Testing Gauge/Gravity Duality: The Eleven-Dimensional PP-Wave Citation Wu, Xinkai (2004) Testing Gauge/Gravity Duality: The Eleven-Dimensional PP-Wave. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/0JV2-RT78. https://resolver.caltech.edu/CaltechETD:etd-05252004-230238 Abstract The gauge/gravity duality in the interaction between M theory objects has taught us a lot about quantum gravity. The eleven-dimensional PP-wave background provides a new arena for exploring this duality beyond flat and almost flat, i.e., weakly curved, backgrounds. In this thesis we discuss the gauge theories that describe the dynamics of interacting M theory objects, the supergravity calculations that capture these dynamics, the comparison of the two sides, and various objects (such as gravitons and membranes) in the eleven-dimensional PP-wave background. We only consider the one-loop gauge theory and linearized supergravity approximations. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: M-theory; Matrix theory; supergravity Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Schwarz, John H. Group: Caltech Theory Thesis Committee: Schwarz, John H. (chair) Weinstein, Alan Jay Kapustin, Anton N. Wise, Mark B. Defense Date: 21 May 2004 Record Number: CaltechETD:etd-05252004-230238 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-05252004-230238 DOI: 10.7907/0JV2-RT78 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 2048 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 28 May 2004 Last Modified: 01 Jul 2025 16:42 Thesis Files Preview PDF - Final Version See Usage Policy. 566kB Repository Staff Only: item control page

Testing Gauge/Gravity Duality: The Eleven-Dimensional PP-wave Thesis by Xinkai Wu In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2004 (Defended May 21, 2004) ii c 2004 Xinkai Wu All Rights Reserved iii TO MY PARENTS iv Acknowledgements I would like to thank my advisor John Schwarz, for introducing me into this field and giving me constant support and encouragement. I would especially like to thank John Schwarz, Anton Kapustin, Alan Weinstein, and Mark Wise for serving on my committee. I also benefited greatly from collaboration and/or discussion with Curtis Callan, Hok Kong Lee, Tristan McLoughlin, Ian Swanson, Vadim Borokhov, Anton Kapustin, Iouri Chepelev, Calin Ciocarlie, Peter Lee, Yi Li, Yuji Okawa, Takuya Okuda, Hirosi Ooguri, Ardadas Ozakin and Jongwon Park. I want to thank Carol Silberstein for all her help. Also, thanks go to Guodong Wang, who help me a lot on formatting the thesis. v Abstract The gauge/gravity duality in the interaction between M theory objects has taught us a lot about quantum gravity. The eleven-dimensional PP-wave background provides a new arena for exploring this duality beyond flat and almost flat, i.e., weakly curved, backgrounds. In this thesis we discuss the gauge theories that describe the dynamics of interacting M theory objects, the supergravity calculations that capture these dynamics, the comparison of the two sides, and various objects (such as gravitons and membranes) in the eleven-dimensional PP-wave background. We only consider the one-loop gauge theory and linearized supergravity approximations. vi Contents 1 Introduction 1 2 Review of the Eleven-Dimensional PP-wave Geometry 6 2.1 Isometries and Supersymmetries of the 11-D PP-wave . . . . . . . . . 7 2.2 How the PP-wave Arises as a Penrose Limit of AdS × S 10 . . . . . . . 3 The Gauge Theories Describing M-theory Objects in an ElevenDimensional PP-wave Geometry 15 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.1.1 Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.1.2 Supersymmetric Classical Vacua . . . . . . . . . . . . . . . . . 26 3.2 3.3 Matrix Theory The Three-Dimensional Theory Describing Multiple Concentric Spherical Membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2.2.1 Supersymmetric Classical Vacua . . . . . . . . . . . 39 3.2.2.2 Euclidean Instantons . . . . . . . . . . . . . . . . . . 40 Two Graviton Interaction without M-momentum Transfer—Gauge Theory Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3.1 Brief Review of Known Results . . . . . . . . . . . . . . . . . 46 3.3.2 The Effective Potential . . . . . . . . . . . . . . . . . . . . . . 47 3.3.3 Background Field Method . . . . . . . . . . . . . . . . . . . . 49 3.3.3.1 Expansion about the Background . . . . . . . . . . . 50 3.3.3.2 The Sum over Mass . . . . . . . . . . . . . . . . . . 51 vii 3.3.4 A Simple Case . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.3.5 Mass Matrix Computation . . . . . . . . . . . . . . . . . . . . 55 3.3.6 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . 57 4 Interaction of M-theory Objects in Eleven-Dimensional Supergravity 4.1 4.2 59 The Light Cone Lagrangian . . . . . . . . . . . . . . . . . . . . . . . 59 4.1.1 Point Particle Probe . . . . . . . . . . . . . . . . . . . . . . . 60 4.1.2 Membrane Probe . . . . . . . . . . . . . . . . . . . . . . . . . 65 Diagonalizing the Supergravity Field Equations for Arbitrary Static Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 71 Two Graviton Interaction Without M-momentum Transfer—Supergravity Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5 Supersymmetric Nonrenormalization Theorems 94 6 Conclusion and Discussion 97 A Proof of Invariance of Lnonabelian under Sixteen Supersymmetries 98 B The One-Loop Effective Potential 105 C An Alternative Derivation of Veff for the Point Particle Probe 113 D Constrained Hamiltonian Dynamics in Field Theory 115 ˜ 117 E The Green’s Function for the x− Fourier Transformed Laplacian  F h−− for a Spherical Source Membrane in the Near-Membrane Limit119 Bibliography 121 1 Chapter 1 Introduction The five superstring theories in ten dimensions, Type IIA, Type IIB, Type I, Heterotic SO(32), Heterotic E8 ×E8 are known to be related to each other via a web of dualities [1]. Underlying these theories, there is believed to be a fundamental M theory whose low energy effective theory is the eleven-dimensional supergravity. Matrix theory was proposed by [2, 3] as a candidate of M theory for flat target space. This is a (0 + 1)-dimensional quantum mechanical theory whose degrees of freedom reside in N by N matrices as a noncommutative generalization of the usual concept of target space coordinates, and it is argued that it should provide a nonperturbative description of quantum gravity in a manifestly unitary way. The generalization of Matrix theory to a generic nonflat background was initiated in [4, 5]. In [6, 7], this generalization was carried out in detail to linear order in the background fields for a weakly curved background (i.e., a background whose metric is gIJ = ηIJ + hIJ , with ηIJ being the flat metric, and |hIJ | << 1) which is independent of the light like direction x− . Not very long ago, another useful technique, different from the weakly curved background approximation, for understanding nonflat backgrounds was put forward in the context of string/M theory. This is the Penrose limit [8, 9, 10], a limit in which one zooms in, roughly speaking, to the close neighborhood of a null geodesic. In general, a space arising from Penrose limit may be highly curved, hence not restricted to the weak background approximation. More importantly, when taking the limit, the resulting background inherits the (super)symmetries of the original space. Hence one 2 expects that upon taking the Penrose limit, the M theory in a nonflat background will become more tractable, provided that the original background has a large amount of (super)symmetries. The Matrix theory in an eleven-dimensional Parallel-Plane-wave (henceforth abbreviated as PP-wave) background proposed in [10] is a good example of such an application of the Penrose limit. As a matter of fact, the main subject of study in this thesis will be the dynamics of this Matrix theory and other theories related to it. Here the original space is the product of four-dimensional Anti de Sitter space and seven-sphere AdS4 × S 7 , or the product of seven-dimensional Anti de Sitter space and four-sphere AdS7 × S 4 , and the eleven-dimensional PP-wave is the space resulting from their Penrose limits. Although the Matrix theory (in a general background) has the form of a (0+1)dimensional quantum mechanics, and can be interpreted as describing N D0-branes in the IIA string theory obtained from compactifying the eleven-dimensional M theory to ten dimensions, it does not only describe point particles. Philosophically, the reason is, since the Matrix theory is supposed to be a realization of the fundamental M theory, it must have the capability to describe the dynamics of all M theory objects (e.g., M2-brane, M5-brane from the perspective of eleven-dimensional supergravity, the fundamental string, NS5-brane, and higher-dimensional D-branes from the tendimensional IIA string perspective), after including all sectors with different values of N . Technically, the reasons are, to name a few, the Matrix theory can be obtained by “discretizing” the supermembrane theory [11, 12], the D0-brane theory can be T-dualized to give theories describing higher-dimensional D-branes, and by taking the “continuum limit” of the Matrix theory, one gets field theories living in higher dimensions. Depending on the problem at hand, e.g., the investigation of membrane scattering, one may choose the (0 + 1)-dimensional theory, or, higher-dimensional field theories as long as it is valid, whichever is more convenient, and the computed physical quantities should be the same. This is one of the points that the discussions in Sections 3.1 and 3.2 try to make. The Matrix theory and related higher-dimensional field theories are nonabelian 3 gauge theories, with which we can do perturbative loop computations in expansion of the gauge coupling. One interesting thing observed is the gauge/gravity duality in the dynamics of interacting M theory objects [13, 14, 15, 16, 17, 18, 19, 20, 7, 21, 22, 23, 24], where results of quantum loop computations on the gauge theories’ side are shown to agree with results of solving classical equations of motion on the supergravity side. This duality is physically motivated recalling that eleven-dimensional supergravity is the low energy limit of M theory, although supersymmetry also seems to be a crucial ingredient to bridge the disparate regions of validity on the two sides. Of course, similar gauge/gravity duality also appears in many other settings, such as the well known AdS/CFT correspondence in IIB string theory (in that case, taking the Penrose limit actually gives us access to full string theory, not just the classical supergravity, on the AdS side [10]). As stated in the title, this thesis is devoted to the study of the gauge/gravity duality for interacting M theory objects in the eleven-dimensional PP-wave background. In this study we shall restrict our attention to one-loop on the gauge theory side, and to first order in κ211 on the supergravity side, that is, linearized supergravity. As our contribution to the subject, this thesis discusses some generalities in the gauge/gravity comparison, the results on two graviton interaction in the absence of transfer of the momentum along the light-like x− direction, the so-called M-momentum, and some initial steps taken towards understanding two membrane interaction with M-momentum transfer, based mainly on works in collaboration with Hok Kong Lee and Tristan McLoughlin. The thesis is organized as follows: In Chapter 2 we briefly review features of the eleven-dimensional PP-wave, giving the expressions for its 38 Killing vectors, 32 Killing spinors, and the (anti)commutators of the symmetry superalgebra [25, 26, 27], also describing how it arises as the Penrose limit of AdS × S space [8, 9, 28]. In Chapter 3 we discuss the gauge theories that describe M-theory objects in the eleven-dimensional PP-wave background. Section 3.1 is devoted to the Matrix theory proposed by [10]. In this section we mainly review three different ways of deriving this matrix theory: the approach taken 4 by [10] which starts with a superparticle and then nonabelianizes the resulting theory; the one taken by [29] where the Matrix theory is obtained by “discretizing” the supermembrane in 11-D PP-wave background; and the one taken by [29] which uses the framework developed in [6, 7] to derive the Matrix theory from the dynamics of multi D0-branes in the IIA supergravity background arising from a space-like compactification of 11-D PP-wave, in the weakly curved background approximation. We pay particular attention to the multi D0-brane approach because firstly it illustrates the connection between the dynamics in eleven dimensions and that in ten dimensions, as well as the appropriate limit [4] in which this connection is made, and secondly it illustrates how, given a large number of (super)symmetries, sometimes the weak background matrix theory actually needs no correction and is exact. Section 3.2 is devoted to the three-dimensional theory describing multi spherical membranes in 11-D PP-wave. This theory was first obtained by [30] as the continuum limit of the matrix theory proposed by [10] expanded around its k-membrane vacuum. In Subsection 3.2.1 we present an alternative derivation which obtains this theory directly from the supermembrane theory, expanding around the single membrane vacuum to get an abelian theory and then carrying out a nonabelian generalization. In Subsection 3.2.2 we discuss the vacua and instantons of this theory in the two membrane case, presenting the instanton solution which interpolates between the flux one vacuum and the flux zero trivial vacuum, and briefly comment on the application of this instanton solution to the study of M-momentum transfer between two spherical membranes in an 11-D PP-wave background. Section 3.3 is an application based on [21]. In short, [21] computes the twograviton one-loop effective action for the Matrix theory in the eleven-dimensional PP-wave background, and compares it to the effective action on the supergravity side in the same background. Agreement is found for the effective action on both sides, to all orders of µ (i.e., beyond the weak background approximation), which provides evidence for the Matrix theory proposed by [10] being the correct description of M theory in the eleven-dimensional PP-wave background, and also points to the existence of a supersymmetric nonrenormalization theorem in this background. This 5 section is mainly the computation on the gauge theory side given in [21]. In Chapter 4 we discuss the treatment of M theory objects’ interaction on the supergravity side. On the supergravity side we use the method of source-probe analysis, which is valid when the source is much heavier than the probe. Section 4.1 works out the light-cone Lagrangian, which is the supergravity quantity that should in the end be compared with the gauge theory one-loop effective Lagrangian, for a point particle and a membrane in an arbitrary (yet static, i.e., x+ independent) background. In doing so, we give a careful treatment of the constrained Hamiltonian mechanics of the systems concerned. Section 4.2 addresses the issue of finding the background fields that the probe feels, describing the diagonalization of the supergravity field equations for arbitrary static sources. The idea here is quite straightforward, and the results given are technical in nature, but since they are necessary for solving the field equations and have not been explicitly given elsewhere to the best of our knowledge, we present them in this section. The above two sections provide the framework for the computation on the supergravity side. In Section 4.3 we apply the general formalism to the specific application of two graviton interaction in the absence of M-momentum transfer, completing our investigation of gauge/gravity duality in this particular physical problem. This section is based on the supergravity computation of [21]. In Chapter 5 we briefly review supersymmetric nonrenormalization theorems in the Matrix theory quantum mechanics in flat space, following [31, 32, 33, 34, 35, 36, 37], and comment on the generalization to a 11-D PP-wave background. In Chapter 6, we make some concluding remarks, discussing possible future directions to pursue. 6 Chapter 2 Review of the Eleven-Dimensional PP-wave Geometry The fields of eleven-dimensional supergravity are the metric gµν , the three-form gauge potential Aµνρ , and the gravitino ψµ . In the eleven-dimensional PP-wave, the gravitino field ψµ vanishes, whereas the nonzero components of the metric gµν and the four-form field strength Fµνρλ are given by " g+− = 1, g++ = −µ2 # 3 9 1 X a 2 1X i 2 (x ) + (x ) , gAB = δAB 9 i=1 36 a=4 F123+ = µ (2.1) (2.2) Here µ is a parameter with the dimension of inverse length. In our conventions, µ, ν, ρ, . . . take the values +, −, 1, . . . , 9; A, B, C, . . . take the values 1, . . . , 9; i, j, k, . . . take the values 1, . . . , 3; and a, b, c, . . . take the values 4, . . . , 9. This solution to eleven-dimensional supergravity was first given by [25], and it also goes under the name KG space (where KG stands for Kowalski-Glikman). 7 2.1 Isometries and Supersymmetries of the 11-D PP-wave The explicit expressions for the Killing vectors (as well as the algebra of the isometry group) and Killing spinors of the 11-D PP-wave were first given in [26]. Here we follow the exposition of the same subject given by [27]. A slight change of the notation of [27] is necessary to bring it in accordance with ours: exchange x+ and x− ; change µ to −µ; also in [27] the transverse index i goes from 1 to 9, thus corresponding to our transverse index A. The isometry group of the 11-D PP-wave has 38 generators, coinciding with the dimension of the isometry algebras of AdS4 × S 7 and AdS7 × S 4 . This is no accident, because the former solution can be obtained from the latter solutions via a Penrose limit [9], which we shall discuss shortly. The Killing vectors of 11-D PP-wave are ξe− = −∂− , ξe+ = −∂+   + i µx+ µx µx ξei = − cos ∂i − sin ∂− 3 3 3  +  + 2 i µx µx µ µx ξe∗i = − sin ∂i + cos ∂− 3 3 3 9  +  + a µx µx µx ξea = − cos ∂− ∂a − sin 6 6 6  +  + 2 a µx µ µx µx ∂a + cos ∂− ξe∗a = − sin 6 6 6 36 ξMij = xi ∂j − xj ∂i  ξMab = xa ∂b − xb ∂a (2.3) A little explanation of the subscripts in the above expressions is in order: a generic so-called Cahen-Wallach space, of which the 11-D PP-wave is a special case, can be written as the coset space G/K, where G is the group whose algebra is spanned by {e− , e+ , eA , e∗A }, and K is the subgroup whose algebra is spanned by e∗A ; Mij , Mab are the generators of SO(3) and SO(6), which are the symmetry transformations 8 preserving the (++) component of the 11-D PP-wave metric. Next let us look at the Killing spinors, which are determined by the condition that the supersymmetry variation of the gravitino field ψµ must vanish ∇M  = 1 S ) FP QRS (ΓP QRSM + 8ΓP QR δM 288 (2.4) where ∇M ≡ ∂M + 41 ωMAB ΓAB and the gamma matrices are those of SO(10, 1). There are 32 independent solutions to this equation, making the 11-D PP-wave a maximally supersymmetric solution of 11-D supergravity. These 32 Killing spinors depend only on x+ and not the other coordinates, and are given by =   +  +    +  +  µx µx µx µx cos + sin Γ123 ψ− + cos + sin Γ123 ψ+ 4 4 12 12 !  +  +  µx µx 1X a µ X i x Γa x Γi − − sin − cos Γ123 Γ− ψ+ + 6 2 a 12 12 i (2.5) where ψ± are arbitrary constant spinors satisfying Γ± ψ± = 0 Now let us look at the symmetry superalgebra. In the superalgebra, the commutator between two bosonic generators is realized as the Lie bracket of the two corresponding Killing vector fields. The commutator between a bosonic generator and a fermionic generator is realized as the spinorial Lie derivative LX  ≡ X M ∇M  + 41 ∇[M XN ] ΓM N , where X and  are the corresponding Killing vector and Killing spinor, respectively. The anti-commutator between two fermionic generators is realized as the bilinear of the corresponding Killing spinors, namely, for two Killing spinors 1 , 2 , one simply takes ¯1 ΓM 2 to get a Killing vector. 9 The commutators of the bosonic generators are µ2 µ2 ei , [e+ , e∗a ] = − ea 9 36 2 2 µ µ [e∗i , ej ] = − e− δij , [e∗a , eb ] = − e− δab 9 36 [MAB , eC ] = −δAC eB + δBC eA [e+ , eA ] = e∗A , [e+ , e∗i ] = − [MAB , e∗C ] = −δAC e∗B + δBC e∗A (2.6) The commutators of the bosonic generators and the fermionic generators, with Q± being the 32 fermionic generators generating shifts proportional to the constant spinors ψ± parametrizing the Killing spinors, are µ µ [e− , Q± ] = 0, [e+ , Q− ] = Γ123 Q− , [e+ , Q+ ] = Γ123 Q+ 4 12 µ µ [ei , Q− ] = Γ123 Γi Γ− Q+ , [ea , Q− ] = Γ123 Γa Γ− Q+ 6 12 2 µ µ2 [e∗i , Q− ] = − Γi Γ− Q+ , [e∗a , Q− ] = − Γa Γ− Q+ 18 72 1 [MAB , Q± ] = ΓAB Q± 2 (2.7) The anticommutators between fermionic generators are (suppressing spinor indices) {Q− , Q− } = −Γ+ C −1 e− X 3X 6X ΓA C −1 eA − {Q− , Q+ } = − Γ123 Γa C −1 e∗a Γ123 Γi C −1 e∗i − µ µ a i A X X µ µ {Q+ , Q+ } = −Γ− C −1 e+ − Γ− Γ123 Γij C −1 Mij + Γ− Γ123 Γab C −1 Mab 6 ij 12 ab (2.8) with C being the charge-conjugation matrix. 10 2.2 How the PP-wave Arises as a Penrose Limit of AdS × S We follow [9, 28] in our review of the Penrose limit in the context of 11-D supergravity (again, we exchange the x+ and x− in their notation). The Penrose limit of a space-time can be thought as a blowup of the space-time along a null geodesic. The starting point for the derivation of the Penrose limit is the metric g and p-form potential (p = 3 in the 11-D supergravity context) Ap in a neighborhood of a conjugate-point-free 1 segment of a null geodesic γ in the original space-time given by ! g = dV dU + αdV + X A βA dY A + X CAB dY A dY B (2.9) AB where α, βA , CAB are functions of all the coordinates, and CAB is a symmetric positivedefinite matrix. In this coordinate system, the null geodesic is parametrized by U , with V = Y A = 0. Also AU B1 B2 ...Bp−1 = AU V B1 B2 ...Bp−2 = 0 (2.10) where this form of Ap can be achieved by using its gauge freedom. Then we rescale the coordinates to U = u, V = Ω2 v, Y A = Ωy A (2.11) with Ω being a positive real constant. Acting with this diffeomorphism on the tensor fields of the theory we obtain the Ω-dependent family of fields g(Ω), Ap (Ω). Then the 1 Recall the following definition of conjugate points [38]: Use γ to denote a geodesic. A solution η of the geodesic deviation equation v a ∇a (v b ∇b η c ) = −Rabdc η b v a v d is called a Jacobi field on γ. A pair of points p, q ∈ γ are said to be conjugate points, if there exists a Jacobi field η a which is not identically zero but vanishes at both p and q. Roughly speaking, two points p and q are conjugate if an “infinitesimally nearby” geodesic intersects γ at both p and q. a 11 Penrose limit of the original space-time is defined as ḡ ≡ lim Ω−2 g(Ω), Āp ≡ lim Ω−p Ap (Ω) Ω→0 (2.12) Ω→0 The limiting fields depend only on the coordinate u, which is the affine parameter along the null geodesic. The space-time resulting from this limit is expressed in Rosen coordinates. One can change to the more usual Brinkman (or harmonic) coordinates (we will not give the details of this coordinate transformation here) in which the metric takes the form ! ḡ = 2dx+ dx− + X ABC (x+ )xB xC BC (dx+ )2 + X dxA dxA (2.13) A When ABC is constant the above metric is a Lorentzian symmetric Cahen-Wallach space. The nice thing about the Penrose limit is that, if the original space-time is a solution to the supergravity field equations, so is its Penrose limit; also, the number of (super)symmetries of the original space-time does not decrease in this limit. Hence, even if the M-theory of concern in the original space-time is beyond our analytical capability, we can still take a Penrose limit, usually making the problem more tractable and still getting physically meaningful data. Now let us see how the Penrose limit works for the original space-times of interest, namely, AdS4 × S 7 and AdS7 × S 4 , which are the near horizon geometries of M2 and M5 branes 2 . If we define ρ ≡ RRAdS as the ratio of the radii of the AdS part and the S S part, then ρ = 12 for AdS4 × S 7 and ρ = 2 for AdS7 × S 4 . 2 A quick review of the near horizon geometries following [39]: The metric created by N M2 branes is ds2 = f −2/3 (−dt2 + d~x2 ) + f 1/3 (dr2 + r2 dΩ27 ), f = 1 + (R/r)6 , where R6 = 32π 2 N lp6 and ~x is a two-dimensional Euclidean vector. In the near horizon region r << R, so f ≈ (R/r)6 . Plugging this  2 3/2 1/2 2 into the metric and letting r = R (u/2) , one gets ds = (R/2) (du2 /u2 ) + u2 (−dt2 + d~x2 ) + R2 dΩ27 , which is of the form AdS4 × S 7 with the AdS part in the Poincaré coordinates, and RAdS = R/2, RS = R. The derivation of the near horizon geometry of N M5 branes is similar. 12 The metric of the original AdS × S space-time is RS−2 g = ρ2    dr2 2 2 2 2 −dτ + (sin τ ) + r dΩp + dψ 2 + (sin ψ)2 dΩ28−p 2 1+r (2.14) with p = 2 for the M2 case, and p = 5 for the M5 case. One can change the coordinates (ψ, τ ) to (u, v) u = ψ + ρτ, v = ψ − ρτ (2.15) in terms of which the metric becomes RS−2 g = dudv + ρ2 sin2  u−v 2ρ     dr2 u+v 2 2 2 dΩ28−p (2.16) + r dΩp + sin 2 1+r 2 One then takes the Penrose limit along the null geodesic parametrized by u. In practice this amounts to dropping the dependence on coordinates other than u, and replacing the spherical metric dΩ28−p with the flat space metric ds2 (E 8−p ) (because in the Penrose limit we are looking at a small region near the null geodesic on the sphere S 8−p , which is effectively flat space). The metric then becomes RS−2 ḡ = dudv + ρ2 sin2  u 2ρ  ds2 (E p+1 ) + sin2 u 2 ds2 (E 8−p ) (2.17) with ds2 (E) denoting the metric of Euclidean flat space. The above ḡ is in Rosen coordinates, so let us change it to Brinkman coordinates. First introduce coordinates y A , A = 1, ..., 9 so that RS−2 ḡ = dudv + X sin2 (λA u) A (2λA )2 dy A dy A (2.18) where λA = 1 1 , A = 1, ..., p + 1; λA = , A = p + 2, ..., 9 2ρ 2 (2.19) 13 then change coordinates to (x+ , x− , xA ) where x+ = 1 X A A sin(2λA u) sin(λA u) u , x− = v − y y , xA = y A 2 4 A 2λA 2λA (2.20) The metric becomes ! RS−2 ḡ = 2dx+ dx− − 4 X λ2A xA xA (dx+ )2 + X dxA dxA (2.21) A A As is easily seen, the ρ = 21 and ρ = 2 cases are isometric, with the explicit diffeomorphism given by x+ → 12 x+ , x− → 2x− , and (x1 , ..., x6 , x7 , ..., x9 ) → (x4 , ..., x9 , x1 , ..., x3 ). Also, this metric is just the 11-D PP-wave, as is seen by rescaling all the coordinates by RS , and then scaling x+ and x− oppositely by µ/3. Finally, let us give an alternative (and easier to remember) way of taking the Penrose limit of AdSp+2 × S 9−p (p = 2 or 5) to get the 11-D PP-wave, which is parallel to the treatment of AdS5 × S 5 in [10]. Let us use the global coordinates (τ, ρ, Ωp ) for the AdS part, and coordinates (β, θ, Ω07−p ) similar to those in [10] for the S part. The metric is 2 ds2 = ρ̃2 R2 (− cosh2 ρdτ 2 + dρ2 + sinh2 ρdΩ2p ) + R2 (cos2 θdβ 2 + dθ2 + sin2 θdΩ07−p ) (2.22) Notice that now we use ρ̃ to denote the ratio RRAdS , with ρ denoting a coordinate. S Consider the null geodesic γ given by ρ = 0, θ = 0, β = ρ̃τ . To zoom in to the neighborhood of γ, we introduce x̃± = τ ±(β/ρ̃) , and perform the rescaling 2 x̃+ = x+ , x̃− = x− r y , ρ= , θ= 2 2 ρ̃ R ρ̃R R (2.23) Now let us take the R → ∞ limit. The infinite terms (i.e., terms proportional to R2 ) 14 in the metric (2.22) cancel out, and we get 2 ) ds2 = −4dx+ dx− − (r2 + ρ˜2 y 2 )(dx+ )2 + (dr2 + r2 dΩ2p ) + (dy 2 + y 2 dΩ07−p (2.24) Notice that in the above metric, the ++ component arises from expanding cosh ρ and cos θ to second order in ρ and θ, respectively. This metric is again just the 11-D 3 PP-wave, as can be seen by rescaling x+ by µ3 and x− by − 2µ . 15 Chapter 3 The Gauge Theories Describing M-theory Objects in an Eleven-Dimensional PP-wave Geometry First, a few remarks on the notations and conventions used in the rest of this thesis: other authors’ results are cited at various places, however different authors use their own notations and conventions, differing by extra minus signs and numerical factors. We do not always bother to recast them in a uniform way, except when necessary, e.g. when comparing gauge theory results with supergravity results where these factors really do matter. 3.1 Matrix Theory The action of Matrix theory in the 11-D PP-wave background was first proposed by Berenstein, Maldacena, and Nastase (BMN) in [10]. It is given by Z S= ( dtTr 9 X 1 9 (M 3 R)2 X I J 2 [X , X ] 2R 4R I=1 I,J=1 " # 9 3 9 X X X 1 µ µ +(M 3 R) ψ T γ J [ψ, X J ] + −( )2 (X i )2 − ( )2 (X a )2 2R 3 6 a=4 i=1 J=1 ) 3 µ (M 3 R)µ X −i ψ T γ123 ψ − i ijk (X i X j X k ) (3.1) 4 R i,j,k=1 (D0 X I )2 + iψ T D0 ψ + 16 where D0 X I = ∂t X I − i[X0 , X I ], and D0 ψ = ∂t ψ − i[X0 , ψ]. Here X0 , X I , and ψ are all N by N hermitian matrices. M is the 11-dimensional Planck mass, and R is the radius of light-like compactification in Discrete Light Cone Quantization (DLCQ) [3]. Eqn. (3.1) is a one-parameter generalization of the flat-space formula of [2]. 3.1.1 Derivations There are many derivations of (3.1). We will explain three of them. Different approaches are motivated by different (although usually related) physical pictures, and they give answers that agree. Being maximally supersymmetric (that is, having 32 supersymmetries, 16 of them linearly realized, and the other 16 nonlinearly realized) is a very restrictive condition, with very few theories satisfying it, so agreement is not surprising. I. The BMN Approach One approach is that taken by [10]. The starting point is the κ-symmetric action of a superparticle in the 11-D PP-wave Z S= A dt e−1 LA t Lt (3.2) where LA t is the pull-back of the 11-D PP-wave supervielbeins, which can be obtained by taking the Penrose limit of the supervielbeins of AdS × S spaces given in [40]. Upon gauge fixing the κ symmetry by choosing the fermionic light-cone gauge, and also fixing the bosonic light-cone gauge by setting e = 1, x+ = t, S takes the form of free massive bosons X and fermions ψ with the masses ∼ µ. Then one generalizes this action to a nonabelian one by promoting the fields X and ψ to N × N hermitian matrices. One then adds the Myers term [41] ∼ µijk Tr(X i X j X k ) and the usual flatspace commutator terms. In this way one gets the matrix theory in 11-D PP-wave background. II. Supermembrane Approach Another approach, given in [29], is also based on the eleven-dimensional perspective. However, the starting point is not the superparticle, but the supermembrane. 17 The basic idea is the same as that in flat space [11]. First one writes out the action for the supermembrane in 11-D PP-wave, which consists of the kinetic term and the Wess-Zumino term Z S=   q 1 ijk A B C r s d σ − − det(Πi Πj ηrs ) −  Πi Πj Πk BCBA 6 3 (3.3) where Z M (σ) = (X µ (σ), θα (σ)) are the target superspace embedding coordinates, M ∂Z A A ΠA i = ∂σ i EM is the pull-back of the supervielbein EM to the membrane world- volume, and BM N P is the three-form superfield whose bosonic part is the three-form Aµνρ . The bosonic part of this action is worked out in Subsection 4.1.2 (see eqn. (4.32)), as a simple example of constructing the light cone Lagrangian for a membrane in an arbitrary eleven-dimensional supergravity background. As for the fermionic part of the action, one again uses the expressions for the supervielbeins in terms of the component world-volume and background fields given in [40], imposes fermionic light cone gauge to fix κ-symmetry, similar to the superparticle approach. The resulting supermembrane Hamiltonian is, in the notation of [29] Z H= ( d2 σ 1 h PA2 1 A B 2 1 + {X , X } + p+ 2 4 2  µp+ 3 2 (X i )2 +  µp+ 6 2 ) i µp+ ijk i j k i i T A A  {X , X }X − Ψ γ {X , Ψ} − µΨT γ 123 Ψ − 6 2 4 ! (X a )2 (3.4) with the Poisson bracket defined as {X A , X B } ≡ (∂1 X A ∂2 X B − ∂2 X A ∂1 X B ), not to be confused with an anti-commutator. Now to pass to the Matrix theory action, one has to “regularize” the above supermembrane theory using the prescription of [11], which, roughly speaking, says one should expand functions on the membrane using a complete set of basis functions, and truncate the basis to a finite subset. The result is that one replaces functions on the membrane with hermitian N × N matrices, integrals with traces, and Poisson 18 brackets with commutators 1 X (σ) → X A , Ψ(σ) → Ψ, p+ N A Z d2 σ → 1 Tr, {, } → −i[, ] R (3.5) This gives the action (3.1). III. Multiple D0-brane Approach Yet another approach is from the ten-dimensional perspective, by considering the dynamics of multi D0-branes in IIA string theory, as given in [29]. That analysis is based on [6, 7], which gives the matrix theory in a general weakly curved background. In [6] it was proposed that up to terms linear in the background metric perturbation hIJ (gIJ = ηIJ + hIJ ) and three-form AIJK , with I, J, K = 0, ..., 10, the matrix theory action in a weakly curved (i.e., |hIJ | << 1, |AIJK | << 1, noting that AIJK is also dimensionless) general 11-D background is given by, in the notation of that paper S = Sflat + Sweak (3.6) with Sflat being the formula for flat space 1 Sflat = − 2R Z  1 i dτ T r −Dτ Xi Dτ Xi + [Xi , Xj ][Xi , Xj ] + Θα Dτ Θα − Θα γαβ [Xi , Θβ ] 2  (3.7) with i, j = 1, ..., 9, and R being the radius of the light-like compactified x− direction as in (3.1). The additional term Z Sweak = ∞ X X 1 1 IJ(i1 ...in ) T ∂i1 ...∂in hIJ (0) + J IJK(i1 ...in ) ∂i1 ....∂in AIJK (0) dτ n 2 n=0 i1 ,...,in ! +M IJKLM N (i1 ...in ) ∂i1 ....∂in AD IJKLM N (0) + fermionic terms (3.8) with AD being the six-form potential dual to the three-form A. Note that Sweak is in the form of the moments of the currents, the moments of the stress tensor 19 T IJ(i1 ...in ) , those of the membrane current J IJK(i1 ...in ) , and those of the five-brane current M IJKLM N (i1 ...in ) , coupling to the background fields. The currents are obtained by taking a trace which is symmetrized over all orderings of terms of the forms Ẋ i , Fij ≡ i[X i , X j ], Θ, and [X i , Θ] (we denote this symmetrized trace as STr). As examples, 1 STr(1) R   1 1 i i 1 1 i i +− T = STr Ẋ Ẋ + Fij Fij + Θγ [X , Θ] R 2 4 2 T ++ = (3.9) There are two types of terms which contribute to higher moments of the currents. For example, the higher moments of the stress tensor are given by
IJ(i1 ...in ) T IJ(i1 ...in ) = Sym T IJ ; X i1 , X i2 , ..., X in + Tfermion (3.10) where the contributions Sym (STr(Y ); X i1 , X i2 , ..., X in ) are defined as the symmetrized average over all possible insertions of the matrices X ik into STr(Y ) (where Y is the IJ(i ...i ) n 1 product of terms of the forms Ẋ i , Fij , Θ, and [X i , Θ]). Tfermion are additional terms containing fermions, with one simple example being +i(j) Tfermion =
1 STr Θγ [ij] Θ 8R (3.11) The higher moments of the membrane current and five-brane current are given in a similar manner. For a more complete list of the moments of the currents, see Appendix A of [7]. Before relating the above matrix theory to multiple D0-brane dynamics in tendimensional IIA string theory and applying the formalism to the IIA background obtained from a space-like compactification of the 11-D PP-wave, we would like to make one remark: as pointed out in [29], using the matrix theory action given in eqn. (3.6), one can directly derive the matrix theory in 11-D PP-wave in the weakly curved approximation. The reason that we would like to present the somewhat indirect 20 10-D approach is that the 10-D viewpoint in term of D0-brane dynamics is quite informative, e.g., it clarifies the subtleties involved in light-like compactification as well as how dynamics in eleven dimensions is related to that in ten dimensions. Also, in [7] this procedure is actually used to predict the previously unknown multi D0brane action in a general weakly curved type IIA supergravity background. As pointed out by Seiberg [4] and Sen [5], light-like compactification and spacelike compactification of M theory are related by a certain limit. More specifically, M theory with Planck scale MP compactified on a light-like circle of radius R and momentum P− = N is the same as M̃ theory with Planck scale M̃P compactified on R a spatial circle of radius Rs with N D0-branes in the limit Rs → 0 M̃P → ∞ Rs M̃P2 = RMP2 = fixed M̃P R̃i = Mp Ri = fixed (3.12) where Ri is the characteristic length of the transverse metric. The IIA string theory resulting from M̃ theory has its string coupling and string scale given by g̃s = 3/4 −1/2 Rs (RMP2 )3/4 and M̃s2 = Rs (RMP2 )3/2 , and is thus a weakly coupled string theory with large string tension in the Seiberg-Sen limit, which is a very simple theory and lies at the root of the simplification of the matrix theory. Also note that the condition Rs M̃P2 = RMP2 = fixed is to ensure that the energies of the states we are interested in remain finite rather than going to zero in this limit. Now let us apply the above Seiberg-Sen limit to the weak 11-D background at hand. Recall that our light-like compactified M theory has the background gIJ = ηIJ + hIJ , with the light-like (in the flat space limit) direction x− being a circle of radius R. This theory can be obtained by an infinite boost of a space-like compactified theory as follows. Consider the M̃ theory with background metric g̃IJ = ηIJ + h̃IJ , where the space-like direction x10 is a circle of radius Rs . These two theories are related by 21 a boost along the x10 direction with the boost parameter being s γ= R2 +1 2Rs2 (3.13) in the Rs → 0 limit. The components h̃IJ and hIJ are related in an obvious manner through this coordinate transformation. The M̃ theory compactified on the space-like circle x10 is equivalent to type IIA string theory with the background metric, Ramond-Ramond (R-R) one-form, and dilaton given to leading order by 1 hIIA µν = h̃µν + ηµν h̃10 10 2 Cµ = h̃10µ 3 φ = h̃10 10 4 (3.14) where µ, ν = 0, ..., 9, with the string coupling and the string scale being gs = 1/2 3/2 (Rs MP )3/2 , Ms = Rs Mp . Combining the above two steps, namely, boosting and compactifying to ten dimensions, one obtains the following relations between the metric components 3 Rs2 R2 hIIA h + h−− ++ 00 = h+− + 2 8R2 2Rs2 R Rs h+i + h−i hIIA 0i = 2R Rs   1 Rs2 R2 IIA hij = hij + δij −h+− + h++ + 2 h−− 2 4R2 Rs 2 2 3 3Rs 3R h++ + h−− φ = − h+− + 2 4 16R 4Rs2 R2 R2 C0 = s2 h++ − 2 h−− 4R Rs Rs R Ci = h+i − h−i 2R Rs (3.15) where on the right-hand side we have kept only the leading term in Rs /R for each of the components of the metric hIJ . In the above we have only considered the 10-D 22 fields resulting from the 11-D metric. The relations for the other 10-D fields, i.e., the Neveu-Schwarz-Neveu-Schwarz (NS-NS) two-form and R-R three-form, can be worked out similarly. Apparently the right-hand sides in the above expressions for the IIA string theory fields diverge when Rs → 0. This is because we are not done with Seiberg-Sen limit yet. In the spirit of that limit, we should make the further rescaling in the IIA string theory action  R→ Rs R 1/2 i x → R,  Rs R 1/2 xi , h(~x) → h(~x) (3.16) with the goal that the energies that we are interested in should remain finite when Rs → 0. Note the above rescaling of the transverse direction xi can be inferred from the rescaling of Ri in eqn. (3.12). Section 2.2.2 of [7] gives a nice illustration of this rescaling in the single D0-brane case. The multi D0-brane action in weak IIA supergravity background is given by S = Sflat + Sweak (3.17) with Sflat being the flat space expression, and Z Sweak = dt ∞ X 1 h1 n! 2 n=0 µν(k1 ...kn ) (∂k1 ...∂kn hIIA µν )Ih µ(k1 ...kn ) + (∂k1 ...∂kn Cµ )I0 (k ...kn ) + (∂k1 ...∂kn φ)Iφ 1 µνλρστ ζ(k1 ...kn ) + (∂k1 ...∂kn C̃µνλρστ ζ )I6 µνλρστ (k1 ...kn ) + (∂k1 ...∂kn Bµν )Isµν(k1 ...kn ) + (∂k1 ...∂kn B̃µνλρστ )I5 (3) µνλ(k1 ...kn ) + (∂k1 ...∂kn Cµνλ )I2 (3) µνλρσ(k1 ...kn ) + (∂k1 ...∂kn C̃µνλρσ )I4 i (3.18) Let us explain the meaning of the various terms in this action. Similar to the 11-D case, the additional term Sweak in the action due to the curved background takes the form of moments of currents coupling to 10-D background fields. Ih is the stress tensor, Iφ is the current coupling to the dilaton, I0 and I2 are D0 brane current and 23 D2 brane current coupling to the R-R one-form C and three-form C (3) , respectively. I6 and I4 are D6 brane current and D4 brane current coupling to the dual seven form C̃ and dual five form C̃ (3) , respectively. And Is and I5 are the currents associated with fundamental string and NS5-brane which couple to NS-NS two form B and its dual B̃, respectively. It is understood that in the above action the Rs → 0 limit is taken, with the rescaling of parameters as described earlier performed to ensure a finite result. By requiring the action (3.17) to reproduce the action (3.6), the various currents I and their moments in (3.17) can be related to the 11-D stress tensor, membrane current, and five-brane current. The results are given in equations (17), (19), (21), (22) of [7], and we do not reproduce those equations here. Finally it is time to apply the above formalism to the 11-D PP-wave [29]. The metric of 11-D PP-wave is given in light-cone coordinates, so we first (un)boost it q R2 with a boost parameter γ = 2R 2 + 1. Under this boost, s √ Rs x → √ x+ , 2R − + x → 2R − x Rs (3.19) As one can see, the 11-D PP-wave metric is invariant under this boost, except that  2 s its (++) component, or equivalently µ2 , is rescaled by √R2R . Combining this with the rescaling of dimensionful parameters R, xA described earlier [7] to ensure finite √ energies, one sees that the overall effect of boost and rescaling is µ → µ/ 2. Hence, in the unboosted frame, the 11-D metric becomes F2 ds = − 1 + 4 2   F2 dt + 1 − 4 2   9 X F2 10 (dx ) − dtdx + dxA dxA 2 A=1 with the positive quantity F 2 ≡ −g++ = µ2 10 2  1 P3 9 1 i 2 i=1 (x ) + 36 P9 (3.20)  a 2 a=4 (x ) . The four- form field strength in the unboosted frame is F123(10) = −F0123 = µ 2 (3.21) 24 Next we perform a space-like compactification along the x10 direction to get a IIA supergravity background −2φ 2φ ds2IIA = −e 3 dt2 + e 3 δAB dxA dxB (3.22) with the dilaton, R-R one-form, R-R three-form, and NS-NS two-form given by 4φ e3 =1− F2 F2 µ µ , C0 = − , C0ij = ijk xk , Bij = ijk xk 2 4 4−F 6 6 (3.23) If we now use the approximation that the background is weakly curved, i.e., F 2 << 1 (recall that −F 2 = g++ = h++ ), then to linear order in F 2 , the dilaton, R-R oneform, and metric perturbation are 3 2 F2 F2 F2 IIA IIA φ ≈ − F , C0 ≈ − , h00 ≈ − , hAB ≈ − δAB 16 4 8 8 (3.24) As a check, these expressions do agree with the general expressions given in eqn. (3.15), with h++ set to −F 2 and all other components of h set to zero (also after rescaling by (Rs /R)2 ). Now consider the multi D0-brane action (3.17) in the above weakly curved background. One finds that the terms arising from the eleven-dimensional metric are 1 1 1 (AB) µν(AB) 0(AB) ∂A ∂B φIφ + ∂A ∂B hIIA + ∂A ∂B C0 I0 µν Ih 2 2 4  3 1 1 1 (AB) 00(AB) CD(AB) 0(AB) 2 + Ih + δCD Ih + 2I0 (3.25) = − (∂A ∂B F ) Iφ 16 2 2 2 S2 = Dropping higher-order terms that vanish in the Seiberg-Sen limit, eqn. (17) in [7] gives the following expressions for the currents Iφ , Ih , I0 1 1 Iφ = T ++ − T +− − δCD T CD 3 3 ++ +− CD 00 Ih = T + T , Ih = T CD I00 = T ++ (3.26) 25 and their (AB) moments are given by just adding superscript (AB) to both sides of the above expressions. Plugging these moments into eqn. (3.25), one finds 1 S2 = − (∂A ∂B F 2 )T ++(AB) 4 (3.27) Using eqn. (3.10), one finds T ++(AB) = R1 Tr(X A X B ). Plugging in the explicit expression for F 2 = −g++ we see that S2 gives the bosonic mass term in the matrix model (3.1). The terms arising from the eleven-dimensional three-form are αβγ(A) S3 = ∂A Bµν Isµν(A) + ∂A Cαβγ I2 µ h i 0ij(m) = ∂m ijk xk Isij(m) + 3I2 6 (3.28) Dropping higher-order terms that vanish in the Seiberg-Sen limit, eqn. (19) in [7] gives following expressions for the currents Is , I2 Isij = 3J +ij , I20ij = J +ij (3.29) and their (m) moments are given by adding superscript (m) to both sides of the above expressions. Plugging these moments into eqn. (3.28), one finds S3 = µijk J +ij(k) (3.30) with J +ij(k)   1 i T ijk i j k = − Tr i[X , X ]X + Ψ γ Ψ 6R 8 (3.31) as given in Appendix A of [7]. Now it is easy to see that S3 gives the fermionic mass term and the Myers term in (3.1). We know that the Matrix model (3.1) is true to all orders in F 2 , since the other two derivations of it described earlier did not require the background to be weakly curved. Hence it is interesting, from the perspective of this third derivation using the formula 26 for multi D0-brane action in weakly curved backgrounds, that there are no corrections to the weak background result. Also, as can be computed, the scalar curvature of the IIA supergravity background obtained by compacting the 11-D PP-wave along x10 is given by # " X X µ2 (xa )2 (xi )2 − cµ2 R = − (1 − F 2 /4)−3/2 a − bµ2 8 a i (3.32) with a, b, c being positive constants. This scalar curvature diverges as F 2 approaches 4, making the IIA background singular. This is due to the fact that g10 10 changes sign when F 2 goes beyond 4, making our space-like compactification along x10 lose its validity. All in all, the reason that the Matrix model derived above turned out to be valid beyond the weak background approximation is likely to be due to the fact that the 11-D PP-wave we started with is maximally supersymmetric. 3.1.2 Supersymmetric Classical Vacua The vacua of the Matrix model in the 11-D PP-wave are given by configurations which minimize the potential term. The nice thing about the potential term is that it can be written as a sum of squares,   2 1  µ 2 R µ i ijk j k a b 2 a i 2 a 2 V = Tr X + i X X + (i[X , X ]) + (i[X , X ]) + (X ) 2 3R 2 6R (3.33) in the notation of [29]. The last term in the above potential requires X a to vanish at a minimum. Then one finds that the whole potential vanishes for Xi = µ i J 3R (3.34) with J i being the generators of an SU (2) algebra in the N -dimensional representation [J i , J j ] = iijk J k (3.35) Hence the classical vacua are labelled by N -dimensional representations of SU (2), 27 which are specified by (N1 , ..., Nk ) : partitions of N into sums of positive integers with the Ni ’s being the dimensions of the individual irreducible representations making up the N -dimensional representation. The interpretation of this result is that each p µ Ni2 − 1 ≈ vacuum corresponds to a set of membrane fuzzy spheres with radii ri = 6R µNi 6R when the Ni ’s are large. The above classical vacua preserve all of the sixteen linearly realized supersymme- tries of the theory [10]. As it turns out these classical vacua are also exact quantum vacua [42]. One interesting observation about these vacua made in [30] is that they actually can also be interpreted as transverse five-branes in M theory. For example, the trivial vacuum X i = 0, labelled by the partition (1, 1, ..., 1), can be interpreted as a single five brane in the large N limit. The evidence for this is provided by comparing the spectrum of fluctuations about the single five-brane in 11-D supergravity with the set of protected excited states about the X i = 0 vacuum on the Matrix theory side and finding precise agreement. 3.2 The Three-Dimensional Theory Describing Multiple Concentric Spherical Membranes 3.2.1 Derivation The three-dimensional theory describing multiple concentric spherical membranes in the 11-D PP-wave background is a SYM-Higgs theory with 16 supersymmetries living on R × S 2 , first derived in [30]. The approach adopted in [30] is taking the Matrix quantum mechanics model in 11-D PP-wave proposed by [10], expanding it around the k-membrane vacuum where there are k copies of the N -dimensional irreducible representations of SU (2), then by letting N → ∞ going to the continuum limit to get a 3d theory with gauge group U (k). However, in a certain sense, the above approach is unnecessarily complicated, because the (0 + 1)d matrix model can be obtained by discretizing the supermembrane 28 theory, and then what the above approach does is taking the continuum limit to get back to a 3d gauge theory. Here we take a different approach that goes directly from supermembrane theory to the 3d gauge theory, without the detour just described (see also Section 7 of [29]). We take the supermembrane action given in [29], which is already a 3d theory, expand it around the single spherical membrane vacuum, throw away higher-order terms, get a U (1) 3d gauge theory, and then by adding adjoint indices to the fields obtain the nonabelian theory which describes multiple concentric membranes. The result of this simpler approach turns out to be the same as that of [30], which is not surprising because both can be regarded as deforming the usual 3d SYM-Higgs theory in flat space while preserving sixteen supersymmetries, and there are not so many such deformations! The following are the details of this approach. I. The Abelian Theory Ref. [29] gives the following supermembrane action in the 11-D PP-wave background L0 1 +  A 2 = p Ẋ 2 " 1 1 A B 2 1 {X , X } + − + p 4 2  µp+ 3 2 X
i 2  + µp+ 6 2 ! a 2 (X ) µp+ ijk i j k  {X , X }X − 6 i +ibΨT Ψ̇ + icΨT γ A {X A , Ψ} + µΨT γ 123 Ψ 4 The indices: A = 1, ..., 9, i = 1, 2, 3, a = 4, ..., 9. {f, g} ≡ rs ∂r f ∂s g is the Poisson bracket on the membrane. Ψ is a 16 component real spinor. γ A are the 16×16 gamma matrices of SO(9), {γ A , γ B } = 2δ AB ; and we choose all the γ A ’s to be real and symmetric. (One explicit representation of the γ A ’s can be found in eqn. (5.B.1) of [43]; however in the following derivation we don’t need to use those explicit expressions). γ 123 = γ 1 γ 2 γ 3 . b and c in the fermionic part of the lagrangian are real constants to be determined later on by requiring supersymmetry. (We don’t take the values for b, c given in [29], since our conventions for spinors and gamma matrices shall differ from [29].) # (3.36) 29 We take the world volume of the membrane to be R × S 2 , with R parametrized by time t (which is the x+ of the 11-D target space) and the unit sphere S 2 parametrized R by (θ, φ) and the action is S = dtdθdφ sin θ L. Note that on a unit sphere the Poisson bracket of two functions f (θ, φ) and g(θ, φ) is given by {f, g} ≡ 1 (∂θ f ∂φ g − ∂φ f ∂θ g) sin θ (3.37) A solution to the classical supermembrane equations of motion is one satisfying + {X i , X j } = ijk µp3 X k , which minimizes the potential. This is a sphere X i = xi , with x1 = µp+ µp+ µp+ sin θ cos φ, x2 = sin θ sin φ, x3 = cos θ 3 3 3 (3.38) Fluctuations about this spherical membrane background give us a U(1) SYM theory. More specifically, let us expand about this background: X i = xi + Y i . Then in terms of the fluctuation fields Y i , X a and Ψ, eqn. (3.36) becomes L0 = L + L0 (3.39) where L consists of terms quadratic in the fluctuation fields, and is given by   1 +  i 2  a 2 L(Y , X , Ψ) = p Ẏ + Ẋ 2 " #  + 2
µp 1 k 2 i a 2 a 2 + {x , X } + (X ) − + F 2p 6 i a i +ibΨT Ψ̇ + icΨT γ i {xi , Ψ} + µΨT γ 123 Ψ 4 with F k ≡ ijk {xi , Y j } − µp+ k Y . 3 (3.40) L0 consists of terms of cubic and quartic orders in the fluctuation fields. Using the shorthand notation x for xi , Y for Y i , X for X a , and suppressing gamma matrices, we see that the cubic terms are of the forms {Y, Y }{x, Y }, {Y, Y }Y , {Y, X}{x, X}, Ψ{Y, Ψ}, and Ψ{X, Ψ}. The quartic terms are of the forms {Y, Y }{Y, Y }, {X, X}{X, X}, and {Y, X}{Y, X}. By looking at the quadratic kinetic terms in L, we determine that Y i and X a 30 both have mass dimension one half, and Ψ has mass dimension one. Noticing that the Poisson bracket {, } contributes two spatial derivatives, and the appearance of x inside the Poisson bracket cancels one spatial derivative, we find that the cubic terms in L0 are of mass dimensions 27 and 29 , and the quartic terms in L0 are of mass dimension 6. Hence all the terms in L0 have mass dimensions greater than 3 and are thus non-renormalizable. So, in what follows we throw L0 away, and only look at L as given in (3.40). Consider the following U (1) gauge transformation, with a time-independent transformation parameter Λ(θ, φ) (the time-independence of Λ is a result of taking the gauge where the temporal component of the U (1) gauge potential At vanishes): δgauge Y i = {Λ, xi }, δgauge X a = 0, δgauge Ψ = 0 (3.41) As can be easily verified using the Jacobi identity of the Poisson bracket and the + fact {xi , xj } = ijk µp3 xk , the F k we introduced above is gauge-invariant. Since Λ is time-independent, Ẏ i is also gauge-invariant. Hence this U (1) gauge transformation leaves L unchanged. Now let’s show that our theory has 16 supersymmetries. Consider the following supersymmetry transformation with the parameter being a 16-component real spinor (t) (Recall that  is time-dependent due to the supersymmetry algebra of the 11-D PP-wave target space) is δY i = iT γ i Ψ + λ{Λ0 , xi }, δX a = iT γ a Ψ   δΨ = e1 Ẏ i γ i + Ẋ a γ a  + e2 F k γ k γ 123  +e3 {xi , X a }γ i γ a  + nµX a γ a γ 123  (3.42) where Λ̇0 = iT Ψ. Also ˙ = mµγ 123 . We will determine the constants λ, e1 , e2 , e3 , n in the supersymmetry transformation, the constant m in the expression for , ˙ and the constants b, c in the action (3.40) simultaneously by requiring supersymmetry. The above supersymmetry transformation basically comes from truncating higher- 31 order terms in the supersymmetry transformation of the original supermembrane theory when expanding around the vacuum background. The Λ0 term in the supersymmetry transformation of Y i is a compensating gauge transformation needed to remain in the At = 0 gauge. Recall that the supersymmetry transformation of At is δAt = pi+ T Ψ, and the gauge transformation is δgauge At = − p1+ ∂t Λ0 . Requiring these two transformation to cancel so that we remain in the At = 0 gauge gives Λ̇0 = iT Ψ. Demonstrating the supersymmetry is fairly easy and not so instructive, so we don’t bother to write out the details and only explain the strategy here: Since the SO(3) and SO(6) parts don’t mix, we’ll look at their variations separately, namely, we consider δL that only depends on the Y i ’s, and δL that only depends on the X a ’s. When there are two time-derivatives on the fields (time-derivatives on  can always be replaced by a term proportional to µγ 123 ), we let one of them act on the boson and the other on the fermion. When there is only one time-derivative, we always move it to act on the bosons by integration by parts. We always let the Poisson bracket act on the bosons using integration by parts. Also, it helps to first look at terms containing two powers of µ explicitly (in comparison, there are many terms containing one power of µ, because the Poisson bracket of two x’s gives a µ). Some useful gamma matrix identities are γ i γ 123 = 12 ijk γ j γ k , 12 jpq γp γq γn = δ jn γ 123 − jnp γp , and γ k γ 123 γ i = δ ki γ 123 − kij γj . The fact {xk , F k } = 0 comes in handy, too. We find the following values for the coefficients: 1 1 p+ 1 1 p+ 1 b = −1, c = − + , λ = − + , e1 = , e2 = − , e3 = − , n = − , m = − p p 2 2 2 12 12 (3.43) plugging these values into the expressions for the Lagrangian and supersymmetry 32 transformation, we get the action   1 +  i 2  a 2 p Ẏ + Ẋ L(Y , X , Ψ) = 2 " #  + 2
1 µp 2 2 − + F k + {xi , X a }2 + (X a ) 2p 6 i a −iΨT Ψ̇ − i 1 T i i i µΨT γ 123 Ψ Ψ γ {x , Ψ} + + p 4 (3.44) and the supersymmetry transformation 1 {Λ0 , xi }, δX a = iT γ a Ψ p+  p+  i i 1 δΨ = Ẏ γ + Ẋ a γ a  − F k γ k γ 123  2 2 + 1 p − {xi , X a }γ i γ a  − µX a γ a γ 123  2 12 δY i = iT γ i Ψ − (3.45) 1 with ˙ = − 12 µγ 123 . II. The Nonabelian Generalization Generalizing the theory to the nonabelian case by adding adjoint indices to the fields is fairly straightforward. We use the latter half of the alphabet, e.g., m, n, p, as group adjoint indices. fmnp is the real, totally antisymmetric structure constant of the gauge group. The gauge transformation of the fields is given by δgauge Ymi = {Λm , xi } + gf mnp Λn Ypi a δgauge Xm = gf mnp Λn Xpa δgauge Ψm = gf mnp Λn Ψp (3.46) where g is the gauge coupling, Λm is the time-independent gauge transformation parameter. Also, the adjoint indices are raised/lowered by the Kronecker delta. 33 Define the following gauge-covariant field strength and covariant-derivatives: Fmk = kij  g {x , Ymj } + f mnp Yni Ypj i 2  − µp+ k Y 3 m a } + gf mnp Yni Xpa Fmia = {xi , Xm Fmab = gf mnp Xna Xpb i = {xi , Ψm } + gf mnp Yni Ψp ξm (3.47) which all have the standard gauge transformation δgauge (...)m = gf mnp Λn (...)p . Now it’s easy to see that, in terms of these gauge-covariant objects, the nonabelian generalization is Lnonabelian = i p+ h i 2 a 2 (Ẏm ) + (Ẋm ) 2 " #  + 2 1 µp 1 a 2 (Xm − + (Fmk )2 + (Fmia )2 + (Fmab )2 + ) 2p 2 6 1 1 1 i − i + gf mnp ΨTm γ a Xna Ψp + i µΨTm γ 123 Ψm −iΨTm Ψ̇m − i + ΨTm γ i ξm p p 4 (3.48) The supersymmetry transformation is δYmi = a δXm = δΨm =
1 i mnp i {(Λ ) , x } + gf (Λ ) Y 0 m 0 n p p+ 1 iT γ a Ψm − + gf mnp (Λ0 )n Xpa p + p 1 a a (Ẏmi γ i + Ẋm γ ) − Fmk γ k γ 123  2 2 1 ab a b p+ 1 ia i a a a 123 µXm γ γ  − Fm γ γ  − Fm γ γ  − 2 4 12 1 − + gf mnp (Λ0 )n Ψp p iT γ i Ψm − (3.49) where (Λ˙0 )m = iT Ψm is the time-dependent compensating gauge transformation 1 parameter, and we still have ˙ = − 12 µγ 123  as in the abelian case. The gauge-invariance of Lnonabelian is easily seen since it is built from gauge- 34 covariant objects. Demonstrating that it is invariant under the supersymmetry transformation given above is straightforward, although a bit tedious. Compared with flat space-time we now have many more terms due to two facts: time and space are now divided into R × S 2 ; the nine transverse directions are now divided into SO(3) and SO(6). We relegate the detailed proof of the invariance of Lnonabelian in eqn. (3.48) under the sixteen supersymmetries to Appendix A. III. Writing the Theory in a More Conventional Form Let’s first restore At . Now the gauge transformation parameter Λm (t, θ, φ) can have time-dependence. The gauge transformation is δgauge Amt = Λ̇m + gf mnp Λn Apt δgauge Ymi = {Λm , xi } + gf mnp Λn Ypi a δgauge Xm = gf mnp Λn Xpa δgauge Ψm = gf mnp Λn Ψp (3.50) The action is Lnonabelian =  p+  (Dt Y i )2m + (Dt X a )2m 2 # "  + 2 1 1 µp a 2 − + (Fmk )2 + (Fmia )2 + (Fmab )2 + (Xm ) 2p 2 6 1 1 1 i − i + gf mnp ΨTm γ a Xna Ψp + i µΨTm γ 123 Ψm −iΨTm (Dt Ψ)m − i + ΨTm γ i ξm p p 4 (3.51) a where (Dt Y i )m ≡ Ẏmi − {Amt , xi } − gf mnp Ant Ypi , (Dt X a )m ≡ Ẋm − gf mnp Ant Xpa , 35 (Dt Ψ)m ≡ Ψ̇m − gf mnp Ant Ψp . The supersymmetry transformations are δYmi = δΨm = a iT γ i Ψm , δXm = iT γ a Ψm , δAmt = i T  Ψm p+ p+ 1 ((Dt Y i )m γ i + (Dt X a )m γ a ) − Fmk γ k γ 123  2 2 1 ab a b p+ 1 ia i a a a 123 µXm γ γ  − Fm γ γ  − Fm γ γ  − 2 4 12 (3.52) So far we have used the Poisson bracket extensively in our proof of gauge symmetry and supersymmetry, because this makes the derivation quite concise (e.g., the Jacobi identity of the Poisson bracket is used at many places of the proof). The fields transform as world volume scalars. This is OK for computation of any physical quantity, e.g., a scattering amplitude. However, let’s also write the theory in the more familiar form where the fields transform as world volume scalars, vectors, and spinors 1 . First let’s look at the bosonic part of (3.51). In the abelian case, the gauge transformation of the components of the Y i ’s in the spherical coordinates is −1 δgauge Yr = 0, δgauge Yθ = ∂φ sin θ    +  µp+ µp Λ , δgauge Yφ = sin θ∂θ Λ 3 3 (3.53) which suggests we should define new fields Φ, Aθ , Aφ as Φ = Yr , A θ = 1 Yφ , Aφ = − sin θYθ sin θ (3.54) which has the simple gauge transformation:  δgauge Φ = 0, δgauge Aθ = ∂θ   +  µp+ µp Λ , δgauge Aφ = ∂φ Λ 3 3 (3.55) In the nonabelian case, we do the same thing, i.e., Φm ≡ Ymr , Amθ ≡ 1 1 Ymφ , Amφ ≡ − sin θYmθ sin θ There is a catch about the spinors, though, as we shall explain later. (3.56) 36 Also, define the rescaled fields √ + √ + µ µ p p a Xm , Φ̃m ≡ Φm X˜a m ≡ 3 3 p p µ(p+ )3/2 Ãmθ ≡ p+ Amθ , Ãmφ ≡ p+ Amφ , Ãmt ≡ Amt 3 (3.57) In terms of these rescaled fields, the bosonic part of the action (3.51) is  2  1 3 1 1  µ 2 µ µν SB = dtdθdφ sin θ − F̃mµν F̃m − (D̃µ Φ̃)m (D̃ Φ̃)m − (Φ̃m )2 µ 4 2 2 3  µ 2 1 1 1 (X̃ a )2 − g̃ 2 f mnp f mrs X̃na X̃pb X̃ra X̃sb − (D̃µ X̃ a )m (D̃µ X̃ a )m − 2 2 6 Z m 4  µ 1 2 mnp mrs f Φ̃n X̃pa Φ̃r X̃sa + Vm ∧ F̃m (3.58) − g̃ f 2 3 Z where g̃ = −   3 µ (p+ )−3/2 g, F̃mµν = ∂µ Ãmν − ∂ν Ãmµ + g̃f mnp Ãnµ Ãpν , (D̃µ Φ̃)m = a ∂µ Φ̃m + g̃f mnp Ãnµ Φ̃p , (D̃µ X̃ a )m = ∂µ X̃m + g̃f mnp Ãnµ X̃pa . and the one-form Vm has the following components: Vmt = Φ̃m , Vmθ = 0, Vmφ = 0. The above action is of the form of a SYM-Higgs theory on a sphere of radius µ3 . Besides the usual flat space terms, the presence of the three-form in the 11-D PP-wave R background adds mass terms for the scalars Φ̃m , X˜a m , and also the term ∝ Vm ∧ F̃m . R The last term, which in components is ∝ Φ̃m F̃mθφ , comes from the corresponding three-form term in the supermembrane action. From the IIA D2 brane view point, it comes from the wedge product of the R-R one-form potential and the field strength of the world-volume gauge field. We see that our SB has the same form of that of eqn. (7) in [30] (their µ is our µ/3.) Now let’s look at the fermionic part of the action (3.51). Let’s start with the i abelian theory. In the abelian case, the ΨTm γ i ξm term in the action (3.51) becomes ΨT γ i {xi , Ψ} = µp+ T 1 Ψ [(cos θ cos φγ 1 + cos θ sin φγ 2 − sin θγ 3 )∂φ Ψ 3 sin θ +(sin θ sin φγ 1 − sin θ cos φγ 2 )∂θ Ψ] (3.59) The above expression contains unwanted explicit-φ-dependence which we will get rid 37 of by applying an SO(3) rotation on Ψ; after that, we will still have to apply another SO(3) rotation on Ψ to bring the θ dependence into the standard form for a field on R × S 2 . Both rotations will have to depend on φ or θ. The coefficients of the ∂φ Ψ term form a vector (cos θ cos φ, cos θ sin φ, − sin θ); and the coefficients of the ∂θ Ψ term form another vector (sin θ sin φ, − sin θ cos φ, 0). It is easily seen that, a rotation around the 3-axis by an angle −φ rotates the first vector into 1-3 plane and the second vector into 2-axis, thus eliminating the φ-dependence in both vectors. The explicit rotation matrix is given by R1 = cos(φ/2) − sin(φ/2)γ 1 γ 2 (recall that rotating the γ i ’s is equivalent to rotating their coefficients). The second rotation is given by R2 = cos(θ/2) − sin(θ/2)γ 3 γ 1 , which rotates the first vector into the 1-axis, keeping the second vector fixed. Define the rotated spinor ψ by Ψ = R1 R2 ψ. In terms of ψ, the fermionic part of the lagrangian (3.51) in the abelian case becomes   µ 1 T cos θ 2 µ 1 T 2 T 1 LF = −iψ ψ̇ − i −ψ γ ∂θ ψ + ψ γ ∂φ ψ − ψ γ ψ − i ψ T γ 123 ψ 3 sin θ 2 sin θ 12 (3.60) T Now let’s compare the above lagrangian with that of a Majorana (real) spinor in 3d η (which is a two-component spinor) with mass m on R × S 2 with the radius of the sphere being µ3  µ  1  τ2  T T L3d = iη η̇ + i −η τ1 ∂θ η + η τ3 ∂φ − i cos θ η + mη T τ2 η (3.61) 3 sin θ 2 T (we have taken the 3d gamma matrices, denoted as γ̃’s to avoid confusion with the SO(9) gamma matrices γ’s, to be all real γ̃ t̂ = iτ2 , γ̃ θ̂ = τ3 , γ̃ φ̂ = τ1 (3.62) with τ1 , τ2 , τ3 being the Pauli matrices). We see that LF and L3d have the same form, with the correspondence γ 1 → 38 µ τ3 , γ 2 → τ1 , 12 → m, except that in the mass terms iγ 123 does not correspond to τ2 (because iγ 123 commutes with γ 1 , γ 2 , while τ2 anti-commutes with τ3 , τ1 ). We would like to show that LF can be written as the sum of 8 copies of L3d . However, this is going to be highly non-trivial. The analog of this in string theory is that, to relate the GS formalism to the RNS formalism, we have to perform a non-trivial bosonization and re-fermionization of the fermions. What we should do is probably to decompose Ψ into 8-component spinors Ψ+ and Ψ− with eigenvalue ±1 under γ 9 . Ψ± transform as 8s ’s of SO(8). Then we should exploit the triality of SO(8) so that after some kind of three-dimensional analog of bosonization and re-fermionization, the Ψ± become S± which transform as 8v ’s. Then we should combine S± to get eight 3d Majorana spinors η M , M = 1, ..., 8 which transform as 8v . This nontrivial nature of relating target space spinors to world-volume spinors is also the reason underlying the mismatch between iγ 123 and τ2 noted above. Carrying out the procedures outlined in the previous paragraph is an interesting exercise in group theory for its own sake. However we are not going to do it here. So we will just leave the fermionic part of the action as it is in eqn. (3.51), in terms of the world-volume scalar Ψ. (The fermions in [30] are also world-volume scalars and space-time spinors.) 3.2.2 Properties In what follows we will mostly work in the static gauge where Amt = 0, and also use the formalism where the fields are world-volume scalars, i.e., we shall use (3.48) as our Lagrangian, because in terms of the Y i ’s, the SO(3) symmetry in this formalism (let’s call it the “Y -formalism”) is manifest. Occasionally we go to the formalism given in (3.58) where the Y i ’s are split into Ã’s and φ̃ (let’s call it the “A-Φ-formalism”). 39 3.2.2.1 Supersymmetric Classical Vacua Let us look at the potential term of (3.48) " #  + 2 1 1 µp a 2 V = + (Fmk )2 + (Fmia )2 + (Fmab )2 + (Xm ) 2p 2 6 (3.63) a has to vanish due to its mass term, hence Fmia and To minimize this potential, Xm Fmab also vanish. Thus vacuum configurations must have Ymi that satisfy Fmk = 0. And by looking at the supersymmetry transformation (3.49), one sees that the vacua preserve all the sixteen supersymmetries. In the A-Φ-formalism the form of these vacuum configurations is most easily seen [30]. In what follows we will focus on the case where the gauge group is SU (2), which describes two membranes. Basically the classical vacua in this theory are just “abelian monopoles” in the U (1) ⊂ SU (2). The following are the details. a Setting X̃m = 0, Ã1µ = Ã2µ = 0, and Φ̃1 = Φ̃2 = 0, with the only nonzero fields being the U (1) part Ã3µ , and Φ̃3 = constant on the sphere, the potential in (3.58) becomes  2 µ 1 Φ̃3 − F̃3θφ 3 sin θ (3.64) because the F̃ 2 term, the mass term for Φ̃, and the Vm ∧ F̃m term nicely form a complete square. Thus we need F̃3θφ = 3 sin θ Φ̃3 µ (3.65) Then, by the Dirac quantization condition, one can work out that Φ̃3 = µn 3 g̃ (3.66) R R (which gives a flux S 2 F̃3 = dθdφF̃3θφ = 2πn ), with the flux number n being integer. g̃ q Recall that (Φ̃m )2 is the radial separation between the two membranes. In flat space 40 the separation can be continuously varied, while here in the PP-wave we see that the radial separation is quantized in units of µ. In the appropriate limit where µ → 0, n → ∞, the radial separation can then be regarded as varying continuously. Going back to the Y -formalism, we find that in the classical vacuum labelled by flux number n Y1k = Y2k = 0 µp+ cos φ(1 ∓ cos θ) 1 Y3 = −n 3g sin θ + sin φ(1 ∓ cos θ) µp Y32 = −n 3g sin θ + µp Y33 = −n (±1) 3g (3.67) where the upper sign is for the northern hemisphere patch and the lower sign for the southern hemisphere patch. As can be easily verified, the above Ymi indeed yields a vanishing Fmi . 3.2.2.2 Euclidean Instantons Just as the theory itself can be regarded as arising from the continuum limit of the Matrix theory, the instantons in this theory can regarded as the continuum limit of the instantons in the Matrix theory (see [44] where instanton solutions in the matrix theory interpolating between an arbitrary vacuum and the trivial vacuum are found). Define the Euclidean time τ ≡ it. Then the Euclidean action is given by, upon setting 41 a = 0, Ψm = 0 Xm Z SE ≡ −iS = − dτ dθdφ sin θL "  # 2 Z p+ ∂Ymk 1 = dτ dθdφ sin θ + + (Fmk )2 2 ∂τ 2p " # 2 Z p+ ∂Ymk Fmk ∂Ymk Fmk = dτ dθdφ sin θ ± + + − 2(±) 2 ∂τ p ∂τ p+  2 Z ∂Ymk Fmk p+ ± + + = dτ dθdφ sin θ ∓ [K(τ → +∞) − K(τ → −∞)] 2 ∂τ p (3.68) k k m Fm term can be where to get to the last line we have used that fact that the ∂Y ∂τ p+ written as a total τ -derivative of K(τ ), where the quantity K(τ ) is defined to be the integral over S 2 1 K(τ ) ≡ 2 Z   µp+ k 2 g kij mnp k i j kij k i j dθdφ sin θ  Ym {x , Ym } − (Ym ) +   Ym Yn Yp (3.69) 3 3 Since we require Ymi to interpolate between two vacua, K(τ → ±∞) can be evaluated using the expression (3.67) worked out earlier for Ymi in the final/initial vacuum. Hence for given initial and final vacua, to minimize SE the following BPS condition has to be satisfied ± ∂Ymk Fmk + + =0 ∂τ p (3.70) and then SE is given just by the boundary term and is a function of the initial and final flux numbers (ni , nf ). It is easy to verify that the BPS condition implies the equation of motion for Ymi . Now let us look at the supersymmetry transformation (3.49), which becomes (us- 42 ing the BPS condition (3.70)) δΨm where the matrix 1∓iγ2 123   + p i i 1 i i 123  = Ẏ γ − Fm γ γ 2 m 2   i 1 ∓ iγ 123 + ∂Ym i = ip γ  ∂τ 2 (3.71) has eight unity eigenvalues and eight zero eigenvalues, and is a projection operator. Thus we see that the instanton breaks eight of the sixteen supersymmetries. As is usually the case, the BPS condition (3.70) is nonlinear in Y (recall that Fmk is quadratic in Y ) and finding its solution is not easy. Below we present the instanton solution interpolating between the n = 1 and n = 0 vacua. + k In the n = 0 vacuum, Ymk (n = 0) = 0. In the n = 1 vacuum, Ymk (n = 1) = µp3g δm . As can be verified, this expression of Ymk (n = 1) indeed gives a zero Fmk , and its radial p + m k component Φm ≡ Ymr = Ymk µpx+ /3 = xg has a gauge invariant length (Φm )2 = µp3g , as it should be for the n = 1 vacuum. This form of Ymk (n = 1) is related to that given in (3.67) by a gauge transformation. We shall call this form of Ymk (n = 1) the hedgehog gauge because of the form of Φm (we’ll call that in (3.67) the sigma-3 gauge). As it turns out, the instanton has a nice form in this hedgehog gauge. Take the trial solution Ymk (τ ) = ω(τ )Ymk (n = 1) (3.72) and plug it into the BPS condition (3.70), one gets dω µ = ± (ω − ω 2 ) dτ 3 (3.73) whose solution is the kink function ω(τ ) = 1  µ  1 + exp ∓ 3 (τ − τ0 ) (3.74) 43 with τ0 being the integration constant which gives the location of the kink. The solution with the upper sign corresponds to ni = 0, nf = 1, and the one with the lower sign corresponds to ni = 1, nf = 0. From now on let us take the one with the lower sign. To summarize, the instanton solution that interpolates between ni = 1 and nf = 0 is given by Ymk (τ ) = µp+ k  δ 1 + exp 3 (τ − τ0 ) 3g m 1 µ (3.75) Note that this instanton solution, in the Y -formalism, is a constant on the sphere. The Euclidean action for this instanton is 2π SE (ni = 1, nf = 0) = 2 g  µp+ 3 3 (3.76) k As a matter of fact, the function obtained from the Ymk in (3.75) by replacing δm with Okm also satisfies the BPS equation, where the matrix Okm is a constant (i.e., (τ, θ, φ)-independent) SO(3) group element. However, this new solution is related to the original one in (3.75) by a gauge transformation, as one can verify. We can expand the action about the above instanton solution, keeping up to terms quadratic in the fluctuations. There are bosonic zero modes and fermionic zero modes for the quadratic part of the action. As can be easily seen, k ym (τ ) = dω µp+ k dYmk δ = dτ0 dτ0 3g m (3.77) is a bosonic zero mode, because it arises from shifting τ0 which is a symmetry operation preserving the action. This bosonic zero mode is square integrable over R × S 2 . k (There are also three bosonic zero modes arising from replacing δm with Okm as de- scribed above. However, since this is just a gauge transformation, and the resulting three zero modes are not square integrable, we usually discount them.) The fermionic zero modes, which we denote as ψm , are given by the solutions to 44 the Dirac equation in the instanton background i ∂ψm 1 − + gmpn Ypi γ i ψn + i µγ 123 ψm = 0 ∂τ p 4 (3.78) where we have assumed that ψm is (θ, φ)-independent. Take ψm = where  1−iγ 123 2  dω m γ  dτ (3.79)  = 0, i.e., iγ 123  = . One can explicitly verify that this satisfies the Dirac equation. There are eight such fermionic zero modes given by the eight choices of . Comparing ψm with the supersymmetry transformation of Ψm given in (3.71) (where we now take the lower sign specific to our instanton), we see that they are just the eight supersymmetries broken by the instanton background, as expected. These eight fermionic zero modes are square integrable. The physical meaning of this instanton: at τ = −∞, the two membranes are separated by one unit of distance (because ni = 1) in the radial direction; at τ = +∞, these two membrane coincide (because nf = 0). Since the radius of each membrane is proportional to its p− momentum, this instanton describes the exchange of one unit of p− momentum (the so called M-momentum) between the two membranes. Given this instanton, one could try to compute the transition amplitude when relative transverse velocity between the two membranes is turned on and compare the result with a supergravity computation, which we are currently investigating, in collaboration with Hok Kong Lee and Tristan McLoughlin. Also, note that for this instanton, the radial p + separation between the two concentric membranes (Φm )2 goes from µp3g to zero, p + which is always much smaller than the size of the membranes (xi )2 = µp3 when the gauge coupling g is large. Hence, if we also require that the separation between + the two membranes in the x4 through x9 directions are much smaller than µp3 , the supergravity computation can be done in the near-membrane limit explained at the end of Section 4.2 and in Appendix F. For discussions on M-momentum transfer between membranes, gravitons, and other M theory objects, see [22, 23, 24]. 45 We would like to make a few more comments on the application of instantons in this 3d gauge theory to the investigation of M-momentum transfer between two membranes. In flat space, because of the SO(7) symmetry for the seven scalars corresponding to the dimensions transverse to a membrane (recall that x− is nondynamical in the light cone gauge), the separation vector between the two membranes for a given instanton can be rotated, thus producing a transverse velocity. Put in other words, suppose the initial separation is along the 4th direction and we want to turn on a velocity along the 5th direction, we can turn on a vev for the scalar of the 5th direction which is in the same gauge as that initial scalar of the 4th direction. In PP-wave, we no longer have this SO(7) symmetry, instead the transverse directions are now divided into the radial direction Yr of the Y i ’s, and the six directions X a . As we have seen, the instantons lie in the Y i ’s. Hence given a instanton specified by Y , if we want to turn on a velocity along, say, the 4th direction, we have to find an X 4 that is in the appropriate gauge, namely, the gauge in accordance with Y . As it turns out, finding this gauge is not so straightforward technically. Also, it is of interest to find instanton solutions with more general ni , nf rather than the simplest one presented above. Given the form of the BPS condition (3.70), one could try to expand the Y ’s in terms of spherical harmonics and consider the resulting nonlinear equations for the coefficients. 3.3 Two Graviton Interaction without M-momentum Transfer—Gauge Theory Computation The content of this section is based on work in collaboration with Hok Kong Lee [21]. In short, [21] computes the two-graviton one-loop effective action for Matrix theory in the 11-D PP-wave background, and compares it to the effective action on the supergravity side in the same background. Agreement is found for the effective action on both sides, to all orders of µ. Besides providing further evidence for Matrix theory as a description of M-theory in the 11-D PP-wave background, this agreement 46 also points to the existence of a supersymmetric nonrenormalization theorem in the 11-D PP-wave background. In this section, we mainly present the computation on the gauge theory side. In their original paper [2], the authors computed graviton scattering in flat space using the Matrix theory and found exact agreement with eleven-dimensional supergravity. Since then, more detailed investigations have been performed in flat space [13, 14, 15, 16, 17, 18, 19, 20]. After the Matrix theory in a weak background was proposed by Taylor and Van Raamsdonk in [6], the case of a space weakly curved in the transverse directions was checked explicitly in [7]. Now that we have the Matrix theory in the 11-D PP-wave proposed by [10] at our disposal, which is exact in this curved background, we expect it to provide further test of the Matrix theory conjecture beyond the weak background approximation proposed by [6]. In addition, the 11-D PP-wave is different from the cases studied in [7], because the metric now has a nontrivial g++ component. Finally a remark about terminology: we use the terms “effective potential”, “effective action”, and “effective Lagrangian” in an interchangeable sense, although strictly speaking we really mean the last one; this should not cause any confusion. 3.3.1 Brief Review of Known Results In [14], the one-loop effective potential for two gravitons in flat spacetime background was computed in Matrix theory to be 1−loop Veff = 15Np Ns v 4 16M 9 R3 r7 (3.80) where Np and Ns are the numbers of D0-branes making up the probe graviton and source graviton, respectively, v and r are the transverse relative velocity and distance between them, M is the eleven-dimensional Planck mass, and R is the radius of compactification in DLCQ. This effective potential agrees precisely with the supergravity result [17]. In [7], the effective potential for a weakly curved background with nontrivial trans- 47 verse metric components was computed. Again agreement was found. In fact, the only modification needed was the replacement of r by d, the geodesic distance between the two gravitons. 3.3.2 The Effective Potential The main object for comparison on both sides of the proposed duality is the effective potential Veff . The computation is carried out in the DLCQ formalism, which was proposed in Susskind’s finite N conjecture [3], and further elucidated by [4, 5]. In this formalism x− and x− + 2πR are identified. p− is therefore quantized in units of 1/R. The implications of such a light-like compactification, however, are far from trivial [45]. One such complication arises from the longitudinal zero modes, which appear to cause perturbative amplitudes to diverge. In addition, there are concerns that the DLCQ of M-theory in the low energy limit is not necessarily the DLCQ of elevendimensional supergravity because some exotic degrees of freedom such as membranes wrapped around the lightlike direction may contribute. Here we are going to take the viewpoint in [46]. Essentially, the presence of a source exerts a pressure that decompactifies the region surrounding it, rendering x− effectively space-like by providing a nonzero g−− component in the metric. In the limit of large N , this bubble of eleven-dimensional space expands, and the approximation of supergravity as a low energy description is justified. This view is further elucidated in [47], and we do not expect new issues to arise in the PP-wave background. One important fact is, the regions of validity for the gauge theory and supergravity are actually disjoint, as can be seen through the following argument by Hok Kong Lee (see also [47]): On the gauge theory side, it can be shown that the loop counting parameter is N M −3 r−3 , which must be small for loop expansion to make sense. This gives the gauge theory’s region of validity r > N 1/3 M −1 . On the supergravity side, the source graviton produces a metric component h−− ∼ N M −9 R−2 r−7 , which gives an p effective space-like compactification radius Rs = h−− R = N 1/2 M −9/2 r−7/2 . Recall 48 −1/2 that the string length is given by ls = M −3/2 Rs = M 3/4 r7/4 N −1/4 . For supergravity to make sense, we must have r > ls (otherwise there is no such notion as spacetime). This gives supergravity’s region of validity r < N 1/3 M −1 . Therefore, in general there is no reason why they should match, as each effective action is valid only within its own validity region. Thus a mismatch does not immediately invalidate the Matrix conjecture. An exact match, however, will point to the existence of a nonrenormalization theorem, which protects the terms evaluated from gaining higher-loop corrections. If such a nonrenormalization theorem does exist, then the agreement of both sides can be viewed as positive evidence for the Matrix conjecture. It is with these points in mind that the comparison of the effective action is made here. On the Matrix theory side, the effective potential is computed up to 1-loop. As in flat space, it should correspond to terms of order κ211 on the supergravity side. The relation κ211 = 16π 5 /M 9 [17] means only terms of order 1/M 9 are relevant on the Matrix theory side for the purpose of such comparison. A natural length scale that arises on the Matrix theory side is 1/(M 3 R)1/2 , which for convenience we will denote as (α)1/2 . 2 In addition to the low velocity and large r approximation necessary to facilitate comparison in flat space, we will also assume that α 2 µ2 << 1 r2 (3.81) where µ is the 123+ component of the four-form field strength. This dimensionless number, as we will see in eqn. (3.82), is simply the relative strength of the new terms in the action arising from the PP-wave background to 2 the quartic terms already present in flat space. In the opposite limit, αr2 µ2 << 1, the effective potential on the Matrix theory side resums to give 1/µ dependence3 , which does not appear possible to be reproduced on the supergravity side. In fact, 2 3 This α should not be confused with the string scale α0 . This can be seen in eqn. (3.94), a typical term in the effective potential. 49 this is nothing new. A similar issue arises already in flat space, where the effective potential only matches when we take the small v and large r limit, or more precisely, by expanding in the small parameter vα/r2 . In other words, even with the existence of a nonrenormalization theorem, the results on both sides should only be compared at very large r, where supergravity is applicable. 3.3.3 Background Field Method We will follow the background field method as reviewed in [48]. X is expanded into a background field B and a fluctuating field Y , i.e., X = B + Y . Only the part of the action that is quadratic in Y will be of interest below. Recall the Matrix theory action in the DLCQ of M-theory in 11-D PP-wave background [10], given in eqn. (3.1) Z S= ( dtTr 9 X 1 9 (M 3 R)2 X I J 2 (D0 X ) + iψ D0 ψ + [X , X ] 2R 4R I=1 I,J=1 " # 3 9 9 X X X µ 1 µ −( )2 (X a )2 ψ T γ J [ψ, X J ] + (X i )2 − ( )2 +(M 3 R) 2R 3 i=1 6 a=4 J=1 ) 3 µ T (M 3 R)µ X −i ψ γ123 ψ − i ijk (X i X j X k ) (3.82) 4 R i,j,k=1 I 2 T Taking the ratios of any of the µ-dependent terms to the µ-independent nonderivative terms gives the parameter in eqn. (3.81). In other words, the assumption stated in the previous section is identical to treating the new terms arising from the PP-wave background as a perturbation of flat space. Note that this is exactly the opposite of the approximation made in [29], where the µ-independent terms are treated as perturbations to the µ-dependent terms. While the computation of the 1-loop effective potential is possible in both limits on the Matrix theory side, an agreement with supergravity is possible only in the large r limit given in eqn. (3.81). In what follows, unless stated otherwise, we will always assume the indices i goes from 1 to 3, a goes from 4 to 9, and I goes from 1 to 9. In addition to the action above, there are terms arising from the ghosts and gauge fixing, which we simply 50 state below: Z Sgf = Z Sghost = " 1 dtTr − (∂t X0 + i[Bi , Xi ])2 2R # " (3.83) # dtTr c∂t2 c − ∂t c[X0 , c] + c[B i , [X i , c]] (3.84) Thus, the complete Matrix theory action is SM = S + Sgf + Sghost (3.85) To simplify the notation, we will put M 3 R = 1/α = 1. This factor can be restored by dimensional analysis. It is also convenient to define g 2 ≡ R, which corresponds to a loop counting parameter in the Matrix theory. 3.3.3.1 Expansion about the Background The fields X, ψ, and c are expanded in the following way, with a purely bosonic background Xµ = Bµ + gYµ   xI 0  BI =  0 0   0 0  B0 =  0 0   η θ  ψ= θ ηe ; ; ; ; µ = 0, 1, 2, ..., 9   ζI zI  YI =  z I ζeI   ζ0 z0  Y0 =  e z 0 ζ0    c1  c= c2 e  (3.86) Here we have set Np = Ns = 1, i.e., we deal with 2 × 2 matrices. We will later restore Np and Ns . The above background has the interpretation of one graviton (the source) sitting at the origin4 , while another graviton (the probe) approaches from the 4 Another possible interpretation is a transverse five brane at the origin [30]. 51 position given by xI in the matrix B. We will use the shorthand r2 = P9 I 2 I=1 (x ) . After a Wick rotation, where we define S = iS (E) and τ = it, and at the same (E) time rotating X0 to iX0 , the quadratic part of the action is 5 (E) Sboson = ( Z dτ 1 1 1 1 − ζ0 ∂τ2 ζ0 − ζe0 ∂τ2 ζe0 + ζi (−∂τ2 + (µ/3)2 )ζi + ζa (−∂τ2 + (µ/6)2 )ζa 2 2 2 2 1 + ζei (−∂τ2 + (µ/3)2 )ζei + ζea (−∂τ2 + (µ/6)2 )ζea 2 + z 0 (−∂τ2 + r2 )z0 − 2i∂τ xI (z I z0 − z 0 zI ) ) + z i (−∂τ2 + r2 + (µ/3)2 )zi + z a (−∂τ2 + r2 + (µ/6)2 )za − iµijk xi z j zk (E) Z Sfermion = (3.87) ( µ µ µ dτ η(i∂τ − i γ123 )η + ηe(i∂τ − i γ123 )e η + 2θ(i∂τ + xI γI − i γ123 )θ 4 4 4 ) (3.88) (E) Sghost 3.3.3.2 Z = ( ) dτ ∂τ2  + e ∂τ2e  + c1 (∂τ2 − r2 )c2 + c2 (∂τ2 − r2 )c1 (3.89) The Sum over Mass The partition function, Z, of the above action can be computed as a product of functional determinants. The one-loop effective action Γ is then simply related to Z via exp(−Γ) = Z (3.90) The one-loop effective potential is defined as 5 For simplicity, all subsequent superscripts (E) on the Euclideanized fluctuation fields will be omitted. 52 Z Γ=− dτ Veff (3.91) To first approximation, however, it is not necessary to compute the functional determinants. As was suggested by Talfjord and Periwal [50], and [49], one could deduce the effective potential by simply evaluating the mass spectrum of the fluctuating fields. From the masses, the one-loop contribution to Veff can be easily deduced using the formula 1−loop =− Veff 1 2 ! X real bosons mb − X real fermions mf − X mg (3.92) real ghosts The physical reason for this is that at large distances, i.e., the limit where supergravity is valid, all strings stretching between the D0-branes can be assumed to lie in their ground state. This result can also be verified using the complete expression for Veff in terms of functional determinants. We provide an argument for this in Appendix B. In what follows, we will omit the superscript “1-loop”, assuming this is understood. The contribution from tree level, which does not concern us here, is simply the Lagrangian with X replaced by B. Both contributions will be put back together at the end in eqn. (3.100). One important point to note is that this method is valid only up to the lowest powers of v, as is already known in the flat space case. In flat space, the above formula reproduces every term predicted by a supergravity computation with the right coefficients, but the Matrix theory corrections to supergravity, i.e., terms with even higher powers of v and 1/r which would not be found in supergravity, will not come out with the correct coefficients. In fact, the parameter α can be treated as the counting parameter for this purpose. All terms of order α3 , which is basically κ211 in the supergravity language, will be found on the supergravity side, but terms on the Matrix theory side with higher powers of α, which represent short distance effects, should be treated as corrections. To compute them correctly, one needs to make use of the complete expression in terms of functional determinants. For our purpose, however, the above approach is sufficient. We are not interested 53 in computing the correction to supergravity, rather we would like to check whether the terms already predicted by supergravity in the PP-wave background can be reproduced by a Matrix theory calculation. 3.3.4 A Simple Case In the next section we will work out a more efficient method to compute Veff without explicitly diagonalizing the mass matrix. Nevertheless, it is instructive to work out the simplest case in a direct approach to get the basic idea of the computation. In this simple case, we put x8 = b and x9 = vτ , while all the other xI are set to zero6 . Here b is a constant, which can be interpreted as the impact parameter of the approaching probe graviton towards the source sitting at the origin. In this case, the mass matrix constructed from eqn.’s (3.87), (3.88) and (3.89) is easily diagonalized to give the mass spectrum listed in Table 3.1. It should be noted that the velocity in the table above is measured in Euclidean time τ , i.e., v = ∂x . In a comparison with ∂τ supergravity, a Wick rotation back into Minkowski time t = −iτ is required, which introduces extra minus signs in Veff . With the mass spectrum at hand, Veff can be evaluated using eqn. (3.92) ( p p 1 µ µ µ 1 p 2 Veff = − (2)(3 + 6 − 8 ) − 6 r + µ2 /32 + 10 r2 + µ2 /62 + 2 r2 + η+ 2 3 6 4 2 ) p p p + 2 r2 + η− − 8 r2 + µ2 /42 + v − 8 r2 + µ2 /42 − v − 4r (3.93) At this point it is useful to restore the factors of M 3 R, which we denote as 1/α. For instance, the first square root term in the about equation becomes r 6 r2 µ2 + α2 32 (3.94) Note that by putting all xi to zero for i = 1, 2, 3, we ensure that in this case the Myers term will not contribute to the mass matrix. 54 m2 Fields 0 ζ0 µ2 /32 ζi ; i = 1, 2, 3 µ2 /62 ζa ζe0 ; a = 4, ..., 9 0 µ2 /32 µ2 /62 ζei ; i = 1, 2, 3 ζea ; a = 4, ..., 9 r2 + µ2 /32 zi, zi ; i = 1, 2, 3 r2 + µ2 /62 za, za ; a = 4, ..., 8 r 2 + η+ z0 + z9, z0 + z9 r 2 + η− z0 − z9, z0 − z9 µ2 /42 η µ2 /42 ηe (8) r2 + µ2 /42 + v θ (8) r2 + µ2 /42 − v θ (8) 0 ,  0 e , e  r2 c I , cI (8) ; I = 1, 2 Table 3.1: The Mass Spectrum for a Simple Case. The numbers inside the round brackets indicate the number of physical q degrees of freedom of the fermions 2 with the given mass. η± is given by 21 [ µ62 ± 2 ( µ62 )2 + 16v 2 ] . 55 This can in turn be written as r α r 1+ 1 α ( )(αµ2 ) 32 r2 (3.95) The expression for Veff given above, being a Matrix theory result, is only expected to match with supergravity in the large r limit (if it does at all!). Defining the large r limit by eqn. (3.81), we can then expand the one-loop effective potential in powers of α2 µ2 /r2 . Thus, expanding Veff gives 15 v 4 7 µ2 v 2 1 µ4 Veff = α ( + + ) + O[α5 ] 7 5 3 16 r 96 r 768 r 3 (3.96) Wick rotating v, and restoring Np , Ns gives Veff = 7 µ2 v 2 1 µ4 Np Ns 15 v 4 ( − + ) + O[α5 ] 9 3 7 5 3 M R 16 r 96 r 768 r (3.97) The α3 terms give the factor 1/M 9 , which translates into κ211 in the supergravity language. This is the order we are interested in. We throw away the higher powers of α (which are always accompanied by powers of 1/r) because they correspond to short distance corrections to supergravity, just as in flat space. Here the first term is just the flat space result. The second and the third term are the interesting ones, with new µ2 v 2 and µ4 dependence created by the PP-wave background. A comparison of their coefficients with supergravity will show exact agreement. 3.3.5 Mass Matrix Computation In the more general cases, when the velocity and the impact parameter point in arbitrary directions, calculating the effective potential Veff by finding the entire m2 spectrum, then taking their square roots and expanding them in powers of µ and v becomes inefficient. In the most general case this involves finding the eigenvalues of 56 mass matrices of very high dimension. Instead, it is possible to make use of the sum over mass formula in eqn. (3.92) without explicitly diagonalizing the mass matrix. Let us denote the square of the mass matrix as W = M 2 . Since there is never any mixing between the bosons, the fermions and the ghosts, we can study their mass matrices separately. In terms of W , the sum over mass formula becomes p p 1 p 1−loop Veff = − tr( Wb − Wf − Wg ) 2 (3.98) The square root of W can be defined unambiguously by its expansion in powers of α/r2 in the supergravity limit, as was discussed in Section 3.3.4. Note that Mb is defined to be the mass matrix for real bosons. If it is taken to be the mass matrix for √ the complex bosons, then there will be an extra factor of two in front of Wb . Simple Recipe for Mass Matrix In this subsection we will give a simple recipe for writing out M 2 for both the bosons and the fermions. The mass for the ghosts is exactly the same as in the simple case of Section 3.3.4. First of all, we should note that the mass of ζ i and ζ a are always µ/3 and µ/6, respectively, for i = 1, 2, 3 and a = 4, ..., 9. The mass of all eight physical degrees in η is always µ/4. These are independent of the background B. Mixing occurs only among the z I and among the θ and θ. Hence in what follows, we will denote the component arising from say z I z J in the bosonic Lagrangian simply as (M 2 )IJ without mentioning z explicitly. Note also that M 2 is symmetric. I. Rules for Bosons 1. (M 2 )00 = r2 ; (M 2 )ii = r2 + µ2 /32 ; (M 2 )aa = r2 + µ2 /62 ; 2. ẋI = vI mixes z 0 and z I ⇒ (M 2 )0I = −2vI 3. x1 = b1 mixes z 2 and z 3 ... etc. ⇒ (M 2 )jk = iµijk bi Note that Rule 3 applies only to z i but not z a . Such mixing is the effect of the Myers term in the Matrix theory action. 57 II. Rules for Fermions The mass matrix for the fermions can be written in a closed form 2 2 2 2 M = r + µ /4 + 9 X vI γI + I=1 3.3.6 3 X iµxi i=1 4 {γi , γ123 } (3.99) The General Case Once the mass matrix squared W = M 2 is known, eqn. (3.98) can be used to compute the one-loop effective potential explicitly. In accordance with our earlier discussions, only terms up to order α3 ∼ 16π 5 /M 9 = κ211 are kept. After restoring all factors of M 3 R, Np , and Ns , the sum of zero and one-loop effective potentials is given by ( P P P 9 X Np Np Ns 15( 9I=1 vI2 )2 µ2 3i=1 vi2 7µ2 9a=4 va2 0,1−loop 2 Veff = v + g++ ) + 9 3 ( − − 2R I=1 I M R 16r7 96r5 96r5 " 3 ! #) 3 9 3 X X X 15µ2 X 2 2 2 2 + x − vi + va + 2( xi vi ) 32r7 i=1 i a=4 i=1 i=1 ) #2 ( " 3 #2 " 9 3 9 X X X X µ4 Np Ns 1 (xi )2 · (xa )2 (3.100) (xa )2 − 12 32 (xi )2 + + 3 9 R M 768r7 a=4 a=4 i=1 i=1 P9 p 2 In (3.100), N ( I=1 vI + g++ ) is the zero-loop potential, and the rest is the one-loop 2R potential. Note that, compared with (3.97), the one-loop potential given in (3.100) contains additional terms arising from the x1 , x2 , x3 directions. This is the equation to be compared with the supergravity result. Notice the effective potential has manifest SO(3) × SO(6) symmetry, as should be expected from the symmetry of the original Matrix theory action. Just as in flat space [6], one should be able to recast this 1-loop effective potential in the form T µν Gµν . A comparison with the supergravity side will indeed confirm this, as this is precisely the form of the effective potential on the supergravity side as derived in Appendix C. Having computed the effective potential on the Matrix theory side, the next step will be to compare it with the result from a supergravity calculation. Before this can 58 be done, the issue of gauge choice has to be addressed. It is necessary to make a gauge choice when solving the Einstein equations. A gauge choice corresponds to a choice of the coordinate system one uses to describe the physics. On the Matrix theory side, such a choice of coordinates was made right from the very beginning: The action in eqn. (3.82) was written in coordinates that made the SO(3) × SO(6) symmetry manifest. Before a comparison is possible, a corresponding choice of coordinates, i.e., a choice of gauge has to be made on the supergravity side. A comparison of the above equation with the general expression for Veff in eqn. (4.113) will in the end determine the correct gauge choice for the supergravity computation. There will be a further discussion about gauge choice in the supergravity section. 59 Chapter 4 Interaction of M-theory Objects in Eleven-Dimensional Supergravity On the supergravity side, we adopt the source-probe viewpoint, which is valid when the source is much heavier than the probe. Section 4.1 deals with the action of the probe via constrained Hamiltonian mechanics with the constraints arising from worldline/worldvolume diffeomorphism, for a point particle probe and a membrane probe. Section 4.2 deals with the determination of the background fields the probe feels, by diagonalizing the linearized supergravity field equations in the presence of the source. These first two sections provide the basis for investigating M-theory objects’ interactions on the supergravity side, at linear κ211 order. Section 4.3 is an application to the two graviton interaction without M-momentum transfer. 4.1 The Light Cone Lagrangian The light cone Lagrangian is the quantity that will be computed on the supergravity side and then compared with the gauge theory result. We shall only consider bosonic degrees of freedom, because we are only concerned with the bosonic coordinates of the probe, and also the background fermionic fields are set to zero. 60 4.1.1 Point Particle Probe Let us start with the case of a point particle (the graviton). By “light-cone Lagrangian” we mean a quantity Llc (x− , p− , xA , ẋA ) which will be defined below. Roughly speaking, it is the original Lagrangian Legendre transformed in the x− degree of freedom. Appendix C gives a quick derivation of it. The dynamics of a point particle is a system with constraint due to time reparametrization symmetry, and has to be dealt with in the Hamiltonian formalism if one wants full rigor. The derivation in Appendix C is in the Lagrangian formalism and not so rigorous. Hence let us now give a careful derivation of the light-cone Lagrangian using the constrained Hamiltonian formalism. This is also a useful warm-up before we derive the same quantity for a membrane. A very quick review about constraints: Constraints are relations between coordinates and momenta. Constraints that arise directly from the definition of momenta, i.e., without using the equations of motion, are called primary constraints. Constraints that arise when imposing the consistency requirement that the primary constraints are preserved in time evolution are called secondary constraints. Constraints that arise when imposing the consistency requirement that the secondary constraints are preserved in time evolution are called tertiary constraints, etc. Secondary, tertiary, etc., constraints are obtained by using the equations of motion. The classification of constraints into primary, secondary, tertiary, etc., constraints is of little importance in the final form of the Hamiltonian formalism. A more fundamental classification of constraints is to define first-class constraints and second-class constraints as follows: a constraint is called a first-class constraint if its Poisson bracket with every constraint vanishes weakly (i.e., vanishes on the submanifold defined by the constraints in phase space); a constraint that is not first-class is called a second-class constraint. This classification of constraints plays a central role in the Hamiltonian formalism, because first-class constraints are generators of gauge transformations. Equalities that hold only on the submanifold defined by the constraints in phase space are called weak equalities and usually denoted with the weak equality symbol “≈”, but in what follows we will simply use the equality symbol “=” for them, expecting no confusion. 61 For a comprehensive discussion on systems with constraints, see, for example, [51] and [52]. Consider a point particle of mass m in a general curved background Gµν (x) (this background includes that generated by the source object, in the context of investigating M-theory objects’ interaction on the supergravity side). The Lagrangian of the particle is q L(x , ẋ ) = −m −Gµν (x)ẋµ ẋν µ µ (4.1) where a dot means a derivative with respect to the world line parameter τ . The momenta are then given by pµ ≡ ∂L m Gµβ (x)ẋβ =p µ ∂ ẋ −Gµν (x)ẋµ ẋν (4.2) As can be easily verified, γ1 (xµ , pµ ) ≡ Gµν (x)pµ pν + m2 = 0, which is the primary constraint of our system. The Hamiltonian is given by H ≡ pµ ẋµ − L = 0, which is expected because in general the Hamiltonian of a system with reparametrization symmetry vanishes. The consistency condition γ̇1 = 0 trivially holds, hence we don’t have any secondary constraint. So the only constraint of our system is γ1 = 0, which is a firstclass constraint and is the generator of the gauge symmetry — the reparametrization freedom of the world line. The extended Hamiltonian is then given by HE = u1 γ1 , where the arbitrary function u1 is the Lagrange multiplier that embodies the gauge degree of freedom (i.e., reparametrization freedom). The evolution of any function F (xµ , pµ ) is given   P ∂F ∂G ∂F ∂G by Ḟ = [F, HE ]PB , where [F, G]PB ≡ µ=+,−,A ∂x − is the Poisson µ ∂p ∂pµ ∂xµ µ 62 bracket. Hence the equations of motion are ẋµ = 2u1 Gµα (x)pα ∂Gαβ (x) ṗµ = −u1 pα pβ ∂xµ γ1 = Gµν (x)pµ pν + m2 = 0 (4.3) The next step is gauge fixing. Let us use the light-cone coordinates xµ = {x+ , x− , xA }, and impose the light-cone gauge condition C1̃ ≡ x+ − τ = 0. We see that [γ1 , C1̃ ]PB = −2G+ν (x)pν which can reasonably be assumed to be nonzero. Hence upon gauge fixing we get two second-class constraints χα = (γ1 , C1̃ ). Then, by the usual story of gauge fixing, now that we have two second-class constraints, we should replace Poisson brackets with the Dirac brackets (which we denote as [, ]D . Recall that the Dirac bracket for two phase space functions is defined as [F, G]D ≡ [F, G]PB −[F, χα ]PB C αβ [χβ , G]PB , where χα , χβ are second-class constraints, and C αβ is the inverse matrix of Cαβ ≡ [χα , χβ ]PB ), and in this case since the gauge fixing condition C1̃ is time-dependent, we should also add a corresponding correction term (for a discussion on time-dependent gauge fixing, see, e.g., exercise 4.8 in [52]). One gets the follow law for time evolution Ḟ = [F, HE ]D − C ab̃ 1 ∂Cb̃ [F, γa ]PB = [F, γ1 ]PB ∂τ 2G+ν (x)pν (4.4) where we have used the fact that [F, HE ]D = 0. In particular, this gives Gµβ (x)pβ G+ν (x)pν  αβ  1 ∂G (x) ṗµ = − pα pβ 2G+ν (x)pν ∂xµ γ1 = 0, C1̃ = 0 ẋµ = (4.5) It easy to verify that this indeed gives ẋ+ = 1, which agrees with the gauge choice x+ = τ . Using Dirac brackets is the longer (and more rigorous) way of fixing the gauge. 63 However, there is also a shortcut which avoids using Dirac brackets. Remember that the gauge freedom is embodied in u1 . Hence gauge fixing amounts to specifying u1 . So we just have to go to the equations of motion before gauge fixing, which contain u1 , and set x+ = τ . Then the x+ equation gives us u1 = 2G+ν1(x)pν , which we can then substitute into other equations and get back the gauge fixed equations of motion obtained using Dirac brackets. This shortcut approach is what we shall use when investigating membranes. Now let’s restrict our attention to situations where the metric is static, i.e., independent of x+ , Gµν = Gµν (x− , xA ). Also, let’s consider functions which are independent of x+ , p+ , i.e., F = F (x− , xA , p− , pA ). In this case, if we define the “light-cone Hamiltonian” as Hlc (x− , xA , p− , pA ) ≡ −p+ , where p+ (x− , xA , p− , pA ) is obtained by solving γ1 = 0 (in which x+ is set to τ ) for p+ in terms of the other variables, then one can verify that  X  ∂F ∂Hlc ∂F ∂Hlc Ḟ = − ∂xµ ∂pµ ∂pµ ∂xµ µ=−,A (4.6) The above expression justifies the name of Hlc . It generates time evolution in the smaller phase space consisting of (x− , xA , p− , pA ). Using the gauge fixed equation of motion for ẋµ and also the expression of p+ (x− , xA , p− , pA ), we can express ẋµ as a function of (x− , xA , p− , pA ). Now let’s make the reasonable assumption that the relation ẋB = ẋB (x− , xA , p− , pA ) is invertible so that we can solve for pB (x− , p− , xA , ẋA ). Now by making a Legendre transformation in the transverse degrees of freedom, we define the “light-cone Lagrangian” as Llc (x− , p− , xA , ẋA ) ≡ pA ẋA − Hlc (4.7) Then it’s easy to see that, in the Llc formalism, the equations of motion are  pA = ∂Llc ∂ ẋA   , x− ,p− ,xB ṗA = ∂Llc ∂xA  (4.8) x− ,p− ,ẋB 64 The above two equations can be combined to give the expected Euler-Lagrange equation d dτ  ∂Llc ∂ ẋA   − ∂Llc ∂xA  =0 (4.9) Also  − ẋ = − ∂Llc ∂p−   , ṗ− = x− ,xA ,ẋA ∂Llc ∂x−  (4.10) p− ,xA ,ẋA One more look at Llc : recall that 0 = H = pµ ẋµ − L = p+ ẋ+ + p− ẋ− + pA ẋA − L, which upon gauge-fixing x+ = τ becomes 0 = p+ + p− ẋ− + pA ẋA − L = −Hlc + p− ẋ− + pA ẋA − L = Llc + p− ẋ− − L, which gives Llc = L − p− ẋ− , which is just the L0 defined in eqn. (C.2) of Appendix C, as expected. (The Pp+ in Appendix C is the p− here.) When investigating graviton interactions in a PP-wave, the background metric is Gµν = gµν + hµν , with gµν being the unperturbed PP-wave, and hµν being the metric perturbation due to the source graviton. Keeping terms up to linear order in hµν , one finds that the light cone Lagrangian for the probe graviton is (    1 2 1 Llc = p− g++ h−− − h+− v + g++ + h++ + g++ 2 4  X X A B 2 A hAB v v [2h+A − h−A (v + g++ )]v + + A,B A  1 1 1 + h−− v 4 − v 2 h+− − g++ h−− 8 2 2 ) (4.11) which is the object one compares to the gauge theory effective Lagrangian (and gets agreement on). Recall that the change of the longitudinal momentum p− is governed by ṗ− =
∂Llc . Hence the x− -dependence of Llc is what’s responsible for the longitu∂x− p− ,xA ,ẋA dinal momentum exchange between the probe graviton and source graviton. In the Llc given in (4.11), x− -dependence comes in only through hµν . Hence what one does is to Fourier transform hµν along the x− direction, and solve the Einstein equations 65 for the Fourier components of hµν . When one is only considering transverse momentum transfer, one plugs the zeroth Fourier component of hµν into Llc ; when one is considering one unit of longitudinal momentum transfer, one plugs the first Fourier component of hµν in, etc. Also, notice that in the Llc given in (4.11), lower powers of velocity are accompanied by hµν with more lower + indices, or equivalently fewer lower − indices (“at higher level”, in the language of Section 4.2). In Section 4.2, we shall see that the metric perturbations at lower level are easier to compute. Hence, in Llc the v 4 term is the easiest to find, next is v 2 , and then the v-independent term is the hardest to compute. In flat space, only the lowest level h−− is nonzero, and there is only the v 4 term. In the PP-wave, we have metric perturbations at all levels, and life is harder. In the membrane case the situation is similar. 4.1.2 Membrane Probe Now we consider the membrane. A brief review of constrained Hamiltonian mechanics in field theory is given in Appendix D. The previous discussion on the point particle may be said to be merely a quest for rigor, but for the membrane the following discussion is a necessity. The reason is, although people have discussed the membrane in a fairly general background [12], the background there is taken to be the special case which is independent of x− , which certainly is not true in the physical situation where there is p− transfer (see discussion near the end of 4.1.1). Also, in [12], the background metric G−− and G−A are set to zero using the target space diffeomorphism freedom. This is not the right gauge to use even in the investigation of only transverse momentum transfer (We know that in the two graviton interaction it is the background G−− that gives the v 4 term in the effective potential of the probe graviton). For the above two reasons, we would like to discuss the derivation of the light cone Lagrangian for a membrane in an arbitrary static background (note: by “static” we mean there is no x+ -dependence, yet x− -dependence is allowed). Denote the background metric and three-form as Gµν (x), Aµνρ (x), respectively, 66 and the membrane embedding coordinates as X µ (σ i ), with σ i , i = 0, 1, 2 being the world-volume coordinates. Our membrane is considered to be a probe membrane, hence it does not have any back reaction on the background geometry. The background geometry can include the contribution from a source, though. The membrane Lagrangian density is given by µ µ L(X , ∂i X ) = T q 1 − det(gij ) − ijk Aµνρ ∂i X µ ∂j X ν ∂k X ρ 6  (4.12) with T being the membrane tension, and gij ≡ Gµν ∂i X µ ∂j X ν being the pullback metric. The momentum density is ∂L Πλ ≡ =T ∂(∂0 X λ ) q 0k µ ν − det(gij ) g (∂k X )Gλµ − Aλνρ ∂1 X ∂2 X ρ  (4.13) Define Π̃λ ≡ Πλ + T Aλνρ ∂1 X ν ∂2 X ρ = T q − det(gij ) g 0k (∂k X µ )Gλµ (4.14) Then it is easy to see we have the following primary constraint φ0 ≡ Gλξ Π̃λ Π̃ξ + T 2 det(grs ) = 0 (4.15) where r, s = 1, 2 label the spatial world-volume coordinates. It is also easily seen that, φ0 only contains Πµ , X µ , and spatial derivatives of X µ , as required for a constraint. One can also verify that there are two more primary constraints: φr ≡ Πλ ∂r X λ , r = 1, 2 (4.16) Of course, we know the gauge freedom these constraints arise from: φ0 comes from world-volume temporal reparametrization freedom and the φr ’s come from worldvolume spatial reparametrization freedom. The Hamiltonian density is given by H ≡ Πλ ∂0 X λ − L = 0 (4.17) 67 as it should be for a system with general covariance. The total Hamiltonian density is given by HT = H + ci φi = ci φi = c0 φ0 + cr φr where the ci ’s denote the Lagrange multiplier fields. HT = (4.18) R dσ 1 dσ 2 HT . Computing the Poisson bracket with HT (see Appendix D), one finds the equations ∂φ0 of motion (using ∂Π = 2Gµξ Π̃ξ , µ ∂φr = ∂r X µ ) ∂Πµ ∂0 X µ = 2c0 Gµξ Π̃ξ + cr ∂r X µ   ∂φi ∂φi i ∂0 Πµ = ∂r c − ci µ ∂(∂r X ) ∂X µ (4.19) where to facilitate the evaluation of ∂0 Πµ we list below some useful expressions ∂ Π̃λ ∂(det(grs )) ∂φ0 = 2Gλξ Π̃ξ + T 2 µ µ ∂(∂r X ) ∂(∂r X ) ∂(∂r X µ ) (4.20) with ∂ Π̃λ = T Aλµρ (δ1r ∂2 X ρ − δ2r ∂1 X ρ ) ∂(∂r X µ ) ∂(det(grs )) = Gµα {2δ1r (∂1 X α )g22 + 2δ2r (∂2 X α )g11 − 2g12 (δ1r ∂2 X α + δ2r ∂1 X α )} ∂(∂r X µ ) (4.21) ∂φ0 ∂Gλξ ∂ Π̃λ ∂(det(grs )) = Π̃λ Π̃ξ + 2Gλξ Π̃ξ + T 2 µ µ µ ∂X ∂X ∂X ∂X µ (4.22) with ∂ Π̃λ ∂Aλνρ =T ∂1 X ν ∂2 X ρ µ µ ∂X ∂X ∂(det(grs )) ∂Gαβ = (∂1 X α ∂1 X β g22 + g11 ∂2 X α ∂2 X β − 2g12 ∂1 X α ∂2 X β ) µ ∂X ∂X µ (4.23) 68 also ∂φs = δsr Πµ , µ ∂(∂r X ) ∂φs = 0 r, s = 1, 2 ∂X µ (4.24) We need to check whether there are any secondary constraints. Using the above equations of motion, one shows after a somewhat tedious manipulation, ∂0 φi = 0 (4.25) for arbitrary ci , which means that there is no secondary constraint, and the φi ’s are all first class constraints with the Lagrange multipliers ci ’s embodying the gauge degrees of freedom. This is an expected result, but it’s reassuring to see that is the case for arbitrary backgrounds, not just the special backgrounds (e.g., flat background, x− -independent backgrounds) investigated previously. Next we should gauge fix the membrane system. In the spirit explained in the point particle case, we will not do so by introducing Dirac brackets, but rather by specifying the ci ’s, which is somewhat shorter. Now use the light-cone coordinates {x+ , x− , xA }, and assume static background Gµν (x− , xA ) and Aµνρ (x− , xA ). Use the light cone gauge X + = σ 0 . Then the equation of motion for X + gives c0 = 1 2G+ξ Π̃ (4.26) ξ So now the constraint φ0 (X − , X A , ∂r X − , ∂r X A , Π+ , Π− , ΠA ) = 0 (in which X + is set to σ 0 ) can be used to solve for Π+ (X − , X A , ∂r X − , ∂r X A , Π− , ΠA ). Then we define the light cone Hamiltonian Hlc (X − , X A , ∂r X − , ∂r X A , Π− , ΠA ) ≡ −Π+ Before defining the light-cone Lagrangian, similar to the point particle case, we need to express ΠA as a function of (X − , X A , ∂r X − , ∂r X A , Π− , ∂0 X A ). This is done by looking at the equations of motion for X A : A GAξ Π̃ξ ∂X A r ∂X = + c ∂σ 0 ∂σ r G+ν Π̃ν (4.27) 69 where we have used the expression for c0 given above. We should fix the gauge by specifying the cr ’s to be functions of (X − , X A , ∂r X − , ∂r X A , Π− , ΠA ). Then using the expression Π+ (X − , X A , ∂r X − , ∂r X A , Π− , ΠA ) obtained earlier, we can invert this equation of motion to find ΠA (X − , X A , ∂r X − , ∂r X A , Π− , ∂0 X A ). Then we can define the light-cone Lagrangian Llc (X − , X A , ∂r X − , ∂r X A , Π− , ∂0 X A ) ≡ ΠA ∂0 X A − Hlc • (4.28) An Easy Example: A Probe Membrane in the Unperturbed PP- wave Background Now, as an easy example, let us derive Llc for a probe membrane in the unperturbed PP-wave background using the above prescription. We will see that we get the expected answer given in [29] In the unperturbed PP-wave, the only nonzero component of Aµνρ is A+ij = µ  xk , and we don’t bother to write down the familiar PP-wave metric here again. 3 ijk Solving the constraint φ0 = 0, we find Π+ = −1  µ −g++ Π2− + ΠA ΠA + T 2 det(grs ) − T ijk (∂1 X i )(∂2 X j )X k 2Π− 3 (4.29) Also easily seen is det(grs ) = 21 (∂1 X A ∂2 X B − ∂1 X B ∂2 X A )2 . Also the equations of motion for X A are ∂0 X A = ΠA + cr ∂r X A Π− (4.30) To motivate a gauge choice, let us note that ∂0 Π− = ∂r (cr Π− ). (This is just equation (2.21) in [12], in the particular case of PP-wave). Hence we choose cr = 0, which means Π− will be σ 0 -independent. This gauge choice gives us ΠA = Π− ∂0 X A (4.31) 70 Hence we find Llc = ΠA ∂0 X A − Hlc = Π− ∂0 X A ∂0 X A + Π+   Π− 2 1 i 2 1 1 A A a 2 µ (X ) + (X ) = Π− ∂0 X ∂0 X − 2 2 9 36 2 T µ − (∂1 X A ∂2 X B − ∂1 X B ∂2 X A )2 − T ijk (∂1 X i )(∂2 X j )X k 4Π− 3 (4.32) which agrees with [29] upon setting T = −1 (this minus sign is to conform to [12]’s convention eqn. (2.16) for the membrane action, which [29] follows.) and identifying our Π− with their p+ . • A Less Trivial Example: A Probe in the PP-wave Background Perturbed by a Source Let us compute Llc in the PP-wave background perturbed by some source. The background is now Gµν = (Gµν )pp + hµν , and Aµνρ = (Aµνρ )pp + aµνρ , with the quantities with subscript pp being those of the unperturbed PP-wave background, and hµν , aµνρ being metric and three-form perturbations caused by the source. We only need the light-cone Lagrangian to linear order in the perturbation. Llc = (Llc )pp + δLlc (4.33) with (Llc )pp being the expression given in the previous example of unperturbed PPwave. The computation is, as in the previous example, quite straightforward, although a bit tedious, because solving for ΠA in terms of (X − , X A , ∂r X − , ∂r X A , Π− , ∂0 X A ) in the perturbed background requires some work. But the four-time-derivative term, i.e., the v 4 term in Llc is not hard to find. Making the gauge choice cr = 0, we find (4) Llc = Π− h−− (δAB ∂0 X A ∂0 X B )2 8 (4.34) 71 (the superscript (4) means it’s the four-time-derivative term) which is quite similar to the graviton case. To know this term, we only need h−− produced by the source. Of course, to get terms with lower powers of the velocity, we need other components of hµν and also the three-form perturbation aµνρ . To summarize, to get the explicit expression for the light-cone Lagrangian, we only need to figure out the hµν and aµνρ produced by the source, which is the subject of the next section. 4.2 Diagonalizing the Supergravity Field Equations for Arbitrary Static Sources Now let us present the diagonalization of the linearized supergravity equations of motions for arbitrary sources. There is, of course, no highbrow knowledge involved here: we are just solving the linearized Einstein equations and Maxwell equations, which are coupled; and by “diagonalization” we basically just mean the prescription using which we get a decoupled Laplace equation for each component of the metric and three-form perturbations. The unperturbed background is the 11-D PP-wave, and we only consider static, i.e., x+ -independent, field configurations, thanks to the fact that the sources considered are taken to be static, i.e., with x+ -independent stress tensor and three-form current. Since we leave the source arbitrary, what we’ll present here are the left-hand side of the linearized equations. These are tensors whose computation is straightforward though a bit tedious: the reason we present them here is because they are necessary when solving the field equations, and to the best of our knowledge have not been explicitly given elsewhere. A somewhat related problem is the diagonalization of the equations of motion when the source is absent. This requires field configurations with x+ -dependence. One good reference along this line is [53]. Roughly speaking, borrowing the language of electromagnetism, what’s considered in [53] are electromagnetic waves in vacuum, 72 while what we are considering here are electrostatics and magnetostatics for arbitrary sources. Denote the metric perturbation δgµν as hµν , and the gauge potential perturbation as δAµνρ = aµνρ . Once again, the nonzero components of the PP-wave background metric and the four-form field strength are given by " g+− = 1, g++ = −µ2 # 3 9 1X i 2 1 X a 2 (x ) + (x ) , gAB = δAB 9 i=1 36 a=4 F123+ = µ (4.35) (4.36) In our conventions, µ, ν, ρ, . . . take the values +, −, 1, . . . , 9; A, B, C, . . . take the values 1, . . . , 9; i, j, k, . . . take the values 1, . . . , 3; and a, b, c, . . . take the values 4, . . . , 9. Also, we follow [54] for the conventions of various tensors. The nonzero components up to (anti)symmetry of the Christoffel symbol, Riemann tensor, etc., of the 11-D PP-wave are 1 1 ∂A g++ , Γ− +A = ∂A g++ 2 2 1 1 R+A+B = − ∂A ∂B g++ , R++ = − ∂C ∂C g++ , 2 2 ΓA ++ = − R=0 (4.37) (We usually don’t substitute the explicit expression of g++ , unless that brings significant simplification to the resulting formula) Now let’s add a source, thus perturbing the background. hµν , aµνρ are treated as rank-two and rank-three tensors, respectively, the covariant derivative ∇ acting on them is defined using the connection coefficient of the unperturbed PP-wave background, and indices are raised/lowered, traces are taken using the background metric gµν . Let’s deal with the Einstein equations first. Define h̄µν ≡ hµν − 12 gµν h, where h ≡ g µν hµν . Without the source, the Einstein equation is 1 Rµν − Rgµν − κ211 [Tµν ]A = 0 2 (4.38) 73 Recall that the stress tensor of the gauge field is 1 [Tµν ]A = 12κ211  Fµλξρ Fν λξρ 1 − gµν F ρσλξ Fρσλξ 8  (4.39) The source perturbs the Einstein equation to  1 δ Rµν − Rgµν 2  − κ211 δ[Tµν ]A = κ211 [Tµν ]S (4.40) with [Tµν ]S standing for the stress tensor of the source. As usual, it helps to proceed in an organized manner, grouping different terms in
the above perturbed Einstein equations. One finds, δ Rµρ − 12 Rgµρ = − 12 ∇σ ∇σ h̄µρ + Kµρ +Qµρ , and κ211 δ[Tµν ]A = Nµν +Lµν , where the explicit expressions of the symmetric tensors ∇σ ∇σ h̄µν , Kµν , Qµν ,Nµν , and Lµν can be obtained after some work. Their definitions and components are given below 1 • ∇σ ∇σ h̄µν σ ∇ ∇σ h̄++ =   1 g ∂µ ∂ν h̄++ + −(∂A ∂A g++ )h̄+− + (∂A g++ ∂A g++ )h̄−− 2   +2 ∂A g++ ∂− h̄+A − ∂A g++ ∂A h̄+− (4.41) µν 1 ∇σ ∇σ h̄+− = g µν ∂µ ∂ν h̄+− − (∂A ∂A g++ )h̄−− + ∂A g++ ∂− h̄−A − ∂A g++ ∂A h̄−− 2 (4.42) 1 ∇σ ∇σ h̄+C = g µν ∂µ ∂ν h̄+C − (∂A ∂A g++ )h̄−C +∂A g++ ∂− h̄AC −∂C g++ ∂− h̄+− −∂A g++ ∂A h̄−C 2 (4.43) ∇σ ∇σ h̄−− = g µν ∂µ ∂ν h̄−− (4.44) 1 Notice that ∂+ will never appear because we only consider the static case; also note g µν ∂µ ∂ν = 2 −g++ ∂− + ∂A ∂A for static configurations. 74 • Kµν ∇σ ∇σ h̄−C = g µν ∂µ ∂ν h̄−C − ∂C g++ ∂− h̄−− (4.45) ∇σ ∇σ h̄CD = g µν ∂µ ∂ν h̄CD − ∂C g++ ∂− h̄−D − ∂D g++ ∂− h̄−C (4.46) Its definition is Kµρ ≡
1 1 1 Rµξ h̄ξρ + Rρξ h̄ξµ + Rσµρ ξ h̄σξ + gµρ Rξσ h̄ξσ − Rh̄µρ 2 2 2 (4.47) Its components are given by  K++ = 1 − ∂A ∂A g++ 2   1 1 h̄+− + g++ h̄−− + (∂A ∂B g++ ) h̄AB 2 2   1 − ∂A ∂A g++ h̄−− 2 (4.49)    1 1 − ∂C ∂C g++ h̄−A + − ∂A ∂B g++ h̄−B 4 2 (4.50) K+− =  K+A = • Qµν (4.48) K−− = 0 (4.51) K−A = 0 (4.52)   1 1 KAB = ∂A ∂B g++ − δAB ∂C ∂C g++ h̄−− 2 2 (4.53) Its definition is Qµρ ≡ 21 (∇µ qρ + ∇ρ qµ ) − 12 gµρ ∇α qα , where qα ≡ ∇β h̄βα . As one can recognize, Qµρ contains the arbitrariness of making different gauge choices when solving the Einstein equation, where one makes a gauge choice by specifying 75 the qµ ’s. The components of Qµρ are 1 1 Q−− = ∂− q− , Q−A = (∂− qA + ∂A q− ), Q−+ = (g++ ∂− q− − ∂A qA ) 2 2 1 1 QAB = (∂A qB + ∂B qA ) − δAB (∂− q+ − g++ ∂− q− + ∂A qA ) 2 2 1 Q+A = [∂A q+ − (∂A g++ )q− ] 2 1 1 Q++ = (∂A g++ )qA − g++ (∂− q+ − g++ ∂− q− + ∂A qA ) (4.54) 2 2 Let’s make a few more remarks about gauge choice here. If one chooses the “Lorentz gauge” where all the qµ ’s vanish, then Qµν all vanish. One can also choose, say, the “harmonic gauge” in which δ(g ρσ Γµρσ ) = 0. (Note that in the unperturbed PP-wave background g ρσ Γµρσ vanishes.) These two gauges are in general different because δ(g ρσ Γµρσ ) = −hρσ Γµρσ +q µ , or more explicitly, δ(g ρσ Γ+ρσ ) = q− , δ(g ρσ Γ−ρσ ) = −h−A ∂A g++ + q+ − g++ q− , and δ(g ρσ ΓAρσ ) = 12 h−− ∂A g++ + qA . One may also choose gauges in between, of which our graviton computation in [21] is an example. We don’t concern ourselves much with the issue of gauge choice here, because in any gauge, provided the qµ ’s are set to some known functions, we shall be able to diagonalize the linearized equations. The gauge choice issue will resurface later when one compares the results of the supergravity calculation with that of the gauge theory calculation. There, for the results from both sides to match, one has to make a “most natural” gauge choice (usually motivated by the symmetry of the problem) on the supergravity side. We will discuss that in Section 4.3 in the specific example of two graviton interactions. • Nµν It is defined to be the part of κ211 δ[Tµν ]A that contains only the metric perturbation, but not the three-form gauge potential perturbation. Its components 76 are given by N++ = µ2 3 9 1 1X 1X 1 h̄+− + g++ h̄−− − h̄ii + h̄aa 3 12 3 i=1 6 a=4 ! µ2 µ2 N+− = h̄−− , N+i = h̄−i , N+b = 0 4 2 N−− = 0, N−i = 0, N−b = 0 µ2 µ2 Nij = − δij h̄−− , Nib = 0, Nab = δab h̄−− 4 4 • Lµν (4.55) This is defined to be the part of κ211 δ[Tµν ]A that contains only the three- form perturbation, but not the metric perturbation. Its components are given by   1 µ µ L++ = µ δF123+ − g++ δF123− , L+− = 0, L+i = ijk δF+jk− , L+b = δF123b 2 4 2 L−− = 0, L−i = 0, L−b = 0 µ µ µ Lij = δij δF123− , Lib = ijk δFbjk− , Lbd = − δbd δF123− (4.56) 2 4 2 Next let us deal with the Maxwell equation. In the absence of the source, it is
√ 1 η̃ µ1 ...µ11 √ ∂λ −g F λµ1 µ2 µ3 − √ Fµ4 ...µ7 Fµ8 ...µ11 = 0 −g 1152 −g (4.57) where η̃ is either +1 or −1 depending on the convention, which we can always fix later by requiring the consistency of the conventions for the equations and the solutions that we consider. (As it turns out, in the two graviton interaction case [21] it does not matter because this F ∧F term has no effect on the final effective potential. Of course, for membrane interactions, that would no longer be the case.) When the source is present, we add its current J µ1 µ2 µ3 to the left-hand side of the above equation, and get 
√ η̃ µ1 ...µ11 1 λµ1 µ2 µ3 √ − Fµ4 ...µ7 Fµ8 ...µ11 = J µ1 µ2 µ3 δ √ ∂λ −g F −g 1152 −g  (4.58) We can write the left-hand side of the above equation as the sum of two totally antisymmetric tensors Z µ1 µ2 µ3 + S µ1 µ2 µ3 , where Z µ1 µ2 µ3 is defined to be the part that 77 contains the metric perturbation only, and S µ1 µ2 µ3 is defined to be the part that contains the three-form perturbation only. One finds Z +−i = µijk ∂j h̄−k , Z +−b = 0, Z +ij = µijk (∂− h̄−k − ∂k h̄−− ), Z +ib = 0, Z +bc = 0 ! # " 3 X 1 h̄ii − ∂b h̄kb , Z −ib = µijk ∂j h̄kb , Z −bc = 0 Z −ij = µijk ∂k h̄ − h̄−− − 3 i=1 " ! # 3 X 1 Z ijk = −µijk ∂− h̄ − h̄−− − h̄ii − ∂b h̄−b , Z ijb = µijk (∂− h̄kb − ∂k h̄−b ) 3 i=1 Z ibc = 0, Z bce = 0 (4.59) and S +−A = g µν ∂µ ∂ν a−+A + ∂B g++ ∂− aBA− − ∂− (∇µ aµ+A ) + ∂A (∇µ aµ+− ) (4.60) S +AB = g µν ∂µ ∂ν a−AB − ∂− (∇µ aµAB ) + ∂A (∇µ aµ−B ) − ∂B (∇µ aµ−A ) (4.61) S −AB = g µν ∂µ ∂ν a+AB − g++ S +AB + {[(∂A g++ )(∂− a−+B ) + ∂A (∇µ aµ+B ) − ∂A (aEB− ∂E g++ )] − [A ↔ B]} η̃ −(∂D g++ )δFD−AB − µ −ABµ4 ...µ7 123+ δFµ4 ...µ7 (4.62) 24 S ABE = g µν ∂µ ∂ν aABE − (∂A g++ )(∂− a−BE ) − (∂B g++ )(∂− a−EA ) − (∂E g++ )(∂− a−AB ) η̃ −∂A (∇µ aµBE ) − ∂B (∇µ aµEA ) − ∂E (∇µ aµAB ) − µ ABEµ4 ...µ7 123+ δFµ4 ...µ7 24 (4.63) Notice that S µ1 µ2 µ3 contains ∇µ aµρλ and its derivatives. Those terms correspond to the gauge freedom for the three-form gauge potential. One could use the “Lorentz gauge” where ∇µ aµρλ = 0. But for the sake of generality, let’s leave the gauge choice for the three-form arbitrary. 78 Now that we have collected the expressions for the various tensors, we are ready to diagonalize the field equations. Recall that the Einstein equation is 1 − ∇σ ∇σ h̄µν + Kµν + Qµν − Nµν − Lµν = κ211 [Tµν ]S 2 (4.64) and the Maxwell equation is Z µ1 µ 2 µ3 + S µ 1 µ2 µ 3 = J µ 1 µ2 µ3 (4.65) The right-hand sides of these equations are given by specifying the source that we consider (recall that the three-form current J is of order κ211 ), hence we only need to concentrate on diagonalizing the left-hand sides. As will be seen shortly, it is useful to define “level” for tensors: lower +/upper − indices contribute +1 to the level; lower −/upper + indices contribute −1 to level; and the upper A/lower A indices contribute zero to the level. We shall see that the field equations should be solved in ascending order of their levels. The following is the detailed prescription of the diagonalization procedure. Let us use the shorthand notation (E.E.)µν for the lower (µν) component of the Einstein equation, and (M.E.)µ1 µ2 µ3 for the upper (µ1 µ2 µ3 ) component of the Maxwell equation. • at level −2 The only field equation at this level is (E.E.)−− , which reads, upon using the expressions of the various tensors ∇σ ∇σ h̄µν , Kµν , Qµν ... etc., that we’ve given above 1 − g µν ∂µ ∂ν h̄−− + Q−− = κ211 [T−− ]S 2 (4.66) This equation can be immediately solved for h̄−− after specifying the source term and the gauge choice term Q−− . • at level −1 We have (E.E.)−A , which reads −  1  µν g ∂µ ∂ν h̄−A − (∂A g++ )(∂− h̄−− ) + Q−A = κ211 [T−A ]S 2 (4.67) 79 which can now be solved for h̄−A , using the h̄−− found previously. Also at this level is (M.E.)+AB , which reads, g µν ∂µ ∂ν a−ij − ∂− (∇µ aµij ) + ∂i (∇µ aµ−j ) − ∂j (∇µ aµ−i ) + µijk (∂− h̄−k − ∂k h̄−− ) = J +ij g µν ∂µ ∂ν a−ib − ∂− (∇µ aµib ) + ∂i (∇µ aµ−b ) − ∂b (∇µ aµ−i ) = J +ib g µν ∂µ ∂ν a−bc − ∂− (∇µ aµbc ) + ∂b (∇µ aµ−c ) − ∂c (∇µ aµ−b ) = J +bc (4.68) from which we can find a−AB , upon specifying the gauge choice ∇µ aµρλ for the threeform and using the h̄−A and h̄−− found previously. • at level 0 At this level we have (E.E.)+− , (M.E.)+−A , (E.E.)AB , and (M.E.)ABE . (E.E.)+− is of the form 1 − g µν ∂µ ∂ν h̄+− = known terms 2 (4.69) (From now on, we will not bother writing down the detailed equations; “known terms” refers to the gauge choice terms Qµν , ∇µ aµρλ , source terms, and terms containing previously found h̄µν ’s and aµνρ ’s, one can write those down by looking up the expressions given earlier for the various tensors.) Hence solving it we get h̄+− . Solving (M.E.)+−A gives a−+A . (E.E.)AB and (M.E.)ABE are coupled, so a little more work is needed. The following are the details. First notice that the only unknown in (M.E.)bce is abce , hence solving this equation we find abce ((M.E.)bce contains the usual term g µν ∂µ ∂ν abce and also a term of the form ∂− adf g which comes from the F ∧ F in the Maxwell equation, hence it is not quite a Laplace equation. But, that being said, one shouldn’t have any difficulty solving it.) (M.E.)ibc is of the form g µν ∂µ ∂ν aibc = known terms, solving which gives aibc . 80 (M.E.)ijb and (E.E.)kb are coupled in the following manner g µν ∂µ ∂ν aijb + µijk ∂− h̄kb = known terms 1 1 − g µν ∂µ ∂ν h̄kb + µklm ∂− almb = known terms 2 4 (4.70) Decoupling these two equations are quite easy. Let us take a12b and h̄3b as the representative case. One sees that these two equations can be recombined to give (g µν ∂µ ∂ν + iµ∂− )(h̄3b + ia12b ) = known terms (g µν ∂µ ∂ν − iµ∂− )(h̄3b − ia12b ) = known terms (4.71) Solving these equations gives (h̄3b + ia12b ) and (h̄3b − ia12b ), and in turn h̄3b and a12b . P (M.E.)ijk is coupled to (E.E.)ij and (E.E.)bd through the quantity H ≡ 23 3i=1 h̄ii − P9 1 a=4 h̄aa in the following manner 3 g µν ∂µ ∂ν a123 + µ∂− H = known terms 1 1 − g µν ∂µ ∂ν h̄ij + µδij ∂− a123 = known terms 2 2 1 1 µν − g ∂µ ∂ν h̄bd − µδbd ∂− a123 = known terms 2 2 (4.72) Combining the last two equations gives −g µν ∂µ ∂ν H + 4µ∂− a123 = known terms (4.73) Recombining this with first equation, we get (g µν ∂µ ∂ν + 2iµ∂− )(H + 2ia123 ) = known terms (g µν ∂µ ∂ν − 2iµ∂− )(H − 2ia123 ) = known terms (4.74) solving which individually gives H and a123 . Using the obtained expression for a123 one can then find h̄ij and h̄bd . Thus we are done with (E.E.)AB and (M.E.)ABE . • at level 1 81 (M.E.)−AB is of the form g µν ∂µ ∂ν a+AB = known terms, solving which gives a+AB . (E.E.)+A is of the form − 12 g µν ∂µ ∂ν h̄+A = known terms, solving which gives h̄+A . • at level 2 (E.E.)++ is of the form − 21 g µν ∂µ ∂ν h̄++ = known terms, solving which gives h̄++ . Thus we have completely diagonalized the whole set of Einstein equations and Maxwell equations! Let us use  to denote g µν ∂µ ∂ν (which we call the Laplacian). In Appendix E we ˜ the Fourier transform of  along x− . Since it does discuss the Green’s function of , not have x+ dependence, this Green’s function is different from the scalar propagator discussed in [55], which has a closed form expression. In Appendix F we give the expression for h−− (again Fourier transformed along the x− direction) when the source is a membrane which is a sphere in the first three transverse directions, a point at the origin of the other six transverse directions, and moving along the trajectory X + = σ 0 , X − = 0, in the near-membrane limit (see Appendix F for what is meant by the “near-membrane limit”). In that case h−− has the form of a massive scalar Green’s function in seven-dimensional Euclidean space, with its “mass” proportional to the k− that the graviton carries. Plugging this h−− into (4.34) gives the v 4 term of the light cone Lagrangian of a probe membrane in the PP-wave background perturbed by this spherical source membrane, in the case when every point on the probe membrane is in the near-membrane limit with respect to the source membrane. 4.3 Two Graviton Interaction Without M-momentum Transfer—Supergravity Computation Completing our investigation of the gauge/gravity duality appearing in two graviton interactions without M-momentum transfer in a PP-wave, the content of this section is basically taken from the computation on the supergravity side given in [21]. This 82 is an application of the general formalism developed in Sections 4.1 and 4.2, although since [21] was done before the writing up of this thesis, the following differs from the general formalism in some nonessential details, e.g., it solves for hµν instead of h̄µν , and the term “level” is not mentioned explicitly. (The “solving the field equations in ascending order of the level” pattern can be very easily recognized though.) Because there is no M-momentum transfer in this application, many terms containing ∂− which couple different equations simply disappear, making the diagonalization easier. To find the two-body effective action, one only needs to solve for the metric perturbation caused by the source graviton at the linear order (∼ κ211 ). The action is given by S = S G + S A + SP (4.75) SG is the Einstein action for the metric, given by 1 SG = 2 κ11 Z d11 x p |g|R (4.76) SA is the action for the three-form, given by 2 SA = − 2 κ11 Z ( p ) |g| η̃ 1 F µνλξ Fµνλξ + µ1 ...µ11 Aµ1 µ2 µ3 Fµ4 ...µ7 Fµ8 ...µ11 d11 x 2 · 2 · 4! 12 3!(4!)2 (4.77) SP is the action for the source graviton (the subscript P means “particle”), given by 1 SP = CP 2 Z +∞  dξ −∞ 1 dy µ dy ν gµν (y) − β(ξ)m2 β(ξ) dξ dξ  (4.78) with CP being some constant. The above action gives the equations of motion for the metric, the three-form field, and the source graviton, listed below. The Einstein equation is
1 Rµν − Rgµν = κ211 [Tµν ]A + [Tµν ]P 2 (4.79) 83 The Maxwell equation is  p µνλξ |g|F − ∂µ η̃ νλξρ1 ...ρ8 Fρ1 ...ρ4 Fρ5 ...ρ8 = 0  1152 (4.80) The geodesic equation is d2 y µ dy ρ dy ν µ =0 + Γ (y) ρν dξ 2 dξ dξ (4.81) [Tµν ]A and [Tµν ]P are the stress tensors obtained by varying SA and SP with respect to the metric 1 [Tµν ]A = 12κ211  1 Fµλξρ Fνλξρ − gµν F ρσλξ Fρσλξ  8 CP 1 p [Tµν ]P (x) = gµρ (x)gνλ (x) 2 |g(x)| Z +∞ dξ −∞ (4.82) 1 dy ρ (ξ) dy λ (ξ) (11) δ (x − y(ξ)) β(ξ) dξ dξ (4.83) Setting CP to zero means the absence of the source graviton. In this case, a solution to the above equations of motion is the 11-D PP-wave background. Recall that the metric gµν and the four-form field strength of the unperturbed 11-D PP-wave background are given by " g+− = 1, g++ = −µ2 # 3 9 1X i 2 1 X a 2 (x ) , gAB = δAB (x ) + 9 i=1 36 a=4 F123+ = µ (4.84) (4.85) As before, in our conventions, µ, ν, ρ, . . . take the values +, −, 1, . . . , 9; A, B, C, . . . take the values 1, . . . , 9; i, j, k, . . . take the values 1, . . . , 3; and a, b, c, . . . take the values 4, . . . , 9 The introduction of a source graviton, i.e., a non-zero CP , perturbs the above PP-wave solution to gµν −→ gµν + hµν ≡ Gµν ; Fµνρσ −→ Fµνρσ + fµνρσ (4.86) It suffices to solve the geodesic equation at the zeroth order of CP , which gives a 84 solution x+ = ξ, x− = 0, xA = 0 (4.87) and the corresponding stress tensor of the source graviton is then + − [Tµν ]P (x) = p gµ+ gν+ δ(x ) 9 Y δ(xA ) (4.88) A=1 CP where p+ = 2β is just the p− of the source graviton (note that β(ξ) is a constant 0 β0 for a geodesic) and in what follows we will use p+ instead of CP . Note that the order of κ211 is the same as the order of p+ . Also note that the only non-vanishing Q component of [Tµν ]P is [T−− ]P = p+ δ(x− ) 9A=1 δ(xA ). In what follows we will integrate everything over the x− direction, thus getting rid of δ(x− ) and derivatives with respect to x− . On the Matrix theory side, the effective potential was only computed up to 1-loop. In supergravity language, that means we are only looking at order κ211 . To find the effective potential on the supergravity side up to this order, we need only the linearized (i.e., to the linear order of p+ ) Einstein equation and Maxwell equation. We consider static solutions which have no x+ dependence. Also, we restrict our attention to metric and gauge field perturbations that go to zero at infinity. The linearized Einstein equation in 11 dimension is δRµν = κ211  
1 αβ αβ δTµν + gµν T hαβ − g δTαβ ≡ Tµν 9 (4.89) where the perturbation to the total stress tensor is given by δTαβ = [δTαβ ]A + [Tαβ ]P (4.90) [δTαβ ]A is the perturbation to the stress tensor of the gauge field, which is to be expressed in terms of the perturbation to the field strength. First look at the (−−) component of the Einstein equation, which is 85 9 1 X ∂ 2 h−− δR−− = − 2 A=1 ∂xA ∂xA (4.91) and T−− = κ211 δT−− = κ211 [T−− ]P = κ211 p+ 9 Y δ(xA ) (4.92) A=1 where [δT−− ]A = 0 (as can be readily verified) has been used. This gives h−− = κ211 p+ 15 1 π 4 16 |~x|7 (4.93) where we use ~x to denote the nine-dimensional vector in the transverse directions. The (−A) component of the Einstein equation is 9 9 1 X ∂ 2 h−A 1 X ∂ 2 h−B δR−A = − + 2 B=1 ∂xB ∂xB 2 B=1 ∂xA ∂xB (4.94) T−A = 0 (4.95) and which gives h−A = 0 (4.96) Now we look at the linearized Maxwell equation, in terms of the gauge potential perturbation aµνρ (note fλµνρ = ∂λ aµνρ − ∂µ aνρλ + ∂ν aρλµ − ∂ρ aλµν ). We choose to P work in the “Lorentz gauge” where 9D=1 ∂D aµνD = 0. The upper (AB+) component of the Maxwell equation gives 9 X D=1 2 ∂D aAB− − 9 X D=1 ∂D [h−− FDAB+ ] = 0 (4.97) 86 Using the expression for h−− that we just found, we have 3 µκ211 p+ 15 X xk  ijk π4 32 k=1 |~x|7 aij− = (4.98) while all other aAB− ’s vanish. Hence we find that the field strength is f−ijk f−ijb " P # 3 i 2 (x ) µκ211 p+ 15 1 = ijk 7 i=1 9 −3 7 π4 32 |~x| |~x|  k b 3 µκ211 p+ 15 X x x = ijk 7 9 4 π 32 k=1 |~x| (4.99) Next consider the upper (ABC) component of the Maxwell equation. Using the fact that h−A = 0 and aAB− = 0 except for aij− , we have 9 X 2 ∂D aABC = 0 (4.100) D=1 hence, all aABC = 0. Now the (A + −) component. Using h−A = 0 we get 9 X 2 ∂D aA−+ = 0 (4.101) D=1 thus aA−+ = 0. Now we go back to look at the (+A) component of the Einstein equation. Using h−A = 0, we get δR+A = − 9 9 1 X ∂ 2 h+A 1 X ∂ 2 h+B + 2 B=1 ∂xB ∂xB 2 B=1 ∂xA ∂xB (4.102) Using aA−+ = 0, aABC = 0, and h−A = 0, we get T+A = 0 (4.103) So we conclude that h+A = 0 (4.104) 87 Now consider the (+−) component of the Einstein equation 9 9 1 X ∂ 2 h+− 1 X ∂g++ ∂h−− δR+− = − + 2 A=1 ∂xA ∂xA 2 A=1 ∂xA ∂xA (4.105) and T+− =
1 2 µ h−− − µf−123 6 (4.106) In writing T+− , we made use of the following equations µ2 h−− 4κ211
1 2 = δ −2µf − µ h ij −123 −− 4κ211
1 δ 2µf−123 + µ2 h−− = 2 bc 4κ11 3 µ X ijk f−jkb = − 2 4κ11 j,k=1 [δT+− ]A = [δTij ]A [δTbc ]A [δTib ]A (4.107) Solving the (+−) component of the Einstein equation, we get " P # µ2 κ211 p+ 5 3i=1 (xi )2 1 1 h+− = − + π4 64 |~x|7 192 |~x|5 (4.108) The (AB) component of the Einstein equation reads δRAB " 9 # 9 9 9 X X X 1 X ∂ 2 hAB ∂ 2 hAC ∂ 2 hBC ∂ 2 hCC ∂ 2 h+− = − − − + +2 A B 2 C=1 ∂xC ∂xC C=1 ∂xB ∂xC C=1 ∂xA ∂xC C=1 ∂xA ∂xB ∂x ∂x   1 ∂ 2 g++ ∂ 2 h−− ∂g++ ∂h−− ∂g++ ∂h−− + 2h−− A B + 2g++ A B + + (4.109) 4 ∂x ∂x ∂x ∂x ∂xA ∂xB ∂xB ∂xA 88 and
1 δij 2µf−123 + µ2 h−− 3
1 = δbc 2µf−123 + µ2 h−− 6 3 µ X = − ijk f−jkb 4 j,k=1 Tij = − Tbc Tib (4.110) So far the need to make a gauge choice for the metric has not arisen. Now to solve for hAB we must make a gauge choice for the metric. Let Gρσ and Γµρσ denote the complete inverse metric and Christoffel symbol, respectively (by “complete”, we mean they include both the unperturbed and perturbed part). We shall fix the gauge by specifying Gρσ Γµρσ . As can be easily verified, G ρσ Γ+ ρσ = Gρσ Γ− ρσ = Gρσ ΓA ρσ 9 X C=1 9 X ∂C h−C = 0 (−h−C ∂C g++ + ∂C h+C − g++ ∂C h−C ) = 0 C=1 9 X 1 = ∂C hAC − ∂A 2 C=1 9 X ! hCC + 2h+− − g++ h−− (4.111) C=1 ρσ A so we need to specify Gρσ ΓA ρσ to fix the gauge. Using the above expressions for G Γρσ , we can rewrite δRAB as " 9

 # ∂ Gρσ ΓA ∂ Gρσ ΓB 1 X ∂ 2 hAB 1 ∂g++ ∂h−− ∂g++ ∂h−− ρσ ρσ δRAB = − − − + + 2 C=1 ∂xC ∂xC ∂xB ∂xA 2 ∂xA ∂xB ∂xB ∂xA (4.112) In general relativity one often uses the “harmonic gauge”, where one sets Gρσ ΓA ρσ = 0 (which is satisfied by the unperturbed PP-wave background). Here, however, we shall opt for a different gauge. 89 As derived in the Appendix C, the effective potential is given by (    1 2 1 v + g++ + h++ + g++ g++ h−− − h+− 2 4  X X 2 A A B + [2h+A − h−A (v + g++ )]v + hAB v v Np Veff = R A A,B  1 1 1 + h−− v 4 − v 2 h+− − g++ h−− 8 2 2 ) (4.113) where Np is the number of D0-branes forming the probe graviton, and v A ≡ ẋA , P9 A 2 v2 ≡ A=1 (v ) . As h+A , h−A all vanish, they simply drop out of the effective potential. The computation on Matrix theory side in section 3.3.6 tells us that in the effective potential there are no terms of the form v a v b for a 6= b, nor are there terms of the form v i v a . This suggests we choose the gauge such that hab ∝ δab , and hia = 0. To make hab ∝ δab , we set 1 Gρσ Γaρσ = h−− ∂a g++ 2 (4.114)
1 µ ∂b Gρσ Γiρσ = ∂i g++ ∂b h−− − ijk f−jkb 2 2 (4.115) then, to make hia = 0, we set which implies Gρσ Γiρσ = 35 µ2 κ211 p+ xi 96 π 4 |~x|7 (4.116) Note that the above expression makes the gauge different from the “Lorentz gauge” where all the qµ ’s vanish (see Section 4.2). Hence our gauge is something in between the harmonic gauge and the Lorentz gauge. In this gauge, the Einstein equation gives " P # µ2 κ211 p+ 1 15 3k=1 (xk )2 1 hab = δab − 5 π 4 96 2 |~x|7 |~x| (4.117) 90 " µ2 κ211 p+ 1 −15 hij = δij π 4 96 # k 2 (x ) 1 µ2 κ211 p+ 15 xi xj 1 k=1 + + |~x|7 2 |~x|5 π 4 64 |~x|7 P3 (4.118) Now let us look at the upper (AB−) component of the Maxwell equation. It gives the following equations 9 X 2 aij+ − g++ ∂D 2 aij− − ∂D 3 X ( ijk − 9 X ∂D hDk + ∂D g++ (∂D aij− + ∂i ajD− + ∂j aDi− ) 3 X " (∂m hmk − ∂k hmm ) + ∂k m=1 D=1 k=1 9 X D=1 D=1 D=1 +µ 9 X 1 2 g++ h−− + 9 X !# ) hDD D=1 =0 9 X D=1 9 X 2 ∂D abc+ = 0 2 ∂D aib+ = 0 (4.119) D=1 Solving them gives aij+ µ3 κ211 p+ = π4 3 X ! ijk xk k=1 # " 9 3 X X 1 (xa )2 (xm )2 + 7 −29 384 |~x| a=4 m=1 abc+ = 0 aib+ = 0 (4.120) Hence the field strength is given by f+ijk f+ijb " # 3 9 3 9 X X X X µ3 κ211 p+ 1 = ijk −58 (xm )2 − 3 (xa )2 + 149 (xm )2 · (xa )2 π4 384 |~x|9 m=1 a=4 m=1 a=4 ! " # 3 3 9 X X µ3 κ211 p+ X 5 xb k m 2 =  x −41 (x ) + (4.121) (xa )2 ijk 9 π4 384 |~ x | m=1 a=4 k=1 As can be easily checked, all the aµνρ we have found indeed satisfy the Lorentz gauge. 91 Finally, we consider the (++) component of the Einstein equation δR++ 9 9 9 1X 2 1 X 1 X = − ∂ h++ + ∂A g++ ∂B hAB − ∂A g++ ∂A hBB 2 A=1 A 2 A,B=1 4 A,B=1 9 9 9 1 X 1X 1X + hAB ∂A ∂B g++ + ∂A g++ ∂A h+− + g++ ∂A g++ ∂A h−− 2 A,B=1 2 A=1 4 A=1 9 1X − h−− (∂A g++ )2 4 A=1 (4.122) and µ T++ = − 2 2f+123 + µ 3 X ! hii i=1 + µ g++ (2f−123 + µh−− ) 6 (4.123) From this we find µ4 κ211 p+ h++ = π4   " 3 #2 " 9 #2 3 9   X X X X 1 a 2 i 2 a 2 i 2 116 (x ) (x ) +2 (x ) − 17 (x ) ·  6912 |~x|7  a=4 a=4 i=1 i=1 (4.124) To summarize, the nonzero components of the metric perturbation are h−− [eqn. (4.93)], h+− [eqn. (4.108)], hab [eqn. (4.117)], hij [eqn. (4.118)], and h++ [eqn. (4.124)]; and the nonzero components of the field strength perturbation are f−ijk , f−ijb [eqn. (4.99)], and f+ijk , f+ijb [eqn. (4.121)]. Substituting the expressions for the metric into our formula for Veff in eqn. (4.113), 5 averaging hµν over x− (i.e., dividing by 2πR), and noting that κ211 = 16π , p+ = NRs , M9 92 we find 15 Np Ns v 4 Np 2 (v + g++ ) + 2R 16 M 9 R3 |~x|7 ( P3  3 µ 2 N p Ns 1 1 15 (x1 )2 + (x2 )2 + (x3 )2 X i 2 15 i,j=1 xi xj v i v j + − − (v ) + R3 M 9 96 |~x|5 32 |~x|7 16 |~x|7 i=1 )  9  7 1 15 (x1 )2 + (x2 )2 + (x3 )2 X a 2 (v ) + − + 96 |~x|5 32 |~x|7 a=4  "  #2 " 9 #2 3 3 9   4 X X X X µ N p Ns 1 i 2 a 2 i 2 a 2 32 (x ) + (x ) − 12 (x ) · (x ) +  R3 M 9 768 |~x|7  Veff = i=1 a=4 i=1 a=4 (4.125) Comparison of the above formula with eqn. (3.100) on the Matrix theory side shows exact agreement. We would like to emphasize the approximation involved once again. We treated the source graviton as a perturbation to the exact PP-wave background, and the calculation was performed to first order in p+ . However, the solution that we found for these linearized equations is exact in µ. So, we have finished our comparison of the effective potentials describing two graviton interactions on the gauge theory side and the supergravity side, and have found precise agreement at order κ211 , up to quantum corrections at short distances. Let us make a few comments on our result: our result at order µ2 agrees with Taylor and Van Raamsdonk’s proposal in [6] for Matrix theory in a weakly curved background up to linear terms. As mentioned in their discussion, their proposal is proven only in the case where the background is produced by well-defined Matrix theory configurations. This is not the case for the PP-wave background, so their proposal for the Matrix theory in this background, while convincing, is not a proven fact. Thus, the result at µ2 , i.e., terms linear in the background, can be treated as additional evidence for their proposal, similar to the explicit calculation in [7], this time with a nontrivial g++ metric component. The result at order µ4 is beyond linear order in the background, and hence is a 93 new result. In fact, our calculation explicitly shows that there are no higher powers of µ in the effective action of the supergravity side at the order κ211 . On the Matrix theory side, higher powers of µ are also not expected for long distances. When they do appear, they are always accompanied by higher powers of α/r2 , which indicates that they are corrections to supergravity at short distances. However, as it stands, the corrections for velocity dependent terms are unreliable because they are computed using the sum over mass formula, which is exact only for terms independent of velocity or terms proportional to κ211 , as is shown in the Appendix B. Evaluating these corrections exactly requires going beyond the sum over mass formula, and an efficient way of handling the mass matrix will be of use. As pointed out by [56, 57], Matrix theory in a generic curved background is not expected to agree with supergravity. In the PP-wave case, however, we do find precise agreement as has been shown above. This is likely to be a result of the large number of supersymmetries of the PP-wave background, or in other words, we can say that this agreement predicts the existence of a supersymmetric nonrenormalization theorem. This leads us naturally to the discussion in Chapter 5. 94 Chapter 5 Supersymmetric Nonrenormalization Theorems The role of supersymmetric nonrenormalization theorems in Matrix theory has been crucial from the very beginning (see [2]). As a matter of fact, they are very likely to be the reason why the two theories, namely, gauge theory and supergravity, with disparate regions of validity (see Subsection 3.3.2), would give agreement in the first place. The most convincing way to address this issue is brute force evaluation of higher loop contributions, as [14] did in showing that there is no correction to the v 4 term from two-loops in the context of graviton scattering in flat space. An alternative approach is to proceed without using detailed knowledge of the underlying theory. One wants to see how sixteen supersymmetries alone, together with the SO(9) invariance of the transverse part of the flat space metric, and CPT invariance, would constrain the effective Lagrangian. Amazingly, to the v 4 order these global symmetries completely fix the effective Lagrangian, up to an overall constant. This approach was pioneered by [31], whose argument we will briefly review below. See also [32, 33, 34, 35, 36, 37]. In [31]’s notation, the bosonic part of the effective action can take the general form Z Sboson = dt(f1 (r)v 2 + f2 (r)v 4 + higher derivative terms) and one then uses supersymmetries to constrain the functions f1 (r), f2 (r). (5.1) 95 At order v 2 , the supersymmetry transformations can be written in the form δxi = −iγ i ψ, δψ = γ i v i  + M  (5.2) where the matrix M contains v and two fermions. The closure of the above supersymmetry transformations then requires M to vanish, which in turn implies that f1 must be a constant, which one normalizes to 12 . Hence adding the fermions the action at this order takes the form  Z S1 = dt 1 2 v + iψ ψ̇ 2  (5.3) At the v 4 order, the bosonic action can be written in the form Z S2 = dt  (0) (8) f2 (r)v 4 + ... + f2 (r)ψ 8  (5.4) where the ellipsis stands for v 3 ψ 2 , v 2 ψ 4 , and vψ 6 terms. The supersymmetry variation in general mixes terms with different numbers of fermions. However, the “top” term, i.e., the eight-fermion term, provides some simplifying clues. Its variation is (8) (8) (8) δ(f2 (r)ψ 8 ) = δf2 (r)ψ 8 + f2 (r)δψ 8 (5.5) where the first term is the only one that contains nine fermions, and thus does not mix with other terms. Hence this nine-fermion term must vanish by itself. After the use of Fierz identities, the most general eight-fermion term of the effective Lagrangian can be written as (ψγ ij ψψγ jk ψψγ lm ψψγ mn ψ)(g1 (r)δin δkl + g2 (r)δkl xi xn + g3 (r)xi xk xl xn ) (5.6) As argued earlier, the nine-fermion term in the supersymmetry variation of the above q d q d expression should vanish. Upon applying the operators γac ∂ and γac x to the dψc q dψc q 96 nine-fermion variation, one gets a set of coupled differential equation for the g’s, whose solution is, unique up to a constant c g1 (r) = 2 c 4 c c , g2 (r) = − , g3 (r) = 15 . 11 13 143 r 13 r r (5.7) This gives precisely the same eight-fermion Lagrangian, up to the overall constant, as obtained in [58] by an explicit loop computation. The other terms with different numbers of fermions in the order v 4 action S2 can be further determined using the eight-fermion result obtained above; for details of that calculation see [34]. In the above proof(s) of supersymmetric nonrenormalization theorems, there are subtleties related to higher derivative terms and integration by parts. After taking those subtleties into account, one finds that the result is unaffected [37]. It is natural to ask whether we can give a similar proof of the supersymmetric nonrenormalization theorem in the 11-D PP-wave. The 11-D PP-wave has a non-vanishing g++ component which breaks the transverse SO(9) symmetry into SO(3) × SO(6). As pointed out by [34] in their discussion section, when the SO(9) is broken, the v 4 order effective action should take a form similar to that in flat space, with the coefficient function f (~x) for the v 4 term now being a harmonic function of the nine-vector ~x, not just its length r, and the coefficient functions of the two, four, six, eight fermion terms being given by partial derivatives ∂i ...∂k f (~x). Unlike in the SO(9) invariant case, where supersymmetry constraints lead to ordinary differential equations with respect to r, now one has to solve partial differential equations with respect to ~x. One would expect that this requires substantially more work. For example, one could introduce the dimensionless quantity ρ/z which is the ratio between the SO(3) radius and the SO(6) radius, and the functions’ dependence on ρ/z would not be so easy to determine. Simplifying facts could appear after scrutinizing the system carefully enough, and it would be an interesting project to give a proof of the supersymmetric nonrenormalization theorem in 11-D PP-wave. 97 Chapter 6 Conclusion and Discussion Gauge/gravity duality for M theory in flat and almost flat (i.e., weakly curved) backgrounds has been investigated extensively, and nice agreements have been found. In generic curved backgrounds, things are less clear, and we certainly want to understand them better. The two graviton results reported in this thesis can be regarded as a step in this direction. We hope that our ongoing investigation of membrane interactions will shed some new light on M theory in generic curved backgrounds in the case of nonzero Mmomentum transfer. There are also many other directions that are very natural to explore. One of them is going away from the Penrose limit towards M theory in the full AdS × S, in the spirit of [59], which is in the IIB string context. To be more specific, we can add 1/R corrections (R → ∞ in the Penrose limit) to the Matrix theory proposed by [10] and investigate the dynamics of that model. Another direction is M theory in backgrounds with fewer supersymmetries, e.g., those preserving sixteen supersymmetries considered in [27]. Since supersymmetric nonrenormalization theorems seem to be crucial for the gauge/gravity duality, those backgrounds that are not maximally supersymmetric should teach us something valuable. Finally, in this thesis we have restricted our attention to one loop in the gauge theory and linearized supergravity. In the future we will try to push our computation to higher loops and nonlinear supergravity. 98 Appendix A Proof of Invariance of Lnonabelian under Sixteen Supersymmetries In this appendix we prove that the action Lnonabelian given in eqn. (3.48) is invariant under sixteen supersymmetries. The supersymmetry transformation given in eqn. (3.49) can be written as δ = δ + δΛ0 (A.1) where δ is the part containing , and δΛ0 is the compensating gauge transformation part containing Λ0 . So we have δLnonabelian = δ Lnonabelian + δΛ0 Lnonabelian (A.2) As we will see in the following, it makes the proof neater to separate the supersymmetry variation δLnonabelian into δ Lnonabelian and δΛ0 Lnonabelian at the very beginning. Carrying out the variation explicitly, we find δΛ0 Lnonabelian = −Ẏmi ({(Λ˙0 )m , xi } + gf mnp (Λ˙0 )n Ypi ) ig a −Ẋm (gf mnp (Λ˙0 )n Xpa ) − + f mnp (Λ˙0 )n ΨTp Ψm p (A.3) Deriving the result given in eqn. (A.3) involves some algebra. For example, to show that the g 3 Y 4 term which arises from δΛ0 (Fmk )2 vanishes, one has to deal with the 99 product of three structure constants and use the Jacobi identity for the structure constants. We do not present the details of the derivation here, because the form of (A.3) is what one would expect, in two aspects. Firstly, it only depends on the timederivative of Λ0 . The reason is, if Λ0 were time-independent, δΛ0 would just be a usual time-independent gauge transformation of the form eqn. (3.46) with Λm = − p1+ (Λ0 )m , and we have already shown that Lnonabelian is invariant under time-independent gauge transformations. Secondly, it only contains g 0 and g 1 powers, but not g 2 and g 3 terms (recall that Lnonabelian has order g 2 terms, and δΛ0 has order g terms, so potentially δΛ0 Lnonabelian could have up to g 3 order terms). The reason is, terms containing the a 2 ) , and δΛ0 (ΨTm Ψ̇m ), so are time-derivative of Λ0 come purely from δΛ0 (Ẏmi )2 , δΛ0 (Ẋm only of orders g 0 and g 1 . Order g 2 and g 3 terms only contain Λ0 not acted on by timederivative, and must vanish, because Lnonabelian is invariant under time-independent gauge transformations order by order in g. Now let us show that δLnonabelian vanishes order by order in g (of course, what we really mean is it vanishes up to total derivatives; in what follows we drop the total derivative terms). δLnonabelian = (δLnonabelian )(0) + (δLnonabelian )(1) + (δLnonabelian )(2) + (δLnonabelian )(3) (A.4) with the superscript denoting the power of g it contains. Also, as in the abelian case, we always move the time-derivative and Poisson bracket (spatial-derivative) to act on the boson through integration by parts. (δLnonabelian )(0) = 0 since this is just a sum of copies of the variation of the abelian Lagrangian. Let us first consider (δLnonabelian )(3) . (We will return to the g 2 and g 1 terms afterwards.) Eqn. (A.3) already tells us that (δΛ0 Lnonabelian )(3) = 0. Also, easily seen δ Lnonabelian does not contain any order g 3 terms. Hence (δLnonabelian )(3) = 0. Next, let us consider (δLnonabelian )(2) . Here eqn. (A.3) again gives (δΛ0 Lnonabelian )(2) = 0, so the contribution comes purely from (δ Lnonabelian )(2) . There are four types of terms in it, which we write schematically as X 3 Ψ, Y X 2 Ψ, Y 2 XΨ, and Y 3 Ψ, and 100 in what follows we’ll write these structures as subscripts to denote the relevant types of variation of the Lagrangian. First look at the X 3 Ψ terms. These come from the variation of two terms in the Lagrangian, one being the Yukawa term ΨXΨ, the other being the (F ab )2 term. It is given by (2) (δLnonabelian )X 3 Ψ = ig 2 mnp mrs a b a b c f f Xr Xs (γ γ )αβ β γαλ Xnc Ψλp 2p+ ig 2 a − + f mnp Xna Xpb f mrs (α γαβ Ψβr )Xsb 2p ig 2 b Ψβs ) − + f mnp Xna Xpb f mrs Xra (α γαβ 2p ig 2 = + f msp f mnr Xra Xsb Xnc T γ a γ b γ c Ψp 4p (A.5) where to get to the last line we have renamed indices at various places and also used the Jacobi identity of the gauge group structure constants. This vanishes because f msp f mnr Xra Xsb Xnc γ a γ b γ c (by using the Jacobi identity for the structure constants) = −(f mnp f mrs + f mrp f msn )Xra Xsb Xnc γ a γ b γ c (renaming n ↔ s, b ↔ c in the 1st term, and r ↔ s, a ↔ b in the 2nd term) = −f msp f mrn Xra Xnc Xsb γ a γ c γ b − f msp f mrn Xsb Xra Xnc γ b γ a γ c = −2f msp f mrn Xra Xnb Xsb γ a + f msp f mrn Xra Xnc Xsb γ a γ b γ c −2f msp f mrn Xsa Xra Xnc γ c + f msp f mrn Xsb Xra Xnc γ a γ b γ c (the two terms with only one gamma matrix cancel upon renaming n ↔ r in the 2nd of them) = −2f msp f mnr Xra Xsb Xnc γ a γ b γ c = 0 (A.6) (2) Thus we have shown (δLnonabelian )X 3 Ψ = 0. (2) (δLnonabelian )Y 3 Ψ = 0 by manipulations similar to the above. Now look at Y X 2 Ψ terms. These come from the variations of three terms in the 101 Lagrangian, one being the Ψξ term, one being the (F ia )2 term, and the other being the Yukawa term ΨXΨ. We have (2) (δLnonabelian )Y X 2 Ψ = ig 2 mnp mrs i a b T a b i f f Yr Xn Xp  γ γ γ Ψs 2p+ ig 2 − + f mnp f mrs Yni Xpa Xsa T γ i Ψr p ig 2 + + f mnp f mrs Yni Xpa Xsb T γ i γ a γ b Ψr p (renaming the indices to bring the fields into the same form − and then using the Jacobi identity) =0 (A.7) (2) (δLnonabelian )Y 2 XΨ = 0 by similar steps. So we have shown that (δLnonabelian )(2) = 0. Next consider (δLnonabelian )(1) . At this order we have contributions from both (δΛ0 Lnonabelian )(1) and (δ Lnonabelian )(1) . There are: terms containing µ, µX 2 Ψ, µY 2 Ψ, µY XΨ; terms containing a Poisson bracket, {x, X}XΨ, {x, Y }Y Ψ, {x, X}Y Ψ, {x, Y }XΨ; terms containing a time-derivative, ẊXΨ, Ẏ Y Ψ, ẊY Ψ, Ẏ XΨ; terms containing four spinors, Ψ3 . Compared with g 2 order, g 1 order is more straightforward because structure constants only appear once in each term, hence it doesn’t involve the Jacobi identity for the structure constants. 102 Let’s first consider terms containing µ. The most complicated term is (1) (δLnonabelian )µY 2 Ψ = igµ [ = = 1 k ijk mnp T i Ym  f ( γ Ψn )Ypj 3 1 + kij f mnp Yni Ypj (T γ k Ψm ) 6 1 kij mnp i j k 123 123 +  f Yn Yp (γ γ )αβ γβλ λ Ψαm 24 1 i − Ymk (γ k γ 123 )αβ β γαλ f mnp Yni Ψλp 3 1 123 − kij f mnp Yni Ypj (γ k γ 123 )αβ β γαλ Ψλm ] 8 (renaming indices to bring the fields into the same form and also using gamma matrix identities given before)   1 1 1 1 1 ijk mnp i j T k + − − − igµ f Yn Yp ( γ Ψm ) 3 6 24 3 8 0 (A.8) The other terms containing µ all vanish by similar manipulations and we omit the details here. The terms containing the Poisson bracket and time-derivative also all vanish by similar steps and we don’t bother to write the details down, either. As always, it is worthwhile to write out the proof for the vanishing of the four spinor term in detail. So let us do this now. We have (1) (δLnonabelian )Ψ3 =  g mnp  T A T A T T f ( γ Ψ )(Ψ γ Ψ ) − ( Ψ )(Ψ Ψ ) n p n p m m p+ (A.9) We just have to work out the Fierz transformation for SO(9) spinors. Any 16 × 16 matrices can be expanded using the complete set of 256 matrices {γ M } = {1, γ A , iγ AB , iγ ABC , γ ABCD } (A.10) which satisfy tr(γ M γ N ) = 16 δ M N . So we can expand γ M Ψn ΨTm γ N = CQ γ Q (A.11) 103 with CQ = − 1 tr(γ N γ Q γ M γ W )(ΨTm γ W Ψn ) 256 (A.12) Using the above result, we find (T γ A Ψn )(ΨTm γ A Ψp ) = 9 T 7 T A (Ψm Ψn )(T Ψp ) + (Ψm γ Ψn )(T γ A Ψp ) 16 16 5 T AB 3 T ABC T AB + (Ψm γ Ψn )( γ Ψp ) − (Ψm γ Ψn )(T γ ABC Ψp ) 16 16 1 T ABCD T ABCD − (Ψ γ Ψn )( γ Ψp ) (A.13) 16 m − Upon contraction with f mnp the γ AB and γ ABC terms vanish because these two types of matrices are antisymmetric. Hence we find 9 mnp T 7 mnp T A f (Ψm Ψn )(T Ψp ) + f (Ψm γ Ψn )(T γ A Ψp ) 16 16 1 mnp T ABCD T ABCD − f (Ψm γ Ψn )( γ Ψp ) (A.14) 16 f mnp (T γ A Ψn )(ΨTm γ A Ψp ) = − Similarly, we find f mnp (T Ψn )(ΨTm Ψp ) = 1 mnp T A 1 mnp T f (Ψm Ψn )(T Ψp ) − f (Ψm γ Ψn )(T γ A Ψp ) 16 16 1 mnp T ABCD T ABCD f (Ψm γ Ψn )( γ Ψp ) (A.15) − 16 − Hence we have (1) (δLnonabelian )Ψ3 = = = =  g  mnp T A f ( γ Ψn )(ΨTm γ A Ψp ) − f mnp (T Ψn )(ΨTm Ψp ) + p (using the Fierz identities derived above)   1 mnp T 1 mnp T A g T T A − f (Ψm Ψn )( Ψp ) + f (Ψm γ Ψn )( γ Ψp ) p+ 2 2 (renaming indices n ↔ p)  1 g  mnp T A f ( γ Ψn )(ΨTm γ A Ψp ) − f mnp (T Ψn )(ΨTm Ψp ) − + 2 p 1 (1) − (δLnonabelian )Ψ3 (A.16) 2 104 which gives (1) (δLnonabelian )Ψ3 = 0 So we have proved (δLnonabelian )(1) = 0. Thus δLnonabelian = 0. (A.17) 105 Appendix B The One-Loop Effective Potential In this appendix we will prove the sum over mass formula in eqn. (3.92). The one-loop effective action Γ of a theory expanded upon a background B in the background field method is given by 1 Γ = [Trboson ln(−∂τ2 +Wb (τ ))−Trfermion ln(−∂τ2 +Wf (τ ))−Trghost ln(−∂τ2 +Wgh (τ ))] 2 (B.1) Here W = M 2 is the mass matrix squared for the fluctuating fields, and the trace Tr is over both the functional space and the field component indices (which are, besides the U (2) indices, the space-time indices 0, 1, .., 9 for the bosons, and the 16 Dirac spinor indices for the fermions). Take the trace of the boson, for example 1 Trboson ln(−∂τ2 + W (τ )) 2 Z ∞ 1 ds = − Tr exp[−s(−∂τ2 + W (τ ))] 2 s 0 Γboson = (B.2) The trace over functional space can be computed by sandwiching the operator between the “plane-wave” basis wave-functions |ωi = √12π e−iωτ and the conjugate 106 wave-functions hω| = √12π eiωτ 1 Γboson = − tr 2 Z Z dτ dω 2π Z ∞ 0 ds iωτ e exp[−s(−∂τ2 + W (τ ))]e−iωτ s (B.3) The trace tr is now over only the field component indices. If we define Veff (τ ) by Z Γ=− dτ Veff (τ ) (B.4) Then, the bosonic part of Veff becomes 1 Veff (boson) = tr 2 Z dω 2π Z ∞ 0 ds iωτ e exp[−s(−∂τ2 + W (τ ))]e−iωτ s (B.5) The operator in the middle can be rewritten as 2 exp[−s(−∂τ2 + W (τ ))] = Xe−sW (τ ) e+s∂τ (B.6) Where X is defined as 2 X ≡ exp[−s(−∂τ2 + W (τ ))]e−s∂τ e+sW (τ ) = 1 + commutator terms (B.7) The commutator terms give corrections to supergravity, so for the purpose of this paper, which is to see whether the Matrix theory can reproduce supergravity results, we can ignore them. This claim will be proven shortly, after the result from approximating X = 1 is examined. In this approximation, we have Z Z dω ∞ ds 1 Veff (boson) = tr exp[−s(ω 2 + W (τ ))] 2 2π 0 s 1 ⇒ Veff = − tr M 2 (B.8) Note that M , the square root of W , can be defined through its expansion in powers 107 of 1/r. Putting everything together, and minding the minus signs for the fermions and the ghosts, we get the sum over mass formula in eqn. (3.92). Now, we return to the claim made above, that the commutator terms in X will not contribute to terms in the supergravity limit. To show this, we first write X in a most general form X= X n m K[m n ](W (τ ))s ∂τ (B.9) n,m Here K[m n ](W (τ )) is a general function of W (τ ) and its τ -derivatives, and is defined by eqn. (B.9). Looking back at the definition of X in eqn. (B.7), we see that n counts the number of terms involved in forming the commutator, and m is the number of derivatives not acting on W . For example, when n = 0, it implies m = 0, and K[00 ] = 1, corresponding to the approximation we made above. All the other values of n correspond to commutator terms in X, and in particular, K = 0 when n = 1, because a commutator takes at least two terms. Putting X in terms of K into Veff , we will encounter the following factor inside the integrand: iωτ e m −iωτ ∂ e = m X [m/2] l m−l (m = l )(−iω) ∂ l=0 X 2 l m−2l (m 2l )(−ω ) ∂ (B.10) l=0 [m/2] is the biggest integer no larger than m/2. In the last line, we made use of the fact that ω will be integrated from −∞ to +∞ so that any odd functions in the integrand will give zero. As a result, only terms with even powers of ω are kept. Therefore, the effective potential becomes 108 1 X tr Veff (boson) = 4π n,m Z ∞ Z dω 0 [m/2] 1 XX = tr 4π n,m l=0 ds m n iωτ m −iωτ −sW −sω2 K[n ]s e ∂ e e e s Z ∞ Z ds m 2 n m−2l −sW (2l )(−ω 2 )l e−sω K[m e n ]s ∂ s 0 r Z ∞ [m/2] 1 XX ds m ∂ l π n m m−2l −sW e = ( )( )s K[n ]∂ tr 4π n,m l=0 s 2l ∂sl s 0 = tr X [m/2] X 1 n,m l=0 = tr 4 X [m/2] X 1 4 n,m l=0 (m 2l ) (m 2l ) dω 1 K[m ]∂ m−2l Γ(1/2 − l) n Z ∞ 0 ds n−l−1/2 −sW s e s Γ(n − l − 1/2) m m−2l 1 K[n ]∂ n−l−1/2 Γ(1/2 − l) W (B.11) Thus, the effective potential can be recast as Veff = tr X [m/2] X 1 n,m l=0 4 (m 2l ) Γ(n − l − 1/2) 2(n−l)−1 m 1 α K[n ](W )∂ m−2l 2 n−l−1/2 (B.12) Γ(1/2 − l) (α W ) As before, α = 1/(M 3 R). The reason these factors of α are inserted will be clear shortly. In a comparison of one-loop Matrix theory with supergravity, the relevant terms on the supergravity side are proportional to κ211 , which is of order α3 on the Matrix theory side. This means that any higher powers of α are irrelevant for such a comparison as they represent Matrix theory corrections to supergravity, and finding them is not the purpose of this paper. In other words, to examine whether the Matrix theory can reproduce supergravity in the appropriate limit, only terms up to order α3 need to be kept. It makes sense, therefore, to examine each factor in Veff and count the powers of α it contains. We begin with the mass matrix squared. By inspection of the explicit expressions given in Subsection 3.3.5, one sees that W can always be schematically 109 written as 1 2 1 r + N α2 α 2 2 ⇒ α W ∼ r + αN W ∼ ⇒ (α2 W )k ∼ 1 + αN + (αN )2 + · · · (B.13) where powers of r and numerical coefficients have been suppressed in the last expression. For example, using eqn. (3.99), we have αNf = α 9 X I=1 vI γI + 3 X iµxi i=1 4 ! {γi , γ123 } + α2 µ2 /42 (B.14) Similarly, αNb can be constructed using the rules given in Subsection 3.3.5 (we omit its explicit expression here), while αNgh = 0. From the explicit expressions for N , it can be shown easily that tr αN = 0, tr [(αN )2 ] = 0 + O[α4 ], and tr [(α∂N )2 ] = 0 + O[α4 ]. These facts are related to the large number of supersymmetries of our system and will be of use shortly. The last line in eqn. (B.13) is a symbolic statement that for any k, whether positive or negative, (α2 W )k will only give non-negative powers of α. Another important point to note is that every α arising from (α2 W )k is accompanied P m m p by a factor of N . Now look at K[m n ]: Let K[n ] = p,q K[n q ] where p is the number τ -derivatives acting on W inside K, and q is the number of W inside K. By definition, we have p+m +q =n 2 For n = 0 ⇒ K = 1; For n = 1 ⇒ K = 0; For n ≥ 2 ⇒ K consists of commutators. In this case, we have   q<n  p+m < n 2 For fixed q and n ≥ 2, m and p have the following extremal values: (B.15) (B.16) 110 mmin = 0 ⇒ pmax = 2(n − q) pmin = 1 ⇒ mmax = 2(n − q) − 1 The reason pmin = 1 is that ∂τ2 must act at least once on W to give non-vanishing commutators like [W, [W, [W, Ẇ ]]] in K. Consider m−2l α2(n−l)−1 K[m n ](W )∂ X 1 1 p m−2l = α2(n−l)−1 K[m n q ](W )∂ 2 n−l−1/2 2 (α W ) (α W )n−l−1/2 p,q X 1 p 2 m−2l = α2(n−q)−1−2l K[m n q ](α W )∂ 2 (α W )n−l−1/2 p,q X 1 p 2 m−2l = αa K[m (B.17) n q ](α W )∂ 2 n−l−1/2 (α W ) p,q where a = 2(n − q) − 1 − 2l. Noting l ≤ [m/2] ≤ [mmax /2] = [n − q − 1/2] = n − q − 1, we must have lmax = n − q − 1 ⇒a≥1 (B.18) From this derivation of the lower bound of a, we see that the equality holds only when m = mmax = 2(n − q) − 1 and l = lmax = n − q − 1. Then, eqn. (B.15) gives m − 2l = 1 p = pmin = 1 (B.19) A comparison of supergravity with one-loop Matrix theory means keeping terms only up to α3 ∼ κ211 . Therefore, we need only consider the range of a to be 1≤a≤3 (B.20) For a = 3: There can be no factors of αN from α2 W because they increase the powers of α beyond 3, hence taking us beyond the limit of supergravity. Without any factors 111 of αN , the effective potential is simply Veff = f (r)tr 1 ⇒ Veff ∼ tr 1 = 0 (B.21) Here, tr is again the trace over the boson minus the trace over the fermions and the ghosts. For a = 2: There can be at most one αN , either from K(α2 W ) or ∂ m−2l (α2 W )−n+l+1/2 But for only one N , we have Veff ∼ tr ∂ k N = ∂ k tr N = 0 (B.22) For a = 1: Now it is possible to have (αN )2 coming from one of the following three cases: (i) Both (αN )2 come from ∂ m−2l (α2 W )−n+l+1/2 Veff ∼ αtr ∂ m−2l (αN )2 ∼ α∂ m−2l tr (αN )2 ∼ αO[α4 ] ∼ O[α5 ] (B.23) (ii) Both (αN )2 come from K(α2 W ): Since we showed that p = 1 when a = 1, there is only one τ -derivative acting on W in K, we must have either: (a) Veff ∼ tr N 2 or (b) Veff ∼ tr (N ∂τ N ) ∼ 21 ∂τ tr (N 2 ). In either case, Veff = 0 + O[α5 ]. (iii) One (αN ) comes from K(α2 W ) and one from ∂ m−2l (α2 W )−n+l+1/2 : We already showed that m − 2l = 1 when a = 1, so we have Veff ∼ K∂(α2 W )−n+l+1/2 This implies either: (a) Veff ∼ tr (N ∂N ) or (b) Veff ∼ tr (∂N ∂N ) (B.24) 112 (a) is identical to case (iib) above. (b) is of order α5 using the fact mentioned before. This exhausts all cases contributing to terms up to order α3 ∼ κ211 in Veff . In particular, we have shown that none of the commutator terms in X, corresponding to K[m n ] with n ≥ 2, contributes to terms relevant to supergravity. This completes the proof of the claim made following eqn. (B.7). 113 Appendix C An Alternative Derivation of Veff for the Point Particle Probe In this appendix, we give an alternative derivation of the effective potential Veff for a point particle probe (a probe graviton). The same quantity is derived in Subsection 4.1.1 under the name of light-cone Lagrangian, with a more rigorous treatment of constraints. The Lagrangian of the probe graviton moving in the perturbed PP-wave with metric Gµν = gµν + hµν , with gµν being the unperturbed PP-wave metric, and hµν being the metric perturbation due to the source graviton, is given by1 L = −m p −Gµν ẋµ ẋν = −m q −(gµν + hµν )ẋµ ẋν (C.1) We make a Legendre transformation L0 = L − Pp+ ẋ− (C.2) δL 1 + h−ν ẋν p = m δ ẋ− −Gµν ẋµ ẋν (C.3) where upon setting x+ = 1 Pp+ = 1 This approach is the one used in [17]. 114 When we let m → 0, this gives Gµν ẋµ ẋν = (gµν + hµν )ẋµ ẋν = 2ẋ− + g++ + v 2 + hµν ẋµ ẋν = 0 (C.4) This is a quadratic equation for ẋ− , which we will solve for ẋ− , keeping only terms up to linear order in hµν . We find (    1 2 1 − ẋ = − v + g++ + h++ + g++ g++ h−− − h+− 2 4  X X 2 A A B + [2h+A − h−A (v + g++ )]v + hAB v v A A,B  1 1 1 + h−− v 4 − v 2 h+− − g++ h−− 8 2 2 ) (C.5) Taking the limit m → 0, the L0 in eqn. (C.2) is simply −Pp+ ẋ− , which is the effective potential Veff that we need. It contains the interaction between the probe and the source up to terms linear in hµν . An alternative way of writing such an interaction is δL hµν = T µν hµν δGµν (C.6) This structure was used by [6] to identify the effective potentials on both sides. 115 Appendix D Constrained Hamiltonian Dynamics in Field Theory For simplicity of notation let us consider field theory in (1 + 1) dimensions. The generalization to field theory in higher dimensions is straightforward. Denote the t η,∂x η) , and the Hamiltonian field as η(t, x), its momentum density π(t, x) ≡ ∂L(η,∂ ∂(∂t η) density H(η, ∂x η, π) ≡ π∂t η − L. The equations of motion are ∂t η = ∂H ∂π ∂t π = − ∂H ∂ − ∂η ∂x  ∂H ∂(∂x η)  (D.1) The Poisson bracket in field theory is defined as follows. In field theory, we can R have density f (t, x) = f (η, ∂x η, π, ∂x π), or integrated density F (t) = dx f (t, x) = R dx f (η, ∂x η, π, ∂x π). Let us denote both densities and integrated densities collectively as ξ. For any two such objects ξ1 and ξ2 , we define their Poisson bracket to be  Z [ξ1 , ξ2 ]PB ≡ dy δξ1 δξ2 δξ2 δξ1 − δη(t, y) δπ(t, y) δη(t, y) δπ(t, y)  (D.2) δ(...) where δ(...) is the functional derivative. δH In terms of the Poisson bracket one can see that ∂t η(t, x) = δπ(t,x) = [η(t, x), H]PB , R δH and ∂t π(t, x) = − δη(t,x) = [π(t, x), H]PB with H ≡ dx H(t, x) being the Hamilto- nian. Using this, one also see that for any function f (t, x) = f (η, ∂x η, π, ∂x π), we 116 have ∂t f (t, x) = [f (t, x), H]PB (D.3) So far we have been talking about an unconstrained system. Now let us introduce constraints. Suppose the system has M primary constraints φm (η, ∂x η, π) = 0, m = 1, 2, ..., M (D.4) Define the total Hamiltonian density as HT ≡ H + um φm , and the total Hamiltonian as Z HT ≡ Z dx HT = H + dx um φm (D.5) with um (t, x) being the Lagrange multiplier fields. Then the time evolution is given by, for any function f (t, x) = f (η, ∂x η, π, ∂x π) ∂t f (t, x) = [f (t, x), HT ]PB (D.6) This completes our brief review of constrained Hamiltonian dynamics in field theory. We didn’t introduce the extended Hamiltonian HE which is different from the total Hamiltonian HT only when there exist secondary constraints, since in the application to the membrane one can show that there is no secondary constraint (see Subsection 4.1.2). 117 Appendix E The Green’s Function for the x− ˜ Fourier Transformed Laplacian  In Section 4.2, upon diagonalizing the supergravity field equations, the components of the metric and three-form perturbations satisfy Laplace equations. (See, for example, eqn. (4.66).) So in this appendix we discuss the Green’s function for the Laplace equation Fourier transformed along the x− direction. Let us use  to denote the Laplacian g µν ∂µ ∂ν , and its x− Fourier transform P P 2 2 2 i 2 ˜ 2 ˜ A where  ˜ i ≡ ∂ 2i 2 − 1 k− ˜ ≡ g++ k− µ (x ) , a ≡ (∂x∂ a )2 − + 9A=1 ∂A2 = 9A=1   (∂x ) 9 1 2 2 a 2 k µ (x ) , 36 − ˜ A is just a harmonic oscillator’s Hamiltonian along the xA direction. Of course, − Let us use φnA and EnA to denote its normalized eigenfunctions and eigenvalues. P Then Φ{n1 ,...,n9 } ≡ φn1 (x1 )...φn9 (x9 ) and E{n1 ,...,n9 } ≡ 9A=1 EnA are the normalized ˜ eigenfunctions and eigenvalues of −. Use ~x to denote the nine vector (x1 , ..., x9 ). Define K(~x, ~y ) ≡ X {n1 ,...,n9 } −1 E{n1 ,...,n9 } Φ{n1 ,...,n9 } (~x)Φ∗{n1 ,...,n9 } (~y ) (E.1) ˜  ˜ ~x K(~x, ~y ) = δ(~x − ~y ) which is the Green’s function of : The above Green’s function doesn’t have closed form. Thus although in principle h̄µν and aµνρ can all be obtained by integrating over the source using this Green’s function, in practice one may have to take some limit to get an answer, e.g., the 118 near-membrane limit in Appendix F. 119 Appendix F h−− for a Spherical Source Membrane in the Near-Membrane Limit In this appendix, we discuss the solution h−− to the field equation (4.66), in the near-membrane limit. Also see the end of Subsection 3.2.2.2 and the end of Section 4.2 for discussions on the near-membrane limit in the investigation of two membrane interactions. Note h̄−− = h−− . In the following we use the same symbol for the metric perturbation and its Fourier transform. Its meaning should be clear from context. For a source membrane extending a sphere of radius r0 in the (X 1 , X 2 , X 3 ) directions, with its center sitting at X − = 0, X 4 = ... = X 9 = 0, and X + = σ 0 , the (−−) component of its stress tensor is − 4 9 [T−− ]S = T δ(x )δ(r − r0 )δ(x )...δ(x ) where r ≡ √  −1 2  µr0 −1 3 (F.1) xi xi . Recall h−− satisfies eqn. (4.66) 1 − h−− + Q−− = κ211 [T−− ]S 2 (F.2) Let us make the gauge choice q− = 0 (thus Q−− = ∂− q− = 0) and take its Fourier 120 transform along the x− direction 1˜ 2 4 9 − h −− = κ11 T δ(r − r0 )δ(x )...δ(x ) 2  −1 2  µr0 −1 3 (F.3) Integrating the Green’s function K over the sphere is not such a great idea. Instead, we consider the near-membrane limit (so that the source membrane looks √ almost flat), where w ≡ r − r0 << r0 , also xa xa << r0 . 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