Equivariant Algebraic Index Theorem
aa r X i v : . [ m a t h . K T ] J a n EQUIVARIANT ALGEBRAIC INDEX THEOREM
ALEXANDER GOROKHOVSKY, NIEK DE KLEIJN, RYSZARD NEST
Abstract.
We prove a Γ-equivariant version of the algebraic index theorem, where Γ isa discrete group of automorphisms of a formal deformation of a symplectic manifold. Theparticular cases of this result are the algebraic version of the transversal index theoremrelated to the theorem of A. Connes and H. Moscovici for hypoelliptic operators and theindex theorem for the extension of the algebra of pseudodifferential operators by a groupof diffeomorphisms of the underlying manifold due to A. Savin, B. Sternin, E. Schroheand D. Perrot.
Contents
1. Introduction 21.1. The main result. 41.2. Structure of the article 52. Algebraic Index Theorem 62.1. Deformed formal geometry 62.2. Fedosov connection and Gelfand-Fuks construction 92.3. Algebraic index theorem in Lie algebra cohomology 112.4. Algebraic index theorem 133. Equivariant Gelfand-Fuks map 144. Pairing with HC per • (cid:0) A ~ c ⋊ Γ (cid:1) Alexander Gorokhovsky was partially supported by an NSF grant. Niek de Kleijn and Ryszard Nest weresupported by the Danish National Research Foundation through the Centre for Symmetry and Deformation(DNRF92). Niek de Kleijn was also partially supported by the IAP “Dygest” of the Belgian Science Policy. Introduction
The term index theorems is usually used to describe the equality of, on one hand,analytic invariants of certain operators on smooth manifolds and, on the other hand, topo-logical/geometric invariants associated to their “symbols”. A convenient way of thinkingabout this kind of results is as follows.One starts with a C ∗ -algebra of operators A associated to some geometric situation anda K -homology cycle ( A, π, H, D ), where π : A → B ( H ) is a ∗ -representation of A on aHilbert space H and D is a Fredholm operator on H commuting with the image of π modulo compact operators K . The explicit choice of the operator D typically has somegeometric/analytic flavour, and, depending on the parity of the K -homology class, H canhave a Z / Z grading such that π is even and D is odd. Given such a (say even) cycle, anindex of a reduction of D by an idempotent in A ⊗ K defines a pairing of K -homology and K -theory, i. e. the group homomorphism(1) K ( C , A ) × K ( A, C ) −→ Z . One can think of this as a Chern character of D , ch ( D ) : K ∗ ( A ) −→ Z , and the goal is to compute it explicitly in terms of some topological data extracted fromthe construction of D .Some examples are as follows. A = C ( X ) , where X is a compact manifold and D is an elliptic pseudodifferential operatoracting between spaces of smooth sections of a pair of vector bundles on X .The number < ch ( D ) , [1] > is the Fredholm index of D, i. e. the integerInd ( D ) = dim(Ker( D )) − dim(Coker( D ))and the Atiyah–Singer index theorem identifies it with the evaluation of the ˆ A -genus of T ∗ X on the Chern character of the principal symbol of D . This is the situation analysedin the original papers of Atiyah and Singer, see [1]. A = C ∗ ( F ) , where F is a foliation of a smooth manifold and D is a transversally ellipticoperator on X . Suppose that a K ( A ) class is represented by a projection p ∈ A , where A is a subalgebraof A closed under holomorphic functional calculus, so that the inclusion A ⊂ A inducesan isomorphism on K-theory. For appropriately chosen A , the fact that D is transversallyelliptic implies that the operator pDp is Fredholm on the range of p and the index theoremidentifies the integer Ind ( pDp ) with a pairing of a certain cyclic cocycle on A with theChern character of p in the cyclic periodic complex of A . For a special class of hypo-ellipticoperators see f. ex. [6] Suppose again that X is a smooth manifold. The natural class of representatives of K -homology classes of C ( X ) given by operators of the form D = P γ ∈ Γ P γ π ( γ ) , where Γ is QUIVARIANT ALGEBRAIC INDEX THEOREM 3 a discrete group acting on L ( X ) by Fourier integral operators of order zero and P γ isa collection of pseudodifferential operators on X , all of them of the same (non-negative)order. The principal symbol σ Γ ( D ) of such a D is an element of the C ∗ -algebra C ( S ∗ X ) ⋊ max Γ,where S ∗ M is the cosphere bundle of M . Invertibility of σ Γ ( D ) implies that D is Fredholmand the index theorem in this case would express Ind Γ ( D ) in terms of some equivariantcohomology classes of M and an appropriate equivariant Chern character of σ Γ ( D ). Forthe case when Γ acts by diffeomorphisms of M , see [25, 19].The typical computation proceeds via a reduction of the class of operators D under con-sideration to an algebra of (complete) symbols, which can be thought of as a ”formaldeformation” A ~ . Let us spend a few lines on a sketch of the construction of A ~ in the casewhen the operators in question come from a finite linear combination of diffeomorphismsof a compact manifold X with coefficients in the algebra D X of differential operators on X . A special case is of course that of an elliptic differential operator on X . Example 1.1.
Let Γ be a subgroup of the group of diffeomorphisms of X viewed as adiscrete group. Γ acts naturally on D X . Let D • X be the filtration by degree of D X . Thenthe associated Rees algebra R = { ( a , a , . . . ) | a k ∈ D kX } with the product( a , a , . . . )( b , b , . . . ) = ( a b , a b + a b , . . . , X i + j = k a i b j , . . . )has the induced action of Γ. The shift ~ : ( a , a , . . . ) → (0 , a , a , . . . )makes R into an C J ~ K -module and R/ ~ R is naturally isomorphic to Q k P ol k ( T ∗ X ) where P ol k ( T ∗ X ) is the space of smooth, fiberwise polynomial functions of degree k on thecotangent bundle T ∗ X . A choice of an isomorphism of R with Q k P ol k ( T ∗ X ) J ~ K induces on Q k P ol k ( T ∗ X ) J ~ K an associative, ~ -bilinear product ⋆ , easily seen to extend to C ∞ ( T ∗ X ).Since Γ acts by automorphisms on R , it also acts on ( C ∞ ( T ∗ X ) J ~ K , ⋆ ).This is usually formalized in the following definition. Definition 1.2.
A formal deformation quantization of a symplectic manifold (
M, ω ) is anassociative C J ~ K -linear product ⋆ on C ∞ ( M ) J ~ K of the form f ⋆ g = f g + i ~ { f, g } + X k ≥ ~ k P k ( f, g );where { f, g } := ω ( I ω ( df ) , I ω ( dg )) is the canonical Poisson bracket induced by the symplec-tic structure, I ω is the isomorphism of T ∗ M and T M induced by ω , and the P k denotebidifferential operators. We will also require that f ⋆ ⋆ f = f for all f ∈ C ∞ ( M ) J ~ K .We will use A ~ ( M ) to denote the algebra ( C ∞ ( M ) J ~ K , ⋆ ). The ideal A ~ c ( M ) in A ~ ( M ), ALEXANDER GOROKHOVSKY, NIEK DE KLEIJN, RYSZARD NEST consisting of power series of the form P k ~ k f k , where f k are compactly supported, has aunique (up to a normalization) trace T r with values in C [ ~ − , ~ K (see f. ex. [10]).It is not difficult to see that the index computations (as in 1) reduce to the computationof the pairing of the trace (or some other cyclic cocycle) with the K -theory of the symbolalgebra, which, in the example above, is identified with a crossed product A ~ c ( M ) ⋊ Γ. Anexample of this reduction is given in [17].
Since the product in A ~ c ( M ) is local, the computation of the pairing of K -theory and cycliccohomology of A ~ c ( M ) reduces to a differential-geometric problem and the result is usuallycalled the “algebraic index theorem ”. Remark . Since cyclic periodic homology is invariant under (pro)nilpotent extensions,the result of the pairing depends only on the ~ = 0 part of the K -theory of A ~ c ( M ) ⋊ Γ. Inour example, the ~ = 0 part of the symbol algebra A ~ c ( M ) ⋊ Γ is just C ∞ c ( M ) ⋊ Γ, hencethe Chern character of D enters into the final result only through a class in the equivariantcohomology H ∗ Γ ( M ).1.1. The main result.
Suppose that Γ is a discrete group acting by continuous automor-phisms on a formal deformation A ~ ( M ) of a symplectic manifold M . Let A ~ ( M ) ⋊ Γ denotethe algebraic crossed product associated to the given action of Γ. For a non-homogeneousgroup cocycle ξ ∈ C k (Γ , C ), the formula below defines a cyclic k -cocycle T r ξ on A ~ c ( M ) ⋊ Γ.(2)
T r ξ ( a γ ⊗ . . . ⊗ a k γ k ) = δ e,γ γ ...γ k ξ ( γ , . . . , γ k ) T r ( a γ ( a ) . . . ( γ γ . . . γ k − )( a k )) . The action of Γ on A ~ ( M ) induces (modulo ~ ) an action of Γ on M by symplectomor-phisms. Let σ be the “principal symbol” map: A ~ ( M ) → A ~ ( M ) / ~ A ~ ( M ) ≃ C ∞ ( M ) . It induces a homomorphism σ : A ~ ( M ) ⋊ Γ −→ C ∞ ( M ) ⋊ Γ , still denoted by σ . Let Φ : H • Γ ( M ) −→ HC • per ( C ∞ c ( M ) ⋊ Γ)be the canonical map (first constructed by Connes) induced by (19), where H • Γ ( M ) denotesthe cohomology of the Borel construction M × Γ E Γ and C ∞ c ( M ) denotes the algebra ofcompactly supported smooth functions on M .The main result of this paper is the following. Theorem 1.4.
Let e , f ∈ M N (cid:0) A ~ ( M ) ⋊ Γ (cid:1) be a couple of idempotents such that thedifference e − f ∈ M N (cid:0) A ~ c ( M ) ⋊ Γ (cid:1) is compactly supported, here A ~ c ( M ) denotes the idealof compactly supported elements of A ~ ( M ) . Let [ ξ ] ∈ H k (Γ , C ) be a group cohomology class.Then [ e ] − [ f ] is an element of K ( A ~ c ( M ) ⋊ Γ) and its pairing with the cyclic cocycle T r ξ is given by (3) < T r ξ , [ e ] − [ f ] > = D Φ (cid:16) ˆ A Γ e θ Γ [ ξ ] (cid:17) , [ σ ( e )] − [ σ ( f )] E . QUIVARIANT ALGEBRAIC INDEX THEOREM 5
Here ˆ A Γ ∈ H • Γ ( M ) is the equivariant ˆ A -genus of M (defined in section 5), θ Γ ∈ H • Γ ( M ) isthe equivariant characteristic class of the deformation A ~ ( M ) (also defined in section 5). In the case when the action of Γ is free and proper, we recover the algebraic version ofConnes-Moscovici higher index theorem.The above theorem gives an algebraic version of the results of [25], without the require-ment that Γ acts by isometries. To recover the analytic version of the index theorem typeresults from [25] and[19] one can apply the methods of [17].1.2.
Structure of the article.Section 2 contains preliminary material, extracted mainly from [3] and [7]. It is includedfor the convenience of the reader and contains the following material. • Deformation quantization of symplectic manifolds and Gelfand–Fuks construction.
Following Fedosov, a deformation quantization of a symplectic manifold A ~ ( M ) can beseen as the space of flat sections of a flat connection ∇ F on the bundle W of Weyl algebrasover M constructed from the bundle of symplectic vector spaces T ∗ M → M . The fiber of W is isomorphic to the Weyl algebra g = W (see definition 2.3) and ∇ F is a connectionwith values in the Lie algebra of derivations of W , equivariant with respect to a maximalcompact subgroup K of the structure group of T ∗ M .Suppose that L is a ( g , K )-module. The Gelfand–Fuks construction provides a complex(Ω( M, L ) , ∇ F ) of L -valued differential forms with a differential ∇ F satisfying ∇ F = 0. Letus denote the corresponding spaces of cohomology classes by H ∗ ( M, L ). An example is theFedosov construction itself, in fact H k ( M, W ) = ( A ~ ( M ) k = 00 k = 0 . The Gelfand–Fuks construction also provides a morphism of complexes GF : C ∗ Lie ( g , K ; L )) −→ Ω( M, L ) • Algebraic index theorem
The Gelfand–Fuks map is used to reduce the algebraic index theorem for a deformationof M to its Lie algebra version involving only the ( g , K )-modules given by the periodiccyclic complexes of W and the commutative algebra O = C J x , . . . , x n , ξ , . . . , ξ n , ~ K . Infact, the following holds. Theorem 1.5.
Let L • = Hom − • ( CC per • ( W ) , ˆΩ − • [ ~ − , ~ K [ u − , u K [2 d ]) . There exist two ele-ments ˆ τ a and ˆ τ t in the hypercohomology group H Lie ( g , K ; L • ) such that the following holds. (1) Suppose that M = T ∗ X for a smooth compact manifold X and A ~ ( M ) is the de-formation coming from the calculus of differential operators. Then whenever p and q are two idempotent pseudodifferential operators with p − q smoothing, Z M GF (ˆ τ a )( σ ( p ) − σ ( q )) = T r ( p − q ) and Z M GF (ˆ τ t )( σ ( p ) − σ ( q )) = Z M ch ( p ) − ch ( q ) ALEXANDER GOROKHOVSKY, NIEK DE KLEIJN, RYSZARD NEST where
T r is the standard trace on the trace class operators on L ( M ) , p (resp. q )are the ~ = 0 components of p (resp. q ) and ch : K ( M ) → H ev ( M ) is the classicalChern character. (2) ˆ τ a = X p ≥ h ˆ A f e ˆ θ i p u p ˆ τ t , where h ˆ A f e ˆ θ i p is the component of degree p of a certain hypercohomology class.In the case of M = T ∗ X as above, GF (cid:16) ˆ A f e ˆ θ (cid:17) coincides with the ˆ A -genus of M . In general, given a group Γ acting on a deformation quantization algebra A ~ ( M ), there doesnot exist any invariant Fedosov connection. As a result, the Gelfand–Fuks map described insection 2 does not extend to this case. The rest of the paper is devoted to the constructionof a Gelfand–Fuks map that avoids this problem and the proof of the main theorem. Section 3 is devoted to a generalization of the Gelfand–Fuks construction to the equi-variant case, where an analogue of the Fedosov construction and Gelfand–Fuks map areconstructed on M × E Γ. Section 4 is devoted to a construction of a pairing of the periodic cyclic homology ofthe crossed product algebra with a certain Lie algebra cohomology appearing in Section 2.The main tool is for this construction is the Gelfand–Fuks maps from Section 3.
Section 5 contains the proof of the main result.The appendix is used to define and prove certain statements about the various coho-mology theories appearing in the main body of the paper. All the results and definitionsin the appendices are well-known and standard and are included for the convenience of thereader. 2.
Algebraic Index Theorem
Deformed formal geometry.
Let us start in this section by recalling the adapta-tion of the framework of Gelfand-Kazhdan’s formal geometry to deformation quantizationdescribed in [16, 18] and [3].
For the rest of this section we fix a symplectic manifold ( M, ω ) of dimension d and itsdeformation quantization A ~ ( M ) . Notation 2.1.
Let m ∈ M .(1) J ∞ m ( M ) denotes the space of ∞ -jets at m ∈ M ; J ∞ m ( M ) := lim ←− C ∞ ( M ) / ( I m ) k ,where I m is the ideal of smooth functions vanishing at m and k ∈ N .(2) Since the product in the algebra A ~ ( M ) is local, it defines an associative, C J ~ K -bilinear product ⋆ m on J ∞ m ( M ). \ A ~ ( M ) m denotes the algebra ( J ∞ m ( M ) J ~ K , ⋆ m ). Notation 2.2.
QUIVARIANT ALGEBRAIC INDEX THEOREM 7 (1) W will denote the algebra \ A ~ ( R d ) , where the deformation A ~ (cid:0) R d (cid:1) has the prod-uct given by the Moyal-Weyl formula( f ⋆ g ) ( ξ, x ) = exp i ~ d X i =1 ( ∂ ξ i ∂ y i − ∂ η i ∂ x i ) ! f ( ξ, x ) g ( η, y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ i = η i x i = y i . (2) Let ˆ x k , ˆ ξ k denote the jets of x k , ξ k – the standard Darboux coordinates on R d respectively. W has a graded algebra structure, where the degree of the ˆ x k ’s andˆ ξ k ’s is 1 and the degree of ~ is 2.(3) W will be endowed with the D ~ , ˆ x , . . . , ˆ x n , ˆ ξ , . . . , ˆ ξ n E -adic topology.(4) We denote the (symbol) map given by P ~ k f k f byˆ σ m : \ A ~ ( M ) m −→ J ∞ m ( M ) . We shall also use the notation J ∞ ( R d ) =: O . Definition 2.3.
For a real symplectic vector space (
V, ω ) we denote W ( V ) := \ T ( V ) ⊗ R C J ~ K h v ⊗ w − w ⊗ v − i ~ ω ( v, w ) i . Here T ( V ) is the tensor algebra of V , \ T ( V ) is its V -adic completion and the topology isgiven by the filtration by assigning elements of V degree 1 and ~ degree 2.The assignment V W ( V ) is clearly functorial with respect to symplectomorphisms. Remark . Suppose (
V, ω ) is a 2 d -dimensional real symplectic vector space. A choice ofsymplectic basis for V induces an isomorphism of C J ~ K algebras: W ( V ∗ ) ≃ W . Notation 2.5.
Let b G := Aut( W ) denote the group of continuous C J ~ K -linear automor-phisms of W . We let g = Der( W ) denote the Lie algebra of continuous C J ~ K -linear deriva-tions of W .For future reference, let us state the following observation Lemma 2.6.
The map W ∋ f → ~ ad f ∈ g is surjective. In particular, the grading of W induces a grading g = Q i ≥− g i on g , namelythe unique grading such that this map is of degree − (note that W = C is central) . Wewill use the notation g ≥ k = Y i ≥ k g i . ALEXANDER GOROKHOVSKY, NIEK DE KLEIJN, RYSZARD NEST
Notation 2.7.
Let ˜ g = { ~ f | f ∈ W } with a Lie algebra structure given by[ 1 ~ f, ~ g ] = 1 ~ [ f, g ]and note that ˜ g is a central extension of g . The corresponding short exact sequence hasthe form(4) 0 −→ ~ C J ~ K −→ ˜ g ad −→ g −→ , where ad ~ f ( g ) = ~ [ f, g ].The extension (4) splits over sp (2 d, C ) and, moreover, the corresponding inclusion sp (2 d, C ) ֒ → ˜ g integrates to the action of Sp(2 d, C ). The Lie subalgebra sp (2 d, R ) ⊂ sp (2 d, C ) gets repre-sented, in the standard basis, by elements of ˜ g given by1 ~ ˆ x k ˆ x j , − ~ ˆ ξ k ˆ ξ j and 1 ~ ˆ x k ˆ ξ j , where k, j = 1 , , . . . , d. Lemma 2.8.
The Lie algebra g ≥ has the structure of a semi-direct product g ≥ = g ≥ ⋊ sp (2 d, C ) . The group b G of automorphisms of W has a structure of a pro-finite dimensional Liegroup with the pro-finite dimensional Lie algebra g ≥ . As such, b G has the structure ofsemi-direct product b G ≃ b G ⋊ Sp(2 d, C ) , where b G = exp g ≥ is pro-unipotent and contractible. The filtration of g induces a filtra-tion of b G : b G = b G ⊃ b G ⊃ b G ⊃ b G ⊃ . . . . Here, for k ≥ b G k = exp( g ≥ k ).Recall that A ~ ( M ) is a formal deformation of a symplectic manifold ( M, ω ). In par-ticular, ω gives an isomorphism T M → T ∗ M and the cotangent bundle T ∗ M will begiven the induced structure of a symplectic vector bundle. For all m ∈ M , there exists anon-canonical isomorphism \ A ~ ( M ) m ≃ W . We collect them together in the following.
Definition 2.9. f M := n ϕ m | m ∈ M, ϕ m : \ A ~ ( M ) m ∼ −→ W o The natural action of b G on f M endows it with the structure of b G -principal bundle. We give f M the pro-finite dimensional manifold structure using the pro-nilpotent group structureof b G/Sp (2 d, C ), see [16] for details. QUIVARIANT ALGEBRAIC INDEX THEOREM 9
Theorem 2.10. [16]
The tangent bundle of f M is isomorphic to the trivial bundle f M × g and there exists a trivialisation given by a g -valued one-form ω ~ ∈ Ω (cid:16) f M (cid:17) ⊗ g satisfyingthe Maurer-Cartan equation dω ~ + 12 [ ω ~ , ω ~ ] = 0 . For later use let us introduce a slight modification of the above construction.
Definition 2.11.
Let b G r = b G ⋊ Sp(2 d, R ). We will use f M r to denote the b G r -principalsubbundle of f M consisting of the isomorphisms ϕ m : \ A ~ ( M ) m ∼ −→ W such that ϕ m , the reduction of ϕ m modulo ~ , is induced by a local symplectomorphism( R d , → ( M, m ).Note that the projection f M r → M factorises through F M , the bundle of symplecticframes in T M , equivariantly with respect to the action of Sp(2 d, R ) ⊂ b G r : b G r / / f M r (cid:15) (cid:15) Sp(2 d, R ) / / F M (cid:15) (cid:15) M. We will use the same symbol for ω ~ and its pull back to f M r .2.2. Fedosov connection and Gelfand-Fuks construction.
Recall that b G is con-tractible, thus in particular the principal b G -bundle f M r → F M admits a section F . Sincethe space K is solid [26], we can choose F to be Sp(2 d, R )-equivariant. Set A F = F ∗ ω ~ ∈ Ω ( F M ; g ) . Since A F is Sp(2 d, R )-equivariant and satisfies the Maurer-Cartan equation,(5) d + A F reduces to a flat g -valued connection ∇ F on M , called the Fedosov connection . Example 2.12.
Consider the case of M = R d with the standard symplectic struc-ture and let A ~ ( R d ) denote the Moyal-Weyl deformation. Then both F R d and g R d are trivial bundles. The trivialization is given by the standard (Darboux) coordinates x , . . . , x d , ξ , . . . , ξ d . So we see, using the construction of ω ~ in [16], that A F ( X ) = i ~ [ ω ( X, − ) , − ], where we consider ω ( X, − ) ∈ Γ( T ∗ M ) ֒ → Γ( M ; W ). Let us denote thegenerators of W corresponding to the standard coordinates by ˆ x i and ˆ ξ i , then we see that A F ( ∂ x i ) = − ∂ ˆ x i and A F ( ∂ ξ i ) = − ∂ ˆ ξ i . Notation 2.13.
Suppose that l ⊂ h is an inclusion of Lie algebras and suppose that thead action of l on h integrates to an action of a Lie group L with Lie algebra l . An h module M is said to be an ( h , L )-module if the action of l on M integrates to a compatible action ofthe Lie group L . If an ( h , L )-module is equipped with a compatible grading and differentialwe will call it an ( h , L )-cochain complex. Definition 2.14.
We setΩ • ( M ; L ) := n η ∈ (Ω • ( F M ) ⊗ L ) Sp (2 d ) | ι X ( η ) = 0 ∀ X ∈ sp (2 d ) o for a ( g , Sp(2 d, R ))-module L . Here the superscript refers to taking invariants for thediagonal action and ι X stands for contraction with the vertical vector fields tangent to theaction of Sp(2 n, R ).Together with ∇ F , (Ω • ( M ; L ) , ∇ F ) forms a cochain complex. The same construction witha ( g , Sp(2 d, R )-cochain complex ( L • , δ ) yields the double complex (Ω • ( M ; L • ) , ∇ F , δ ) Remark . Ω ( M ; L ) is the space of sections of a bundle which we will denote by L ,whose fibers are isomorphic to L . (Ω • ( M ; L ) , ∇ F ) is the de Rham complex of differentialforms with coefficients in L . Definition 2.16.
Suppose that ( L • , δ ) is a ( g , Sp(2 d ))-cochain complex. The Gelfand-Fuksmap C • Lie ( g , sp (2 d ); L • ) −→ Ω • ( M ; L • ) is defined as follows. Given ϕ ∈ C nLie ( g , sp (2 d ); L • )and vector fields { X i } i =1 ,...,n on F M set GF ( ϕ )( X , . . . , X n )( p ) = ϕ ( A F ( X )( p ) , . . . , A F ( X n )( p )) . Direct calculation using the fact that ω ~ satisfies the Maurer-Cartan equation gives thefollowing theorem: Theorem 2.17.
The map GF is a morphism of double complexes GF : ( C • Lie ( g , sp (2 d ); L • ) , ∂ Lie , δ ) −→ (Ω • ( M ; L • ) , ∇ F , δ ) . The change of Fedosov connection, i.e. of the section F , gives rise to a chain homotopicmorphism of the total complexes. Example 2.18. (1) Suppose that L = C . The associated complex is just the de Rham complex of M .(2) Suppose that L = W . The associated bundle W ( T ∗ M ), the Weyl bundle of M , isgiven by applying the functor W to the symplectic vector bundle T ∗ M . Moreover,the choice of F determines a canonical quasi-isomorphism J ∞ F : ( A ~ , −→ (Ω • ( M ; W ) , ∇ F ) . (3) Suppose that L = ( CC per • ( W ) , b + uB ), the cyclic periodic complex of W . Thecomplex (Ω • ( M ; CC per • ( W )) , ∇ F + b + uB ) is a resolution of the jets at the diagonalof the cyclic periodic complex of A ~ ( M ). Example 2.19.
QUIVARIANT ALGEBRAIC INDEX THEOREM 11 (1) Let ˆ θ ∈ C Lie ( g , sp (2 d ); C ) denote a representative of the class of the extension (3).The class of θ = GF (ˆ θ ) belongs to ωi ~ + H ( M ; C ) J ~ K and classifies the deformationsof M up to gauge equivalence (see e.g. [18]).(2) The action of sp (2 d ) on g is semisimple and sp (2 d ) admits a Sp(2 d, R )-equivariantcomplement. Let Π be the implied Sp(2 d, R )-equivariant projection g → sp (2 d ).Set R : g ∧ g −→ sp (2 d ) to be the two-cocycle R ( X, Y ) = [Π( X ) , Π( Y )] − Π([
X, Y ]) . The Chern-Weil homomorphism is the map CW : S • ( sp (2 d ) ∗ ) Sp (2 d ) −→ H • Lie ( g , sp (2 d ))given on the level of cochains by CW ( P )( X , . . . , X n ) = P ( R ( X , X ) , . . . , R ( X n − , X n )) . An example is the ˆ A -power seriesˆ A f = CW det ad ( X )exp( ad ( X )) − exp( ad ( − X )) !! .GF ( ˆ A f ) = ˆ A ( T M ), the ˆ A -genus of the tangent bundle of M .2.3. Algebraic index theorem in Lie algebra cohomology.Notation 2.20.
We denote W ( ~ ) := W [ ~ − ]. Notation 2.21.
Our convention for shifts of complexes is as follows:( V • [ k ]) p = V p + k . Theorem 2.22 ([3],[4]) . Let ( ˆΩ • , ˆ d ) denote the formal de Rham complex in d dimensions,and let (cid:0) C • (cid:0) W ( ~ ) (cid:1) , b (cid:1) denote the Hochschild complex of W ( ~ ) . (1) There exists a unique (up to homotopy) quasi-isomorphism µ ~ : (cid:0) C Hoch • (cid:0) W ( ~ ) (cid:1) , b (cid:1) −→ (cid:16) ˆΩ − • [ ~ − , ~ K [2 d ] , ˆ d (cid:17) . which maps the Hochschild d -chain ϕ = 1 ⊗ Alt (cid:16) ˆ ξ ⊗ ˆ x ⊗ ˆ ξ ⊗ ˆ x ⊗ . . . ⊗ ˆ ξ d ⊗ ˆ x d (cid:17) , where Alt ( z ⊗ . . . ⊗ z n ) := P σ ∈ Σ n ( − sgnσ z σ (1) ⊗ . . . ⊗ z σ ( n ) , to the -form . µ ~ extends to a quasi-isomorphism µ ~ : ( CC per • ( W ( ~ ) ) , b + uB ) −→ ( ˆΩ − • [ ~ − , ~ K [ u − , u K [2 d ] , ˆ d ) . (2) The principal symbol map σ : W → W / ~W ≃ O together with the Hochschild-Kostant-Rosenberg map HKR given by f ⊗ f ⊗ . . . ⊗ f n n ! f ˆ df ∧ ˆ df ∧ . . . ∧ ˆ df n induces a C -linear quasi-isomorphism ˆ µ : CC per • ( W ) −→ (cid:16) ˆΩ • [ u − , u K , u ˆ d (cid:17) . (3) The map of complexes J : ( ˆΩ • [ u − , u K , u ˆ d ) → ( ˆΩ − • [ ~ − , ~ K [ u − , u K [2 d ] , ˆ d ) given by f ˆ df ∧ . . . ∧ ˆ df n u − d − n f ˆ df ∧ . . . ∧ ˆ df n . makes the following diagram commute up to homotopy,. (6) CC per • ( W ) ι / / σ (cid:15) (cid:15) CC per • ( W ( ~ ) ) µ ~ / / ( ˆΩ − • [ ~ − , ~ K [ u − , u K [2 d ] , ˆ d ) CC per • ( O ) HKR / / (cid:16) ˆΩ • [ u − , u K , u ˆ d (cid:17) J O O . Here the complex CC per • ( W ) at the leftmost top corner is that of W as an algebraover C :Remark . One can in fact extend the above C -linear ”principal symbol map” σ : CC per • ( W ) → CC per • ( O )to a C J ~ K -linear map of complexes CC per • ( W ) → CC per • ( O J ~ K ) , but we will not need itbelow. Notation 2.24. (1) Action of g by derivations on W extends to the complex CC per • ( W )and we give it the corresponding ( g , Sp(2 d, R ))-module structure.(2) The action of g on W taken modulo ~W , induces an action of g (by Hamilton-ian vector fields) on ( ˆΩ − • , d ) and hence on ( ˆΩ − • [ ~ − , ~ K [ u − , u K [2 d ] , d ). We give( ˆΩ − • [ ~ − , ~ K [ u − , u K [2 d ] , d ) the induced structure of ( g , Sp(2 d, R ))-module.(3) We set L • := Hom − • ( CC per • ( W ) , ˆΩ − • [ ~ − , ~ K [ u − , u K [2 d ]) . L inherits the ( g , Sp(2 d, R ))-module structure from the actions of g described above.The composition J ◦ HKR ◦ ˆ σ is equivariant with respect to the action of g , hence thefollowing definition makes sense. Definition 2.25. ˆ τ t is the cohomology class in the hypercohomology H Lie (cid:0) g , sp (2 d ); L ( ~ ) (cid:1) given by the cochain(7) J ◦ HKR ◦ σ ∈ C Lie ( g , Sp(2 d, R ); L ) . Lemma 2.26.
The cochain µ ~ ◦ ι ∈ C Lie ( g , Sp (2 d, R ); L ) extends to a cocycle in the complex ( C • Lie ( g , Sp (2 d, R ); L • ) , ∂ Lie + ∂ L ) . QUIVARIANT ALGEBRAIC INDEX THEOREM 13
The cohomology class of this cocycle is independent of the choice of the extension. We willdenote the corresponding class in H Lie (cid:0) g , sp (2 d ); L ( ~ ) (cid:1) by ˆ τ a . For a proof of the next result see e.g. [3].
Theorem 2.27 (Lie Algebraic Index Theorem) . We have ˆ τ a = X p ≥ h ˆ A f e ˆ θ i p u p ˆ τ t , where h ˆ A f e ˆ θ i p is the component of degree p of the cohomology class of ˆ A f e ˆ θ . Algebraic index theorem.
An example of an application of the above is the alge-braic index theorem for a formal deformation of a symplectic manifold M . Note that wecan view A ~ as a complex concentrated in degree 0 and with trivial differential. Then,using the notation of remark 2.18, we find the quasi-isomorphism J ∞ F : ( A ~ , −→ (Ω • ( M ; W ) , ∇ F ) . Similarly we find the quasi-isomorphism J ∞ F : CC per • ( A ~ ) −→ Ω • ( M ; CC per • ( W )) . For future reference let us record the following observation.
Lemma 2.28.
The quasi-isomorphic inclusion C [ ~ − , ~ K [ u − , u K ֒ → ˆΩ − • [ ~ − , ~ K [ u − , u K in-duces a quasi-isomorphism ι : (cid:0) Ω • ( M )[ ~ − , ~ K [ u − , u K [2 d ] , d dR (cid:1) −→ (cid:16) Ω • ( M ; ˆΩ − • [ ~ − , ~ K [ u − , u K [2 d ]) , ∇ F + ˆ d (cid:17) . Notation 2.29.
We denote the inverse (up to homotopy) of ι by T : (cid:16) Ω • ( M ; ˆΩ − • [ ~ − , ~ K [ u − , u K [2 d ]) , ∇ F + ˆ d (cid:17) −→ (cid:0) Ω • ( M )[ ~ − , ~ K [ u − , u K [2 d ] , d dR (cid:1) . For each Q ∈ Ω • ( M ; L • ) of total degree zero, we can define the map C Q : CC per ( A ~ ) −→ Ω • ( M ; CC per • ( W )) Q −→ Ω • ( M ; ˆΩ −∗ [ ~ − , ~ K [ u − , u K [2 d ]) T −→ Ω • −∗ ( M ; C )[ ~ − , ~ K [ u − , u K [2 d ] u − d R M −→ C [ ~ − , ~ K . Clearly C Q is a periodic cyclic cocycle if Q is a cocycle. We will apply this construction tothe two cocycles ˆ τ t and ˆ τ a .Let us start with C ˆ τ t . Tracing the definitions we get the following result. Proposition 2.30. C ˆ τ t is given by u n w ⊗ . . . ⊗ w n u n − d (2 n )! Z M σ ( w ) dσ ( w ) ∧ . . . ∧ dσ ( w n ) . To get the corresponding result for C ˆ τ a recall first that the algebra A ~ ( M ) has a unique C J ~ K -linear trace, up to a normalisation factor. This factor can be fixed as follows. Locallyany deformation of a symplectic manifold is isomorphic to the Weyl deformation. Let U be such a coordinate chart and let ϕ : A ~ ( U ) → A ~ ( R d ) be an isomorphism. Then thetrace T r is normalized by requiring that for any f ∈ A ~ c ( U ) we have T r ( f ) = 1( i ~ ) d Z R d ϕ ( f ) ω d d ! . Proposition 2.31.
We have C ˆ τ a = T r.
Comments about the proof.
First one checks that C ˆ τ a is a 0-cocycle and therefore a trace.Hence it is a C J ~ K -multiple of T r and it is sufficient to evaluate it on elements supportedin a coordinate chart. Moreover, the fact that the Hochschild cohomology class of C ˆ τ a is independent of the Fedosov connection implies that C ˆ τ a is independent of it. Thus itis sufficient to verify the statement for R d with the standard Fedosov connection. Let f ∈ A ~ c ( R d ), one checks that J ∞ F ( f ) ∈ Ω ( R d ; C ( W )) is cohomologous to the element i ~ ) d d ! f ϕ ω d ∈ Ω dc ( R d ; C d ( W )) in Ω • c ( R d ; C − • ( W )) . It follows that the GF (ˆ τ a ) J ∞ F ( f ) iscohomologous to i ~ ) d d ! f ϕ ω d (see the theorem 2.22) and therefore C ˆ τ a ( f ) = T r ( f ) = 1( i ~ ) d Z R d f ω d d !and the statement follows. (cid:3) Given above identifications of C ˆ τ a and C ˆ τ t , the theorem 2.27 implies the following result. Theorem 2.32 (Algebraic Index Theorem) . Suppose a ∈ CC per (cid:0) A ~ c (cid:1) is a cycle, then T r ( a ) = u − d Z M X p ≥ HKR ( σ ( a )) (cid:16) ˆ A ( T C M ) e θ (cid:17) p u p . Equivariant Gelfand-Fuks map
Suppose Γ is a discrete group acting by automorphisms on A ~ . Then we can extendthe action to f M r as follows. First note that the action on A ~ induces an action on ( M, ω ).Now suppose ( m, ϕ m ) ∈ f M r and γ ∈ Γ, then let γ ( m, ϕ m ) = ( γ ( m ) , ϕ γm ), here ϕ γm is givenby \ A ~ ( M ) γ ( m ) −→ \ A ~ ( M ) m −→ W , where the first arrow is given by the action of Γ on A ~ and the second arrow is given by ϕ m .Note that, since b G r acts on f M r by postcomposition and Γ acts on f M r by precomposition,we find that the two actions commute.It should be apparent from the preceding section that we will need an equivariantversion of the Gelfand-Fuks map in the deformed setting. This will allow us to derive the QUIVARIANT ALGEBRAIC INDEX THEOREM 15 equivariant algebraic index theorem from the Lie algebraic one. To do this we will extendthe definition of the one-form A F to the Borel construction E Γ × Γ M . Explicitly this isdone by defining connection one-forms A F k on the manifolds ∆ k × Γ k × F M which satisfycertain boundary conditions. This connection one-form will then serve the usual purposein the Gelfand-Fuks map, only now the range of the map will be a model for the equivariantcohomology of the manifold.Assume that Γ is a discrete group acting on a manifold X . Set X k := X × Γ k . Definethe face maps ∂ ki : X k → X k − by ∂ ki ( x, γ , . . . , γ k ) = ( γ − ( x ) , γ , . . . , γ k ) if i = 0( x, γ , . . . , γ i γ i +1 , . . . , γ k ) if 0 < i < k ( x, γ , . . . , γ k − ) if i = k We denote the standard k -simplex by∆ k := ( ( t , . . . , t k ) ≥ | k X i =0 t i = 1 ) ⊂ R k +1 and define by ǫ ki : ∆ k − → ∆ k ǫ ki ( t , . . . , t k − ) = ( (0 , t , . . . , t k − ) if i = 0( t , . . . , t i − , , t i , . . . , t k − ) if 0 < i ≤ k Definition 3.1. A de Rham-Sullivan , or compatible form ϕ of degree p is a collection offorms ϕ k ∈ Ω p (∆ k × X k ), k = 0 , , . . . , satisfying(8) ( ǫ ki × id) ∗ ϕ k = (id × ∂ ki ) ∗ ϕ k − ∈ Ω p (∆ k − × X k )for 0 ≤ i ≤ k and any k > ϕ = { ϕ k } is a compatible form, then dϕ := { dϕ k } is also a compatible form; fortwo compatible forms ϕ = { ϕ k } and ψ = { ψ k } their product ϕψ := { ϕ k ∧ ψ k } is anothercompatible form. We denote the space of de Rham-Sullivan forms by Ω • ( M × Γ E Γ) in viewof the following
Theorem 3.2.
We have H • (Ω • ( M × Γ E Γ) , d ) ≃ H • Γ ( M ) where the left hand side is the cohomology of the complex Ω • ( M × Γ E Γ) and the right handside is the cohomology of the Borel construction M × Γ E Γ . See for instance [8] for the proof.More generally, let V be a Γ-equivariant bundle on X . Let π k : X k → X be theprojection and let V k := π ∗ k V . Notice that we have canonical isomorphisms(9) ( ∂ ki ) ∗ V k − ∼ = V k . Definition 3.3.
Let V be Γ-equivariant vector bundle. A V valued de Rham-Sullivan(compatible) form ϕ is a collection ϕ k ∈ Ω p (∆ k × X k ; V k ), k = 0 , , . . . , satisfying theconditions (8), where we use the isomorphisms (9) to identify ( ∂ ki ) ∗ V k − with V k .We let Ω • ( M × Γ E Γ; V ) denote the space of V -valued de Rham-Sullivan (compatible)forms.For equivariant vector bundles V and W there is a productΩ • ( M × Γ E Γ; V ) ⊗ Ω • ( M × Γ E Γ; W ) −→ Ω • ( M × Γ E Γ; V ⊗ W )defined as for the scalar forms by ϕψ := { ϕ k ∧ ψ k } .Assume that we have a collection of connections ∇ k on the bundles V k satisfying thecompatibility conditions(10) ( ǫ ki × id) ∗ ∇ k = (id × ∂ ki ) ∗ ∇ k − . Then for a compatible form ϕ = { ϕ k }∇ ϕ := {∇ k ϕ k } is again a compatible form. Notation 3.4.
Now let M be a symplectic manifold and Γ a discrete group acting bysymplectomorphisms on M . We introduce the following notations: P k Γ := ∆ k × ( F M ) k = ∆ k × F M × Γ k , and similarly M k Γ := ∆ k × M k = ∆ k × M × Γ k and f M k Γ := ∆ k × ( M r ) k = ∆ k × f M r × Γ k . Note that P k Γ → M k Γ is a principal Sp(2 d )-bundle, namely the pull-back of F M → M viathe obvious projection. Similarly f M k Γ is the pull-back of f M r → M . We define Ω • ( M k Γ ; L )for a ( g , Sp(2 d ))-module L as we did for M above only replacing F M by P k Γ and consideringthe trivial action of the symplectic group on ∆ k × Γ k . We shall denote the b G -principalbundle f M r → F M by π , the b G r -principal bundle f M r → M by π r and the Sp(2 d )-principalbundle F M → M by π .Note that, for all i and k , the (obvious) fibration of ∆ k − × ( f M r ) k over ∆ k − × ( F M ) k is canonically isomorphic to the pull back by id × ∂ ki of the fibration of f M k − over P k − .Hence, for any section F of the projection f M k − → P k − , there is a natural pull back(id × ∂ ki ) ∗ F, the unique section of ∆ k − × ( f M r ) k → ∆ k − × ( F M ) k making the followingdiagram commutative:(11) ∆ k − × ( f M r ) k id × ∂ ki / / f M k − ∆ k − × ( F M ) k id × ∂ ki / / (id × ∂ ki ) ∗ F O O P k − F O O . QUIVARIANT ALGEBRAIC INDEX THEOREM 17
Lemma 3.5.
There exist sections f M k Γ (cid:15) (cid:15) P k Γ F k : : of the projections f M k Γ → P k Γ satisfying the compatibility conditions ( ǫ ki × id) ∗ F k = (id × ∂ ki ) ∗ F k − Proof.
We construct the sections F k recursively. In section 2 we constructed the inital section F r : P = F M −→ f M r . Set F := F r .Now suppose we have found F l satisfying the compatibility conditions for all l < k .Notice that the principal bundle f M k Γ → P k Γ is trivial, and thus, by fixing a trivialization,we can view its sections as functions on P k Γ with values in b G . which we identify witha vector space g via the exponential map. The compatibility conditions require that F k takes on certain values determined by F k − on ( ∂ ∆ k ) × F M × Γ k ⊂ P k Γ . Since b G can beidentified with a vector space g via the exponential map, F k can be extended smoothlyfrom ( ∂ ∆ k ) × F × Γ k to P k Γ (cid:3) As before we can construct the sections in the lemma above Sp(2 d ) equivariantly andwe will fix a system of such equivariant sections { F k } k ≥ from now on. Now, as before,we can use the sections F k to pull back the canonical connection form from f M k Γ (whichwas itself pulled back from f M through the composition of the projection onto f M r and aninclusion), to define a g -valued differential form A F k on P k Γ for each k . Notation 3.6.
Suppose ( L • , ∂ L ) is a ( g , b G r )-cochain complex. Then we denote by (cid:0) L • π r , ∂ L (cid:1) the bundle of cochain complexes over M associated to f M r , i.e. with total space f M r × b G r L .We will denote the pull-back to the M k Γ by the same symbol. Note that the pullback π ∗ L π r is exactly the bundle associated to f M r → F M with fiber the ( g , b G )-cochain complex L given by b G ֒ → b G r , i.e the pull-back has total space f M r × b G L . Thus we will denote π ∗ L π r = L π . Again we will use the same notation for the pull-backs over the P k Γ . Remark . Note that since Γ acts on the b G r -bundle f M r → M we find that Γ also actson L • π r and, since this action lifts to an Sp(2 d )-equivariant action on f M r → F M , we finda corresponding action on the space Ω • ( M ; L • ). Let us be a bit more precise about thisaction. Note first that the section F yields a trivialization of f M r → F M . This means alsothat it yields a trivialization (denoted by the same symbol) F : F M × L ∼ −→ f M r × G L explicitly given by ( p, ℓ ) [ F ( p ) , ℓ ] with the inverse given by mapping [ ϕ m , ℓ ] to( π ( ϕ m ) , ( ϕ m ◦ F ( π ( ϕ m )) − ) ( ℓ )). In these terms the action is given by γ ( η ⊗ ℓ ) = ( γ ∗ η ) ⊗ ( F − ) ∗ γ ∗ F ∗ ℓ where ( F − ) ∗ γ ∗ F ∗ ℓ is the section given by p ( γ ( p ) , F ( p ) γF ( γ ( p )) − ℓ ) . We consider the corresponding action of Γ on the spaces Ω • (cid:0) M k Γ ; L • (cid:1) , where we use F k instead of F (or in fact on Ω • ( N × M ; L • ) for any N ).Note that the differential forms A F k define flat connections ∇ F k on Ω • (cid:0) M k Γ ; L • (cid:1) forall k and so we can consider the product complex Q k Ω • (cid:0) M k Γ ; L • (cid:1) with the differential˜ ∇ + ∂ L , where ˜ ∇ = Q k ∇ F k . Note also that the connections ∇ F k satisfy the compatibilityconditions of the equation (10).Now we can consider the equivariant Gelfand–Fuks map GF Γ : C • Lie ( g , sp (2 d ) ; L • ) −→ Y k Ω • (cid:0) M k Γ ; L • (cid:1) given by(12) GF Γ ( χ ) k = χ ◦ A ⊗ pF k , where χ ∈ C pLie ( g , sp (2 d ) ; L • ) and the subscript k refers to taking the k -th coordinate inthe product. In other words the definition is the same as in definition 2.16 only we nowuse the compatible system of connections A F k . Lemma 3.8.
For all χ ∈ C pLie ( g , sp (2 d ); L • ) we have that GF Γ ( χ ) ∈ Ω • ( M × Γ E Γ , L • ) .Proof. The boundary conditions put on the sections F k in lemma 3.5 are meant exactly to ensurethis property of the Gelfand-Fuks map GF Γ . The lemma follows straightforwardly fromthese boundary conditions. (cid:3) Theorem 3.9.
The equivariant Gelfand-Fuks map is a morphism of complexes GF Γ : C • Lie ( g , sp (2 d ) ; L • ) → Ω • ( M × Γ E Γ , L • ) . Proof.
This proof is exactly the same as in the non-equivariant setting, carried out coordinate-wisein the product. Since we do not give the usual proof in this article let us be a bit moreexplicit. Note that GF Γ : C • Lie ( g , sp (2 d ) ; L • ) −→ Y k Ω • (cid:0) M k Γ ; L • (cid:1) is given by GF Γ ( χ ) m k ( X , . . . X p ) = χ (cid:16) ( A F k ) m k X , . . . , ( A F k ) m k X p (cid:17) , QUIVARIANT ALGEBRAIC INDEX THEOREM 19 for χ ∈ C pLie ( g , sp (2 d ) ; L • ), m k ∈ P k Γ and X i ∈ T m k P k Γ . The differential ∂ L and GF Γ clearlycommute so it is left to show that GF Γ ◦ ∂ Lie = ˜ ∇ ◦ GF Γ . This follows by direct computation using the facts that dA F k + 12 [ A F k , A F k ] = 0and ˜ ∇ = Y k d P k Γ + A F k , here d P k Γ refers to the de Rham differential on P k Γ . (cid:3) Pairing with HC per • (cid:0) A ~ c ⋊ Γ (cid:1) In order to derive the equivariant version of the algebraic index theorem we should showthat the universal class ˆ τ a maps to the class of the trace supported at the identity underthe equivariant Gelfand-Fuks map GF Γ constructed in the previous section. The class ˆ τ a lives in the Lie algebra cohomology with values in the ( g , b G r )-cochain complex L • := Hom − • ( CC per • ( W ~ ) , ˆΩ − • [ u − , u K [ ~ − , ~ K [2 d ]) . Here the action is induced (through conjugation) by the action on W and by the action(by modding out ~ ) on ˆΩ. From now on the notation L • will refer to this complex. Thedifferential ∂ L on L • is given by viewing it as the usual morphism space internal to chaincomplexes. In order to derive the equivariant algebraic index theorem we shall have to pairclasses in Lie algebra cohomology with values in L • with periodic cyclic chains of A ~ c ⋊ Γusing the equivariant Gelfand–Fuks map. Since the trace on A ~ c ⋊ Γ is supported at theidentity we only need to consider the component of the cyclic complexes supported at theidentity.
Definition 4.1 (Homogeneous Summand) . Let CC per • ( A ~ ⋊ Γ) e be the subcomplex spanned (over C [ u − , u K ) by the chains a γ ⊗ . . . ⊗ a n γ n such that γ γ . . . γ n = e ∈ Γ , where e denotes the neutral element of Γ. Proposition 4.2.
The map D : CC per • ( A ~ ⋊ Γ) −→ C • (Γ; CC per • ( A ~ )) given by composing the projection CC per • ( A ~ ⋊ Γ) −→ CC per • ( A ~ ⋊ Γ) e with the quasi-isomorphism of theorem A.13 in the appendix is a morphism of complexes. The proof is contained in the appendix.As in lemma 2.28, the canonical inclusion ι (cid:0) Ω • ( M × Γ E Γ)[ ~ − , ~ K [ u − , u K [2 d ] , d (cid:1) → (cid:16) Ω • ( M × Γ E Γ; ˆΩ − • [ ~ − , ~ K [ u − , u K [2 d ]) , ∇ F + ˆ d (cid:17) is a quasi-isomorphism and we will denote its quasi-inverse by TT : Ω • ( M × Γ E Γ; ˆΩ − • [ ~ − , ~ K [ u − , u K [2 d ]) −→ Ω • ( M × Γ E Γ)[ ~ − , ~ K [ u − , u K [2 d ] Definition 4.3.
We define the pairing h· , ·i : Ω • ( M × Γ E Γ; L • ) × C • (Γ; CC per • ( A ~ c ( M ))) −→ C [ ~ − , ~ K [ u − , u K as follows. In the following let α = a ⊗ ( g ⊗ g ⊗ . . . ⊗ g p ) ∈ CC perk − p ( A ~ ( M )) ⊗ ( C Γ) ⊗ p andlet ϕ ∈ Ω • ( M × Γ E Γ; L • ). We define h ϕ, α i := Z ∆ p × M × g × ... × g p T ϕ p ( J ∞ F p ( a )) , where J ∞ F p is the map given by taking the ∞ -jets of elements of A ~ ( M ) relative to theFedosov connection ∇ F p as in the example 2.18 using the section F p over M k Γ . Since theintegral of ξ ∈ Ω k ( M × Γ E Γ) over any simplex ∆ p for p > k will vanish, the pairing h· , ·i extends by linearity to C • (Γ; CC per • ( A ~ c ( M ))). Lemma 4.4.
We have: h ( ˜ ∇ F + ∂ L ) ϕ, α i = ( − | ϕ | +1 h ϕ, ( δ Γ + b + uB ) α i Proof. h ( ˜ ∇ F + ∂ L ) ϕ, α i = Z ∆ p × M × g × ... × g p T (( ˜ ∇ F + ∂ L ) ϕ p )( J ∞ F p ( a ))Notice that ( ∂ L ϕ p )( J ∞ F p ( a )) = ˆ d (( ϕ p )( J ∞ F p ( a )) − ( − | ϕ | ϕ p ( J ∞ F p (( b + uB ) a )). Also, since˜ ∇ F ( J ∞ F p ( a )) = 0 we have ( ˜ ∇ F ϕ p )( J ∞ F p ( a )) = ˜ ∇ F ( ϕ p )( J ∞ F p ( a )). Combining these formulas weobtain that h ( ˜ ∇ F + ∂ L ) ϕ, α i equals(13) Z ∆ p × M × g × ... × g p T (cid:16) ( ˜ ∇ F + ˆ d )( ϕ p ( J ∞ F p ( a )) − ( − | ϕ | ϕ p ( J ∞ F p (( b + uB ) a ))) (cid:17) = Z ∆ p × M × g × ... × g p dT ( ϕ p ( J ∞ F p ( a )) − ( − | ϕ | h ϕ, ( b + uB ) α i Applying Stokes’ formula to R ∆ p × M × g × ... × g p dT ( ϕ p ( J ∞ F p ( a )) and noticing that the collectionof forms { T ( ϕ p ( J ∞ F p ( a )) } is compatible we see that(14) Z ∆ p × M × g × ... × g p dT ( ϕ p ( J ∞ F p ( a )) = ( − | ϕ | +1 h ϕ, δ Γ α i QUIVARIANT ALGEBRAIC INDEX THEOREM 21
The statement of the lemma now follows from (13) and (14). (cid:3)
Recall that we have a cap-product C • (Γ; CC per • ( A ~ c ( M ))) ⊗ C • (Γ , C ) ∩ −→ C • (Γ; CC per • ( A ~ c ( M ))) . Definition 4.5.
Let ξ ∈ C • (Γ , C ) be a cocycle. Define I ξ : C • Lie ( g , sp (2 d ); L • ) −→ CC • + | ξ | per ( A ~ c ⋊ Γ)by I ξ ( λ )( a ) = ǫ ( | λ | ) h GF Γ ( λ ) , D ( a ) ∩ ξ i for all λ ∈ C • Lie ( g , sp (2 d ); L • ) and a ∈ CC per • ( A ~ c ⋊ Γ), where ǫ ( m ) = ( − m ( m +1) / . Proposition 4.6.
The map I ξ : ( C • Lie ( g , sp (2 d ); L • ) , ∂ Lie + ∂ L ) −→ (cid:0) CC • + | ξ | per ( A ~ c ⋊ Γ) , ( b + uB ) ∗ (cid:1) is a morphism of complexes.Proof. Using Theorem 3.9 and Lemma 4.4 we have I ξ (( ∂ Lie + ( − r ∂ L ) λ ))( a ) = ǫ ( | λ | + 1) h GF Γ (( ∂ Lie + ( − r ∂ L ) λ )) , D ( a ) ∩ ξ i = ǫ ( | λ | +1) h ( ˜ ∇ F + ∂ L ) GF Γ ( λ ) , D ( a ) ∩ ξ i = ( − | λ | +1 ǫ ( | λ | +1) h GF Γ ( λ ) , ( δ Γ + b + uB )( D ( a )) ∩ ξ i = ǫ ( | λ | ) h GF Γ ( λ ) , ( D (( b + uB ) a )) ∩ ξ i = I ξ ( λ )(( b + uB ) a )and the statement follows. (cid:3) Remark . The induced map on cohomology I ξ : H • ( g , sp (2 d ); L • ) −→ HC • + | ξ | per ( A ~ c ⋊ Γ) iseasily seen to depend only on the cohomology class [ ξ ] ∈ H • (Γ , C ).5. Evaluation of the equivariant classes
In the previous sections we defined the map I ξ : H ( g , sp (2 d ); L • ) −→ HC kper ( A ~ c ⋊ Γ) , where k = | ξ | . The last step in proving the main result of this paper is to evaluate theclasses appearing in Lie algebraic index theorem 2.27.First of all we consider the image under I ξ of the trace density ˆ τ a . Consider the map h GF Γ (ˆ τ a ) , ·i : C (Γ; C ( A ~ c )) −→ C [ ~ − , ~ K . Since in degree 0 the equivariant Gelfand–Fuks map is given by the ordinary Gelfand–Fuksmap on M , this map coincides with the canonical trace T r (cf. the proof of theorem 2.32).It follows that h GF Γ (ˆ τ a ) , α ⊗ ( γ ⊗ . . . γ k ) ∩ ξ i = ξ ( γ , . . . , γ k ) T r ( α )From this discussion we obtain the following: Proposition 5.1.
We have I ξ (ˆ τ a ) = T r ξ where T r ξ is a cocycle on A ~ c ( M ) ⋊ Γ given by (15) T r ξ ( a γ ⊗ . . . ⊗ a k γ k ) = ξ ( γ , . . . , γ k ) T r ( a γ ( a ) . . . ( γ γ . . . γ k − ( a k )) if γ γ . . . γ k = e and otherwise. Definition 5.2.
The equivariant Weyl curvature θ Γ is defined as the image of ˆ θ under GF Γ followed by ( C J ~ K -linear extension of) the map in Theorem 3.2. Similarly, the equivariantˆ A -genus of M , denoted ˆ A ( M ) Γ , is defined as the image of ˆ A under the equivariant Gelfand–Fuks map followed by ( C J ~ K -linear extension of) the isomorphism in Theorem 3.2. Example 5.3.
Let us provide an example of the characteristic class θ Γ . To do this con-sider the example of group actions on deformation quantization given in [14]. Namely, weconsider the symplectic manifold R / Z = T , the 2-torus, with the symplectic structure ω = dy ∧ dx induced from the standard one on R , where x , y ∈ R / Z are the standardcoordinates on T . We then consider the action of Z on T by symplectomorphisms wherethe generator of Z acts by T : ( x, y ) ( x + x , y + y ). Note that, for a generic pair ( x , y ),the quotient space is not Hausdorff.The Fedosov connection ∇ F given as in Example 2.12 descends to the connection on T which is, moreover, Z -invariant (where we endow C ∞ ( T , W ) with the action of Z induced by the symplectic action on T ). It follows that A ~ = Ker ∇ F is a Z -equivariantdeformation with the characteristic class ωi ~ .We can obtain a more interesting example by modifying the previous one as follows (cf.[14]). Let u ∈ C ∞ ( T , W ) be an invertible element such that u − ( ∇ F u ) is central. Definea new action of Z on C ∞ ( T , W ) where the generator acts by w u − ( T w ) u. Ker ∇ F is again invariant under this action and we thus obtain an action of Z on A ~ .To describe its characteristic class note that, since E Z ∼ = R , we find that the cohomologyH • Z ( T ) = H • ( R × Z T ) ∼ = H • ( T ). Let ν be a compactly supported 1-form on R with R R ν = 1. Denote by τ the translation t → t −
1. Then˜ α = X n ∈ Z ( τ ∗ ) n ( ν ) ∧ ( T ∗ ) n ( U − ∇ F U )is a Z -invariant form on R × T , hence a lift of a form, say α , on R × Z T = T . Thecharacteristic class of the associated Z -equivariant deformation is equal to θ Z = ωi ~ + α. Finally we arrive at the main theorem of this paper. Let R : H even Γ ( M ) → H • Γ ( M )[ u ]be given by R ( a ) = u deg a/ a and recall the morphism defined in (19)Φ : H • Γ ( M ) −→ HC • per ( C ∞ c ( M ) ⋊ Γ) . Theorem 5.4 (Equivariant Algebraic Index Theorem) . Suppose a ∈ CC per ( A ~ c ⋊ Γ) is a cycle, then we have T r ξ ( a ) = D Φ (cid:16) R (cid:16) ˆ A ( M ) Γ e θ Γ (cid:17) [ ξ ] (cid:17) , σ ( a ) E where h· , ·i denotes the pairing of CC • per and CC per • .Proof. The theorem follows from Theorem 2.27 by applying the morphism I ξ . The imageof τ a under I ξ is T r ξ (cf. Proposition 5.1). On the other hand, by equation (19), " I ξ X p ≥ (cid:16) ˆ A f e ˆ θ (cid:17) p u p ˆ τ t ! = Φ (cid:16) R (cid:16) ˆ A ( M ) Γ e θ Γ (cid:17) [ ξ ] (cid:17) . (cid:3) Note that the form of the theorem 1.4 stated in the introduction follows by consideringthe pairing of periodic cyclic cohomology and K -theory using the Chern–Connes character[15]. Appendices
Below we shall fix our conventions with regard to cyclic/simplicial structures and homolo-gies. We will also define the complexes we use to describe group (co)homology, Lie algebracohomology and cyclic (co)homology. The general reference for this section is [15].Fix a field k of characteristic 0. Appendix A. Cyclic/simplicial structure
Let Λ denote the cyclic category. Instead of giving the intuitive definition let us simplygive a particularly useful presentation. The cyclic category Λ has objects [ n ] for each n ∈ Z ≥ and is generated by δ ni ∈ Hom([ n − , [ n ]) and σ ni ∈ Hom([ n + 1] , [ n ]) for 0 ≤ i ≤ nt n ∈ Hom([ n ] , [ n ]) for all n ∈ Z ≥ with the relations δ nj ◦ δ n − i = δ ni ◦ δ n − j − if i < j σ nj ◦ σ n +1 i = σ ni ◦ σ n +1 j +1 if i ≤ jσ nj ◦ δ n +1 i = δ ni ◦ σ n − j − if i < j σ nj ◦ δ n +1 i = Id [ n ] if i = j, j + 1 σ nj ◦ δ n +1 i = δ ni − ◦ σ n − j if i > j + 1 t n +1 n = Id [ n ] t n ◦ δ ni = δ ni +1 ◦ t n − if 0 ≤ i < n t n ◦ δ nn = δ n t n ◦ σ ni = σ ni +1 ◦ t n +1 if 0 ≤ i < n t n ◦ σ nn = σ n ◦ t n +1 ◦ t n +1 . Using only the generators δ ni and σ ni and relations not involving t n ’s gives a presentationof the simplicial category △ . A contravariant functor from Λ ( △ ) to the category of k -modules is called a cyclic (simplicial) k -module. Definition A.1.
Given a unital associative k -algebra A we shall denote by A ♮ the functorΛ op → k − M od given by A ♮ ([ n ]) = A ⊗ n +1 and δ ni ( a ⊗ . . . ⊗ a n ) = a ⊗ . . . ⊗ a i a i +1 ⊗ . . . ⊗ a n if 0 ≤ i < nδ nn ( a ⊗ . . . ⊗ a n ) = a n a ⊗ a ⊗ . . . ⊗ a n − σ ni ( a ⊗ . . . ⊗ a n ) = a ⊗ . . . ⊗ a i ⊗ ⊗ a i +1 ⊗ . . . ⊗ a n for all 0 ≤ i ≤ nt n ( a ⊗ . . . ⊗ a n ) = a ⊗ . . . ⊗ a n ⊗ a Note that if A admits a group action of the group G by unital algebra homomorphismsthen G also acts on A ♮ (diagonally). Definition A.2.
Given a group G we shall denote by G k♮ the functor Λ op → k − M od given by G k♮ ([ n ]) = ( kG ) ⊗ n +1 and δ ni ( g ⊗ . . . ⊗ g n ) = g ⊗ . . . ⊗ ˆ g i ⊗ . . . ⊗ g n for all 0 ≤ i ≤ nσ ni ( g ⊗ . . . ⊗ g n ) = g ⊗ . . . ⊗ g i ⊗ g i ⊗ g i +1 ⊗ . . . ⊗ g n for all 0 ≤ i ≤ nt n ( g ⊗ . . . ⊗ g n ) = g ⊗ g ⊗ . . . ⊗ g n ⊗ g . Note that G acts on G k♮ from the right by g · ( g ⊗ . . . ⊗ g n ) = g − g ⊗ . . . ⊗ g − g n . Definition A.3.
Given two cyclic k -modules A ♮ and B ♮ we shall denote by A♮B the cyclic k -module given by A♮B ([ n ]) = A ♮ ([ n ]) ⊗ B ♮ ([ n ]) with the diagonal cyclic structure.A.1. Cyclic homologies.
Given a cyclic k -module M ♮ we can consider four differentcomplexes associated to the simplicial/cyclic structure. To define them we shall first definetwo operators: b and B .The first is induced through the Dold–Kan correspondence and uses only the simplicialstructure. It is given by b n = n X i =0 ( − i δ ni : M ♮ ([ n ]) −→ M ♮ ([ n − . By using the simplicial identities above it is easily verified that b n − b n = 0. To define the“Hochschild” complex it is enough to have just the operators b n .To define the three cyclic complexes we shall use the operator B n = ( t − n +1 + ( − n ) ◦ σ nn ◦ n X i =0 ( − in t in ! : M ♮ ([ n ]) −→ M ♮ ([ n + 1]) . Note that B n +1 B n = 0 since(16) n +1 X i =0 ( − i ( n +1) t in +1 ◦ ( t − n +1 + ( − n ) = n +1 X i =0 ( − i ( n +1) ( t i − n +1 + ( − n t in +1 ) = 0 . Vanishing of the above expression follows since the sum telescopes except for the first term t − n +1 and the last term − ( − n ( n +1) t n +1 n +1 , which also cancel each other. Note also that(17) b n +1 B n + B n − b n = 0 , this can be seen by writing out both operators as sums of operators in the normal form δ nk σ n − l t in .From now on we will drop the subscripts of the b and B operators. The cyclic module M ♮ gives rise to a graded module { M ♮n } n ∈ Z ≥ by M ♮n = M ♮ ([ n ]). Then we see that theoperator b turns M ♮ into a chain complex. Definition A.4.
The
Hochschild complex ( C Hoch • ( M ♮ ) , b ) of the cyclic module M ♮ is definedas C Hochn ( M ♮ ) := M ♮ ([ n ]) equipped with the boundary operator b (of degree − • ( M ♮ ).Note that we have not used the full cyclic structure of M ♮ to construct the Hochschildcomplex. In fact one can form the Hochschild complex (cid:0) C • ( M △ ) , b (cid:1) of any simplicial k -module M △ in exactly the same way.Note that by (17) and (16) we find that ( b + B ) = 0. This implies that we couldconsider a certain double complex with columns given by the Hochschild complex. Notehowever that, if b is of degree − B is naturally ofdegree +1. We can consider a new grading for which the operator b + B is of homogeneousdegree −
1. In order to make this grading easy to see, it will be useful to introduce theformal variable u of degree −
2. This leads us to several choices of double complexes.
Definition A.5.
We define the cyclic complex by( CC • ( M ♮ ) , δ ♮ ) := (cid:16) C Hoch • ( M ♮ )[ u − , u K . C Hoch • ( M ♮ ) J u K , b + uB (cid:17) , the negative cyclic complex by( CC −• ( M ♮ ) , δ ♮ − ) := (cid:0) C Hoch • ( M ♮ ) J u K , b + uB (cid:1) and finally the periodic cyclic complex by( CC per • ( M ♮ ) , δ ♮per ) := (cid:0) C Hoch • ( M ♮ )[ u − , u K , b + uB (cid:1) . Here u denotes a formal variable of degree −
2. The corresponding homologies will bedenoted HC • ( M ♮ ), HC −• ( M ♮ ) and HC per • ( M ♮ ) respectively. The cyclic cochain complexes CC • per ( M ♮ ), CC • − ( M ♮ ) and CC • ( M ♮ ) are defined as the k -duals of the chain complexes.We shall often omit the superscripts ♮ when there can be no confusion as to what thecyclic structures are. Remark
A.6 . Note that every “flavor” of cyclic homology comes equipped with spectralsequences induced from the fact that they are realized as totalizations of a double complex.The double complex corresponding to cyclic homology is bounded (second octant) andtherefore the spectral sequence which starts by taking homology on columns converges toHC • . The negative (or periodic) cyclic double complex is unbounded, but concentratedin the (second,) third, fourth and fifth octant. This means that the spectral sequencestarting with taking homology in the columns converges again to HC −• (or HC per • ). Notehowever that in this case the negative (or periodic) cyclic homology is given by the producttotalization.The remark A.6 provides the proof of the following proposition. Proposition A.7.
Suppose M ♮ and N ♮ are two cyclic k -modules and ϕ : N ♮ −→ M ♮ isa map of cyclic modules that induces an isomorphism on Hochschild homologies. Then ϕ induces an isomorphism on cyclic, negative cyclic and periodic cyclic homologies as well.Proof. The proof follows since ϕ induces isomorphisms on the first pages of the relevant spectralsequences, which converge. (cid:3) A.2.
Replacements for cyclic complexes.
It will often be useful to consider differentcomplexes that compute the various cyclic homologies. We shall give definitions of thecomplexes that are used in the main body of the article here.A.2.1.
Crossed product.
Suppose A is a unital k -algebra and G is a group acting on theleft by unital algebra homomorphisms. We denote by A ⋊ G the crossed product algebragiven by A ⊗ kG as a k -vector space and by the multiplication rule ( ag )( bh ) = ag ( b ) gh for all a, b ∈ A and g, h ∈ G . Note that the cyclic structure of ( A ⋊ G ) ♮ splits over theconjugacy classes of G . Namely, given a tensor a g ⊗ a g ⊗ . . . ⊗ a n g n , the conjugacy classof the product g · . . . · g n is invariant under δ ni , σ ni and t n for all i and n . So we have( A ⋊ G ) ♮ = M x ∈h G i ( A ⋊ G ) ♮x where we denote the set of conjugacy classes of G by h G i and the span of all tensors a g ⊗ . . . ⊗ a n g n such that g · . . . · g n ∈ x by ( A ⋊ G ) ♮x . The summand ( A ⋊ G ) ♮e , here e = { e } the conjugacy class of the neutral element, is called the homogeneous summand.We shall use the specialized notation A♮G := A ♮ ♮G k♮ . Note that A♮G carries a right G action given by the diagonal action (the left action on A is converted to a right action byinversion, i.e. G ≃ G op ). Thus the co-invariants ( A♮G ) G = A♮G . h a − g ( a ) i form anothercyclic k -module. Proposition A.8.
The homogeneous summand of ( A ⋊ G ) ♮ is isomorphic to the co-invariants of A♮G . ( A ⋊ G ) ♮e ∼ −→ ( A♮G ) G . Proof.
Consider the map given by a g ⊗ . . . ⊗ a n g n ( g − ( a ) ⊗ a ⊗ g ( a ) ⊗ . . . ⊗ g . . . g n − ( a n )) ♮ ( e ⊗ g ⊗ g g ⊗ . . . ⊗ g · . . . · g n ) , it is easily checked to commute with the cyclic structure and allows the inverse given by( a ⊗ . . . ⊗ a n ) ♮ ( g ⊗ . . . ⊗ g n ) g − n ( a ) g − n g ⊗ g − ( a ) g − g ⊗ . . . ⊗ g − n − ( a n ) g − n − g n this last tensor can also be expressed as g − n a g ⊗ g − a g ⊗ . . . ⊗ g − n − a n g n . (cid:3) Definition A.9.
Suppose ( M • , ∂ ) is a right kG -chain complex. Then we define the grouphomology of G with values in M as( C • ( G ; M ) , δ ( G,M ) ) := Tot Q M • ⊗ kG C Hoch • ( G )where we consider the tensor product of kG -chain complexes with the obvious structure ofleft kG -chain complex on C Hoch • ( G ). Note that this means that C n ( G ; M ) = Y p + q = n M p ⊗ kG C Hochq ( G )and δ ( G,M ) = ∂ ⊗ Id + Id ⊗ b where we use the Koszul sign convention. Proposition A.10.
Suppose M is a right kG -module. Then M ⊗ kG with the diagonalright action is a free kG -module.Proof. Let us denote the k -module underlying M by F ( M ), then F ( M ) ⊗ kG denotes thefree (right) kG -module induced by the k -module underlying M . Consider the map M ⊗ kG −→ F ( M ) ⊗ kG given by m ⊗ g mg − ⊗ g . It is obviously a map of kG -modules and allows for the inverse m ⊗ g mg ⊗ g . (cid:3) Proposition A.11.
Suppose F is a free right kG -module (we view it as a chain complexconcentrated in degree with trivial differential) then there exists a contracting homotopy H F : C • ( G ; F ) −→ C • +1 ( G ; F ) . Suppose ( F • , ∂ ) is a quasi-free right kG -chain complex (i.e. F n is a free kG -module for all n ) then the homotopies H F n give rise to a quasi-isomorphism (( F • ) G , ∂ ) ∼ −→ ( C • ( G ; F ) , δ ( G,F ) ) . Proof.
Note that F ≃ M ⊗ kG since it is a free module. So we find that C p ( G ; F ) = ( M ⊗ kG ) ⊗ kG ( kG ) ⊗ p +1 ≃ M ⊗ ( kG ) ⊗ p +1 by the map m ⊗ g ⊗ g ⊗ . . . ⊗ g p m ⊗ gg ⊗ . . . ⊗ gg p . Using this normalization weconsider the map H M given by m ⊗ g ⊗ . . . ⊗ g p m ⊗ e ⊗ g ⊗ . . . ⊗ g p and note that indeed δ p +1 G H M + H M δ pG = Id(we denote δ G := δ ( G,M ) = Id ⊗ b ) for all p > F n ≃ M n ⊗ kG for each n since it is quasi-free.For each n we have the homotopy H n := H F n given by the formula above on C • ( G ; F n ).Then we consider the map Q H : ( F p ) G −→ C p ( G ; F )given by Q F ([ f ]) = f − δ G Hf + ∞ X q =1 ( − H∂ ) q f − ∂ ( − H∂ ) q − Hf − δ q +1 G ( − H∂ ) q Hf where we have dropped the subscript from H and we denote the class of f in the co-invariants F G by [ f ] . One may check by straightforward computation that Q F is a well-defined morphism of complexes. Now we note that the double complex defining C • ( G ; F )is concentrated in the upper half plane and therefore comes with a spectral sequencewith first page given by H p ( G ; F q ) which converges to H( C p + q ( G ; F )) (group homology).Note however that since F • is quasi-free we find that H p ( G, F q ) = 0 unless p = 0 and H ( G, F q ) = ( F q ) G . Thus, since Q F induces an isomorphism on the first page and thespectral sequence converges, we find that Q F is a quasi-isomorphism. (cid:3) As a kG -module we see that A♮G ([ n ]) = A ♮ ([ n ]) ⊗ G k♮ ([ n ]) = B ([ n ]) ⊗ kG with thediagonal action, where B ([ n ]) = A ⊗ n +1 ⊗ kG ⊗ n . So by proposition A.10 we find that theHochschild and various cyclic chain complexes corresponding to A♮G are quasi-free. Thuswe can construct the quasi-isomorphisms from proposition A.11 for each chain complexassociated to the cyclic module
A♮G . So we find four quasi-isomorphisms which we shalldenote Q Hoch , Q , Q − and Q per corresponding to the Hochschild, cyclic, negative cyclic andperiodic cyclic complexes respectively. Proposition A.12.
The map
A♮G −→ A ♮ given by ( a ⊗ . . . ⊗ a n ) ♮ ( g ⊗ . . . ⊗ g n ) a ⊗ . . . ⊗ a n induces a quasi-isomorphism on all associated complexes.Proof. Note that, by proposition A.7, it is sufficient to prove the statement for the Hochschildcomplexes. Let us denote the standard free resolution of G by F ( G ), note that F ( G ) = ( C Hoch • ( G k♮ ) , b ) . The map given above is obtained by first applying the Alexander–Whitney map C Hochn ( A ♮ ) ⊗ C Hochn ( G k♮ ) −→ M p + q = n C Hochp ( A ♮ ) ⊗ C Hochq ( G k♮ ) , which yields a quasi-isomorphism C Hoch • ( A♮G ) ∼ −→ C Hoch • ( A ♮ ) ⊗ C Hoch • ( G k♮ ) , where we consider the tensor product of chain complexes on the right-hand side. Thenone simply takes the cap product with the generator in H ∗ ( F ( G ) ∗ ) ≃ k , which is also aquasi-isomorphism. So we find that the map is a quasi-isomorphism for the Hochschildcomplexes. (cid:3) Note that the map given in proposition A.12 is also G -equivariant and therefore itinduces a map C • ( G ; A♮G ) −→ C • ( G ; A ♮ )which is a quasi-isomorphism when we consider the group homology complex with valuesin the various complexes associated to A ♮ . Theorem A.13.
The composite maps from the Hochschild and various cyclic complexesassociated to ( A ⋊ Γ) ♮e to the group homology with values in the various Hochschild and cycliccomplexes associated to A ♮ implied by propositions A.8 and A.12 are quasi-isomorphisms,i.e. there are quasi-isomorphisms (cid:0) C Hoch • (cid:0) ( A ⋊ G ) ♮e (cid:1) , b (cid:1) ∼ −→ C • ( G ; C Hoch • ( A )) (cid:0) CC • (cid:0) ( A ⋊ G ) ♮e (cid:1) , δ ♮ (cid:1) ∼ −→ C • ( G ; CC • ( A )) (cid:16) CC −• (cid:0) ( A ⋊ G ) ♮e (cid:1) , δ ♮ − (cid:17) ∼ −→ C • ( G ; CC −• ( A )) and (cid:0) CC per • (cid:0) ( A ⋊ G ) ♮e (cid:1) , δ ♮per (cid:1) ∼ −→ C • ( G ; CC per • ( A )) . Remark
A.14 . Note that since the cyclic and Hochschild complexes are bounded below theproduct totalizations in our definition of group homology agrees with the (usual) directsum totalizations. In the periodic cyclic and negative cyclic cases they do not agree ingeneral.
Remark
A.15 . Suppose that a discrete group Γ acts on a smooth manifold M by diffeo-morphisms. The above produces a morphism of complexes CC per • ( C ∞ ( M ) c ⋊ Γ) → C • (Γ , CC per • ( C ∞ c ( M ))Composing it with the morphism CC per • ( C ∞ c ( M )) −→ Ω • c ( M )[ u − , u K , induced by the map f ⊗ f ⊗ . . . ⊗ f n n ! f df . . . df n we get a morphism of complexes(18) CC per • ( C ∞ ( M ) c ⋊ Γ) → C • (Γ , Ω • c ( M )[ u − , u K ) . In the case when M is oriented and the elements of Γ preserve orientation, the transposeof this map can be interpreted as a morphism of complexes(19) Φ : C • (Γ , Ω dim( M ) −• ( M )[ u − , u K ) −→ CC • per ( C ∞ c ( M ) ⋊ Γ) , compare [5] section 3.2. δ .A.2.2. Group Homology.
It is often useful to consider instead of the above complex forgroup homology an isomorphic complex, which we will call the non-homogeneous complex . Definition A.16.
Suppose ( M • , ∂ ) is a right kG -chain complex, then we set˜ C n ( G ; M ) := Y p + q = n M q ⊗ ( kG ) ⊗ p . We define the operators δ pi : M • ⊗ ( kG ) ⊗ p → M • ⊗ ( kG ) ⊗ p − by δ p ( m ⊗ g ⊗ . . . ⊗ g p ) := g ( m ) ⊗ g ⊗ . . . ⊗ g p δ pi ( m ⊗ g ⊗ . . . ⊗ g p ) := m ⊗ g ⊗ . . . ⊗ g i g i +1 ⊗ . . . ⊗ g p for all 0 < i < p and finally δ pp ( m ⊗ g ⊗ . . . ⊗ g p ) := m ⊗ g ⊗ . . . ⊗ g p − . We define ( ˜ C • ( G ; M ) , ˜ δ ( G,M ) ) to be the chain complex given by˜ δ ( G,M ) = ∂ ⊗ Id + Id ⊗ δ G where δ pG = P pi =0 δ pi . Proposition A.17.
There is an isomorphism of chain complexes C • ( G ; M ) −→ ˜ C • ( G ; M ) . Proof.
Consider the map C n ( G ; M ) −→ ˜ C n ( G ; M ) , given by m ⊗ g ⊗ . . . ⊗ g p g ( m ) ⊗ g − g ⊗ g − g ⊗ . . . ⊗ g − p − g p . Note that it commutes with the differentials and allows for the inverse given by m ⊗ g ⊗ . . . ⊗ g p m ⊗ e ⊗ g ⊗ g g ⊗ . . . ⊗ g · . . . · g p . (cid:3) We will usually use this chain complex when dealing with group homology and thus wewill drop the tilde in the main body of this article. A.3.
Lie algebra cohomology.
In this section let us describe the Lie algebra cohomology.Although there are various deep relations between Lie algebra cohomology and cyclic andHochschild homologies we have chosen to present the complex in a separate manner. Onecould compute the Lie algebra cohomology using a Hochschild complex, however in therelative case (which we need in this article) there are several subtleties that we wouldrather avoid by considering a different complex.
Definition A.18.
Suppose g is a Lie algebra over k , h ֒ → g a subalgebra and ( M • , ∂ ) is a g -chain complex. Then we denote C pLie ( g , h ; M q ) := Hom h (cid:0) ∧ p g / h , M q (cid:1) . We define operators ∂ pLie : C pLie ( g , h ; M q ) −→ C p +1 Lie ( g , h ; M q )by ∂ pLie ϕ ( X , . . . , X p ) = p X i =0 ( − i X i ϕ ( X , . . . , ˆ X i , . . . , X p )+ X ≤ i A.19 . We actually only consider the Lie algebra cohomology of infinite dimensionalLie algebras here. For these the complex above is not very useful. The Lie algebras weconsider come with a topology (induced by filtration) however and so do the coefficients.Using this fact we can consider in the above not simply anti-symmetric linear maps, butcontinuous anti-symmetric linear maps from the completed tensor products. References [1] Michael Atiyah and Isadore Singer, The Index Of Elliptic Operators on Compact Manifolds , Bull.Amer. Math. Soc., 69, 422-433, 1963.[2] Paul Bressler, Ryszard Nest and Boris Tsygan, Riemann-Roch Theorems Via Deformation Quantiza-tion I Adv. Math. 167, 1–25, 2002.[3] Paul Bressler, Ryszard Nest and Boris Tsygan, Riemann-Roch Theorems Via Deformation Quantiza-tion II Adv. Math. 167, 26–73, 2002.[4] Alain Connes, Non-Commutative Differential Geometry , IHES Publ. Math., 62, 257–360, 1985.[5] Alain Connes, Non-commutative Geometry , Academic Press, San Diego, 1990.[6] Alain Connes and Henri Moscovici, Hopf Algebras, Cyclic Cohomology and the Transverse IndexTheorem Commun. Math. Phys., 198, 199-246, 1998.[7] Paul Bressler, Alexander Gorokhovsky, Ryszard Nest and Boris Tsygan, Algebraic Index Theorem forSymplectic Deformations of Gerbes , Contemporary Mathematics vol. 546 (Non-commutative Geome-try and Global analysis), 2011.[8] Johan Dupont, Curvature and Characteristic Classes , Springer, LNM 640, Heidelberg, 1978. [9] Boris Fedosov, The Index Theorem for Deformation Quantization , in Boundary Value Problems,Schr¨odinger Operators, Deformation Quantization; Akademie, Advances in Partial Differential Equa-tions, 319-333, Berlin, 1995.[10] Boris Fedosov, Deformation Quantization and Index Theory , Akademie Verlag, Berlin, 1st Edition,1996, chapter 5 and 6.[11] Israel Gelfand, Cohomology of Infinite-Dimensional Lie Algebras. Some Questions in Integral Geom-etry , Lecture given at ICM, Nice, 1970.[12] Israel Gelfand and David Kazhdan, Certain Questions of Differential Geometry and the Computationof the Cohomologies of the Lie Algebras of Vector Fields , Soviet Math. Doklady, 12, 1367-1370, 1971.[13] Simone Gutt, Deformation Quantization: an Introduction , 3rd edition, HAL, Monastir, 2005.[14] Niek de Kleijn Extension and Classification of Group Actions on Formal Deformation Quantizationsof Symplectic Manifolds , preprint, arXiv:1601.05048, 2016.[15] Jean-Louis Loday, Cyclic Homology , 2nd edition, Springer, GMW 301, Heidelberg, 1998.[16] Ryszard Nest and Boris Tsygan, Algebraic Index Theorem Commun. Math. Phys. 172, 223–262, 1995.[17] Ryszard Nest and Boris Tsygan, Formal Versus Analytic Index Theorems , Internat. Math. Res. No-tices, 11, 1996.[18] Ryszard Nest and Boris Tsygan, Deformations of Symplectic Lie Algebroids, Deformations of Holo-morphic Symplectic Structures, and Index Theorems , Asian J. Math. 5 (2001), no. 4, 599–635.[19] Denis Perrot, Rudy Rodsphon, An equivariant index theorem for hypo-elliptic operators arXiv:1412.5042[20] Marcus Pflaum, Hessel Posthuma and Xiang Tang, An Algebraic Index Theorem for Orbifolds , Adv.Math., 210, 83-121, 2007.[21] Markus Pflaum, Hessel Posthuma and Xiang Tang, On the Algebraic Index for Riemannian ´EtaleGroupoids , Lett. Math. Phys., 90, 287-310, 2009.[22] Anton Savin, Elmar Schrohe, Boris Sternin, Uniformization and index of elliptic operators associatedwith diffeomorphisms of a manifold. Russ. J. Math. Phys. 22 (2015), no. 3, 410–420.[23] Anton Savin, Elmar Schrohe, Boris Sternin, On the index formula for an isometric diffeomorphism.(Russian) Sovrem. Mat. Fundam. Napravl. *46 * (2012), 141–152; translation in J. Math. Sci. (N.Y.)201:818829 (2014)[24] Anton Savin, Elmar Schrohe, Boris Sternin, The index problem for elliptic operators associated witha diffeomorphism of a manifold and uniformization. (Russian) Dokl. Akad. Nauk 441 (2011), no. 5,593–596; translation in Dokl. Math. 84 (2011), no. 3, 846–849[25] Anton Savin, Boris Sternin Elliptic theory for operators associated with diffeomorphisms of smoothmanifolds , arXiv:1207.3017[26] Norman Steenrod,