Projective Dirac Operators, Twisted K-Theory, and Local Index Formula Citation Zhang, Dapeng (2011) Projective Dirac Operators, Twisted K-Theory, and Local Index Formula. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/B2Z4-P206. https://resolver.caltech.edu/CaltechTHESIS:05272011-153933399 Abstract We construct a canonical noncommutative spectral triple for every oriented closed Riemannian manifold, which represents the fundamental class in the twisted K-homology of the manifold. This so-called "projective spectral triple" is Morita equivalent to the well-known commutative spin spectral triple provided that the manifold is spin-c. We give an explicit local formula for the twisted Chern character for K-theories twisted with torsion classes, and with this formula we show that the twisted Chern character of the projective spectral triple is identical to the Poincare dual of the A-hat genus of the manifold. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Twisted K-theory; spectral triple; Chern character Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Marcolli, Matilde Thesis Committee: Marcolli, Matilde (chair) Agarwala, Susama Calegari, Danny C. Kitaev, Alexei Defense Date: 25 May 2011 Record Number: CaltechTHESIS:05272011-153933399 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:05272011-153933399 DOI: 10.7907/B2Z4-P206 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 6466 Collection: CaltechTHESIS Deposited By: Dapeng Zhang Deposited On: 31 May 2011 21:36 Last Modified: 21 May 2025 22:49 Thesis Files Preview PDF (Complete Thesis) - Final Version See Usage Policy. 442kB Repository Staff Only: item control page

Projective Dirac Operators, Twisted K-Theory, and Local Index Formula Thesis by Dapeng ZHANG In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2011 (Defended May 25, 2011) ii c 2011 Dapeng ZHANG All Rights Reserved iii Acknowledgements I am very grateful to Bai-Ling Wang. He foresaw the possibility that the projective spin Dirac operator defined by [19] in formal sense can be realized by a certain spectral triple, and introduced his interesting research project to me in 2008. The spectral triple in his mind turned out to be the projective spectral triple constructed in this paper. Without his insight, I wouldn’t have been writing this thesis. I also wish to thank my advisor, Matilde Marcolli, for her many years of encouragement, support, and many helpful suggestions on both this research and other aspects. iv Abstract We construct a canonical noncommutative spectral triple for every oriented closed Riemannian manifold, which represents the fundamental class in the twisted K-homology of the manifold. This socalled “projective spectral triple” is Morita equivalent to the well-known commutative spin spectral triple provided that the manifold is spin-c. We give an explicit local formula for the twisted Chern character for K-theories twisted with torsion classes, and with this formula we show that the twisted Chern character of the projective spectral triple is identical to the Poincaré dual of the A-hat genus of the manifold. Keywords. Twisted K-theory, spectral triple, Chern character. v Contents Acknowledgements iii Abstract iv Introduction 1 1 Azumaya Bundles, Twisted K-theory, and Twisted Cohomology 6 2 Generalized Connes-Hochschild-Kostant-Rosenberg Theorem 11 3 Spectral Analysis of Spectral Triples 18 4 Morita Equivalence of Spectral Triples 23 5 Gluing Local Spin Structures via Morita Equivalence 30 6 Projective Spectral Triple as Fundamental Class in K0 (M, W3 ) 37 7 Local Index Formula for Projective Spectral Triples 39 Bibliography 43 1 Introduction The notion of spectral triple in Connes’ noncommutative geometry arises from extracting essential data from the K-homology part of index theory in differential geometry. The following are basic examples of commutative spectral triples: i) The spin spectral triple for a spinc manifold M with a spinor bundle S: / ω), ς1 = (C ∞ (M ), Γ(S), D, where ω is the grading operator on S. The identity between the analytic and topological / is the Atiyah-Singer index formula for spinc manifold. indexes of D ii) The spectral triple for the signature for a Riemannian manifold M : ς2 = (C ∞ (M ), Ω(M ), d + d∗ , ∗(−1) deg(deg −1) − dim4 M 2 ). The index formula corresponding to this spectral triple is the Hirzebruch signature formula. iii) The spectral triple for Euler characteristic for a Riemannian manifold M : ς3 = (C ∞ (M ), Ω(M ), d + d∗ , (−1)deg ). The local index formula corresponding to this spectral triple is the Gauss-Bonnet-Chern theorem. 2 In fact, every special case of Atiyah-Singer Index theorem corresponds to an instance of commutative spectral triple (with additional structures when necessary). These spectral triples, like ς1 , ς2 , ς3 , have many nice properties such as “the five conditions” in Connes [8], and conversely, it is proved that [8] any commutative spectral triple (A, H, D, γ) satisfying those five conditions is equivalent to a spectral triple consisting of the algebra of smooth functions on a Riemannian manifold M , the module of sections of a Clifford bundle over M and a Dirac type operator on it. Furthermore, if (A, H, D, γ) satisfies an additional important property – the Poincaré duality in K-theory – which means (A, H, D, γ) represents the fundamental class (i.e., a K-orientation) in K 0 (A), then it is equivalent to a spin spectral triple ς1 for some spinc manifold. The spectral triple for Hirzebruch signature ς2 (as well as ς3 ) does not have the property of Poincaré duality; however, we show in this paper (Corollary 5.4, Theorem 6.1) that for every closed oriented Riemannian manifold there is a canonical noncommutative spectral triple having the property of Poincaré duality in K 0 (M, W3 (M )), the twisted K-theory of M with local coefficient W3 (M ) - the third integral Stiefel-Whitney class. This canonical spectral triple is called the projective spectral triple on M , and its center is unitarily equivalent to ς2 . The projective spectral triple is Morita equivalent to the spin spectral triple provided the underlying manifold is spinc . On the other hand, in the paper of Mathai-MelroseSinger [19], a so-called projective spin Dirac operator was defined for every Riemannian manifold; however, this operator is in a formal sense. It turns out that the projective spectral triple, in which the Dirac operator is really an operator acting on a Hilbert space, just plays the role of the projective spin Dirac operator. A spectral triple that gives rise to Poincaré duality in KK-theory first appeared in Kasparov [17]. In Kasparov’s spectral triple (although there was no such terminology at that time), the algebra is noncommutative and Z2 -graded, but in many cases it would be much easier if the algebra is ungraded, especially when considering its Dixmier-Douady class or passing it to cyclic cohomology class via Connes-Chern character. The projective spectral triple constructed in this paper (Corollary 5.4) (AW3 , HW3 , DW3 , γW3 ) 3 has a noncommutative but ungraded algebra, and it is in fact Morita equivalent to that of Kasparov’s. To construct such a spectral triple, we first introduce in chapter 4 the notion of Morita equivalence between spectral triples, then find in chapter 5 that the local spin spectral triples on small open subsets of the manifold can be glued together, via Morita equivalence, to form a globally defined spectral triple. The noncommutative algebras underlying projective spectral triples are examples of Azumaya algebras. In chapter 1 we review some basic theory on Azumaya algebras, such as the fact that Morita equivalent classes of Azumaya algebras are classified by their Dixmier-Douady classes, and that the K-theory of an Azumaya algebra A coincides with the twisted K-theory of the manifold with the Dixmier-Douady class of A. Mathai-Stevenson [20] showed that the K-theory (tensoring with C) of an Azumaya algebra A is isomorphic to the periodic cyclic homology group of A via Connes-Chern character, and that the latter is isomorphic to the twisted de-Rham cohomology of the manifold with the Dixmier-Douady class of A via a generalized Connes-Hochschild-Kostant-Rosenberg (CHKR) map. / HP0 (A) ch ∼ = K 0 (M, δ(A)) ⊗ C ∼ = chδ(A) ∼ = ) w Chkr ev HdR (M, δ(A)) In chapter 2, we find an alternative CHKR map (Theorem 2.5) for the special case that the DixmierDouady class of A is torsion. For an algebra A, a finite projective A-module E as a K-cocycle in the K-theory of A has a Connes-Chern character ch([E]) as a cyclic homology class, whereas a spectral triple on A as a K-cycle in the K-homology group of A has also a Connes-Chern character as a cyclic cohomology class, and the index pairing of a K-cocyle and a K-cycle is identical to the index pairing of their Connes-Chern characters ([6, 7]). The main purpose of this paper is to compute the Connes-Chern character of the projective spectral triple and identify it with the Poincaré dual of the A-hat genus of the manifold. 4 In chapter 7, with the help of the alternative CHKR map ρ and applying the Poincaré duality, we obtain our main result, a local formula for the Connes-Chern character of the projective spectral triple, ch(AW3 , HW3 , DW3 , γW3 )(·) = 1 n (2m)! 2 m X Z Â(M ) ◦ ρ2m (·). M 5 Some conventions and notations Throughout this paper, we assume the following: Unless otherwise stated explicitly, all vector spaces, algebras, differential forms, and vector bundles except cotangent bundles are considered over the field C of complex numbers. The notation Γ(X, E) or Γ(E) for a fibre bundle E over X always stands for the space of smooth sections of E. For each function f : N → C, we define the operator f (deg) : Ω(X) → Ω(X) acting on the space of differential forms on X to be the linear map given by f (deg) ω = f (k) ω, ∀ω ∈ Ωk (X). 6 Chapter 1 Azumaya Bundles, Twisted K-theory, and Twisted Cohomology Suppose X is a closed oriented manifold. We use the notations Mn =     M (C), n = 1, 2, . . .    K(H), n=∞ n , Un =     U(n), n = 1, 2, . . .    U(H), n=∞ , where n could either be a positive integer or infinity, H is an infinite dimensional separable Hilbert space, K(H) is the C ∗ -algebra of compact operators on H, and U(H) is the topological group of unitary operators with the operator norm topology. Kuiper’s theorem states that U(H) is contractible. Let PUn = Un /U(1) be the projective unitary groups. In particular PU(H) = PU∞ is endowed with the topology induced from the norm topology of U(H). Let Aut(Mn ) be the group of automorphisms the C ∗ -algebra Mn . Fact 1. For every element g ∈ Aut(Mn ), there exists g̃ ∈ Un , such that g = Adg̃. For every u ∈ Un , Ad u = 1 if and only if u is scalar. In other words, as groups PUn ∼ = Aut(Mn ). Fact 2. If n is finite, Un /U(1) ∼ = SU(n)/{z ∈ C | z n = 1}. Definition 1.1. An Azumaya bundle over X of rank n (possibly n = ∞) is a vector bundle over X with fibre Mn and structure group PUn . Every Azumaya bundle of rank n is associated with a principal PUn -bundle and vice versa. 7 Definition 1.2. The space A = Γ0 (A) of continuous sections of an Azumaya bundle A over X forms a C ∗ -algebra called an Azumaya algebra over X. The following are examples of Azumaya algebras over X: i) the algebra of complex valued continuous functions C 0 (X); ii) C 0 (X) ⊗ Mn ; iii) if E is a finite rank vector bundle over X, the algebra of continuous sections of End(E), Γ0 (End(E)); iv) if X is an even dimensional Riemannian manifold, the algebra of continuous sections of the Clifford bundle Cl(T ∗ X), Γ0 (Cl(T ∗ X)); v) if E is a real vector bundle over X of even rank with a fiberwise inner product, the algebra of continuous sections of the Clifford bundle Cl(E), Γ0 (Cl(E)). Note that examples i), ii), iii) are Morita equivalent (in the category of C∗ -algebras, i.e., strongly Morita equivalent) to C 0 (X), while example iv) or v) is Morita equivalent to C 0 (X) if and only if X or E is spinc respectively. Fact 3. The center of a finite Azumaya algebra over X is C 0 (X). Fact 4. An Azumaya algebra A over X is locally Morita equivalent to C 0 (X). The obstruction to an Azumaya algebra being (globally) Morita equivalent to its “center” is characterized by its Dixmier-Douady class: Definition 1.3. Every Azumaya bundle π : A → X of rank n is associated with a cohomology class δ(A) in H 3 (X, Z), called the Dixmier-Douady class of A, constructed as follows: Let {Ui }i∈I be a good covering of X, and write Ui1 ···in for the intersection of Ui1 , Ui2 , · · · , Uin . Suppose ≃ ψi : Ui × Mn − → π −1 (Ui ), ∀i ∈ I, 8 provide a local trivialization of A. Then ψi−1 ψj : Uij × Mn → Uij × Mn give rise to the transition −1 functions gij ∈ C 0 (Uij , Aut(Mn )). Pick g̃ij ∈ C 0 (Uij , Un ) such that Adg̃ij = gij and g̃ij = g̃ji . Thus Ad(g̃ij g̃jk g̃ki ) = gij gjk gki = 1, which implies µijk := g̃ij g̃jk g̃ki ∈ C 0 (Uijk , U(1)). Therefore µ is a Čech 2-cocycle with coefficient sheaf U (1) : U 7→ C 0 (U, U(1)), since (∂µ)ijkl = −1 µjkl µ−1 ikl µijl µijk = 1. The Čech 2-cocycle µ is also called the bundle gerbe structure of A. The short exact sequence of sheaves exp 2πi· 0 −→ Z −→ R −−−−−→ U (1) −→ 0, where R is the sheaf U 7→ C 0 (U, R), induces an isomorphism of Čech cohomology groups ∼ = ∂ : Ȟ 2 (X, U (1)) − → Ȟ 3 (X, Z). Define the Dixmier-Douady class by δ(A) := ∂[µ]. More explicitly, pick νijk ∈ C 0 (Uijk , R) such that exp 2πiνijk = µijk . Then exp 2πi(∂ν)ijkl = (∂µ)ijkl = 1, which implies (∂ν)ijkl = νjkl − νikl + νijl − νijk ∈ Z are locally constant integers on Uijkl . In fact, δ(A) = [∂ν] ∈ Ȟ 3 (X, Z). Definition 1.4. Suppose that A is an Azumaya bundle, that A is the Azumaya algebra corresponding to A, and that P is the principal PU-bundle associated to A. We say δ(A) = δ(P ) = δ(A) are the Dixmier-Douady class of A and P respectively. As a consequence of Kuiper’s theorem, Proposition 1.5. For every cohomology class δ in H 3 (X, Z), there is a unique (up to isomorphism) infinite rank Azumaya bundle (or algebra) with Dixmier-Douady class δ. Proposition 1.6. Let A be an Azumaya bundle. If δ(A) = 0, then one can choose g̃ij so that g̃ij 9 are the transition functions of a certain Hermitian bundle E over X, and A is isomorphic to K(E), the bundle over X with fibres K(Ex ). Corollary 1.7. An Azumaya algebra A over X is Morita equivalent to C 0 (X) if and only if δ(A) = 0. Corollary 1.8. Two Azumaya algebras A1 , A2 over X are Morita equivalent if and only if δ(A1 ) = δ(A2 ). Namely, Morita equivalence classes of Azumaya algebras are parameterized by H 3 (X, Z). As a consequence of Fact 2, Proposition 1.9. If A is an Azumaya bundle of finite rank n, then nδ(A) = 0. For example, suppose X is a 2m-dimensional smooth manifold. The Clifford bundle Cl(T ∗ X) is an Azumaya bundle of rank 2m . Its Dixmier-Douady class δ(Cl(T ∗ X)) = W3 (X) is the third integral Stiefel-Whitney class of X, and 2W3 (X) = 0. Proposition 1.10. If A1 , A2 are two Azumaya bundles over X, then δ(A1 ⊗ A2 ) = δ(A1 ) + δ(A2 ). Proposition 1.11. If A is an Azumaya algebra, then its opposite algebra Aop is also an Azumaya algebra and δ(Aop ) = −δ(A). Let δ be a cohomology class in H 3 (X, Z). Recall that (Rosenberg [23], Atiyah-Segal [1]) the twisted K-theory K 0 (X, δ) can be defined by K 0 (X, δ) = [P → Fred(H)]PU(H) , the abelian group of homotopy classes of maps P → Fred(H) that are equivariant under the natural action of PU(H), where P is a principal PU(H)-bundle over X with Dixmier-Douady class δ(P ) = δ; and where Fred(H) is the space of Fredholm operators on H. Twisted K-theory can also be defined 10 with K-theory of a C∗ -algebra: K 0 (X, δ) = K0 (A), where A is an (infinite rank) Azumaya algebra over X with Dixmier-Douady class δ(A) = δ. One can also define the twisted K 1 -group by K 1 (X, δ) = K1 (A). The above two definitions of twisted K-theory are equivalent (Rosenberg [23]). We will always use the second definition in this paper. Proposition 1.12. The direct sum of twisted K-groups of X M K • (X, δ) δ∈H 3 (X,Z) forms a Z2 × H 3 (X, Z)-bigraded ring. The product K i (X, δ1 ) × K j (X, δ2 ) → K i+j (X, δ1 + δ2 ) is naturally defined. Definition 1.13. Let c ∈ Ω3 (X) be a closed 3-form, the twisted de Rham complex is the following periodic sequence d d d c c c −→ Ωev (X) −→ Ωodd (X) −→, ∗ where dc ω = dω + c ∧ ω. The twisted de Rham cohomology is HdR (X, c) = H ∗ (Ω∗ (X), dc ). ∗ ∗ Proposition 1.14. If c is a closed 3-form, then HdR (X, c) ∼ (X, zc) as isomorphic vector = HdR spaces for all nonzero z ∈ C. ∗ ∗ In particular, HdR (X, c) ∼ (X, −c) as vector spaces. In fact, in some literatures such as [20], = HdR the twisted coboundary dc ω of ω is defined by dω − c ∧ ω. Proposition 1.15. If a closed 3-form c1 = c2 + dβ for some β ∈ Ω2 (X), then dc 1 −−−−→ Ωev (X) −−−− → Ωodd (X) −−−−→   ∧ exp β ∧ exp β y y dc 2 −−−−→ Ωev (X) −−−− → Ωodd (X) −−−−→ ∗ ∗ is a chain isomorphism. Therefore HdR (X, c1 ) ∼ (X, c2 ) as vector spaces. = HdR 11 Chapter 2 Generalized Connes-HochschildKostant-Rosenberg Theorem In this section, we assume that M is a smooth oriented closed manifold, and that A is an Azumaya bundle over M with a smooth structure in the sense that the transition functions for the vector bundle A are smooth functions valued in the (Banach) Lie group PUn . Let A be the space of trace class smooth sections of A, then A is a Fréchet pre-C∗ -algebra densely embedded in A = Γ0 (A). In particular, if the rank n of A is finite, then A = Γ(A). Given a PUn -connection ∇ : Ωk (M, A) → Ωk+1 (M, A) on A, the image of the Dixmier-Douady 3 class δ(A) in HdR (M, R) can be represented by a differential 3-form in terms of the connection and curvature (e.g., Freed [13]) as follows: Let {Ui } be a good covering of M , and ψi : Ui × Mn → A|Ui be a local trivialization compatible with the smooth structure on A. Denote by gji ∈ C ∞ (Uij , PUn ) the transition function corresponding to ψj−1 ψi . Pick g̃ji ∈ C ∞ (Uij , Un ) so that Adg̃ji = gji . Let θi be the local connection forms of ∇ on Ui , ∇(ψi (O)) = ψi (dO + θi (O)), ∀O ∈ C ∞ (Ui , Mn ). −1 −1 Then θi = gji θj gji + gji dgji . Pick θ̃i ∈ Ω1 (Ui , Mn ) if n 6= ∞, or pick θ̃i ∈ Ω1 (Ui , B(H)) if n = ∞, so that θi = adθ̃i . Thus −1 −1 θ̃i = g̃ji θ̃j g̃ji + g̃ji dg̃ji + αij , 12 for some scalar valued 1-form αij ∈ Ω1 (Uij ). Let ωi be the local curvature forms of Ω = ∇2 : Γ(A) → Ω2 (X, A) on Ui , Ω(ψi (O)) = ψi (ωi (O)), ∀O ∈ C ∞ (Ui , Mn ). −1 So ωi = dθi + θi ∧ θi , and ωi = gji ωj gji . Let ω̃i = dθ˜i + θ̃i ∧ θ̃i , then adω̃i = ωi , and ω̃i = −1 g̃ji ω̃j g̃ji + dαij . Let Ω̃i = ψi ω̃i ψi−1 , then Ω̃i = Ω̃j + dαij . Since dαij + dαjk + dαki = 0, dα forms a 2-form valued cocycle, and since the sheaf of 2-forms is fine (or because of the existence of partition of unity on M ), there exist βi ∈ Ω2 (Ui ) so that 2πi(βi − βj ) = dαij . We can define a generalized 2-form by Ω̃i − 2πiβi on Ui , and it is globally well-defined. Theorem 2.1. 1). If A is a finite rank Azumaya bundle with connection ∇ and curvature Ω, then there is a unique traceless σ(Ω) ∈ Ω2 (M, A) such that adσ(Ω) = Ω. 2). If A is an infinite rank Azumaya bundle associated to a principal PU(H)-bundle P , with connection ∇ and curvature Ω, then there is a Γ(P ×PU(H) B(H))-valued 2-form σ(Ω) so that adσ(Ω) = Ω. Here PU(H) acts on B(H) the same way as on K(H). Proof. σ(Ω), up to a scalar valued 2-form, can be defined by Ω̃i − 2πiβi as above the theorem. Theorem 2.2. If A is an Azumaya bundle over M with connection ∇ and curvature Ω, then 3 − ∇(σ(Ω)) represents the image of Dixmier-Douady class δ(A) in HdR (M ). 2πi Proof. First recall that the Čech-de Rham isomorphism between the third de Rham cohomology 3 (M ) and Čech cohomology Ȟ 3 (M, C) with constant coefficient sheaf C can be constructed as HdR follows. For any closed 3-form c ∈ Ω3 (M ), one can find β(c)i ∈ Ω2 (Ui ) so that dβ(c)i = c|Ui . Since dβ(c)i − dβ(c)j = 0 one can find α(c)ij ∈ Ω1 (Uij ) so that β(c)i − β(c)j = dα(c)ij . Since dα(c)ij + dα(c)jk + dα(c)ki = 0 one can find ν(c)ijk ∈ C ∞ (Uijk ) so that (∂α(c))ijk = dν(c)ijk on Uijk . Here ∂ denotes the coboundary operator on Čech cocycles. Likewise, since d(∂ν(c))ijkl = (∂dν(c))ijkl = 0, one can find δ(c)ijkl ∈ C so that (∂ν(c))ijkl = δ(c)ijkl . The Čech-de Rham 3 isomorphism HdR (M ) → Ȟ 3 (M, C) is provided by the correspondence c 7→ δ(c). ∇(σ(Ω)) 1 Now let c be the 3-form − ∇(σ(Ω)) = − 2πi ∇(Ω̃i −2πiβi ) = dβi , 2πi , then by Bianchi identity − 2πi thus we can choose β(c)i = βi , α(c)ij = αij , ν(c)ijk = νijk , and δ(c)ijkl = δijkl . Therefore it follows 13 3 that c represents the image of δ(A) in HdR (M ). Recall that if B is a pre-C∗ -algebra densely embedded in a C∗ -algebra B, K0 (B) = K0alg (B) is naturally isomorphic to K0 (B). If B is a unital Fréchet algebra, K1 (B) is defined to be the abelian group of the equivalence classes of GL∞ (B). We say that u, v ∈ GL∞ (B) are equivalent if there is a piecewise C 1 -path in GL∞ (B) joining u and v. The definition of K1 (B) can be extended to the case of non-unital algebras so that the six-term exact sequence property always holds. For Azumaya algebras, K∗ (A) is naturally isomorphic to K∗ (A) = K ∗ (M, δ(A)). We refer to [6, 7, 18] for the definitions of Hochschild, cyclic and periodic cyclic homologies and cohomologies. Definition 2.3. Following Gorokhovsky [15], define two maps Chkr : red M C k (A) → Ωev (M ) and Chkr : M red C k (A) → Ωodd (M ) k odd k even by the JLO-type ([16]) formula Z tr(a0 e−s0 σ(Ω) (∇a1 )e−s1 σ(Ω) · · · (∇ak )e−sk σ(Ω) )ds. Chkr(a0 , a1 , ..., ak ) = (2.1) s∈∆k red red ˆ ˆ ⊗j Here C 0 (A) = A and C j (A) = A+ ⊗A , for all j 6= 0, with A+ being the unitalization of A and ˆ the projective tensor product of locally convex topological algebras. ⊗ With the assumptions and notations above, the generalized CHKR theorem of Mathai-Stevenson’s states that Proposition 2.4 (Mathai-Stevenson [20]). 1). The map Chkr in (2.1) induces a quasi-isomorphism between the two complexes [ red [ red [ → − C ev (A) → − C odd (A) → −, 0 0 0 − → Ωev (M ) − → Ωodd (M ) − →; and hence isomorphisms HHev (A) ∼ = Ωev (M ), HHodd (A) ∼ = Ωodd (M ). 14 2). The map Chkr induces a quasi-isomorphism between the complex [+b red [+b red [+b b b b −−→ C ev (A) −−→ C odd (A) −−→, or equivalently, − → HHev (A) − → HHodd (A) − → and the twisted de Rham complex d d d c c c −→ Ωev (M ) −→ Ωodd (M ) −→; and hence an isomorphism ∼ = ∗ Chkr : HP∗ (A) − → HdR (M, c), (2.2) 3 where c = − ∇(σ(Ω)) is a representative of the image of δ(A) in HdR (M ). 2πi 3). The Connes-Chern character ch : K∗ (A)⊗C → HP∗ (A) and the twisted Chern character ∗ chδ(A) = Chkr ◦ ch : K ∗ (M, δ(A)) ⊗ C → HdR (M, c) are isomorphisms. If δ(A) is a torsion class, then on the cyclic cycles level, there is an alternative way of constructing the maps Chkr, which we will see, is closely related to the relative Chern character ([3]) of Clifford modules. ˆ ˆ C ∞ (M ) Ωk (M ) by letting Define ψk : A⊗k → A⊗ ψ−1 = 0, ψ0 = 1, ψ1 (a1 ) = ∇a1 , ψ2 (a1 , a2 ) = (∇a1 )(∇a2 ) + a1 σ(Ω)a2 , ψk (a1 , ..., ak ) = (∇a1 )ψk−1 (a2 , ..., ak ) + a1 σ(Ω)a2 ψk−2 (a3 , ..., ak ), ∀k ≥ 2. (2.3) In other words, ψk (a1 , ..., ak ) is obtained as follows: Consider all partitions π of the ordered set {a1 , ..., ak } into blocks, where each block contains either one or two elements. Assign to each block {ai } of π a term of the form ∇ai , and to each block {aj , aj+1 } of π a term of the form aj σ(Ω)aj+1 . 15 Then let ψk,π be the product of these terms, and ψk (a1 , ..., ak ) be the sum of ψk,π over all such partitions. So in its expansion, ψk (a1 , ..., ak ) consists of a Fibonacci number of summands. Then let ρ : Ck (A) → Ωk (M ) be given by ρk (a0 , ..., ak ) = tr(a0 ψk (a1 , ..., ak )). (2.4) Theorem 2.5. If δ(A) is a torsion class, then the map ρk in (2.4) induces a homomorphism ρ : C∗λ (A) → Ω∗ (M )/d(Ω∗−1 (M )), where C∗λ (A) is the Connes complex of A (cf. [6], [7]), and an isomorphism ∼ = ∗ ρ : HP∗ (A) − → HdR (M ) which coincides with Chkr in (2.2). Proof. To see that the induced homomorphism C∗λ (A) → Ω∗ (M )/d(Ω∗−1 (M )) is well-defined, we show that for all k ≥ 0,
(−1)k−1 ρk (a0 , . . . , ak ) + ρk (ak , a0 , . . . , ak−1 ) = d tr a0 ψk−1 (a1 , ...ak−1 ) ak , for all ai ∈ A. Noticing that d ◦ tr = tr ◦ ∇, it suffices to show
(−1)k−1 a0 ψk (a1 , .., ak ) + ψk (a0 , ..., ak−1 )ak = ∇ a0 ψk−1 (a1 , ..., ak−1 )ak , (2.5) for all ai ∈ A+ . In fact, it is easy to see (2.5) is true for k = 0, 1, 2. Suppose (2.5) holds for all k ≤ m for some m. Then using ∇2 = adσ(Ω) and the Bianchi identity ∇(σ(Ω)) = 0, we have 16 ∇(a0 ψm (a1 , ..., am )am+1 ) = ∇a0 ψm (a1 , ..., am )am+1 + a0 ∇ψm (a1 , ..., am )am+1 +(−1)m a0 ψm (a1 , ..., am )∇am+1
= ∇a0 ψm (a1 , ..., am )am+1 + a0 ∇ ∇a1 ψm−1 (a2 , ..., am ) am+1
+a0 ∇ a1 σ(Ω)a2 ψm−2 (a3 , ..., am ) am+1 + (−1)m a0 ψm (a1 , ..., am )∇am+1 = ψm+1 (a0 , ..., am )am+1 − a0 a1 σ(Ω)ψm−1 (a2 , ..., am )am+1 −a0 ∇a1 ∇ψm−1 (a2 , ..., am )am+1 + a0 ∇a1 σ(Ω)a2 ψm−2 (a3 , ..., am )am+1 +a0 a1 σ(Ω)∇a2 ψm−2 (a3 , ..., am )am+1 + a0 a1 σ(Ω)a2 ∇ψm−2 (a3 , ..., am )am+1 +(−1)m a0 ψm (a1 , ..., am )∇am+1 = ψm+1 (a0 , ..., am )am+1 − a0 a1 σ(Ω)ψm−1 (a2 , ..., am )am+1 +(−1)m a0 ∇a1 ψm (a2 , ..., am , 1)am+1 − a0 ∇a1 ψm (1, a2 , ..., am )am+1 +a0 ∇a1 σ(Ω)a2 ψm−2 (a3 , ..., am )am+1 + a0 a1 σ(Ω)∇a2 ψm−2 (a3 , ..., am )am+1 +(−1)m a0 a1 σ(Ω)a2 ψm−1 (a3 , ..., am , 1)am+1 +a0 a1 σ(Ω)a2 ψm−1 (1, a3 , ..., am )am+1 +(−1)m a0 ψm (a1 , ..., am )∇am+1 = ψm+1 (a0 , ..., am )am+1 − a0 a1 σ(Ω)ψm−1 (a2 , ..., am )am+1 +(−1)m a0 ∇a1 ψm−2 (a2 , ..., am−1 )am σ(Ω)am+1 +a0 a1 σ(Ω)∇a2 ψm−2 (a3 , ..., am )am+1 +(−1)m a0 a1 σ(Ω)a2 ψm−3 (a3 , ..., am−1 )am σ(Ω)am+1 +a0 a1 σ(Ω)a2 σ(Ω)a3 ψm−3 (a4 , ..., am )am+1 +(−1)m a0 ψm (a1 , ..., am )∇am+1 = ψm+1 (a0 , ..., am )am+1 + (−1)m a0 ∇a1 ψm−2 (a2 , ..., am−1 )am σ(Ω)am+1 +(−1)m a0 a1 σ(Ω)a2 ψm−3 (a3 , ..., am−1 )am σ(Ω)am+1 +(−1)m a0 ψm (a1 , ..., am )∇am+1 = ψm+1 (a0 , ..., am )am+1 + (−1)m a0 ψm+1 (a1 , ..., am+1 ). 17 Thus, by induction, identity (2.5) is proved. ∗ To show that the induced map ρ : HP∗ (A) → HdR (M ) is well-defined: First note that ρk ◦ [ = 0 λ on Ck+1 (A). This means that the map C∗λ (A)/[(C∗+1 (A)) → Ω∗ (M )/d(Ω∗−1 (M )) is well-defined. Then it suffices to show that the images of HP∗ (A) under the map ρ are represented by closed forms. We prove this only for the even case, and the odd case is similar. Since ch : K0 (A) → HP0 (A) is an isomorphism, elements of HP0 (A) are generated by ch[p] for [p] ∈ K0 (A). Observe that p(∇p)2i+1 p = 0 for all idempotent p and i ≥ 0, then ρ2k (chλ2k (p)) = (−1)k

(2k)! (2k)! tr p ψ2k (p, ..., p) = (−1)k tr pψ2 (p, p)k , k! k! because any term in the expansion of p ψ2k (p, ..., p)p that has a factor p(∇p)2i+1 p vanishes. Since ∇(pψ2 (p, p)) = 0, it follows that ρ(ch[p]) is a closed form for all [p] ∈ K0 (A). ∗ Finally, we can prove that ρ : HP∗ (A) → HdR (M ) is an isomorphism identified with Chkr by an argument on the Čech-de Rham bicomplex of M , just similar to the argument used in MathaiStevenson [20]. 18 Chapter 3 Spectral Analysis of Spectral Triples In this section we review the definition and some analytical properties of spectral triples. Note that a slight modification to the standard definition of spectral triple (cf. [10]) is made so that it will be more convenient to develop the theory in this paper. In fact, in definition 3.6 we require that the second entry H of a spectral triple (A, H, D) to be an A-module as well as the smooth Sobolev domain of D, instead of the Hilbert space H̄. So in application in differential geometry, spectral triples defined this way operate directly with smooth sections of vector bundles. For a spectral triple in the conventional sense, that would be a strong requirement, as strong as the smoothness condition in Appendix B in [11]. Suppose that D is a densely defined self-adjoint operator on a Hilbert space H, and that D has compact resolvent. Let µ1 > µ2 > · · · be the list of eigenvalues of (D2 + 1)−1 in decreasing order, and Vi ⊂ H be the eigenspace corresponding to µi for each i. Then every vector v ∈ H can be uniquely represented as a sequence (v1 , v2 , . . . ) with vi ∈ Vi and 2 i kvi k < ∞, and vice versa. P For every s ≥ 0, consider the following subspaces of H, W s (D) = {(v1 , v2 , . . . ) ∈ H | X i 2 µ−s i kvi k < ∞}, 19 with the norm k(v1 , v2 , . . . )ks = qP −s 2 i µi kvi k ; W s,p (D) = {(v1 , v2 , . . . ) ∈ H | X −sp/2 µi kvi kp < ∞}, ∀p > 0, i with the norm k(v1 , v2 , . . . )ks,p = P −sp/2 i µi kvi kp 1/p ; and −s/2 W s,∞ (D) = {(v1 , v2 , . . . ) ∈ H | sup µi kvi k < ∞}, i −s/2 with the norm k(v1 , v2 , . . . )ks,∞ = supi µi kvi k. W s = W s,2 has a natural Hilbert space structure and Proposition 3.1 (Rellich). For each  > 0, the inclusion W s+ → W s is compact. Proposition 3.2. W 1 ⊂ H is the domain of the self-adjoint operator D, and D : W 1 → H is a Fredholm operator. Let W ∞ = T s>0 W s , then W ∞ is a Fréchet space with a family of norms k · ks . It is easy to see that restricted to W ∞ , the mapping D : W ∞ → W ∞ is continuous with respect to the Fréchet space topology. We say the operator D is finitely summable or has spectral dimension less than 2d (for some real number d > 0), if (D2 + 1)−d is a trace class operator. Theorem 3.3. Suppose D has finite spectral dimension. If T ∈ B(H) is a bounded operator that maps W ∞ into W ∞ , then the restricted mapping T : W ∞ → W ∞ is also continuous. The theorem can be proved by the following lemmas. Lemma 3.4 (Sobolev embeddings). If D has spectral dimension less than 2d, then we have the following obvious estimate: kvks,∞ ≤ kvks,p ≤ X µdj 1/p kvks+ 2d ,∞ , p ∀v ∈ H, ∀s ≥ 0, ∀p > 0, j 2d i.e., there are bounded embeddings W s+ p ,∞ ⊂ W s,p ⊂ W s,∞ . 20 Lemma 3.5. Suppose D has finite spectral dimension. i) Let T : W ∞ → W ∞ be a continuous operator. Suppose for each j, ∀vj ∈ Vj , T (0, . . . , 0, vj , 0, . . . ) = (t1j vj , t2j vj , . . . ), where (tij ) is an infinite matrix with entries tij ∈ Hom(Vj , Vi ). Then (tij ) satisfies the property: for any s > 0 there exist C and r > 0 such that k X −r µ−s i tij k < C + µj , ∀j. i ii) Conversely any matrix (tij ) with entries tij ∈ Hom(Vj , Vi ) satisfying the above property represents a continuous operator T : W ∞ → W ∞ . Proof. i) For any v ∈ W ∞ , we see that as n → ∞, Pn W ∞ . Because T is continuous, it follows that T ( T v, hence (T v)i = Pn j=0 vj ) = ( Pn s , therefore Pn Pn j=0 vj → v in W j=0 t1j vj , j=0 vj → v in j=0 t2j vj , . . . ) → P j tij vj . Suppose the claim is not true, then there must exist s > 0, such that for any C and n, there is j(n, C) satisfying k X −n µ−s i tij(n,C) k > C + µj(n,C) . i Thus one may find an increasing sequence {j(n)}, such that k X −n µ−s i tij(n) k > µj(n) . i For each j, pick uj ∈ Vj so that kuj(n) k = k P −s −1 < µnj(n) and k i µ−s i tij(n) uj(n) k = i µi tij(n) k P 1, while uj = 0 if j 6= j(n), ∀n. Then, because of the finite spectral dimension, u = W ∞ , but T ( Pn j=0 uj ) does not converge in W 2s P j uj ∈ as n → ∞, which yields a contradiction. 21 ii) Suppose the matrix (tij ) has that property. Define T : W ∞ → W ∞ by (T v)i = X tij vj , ∀v ∈ W ∞ . j For any sequence u(n) ∈ W ∞ , we now prove that if u(n) → 0 as n → ∞ then T u(n) → 0. For any s > 0, kT u(n)k2s = k X i ≤ X µ−s i X j tij uj (n)k ≤ X X k µ−s i tij uj (n)k j i (C + µ−r j )kuj (n)k = Cku(n)k0,1 + ku(n)kr,1 . j Since u(n) → 0 in W ∞ implies ku(n)ks0 → 0 for all s0 , using Lemma 3.4, it follows that kT u(n)k2s → 0 for all s. So this implies T u(n) → 0, and therefore T : W ∞ → W ∞ is a continuous operator. For any pre-Hilbert space H, we define the ∗-algebra B(H) = {T ∈ B(H̄) | T (H) ⊂ H, T ∗ (H) ⊂ H}. Definition 3.6. A triple (A, H, D) is said to be a unital spectral triple if it is given by a unital pre-C ∗ -algebra A, a pre-Hilbert space H with a norm-continuous unital ∗-representation A → B(H), and a self-adjoint operator D on H̄ called Dirac operator, with the following properties: i) D has compact resolvent, ii) W ∞ (D) = H, iii) under the representation of A, the commutator [D, a] : H → H is norm-bounded for each a ∈ A. Besides, we always assume that A is a locally convex topological ∗-algebra with a topology finer than the norm topology of A, and the representation A × H → H is jointly continuous with respect to the locally convex topology of A and the Fréchet topology of H. 22 Note that if (A, H, D) is a unital spectral triple, then W 1 (D), the domain of D, also forms a left A-module because of the last condition in the definition. A spectral triple (A, H, D) is said to be even if there is a Z2 -grading on H: H = H+ ⊕ H− , so that the grading operator   idH+ γ=  0 0 −idH−  :H→H  commutes with all a ∈ A and anti-commutes with D. Spectral triples equipped with no such gradings are said to be odd . Two spectral triples (A1 , H1 , D1 ) and (A2 , H2 , D2 ) with the isomorphic algebras are said to be unitarily equivalent if there is a unitary operator U : H̄1 → H̄2 intertwining the two representations of Ai and the two Dirac operators Di in an obvious way. For the even case the unitary operator U also needs to be grade preserving. 23 Chapter 4 Morita Equivalence of Spectral Triples In this section we introduce the notion of Morita equivalence of spectral triples. Let A be a preC ∗ -algebra. Recall that a right A-module S is called a pre-Hilbert (or Hermitian) right A-module if there is an A-valued inner product (·, ·) : S × S → A, such that for all x, y ∈ S, a ∈ A, i) (x, y) = (y, x)∗ , ii) (x, ya) = (x, y)a, iii) (x, x) ≥ 0, and (x, x) = 0 only if x = 0. 1 The norm on S is given by kxk = k(x, x)k 2 , and its norm completion S̄ (which is automatically a pre-Hilbert right Ā-module) is called a Hilbert right Ā-module. Hilbert left modules can be defined in the same manner. In particular, every Hilbert space is a Hilbert C-module. If S is a pre-Hilbert right A-module, then its conjugate space S ∗ = {fx := (x, ·) | x ∈ S} is a pre-Hilbert left A-module, and for all x, y ∈ S, a ∈ A, afx := a(x, ·) = fx(a∗ ) , (fx , fy ) := (y, x), (fx , afy ) = a(fx , fy ). 24 ¯ denotes the ∗-algebra of all module homomorphisms If S is a pre-Hilbert right A-module, BA (S) T : S̄ → S̄ for which there is an adjoint module homomorphism T ∗ : S̄ → S̄ with (T x, y) = (x, T ∗ y) for all x, y ∈ S̄. Define BA (S) = {T ∈ BA (S̄) | T (S) ⊂ S, T ∗ (S) ⊂ S}. In particular, if A = C, then BC (S)=B(S). If A is unital and S is unital and finitely generated, then BA (S) in fact consists of all A-endomorphisms of S, i.e., BA (S) = EndA (S). Suppose A and B are unital pre-C ∗ -algebras. Let E be a unital finitely generated projective right A-module, then as a summand of a free module, one can endow E with a pre-Hilbert module structure (which is unique up to unitary A-isomorphism). Suppose B acts on E on the left, and the representation B → BA (E) is unital, ∗-preserving and norm-continuous. We also assume that E is also endowed with a topology induced from the locally convex topology of A, that B is locally convex topological ∗-algebra with a topology finer than the norm topology, and that the representation B × E → E is jointly continuous with respect to the locally convex topologies on B and E. We call such a B-A-bimodule with the above structure a finite Kasparov B-A-module, then we introduce the following Definition 4.1. Suppose σ = (A, H, D) is a unital spectral triple. A σ-connection on a finite Kasparov B-A-module E is a linear mapping ∇ : E → E ⊗A B(H) with the following properties: i) ∇(ξa) = (∇ξ)a + ξ ⊗A [D, a], ii) (ξ, ∇ε) − (∇ξ, ε) = [D, (ξ, ε)], for all a ∈ A, and ξ, ε ∈ E. Note that by the notation (∇ξ, ε), which is a bit ambiguous, we really mean (ε, ∇ξ)∗ ∈ B(H). 25 Since E is finitely generated, it follows that for each b ∈ B, the commutator [∇, b] corresponds to an element in B(E ⊗A H). It also follows that any two σ-connections on E differ by a Hermitian element in B(E ⊗A H). From the above data (A, H, D, B, E, ∇), one can construct a new spectral triple σ E = (B, HE , DE ) for B (cf. Connes [7, §VI.3]). The pre-Hilbert space HE is E ⊗A H with inner product given by < ξ1 ⊗A h1 , ξ2 ⊗A h2 >=< h1 , (ξ1 , ξ2 )h2 >, ∀ξi ∈ E, hi ∈ H. The Dirac operator DE on HE is given by DE (ξ ⊗A h) = ξ ⊗A Dh + (∇ξ)h, ∀ξ ∈ E, h ∈ W 1 (D). It is easy to see that the commutator [DE , b] = [∇, b] ∈ B(E ⊗A H) is bounded. To verify that DE is self-adjoint with domain E ⊗A W 1 (D), that W ∞ (DE ) = HE , and that DE has compact resolvent, it is adequate to check choosing one particular σ-connection on E, because bounded perturbations do not affect the conclusions. Recall that a universal connection on a pre-Hilbert right A-module S is a linear mapping ∇ : S → S ⊗A Ω1u (A) that satisfies the Leibniz rule ∇(sa) = (∇s)a + δu a, and ∇ is said to be Hermitian if (s, ∇ε) − (∇s, ε) = δu (s, ε). Here Ω1u (A) is the space of universal 1-forms of A, and involutions (δu a)∗ are set to be −δu (a∗ ). Cuntz and Quillen [12] showed that only projective modules admit universal connections. Given a universal Hermitian connection on a finite Kasparov module E, by sending 1-forms δu a to [D, a], one 26 can associate with the universal connection a σ-connection on E for any spectral triple σ = (A, H, D). Let p ∈ Mn (A) be a projection (i.e., a self-adjoint idempotent n × n matrix). Now we consider the right A-module pAn . There is a canonical universal connection on pAn which is given by the matrix pdiag{δu , . . . , δu } or simply written as ∇(pa) = pδu (pa), ∀a ∈ An , and this connection is Hermitian. Definition 4.2. A universal connection ∇ on a pre-Hilbert right A-module S is said to be projectional if there is a unitary A-isomorphism φ : S → pAn for some n and some projection p, such that ∇ = φ−1 ◦ pδu ◦ φ. A projectional universal connection on each finite projective pre-Hilbert A-module is unique up to unitary A-isomorphism. If E is a finite Kasparov B-A-module admitting a universal connection ∇A and F is a finite Kasparov C-B-module admitting a universal connection ∇B , then there is a twisted universal connection ∇B ◦ ∇A defined in an obvious way on the finite Kasparov C-A-module F ⊗B E. Furthermore if ∇A and ∇B are projectional, then so is ∇B ◦ ∇A . This leads to a category of noncommutative differential geometry NDG, consisting of formal objects XA for all unital pre-C ∗ algebras A. The morphisms XA → XB in NDG are (isomorphism classes of) finite Kasparov B-Amodules with projectional universal connections. It is not difficult to extend morphisms of NDG to graded modules with super-connections in the sense of Quillen [22]; however in this paper, we only focus on finite Kasparov modules with trivial gradings. For an even spectral triple σ = (A, H, D, γ) with grading γ, σ-connections ∇ are required to be odd operators, then for each (E, ∇), σ E is also an even spectral triple with an obvious grading γ E . Denote by Sptr(A) = Sptr0 (A)qSptr1 (A) the set of even spectral triples and odd spectral triples for A up to unitary equivalence, then Sptr yields a functor NDG → Set given by XA 7→ Sptr(A). Remark 4.3. Baaj-Julg [2] established the theory of unbounded Kasparov modules and showed that every element in the bivariant KK-theory can be represented by an unbounded Kasparov module. 27 It would be nice if morphisms of NDG could be enlarged to unbounded (even) Kasparov modules (E, D, Γ) with a grading Γ and an appropriate universal connection ∇, and thereby DE,D,Γ (ξ⊗A h) = Dξ ⊗A h + Γξ ⊗A Dh + Γ(∇ξ)h. The notion of connection for bounded Kasparov modules introduced by Connes-Skandalis was well-known, whereas theory of connection for unbounded Kasparov modules has been developed in a recent work by Mesland [21]. Definition 4.4. Two unital spectral triples σ1 = (A1 , H1 , D1 ) and σ2 = (A2 , H2 , D2 ) are said to be Morita equivalent if A1 and A2 are Morita equivalent as algebras and there is a finite Kasparov A2 -A1 -module E with a σ1 -connection, such that E is an equivalence bimodule and σ2 is unitarily equivalent to σ1E . Theorem 4.5. The Morita equivalence between spectral triples is an equivalence relation. Proof. Reflexivity: (A, H, D) is Morita equivalent to itself via the trivial connection ∇a = [D, a] on A. Transitivity: it is straight forward by definition. Symmetry: suppose (B, HE , DE ) is Morita equivalent to σ = (A, H, D) via a σ-connection ∇ on E. Let F = E ∗ . Define ∇F : F → F ⊗B B(HE ) by (∇F f )(ξ ⊗A h) = −f (∇ξ) ⊗A h + [D, f ξ]h, ∀f ∈ F, ξ ∈ E, h ∈ H. Here we use a map HE → H to represent an element in F ⊗B B(HE ). One can verify ∇F is a (B, HE , DE )-connection. Now we check (DE )F = D as follows, (DE )F (f ⊗B ξ ⊗A h) = f ⊗B DE (ξ ⊗A h) + (∇F f )(ξ ⊗A h) = f ⊗B ξDh + f (∇ξ)h + (∇F f )(ξ ⊗A h) = f ξDh + f (∇ξ)h − f (∇ξ)h + [D, f ξ]h = D(f ξh). (An alternative way to prove the symmetry property is using bounded perturbations of the σ- 28 connection and σ E -connection that are associated to projectional universal connections.) In conclusion, Morita equivalence of spectral triples is an equivalence relation. Remark 4.6. As a special case, A is Morita equivalent to itself via E = A. Using universal connections on A, one can construct new spectral triples which are called inner fluctuations of spectral triples [9]. Morita equivalence of spectral triples (A, H, D1 ) and (A, H, D2 ) with the same A and H is different from inner fluctuation of spectral triples, as the latter is generally not an equivalence relation when A is noncommutative. For instance, any spectral triple (A, H, D) with a finite dimensional matrix algebra A and a finite dimensional Hilbert space H, has an inner fluctuation (A, H, 0), whereas (A, H, D) is not an inner fluctuation of (A, H, 0) as long as D 6= 0. In general, if we confine ourselves to equivalence bimodules with universal connections, the symmetry property in the above proof does not hold. However if we restrict the universal connections on equivalence bimodules to the projectional ones, the symmetry property holds again. Similar to Sptr(A), let SptrM (A) denote the set of spectral triples over A up to Morita equivalence. In other words, SptrM (A) is the set of spectral triples over A up to unitary equivalence and bounded perturbations of Dirac operators. SptrM yields a functor NDG → Set, XA 7→ SptrM (A), and the induced morphisms SptrM (E, ∇) do not depend on the choice of spectral-triple-connections ∇ on E. Suppose (E, ∇) is a morphism in HomNCG (XA , XB ), then there is a free right A-module Am and a projection p in Mm (A) such that E ∼ = pAm . Let {e1 , ..., em } be the standard generators of Am . For each b in B, one can find a matrix α(b) in pMm (A)p, such that bpei = P ej αji (b). j The K-theory, Hochschild (co)homology and (periodic) cyclic (co)homology are all functors on the category NDG. For instance, suppose (b0 , ..., bn ) is a Hochschild n-cycle representing an element 29 Φ ∈ HH∗ (B), then E(Φ) ∈ HH∗ (A) is the Hochschild n-cycle given by the Dennis trace map X tr(α(b0 ) ⊗ · · · ⊗ α(bn )) = (αi0 i1 (b0 ), αi1 i2 (b1 ), ..., αin i0 (bn )). i0 ,...,in Note that α depends on the choice of the isomorphism E ∼ = pAm ; however, it does induce the well-defined morphisms E : HH∗ (B) → HH∗ (A). Furthermore, the Connes-Chern characters [6] are natural transformations from the K-theory functor XA 7→ K0 (A) to periodic cyclic homology functor XA 7→ HP0 (P ), and from functors XA 7→ Sptr0 (A), Sptr0M (A) and K 0 (Ā) to the periodic cyclic cohomology functor XA 7→ HP 0 (A). The naturalness of Connes-Chern characters is illustrated in the following commutative diagrams K0 (B) −−−−→ K0 (A)     chy chy E Sptr0 (A) −−−−→ Sptr0 (B)     chy chy E HP 0 (A) −−−−→ HP0 (B). HP0 (B) −−−−→ HP0 (A), (E,∇) E Proposition 4.7. The following diagrams Ind HP0 (A) x  E × HP 0 (A) −−−−→ C   yE Ind HP0 (B) × HP 0 (B) −−−−→ C, K0 (A) x  E × Sptr0 (A) −−−−→ Z  (E,∇) y K0 (B) × Sptr0 (B) −−−−→ Z, and commute. Ind K0 (A)   ych × Sptr0 (A) −−−−→ Z     ych y HP0 (A) × HP 0 (A) −−−−→ C, 30 Chapter 5 Gluing Local Spin Structures on Riemannian Manifolds via Morita Equivalence We can apply the above theory to spectral triples on Riemannian manifolds. By gluing local pieces of spectral triples via Morita equivalence, we construct a so called projective spectral triple, the Dirac operator of which was defined in a formal sense by Mathai-Melrose-Singer [19]. Let X be a closed oriented Riemannian manifold of dimension n. Suppose X is spin or spinc . Let Cln denote the complex Clifford algebra of Rn . In this paper we use the following convention for the definition of Clifford algebras Cln :=< u ∈ Rn |uv + vu = −2(u, v), ∀u, v ∈ Rn > . Decomposed by the parity of degree, Cln = Cln0 ⊕ Cln1 . Write Bn =     Cln ∼ = M2m (C)    Cln0 ∼ = M2m (C) and Bx =     Cl(T ∗ X) when n = 2m is even    Cl0 (Tx∗ X) when n = 2m + 1 is odd. x (5.1) Denote by Cl(X) and B(X) the vector bundles over X whose fibers at a point x ∈ X are Cl(Tx∗ X) and Bx . Let Sn = C2 m be the standard spinor vector space, and we fix a specific isomorphism c : Bn → EndC (Sn ). 31 n+1 Let ωn = i[ 2 ] e1 · · · en ∈ Cln , then when n is odd, Bn = ωn Cln1 . This indicates a homomorphism c : Cln → EndC (Sn ). When n is even, Sn = Sn+ ⊕ Sn− is graded by the eigenspaces of ωn , which are invariant under the action of Spin(n). Let PFr (X) and PSpin(c ) (X) denote the orthonormal oriented frame bundle and the principal Spin(n) or Spinc (n)-bundle over X respectively. Then Cl(X) = PFr (X) ×SO(n) Cln = PSpin(c ) (X) ×Ad Cln . The spinor bundle over X is the associated Spin(c )(n)-bundle SX = PSpin(c ) (X) ×c Sn . The Clifford algebra bundle acts naturally on the spinor bundle, c : Cl(X) ×X SX → SX , which is given by (p, ξ) × (p, s) 7→ (p, c(ξ)s), ∀p ∈ PSpin(c ) (X), ξ ∈ Cln , s ∈ Sn . When n is even, ωn induces a grading operator ω on SX = SX+ ⊕ SX− . / the Dirac operator on SX . Let E be a Hermitian vector bundle over M with a Denote by D Hermitian connection ∇E . Let A = C ∞ (X), B = Γ(B(X)), and E = Γ(E). Then the well-known spin spectral triple / (A, H, D) = (C ∞ (X), Γ(SX ), D) (5.2) is Morita equivalent to the following spectral triple with a noncommutative algebra E / ), (B, HE , DE ) = (Γ(End(E)), Γ(E ⊗ SX ), D / via the finite Kasparov module E with (A, H, D)-connection associated to ∇E . Here D (5.3) E denotes the twisted Dirac operator on the vector bundle E ⊗ SX . Now interesting things happen when a manifold has no spinc structure: The spinor bundle does 32 not exist, and neither does the spin spectral triple. However, as constructed later in this section, for any closed oriented Riemannian manifold, not necessarily spinc , there is a canonical noncommutative spectral triple: Definition 5.1. The projective spectral triple of a closed oriented Riemannian manifold M is defined to be (AW3 , HW3 , DW3 ) := (Γ(B(M )), (1 + ∗)Ω(M ), (d − d∗ )(−1)deg ), (5.4) if M is odd dimensional, and (AW3 , HW3 , DW3 , γW3 ) := (Γ(B(M )), Ω(M ), (d − d∗ )(−1)deg , ∗(−1) deg(deg +1) −n 2 4 ), (5.5) if M is even dimensional. We can consider that projective spectral triples are obtained by gluing local spin spectral triples in the following way: Let M be a closed oriented Riemannian manifold of dimension n, not necessarily spinc . Let {Ui } be a good covering of M . Then on each local piece we have the principal Spin(n)-bundle Pi = PSpin (Ui ), the associated spinor bundle Si = SUi , the spin connection ∇i , and the Dirac / i . Over each intersection Uij = Ui ∩ Uj 6= ∅, there is up to ±1 ∈ Spin(n) a unique operator D homeomorphism of principal bundles αij : Pi |Uij → Pj |Uij , such that αij (pi g) = αij (pi )g, ∀pi ∈ Pi |Uij , g ∈ Spin(n). This homeomorphism induces up to ±1 a Clifford module homomorphism αij : Si |Uij → Sj |Uij , αij (p, s) = (αij (p), s), ∀p ∈ Pi |Uij , s ∈ Sn . 33 For each triple overlap Uijk = Ui ∩ Uj ∩ Uk 6= ∅, write σijk = αki ◦ αjk ◦ αij |Uijk , then {σijk } represents a Čech cocycle in Ȟ 2 (M, Z2 ), and the corresponding singular class w2 (M ) ∈ H 2 (M, Z2 ) is the second Stiefel-Whitney class. Let Lij = HomCl(Uij ) (Si |Uij , Sj |Uij ) denote the line bundle of the Clifford module homomorphisms of Si and Sj over Uij , then αij is the canonical section of Lij . {Lij } forms a gerbe of line bundles over M characterized by the Dixmier-Douady class δ(B(M )) = W3 (M ) (the third integral Stiefel-Whitney class). We refer [4] for (twisted) K-theory of bundle gerbes. If M is spin, then for each Uij there is βij ∈ Z2 , such that {βij αij } satisfies the cocycle condition. That means the local spinor bundles Sij can be glued together as a global spinor bundle via the Clifford module isomorphisms βij αij . The difference between two different sets of such {βij } is a cocycle in Ȟ 1 (M, Z2 ) which parameterizes distinct spin structures on M . If M is spinc , then for each Uij there is βij ∈ C ∞ (Uij , U (1)), such that {βij αij } satisfies the cocycle condition. That means the local spinor bundles Sij can be glued together as a global 2 spinor bundle via the Clifford module isomorphisms βij αij . βij also satisfies the cocycle condition, 2 and the Čech cocycle {βij } ∈ Ȟ 1 (M, U (1)) corresponds to the canonical line bundle L of a spinc structure. The difference between two different sets of such {βij } is a cocycle in Ȟ 1 (M, U (1)) which parameterizes distinct spinor bundles on M . Denote by C ∞ (Ūi ) the space of smooth functions on Ui that can be extended to a smooth function on a small open neighborhood Vi of Ūi , and likewise denote by Γ(Ūi , ·) for extendable smooth sections. Take Ei = Γ(Ūi , Si ), then the “local spin spectral triple” / i) (Ai , Hi , Di ) = (C ∞ (Ūi ), Γ(Ūi , Si ), D (5.6) 34 is Morita equivalent to the local spectral triple S / i i ). (Bi , HiEi , DiEi ) = (Γ(Ūi , B(Ui )), Γ(Ūi , Si ⊗ Si ), D (5.7) Remark 5.2. We use here the triple (5.6) to formulate the spin structure of the open subspace Ui , but it is by definition not a spectral triple, for the compact resolvent condition fails. However, it can naturally act on the relative K-theory for the pair of spaces (Vi , Ui ) or (Y, ι(Ui )) to get an index, where Y is any compact Riemannian spin manifold that admits an isometric embedding ι : Vi → Y . By excision property, the choice of Vi is irrelevant. In this sense the triple (5.6) represents a relative K-cycle. One may also think of the standard treatment using the nonunital / i ) with the algebra of smooth functions with compact support. spectral triple (Cc∞ (Ui ), L2 (Ui , Si ), D See Gayral-Gracia-Bondı́a-Iochum-Schücker-Várilly [14] for a set of axioms for nonunital spectral triples which is proposed to set up the notion of noncompact noncommutative spin manifolds. This, however, will cause some subtleties when considering Morita equivalence and smoothness condition. Because the collection of maps {αij ⊗αij } satisfy the cocycle condition, the vector bundles Si ⊗Si and Dirac operators DiEi can be glued together to form a vector bundle N over M and a Dirac operator DN on N , so that (B|Ui , Γ(M, N )|Ui , DN |Ui ) are unitarily equivalent to (Bi , HiEi , DiEi ), where B = Γ(M, B(M )). Thus we succeed to construct a globally well-defined spectral triple (B, Γ(N ), DN ) on M . Proposition 5.3. The global vector bundle N is isomorphic to B(M ), and Γ(N ) is isomorphic to Γ(B(M )) as both Γ(B(M ))-modules and pre-Hilbert spaces. Proof. Let Sn∗ be the dual vector space of the standard spinor vector space Sn . We endow Sn∗ with a left Bn -module structure by γ ∗ (b)fx := fb̄x = (x, b̄∗ ·), for all x ∈ Sn , b ∈ Bn , where b̄ is the complex conjugation of b, and b∗ is the adjoint of b ∈ Bn ∼ = M2m (C). Since Bn is a simple algebra, up to a scalar there is a unique Bn -module isomorphic from Sn to Sn∗ . We fix one specific unitary Bn -module isomorphism Tn : Sn → Sn∗ . Then Tn induces a Clifford module isomorphism of bundles T : Si → Si∗ , ∗ given by (p, s) 7→ (p, Tn s), for all p ∈ Pi , s ∈ Sn . Let Si∗ = Pi ×γ ∗ Sn∗ , and let αij : Si∗ → Sj∗ denote 35 ∗ (p, f )) = (αij p, f ), for all f ∈ Si∗ . The mappings T on the Clifford module isomorphism given by αij Ui and Uj are compatible, namely the diagram below commutes on Uij . Si T i α∗ ij αij  Sj / S∗ T  / S∗ j Then one can glue the bundles Si∗ ⊗ Si ∼ = EndC (Si∗ ) together as a global bundle B(M ) via the ∗ ⊗ αij , and T induces an isomorphism from N to B(M ). Then it is easy to verify the maps αij proposition. Corollary 5.4. When M is odd dimensional, one can take the bundle N = (1 + ∗) V∗ (TC∗ M ), then the spectral triple (B, Γ(N ), DN ) is the projective spectral triple (AW3 , HW3 , DW3 ) = (B, (1 + ∗)Ω(M ), (d − d∗ )(−1)deg ); and when M is even dimensional, take N = V∗ (5.8) (TC∗ M ), then the spectral triple (B, Γ(N ), DN ) with the grading on Γ(N ) obtained from the grading on Si is the even projective spectral triple (AW3 , HW3 , DW3 , γW3 ) = (B, Ω(M ), (d − d∗ )(−1)deg , ∗(−1)deg(deg +1)/2−n/4 ), (5.9) and its center (A, HW3 , DW3 , γW3 ) is unitarily equivalent to the spectral triple for Hirzebruch signature. For any a ∈ A, the commutator [DW3 , a] is just the right Clifford action of da on HW3 . Theorem 5.5. If M is spinc , the projective spectral triple on M is Morita equivalent to the spin spectral triple. 1/2 Proof. Let S be the global spinor bundle over M . There exist local half line bundles Li 1/2 1/2 that Li are characterized by {βij } as local transition functions, Li 1/2 S|Ui = Li 1/2 ⊗ Li , such = L |Ui , and ⊗ Si . There also exists a hermitian connection ∇1 on L 1/2 such that the Spinc - connection ∇|Ui = 1 ⊗ ∇i + ∇1 ⊗ 1, where ∇i is the Spin-connection on Si . Then follows the Morita 36 equivalence of the spectral triples / = (A, H, D) ∼ (A, HL (C ∞ (M ), Γ(M, S), D) ∼ (B, (HL −1 )S , (DL −1 −1 , DL −1 ) )S ) = (AW3 , HW3 , DW3 ). We see that the projective spectral triple is defined for any closed oriented Riemannian manifold regardless of whether the manifold is spinc or not. The projective spectral triple depends only on the metric and orientation of M and does not depend on the choice of the local spinor bundles Si . 37 Chapter 6 Projective Spectral Triple as Fundamental Class in K0(M, W3) In this section we see how projective spectral triples represent the fundamental classes in the twisted K-homology K 0 (AW3 ) ∼ = K0 (M, W3 ). Denote by Cgr the Z2 -graded algebra of sections of Clifford bundle Cl(T ∗ M ) over M (even dimensional only), then every Clifford module E over M can be considered as a finitely generated E E projective right Cop gr -module, and a Clifford connection ∇ on E gives rise to a Dirac operator D on E. Then E 7→ Ind DE defines a canonical homomorphism Ind K0 (Cop −→ Z. gr ) − By Morita equivalence, K0 (Cop gr ) can be replaced by the K-theory of an ungraded algebra, K0 (AW3 ), and the homomorphism Ind becomes the operation of pairing with the projective spectral triple. Theorem 6.1 (Poincaré duality). For an even dimensional closed oriented manifold M , the projective spectral triple ς = (AW3 , HW3 , DW3 , γW3 ) represents the twisted K-orientation as a cycle of the twisted K-homology K 0 (AW3 ) ∼ = K0 (M, W3 ), and hence gives rise to the Poincaré duality _[ς] K 0 (M, W3 − c) −−∼−→ K0 (M, c), = or nondegenerate K ∗ (M, c) × K ∗ (M, W3 − c) −−−−−−−−−→ Z, pairing for all c ∈ H 3 (M, Z). Here the cap product can be defined by [E] _ [ς] = [ς E ] for any finite Kasparov 38 C ∞ (M )-AW3 -module E. For odd dimensional M , the Poincaré duality reads nondegenerate K ∗ (M, c) × K ∗+1 (M, W3 − c) −−−−−−−−−→ Z, pairing ∀c ∈ H 3 (M, Z). See Kasparov [17], Carey-Wang [5], and Wang [24] for details. When c is 0, this is a special case of the second Poincaré duality theorem [17] in KK-theory. 39 Chapter 7 Local Index Formula for Projective Spectral Triples In this section we present a local index formula associated to the projective spectral triple for every closed oriented Riemannian manifold M of dimension 2n. Let A = C ∞ (M ). Denote by ς = (B, H, D, γ) = (AW3 , HW3 , DW3 , γW3 ) the projective spectral triple of M defined in the preceding sections. Suppose a K-class [p] or [E] in K0 (B) is represented by a projection matrix p = (pij ) ∈ Mm (B) or by a right B-module E = pB m . Let DE denote the twisted Dirac operator on HE = E ⊗B H = pHm associated to the projective universal connection ∇E : E → E ⊗B Ω1u (B) on E, namely ∇E (pb) = pδu (pb) and DE (ph) = pD(ph), ∀b ∈ B m , ∀h ∈ Hm . E E E the . Denote by D± The left B-module H = H+ ⊕ H− is Z2 -graded and so is HE = H+ ⊕ H− E E restrictions of DE to H± → H∓ . The index of DE is E E Ind(DE ) = dim ker D+ − dim ker D− . Using the well-known local index formula (cf.[3]), we have Ind(DE ) = Z M Â(M )ch(HE /S). 40 Â(M ) is the Â-genus of the manifold M , Â(M ) = det1/2  R/2 sinh(R/2)  ∈ Ωev (M ). The relative Chern character ch(HE /S) is explained as follows. We consider H, and HE as well, as right Clifford modules with right Clifford actions cR . The connection ∇ : H → H ⊗A Ω1 (M ) on H induced by the Levi-Civita connection on M is a right Clifford connection. We can define a E E right Clifford connection ∇H : HE → HE ⊗A Ω1 (M ) on HE by ∇H (ph) = p∇(ph). Denote by E E RH ∈ EndA (HE ) ⊗A Ω2 (M ) the curvature of the connection ∇H , E E E RH = ∇H ∇H = p(∇p)(∇p) + p∇2 ◦ p, E and denote by T the twisting curvature, that is T = RH − RS , where RS = cR (R) = 1 Rijkl cR (el )cR (ek )ei ∧ ej , 4 and Rijkl are the components of the Riemannian curvature tensor on M under an orthonormal frame {ei }. One can verify that T = p(∇p)(∇p) − pcL (R)p. With the above notations, the relative Chern character is ch(HE /S) = 2−n tr exp(−T ). So we have an explicit local index formula E −n Z Ind(D ) = 2 Â(M )tr exp(−p(∇p)(∇p) + pcL (R)p). M From the viewpoint of noncommutative geometry, Ind(DE ) =< [p], [ς] >=< ch[p], ch[ς] >, (7.1) 41 where ch[p] ∈ HP0 (B) and ch[ς] ∈ HP 0 (B) are the periodic Connes-Chern characters of [p] and [ς] respectively. On the other hand, in terms of twisted Chern characters, as defined below, chW3 [ς] := (Chkr∗ )−1 (ch[ς]) ∈ Hev (M, C), chW3 [p] := Chkr(ch[p]) ∈ H ev (M, C), (7.2) the index pairing can be written as Ind(DE ) =< [p], [ς] >=< chW3 [p], chW3 [ς] > . We now try to give local expressions of ch[p], ch[ς], chW3 [p], and chW3 [ς] as well as their relation (7.2) explicitly. The periodic Connes-Chern character ch[p] is represented by a sequence of cyclic cycles {chλ0 (p), chλ2 (p), ...}, where chλ2m (p) = (−1)m (2m)! λ tr(p⊗2m+1 ) ∈ C2m (B). m! This sequence satisfies the periodicity condition S(chλ2m+2 (p)) = chλ2m (p). An alternate way to represent ch[p] is to use normalized ([, b)-cycles, that is ([,b) ch2m (p) = (−1)m 1 (2m)! tr((p − ) ⊗ p⊗2m ). m! 2 As for the Connes-Chern character of ς, one can apply Connes-Moscovici [11] local index formula to get a normalized ([, b)-cocycle. However, when trying to derive from Connes-Moscovici’s formula an expression in terms of integrals of differential forms on M , one will be confronted with a very much involved calculation of Wodzicki residues of various pseudo-differential operators. On the other hand, based on the appearance of formula (7.1), one can get a Cλ -cocycle chλ (ς) = P m follows: Let T (b1 , b2 ) = (∇b1 )(∇b2 ) − b1 cL (R)b2 , and define ρ02m : B ⊗2m+1 → Ω2m (M ) by ρ02m (b0 , ..., b2m ) = tr(b0 T (b1 , b2 ) · · · T (b2m−1 , b2m )). ch2m λ (ς) as 42 1 Then the relative Chern character ch(HE /S) = 2n (2m)! ρ02m (chλ2m [p]). 1 It is easily seen that 2n (2m)! R M Â(M )ρ02m (b0 , ..., b2m ) is a Hochschild cocycle but not cyclic cocycle if m ≥ 2. By applying Theorem 2.5, we know that ρ2m (b0 , ..., b2m ) = tr(b0 ψ2m (b1 , ..., b2m )) (2.4) is a cyclic cocycle, and that ρ2m (chλ2m (p)) = ρ02m (chλ2m (p)) for all p with [p] ∈ K0 (B). Thus by Theorem 2.5 and the duality theorem (Thm. 6.1), we have the following conclusions: Theorem 7.1. The cyclic cocycle chλ (ς) = P m ch2m λ (ς)(b0 , ..., b2m ) = 1 n 2 (2m)! ch2m λ (ς), where Z Â(M )ρ2m (b0 , ..., b2m ), ∀bi ∈ B, M represents the Connes-Chern character ch[ς] of the projective spectral triple ς. Theorem 7.2. The Connes-Chern character and the twisted Chern character are related by ch[ς] = chW3 [ς] ◦ X ρ2m , and chW3 [p] = X m ρ2m (chλ2m [p]) m as identical periodic cyclic cohomology classes and de Rham cohomology classes respectively. Corollary 7.3. The twisted Chern characters of [p] and [ς] can be represented by chW3 [p] = 2n (deg)! ch(HE /S), respectively. chW3 [ς] = 1 2n (deg)! 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