Projective Dirac Operators, Twisted K-Theory and Local Index Formula
aa r X i v : . [ m a t h . DG ] N ov Projective Dirac operators, twisted K-theory, andlocal index formula ∗ Dapeng Zhang † California Institute of Technology, MC 253-37, Pasadena, CA 91125, USAE-mail: [email protected]
Abstract:
We construct a canonical noncommutative spectral triple for everyoriented closed Riemannian manifold, which represents the fundamental class inthe twisted K-homology of the manifold. This so-called “projective spectral triple”is Morita equivalent to the well-known commutative spin spectral triple providedthat the manifold is spin-c. We give an explicit local formula for the twisted Cherncharacter for K-theories twisted with torsion classes, and with this formula we showthat the twisted Chern character of the projective spectral triple is identical tothe Poincar´e dual of the A-hat genus of the manifold.
Mathematics Subject Classification (2010). 19K56, 19L50, 58J20.
Keywords.
Twisted K-theory, spectral triple, Chern character.
Introduction
The notion of spectral triple in Connes’ noncommutative geometry arises fromextracting essential data from the K-homology part of index theory in differentialgeometry. The following are basic examples of commutative spectral triples:i) The spin spectral triple for a spin c manifold M with a spinor bundle S : ς = ( C ∞ ( M ) , Γ( S ) , /D, ω ) , where ω is the grading operator on S .(Throughout this paper, unless otherwise stated explicitly, all vector spaces,algebras, differential forms, and vector bundles except cotangent bundles areconsidered over the field C of complex numbers. For example, the notation C ∞ ( M ) is the same as C ∞ ( M, C ). The notation Γ( M, E ) or Γ( E ) for a fibrebundle E over M always stands for the space of smooth sections of E .)The identity between the analytic and topological indices of /D is the Atiyah-Singer index formula for the spin c manifold. ∗ This article is the author’s dissertation in publication form. † Supported in part by International Max-Planck Research School (IMPRS). M : ς = ( C ∞ ( M ) , Ω( M ) , d + d ∗ , ∗ ( − deg(deg − − dim M ) . (For each function f : N → C , we denote by f (deg) : Ω( M ) → Ω( M ) thelinear operator given by f (deg) ω = f ( k ) ω, ∀ ω ∈ Ω k ( M ) . )The index formula corresponding to this spectral triple is the Hirzebruchsignature formula.iii) The spectral triple for the Euler characteristic for a Riemannian manifold M : ς = ( C ∞ ( M ) , Ω( M ) , d + d ∗ , ( − deg ) . The local index formula corresponding to this spectral triple is the Gauss-Bonnet-Chern theorem.In fact, every special case of Atiyah-Singer Index theorem corresponds to an in-stance of commutative spectral triple (with additional structures when necessary).These spectral triples, like ς , ς , ς , have many nice properties such as “the fiveconditions” in Connes [8], and conversely, it is proved that [8] any commutativespectral triple ( A , H , D, γ ) satisfying those five conditions is equivalent to a spec-tral 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 operatoron it. Furthermore, if ( A , H , D, γ ) satisfies an additional important property –the Poincar´e duality in K-theory – which means ( A , H , D, γ ) represents the fun-damental class (i.e., a K-orientation) in K ( A ), then it is equivalent to a spinspectral triple ς for some spin c manifold. The spectral triples ς and ς do nothave the property of Poincar´e duality; however, we show in this paper (Corollary5.4, Theorem 7.1) that for every closed oriented Riemannian manifold there is acanonical noncommutative spectral triple having the property of Poincar´e dual-ity in K ( M, W ( M )), the twisted K-theory of M with local coefficient W ( M )- the third integral Stiefel-Whitney class. This canonical spectral triple is calledthe projective spectral triple on M , and its center is unitarily equivalent to ς .The projective spectral triple is Morita equivalent to the spin spectral triple pro-vided that the underlying manifold is spin c . On the other hand, in the paper ofMathai-Melrose-Singer [20], a so-called projective spin Dirac operator was definedfor every Riemannian manifold; however, this operator is in a formal sense. Itturns out that the projective spectral triple, in which the Dirac operator is reallyan operator acting on a Hilbert space, just plays the role of the projective spinDirac operator.A spectral triple that gives rise to Poincar´e duality in KK-theory first appearedin Kasparov [17]. Kasparov’s fundamental class, namely the Dirac element inDefinition-Lemma 4.2 in [17], is essentially a spectral triple, although there was nosuch terminology at that time. The algebra underlying Kasparov’s spectral tripleis noncommutative and Z -graded, but in many situations things would become2uch less complicated if the algebra were ungraded, especially when considering itsDixmier-Douady class or passing it to cyclic cohomology class via Connes-Cherncharacter. The projective spectral triple constructed in this paper (Corollary 5.4)has a noncommutative but ungraded algebra, and we show in Section 6 that itis in fact Morita equivalent to that of Kasparov’s. To construct such a spectraltriple, we first review in Section 3 the definition of spectral triples with certainsmoothness property, then introduce in Section 4 the notion of Morita equivalencebetween them, and then find in Section 5 that local spin spectral triples on smallopen subsets of a manifold can be glued together, via Morita equivalence, to forma globally defined spectral triple.The noncommutative algebras underlying projective spectral triples are exam-ples of Azumaya algebras. In Section 1 we review some basic theory on Azumayaalgebras, such as the fact that Morita equivalent classes of Azumaya algebras areclassified by their Dixmier-Douady classes, and that the K-theory of an Azumayaalgebra A coincides with the twisted K-theory of the underlying manifold withthe Dixmier-Douady class of A .Mathai-Stevenson [21] showed that the K-theory (tensoring with C ) of an Azu-maya algebra A is isomorphic to the periodic cyclic homology group of A viaConnes-Chern character, and that the latter is isomorphic to the twisted de-Rhamcohomology of the underlying manifold with the Dixmier-Douady class of A via ageneralized Connes-Hochschild-Kostant-Rosenberg (CHKR) map. K ( M, δ ( A )) ⊗ C ch ∼ = / / ch δ ( A ) ∼ = ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘ HP ( A ) Chkr ∼ = w w ♦♦♦♦♦♦♦♦♦♦♦ H evdR ( M, δ ( A ))In Section 2, we find an alternative CHKR map (Theorem 2.12) for the specialcase that the Dixmier-Douady class of A is torsion.For each 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 also hasa Connes-Chern character as a cyclic cohomology class, and the index pairing ofa K-cocycle and a K-cycle is identical to the index pairing of their Connes-Cherncharacters [6, 7]. The main purpose of this paper is to compute the Connes-Cherncharacter of the projective spectral triple and identify it with the Poincar´e dual ofthe A-hat genus of the underlying manifold.In Section 8, with the help of the alternative CHKR map ρ and by applyingPoincar´e duality, we obtain our main result, a local formula for the Connes-Cherncharacter of the projective spectral triple for every even dimensional oriented closedRiemannian manifold. Acknowledgements.
I am very grateful to Bai-Ling Wang. He foresaw the pos-sibility that the projective spin Dirac operator defined by [20] in formal sense canbe realized by a certain spectral triple, and introduced his interesting researchproject to me in 2008. The spectral triple in his mind turned out to be the pro-jective spectral triple constructed in this paper. Without his insight, I wouldn’t3ave been writing this thesis. I would like to thank Adam Rennie for his veryhelpful remarks about Morita equivalence between various presentations of Kas-parov’s fundamental class on reading a draft of this paper. I also wish to thankmy advisor, Matilde Marcolli, for her many years of encouragement, support, andmany helpful suggestions on both this research and other aspects.
Suppose X is a closed oriented manifold. LetM n = (cid:26) M n ( C ) , n = 1 , , . . . K( H ) , n = ∞ , U n = (cid:26) U( n ) , n = 1 , , . . . U( H ) , n = ∞ , where n could be either a positive integer or infinity, H is an infinite dimensionalseparable Hilbert space, K( H ) is the C ∗ -algebra of compact operators on H , andU( H ) is the topological group of unitary operators with the operator norm topol-ogy. Kuiper’s theorem states that U( H ) is contractible. Let PU n = U n / U(1) bethe projective unitary groups. In particular PU( H ) = PU ∞ is endowed with thetopology induced from the norm topology of U( H ).Let Aut(M n ) be the group of automorphisms of the C ∗ -algebra M n . Fact 1.
For every element g ∈ Aut(M n ) , there exists ˜ g ∈ U n , such that g = Ad˜ g .For every u ∈ U n , Ad u = 1 if and only if u is scalar. In other words, as groups PU n ∼ = Aut(M n ) . Fact 2. If n is finite, U n / U(1) ∼ = SU( n ) / { z ∈ C | z n = 1 } . Definition 1.1. An Azumaya bundle over X of rank n (possibly n = ∞ ) is avector bundle over X with fibre M n and structure group PU n .Every Azumaya bundle of rank n is associated with a principal PU n -bundleand vice versa. Definition 1.2.
The space A = Γ ( A ) of continuous sections of an Azumayabundle 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 ( X );ii) C ( X ) ⊗ M n ;iii) if E is a finite rank vector bundle over X , the algebra of continuous sectionsof End( E ), Γ (End( E ));iv) if X is an even dimensional Riemannian manifold, the algebra of continuoussections of the Clifford bundle C l ( T ∗ X ), Γ ( C l ( T ∗ X ));4) 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 C l ( E ), Γ ( C l ( E )).Note that examples i), ii) and iii) are Morita equivalent (in the category of C ∗ -algebras, i.e., strongly Morita equivalent [25, 24]) to C ( X ), while example iv) orv) is Morita equivalent to C ( X ) if and only if X or E is spin c respectively. Fact 3.
The center of a finite Azumaya algebra over X is C ( X ) . Fact 4.
An Azumaya algebra A over X is locally Morita equivalent to C ( X ) . The obstruction to an Azumaya algebra being (globally) Morita equivalent toits “center” is characterized by its Dixmier-Douady class:
Definition 1.3.
For every Azumaya bundle π : A → X of rank n , there is a coho-mology class δ ( A ) in H ( X, Z ), called the Dixmier-Douady class of A , constructedas follows:Let { U i } i ∈ I be a good covering of X , and write U i ··· i n for the intersection of U i , U i , · · · , U i n . Suppose ψ i : U i × M n → π − ( U i ) , ∀ i ∈ I, provide a local trivialization of A . Then ψ − i ψ j : U ij × M n → U ij × M n give rise tothe transition functions g ij ∈ C ( U ij , Aut(M n )). Pick ˜ g ij ∈ C ( U ij , U n ) such thatAd˜ g ij = g ij and ˜ g ij = ˜ g − ji . Thus Ad(˜ g ij ˜ g jk ˜ g ki ) = g ij g jk g ki = 1, which implies µ ijk := ˜ g ij ˜ g jk ˜ g ki ∈ C ( U ijk , U(1)) . Therefore µ is a ˇCech 2-cocycle with coefficient sheaf U (1) : U C ( U, U(1)),since ( ∂µ ) ijkl = µ jkl µ − ikl µ ijl µ − ijk = 1. The ˇCech 2-cocycle µ is also called the bundle gerbe structure of A . The short exact sequence of sheaves0 −→ Z −→ R exp 2 πi · −−−−−→ U (1) −→ , where R is the sheaf U C ( U, R ), induces an isomorphism of ˇCech cohomologygroups ∂ : ˇ H ( X, U (1)) ∼ = −→ ˇ H ( X, Z ) . Define the Dixmier-Douady class by δ ( A ) := ∂ [ µ ]. More explicitly, pick ν ijk ∈ C ( U ijk , R ) such that exp 2 πiν ijk = µ ijk . Then exp 2 πi ( ∂ν ) ijkl = ( ∂µ ) ijkl = 1, which implies ( ∂ν ) ijkl = ν jkl − ν ikl + ν ijl − ν ijk ∈ Z are locally constant integers on U ijkl . Thus δ ( A ) = [ ∂ν ] ∈ ˇ H ( X, Z ). Definition 1.4.
Suppose that A is an Azumaya bundle, that A is the Azumayaalgebra corresponding to A , and that P is the principal PU-bundle associatedto 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,5 roposition 1.5. For every cohomology class δ in H ( X, Z ) , there is a unique ( upto isomorphism ) infinite rank Azumaya bundle ( or algebra ) with Dixmier-Douadyclass δ . Proposition 1.6.
Let A be an Azumaya bundle. If δ ( A ) = 0 , then one can choose ˜ g ij so that ˜ g ij 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( E x ) for all x ∈ X . Corollary 1.7.
An Azumaya algebra A over X is Morita equivalent to C ( X ) ifand only if δ ( A ) = 0 . Corollary 1.8.
Two Azumaya algebras A and A over X are Morita equivalentif and only if δ ( A ) = δ ( A ) . Namely, Morita equivalence classes of Azumayaalgebras are parameterized by H ( 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 2 m -dimensional smooth manifold. The Clif-ford bundle C l ( T ∗ X ) is an Azumaya bundle of rank 2 m . Its Dixmier-Douadyclass δ ( C l ( T ∗ X )) = W ( X ) is the third integral Stiefel-Whitney class of X , and2 W ( X ) = 0. Proposition 1.10. If A and A are two Azumaya bundles over X , then δ ( A ⊗ A ) = δ ( A ) + δ ( A ) . Proposition 1.11. If A is an Azumaya algebra, then its opposite algebra A op isalso an Azumaya algebra and δ ( A op ) = − δ ( A ) . Let δ be a cohomology class in H ( X, Z ). Recall that (Rosenberg [26], Atiyah-Segal [1]) the twisted K-theory K ( X, δ ) can be defined by K ( X, δ ) = [ P → Fred( H )] PU( H ) , the abelian group of homotopy classes of maps P → Fred( H ) that are equivariantunder 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 Fredholmoperators on H . Twisted K-theory can also be defined with K-theory of a C ∗ -algebra: K ( X, δ ) = K ( 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 -group by K ( X, δ ) = K ( A ). Theabove two definitions of twisted K-theory are equivalent (Rosenberg [26]). We willalways use the second definition in this paper.6 roposition 1.12. The direct sum of twisted K-groups of X M δ ∈ H ( X, Z ) K • ( X, δ ) forms a Z × H ( X, Z ) -bigraded ring. The product K i ( X, δ ) × K j ( X, δ ) → K i + j ( X, δ + δ ) is naturally defined. Definition 1.13.
Let c ∈ Ω ( X ) be a closed 3-form, the twisted de Rham complex is the following periodic sequence d c −→ Ω ev ( X ) d c −→ Ω odd ( X ) d c −→ , where d c ω = dω + c ∧ ω . The twisted de Rham cohomology is H ∗ dR ( X, c ) = H ∗ (Ω ∗ ( X ) , d c ). Proposition 1.14. If c is a closed -form, then H ∗ dR ( X, c ) ∼ = H ∗ dR ( X, zc ) asisomorphic vector spaces for all nonzero z ∈ C . In particular, H ∗ dR ( X, c ) ∼ = H ∗ dR ( X, − c ) as vector spaces. In fact, in someliteratures such as [21], the twisted coboundary d c ω of ω is defined by dω − c ∧ ω . Proposition 1.15.
If a closed -form c = c + dβ for some β ∈ Ω ( X ) , then −−−−→ Ω ev ( X ) d c −−−−→ Ω odd ( X ) −−−−→ y ∧ exp β y ∧ exp β −−−−→ Ω ev ( X ) d c −−−−→ Ω odd ( X ) −−−−→ is a chain isomorphism. Therefore H ∗ dR ( X, c ) ∼ = H ∗ dR ( X, c ) as vector spaces. 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 allthe transition functions for the vector bundle A are smooth functions valued inthe (Banach) Lie group PU n . Let A be the space of smooth trace class sectionsof A , then A is a Fr´echet pre-C ∗ -algebra densely embedded in A = Γ ( A ). Inparticular, if the rank n of A is finite, then A = Γ( A ).Given a PU n -connection ∇ : Ω k ( M, A ) → Ω k +1 ( M, A ) on A , the image ofthe Dixmier-Douady class δ ( A ) in H ( M, R ) can be represented by a differential3-form in terms of the connection and curvature (e.g., Freed [13]) as follows:Let { U i } be a good open covering of M , and ψ i : U i × M n → A | U i bea local trivialization compatible with the smooth structure on A . Denote by g ji ∈ C ∞ ( U ij , PU n ) the transition function corresponding to ψ − j ψ i . Pick ˜ g ji ∈ ∞ ( U ij , U n ) so that Ad˜ g ji = g ji and ˜ g ji = ˜ g − ij . Let θ i be the local connectionforms of ∇ on U i , ∇ ( ψ i ( O )) = ψ i ( dO + θ i ( O )) , ∀ O ∈ C ∞ ( U i , M n ) . Then θ i = g − ji θ j g ji + g − ji dg ji . Pick ˜ θ i ∈ Ω ( U i , M n ) if n = ∞ , or pick ˜ θ i ∈ Ω ( U i , B( H )) if n = ∞ , so that ˜ θ i is skew-Hermitian and θ i = ad˜ θ i . Thus˜ θ i = ˜ g − ji ˜ θ j ˜ g ji + ˜ g − ji d ˜ g ji + α ij , for some scalar valued 1-form α ij ∈ Ω ( U ij ). Let ω i be the local curvature formsof Ω = ∇ : Γ( A ) → Ω ( X, A ) on U i ,Ω( ψ i ( O )) = ψ i ( ω i ( O )) , ∀ O ∈ C ∞ ( U i , M n ) . So ω i = dθ i + θ i ∧ θ i , and ω i = g − ji ω j g ji . Let ˜ ω i = d ˜ θ i + ˜ θ i ∧ ˜ θ i , then ad˜ ω i = ω i ,and ˜ ω i = ˜ g − ji ˜ ω j ˜ g ji + dα ij . Let ˜Ω i = ψ i ˜ ω i ψ − i , then ˜Ω i = ˜Ω j + dα ij . Since dα ij + dα jk + dα ki = 0, dα forms a 2-form valued cocycle, and since the sheaf of2-forms is fine (or because of the existence of partition of unity on M ), there exist β i ∈ Ω ( U i ) so that 2 πi ( β i − β j ) = dα ij . We can define a generalized 2-form by˜Ω i − πiβ i on U i , and it is globally well-defined. Lemma 2.1. If A is an Azumaya bundle over M with connection ∇ and curvature Ω , then − πi ∇ ( ˜Ω i − πiβ i ) represents the image of the Dixmier-Douady class δ ( A ) in H ( M ) .Proof. First recall that the ˇCech-de Rham isomorphism between the third de Rhamcohomology H ( M ) and ˇCech cohomology ˇ H ( M, C ) with constant coefficientsheaf C can be constructed as follows. For any closed 3-form τ ∈ Ω ( M ), one canfind β ( τ ) i ∈ Ω ( U i ) so that dβ ( τ ) i = τ | U i . Since dβ ( τ ) i − dβ ( τ ) j = 0 one canfind α ( τ ) ij ∈ Ω ( U ij ) so that β ( τ ) i − β ( τ ) j = dα ( τ ) ij . Since dα ( τ ) ij + dα ( τ ) jk + dα ( τ ) ki = 0 one can find ν ( τ ) ijk ∈ C ∞ ( U ijk ) so that ( ∂α ( τ )) ijk = dν ( τ ) ijk on U ijk . Here ∂ denotes the coboundary operator on ˇCech cocycles. Likewise, since d ( ∂ν ( τ )) ijkl = ( ∂dν ( τ )) ijkl = 0, one can find δ ( τ ) ijkl ∈ C so that ( ∂ν ( τ )) ijkl = δ ( τ ) ijkl . The ˇCech-de Rham isomorphism H ( M ) → ˇ H ( M, C ) is given by thecorrespondence τ δ ( τ ).Now let τ = − πi ∇ ( ˜Ω i − πiβ i ). By the Bianchi identity − πi ∇ ( ˜Ω i − πiβ i ) = dβ i . Thus we can choose β ( τ ) i = β i , α ( τ ) ij = α ij , ν ( τ ) ijk = ν ijk , and δ ( τ ) ijkl = δ ijkl . Therefore it follows that τ represents the image of δ ( A ) in H ( M ). Theorem 2.2. . If A is a finite rank Azumaya bundle with connection ∇ and curvature Ω , then there is a unique traceless σ (Ω) ∈ Ω ( M, A ) such that ad σ (Ω) = Ω and ∇ ( σ (Ω)) = 0 . . Suppose that A is an infinite rank Azumaya bundle associated to a principal PU( H ) -bundle P with connection ∇ and curvature Ω , and that c ∈ Ω ( M ) isa differential form representing the image of the Dixmier-Douady class δ ( A ) in H ( M ) . Then up to a closed scalar -form, there is a unique Γ( P × PU( H ) B( H )) -valued -form σ (Ω) such that ad σ (Ω) = Ω and − ∇ ( σ (Ω))2 πi = c . Here PU( H ) actson B( H ) the same way as on K( H ) . roof. Up to a scalar valued 2-form, σ (Ω) can be defined by ˜Ω i − πiβ i as inthe above lemma. In fact, ad( ˜Ω i − πiβ i ) = ad ˜Ω i = Ω, and − πi ∇ ( ˜Ω i − πiβ i )represents the image of the Dixmier-Douady class δ ( A ) in H ( M ).1). If A is of rank m < ∞ , we just take σ (Ω) = ˜Ω i − πiβ i − tr( ˜Ω i − πiβ i ) /m .The scalar 3-form ∇ ( σ (Ω)) must be 0 since it is traceless. The uniqueness of σ (Ω)is obvious.2). Suppose A is of infinite rank. There exists a scalar 2-form η such that − πi ∇ ( ˜Ω i − πiβ i ) − c = dη. We can take σ (Ω) = ˜Ω i − πiβ i − πiη .Recall that if B is a pre-C ∗ -algebra densely embedded in a C ∗ -algebra B , then K ( B ) = K alg0 ( B ) is naturally isomorphic to K ( B ). If B is a unital Fr´echetalgebra, we define K ( B ) to be the abelian group of the equivalence classes ofGL ∞ ( B ). We say that u, v ∈ GL ∞ ( B ) are equivalent if there is a piecewise C -path in GL ∞ ( B ) joining u and v . The definition of K ( B ) can be extended to thecase of non-unital algebras so that the six-term exact sequence property 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 cyclichomologies and cohomologies. We denote the Hochschild boundary map by ♭ , anddenote the Connes boundary map by b . Definition 2.3.
Following Gorokhovsky [15], define two mapsChkr : M k even C red k ( A ) → Ω ev ( M ) and Chkr : M k odd C red k ( A ) → Ω odd ( M )by the JLO-type ([16]) formulaChkr( a , a , ..., a k ) = Z s ∈ ∆ k tr( a e − s σ (Ω) ( ∇ a ) e − s σ (Ω) · · · ( ∇ a k ) e − s k σ (Ω) ) d s . (1)Here C red0 ( A ) = A and C red j ( A ) = A + ˆ ⊗A ˆ ⊗ j , for all j = 0, with A + being theunitalization of A and ˆ ⊗ the projective tensor product of locally convex topologicalalgebras. Here and subsequently, if V = L i V i is a Z -graded vector space, we write V ev = L k V k and V odd = L k V k +1 .The generalized CHKR theorem of Mathai-Stevenson’s is as follows. Proposition 2.4 (Mathai-Stevenson [21]) . . The map Chkr in (1) induces aquasi-isomorphism between the two complexes ♭ −→ C redev ( A ) ♭ −→ C redodd ( A ) ♭ −→ and −→ Ω ev ( M ) −→ Ω odd ( M ) −→ , and hence isomorphisms HH ev ( A ) ∼ = Ω ev ( M ) and HH odd ( A ) ∼ = Ω odd ( M ) . . The map Chkr induces a quasi-isomorphism between the complex ♭ + b −−→ C redev ( A ) ♭ + b −−→ C redodd ( A ) ♭ + b −−→ , or equivalently b −→ HH ev ( A ) b −→ HH odd ( A ) b −→ nd the twisted de Rham complex d c −→ Ω ev ( M ) d c −→ Ω odd ( M ) d c −→ ; and hence an isomorphism Chkr : HP • ( A ) ∼ = −→ H • dR ( M, c ) , ( • = ev , odd) , (2) where c = − ∇ ( σ (Ω))2 πi is a representative of the image of δ ( A ) in H dR ( M ) . . The Connes-Chern character ch : K • ( A ) ⊗ C → HP • ( A ) and the twistedChern character ch δ ( A ) = Chkr ◦ ch : K • ( M, δ ( A )) ⊗ C → H • dR ( M, c ) are isomorphisms. If δ ( A ) is a torsion class, then there is an alternative formula for the map Chkr,which we will see, is closely related to the relative Chern character ([3]) of Cliffordmodules. From now on we take c = 0, then by Theorem 2.2, up to a closed 2-form,there is a unique σ (Ω) such that ad σ (Ω) = Ω and ∇ ( σ (Ω)) = 0.Define ψ k : A ˆ ⊗ k → A ˆ ⊗ C ∞ ( M ) Ω k ( M ) by letting ψ − = 0 , ψ = 1 , ψ ( a ) = ∇ a , ψ ( a , a ) = ( ∇ a )( ∇ a ) + a σ (Ω) a ,ψ k ( a , ..., a k ) = ( ∇ a ) ψ k − ( a , ..., a k ) + a σ (Ω) a ψ k − ( a , ..., a k ) , ∀ k ≥ . (3)In other words, ψ k ( a , ..., a k ) is obtained as follows: Consider all partitions π ofthe ordered set { a , ..., a k } into blocks, where each block contains either one ortwo elements. Assign to each block { a i } of π a term of the form ∇ a i , and to eachblock { a j , a j +1 } of π a term of the form a j σ (Ω) a j +1 . Then let ψ k,π be the productof these terms, and ψ k ( a , ..., a k ) be the sum of ψ k,π over all such partitions. So inits expansion, ψ k ( a , ..., a k ) consists of a Fibonacci number of summands. Thendefine ρ k : A ˆ ⊗ k +1 → Ω k ( M ) by ρ k ( a , ..., a k ) = 1 k ! tr( a ψ k ( a , ..., a k )) (4)for all k ≥
0. Note that ρ depends on the choice of connection ∇ and, for infiniterank Azumaya bundles, the choice of σ (Ω), so we will write the complete form ρ ∇ σ for ρ when we need to specify ∇ and σ . We have the following results about ρ . Lemma 2.5. ρ ◦ ♭ = 0 . Lemma 2.6. If δ ( A ) is a torsion class, then for all k ≥ and a i ∈ A , ( − k − ρ k ( a , . . . , a k ) + ρ k ( a k , a , . . . , a k − ) = d tr (cid:0) a ψ k − ( a , ...a k − ) a k (cid:1) . Proof.
Noticing that d ◦ tr = tr ◦ ∇ , it suffices to show( − k − a ψ k ( a , .., a k ) + ψ k ( a , ..., a k − ) a k = ∇ (cid:0) a ψ k − ( a , ..., a k − ) a k (cid:1) , (5)10or all a i ∈ A + . In fact, it is easy to see (5) is true for k = 0 , ,
2. Suppose (5)holds for all k ≤ m for some m . Then using ∇ = ad σ (Ω) and the Bianchi identity ∇ ( σ (Ω)) = 0, we have ∇ ( a ψ m ( a , ..., a m ) a m +1 )= ∇ a ψ m ( a , ..., a m ) a m +1 + a ∇ ψ m ( a , ..., a m ) a m +1 +( − m a ψ m ( a , ..., a m ) ∇ a m +1 = ∇ a ψ m ( a , ..., a m ) a m +1 + a ∇ (cid:0) ∇ a ψ m − ( a , ..., a m ) (cid:1) a m +1 + a ∇ (cid:0) a σ (Ω) a ψ m − ( a , ..., a m ) (cid:1) a m +1 + ( − m a ψ m ( a , ..., a m ) ∇ a m +1 = ψ m +1 ( a , ..., a m ) a m +1 − a a σ (Ω) ψ m − ( a , ..., a m ) a m +1 − a ∇ a ∇ ψ m − ( a , ..., a m ) a m +1 + a ∇ a σ (Ω) a ψ m − ( a , ..., a m ) a m +1 + a a σ (Ω) ∇ a ψ m − ( a , ..., a m ) a m +1 + a a σ (Ω) a ∇ ψ m − ( a , ..., a m ) a m +1 +( − m a ψ m ( a , ..., a m ) ∇ a m +1 = ψ m +1 ( a , ..., a m ) a m +1 − a a σ (Ω) ψ m − ( a , ..., a m ) a m +1 +( − m a ∇ a ψ m ( a , ..., a m , a m +1 − a ∇ a ψ m (1 , a , ..., a m ) a m +1 + a ∇ a σ (Ω) a ψ m − ( a , ..., a m ) a m +1 + a a σ (Ω) ∇ a ψ m − ( a , ..., a m ) a m +1 +( − m a a σ (Ω) a ψ m − ( a , ..., a m , a m +1 + a a σ (Ω) a ψ m − (1 , a , ..., a m ) a m +1 +( − m a ψ m ( a , ..., a m ) ∇ a m +1 = ψ m +1 ( a , ..., a m ) a m +1 − a a σ (Ω) ψ m − ( a , ..., a m ) a m +1 +( − m a ∇ a ψ m − ( a , ..., a m − ) a m σ (Ω) a m +1 + a a σ (Ω) ∇ a ψ m − ( a , ..., a m ) a m +1 +( − m a a σ (Ω) a ψ m − ( a , ..., a m − ) a m σ (Ω) a m +1 + a a σ (Ω) a σ (Ω) a ψ m − ( a , ..., a m ) a m +1 +( − m a ψ m ( a , ..., a m ) ∇ a m +1 = ψ m +1 ( a , ..., a m ) a m +1 + ( − m a ∇ a ψ m − ( a , ..., a m − ) a m σ (Ω) a m +1 +( − m a a σ (Ω) a ψ m − ( a , ..., a m − ) a m σ (Ω) a m +1 +( − m a ψ m ( a , ..., a m ) ∇ a m +1 = ψ m +1 ( a , ..., a m ) a m +1 + ( − m a ψ m +1 ( a , ..., a m +1 ) . Thus identity (5) is proved by induction.
Lemma 2.7. If A is unital, then ρ k +1 ◦ b k = d ◦ ρ k on A ˆ ⊗ k +1 .Proof. Recall that b = (1 − λ ) ◦ s ◦ N, where λ is the cyclic permutation λ ( a , ..., a n ) = ( − n ( a n , a , ..., a n − ) ,N is the sum of all cyclic permutations N k = 1 + λ + · · · + λ k , s k : C k ( A ) → C k +1 ( A ) is the extra degeneracy operator s ( a , ..., a k ) = (1 , a , ..., a k ) . Thus we have ρ ◦ b ( a , ..., a k ) = ρ ◦ (1 − λ ) ◦ k X i =0 ( − ki (1 , a i , ..., a i − )(By Lemma 2.6.) = 1( k + 1)! k X i =0 ( − ki ( − k d tr( ψ ( a i , ..., a i − ) a i − )= 1( k + 1) k X i =0 ( − k ( i − dρ ( a i − , ..., a i − )(By Lemma 2.6 again.) = d ◦ ρ ( a , ..., a k ) . Corollary 2.8. ρ k +1 ◦ b k = d ◦ ρ k on C red k ( A ) . We now take a look at what happens for formula (4) when we choose differentconnections and different σ ’s. Let ∇ t = ∇ + tθ be a family of PU n -connections onthe Azumaya bundle A , where t ∈ I = [0 , θ is a generalized 1-form. Denoteby Ω t the curvature of ∇ t . If A has finite rank, then the choice of σ is unique,and σ (Ω t ) is a smooth family of traceless generalized 2-forms. If A has infiniterank, then given σ (Ω ) and σ (Ω ) with ∇ i ( σ i (Ω i )) = 0 and ad σ i (Ω i ) = Ω i ,where i = 0 ,
1, we can always find a smooth family σ t (Ω t ) with ∇ t ( σ t (Ω t )) = 0and ad σ t (Ω t ) = Ω t . In fact, consider the projection e π : M × I → I and thepull-back bundle e π ∗ A = A × I over M × I with connection e ∇ = ∇ t + dt∂ t . Thecurvature on e π ∗ A is e Ω = Ω t + θdt . Since e π ∗ A is also an Azumaya bundle withtorsion Dixmier-Douady class, we can choose some σ ( e Ω) with e ∇ ( σ ( e Ω)) = 0 andad σ ( e Ω) = e Ω. Suppose σ ( e Ω) = Λ t + η t dt, where Λ t and η t are a smooth familyof generalized 2-forms and a smooth family of generalized 1-forms on M , thenadΛ t = Ω t , ad η t = θ , and ∇ t Λ t = 0. Therefore, by Theorem 2.2, σ (Ω ) − Λ and σ (Ω ) − Λ are closed scalar 2-forms. Then we can take a smooth family ofgeneralized 2-forms σ t (Ω t ) = Λ t + (1 − t )( σ (Ω ) − Λ ) + t ( σ (Ω ) − Λ ) . We have ∇ t ( σ t (Ω t )) = 0 and ad σ t (Ω t ) = Ω t .Consider the good open covering { U i } of M again. Suppose( σ (Ω ) − Λ ) − ( σ (Ω ) − Λ ) = dη i on U i for some scalar 1-form η i , and let e σ ( e Ω) = σ t (Ω t ) + η t dt + η i dt. Then we have e ∇ ( e σ ( e Ω)) = 0 and ad e σ ( e Ω) = e Ω.12ollowing Mathai-Stevenson [21], let K ( a , ..., a k ) = Z I dt ∧ ( ι ∂ t ρ e ∇ e σ ( a , ..., a k ))on U i . Here the functions a , ..., a k inside ρ e ∇ e σ () are considered as functions on U i × I which are constant in the t direction. We have the following lemma. Lemma 2.9.
The map K defined above is a chain homotopy – we have the formula ρ ∇ σ − ρ ∇ σ = K ◦ ( ♭ + b ) − d ◦ K on C red ∗ ( A| U i ) .Proof. By Lemma 2.5 and Corollary 2.8, K ◦ ( ♭ + b ) − d ◦ K = K ◦ b − d ◦ K = Z I dt ∧ ι ∂ t ◦ d ◦ ρ e ∇ e σ − d Z I dt ∧ ι ∂ t ◦ ρ e ∇ e σ = Z I dt ∧ ( ι ∂ t ◦ d + d ◦ ι ∂ t ) ◦ ρ e ∇ e σ = Z I dt ∧ ∂∂t ◦ ρ e ∇ e σ = ρ ∇ σ − ρ ∇ σ . Theorem 2.10. If δ ( A ) is a torsion class, then the map ρ k in (4) induces aquasi-isomorphism between the complexes ♭ + b −−→ C redev ( A ) ♭ + b −−→ C redodd ( A ) ♭ + b −−→ and d −→ Ω ev ( M ) d −→ Ω odd ( M ) d −→ , and its induced isomorphism ρ : HP • ( A ) ∼ = −→ H • dR ( M ) (6) coincides with the map Chkr in (2) .Proof. First, by Lemma 2.5 and Corollary 2.8, we see that ρ is a chain map.Then the homotopy formula in Lemma 2.9 and Mathai-Stevenson [21]’s spectralsequence argument on the ˇCech-de Rham bicomplex prove the theorem.Because the map ρ is degree-preserving, we have Corollary 2.11. If δ ( A ) is a torsion class, then the map ρ in (4) induces a chainisomorphism ( HH ∗ ( A ) , b ) → (Ω ∗ ( M ) , d ) . heorem 2.12. If δ ( A ) is a torsion class, then the map ρ in (4) induces a ho-momorphism ρ : C λ ∗ ( A ) → Ω ∗ ( M ) /d (Ω ∗− ( M )) , where C λ ∗ ( A ) is the Connes complex of A (cf. [6], [7]) , and an isomorphism ρ : HP • ( A ) ∼ = −→ H • dR ( M ) , which coincides with Chkr in (2) .Proof. First, by Lemma 2.6 we see that the induced homomorphism C λ ∗ ( A ) → Ω ∗ ( M ) /d (Ω ∗− ( M ))is well-defined.Next, we show that the induced map ρ : HP • ( A ) → H • dR ( M ) is well-defined.Note that ρ k ◦ ♭ = 0 on C k +1 ( A ). This means that the map C λ ∗ ( A ) /♭ ( C λ ∗ +1 ( A )) → Ω ∗ ( M ) /d (Ω ∗− ( M )) is well-defined. Then it suffices to show that the im-ages of HP ∗ ( A ) under the map ρ are represented by closed forms. We provethis only for the case of even degree, and the odd degree case is similar. Sincech : K ( A ) → HP ( A ) is an isomorphism, elements of HP ( A ) are generated bych[ p ] for [ p ] ∈ K ( A ) (here we may suppose A is unital for simplicity). The pe-riodic Connes-Chern character ch[ p ] is represented by a sequence of cyclic cycles { ch λ ( p ) , ch λ ( p ) , ... } , wherech λ m ( p ) = ( − m (2 m )! m ! tr( p ⊗ m +1 ) ∈ C λ m ( A ) . Observe that p ( ∇ p ) i +1 p = 0 for all idempotent p and i ≥
0, then ρ k (ch λ k ( p )) = ( − k (2 k )! k ! tr (cid:0) p ψ k ( p, ..., p ) (cid:1) = ( − k (2 k )! k ! tr (cid:0) pψ ( p, p ) k (cid:1) , because in the expansion of p ψ k ( p, ..., p ) p , any term that has a factor p ( ∇ p ) i +1 p vanishes. Since ∇ ( pψ ( p, p )) = 0, it follows that ρ (ch[ p ]) is a closed form for all[ p ] ∈ K ( A ).Finally, since the periodic cyclic homology of A defined from the C λ -complex isthe same as that defined from the ( ♭, b )-bicomplex, by Theorem 2.10, the inducedhomomorphism ρ : HP • ( A ) → H • dR ( M ) is an isomorphism identical to Chkr in(2). In this section we review the definition and some analytical properties of spectraltriples. 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 thispaper. In fact, in definition 3.6 we require that the second entry H of a spectraltriple ( A , H , D ) to be an A -module as well as the smooth Sobolev domain of D ,14nstead of the Hilbert space ¯ H , the norm completion of H . So in applicationin differential geometry, spectral triples defined this way operate directly withsmooth sections of vector bundles. For a spectral triple in the conventional sense,that would be a strong requirement, as strong as the smoothness condition inAppendix 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 µ > µ > · · · be the list of eigenvaluesof ( D + 1) − in decreasing order, and V i ⊂ H be the eigenspace corresponding to µ i for each i . Then every vector v ∈ H can be uniquely represented as a sequence( v , v , . . . ) with v i ∈ V i and P i k v i k < ∞ , and vice versa.For every s ≥
0, consider the following subspaces of H , W s ( D ) = { ( v , v , . . . ) ∈ H | X i µ − si k v i k < ∞} , with the norm k ( v , v , . . . ) k s = qP i µ − si k v i k ; W s,p ( D ) = { ( v , v , . . . ) ∈ H | X i µ − sp/ i k v i k p < ∞} , ∀ p > , with the norm k ( v , v , . . . ) k s,p = (cid:16)P i µ − sp/ i k v i k p (cid:17) /p ; and W s, ∞ ( D ) = { ( v , v , . . . ) ∈ H | sup i µ − s/ i k v i k < ∞} , with the norm k ( v , v , . . . ) k s, ∞ = sup i µ − s/ i k v i k . W s = W s, has a naturalHilbert space structure. Proposition 3.1 (Rellich) . For each ǫ > , the inclusion W s + ǫ → W s is compact. Proposition 3.2. W ⊂ H is the domain of the self-adjoint operator D , and D : W → H is a Fredholm operator. Let W ∞ = T s> W s , then W ∞ is a Fr´echet space with a family of norms k · k s .It is easy to see that restricted to W ∞ , the mapping D : W ∞ → W ∞ is continuouswith respect to the Fr´echet space topology.We say the operator D is finitely summable or has spectral dimension less than2 d (for some real number d > D + 1) − d is a trace class operator. Theorem 3.3.
Suppose D has finite spectral dimension. If T ∈ B ( H ) is a boundedoperator that maps W ∞ into W ∞ , then the restricted mapping T : W ∞ → W ∞ isalso continuous. This theorem can be proved by the following lemmas.
Lemma 3.4 (Sobolev embeddings) . If D has spectral dimension less than d , thenwe have the following obvious estimate: k v k s, ∞ ≤ k v k s,p ≤ (cid:18) X j µ dj (cid:19) /p k v k s + dp , ∞ , ∀ v ∈ H, ∀ s ≥ , ∀ p > , i.e., there are bounded embeddings W s + dp , ∞ ⊂ W s,p ⊂ W s, ∞ . emma 3.5. Suppose D has finite spectral dimension.i) Let T : W ∞ → W ∞ be a continuous operator. Suppose for each j , T (0 , . . . , , v j , , . . . ) = ( t j v j , t j v j , . . . ) , ∀ v j ∈ V j , where ( t ij ) is an infinite matrix with entries t ij ∈ Hom( V j , V i ) . Then ( t ij ) satisfies the property: for any s > there exist C and r > such that k X i µ − si t ij k < C + µ − rj , ∀ j. ii) Conversely any matrix ( t ij ) with entries t ij ∈ Hom( V j , V i ) satisfying theabove property represents a continuous operator T : W ∞ → W ∞ .Proof. i) For any v ∈ W ∞ , we see that as n → ∞ , P nj =0 v j → v in W s ,therefore P nj =0 v j → v in W ∞ . Because T is continuous, it follows that T ( P nj =0 v j ) = ( P nj =0 t j v j , P nj =0 t j v j , . . . ) → T v , hence (
T v ) i = P j t ij v j .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 i µ − si t ij ( n,C ) k > C + µ − nj ( n,C ) . Thus one may find an increasing sequence { j ( n ) } , such that k X i µ − si t ij ( n ) k > µ − nj ( n ) . For each j , pick u j ∈ V j so that k u j ( n ) k = k P i µ − si t ij ( n ) k − < µ nj ( n ) and k P i µ − si t ij ( n ) u j ( n ) k = 1, while u j = 0 if j = j ( n ) , ∀ n . Then, because ofthe finite spectral dimension, u = P j u j ∈ W ∞ , but T ( P nj =0 u j ) does notconverge in W s as n → ∞ , and this yields a contradiction.ii) Suppose the matrix ( t ij ) has that property. Define T : W ∞ → W ∞ by( T v ) i = X j t ij v j , ∀ v ∈ W ∞ . For any sequence u ( n ) ∈ W ∞ , we now prove that if u ( n ) → n → ∞ then T u ( n ) →
0. For any s > k T u ( n ) k s = k X i µ − si X j t ij u j ( n ) k ≤ X j k X i µ − si t ij u j ( n ) k≤ X j ( C + µ − rj ) k u j ( n ) k = C k u ( n ) k , + k u ( n ) k r, . Since u ( n ) → W ∞ implies k u ( n ) k s ′ → s ′ , using Lemma 3.4,it follows that k T u ( n ) k s → s . So this implies T u ( n ) →
0, andtherefore T : W ∞ → W ∞ is a continuous operator.16or any pre-Hilbert space H , we define the ∗ -algebra B ( H ) = { T ∈ B ( ¯ H ) | T ( H ) ⊂ H , T ∗ ( H ) ⊂ H} . Definition 3.6.
A triple ( A , H , D ) is said to be a unital spectral triple if it isgiven by a unital pre- C ∗ -algebra A , a pre-Hilbert space H with a norm-continuousunital ∗ -representation A → B ( H ), and a self-adjoint operator D on ¯ H 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 atopology finer than the norm topology of A , and the representation A × H → H isjointly continuous with respect to the locally convex topology of A and the Fr´echettopology of H .Note that if ( A , H , D ) is a unital spectral triple, then W ( 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 Z -grading on H : H = H + ⊕ H − , so that the grading operator γ = (cid:20) id H + − id H − (cid:21) : H → H commutes with all a ∈ A and anti-commutes with D . Spectral triples equippedwith no such gradings are said to be odd .Two spectral triples ( A , H , D ) and ( A , H , D ) with isomorphic algebras A ∼ = A are said to be unitarily equivalent if there is a unitary operator U : ¯ H → ¯ H intertwining the two representations of A i and the two Dirac operators D i inan obvious way. For the even case the unitary operator U also needs to be gradepreserving. In this section we introduce the notion of Morita equivalence of spectral triples. Let A be a pre- C ∗ -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 , 17ii) ( x, x ) ≥
0, and ( x, x ) = 0 only if x = 0.The norm on S is given by k x k = k ( x, x ) k , and its norm completion ¯ S (whichis automatically a pre-Hilbert right ¯ A -module) is called a Hilbert right ¯ A -module .Hilbert left modules can be defined in the same manner. In particular, everyHilbert space is a Hilbert C -module.If S is a pre-Hilbert right A -module, then its conjugate space S ∗ = { f x := ( x, · ) | x ∈ S} is a pre-Hilbert left A -module, and for all x, y ∈ S , a ∈ A , af x := a ( x, · ) = f x ( a ∗ ) , ( f x , f y ) := ( y, x ) , ( f x , af y ) = a ( f x , f y ) . If S is a pre-Hilbert right A -module, B A ( ¯ S ) denotes the ∗ -algebra of all modulehomomorphisms T : ¯ S → ¯ S for which there is an adjoint module homomorphism T ∗ : ¯ S → ¯ S with ( T x, y ) = ( x, T ∗ y ) for all x, y ∈ ¯ S . Define B A ( S ) = { T ∈ B A ( ¯ S ) | T ( S ) ⊂ S , T ∗ ( S ) ⊂ S} . In particular, if A = C , then B C ( S )= B ( S ). If A is unital and S is unital andfinitely generated, then B A ( S ) in fact consists of all A -endomorphisms of S , i.e., B A ( S ) = End A ( S ).Suppose A and B are unital pre- C ∗ -algebras. Let E be a unital finitely gen-erated projective right A -module, then as a summand of a free module, one canendow 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 → B A ( E )is unital, ∗ -preserving and norm-continuous. We also assume that E is endowedwith the topology induced from the locally convex topology of A , that B is a lo-cally convex topological ∗ -algebra with a topology finer than the norm topology,and that the representation B × E → E is jointly continuous with respect to thelocally convex topologies on B and E . We call such a B - A -bimodule with theabove structure a finite Kasparov B - A -module , then we introduce the followingdefinition. 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 . 18ote that by the notation ( ∇ ξ, ε ), which is a bit ambiguous, we actually mean( ε, ∇ ξ ) ∗ ∈ B ( H ).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 , H E , D E ) for B (cf. Connes [7, § VI.3]). The pre-Hilbert space H E is E ⊗ A H with inner product given by < ξ ⊗ A h , ξ ⊗ A h > = < h , ( ξ , ξ ) h >, ∀ ξ i ∈ E , h i ∈ H . The Dirac operator D E on H E is given by D E ( ξ ⊗ A h ) = ξ ⊗ A Dh + ( ∇ ξ ) h, ∀ ξ ∈ E , h ∈ W ( D ) . It is easy to see that the commutator [ D E , b ] = [ ∇ , b ] ∈ B ( E ⊗ A H ) is bounded.To verify that D E is self-adjoint with domain E ⊗ A W ( D ), that W ∞ ( D E ) = H E ,and that D E has compact resolvent, it is adequate to check choosing one particular σ -connection on E , because bounded perturbations do not affect these conclusions.Recall that a universal connection on a pre-Hilbert right A -module S is a linearmapping ∇ : S → S ⊗ A Ω u ( A ) that satisfies the Leibniz rule ∇ ( sa ) = ( ∇ s ) a + δ u a, and ∇ is said to be Hermitian if( s, ∇ ε ) − ( ∇ s, ε ) = δ u ( s, ε ) . Here Ω u ( A ) is the space of universal 1-forms of A , and involutions ( δ u a ) ∗ areset to be − δ u ( a ∗ ). Cuntz and Quillen [12] showed that only projective modulesadmit universal connections. Given a universal Hermitian connection ∇ on afinite Kasparov module E , by sending 1-forms δ u a to [ D, a ], one can associatewith the universal connection ∇ a σ -connection ∇ D on E for any spectral triple σ = ( A , H , D ).Let p ∈ M n ( A ) be a projection (i.e., a self-adjoint idempotent n × n ma-trix). Now we consider the right A -module p A n . There is a canonical univer-sal connection on p A n which is given by the matrix p diag { δ u , . . . , δ u } or simply ∇ ( p a ) = pδ u ( p a ) , ∀ a ∈ A n , and this connection is Hermitian. Definition 4.2.
A universal connection ∇ on a pre-Hilbert right A -module S issaid to be projectional if there is a unitary A -isomorphism φ : S → p A n for some n and some projection p such that ∇ = φ − ◦ pδ u ◦ 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 ∇ E and F is a finite Kasparov C - B -module admitting a universal connection ∇ F , then there is a twisted universalconnection ∇ F ◦ ∇ E defined in an obvious way on the finite Kasparov C - A -module19 ⊗ B E . Furthermore if ∇ E and ∇ F are projectional then so is ∇ F ◦ ∇ E . Thisgives rise to a category – the category of noncommutative differential spaces NDG defined as follows. Objects in
NDG consist of all unital pre- C ∗ algebras. If A is a unital pre- C ∗ algebra, we denote by X A the corresponding object in NDG .Morphisms from X A to X B in NDG are isomorphism classes of finite Kasparov B - A -modules with projectional universal connections. If E is an finite Kasparov B - A -module with a projective universal connection ∇ , we denote by ( E , ∇ ) thecorresponding morphism in Hom NDG ( X A , X B ).Denote by Sptr ( A ) the set of unitary equivalence classes of even spectral triplesfor A . The set Sptr ( A ) has an abelian monoid structure, the binary operation ofwhich is the direct sum operation for spectral triples with algebra A . Thus Sptr yields a functor from NDG to AbMonoid , the category of abelian monoids, givenby X A Sptr ( A ) and Sptr (( E , ∇ )) : σ σ E , ∀ σ ∈ Sptr ( A ) . Remark . It is not difficult to extend morphisms of
NDG to graded mod-ules with super-connections in the sense of Quillen [23]; however, in this paperexcept Section 6, we only focus on finite Kasparov modules with trivial grad-ings. Baaj-Julg [2] established the theory of unbounded Kasparov modules andshowed that every element in the bivariant KK-theory can be represented by anunbounded Kasparov module. It would be nice if morphisms of
NDG couldbe enlarged to unbounded (even) Kasparov modules ( E, D , Γ) with a gradingΓ and an appropriate universal connection ∇ , and thereby D E, D , Γ ( ξ ⊗ A h ) = D ξ ⊗ A h + Γ ξ ⊗ A Dh + Γ( ∇ ξ ) h . The notion of connection for bounded Kasparovmodules introduced by Connes-Skandalis was well-known, whereas theory of con-nection for unbounded Kasparov modules has been developed in a recent work byMesland [22]. Definition 4.4.
Two unital spectral triples σ = ( A , H , D ) and σ = ( A , H ,D ) are said to be Morita equivalent if A and A are Morita equivalent as algebrasand there is a finite Kasparov A - A -module E with a σ -connection, such that E is an equivalence bimodule and σ is unitarily equivalent to σ E .Notice that the definition of Morita equivalence for spectral triples here, us-ing σ -connections rather than the usual universal connections, will give rise to asymmetric relation: Theorem 4.5.
The Morita equivalence between spectral triples is an equivalencerelation.Proof.
Reflexivity: ( A , H , D ) is Morita equivalent to itself via the trivial connec-tion ∇ a = [ D, a ] on A .Transitivity: It is straight forward by definition.Symmetry: Suppose ( B , H E , D E ) is Morita equivalent to σ = ( A , H , D ) via a σ -connection ∇ on E . Let F = E ∗ . Define ∇ F : F → F ⊗ B B ( H E ) by( ∇ F f )( ξ ⊗ A h ) = − f ( ∇ ξ ) ⊗ A h + [ D, f ξ ] h, ∀ f ∈ F , ξ ∈ E , h ∈ H . H E → H to represent an element in F ⊗ B B ( H E ). One canverify that ∇ F is a ( B , H E , D E )-connection. Now we check ( D E ) F = D as follows,( D E ) F ( f ⊗ B ξ ⊗ A h ) = f ⊗ B D E ( 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 perturba-tions of the σ -connection and σ E -connection that are associated to projectionaluniversal connections.)In conclusion, Morita equivalence of spectral triples is an equivalence relation. Remark . There are at least three possible spaces of “connections” for spectraltriples:i) The space of σ -connections.ii) The space of , i.e., connections induced from universalconnections, which is a subspace of type i).iii) The space of connections induced from projectional universal connections,which is a subspace of type ii).There are several reasons not restricting ourselves to the 1-form connections whentrying to define Morita relation between spectral triples:i) The Morita relation defined using 1-form connections is not symmetric.ii) For the Z -graded cases (see Section 6), one need to use super-connectionswhich are in general high order form connections [3].iii) The Morita relation defined using 1-form connections does not include allbounded perturbations of Dirac operators.As a special case, A is Morita equivalent to itself via E = A . Using univer-sal connections on A , one can construct new spectral triples which are calledinner fluctuations of spectral triples [9]. Morita equivalence of spectral tripleswith common algebra A and pre-Hilbert space H is exactly bounded perturbationof Dirac operators. In fact, ( A , H , D ) is Morita equivalent to ( A , H , D + B ) forany bounded self-adjoint operator B (or bounded odd self-adjoint operator B inthe even case) through the bimodule A and the ( A , H , D )-connection ∇ given by ∇ (1) := B . Conversely, ( A , H , D + B ) is Morita equivalent to ( A , H , D ) throughthe ( A , H , D + B )-connection ∇ ∗ given by ∇ ∗ (1) = − B . This is not the case forinner fluctuation of spectral triples, as the latter is generally not an equivalencerelation when A is noncommutative. In general, if we confine ourselves to equiva-lence bimodules with universal connections, the symmetry property in the aboveproof does not hold. However, if we restrict the universal connections on equiv-alence bimodules to the projectional ones, the symmetry property holds again.Unfortunately, Levi-Civita connections are not projectional connections.21uppose ( E , ∇ ) is a morphism in Hom NDG ( X A , X B ), then there is a free right A -module A m and a projection p in M m ( A ) such that E ∼ = p A m . Let { e , ..., e m } be the standard generators of A m . For each b in B , one can find a matrix α ( b ) in p M m ( A ) p , such that bpe i = P j e j α ji ( b ).The K-theory, Hochschild (co)homology and (periodic) cyclic (co)homologyare all functors on the category NDG . For instance, suppose ( b , ..., b n ) is aHochschild n -cycle representing an element Φ ∈ HH ∗ ( B ), then E (Φ) ∈ HH ∗ ( A ) isthe Hochschild n -cycle given by the Dennis trace maptr( α ( b ) ⊗ · · · ⊗ α ( b n )) = X i ,...,i n ( α i i ( b ) , α i i ( b ) , ..., α i n i ( b n )) . Note that α depends on the choice of the isomorphism E ∼ = p A m ; however, it doesinduce the well-defined morphisms E : HH ∗ ( B ) → HH ∗ ( A ).Furthermore, the Connes-Chern characters [6] are natural transformations fromthe K-theory functor X A K ( A ) to the periodic cyclic homology functor X A HP ( P ), and from functors X A Sptr ( A ) and X A K ( ¯ A ) to the periodiccyclic cohomology functor X A HP ( A ). The naturalness of Connes-Cherncharacters is illustrated in the following commutative diagrams K ( B ) E −−−−→ K ( A ) ch y ch y HP ( B ) E −−−−→ HP ( A ) , Sptr ( A ) ( E , ∇ ) −−−−→ Sptr ( B ) ch y ch y HP ( A ) E −−−−→ HP ( B ) . Proposition 4.7.
The following diagrams K ( A ) × Sptr ( A ) Ind −−−−→ Z x E y ( E , ∇ ) (cid:13)(cid:13)(cid:13) K ( B ) × Sptr ( B ) Ind −−−−→ Z , HP ( A ) × HP ( A ) −−−−→ C x E y E (cid:13)(cid:13)(cid:13) HP ( B ) × HP ( B ) −−−−→ C , and K ( A ) × Sptr ( A ) Ind −−−−→ Z y ch y ch y HP ( A ) × HP ( A ) −−−−→ C , commute. We can apply the above theory to spectral triples on Riemannian manifolds. Bygluing local pieces of spectral triples via Morita equivalence, we construct a so22alled projective spectral triple, the Dirac operator of which was defined in aformal sense by Mathai-Melrose-Singer [20].Let X be a closed oriented Riemannian manifold of dimension n . Suppose X is spin or spin c . Let C l n denote the complex Clifford algebra of R n . In this paperwe use the following convention for the definition of Clifford algebras C l n := < u ∈ R n | uv + vu = − u, v ) , ∀ u, v ∈ R n > . Decomposed by the parity of the degree, C l n = C l n ⊕ C l n . Write B n = (cid:26) C l n ∼ = M m ( C ) C l n ∼ = M m ( C ) and B x = (cid:26) C l ( T ∗ x X ) C l ( T ∗ x X ) when n = 2 m is evenwhen n = 2 m + 1 is odd.(7)Denote by C l ( X ) and B ( X ) the vector bundles over X whose fibers at a point x ∈ X are C l ( T ∗ x X ) and B x . Let S n = C m be the standard spinor vector space,and we fix a specific isomorphism c : B n → End C ( S n ) . Let ω n = i [ n +12 ] e · · · e n ∈ C l n . When n is odd, B n = ω n C l n . This indicates ahomomorphism c : C l n → End C ( S n ) . When n is even, S n = S n + ⊕ S n − is gradedby the eigenspaces of ω n , which are invariant under the action of Spin( n ).Let P Fr ( X ) and P Spin( c ) ( X ) denote the orthonormal oriented frame bundle andthe principal Spin( n ) or Spin c ( n )-bundle over X respectively. Then C l ( X ) = P Fr ( X ) × SO( n ) C l n = P Spin( c ) ( X ) × Ad C l n . The spinor bundle over X is the associated Spin( c )( n )-bundle S X = P Spin( c ) ( X ) × c S n . The Clifford algebra bundle acts naturally on the spinor bundle, c : C l ( X ) × X S X → S X , which is given by( p, ξ ) × ( p, s ) ( p, c ( ξ ) s ) , ∀ p ∈ P Spin( c ) ( X ) , ξ ∈ C l n , s ∈ S n . When n is even, ω n induces a grading operator ω on S X = S X + ⊕ S X − .Denote by /D the Dirac operator on S X . Let E be a Hermitian vector bundleover M with a Hermitian connection ∇ E . Let A = C ∞ ( X ) , B = Γ( B ( X )), E = Γ( E ). Then the well-known spin spectral triple( A , H , D ) = ( C ∞ ( X ) , Γ( S X ) , /D ) (8)is Morita equivalent to the following spectral triple with a noncommutative algebra( B , H E , D E ) = (Γ(End( E )) , Γ( E ⊗ S X ) , /D E ) , (9)via the finite Kasparov module E with ( A , H , D )-connection associated with ∇ E .Here /D E denotes the twisted Dirac operator on the vector bundle E ⊗ S X .Now interesting things happen when a manifold has no spin c structure: Thespinor bundle does not exist, and neither does the spin spectral triple. However,as constructed later in this section, for any closed oriented Riemannian manifold,not necessarily spin c , there is a canonical noncommutative spectral triple:23 efinition 5.1. The projective spectral triple of a closed oriented Riemannianmanifold M is defined to be( A W , H W , D W ) := (Γ( B ( M )) , (1 + ∗ )Ω( M ) , ( d − d ∗ )( − deg ) , (10)if M is odd dimensional, and( A W , H W , D W , γ W ) := (Γ( B ( M )) , Ω( M ) , ( d − d ∗ )( − deg , ∗ ( − deg(deg +1)2 − n ) , (11)if M is even dimensional.We can consider that projective spectral triples are obtained by gluing localspin spectral triples in the following way:Let M be a closed oriented Riemannian manifold of dimension n , not neces-sarily spin c . Let { U i } be a good covering of M . Then on each local piece wehave the principal Spin( n )-bundle P i = P Spin ( U i ), the associated spinor bundle S i = S U i , the spin connection ∇ i , and the Dirac operator /D i . Over each intersec-tion U ij = U i ∩ U j = ∅ , there is up to ± ∈ Spin( n ) a unique homeomorphism ofprincipal bundles α ij : P i | U ij → P j | U ij , such that α ij ( p i g ) = α ij ( p i ) g, ∀ p i ∈ P i | U ij , g ∈ Spin( n ) . This homeomorphism induces up to ± α ij : S i | U ij → S j | U ij , α ij ( p, s ) = ( α ij ( p ) , s ) , ∀ p ∈ P i | U ij , s ∈ S n . For each triple overlap U ijk = U i ∩ U j ∩ U k = ∅ , write σ ijk = α ki ◦ α jk ◦ α ij | U ijk , then { σ ijk } represents a ˇCech cocycle in ˇ H ( M, Z ), and the corresponding sin-gular class w ( M ) ∈ H ( M, Z ) is the second Stiefel-Whitney class. Let L ij =Hom C l ( U ij ) ( S i | U ij , S j | U ij ) denote the line bundle of the Clifford module homomor-phisms of S i and S j over U ij , then α ij is the canonical section of L ij . { L ij } forms a gerbe of line bundles over M characterized by the Dixmier-Douady class δ ( B ( M )) = W ( M ) (the third integral Stiefel-Whitney class). We refer [4] for(twisted) K-theory of bundle gerbes.If M is spin, then for each U ij there is β ij ∈ Z such that { β ij α ij } satisfiesthe cocycle condition. That means the local spinor bundles S ij can be gluedtogether as a global spinor bundle via the Clifford module isomorphisms β ij α ij .The difference between two different choices of such collection { β ij } is a cocyclein ˇ H ( M, Z ) which parameterizes distinct spin structures on M .If M is spin c , then for each U ij there is β ij ∈ C ∞ ( U ij , U (1)) such that { β ij α ij } satisfies the cocycle condition. That means that the local spinor bundles S ij canbe glued together as a global spinor bundle via the Clifford module isomorphisms β ij α ij . The collection { β ij } also satisfies the cocycle condition, and the ˇCechcocycle { β ij } ∈ ˇ H ( M, U (1)) corresponds to the canonical line bundle L of a24pin c structure. The difference between two different choices of such collection { β ij } is a cocycle in ˇ H ( M, U (1)) which parameterizes distinct spinor bundles on M . Denote by C ∞ ( ¯ U i ) the space of smooth functions on U i that can be extendedto a smooth function on a small open neighborhood V i of ¯ U i , and likewise denoteby Γ( ¯ U i , · ) for extendable smooth sections. Take E i = Γ( ¯ U i , S i ), then the “localspin spectral triple” ( A i , H i , D i ) = ( C ∞ ( ¯ U i ) , Γ( ¯ U i , S i ) , /D i ) (12)is Morita equivalent to the local spectral triple( B i , H E i i , D E i i ) = (Γ( ¯ U i , B ( U i )) , Γ( ¯ U i , S i ⊗ S i ) , /D S i i ) . (13) Remark . We use here the triple (12) to formulate the spin structure of the opensubset U i , but it is by definition not a spectral triple, for the compact resolventcondition fails. However, it can naturally act on the relative K-theory for the pairof spaces ( V i , U i ) or ( Y, ι ( U i )) to get an index, where Y is any compact Riemannianspin manifold that admits an isometric embedding ι : V i → Y . By the excisionproperty, the choice of V i is irrelevant. In this sense the triple (12) represents arelative K-cycle. One may also think of the standard treatment using the nonunitalspectral triple ( C ∞ c ( U i ) , L ( U i , S i ) , /D i ) with the algebra of smooth functions withcompact support. See Gayral-Gracia-Bond´ıa-Iochum-Sch¨ucker-V´arilly [14] for aset of axioms for nonunital spectral triples which is proposed to set up the notionof noncompact noncommutative spin manifolds. This, however, will cause somesubtleties when considering Morita equivalence and the smoothness condition.Because the collection of maps { α ij ⊗ α ij } satisfy the cocycle condition, the vec-tor bundles S i ⊗ S i and Dirac operators D E i i can be glued together to form a vectorbundle N over M and a Dirac operator D N on N , so that ( B| U i , Γ( M, N ) | U i , D N | U i )are unitarily equivalent to ( B i , H E i i , D E i i ), where B = Γ( M, B ( M )). Thus we suc-ceed to construct a globally well-defined spectral triple ( B , Γ( N ) , D N ) 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 S ∗ n be the dual vector space of the standard spinor vector space S n .We endow S ∗ n with a left B n -module structure by γ ∗ ( b ) f x := f ¯ bx = ( x, ¯ b ∗ · ), for all x ∈ S n , b ∈ B n , where ¯ b is the complex conjugate of b , and b ∗ is the adjoint of b ∈ B n ∼ = M m ( C ). Since B n is a simple algebra, up to a scalar there is a unique B n -module isomorphism from S n to S ∗ n . We fix one specific unitary B n -moduleisomorphism T n : S n → S ∗ n . Then T n induces a Clifford module isomorphism ofbundles T : S i → S ∗ i , given by ( p, s ) ( p, T n s ), for all p ∈ P i , s ∈ S n . Let S ∗ i = P i × γ ∗ S ∗ n , and let α ∗ ij : S ∗ i → S ∗ j denote the Clifford module isomorphismgiven by α ∗ ij ( p, f )) = ( α ij p, f ), for all f ∈ S ∗ i . The mappings T on U i and U j are25ompatible, namely the diagram below commutes on U ij . S iα ij (cid:15) (cid:15) T / / S ∗ iα ∗ ij (cid:15) (cid:15) S j T / / S ∗ j Then one can glue the bundles S ∗ i ⊗ S i ∼ = End C ( S ∗ i ) together as a global bundle B ( M ) via the maps α ∗ ij ⊗ α ij , and T induces an isomorphism from N to B ( M ).Then it is easy to verify the proposition. Corollary 5.4.
When M is odd dimensional, one can take the bundle N = (1 + ∗ ) V ∗ ( T ∗ C M ) , then the spectral triple ( B , Γ( N ) , D N ) is the projective spectral triple ( A W , H W , D W ) = ( B , (1 + ∗ )Ω( M ) , ( d − d ∗ )( − deg ); (14) and when M is even dimensional, take N = V ∗ ( T ∗ C M ) , then the spectral triple ( B , Γ( N ) , D N ) with the grading on Γ( N ) obtained from the grading on S i is theeven projective spectral triple ( A W , H W , D W , γ W ) = ( B , Ω( M ) , ( d − d ∗ )( − deg , ∗ ( − deg(deg +1) / − n/ ) , (15) and its center ( A , H W , D W , γ W ) is unitarily equivalent to the spectral triple forHirzebruch signature. For any a ∈ A , the commutator [ D W , a ] is just the rightClifford action of da on H W . Theorem 5.5. If M is spin c , the projective spectral triple on M is Morita equiv-alent to the spin spectral triple.Proof. Let S be the global spinor bundle over M . There exist local half line bundles L / i , such that L / i are characterized by { β ij } as local transition functions, L / i ⊗ L / i = L | U i , and S | U i = L / i ⊗ S i . There also exists a hermitianconnection ∇ ′ i on L / i such that the Spin c -connection ∇| U i = 1 ⊗ ∇ i + ∇ ′ i ⊗ ∇ i is the Spin-connection on S i . Then follows the Morita equivalence of thespectral triples( C ∞ ( M ) , Γ( M, S ) , /D ) = ( A , H , D ) ∼ ( A , H L − , D L − ) ∼ ( B , ( H L − ) S , ( D L − ) S ) = ( A W , H W , D W ) . We see that the projective spectral triple is defined for any closed orientedRiemannian manifold regardless of whether the manifold is spin c or not. Theprojective spectral triple depends only on the metric and orientation of M anddoes not depend on the choice of the local spinor bundles S i .26 Kasparov’s spectral triple
In Introduction we mentioned that Kasparov’s fundamental class, the Dirac ele-ment [17], is essentially an even spectral triple with a Z -graded underlying alge-bra, and claimed that the projective spectral triple is in fact Morita equivalent toKasparov’s spectral triple. In this section we see in details how these two spectraltriples are related by Morita equivalence.It is very easy to make sense of even spectral triples with Z -graded algebrasand Morita equivalence between them. Replacing all the commutator relationsfor even spectral triples by graded commutator relations, the definition of evenspectral triples with Z -graded algebras will be the same as that of even spec-tral triples with ungraded algebras. Similarly, we can generalize σ -connectionsto σ -supper-connections. To be compatible with graded commutator relations, σ -supper-connections are required to be odd operators. Two Z -graded algebras A and A are Morita equivalent through a Z -graded A - A -bimodule E withgrading γ , if A and A as ungraded algebras are Morita equivalent through E as an ungraded module and A ∼ = End A ( E ) as Z -graded algebras. Suppose σ = ( A , H , D , γ ) is an even spectral triple with Z -graded algebra A , E is a Z -graded A - A -bimodule with grading γ , and ∇ is a σ -super-connection on E ,we define D E ( ξ ⊗ A h ) = γ ξ ⊗ A D h + γ ( ∇ ξ ) h, ∀ h ∈ H , ξ ∈ E ,γ E = γ ⊗ A γ . Then we can straightforwardly extend the notion of Morita equivalence of spectraltriples to the case of even spectral triples with Z -graded algebras.Recall that the projective spectral triple for an even dimensional closed Rie-mannian manifold M is an even spectral triple( A , H , D , γ ) = ( B , Ω( M ) , ( d − d ∗ )( − deg , ∗ ( − deg(deg +1) / − dim M/ ) , where B is the algebra of smooth sections of the Clifford bundle of M with trivialgrading.We define a variant of Kasparov’s spectral triple to be the even spectral triplewith a Z -graded algebra( A , H , D , γ ) = ( B gr , Ω( M ) , √− d − d ∗ ) , ( − deg ) , where B gr is the algebra of smooth sections of the Clifford bundle of M graded bythe degree of its elements. Proposition 6.1. ( A , H , D , γ ) and ( A , H , D , γ ) are Morita equivalent.Proof. First note that ( A , H , D , γ ) is unitarily equivalent to ( A , H , D ′ , γ ),where D ′ = √− D γ . Then the Morita equivalence between ( A , H , D , γ )and ( A , H , D , γ ) will be given by a graded A - A -bimodule E with grading γ , together with a super-connection b ∇ on E , such that( A , H , D , γ ) = (End A ( E ) , E ⊗ A H , D ′E , γ ⊗ A γ ) . E , b ∇ ) is as follows.Denote by c L and c R the left and right Clifford action of B on Ω( M ) respec-tively. B is a complex algebra generated by Γ( M, T ∗ R M ). For any α ∈ Γ( M, T ∗ R M ),and ω ∈ Ω k ( M ), we have c L ( α ) ω = α ∧ ω − ι ( α ) ω, (16) c R ( α ) ω = ( − k ( α ∧ ω + ι ( α ) ω ) , (17)where ι is the contraction determined by the Riemannian metric g . Note that c L ( α ) c R ( β ) = c R ( β ) c L ( α ) and that ∗ ( a ∧ ω ) = ( − k ι ( α )( ∗ ω ) , (18) ∗ ( ι ( α ) ω ) = ( − k +1 α ∧ ( ∗ ω ) . (19)Let E := Ω( M ) be the B gr - B -bimodule with actions of B gr and B being theleft and right Clifford action respectively. Now we consider three different gradings γ , γ and γ on Ω( M ). We define γ ( ω ) := ( − k ( k − / − n/ ∗ ω, ∀ ω ∈ Ω k ( M ) , where n = dim M is an even number. One can easily check that γ = γ = γ =id . By (16)(17)(18)(19) we have c L ( α ) γ ( ω ) = γ ( c L ( α ) ω ) , (20) c R ( α ) γ ( ω ) = γ ( c R ( α ) ω ) . (21)We consider Ω( M ) ∼ = −→ E ⊗ B H via ω ω ⊗ B , and this is an isomorphism ofleft B -modules. By (20)(21), γ ⊗ B γ is well-defined. Hence( γ ⊗ B γ ) ω = γ ( ω ) ⊗ B γ (1)= ( − k ( k − / − n/ ( ∗ ω ) ⊗ B ( e ∧ · · · ∧ e n )= ( − k ( k − / − n/ ( ∗ ω ) ⊗ B c L ( e ) · · · c L ( e n )1= ( − k ( k − / − n/ c R ( e n ) · · · c R ( e )( ∗ ω ) ⊗ B
1= ( − k ( k − / − n/ ( − ( n − k )( n − k +1) / ( ∗ ∗ ω ) ⊗ B
1= ( − k ω ⊗ B γ ( ω ) . Again by (16)(17)(18)(19) we have c L ( α ) γ ( ω ) = − γ ( c L ( α ) ω ) , so the grading γ on E is compatible with the grading on B gr , and hence B gr =End B ( E ) as Z -graded algebras.Let ∇ be the connection on Ω( M ) induced by the Levi-Civita connection on T ∗ R M . Note that if ∇ ( ω ) = X i ν i ⊗ µ i , µ i ∈ Ω k ( M ) , ν i ∈ Ω ( M ), then dω = P i ν i ∧ µ i , d ∗ ω = − P i ι ( ν i ) µ i ,therefore ( d + d ∗ ) ω = X i c L ( ν i ) µ i , ( d − d ∗ ) ω = ( − k X i c R ( ν i ) µ i , namely d + d ∗ = c L ◦ ∇ ,D = ( − k ( d − d ∗ ) = c R ◦ ∇ . Define b ∇ : E
7→ E ⊗ B B ( H ) by( b ∇ ( ω ))( h ) = √− X i γ ( µ i ) ⊗ B c R ( ν i ) h, that is b ∇ ( ω ) = √− c R ◦ ∇ ( γ ( ω )) . So b ∇ is a ( A , H , D ′ , γ )-connection, and we have D ′E ( ω ) = D ′E ( ω ⊗ B
1) = ( γ ⊗ B id)( b ∇ ( ω ))(1) + γ ( ω ) ⊗ B D ′ (1)= √− γ ⊗ B id) X i γ ( µ i ) ⊗ B c R ( ν i )1 + 0= √− − k X i c R ( ν i ) µ i = √− d − d ∗ ) ω. So D = D ′E .Therefore we have the Morita equivalence( A , H , D , γ ) = (End A ( E ) , E ⊗ A H , D ′E , γ ⊗ A γ ) ∼ ( A , H , D , γ ) . Now let D = d + d ∗ and γ = γ = ( − deg . Note that for even dimensionalmanifolds, ∗ − √− d − d ∗ ) ∗ = ( d + d ∗ )( − deg +1 / = √− D γ . If we write γ = (cid:20) id 00 − id (cid:21) and D = (cid:20) TT ∗ (cid:21) , then ∗ − D ∗ = √− D γ = (cid:20) √− (cid:21) D (cid:20) −√− (cid:21) , so we have D = (cid:20) −√− (cid:21) ∗ − D ∗ (cid:20) √− (cid:21) . (22)29et C gr be a Z -graded complex algebra generated by Γ( M, T ∗ R M ) with gener-ator relations as follows. C gr := < u ∈ Γ( M, T ∗ R M ) | uv + uv = 2 g ( u, v ) , ∀ u, v ∈ Γ( M, T ∗ R M ) > . Although C gr is isomorphic to B gr through the map B gr → C gr : u
7→ √− u, ∀ u ∈ Γ( M, T ∗ R M ) , we consider a different representation on Ω( M ). Let c ′ L be the representation of C gr on Ω( M ) given by c ′ L ( √− u ) = (cid:20) −√− (cid:21) ∗ − c L ( u ) ∗ (cid:20) √− (cid:21) , ∀ u ∈ Γ( M, T ∗ R M ) . (23)Then c ′ L ( u )( ω ) = ( − k − ∗ − c L ( u ) ∗ ω, ∀ u ∈ Γ( M, T ∗ R M ) , ∀ ω ∈ Ω k ( M ) , and therefore c ′ L ( u ) = ( − deg c R ( u ) , ∀ u ∈ Γ( M, T ∗ R M ) . From Kasparov [17], the spectral triple representing Kasparov’s fundamental class is supposed to be( A , H , D , γ ) = ( C gr , Ω( M ) , d + d ∗ , ( − deg ) , which, because of (22)(23), is unitarily equivalent to ( A , H , D , γ ). Thus wehave the following theorem. Theorem 6.2.
The projective spectral triple ( A , H , D , γ ) is Morita equivalentto Kasparov’s spectral triple ( A , H , D , γ ) . For a recent account of Kasparov’s fundamental class on noncommutative Rie-mannian manifolds, we refer the reader to Lord-Rennie-Varilly [19]. K ( M, W ) In this section we see how projective spectral triples represent the fundamentalclasses in the twisted K-homology K ( A W ) ∼ = K ( M, W ).Denote by B gr the Z -graded algebra of sections of Clifford bundle C l ( T ∗ M )over M (even dimensional only), then every Clifford module E over M can beconsidered as a finitely generated projective right B opgr -module, and a Cliffordconnection ∇ E on E gives rise to a Dirac operator D E on E . Then E Ind D E defines a canonical homomorphism K ( B opgr ) Ind −−→ Z . By Morita equivalence, K ( B opgr ) can be replaced by the K-theory of an ungradedalgebra, K ( A W ), and the homomorphism Ind becomes the operation of pairingwith the projective spectral triple. 30 heorem 7.1 (Poincar´e duality) . For an even dimensional closed oriented man-ifold M , the projective spectral triple ς = ( A W , H W , D W , γ W ) represents thetwisted K-orientation as a cycle of the twisted K-homology K ( A W ) ∼ = K ( M, W ) ,and hence gives rise to the Poincar´e duality K ( M, W − c ) ⌢ [ ς ] −−−→ ∼ = K ( M, c ) , or K • ( M, c ) × K • ( M, W − c ) nondegenerate −−−−−−−−−→ pairing Z , for all c ∈ H ( M, Z ) . Here the cap product can be defined by [ E ] ⌢ [ ς ] = [ ς E ] forany finite Kasparov C ∞ ( M ) - A W -module E .For odd dimensional M , the Poincar´e duality can be formulated as K • ( M, c ) × K • +1 ( M, W − c ) nondegenerate −−−−−−−−−→ pairing Z , ∀ c ∈ H ( M, Z ) . See Kasparov [17], Carey-Wang [5], and Wang [27] for details. When c is 0,this is a special case of the second Poincar´e duality theorem [17] in KK-theory. In this section we present a local index formula associated to the projective spectraltriple for every closed oriented Riemannian manifold M of dimension 2 n . Let A = C ∞ ( M ). Denote by ς = ( B , H , D, γ ) = ( A W , H W , D W , γ W )the projective spectral triple of M defined in the preceding sections. Suppose a K-class [ p ] or [ E ] in K ( B ) is represented by a projection matrix p = ( p ij ) ∈ M m ( B )or by a right B -module E = p B m respectively. Let D E denote the twisted Diracoperator on H E = E ⊗ B H = p H m associated to the projective universal connection ∇ E : E → E ⊗ B Ω u ( B ) on E , namely ∇ E ( p b ) = pδ u ( p b ) and D E ( p h ) = pD ( p h ), ∀ b ∈ B m , ∀ h ∈ H m .The left B -module H = H + ⊕ H − is Z -graded and so is H E = H E + ⊕ H E− .Denote by D E± the restrictions of D E to H E± → H E∓ . The index of D E isInd( D E ) = dim ker D E + − dim ker D E− . Using the well-known local index formula (cf. [3]), we haveInd( D E ) = Z M ˆ A ( M )ch( H E / S ) . ˆ A ( M ) is the ˆ A -genus of the manifold M ,ˆ A ( M ) = det / (cid:18) R/ R/ (cid:19) ∈ Ω ev ( M ) . The relative Chern character ch( H E / S ) is explained as follows. We consider H and H E as right Clifford modules with right Clifford actions c R . The connection31 : H → H ⊗ A Ω ( M ) on H induced by the Levi-Civita connection on M is a rightClifford connection. We can define a right Clifford connection ∇ H E : H E → H E ⊗ A Ω ( M ) on H E by ∇ H E ( p h ) = p ∇ ( p h ). Denote by R H E ∈ End A ( H E ) ⊗ A Ω ( M )the curvature of the connection ∇ H E , R H E = ∇ H E ∇ H E = p ( ∇ p )( ∇ p ) + p ∇ ◦ p, and denote by T the twisting curvature, that is T = R H E − R S , where R S = c R ( R ) = 14 R ijkl c R ( e l ) c R ( e k ) e i ∧ e j , and R ijkl are the components of the Riemannian curvature tensor on M under anorthonormal frame { e i } . One can verify that T = p ( ∇ p )( ∇ p ) − pc L ( R ) p . Withthe above notations, the relative Chern character isch( H E / S ) = 2 − n tr exp( − T ) . So we have an explicit local index formulaInd( D E ) = 2 − n Z M ˆ A ( M )tr exp( − p ( ∇ p )( ∇ p ) + pc L ( R ) p ) . (24)From the viewpoint of noncommutative geometry,Ind( D E ) = < [ p ] , [ ς ] > = < ch[ p ] , ch[ ς ] >, where ch[ p ] ∈ HP ( B ) and ch[ ς ] ∈ HP ( B ) are the periodic Connes-Chern char-acters of [ p ] and [ ς ] respectively. On the other hand, in terms of twisted Cherncharacters, as defined below,ch W [ p ] := Chkr(ch[ p ]) ∈ H ev ( M, C ) , ch W [ ς ] := (Chkr ∗ ) − (ch[ ς ]) ∈ H ev ( M, C ) , (25)the index pairing can be written asInd( D E ) = < [ p ] , [ ς ] > = < ch W [ p ] , ch W [ ς ] > . We now try to give local expressions of ch[ p ], ch[ ς ], ch W [ p ], and ch W [ ς ] as wellas their relation (25) explicitly. The periodic Connes-Chern character ch[ p ] isrepresented by a sequence of cyclic cycles { ch λ ( p ) , ch λ ( p ) , ... } , wherech λ m ( p ) = ( − m (2 m )! m ! tr( p ⊗ m +1 ) ∈ C λ m ( B ) . This sequence satisfies the periodicity condition S (ch λ m +2 ( p )) = ch λ m ( p ). Analternative way to represent ch[ p ] is to use normalized ( ♭, b )-cycles, that isch ( ♭, b )2 m ( p ) = ( − m (2 m )! m ! tr(( p −
12 ) ⊗ p ⊗ m ) .
32s for the Connes-Chern character of ς , one can apply the Connes-Moscovici[11] local index formula to get a normalized ( ♭, b )-cocycle. However, when tryingto derive from Connes-Moscovici’s formula an expression in terms of integralsof differential forms on M , one will be confronted with a very much involvedcalculation of Wodzicki residues of various pseudo-differential operators. On theother hand, based on the appearance of formula (24), one can get a C λ -cocyclech λ ( ς ) = P m ch mλ ( ς ) as follows:Let T ( b , b ) = ( ∇ b )( ∇ b ) − b c L ( R ) b , and define ρ m : B ⊗ m +1 → Ω m ( M )by ρ m ( b , ..., b m ) = 1(2 m )! tr( b T ( b , b ) · · · T ( b m − , b m )) . Then the relative Chern character ch( H E / S ) = 2 − n ρ m (ch λ m [ p ]).It is easily seen that 2 − n R M ˆ A ( M ) ρ m ( b , ..., b m ) is a Hochschild cocycle butnot cyclic cocycle if m ≥
2. By applying Theorem 2.12, we know that ρ m ( b , ..., b m ) = 1(2 m )! tr( b ψ m ( b , ..., b m )) (4)is a cyclic cocycle, and that ρ m (ch λ m ( p )) = ρ m (ch λ m ( p )) for all p with [ p ] ∈ K ( B ). Thus by Theorem 2.12 and the duality theorem (Thm. 7.1), we have thefollowing conclusions: Theorem 8.1.
The cyclic cocycle ch λ ( ς ) = P m ch mλ ( ς ) , where ch mλ ( ς )( b , ..., b m ) = 2 − n Z M ˆ A ( M ) ρ m ( b , ..., b m ) , ∀ b i ∈ B , represents the Connes-Chern character ch[ ς ] of the projective spectral triple ς . Theorem 8.2.
The Connes-Chern character and the twisted Chern character arerelated by ch[ ς ] = ch W [ ς ] ◦ X m ρ m and ch W [ p ] = X m ρ m (ch λ m [ p ]) as identical periodic cyclic cohomology classes and de Rham cohomology classesrespectively. Corollary 8.3.
The twisted Chern characters of [ p ] and [ ς ] can be represented by ch W [ p ] = 2 n ch( H E / S ) and ch W [ ς ] = 2 − n [ ˆ A ( M )] ⌢ [ M ] respectively. References [1] M. Atiyah and G. Segal, Twisted K-theory.
Ukr. Math. Bull. (2004), 291-334. 332] S. Baaj and P. Julg, Th´eorie bivariante de Kasparov et op´erateurs non born´esdans les C ∗ -modules hilbertiens. C. R. Acad. Sci. Paris Sr. I Math. (1983), 875-878.[3] N. Berline, E. Getzler and M. Vergne,
Heat kernels and Dirac operators .Springer-Verlag, Berlin 2004.[4] P. Bouwknegt, A.L. Carey, V. Mathai, M.K. Murray and D. Stevenson,Twisted K-theory and K-theory of bundle gerbes.
Comm. Math. Phys. (2002), 17-45.[5] A.L. Carey and B-L. Wang, Thom isomorphism and push-forward map intwisted K-theory.
J. K-theory (2008), 357-393.[6] A. Connes, Non-commutative differential geometry. Inst. Hautes ´Etudes Sci.Publ. Math. (1985), 257-360.[7] A. Connes, Noncommutative geometry . Academic Press, San Diego, CA 1994.[8] A. Connes, On the spectral characterization of manifolds. arXiv:0810.2088.[9] A. Connes and A. Chamseddine, Inner fluctuations of the spectral action.
J.Geom.Phys. (2006), 1-21.[10] A. Connes and M. Marcolli, Noncommutative geometry, quantum fields andmotives . American Mathematical Society, Providence, RI; Hindustan BookAgency, New Delhi 2008.[11] A. Connes and H. Moscovici, The local index formula in noncommutativegeometry.
Geom. Funct. Anal. (1995), 174-243.[12] J. Cuntz and D. Quillen, Algebra exensions and nonsingularity. J. Amer.Math. Soc. (1995), 251-289.[13] D.S. Freed, Twisted K-theory and loop groups. Proceedings of the Interna-tional Congress of Mathematicians, Vol. III (Beijing 2002) . Higher Ed. Press,Beijing 2002, pp 419-430.[14] V. Gayral, J.Gracia-Bond´ıa, B. Iochum, T. Sch¨ucker and J. V´arilly, Moyalplanes are spectral triples.
Comm. Math. Phys. (2004), 569–623.[15] A. Gorokhovsky, Explicit formulae for characteristic classes in noncommuta-tive geometry. Ph.D. thesis, Ohio State Universtity 1999.[16] A. Jaffe, A. Lesniewski and K. Osterwalder, Quantum K-theory, I. The Cherncharacter.
Commun. Math. Phys. (1998), 1-14.[17] G.G. Kasparov, Equivariant KK-theory and the Novikov conjecture.
Invent.Math. (1998), 147-201.[18] J-L. Loday, Cyclic Homology.
Springer-Verlag, Berlin 1998.3419] S. Lord, A. Rennie and J.C. Varilly, Riemannian manifolds in noncommuta-tive geometry. Preprint 2011. arXiv:1109.2196[20] V. Mathai, R.B. Melrose and I.M. Singer, Fractional analytic index.
J. Diff.Geom. (2006), 265-292.[21] V. Mathai and D. Stevenson, On a generalized Connes-Hochschild-Kostant-Rosenberg theorem. Adv. Math. (2006), 303-335.[22] B. Mesland, Bivariant K-theory of groupoids and the noncommutative ge-ometry of limit sets. Ph.D. thesis, Rheinische Friedrich-Wilhelms-Universit¨atBonn 2009.[23] D. Quillen, Superconnections and the Chern character.
Topology (1985),89-95.[24] I. Raeburn and D.P. Williams, Morita equivalence and continuous-trace C*-algebras.
American Mathematical Society, Providence, RI 1998.[25] M.A. Rieffel, Morita equivalence for C*-algebras and W*-algebras.
J. PureAppl. Algebra (1974), 51-96.[26] J. Rosenberg, Continuous trace C*-algebras from the bundle theoretical pointof view. J. Aust. Math. Soc. A (1989), 368-381.[27] B-L. Wang, Geometric cycles, index theory and twisted K-homology. J. Non-commut. Geom.2