: 10.1007/JHEP09(2022)234. arXiv: 2206.05544 [hep-th]. [132] Christoph A. Keller and Alexander Maloney. “Poincare Series, 3D Gravity and CFT Spectroscopy”. In: JHEP 02 (2015), p. 080. doi: 10 . 1007 / JHEP02(2015)080. arXiv: 1407.6008 [hep-th]. [133] Seok Kim et al. “‘Grey Galaxies’ as an endpoint of the Kerr-AdS superradiant instability”. In: (May 2023). arXiv: 2305.08922 [hep-th]. [134] Igor R. Klebanov, Silviu S. Pufu, and Benjamin R. Safdi. “F-Theorem without Supersymmetry”. In: JHEP 10 (2011), p. 038. doi: 10 . 1007 / JHEP10(2011)038. arXiv: 1105.4598 [hep-th]. [135] Murat Kologlu et al. “Shocks, Superconvergence, and a Stringy Equivalence Principle”. In: JHEP 11 (2020), p. 096. doi: 10.1007/JHEP11(2020)096. arXiv: 1904.05905 [hep-th]. [136] Murat Kologlu et al. “The light-ray OPE and conformal colliders”. In: JHEP 01 (2021), p. 128. doi: 10.1007/JHEP01(2021)128. arXiv: 1905.01311 [hep-th]. [137] Zohar Komargodski and Adam Schwimmer. “On Renormalization Group Flows in Four Dimensions”. In: JHEP 12 (2011), p. 099. doi: 10.1007/ JHEP12(2011)099. arXiv: 1107.3987 [hep-th]. [138] Zohar Komargodski and Alexander Zhiboedov. “Convexity and Liberation at Large Spin”. In: JHEP 1311 (2013), p. 140. doi: 10.1007/JHEP11(2013) 140. arXiv: 1212.4103 [hep-th]. 218 [139] Filip Kos, David Poland, and David Simmons-Duffin. “Bootstrapping Mixed Correlators in the 3D Ising Model”. In: JHEP 11 (2014), p. 109. doi: 10.1007/JHEP11(2014)109. arXiv: 1406.4858 [hep-th]. [140] Filip Kos, David Poland, and David Simmons-Duffin. “Bootstrapping the 𝑂 (𝑁) vector models”. In: JHEP 1406 (2014), p. 091. doi: 10 . 1007 / JHEP06(2014)091. arXiv: 1307.6856 [hep-th]. [141] Per Kraus and Alexander Maloney. “A cardy formula for three-point coefficients or how the black hole got its spots”. In: JHEP 05 (2017), p. 160. doi: 10.1007/JHEP05(2017)160. arXiv: 1608.03284 [hep-th]. [142] Petr Kravchuk and David Simmons-Duffin. “Counting Conformal Correlators”. In: JHEP 02 (2018), p. 096. doi: 10.1007/JHEP02(2018)096. arXiv: 1612.08987 [hep-th]. [143] M. Krech and D. P. Landau. “Casimir effect in critical systems: A Monte Carlo simulation”. In: Phys. Rev. E 53 (5 1996), pp. 4414–4423. doi: 10. 1103/PhysRevE.53.4414. [144] M. Krech and D. P. Landau. “Casimir effect in critical systems: A Monte Carlo simulation”. In: Phys. Rev. E 53 (5 1996), pp. 4414–4423. doi: 10. 1103/PhysRevE.53.4414. url: https://link.aps.org/doi/10. 1103/PhysRevE.53.4414. [145] Michael Krech. “Casimir forces in binary liquid mixtures”. In: Phys. Rev. E 56 (2 1997), pp. 1642–1659. doi: 10.1103/PhysRevE.56.1642. arXiv: cond-mat/9703093 [cond-mat.soft]. url: https://link.aps.org/ doi/10.1103/PhysRevE.56.1642. [146] Yuya Kusuki. “Modern Approach to 2D Conformal Field Theory”. In: (Dec. 2024). arXiv: 2412.18307 [hep-th]. [147] David Kutasov and Finn Larsen. “Partition sums and entropy bounds in weakly coupled CFT”. In: JHEP 01 (2001), p. 001. doi: 10.1088/11266708/2001/01/001. arXiv: hep-th/0009244. [148] Pedro Liendo, Leonardo Rastelli, and Balt C. van Rees. “The Bootstrap Program for Boundary CFT𝑑 ”. In: JHEP 07 (2013), p. 113. doi: 10.1007/ JHEP07(2013)113. arXiv: 1210.4258 [hep-th]. [149] Aike Liu et al. “Skydiving to Bootstrap Islands”. In: (July 2023). arXiv: 2307.13046 [hep-th]. [150] Hong Liu and Paolo Glorioso. “Lectures on non-equilibrium effective field theories and fluctuating hydrodynamics”. In: PoS TASI2017 (2018), p. 008. doi: 10.22323/1.305.0008. arXiv: 1805.09331 [hep-th]. [151] Junyu Liu et al. “𝑑-dimensional SYK, AdS Loops, and 6 𝑗 Symbols”. In: JHEP 03 (2019), p. 052. doi: 10.1007/JHEP03(2019)052. arXiv: 1808. 00612 [hep-th]. 219 [152] Junyu Liu et al. “The Lorentzian inversion formula and the spectrum of the 3d O(2) CFT”. In: JHEP 09 (2020). [Erratum: JHEP 01, 206 (2021)], p. 115. doi: 10.1007/JHEP09(2020)115. arXiv: 2007.07914 [hep-th]. [153] R. Loganayagam. “Anomalies and the Helicity of the Thermal State”. In: JHEP 11 (2013), p. 205. doi: 10.1007/JHEP11(2013)205. arXiv: 1211. 3850 [hep-th]. [154] R. Loganayagam and Piotr Surówka. “Anomaly/Transport in an Ideal Weyl gas”. In: JHEP 04 (2012), p. 097. doi: 10 . 1007 / JHEP04(2012 ) 097. arXiv: 1201.2812 [hep-th]. [155] LAJ London. “Arbitrary dimensional cosmological multi-black holes”. In: Nuclear Physics B 434.3 (1995), pp. 709–735. [156] Conghuan Luo and Yifan Wang. “Casimir energy and modularity in higherdimensional conformal field theories”. In: JHEP 07 (2023), p. 028. doi: 10.1007/JHEP07(2023)028. arXiv: 2212.14866 [hep-th]. [157] M. Luscher. “Volume Dependence of the Energy Spectrum in Massive Quantum Field Theories. 1. Stable Particle States”. In: Commun. Math. Phys. 104 (1986), p. 177. doi: 10.1007/BF01211589. [158] M. Luscher. “Volume Dependence of the Energy Spectrum in Massive Quantum Field Theories. 2. Scattering States”. In: Commun. Math. Phys. 105 (1986), pp. 153–188. doi: 10.1007/BF01211097. [159] M. Lüscher. “Volume dependence of the energy spectrum in massive quantum field theories”. In: Communications in Mathematical Physics 104.2 (June 1986), pp. 177–206. issn: 1432-0916. doi: 10.1007/BF01211589. url: https://doi.org/10.1007/BF01211589. [160] Javier M. Magan. “Proof of the universal density of charged states in QFT”. In: JHEP 12 (2021), p. 100. doi: 10.1007/JHEP12(2021)100. arXiv: 2111.02418 [hep-th]. [161] Juan Maldacena, David Simmons-Duffin, and Alexander Zhiboedov. “Looking for a bulk point”. In: JHEP 01 (2017), p. 013. doi: 10.1007/JHEP01(2017) 013. arXiv: 1509.03612 [hep-th]. [162] Alexander Maloney and Edward Witten. “Quantum Gravity Partition Functions in Three Dimensions”. In: JHEP 02 (2010), p. 029. doi: 10.1007/ JHEP02(2010)029. arXiv: 0712.0155 [hep-th]. [163] Enrico Marchetto, Alessio Miscioscia, and Elli Pomoni. “Sum rules & Tauberian theorems at finite temperature”. In: (Dec. 2023). arXiv: 2312. 13030 [hep-th]. [164] Eric Mefford, Edgar Shaghoulian, and Milind Shyani. “Sparseness bounds on local operators in holographic CFT𝑑 ”. In: JHEP 07 (2018), p. 051. doi: 10.1007/JHEP07(2018)051. arXiv: 1711.03122 [hep-th]. 220 [165] Tom Melia and Sridip Pal. “EFT Asymptotics: the Growth of Operator Degeneracy”. In: SciPost Phys. 10.5 (2021), p. 104. doi: 10 . 21468 / SciPostPhys.10.5.104. arXiv: 2010.08560 [hep-th]. [166] Alexander A. Migdal. “Recursion Equations in Gauge Theories”. In: Sov. Phys. JETP 42 (1975). Ed. by I. M. Khalatnikov and V. P. Mineev, p. 413. [167] Alexander Monin et al. “Semiclassics, Goldstone Bosons and CFT data”. In: JHEP 06 (2017), p. 011. doi: 10.1007/JHEP06(2017)011. arXiv: 1611.02912 [hep-th]. [168] Baur Mukhametzhanov and Sridip Pal. “Beurling-Selberg Extremization and Modular Bootstrap at High Energies”. In: SciPost Phys. 8.6 (2020), p. 088. doi: 10 . 21468 / SciPostPhys . 8 . 6 . 088. arXiv: 2003 . 14316 [hep-th]. [169] Baur Mukhametzhanov and Alexander Zhiboedov. “Analytic Euclidean Bootstrap”. In: JHEP 10 (2019), p. 270. doi: 10.1007/JHEP10(2019)270. arXiv: 1808.03212 [hep-th]. [170] Baur Mukhametzhanov and Alexander Zhiboedov. “Modular invariance, tauberian theorems and microcanonical entropy”. In: JHEP 10 (2019), p. 261. doi: 10.1007/JHEP10(2019)261. arXiv: 1904.06359 [hep-th]. [171] Gim Seng Ng and Piotr Surówka. “One-loop effective actions and 2D hydrodynamics with anomalies”. In: Phys. Lett. B 746 (2015), pp. 281–284. doi: 10.1016/j.physletb.2015.05.011. arXiv: 1411.7989 [hep-th]. [172] Sridip Pal. “Bound on asymptotics of magnitude of three point coefficients in 2D CFT”. In: JHEP 01 (2020), p. 023. doi: 10.1007/JHEP01(2020)023. arXiv: 1906.11223 [hep-th]. [173] Sridip Pal, Jiaxin Qiao, and Slava Rychkov. “Twist Accumulation in Conformal Field Theory: A Rigorous Approach to the Lightcone Bootstrap”. In: Commun. Math. Phys. 402.3 (2023), pp. 2169–2214. doi: 10.1007/ s00220-023-04767-w. arXiv: 2212.04893 [hep-th]. [174] Sridip Pal and Zhengdi Sun. “High Energy Modular Bootstrap, Global Symmetries and Defects”. In: JHEP 08 (2020), p. 064. doi: 10 . 1007 / JHEP08(2020)064. arXiv: 2004.12557 [hep-th]. [175] Sridip Pal and Zhengdi Sun. “Tauberian-Cardy formula with spin”. In: JHEP 01 (2020), p. 135. doi: 10.1007/JHEP01(2020)135. arXiv: 1910.07727 [hep-th]. [176] Duccio Pappadopulo et al. “OPE Convergence in Conformal Field Theory”. In: Phys. Rev. D 86 (2012), p. 105043. doi: 10 . 1103 / PhysRevD . 86 . 105043. arXiv: 1208.6449 [hep-th]. [177] João Penedones, Emilio Trevisani, and Masahito Yamazaki. “Recursion Relations for Conformal Blocks”. In: JHEP 09 (2016), p. 070. doi: 10. 1007/JHEP09(2016)070. arXiv: 1509.00428 [hep-th]. 221 [178] Vasily Pestun et al. “Localization techniques in quantum field theories”. In: J. Phys. A 50.44 (2017), p. 440301. doi: 10.1088/1751-8121/aa63c1. arXiv: 1608.02952 [hep-th]. [179] A. M. Polyakov. “Nonhamiltonian approach to conformal quantum field theory”. In: Zh. Eksp. Teor. Fiz. 66 (1974), pp. 23–42. [180] Jiaxin Qiao and Slava Rychkov. “A tauberian theorem for the conformal bootstrap”. In: JHEP 12 (2017), p. 119. doi: 10.1007/JHEP12(2017)119. arXiv: 1709.00008 [hep-th]. [181] Riccardo Rattazzi et al. “Bounding scalar operator dimensions in 4D CFT”. In: JHEP 12 (2008), p. 031. doi: 10.1088/1126-6708/2008/12/031. arXiv: 0807.0004 [hep-th]. [182] Shlomo S. Razamat and Brian Willett. “Global Properties of Supersymmetric Theories and the Lens Space”. In: Commun. Math. Phys. 334.2 (2015), pp. 661–696. doi: 10.1007/s00220- 014- 2111- 0. arXiv: 1307.4381 [hep-th]. [183] L. J. Romans. “Supersymmetric, cold and lukewarm black holes in cosmological Einstein-Maxwell theory”. In: Nucl. Phys. B 383 (1992), pp. 395– 415. doi: 10.1016/0550-3213(92)90684-4. arXiv: hep-th/9203018. [184] B. E. Rusakov. “Loop averages and partition functions in U(N) gauge theory on two-dimensional manifolds”. In: Mod. Phys. Lett. A 5 (1990), pp. 693– 703. [185] Slava Rychkov and Pierre Yvernay. “Remarks on the Convergence Properties of the Conformal Block Expansion”. In: Phys. Lett. B753 (2016), pp. 682– 686. doi: 10.1016/j.physletb.2016.01.004. arXiv: 1510.08486 [hep-th]. [186] Subir Sachdev. “Polylogarithm identities in a conformal field theory in three-dimensions”. In: Phys. Lett. B 309 (1993), pp. 285–288. doi: 10 . 1016/0370-2693(93)90935-B. arXiv: hep-th/9305131. [187] A. Schwimmer and S. Theisen. “Spontaneous Breaking of Conformal Invariance and Trace Anomaly Matching”. In: Nucl. Phys. B 847 (2011), pp. 590– 611. doi: 10.1016/j.nuclphysb.2011.02.003. arXiv: 1011.0696 [hep-th]. [188] Edgar Shaghoulian. “Black hole microstates in AdS”. In: Phys. Rev. D 94.10 (2016), p. 104044. doi: 10.1103/PhysRevD.94.104044. arXiv: 1512.06855 [hep-th]. [189] Edgar Shaghoulian. “Modular forms and a generalized Cardy formula in higher dimensions”. In: Phys. Rev. D 93.12 (2016), p. 126005. doi: 10. 1103/PhysRevD.93.126005. arXiv: 1508.02728 [hep-th]. 222 [190] Edgar Shaghoulian. “Modular Invariance of Conformal Field Theory on 𝑆 1 𝑆 3 and Circle Fibrations”. In: Phys. Rev. Lett. 119.13 (2017), p. 131601. doi: 10.1103/PhysRevLett.119.131601. arXiv: 1612.05257 [hep-th]. [191] Ben L. Shepherd and Elizabeth Winstanley. “Dyons and dyonic black holes in SU(N) Einstein-Yang-Mills theory in anti de Sitter spacetime”. In: Physical Review D 93.6 (Mar. 2016). doi: 10.1103/physrevd.93.064064. url: https://doi.org/10.1103%2Fphysrevd.93.064064. [192] David Simmons-Duffin. PH230A: Quantum Field Theory Notes. 2023. url: https : / / gitlab . com / davidsd / ph230a - notes (visited on 01/09/2025). [193] David Simmons-Duffin. “Projectors, Shadows, and Conformal Blocks”. In: JHEP 04 (2014), p. 146. doi: 10.1007/JHEP04(2014)146. arXiv: 1204. 3894 [hep-th]. [194] David Simmons-Duffin. “The Conformal Bootstrap”. In: Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings (TASI 2015): Boulder, CO, USA, June 1-26, 2015. 2017, pp. 1–74. doi: 10 . 1142 / 9789813149441 _ 0001. arXiv: 1602 . 07982 [hep-th]. url: http : / / inspirehep . net / record / 1424282/files/arXiv:1602.07982.pdf. [195] David Simmons-Duffin. “The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT”. In: JHEP 03 (2017), p. 086. doi: 10.1007/JHEP03(2017) 086. arXiv: 1612.08471 [hep-th]. [196] Robert Southey. “The Story of the Three Bears”. In: The Doctor & C. Longman, Rees, Orme, Brown, Green and Longman, 1837, pp. 318–326. [197] Andrew Strominger. “Black hole entropy from near horizon microstates”. In: JHEP 02 (1998), p. 009. doi: 10.1088/1126-6708/1998/02/009. arXiv: hep-th/9712251. [198] O. Vasilyev et al. “Universal scaling functions of critical Casimir forces obtained by Monte Carlo simulations”. In: Phys. Rev. E 79 (4 2009), p. 041142. doi: 10 . 1103 / PhysRevE . 79 . 041142. arXiv: 0812 . 0750 [cond-mat.stat-mech]. [199] Erik P. Verlinde. “On the holographic principle in a radiation dominated universe”. In: (Aug. 2000). arXiv: hep-th/0008140. [200] M.S Volkov et al. “The number of sphaleron instabilities of the BartnikMcKinnon solitons and non-Abelian black holes”. In: Physics Letters B 349.4 (Apr. 1995), pp. 438–442. doi: 10.1016/0370-2693(95)00293-t. url: https://doi.org/10.1016%2F0370-2693%2895%2900293-t. 223 [201] Mikhail S. Volkov. “Hairy black holes in the XX-th and XXI-st centuries”. In: 14th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories. Vol. 2. 2017, pp. 1779–1798. doi: 10.1142/9789813226609_ 0184. arXiv: 1601.08230 [gr-qc]. [202] Mikhail S. Volkov and Dmitri V. Gal’tsov. “Gravitating nonAbelian solitons and black holes with Yang-Mills fields”. In: Phys. Rept. 319 (1999), pp. 1– 83. doi: 10.1016/S0370-1573(99)00010-1. arXiv: hep-th/9810070. [203] Elizabeth Winstanley. “A menagerie of hairy black holes”. In: Springer Proc. Phys. 208 (2018). Ed. by Piero Nicolini et al., pp. 39–46. doi: 10. 1007/978-3-319-94256-8_3. arXiv: 1510.01669 [gr-qc]. [204] Elizabeth Winstanley. “Classical Yang-Mills black hole hair in anti-de Sitter space”. In: Lect. Notes Phys. 769 (2009). Ed. by Eleftherios Papantonopoulos, pp. 49–87. arXiv: 0801.0527 [gr-qc]. [205] Edward Witten. “Two-dimensional gauge theories revisited”. In: J. Geom. Phys. 9 (1992), pp. 303–368. arXiv: hep-th/9204083. [206] Al.B Zamolodchikov. “Conformal symmetry in two-dimensional space: Recursion representation of conformal block”. English. In: Theoretical and Mathematical Physics 73.1 (1987), pp. 1088–1093. issn: 0040-5779. doi: 10.1007/BF01022967. [207] Wei Zhu et al. “Uncovering Conformal Symmetry in the 3D Ising Transition: State-Operator Correspondence from a Quantum Fuzzy Sphere Regularization”. In: Phys. Rev. X 13.2 (2023), p. 021009. doi: 10.1103/PhysRevX. 13.021009. arXiv: 2210.13482 [cond-mat.stat-mech].