Cyclic cocycles on deformation quantizations and higher index theorems

CYCLIC COCYCLES ON DEFORMA TION QUANTIZA TIONS AND HIGHER INDEX THEOREMS M.J. PFLAUM, H. POSTHUMA, AND X . T ANG A B S T R AC T . W e constr uct a nontrivial cyclic cocycle on t he W eyl algebra of a sym- plectic vector space. Using this cyclic cocycle we construct an explicit, local, quasi- isomorphism from the complex of differential forms on a symplectic manifold to the complex of c yclic cochains of a ny formal deformation qu antization thereof. W e give a ne w proof of Nest -T sygan’s algebraic h igher index theorem by computing the pairing between such cycli c cocycles and t he K -theory of t h e formal d eforma- tion quantization. Furthermore, we extend this approac h to derive an algebraic higher index theorem on a symplectic or bifold. As an applic ation, we obtain the analytic higher index theorem of Connes–Moscovici and its extension to orbifolds. C O N T E N T S 1. Introduction 2 2. Cyclic cohomo logy of the W eyl algebra 5 2.1. The W eyl algebra 5 2.2. Cyclic cocycles on the W eyl algebra 6 2.3. The sp 2 n -action 10 3. Cyclic cocycles on symplectic manifolds 10 3.1. Deformation quantization of symplectic manifolds 10 3.2. Shufﬂe p roduct on Hochschild chains 11 3.3. Cyclic cocycles on deformation quantizations of symplectic manifolds 12 4. Algebraic index theorems 17 4.1. The pairing between cyclic cohomolog y and K -theory 17 4.2. Quantization twisted by a vector bundle 18 4.3. Lie algebra cohomolo gy 19 4.4. Local Riemann-Roch theorem 20 4.5. Higher algebraic index theorem 22 5. Generalization to orbifolds 23 5.1. Preliminaries 23 5.2. Adding an automorphism to the W eyl algebra 24 5.3. Cyclic cocycles on formal deformations of proper ´ etale groupoids 25 5.4. T wisting by vector bundles 27 5.5. A twisted Riemann–Roch theorem 28 5.6. The higher index theorem f or proper ´ etale groupoids 29 6. The Higher analytic index theorem o n manifolds 29 6.1. Alexander–Spanier cohomology 30 6.2. Higher indices 36 7. A higher analytic index theorem for orbifolds 43 7.1. Orbifold Alexander-Spanier c ohomolo gy 43 1 2 M.J. PFLAUM, H. POSTHUMA, AND X. T AN G 7.2. Explicit realization of Alexander–Spanier c ocycles 45 7.3. Relating orbifold Alexander–Spanier cohomology with cyclic cohomolog y 48 7.4. Pairing with localized K -theory 51 7.5. Operator-Symbol calculus on orbifolds and the higher analytic index 53 Appendix A. Cyclic cohomo logy 56 A.1. The cyclic bicomplex 56 A.2. Localization 57 References 58 1. I N T R O D U C T I O N Let D be an elliptic differential operator on a compact manifold M . As is well- known ellipticity implies that D is a Fredholm operator and the Atiyah-Singer index theorem [ A T S I ] expresses the index of D as a topological formula involv- ing the Chern character o f th e symbol σ ( D ) and the T odd class of the manifold M . In [ C O M O ], C O N N E S – M O S C O V I C I pro ved a far reaching generaliza tion of the Atiyah–Singer index theorem, the so-called higher index theor em. In subsequent work [ M O W U ], M O S C O V I C I – W U pro vided an abstract setting to c onstruct higher indices. The essential idea hereby is as f ollows. Let Ψ DO − ∞ ( M ) b e the algebra of smoothing pseudodifferen tial operators on M . The oper a tor D deﬁnes an element e D in the K 0 -group of the algebra of smooth- ing pseudodifferenti al o perators Ψ DO − ∞ ( M ) , and its im age under the Chern- Connes chara cter an element Ch ( e D ) in the cyclic homolo gy of Ψ DO − ∞ ( M ) . Since smoothing operators act by trace class operators, the operator tra ce gives rise to a cyclic cocycle tr on Ψ DO − ∞ ( M ) of degree 0. Pairing this cocycle with the cy- cle Ch ( e D ) o ne recovers the analytic index of D as in d ( D ) = h tr, Ch ( e D ) i . As has been explained in [C O M O , § 2], th e local information cont ained in D r espec- tively its symbol σ ( D ) is not fully ca ptured by this index pair ing. T o remedy this, C O N N E S – M O S C O V I C I constructed a localized index which in the li terature and also in this work is called the high er index. Accor ding to [ M O W U ], one can un- derstand the higher index of D as a pairing ind [ f ] ( D ) = h [ f ] , Ch loc ( e D ) i , where f is a given Alexander-Spanier cocycle on M (on which on e localizes the index) , and Ch loc ( e D ) is an Alexander-Spanier homology class associated e D which her e is regarded a s a difference of projections in Ψ DO − ∞ ( M ) . The higher index theorem in [ C O M O ] compute the localized index - which no longer is integral - in te rms of topological data generalizing the Atiyah–Singer index theorem. In this paper , we prove a n a lgebraic generaliza tion of the higher inde x theo- rem to symplectic manifolds. Applying our theor e m to co tangent bundles, we recover the theorem of C O N N E S – M O S C O V I C I . Furthermore, we extend our the- orem to gen eral symplectic orbifolds a nd obtain an analog of the higher index theorem on orbifolds genera lizing K AWA S A K I ’s orbifold index theor e m [ K A ] and also M A R C O L L I – M AT H A I ’s higher index theorem for good orbifolds [M A M A ] . Our approach to an algebraic higher index theorem for symplectic manifolds is inspired by the work [ F E F E S H ]. There, F E I G I N – F E L D E R – S H O I K H E T proved an algebra ic index theorem for symplectic manifolds based on a formula for a CYCLIC COCYCL E S O N DE FORMA TIO N QUANTIZA TIO NS 3 Hochschild 2 n -cocycle τ 2 n on the W eyl algebra W 2 n over R 2 n with its canonical symplectic structur e. In this pa per , we constru ct an extension of the Hochschild cocycle τ 2 n to a sequence of cochains ( τ 0 , τ 2 , . . . , τ 2 n ) which forms a cocycle in the total cyclic bicomplex  T ot 2 n B C • ( W poly 2 n ) , b + B  . Using this ( b + B ) -cocycle and Fedosov’s cons truction of a deformation quantization ( A ( ( ¯ h ) ) cpt , ⋆ ) on a symplectic manifold, we construct a quasi-isomorphism Q f rom the cyclic de Rham complex to the b + B total complex of ( A ( ( ¯ h )) cpt , ⋆ ) , Q :  T ot • B Ω • ( M ) ( ( ¯ h )) , d  →  T ot • B C •  A ( ( ¯ h )) cpt  , b + B  . If one views ( A ( ( ¯ h ) ) cpt , ⋆ ) as the generalization of the algebra of pseudodifferential operators, one can try to compute the pairing between a cyclic c ocycle on A ( ( ¯ h ) ) cpt with the Chern-Connes char a cter of an element in K 0 ( A ( ( ¯ h )) cpt ) . Fedosov proved in [ F E 9 5 ] that K 0 ( A ( ( ¯ h )) cpt ) can be represented by pairs of proj ectors ( P 1 , P 2 ) on A ( ( ¯ h ) ) cpt with P 1 − P 2 compactly supported modulo stabilization. U sing methods from Lie algebra cohomol ogy , we obtain the followin g formula for this pairing, h Q ( α ) , P 1 − P 2 i = k ∑ l = 0 1 ( 2 π √ − 1 ) l Z M α 2 l ∧ ˆ A ( M ) Ch ( V 1 − V 2 ) exp  − Ω 2 π √ − 1 ¯ h  , where α = ( α 0 , · · · , α 2 k ) ∈ T ot 2 k B Ω • ( M ) ( ( ¯ h )) is a sequence of closed differential forms on M , and V 1 and V 2 are vector bundles on M deter mined by the 0-th order terms of P 1 and P 2 , and Ω ∈ ω + ¯ h H 2 ( M ) [ [ ¯ h ]] is the characte ristic class of the deformation quantization ( A ( ( ¯ h )) cpt , ⋆ ) . That the right hand side of the above algebraic higher index formula coin- cides with the algebraic localize d index was originally proved by N E S T - T S Y G A N in [ N E T S 9 6], [B R N E T S ], and A A S T R U P in [ A A ] by a different approach. U sing ˇ Cech- methods, N E S T - T S Y G A N computed the Chern-Connes charac ter of an e lement in K 0 ( A ( ( ¯ h ) ) ) by cons tructing a morphism fro m the cyclic homology of A ( ( ¯ h ) ) to the cohomolog y of M . Our construction is exactly in the opposite direction and lifted to the (co)chain level: B y means of the formula for the ( b + B ) -cocycles ( τ 0 , . . . , τ 2 n ) we are able to constr uct an ex plicit quasi-isomorphism Q from the sheaf c omplex of differential forms to the sheaf complex of c yclic cochains of A ( ( ¯ h ) ) . This a l- lows us to write down explicit expressions for cyclic cocycles on A ( ( ¯ h ) ) . W ith this new constr uction, we give a more transparent proo f of the above index theorem using differential forms a nd Lie algebra cohomology , which is closer to C O N N E S – M O S C O V I C I ’s original approach. Let us mention that the b + B cycle ( τ 0 , · · · , τ 2 n ) has bee n discovered inde- pendently by W I L LW A C H E R [W I L ]. He used this cocycle to compute a higher Riemann-Roch formula. By a similar idea as above, we extend the a lgebraic index theorem of [ P F P O T A ] for orbifolds to the above higher version. W e represent an orbifold by a proper ´ etale groupoid, and consider A ( ( ¯ h ) ) ⋊ G as a de formation quantization of a sym- plectic orbifold M = ( G 0 / G , ω ) , as it has been constructed by the third author in [ T A ]. Using Fedosov’s ide a [ F E 0 2 ], we generalize the above b + B cocycle 4 M.J. PFLAUM, H. POSTHUMA, AND X. T ANG ( τ 0 , · · · , τ 2 n ) on the W eyl algebra W 2 n to a γ - twisted b + B cocycle with γ a lin- ear symplectic isomorphism on V of ﬁnite order . Analogously to the manifold case, we use the γ -twisted cocycle and Fedosov’s connection to deﬁne a S -quasi- isomorphism Q from the cyclic de Rham differential complex on the correspond- ing inertia orbifold e M to the b + B total complex of the algebra A ( ( ¯ h ) ) ⋊ G . For α = ( α 2 k , · · · , α 0 ) ∈ T ot 2 k B Ω • ( e M ) ( ( ¯ h )) , and P 1 , P 2 two pro jectors in the matrix al- gebra over A ( ( ¯ h ) ) ⋊ G with P 1 − P 2 compactly supported, we obtain the following formula as Thm. 5.13. h Q ( α ) , P 1 − P 2 i = k ∑ j = 0 Z e M 1 ( 2 π √ − 1 ) j m α 2 j ∧ ˆ A ( e M ) C h θ ( ι ∗ V 1 − ι ∗ V 2 ) exp ( − ι ∗ Ω 2 π √ − 1 ¯ h ) Ch θ ( λ − 1 N ) , where V 1 and V 2 are the orbifold vec tor bundles on M de termined by the 0-th order ter ms of P 1 and P 2 , Ω is the characteristic cla ss of ( A ( ( ¯ h ) ) ⋊ G , ⋆ ) , ι is the canonical map from e M to M , and m is deﬁned in terms of the order of the local isotopy gro ups. As an application of our a lgebraic f ormulas, we derive higher a nalytic index theorems for elliptic operators using an asymptotic symbol calculus. T o this e nd we ﬁrst consider a cotangent bundle of a manifold Q . It was shown by the ﬁrst author [ P F 98] that the asymptotic symbol calculus on pseudodiffer- ential operators on Q naturally de ﬁnes a deformation quantization ( A ( ( ¯ h )) cpt , ⋆ op ) of T ∗ Q and the operator trace induces a ca nonical trace on ( A ( ( ¯ h )) cpt , ⋆ op ) . T o de- rive C O N N E S – M O S C O V I C I ’s higher index from the higher algebra ic index theo- rem, we prove that the algebra ic pairing h Q ( α ) , P 1 − P 2 i coincides a symptotically with the pairing h X [ f ] , Ch ( e D ) i deﬁned in [ C O M O ]. Mor e precisely , we prove that the cyclic cocycles Q ( α ) a nd X [ f ] on A ( ( ¯ h )) cpt ( T ∗ Q ) are cohomologous, if the Alexander–Spanier cocycle f and the closed form α induce the same cohomology class on Q . W e prove the claimed relation by using sheaf theoretic methods and by applying inherent pro perties of the calculus of asymptotic pseudodifferential operators. Let us mention that a sketch of how to derive the analytic higher index theorem from the a lgebraic one has already been outlined in [ N E T S 96]. Here, we take a different a pproach by elaborating more on the nature of A lexander–Spanier cohomolog y a nd its use for constructing cyclic cocycles on a de f ormation quanti- zation in genera l. In particular , thi s enables us to dir ectly compare the algebraic higher index with the deﬁnition of the localized index by C O N N E S – M O S C O V I C I . Secondly , we consider the cota ngent bundle of an orbifold Q . The way we ad- dress this problem is similar to the above manifold c a se. T o deﬁne a higher index for an elliptic operator D on Q , we need to d eﬁne a localized index of an elliptic operator D on Q . This leads us to introduce a new notion of orbifold Alexander- Spanier cohomology , whose cohomology is equal to the cohomology of the cor- responding inertia orbifold e Q . Next, we introduce a notion of localized K - theory of an orbifold, and show that there is a well deﬁned pairing between localized K -theory and orbifold Alexa nder-Spanier cohomolog y of Q . W ith these natural deﬁnitions a nd constructions, we follow the same ideas as in the manifold case to prove a higher index theorem on a reduced orbifold. W e would like to remark CYCLIC COCYCL E S O N DE FORMA TIO N QUANTIZA TIO NS 5 that our deﬁnition of orbifold Alexander–Spanier cohomology is new and differ- ent f rom the standard deﬁnition of Ale x ander-Spanier cohomology of a topolog- ical space. In particular , we ha ve view an orbifold as a sta c k more than just a topological space. For this reason, our higher index theor e m on orbifolds detects the topological information of an orbifold as a stack. Recall that C O N N E S – M O S C O V I C I [ C O M O ] used their higher index theor em to prove a covering in dex theor em, which was used to p rove the Novikov conjecture in the ca se of hyperbolic groups. W e woul d like to view this pap e r as a seed for the study of c over ing index theorems (cf. [ M A M A ]) for orbifolds and the equivariant Novikov conjecture [R O W E ]. W e plan to study these questions in the future. This paper is organized as follows. In Section 2 , we introduce and prove that ( τ 0 , · · · , τ 2 n ) deﬁnes a b + B cocycle on the W eyl algebra W 2 n . In Section 3, we use a Fedosov connection to construct a quasi-isomorphism from the sheaf complex of differential forms to the sheaf complex of cyclic cochains on the algebra of the deformation quantization ( A ( ( ¯ h ) ) cpt , ⋆ ) of a symplectic manifold corresponding to the Fedosov connection. Then, in Section 4, we use Lie algebra Chern-W eil theory technique to pr ove a a lgebraic higher index theorem. Afterwards, in Section 5, we extend the constructions from Sections 2-4 to orbifolds and obtain a higher alge- braic index theo rem for orbifolds. In Sections 6 a nd 7, we discuss how to apply the higher algebraic index theorem to prove C O N N E S and M O S C O V I C I ’s higher index theorem on manifolds and its generalization to orbifolds. Acknowledgme nts: The authors would like to thank M . Crainic, G. Felder , A. Gorokh ovsky , a nd H. Moscovici f or helpful discussions. H. Posthuma and X. T ang would like to thank the Department of Mathematics, University of Colorado for hosting their visits at CU Boulder , where part of the work has been completed. The research of H. Posthuma is supported by NWO, and X. T ang is pa rtially supported by NSF grant 0703 7 75. 2. C Y C L I C C O H O M O L O G Y O F T H E W E Y L A L G E B R A 2.1. The W eyl algeb ra . Let ( V , ω ) be a ﬁn ite dimensional symplectic vector spac e. In canonical coordinates ( p 1 , . . . p n , q 1 , . . . q n ) the symplectic f orm simply reads ω = ∑ i d p i ∧ dq i . The polyn omial W eyl algebra W poly ( V ) over the ring C [ ¯ h , ¯ h − 1 ] is the spac e of polynomials S ( V ∗ ) ⊗ C [ ¯ h , ¯ h − 1 ] with algebra structure given by the Moyal–W e yl product f ⋆ g = ( m ◦ exp ( ¯ h 2 α ) ) ( f ⊗ g ) where m is the commutative multiplication and α ∈ E nd  W poly ( V ) ⊗ W poly ( V )  is basically the Poisson bracket associated to ω : α ( f ⊗ g ) = n ∑ i = 1  ∂ f ∂ p i ⊗ ∂ g ∂ q i − ∂ f ∂ q i ⊗ ∂ g ∂ p i  . 6 M.J. PFLAUM, H. POSTHUMA, AND X. T ANG In the formula for the Moyal product, the exponential is d eﬁned b y means of its power series expansion, which terminates af te r ﬁnitely many terms f or the poly- nomial W eyl algebra. For the particular ca se where V = R 2 n with its natural symplectic structure, we write W poly 2 n for W poly ( V ) . The symplectic group Sp 2 n acts on W poly 2 n by a utomorphisms. Inﬁnitesimally , this leads to an action of the Lie algebra sp 2 n by d e rivations. It is known tha t all derivations of W poly 2 n are inner , in fact there is a short exact sequence of Lie a lgebras 0 → C [ ¯ h , ¯ h − 1 ] → W poly 2 n → Der  W poly 2 n  → 0. The action of sp 2 n is explicitly given by identifying sp 2 n with the quadratic homo- geneous polynomials in S ( V ∗ ) . Finally , using the spectral sequence associated to the ¯ h -ﬁltration on W poly 2 n , one proves the follo wing well-known result: Proposition 2.1. [ F E T S ] The cy clic cohomology of the W eyl algebra is given by H C k ( W poly 2 n ) = ( C [ ¯ h , ¯ h − 1 ] if k = 2 n + 2 p , p ≥ 0, 0 else . 2.2. Cyclic c ocycl es on t he W eyl algebra. The aim of this section is to deﬁne an explicit cocycle in the ( b , B ) -complex that generates the nontrivial cyclic cohomol- ogy class at degree 2 n as is suggested in Proposition 2.1 a bove. W e ﬁrst need a couple of deﬁnitions. For 1 ≤ i 6 = j ≤ 2 k ≤ 2 n we de ﬁne α i j ∈ End  ( W poly 2 n ) ⊗ 2 k + 1  by α i j ( a 0 ⊗ . . . ⊗ a 2 k ) = n ∑ s = 1  a 0 ⊗ . . . ⊗ ∂ a i ∂ p s ⊗ . . . ⊗ ∂ a j ∂ q s ⊗ . . . ⊗ a 2 k − a 0 . . . ⊗ ∂ a i ∂ q s ⊗ . . . ⊗ ∂ a j ∂ p s ⊗ . . . ⊗ a 2 k  , i.e., the Poisso n tensor acting on i ’th and j ’th slot of the tensor product. W e also need π 2 k = 1 ⊗ ( ¯ h α ) ∧ k ∈ End  ( W poly 2 n ) ⊗ ( 2 k + 1 )  , and ﬁnally µ i : ( W poly 2 n ) ⊗ ( i + 1 ) → C [ ¯ h , ¯ h − 1 ] is given by µ i ( a 0 ⊗ . . . ⊗ a i ) = a 0 ( 0 ) · · · a i ( 0 ) . In the f ollowing, ∆ k ⊂ R k is the sta nda rd simplex given by 0 ≤ u 1 ≤ . . . ≤ u k ≤ 1. Deﬁnition 2.2. Let W poly 2 n be the W eyl algebra. For all i with 0 ≤ i ≤ 2 n deﬁne the cochains τ i ∈ C i ( W poly 2 n ) as follows. For even degrees put τ 2 k ( a ) = ( − 1 ) k µ 2 k Z ∆ 2 k ∏ 0 ≤ i < j ≤ 2 k e ¯ h ( u i − u j + 1 2 ) α i j      u 0 = 0 π 2 k ( a ) d u 1 · · · d u 2 k . In the odd case, we put τ 2 k − 1 ( a ) : = ( − 1 ) k − 1 µ 2 k − 1 Z ∆ 2 k − 1 ∏ 0 ≤ i < j ≤ 2 k − 1 e ¯ h ( u i − u j + 1 2 ) α i j      u 0 = 0 ( ¯ h α ) ∧ k ( a ) d u 1 · · · d u 2 k − 1 , CYCLIC COCYCL E S O N DE FORMA TIO N QUANTIZA TIO NS 7 Remark 2. 3. The cocycle τ 2 n ∈ C 2 n ( W poly 2 n ) is the Hochschild cocycle of [ F E F E S H ] up to a sign ( − 1 ) n . The sign is needed for ( τ 0 , · · · , τ 2 n ) to be b + B closed, as in the theorem be low . Theorem 2.4. The cochains τ i ∈ C i ( W poly 2 n ) satisfy the relation − B τ 2 k = τ 2 k − 1 = b τ 2 k − 2 . Remark 2.5 . For n = 1, the proof of this theorem is quite easy since the cocycles can be written down explicitly . W e have τ 2 ( a 0 ⊗ a 1 ⊗ a 2 ) : = − ¯ h µ 2 Z ∆ 2 e ¯ h ( 1 2 − u 1 ) α 01 e ¯ h ( 1 2 − u 2 ) α 02 e ¯ h ( u 1 − u 2 + 1 2 ) α 12 ( 1 ⊗ α ) ( a 0 ⊗ a 1 ⊗ a 2 ) d u 1 d u 2 τ 0 ( a 0 ) : = a 0 ( 0 ) W ith this we compute: B τ 2 ( a 0 ⊗ a 1 ) = − τ 2 ( 1 ⊗ a 0 ⊗ a 1 ) + τ 2 ( 1 ⊗ a 1 ⊗ a 0 ) = − ¯ h µ 2 Z ∆ 2 e ¯ h ( u 1 − u 2 + 1 2 ) α α ( a 0 ⊗ a 1 − a 1 ⊗ a 0 ) d u 1 ∧ d u 2 = − ¯ h µ 2 Z 1 0 d u 2 Z u 2 0 d u 1 e ¯ h ( u 1 − u 2 + 1 2 ) α α ( a 0 ⊗ a 1 − a 1 ⊗ a 0 ) = − µ 2 Z 1 0 d u 2 Z u 2 0 d u 1  d d u 1 e ¯ h ( u 1 − u 2 + 1 2 ) α  ( a 0 ⊗ a 1 − a 1 ⊗ a 0 ) = − µ 2 Z 1 0 d u 2  e ¯ h 2 α − e ¯ h ( 1 2 − u 2 ) α  ( a 0 ⊗ a 1 − a 1 ⊗ a 0 ) = − µ 2 e ¯ h 2 α ( a 0 ⊗ a 1 − a 1 ⊗ a 0 ) + µ 2 Z 1 0 d u 2  e ¯ h ( 1 2 − u 2 ) α − e − ¯ h ( 1 2 − u 2 ) α  ( a 0 ⊗ a 1 ) = − µ 2 e ¯ h 2 α ( a 0 ⊗ a 1 − a 1 ⊗ a 0 ) = − ( a 0 ⋆ a 1 ( 0 ) − a 1 ⋆ a 0 ( 0 ) ) = − b τ 0 ( a 0 ⊗ a 1 ) . The integral in the sixth line can be seen to be ze ro by using the antisymmetry of the integrand under reﬂection in the point u 2 = 1 /2 . This gives a n easy proof of the n = 1 case. In the general case, the proof of Theor em 2.4 proceeds in two steps: Lemma 2.6. One has τ 2 k − 1 ( a 0 ⊗ . . . ⊗ a 2 k − 1 ) = − B τ 2 k ( a 0 ⊗ . . . ⊗ a 2 k − 1 ) . 8 M.J. PFLAUM, H. POSTHUMA, AND X. T ANG Proof. First we write out the left hand side: τ 2 k − 1 ( a 0 ⊗ . . . ⊗ a 2 k − 1 ) = = ( − 1 ) k − 1 Z ∆ 2 k − 1 ∏ 0 ≤ i < j ≤ 2 k − 1 e ¯ h ( u i − u j + 1 2 ) α i j ( ¯ h α ) ∧ k ( a 0 ⊗ . . . ⊗ a 2 k − 1 ) d u 1 · · · d u 2 k − 1 = ( − 1 ) k − 1 Z 1 0 d s Z ∆ 2 k − 1 ∏ 0 ≤ i < j ≤ 2 k − 1 e ¯ h ( u i − u j + 1 2 ) α i j ( ¯ h α ) ∧ k ( a 0 ⊗ . . . ⊗ a 2 k − 1 ) d u 1 · · · d u 2 k − 1 = ( − 1 ) k − 1 2 k − 1 ∑ l = 0 Z u l + 1 u l d s Z ∆ 2 k − 1 ∏ 0 ≤ i < j ≤ 2 k − 1 e ¯ h ( u i − u j + 1 2 ) α i j ( ¯ h α ) ∧ k ( a 0 ⊗ . . . ⊗ a 2 k − 1 ) d u 1 · · · d u 2 k − 1 = ( − 1 ) k − 1 2 k − 1 ∑ l = 0 ( − 1 ) l Z ∆ 2 k ∏ 0 ≤ i < j ≤ 2 k − 1 e ¯ h ( u i − u j + 1 2 ) α i j ( ¯ h α ) ∧ k ( a 0 ⊗ . . . ⊗ a 2 k − 1 ) d u 1 · · · d u l d sd u l + 1 · · · d u 2 k − 1 . In the l -th term of the sum we now change variables v 1 = s v 2 = u l + 1 + s . . . v 2 k − l = u 2 k − 1 + s v 2 k − l + 1 = u 1 + s . . . v 2 k = u l + s . Now let σ l ∈ S 2 k be the cyclic permutation σ l ( 1, . . . 2 k ) = ( l , . . . , 2 k , 1 , . . . l − 1 ) . W ith this, the l -th term can be written as ( − 1 ) l Z σ l ( ∆ 2 k ) ∏ 1 ≤ i < j ≤ 2 k e ¯ h ψ ( v σ l ( i ) − v σ l ( j ) ) α i j ( ¯ h α ) ∧ k ( a 0 ⊗ . . . ⊗ a 2 k − 1 ) d v 1 · · · d v 2 k , where ψ : R → [ − 1, 1 ] is the function introduced in [ F E F E S H , § 2.4 ]. A s in the proof of Lemma 2.2. of loc. cit., the expression above is equal to ( − 1 ) l Z ∆ 2 k ∏ 1 ≤ i < j ≤ 2 k e ¯ h ( v i − v j + 1 2 ) α i j ( ¯ h α ) ∧ k ( a l ⊗ . . . ⊗ a 2 k − 1 ⊗ a 0 ⊗ . . . a l − 1 ) d v 1 · · · d v 2 k . CYCLIC COCYCL E S O N DE FORMA TIO N QUANTIZA TIO NS 9 On the other hand we have B τ 2 k ( a 0 ⊗ . . . ⊗ a 2 k − 1 ) = 2 k − 1 ∑ l = 0 ( − 1 ) l τ 2 k ( 1 ⊗ a l ⊗ . . . ⊗ a 2 k − 1 ⊗ a 0 ⊗ . . . ⊗ a l − 1 ) = ( − 1 ) k 2 k − 1 ∑ l = 0 ( − 1 ) l µ 2 k − 1 Z ∆ 2 k ∏ 1 ≤ j < l ≤ 2 k e ¯ h ( u i − u j + 1 2 ) α i j × ( ¯ h α ) ∧ k ( a l ⊗ . . . ⊗ a 2 k − 1 ⊗ a 0 ⊗ . . . ⊗ a l − 1 ) d u 1 . . . d u 2 k . One ﬁnally concludes that the two sides of the claimed equality coincide.  Lemma 2.7. One has b τ 2 k = τ 2 k + 1 . Proof. The proof of the cla im pro ceeds along the lines of the proof of Pro position 2.1. in [F E F E S H ]. Introduce the d ifferential form η ∈ Ω 2 k ( ∆ 2 k + 1 , C 2 k + 1 ( W 2 n ) ) by η : = ( − 1 ) k µ 2 k + 1 2 k + 1 ∑ i = 1 ∏ 0 ≤ j < l ≤ 2 k + 1 e ¯ h ( u j − u l + 1 2 ) α j l      u 0 = 0 d u 1 ∧ . . . ∧ c d u i ∧ . . . ∧ du 2 k + 1 π i 2 k , where π i 2 k ∈ End  W ⊗ 2 k + 2 2 n  is ( ¯ h α ) ∧ k acting on all slots in the tensor product except the zero-th a nd the i -th. It then follows that b τ 2 k = Z ∂ ∆ 2 k + 1 η = Z ∆ 2 k + 1 d η . W e have d η = ( − 1 ) k µ 2 k + 1 2 k + 1 ∑ i = 1 ( − 1 ) i 2 k + 1 ∑ s = 0 ¯ h α i s π i 2 k ∏ 0 ≤ j < l ≤ 2 k + 1 e ¯ h ( u j − u l + 1 2 ) α j l d u 1 ∧ . . . ∧ d u 2 k + 1 . W e now claim that 2 k + 1 ∑ i = 1 ( − 1 ) i 2 k + 1 ∑ s = 0 ¯ h α i s π i 2 k = ( ¯ h α ) ∧ ( k + 1 ) ∈ End  W ⊗ ( 2 k + 2 ) 2 n  . Indeed one can split the sum as 2 k + 1 ∑ i = 1 ( − 1 ) i 2 k + 1 ∑ s = 0 ¯ h α i s π i 2 k = 2 k + 1 ∑ i = 1 ( − 1 ) i ¯ h α i 0 π i 2 k + 2 k + 1 ∑ i = 1 ( − 1 ) i ¯ h 2 k + 1 ∑ s = 1 α i s π i 2 k . The ﬁrst part equals ( ¯ h α ) ∧ ( k + 1 ) , whereas the second equals ze ro: the α i j all c om- mute among each other , the number of te rms 2 k ( 2 k − 1 ) is e ven, they cancel p a ir- wise.  This completes the proof of Theorem 2 .4. As a cor ollary we have of course: Corollary 2.8. T he cochains τ 2 k , 0 ≤ k ≤ n combine to deﬁne a cocy cle ( τ 0 , τ 2 , . . . , τ 2 n ) ∈  T ot 2 n B C • ( W poly 2 n ) , b + B  , Remark 2.9. In particular , b τ 2 n = 0 , wh ich is the statement in [ F E F E S H ] that τ 2 n is a Hochschild cocycle, genera ting the Hochschild cohomology in degree 2 n . In other words, we have completed this Hochschild c ocycle τ 2 n to a full cyclic cocycle ( τ 0 , τ 2 , . . . , τ 2 n ) in the ( b , B ) -complex. Notice the similarity of this cocycle with the so-called JLO-cocycle [ J A L E O S ]. 10 M.J. PFLAUM, H. POSTHUMA, AND X. T ANG 2.3. The sp 2 n -action. For a ny algebra A , denote by gl ( A ) the a ssociated Lie al- gebra given by A equipped with the Lie bra cket [ a 1 , a 2 ] = a 1 a 2 − a 2 a 1 . This Lie algebra acts on the Hochschild chains by L a ( a 0 ⊗ . . . ⊗ a k ) = k ∑ i = 0 ( a 0 ⊗ . . . ⊗ [ a , a i ] ⊗ . . . ⊗ a k ) . The Cartan formula L a = b ◦ ι a + ι a ◦ b holds wi th respect to the Hochschild differ- ential, if we deﬁne ι a : C k ( A ) → C k + 1 ( A ) by ι a ( a 0 ⊗ . . . ⊗ a k ) = k ∑ i = 0 ( − 1 ) i + 1 ( a 0 ⊗ . . . ⊗ a i ⊗ a ⊗ a i + 1 ⊗ . . . ⊗ a k ) . Dually , these formulas induce Lie algebra actions of gl ( A ) on C • ( A ) and C • ( A ) . Recall that sp 2 n acts on W poly 2 n by inner derivations where we identify sp 2 n with the homogeneous quadratic polynomials in W poly 2 n . Proposition 2.1 0. Th e cochains τ 2 k ∈ C 2 k ( W poly 2 n ) , 0 ≤ k ≤ n are invariant and basic with respect to s p 2 n , i.e., L a τ 2 k = 0 and ι a τ 2 k = 0 for all a ∈ sp 2 n . Proof. The proof is literally the same a s for the Hochschild cocycle τ 2 n , cf. [ F E F E S H , Thm 2.2.].  This property of the cocycle ( τ 0 , . . . , τ 2 n ) ∈ T ot 2 n B C • ( W poly 2 n ) is important in the next section where we apply the Fedosov construction to globalize these cocycles to deformed algebras over arbitrary symplectic manifolds. 3. C Y C L I C C O C Y C L E S O N S Y M P L E C T I C M A N I F O L D S Let ( M , ω ) be a symplectic manifold with symplectic form ω . W e study in this section the c yclic cohomology of a def ormation quantization A ¯ h of ( M , ω ) . In particular , we construct an explicit chain map from the space of differential forms on M to the space of cyclic cochains on the quantum algebra A ¯ h . 3.1. Deformation quant ization of symplectic manifolds. For the convenience of the reader let us brieﬂy review Fedosov’s c onstruction of a deformation quantiza- tion of a symplectic manifold ( M , ω ) . W e ﬁrst extend the W eyl algebra W poly ( V ) for a symplectic v e ctor space ( V , ω ) to W + ( V ) and W ( V ) . Le t y 1 , · · · , y 2 n be a symplectic basis of V with y 2 i − 1 = p i , y 2 i = q i for 1 ≤ i ≤ 2 n . Then W + ( V ) consists of e lements of the form ∑ i 1 , · · · , i 2 n , i ≥ 0 ¯ h i a i , i 1 , · · · , i 2 n y i 1 · · · y 2 n with a i , i 1 , · · · , i 2 n constant. It is ea sy to check that the product ⋆ on W extends to a well deﬁned associative product on W + ( V ) . Furthermore, we deﬁne W ( V ) to be W + ( V ) [ ¯ h − 1 ] . Observe that the standard symplectic Lie group S p ( 2 n , R ) lifts to act on W ( V ) and W + ( V ) . Let F M be the symplectic f rame bundle of T M , which is a principal Sp ( 2 n , R ) -b undle. W e consider the following associated bundle W = F M × Sp 2 n W + V , which is usually called the W eyl algebra b undle. W e ﬁx a symplectic con- nection ∇ on T M , which lifts to a conn ection ˜ ∇ on W . L et R ∈ Ω 2  M ; sp ( T M )  CYCLIC COCYCL E S O N DE FORMA TIO N QUANTIZA TIO NS 11 be the curvature of ∇ . Then ˜ ∇ 2 is equal to 1 ¯ h [ ˜ R , − ] ∈ Ω 2  M ; End ( W )  , where ˜ R is obtained from R via the embedding sp 2 n ֒ → W + 2 n . Assign d e g ( y i ) = 1 , and d e g ( ¯ h ) = 2, and denote W ≥ k be subset of W with degree greater than or equal k . Fedosov proved in [ F E 9 5 ] that there exists a smooth section ˜ A ∈ Ω 1 ( M ; W ≥ 3 ) such that D = ˜ ∇ + 1 ¯ h [ A , − ] deﬁnes a ﬂat connection on W , which means that D 2 = 0 ∈ Ω 2  M ; End ( W )  . This implies that the W eyl curvature Ω of D , which is deﬁned b y Ω = ˜ R + ˜ ∇ ( A ) + 1 2 ¯ h [ A , A ] is in the center of W since D 2 = 1 ¯ h [ Ω , − ] . S ince the center of W + 2 n is given by C [ [ ¯ h ] ] , Ω = − ω + ¯ h ω 1 + · · · is a closed form in Ω 2  M ; C [ [ ¯ h ] ]  . By [ F E 9 5 ] it follows that the sh eaf A ¯ h D of ﬂ at sections with respect to D is isomorphi c to C ∞ M [[ ¯ h ] ] as a C [ [ ¯ h ] ] -module sheaf. Moreover , the induced product on C ∞ ( M ) [ [ ¯ h ]] d eﬁnes a star product on M . The connection D is usually called a Fedosov connection on W . In the following we will refer to A ¯ h D ( M ) a s the quantum algebra a ssociated to D , a nd will often de note it for short by A ¯ h D or A ¯ h if no c onfusion can arise. The algebra of sections with compact support of the sheaf A ¯ h D will be denoted by A ¯ h cpt . Finally , let us remark that gauge equivalent D a nd D ′ deﬁne isomorphic sheaves of algebras A ¯ h D and A ¯ h D ′ , and that any formal deformation quantiza tion of M can be obtained in this way . 3.2. Shufﬂe product on Hochschi ld chains. In this part, we review the construc- tion of shufﬂ e pr oduct on Hochschild chains. Let A be a graded algebra with a de- gree 1 derivation ∇ . Recall that the shuf ﬂe product between a 0 ⊗ · · · ⊗ a p ∈ C p ( A ) and b 0 ⊗ · · · ⊗ b q ∈ C q ( A ) is deﬁned to be ( a 0 ⊗ · · · ⊗ a p ) × ( b 0 ⊗ · · · ⊗ b q ) = = ( − 1 ) deg ( b 0 ) ( ∑ j deg ( a j ) ) Sh p , q ( a 0 b 0 ⊗ a 1 ⊗ · · · ⊗ a p ⊗ b 1 ⊗ · · · ⊗ b q ) , where Sh p , q ( c 0 ⊗ · · · ⊗ c p + q ) = ∑ σ ∈ S p , q sgn ( σ ) c 0 ⊗ c σ ( 1 ) ⊗ · · · ⊗ c σ ( p + q ) with sum over all ( p , q ) -shufﬂes in S p + q . In [ E N F E , Sec. 2], E N G E L I – F E L D E R considered differential graded a lgebras, a nd studied the properties of the shufﬂe p roduct of a Hochschild chain with a Maurer- Cartan element in the differential graded algebra . Due to the needs of our applica- tion here to deformation quantization, we consider a generalized Maurer-Cartan element ω which means a degree 1 element of A such that ∇ ω + ω 2 / ¯ h + ˜ R = Ω is in the ce nter of A a nd ˜ R is a degree 2 element. W e prove the following analogous properties of shufﬂe products with ω as in [ E N F E ]. Lemma 3.1. Let ω ∈ A be such that Ω − ˜ R = ∇ ω + ω 2 / ¯ h. Put ( ω ) k : = 1 ⊗ ω ⊗ · · · ⊗ ω ∈ C k ( A ) . Then one has for all a = a 0 ⊗ · · · ⊗ a p ∈ C p ( A ) b ( a × ( ω ) k ) = b ( a ) × ( ω ) k + ( − 1 ) p a × b ( ω ) k − ( − 1 ) p k ∑ i = 0 ( a 0 ⊗ · · · ⊗ [ ω , a i ] ⊗ · · · ⊗ a p ) × ( ω ) k − 1 , (3.1) where [ a , a ′ ] for a , a ′ ∈ A is th e graded commutator between a a nd a ′ . 12 M.J. PFLAUM, H. POSTHUMA, AND X. T ANG Proof. This is literally the same as the proof of [E N F E , Lemma 2.6 ].  Lemma 3.2. For ω as in Lemma 3.1 and k ≥ 1 b ( ω ) 0 = 0 and b ( ω ) k = ¯ h ∇ ( ( ω ) k − 1 ) + ¯ h k − 1 ∑ j = 1 ( − 1 ) j 1 ⊗ ω ⊗ · · · ⊗ ( Ω − ˜ R ) ⊗ · · · ⊗ ω . Let us remark a t this point that Lemma 3.2 is slightly different from [ E N F E , Le mma 2.5] because of the existence of Ω . Proof. First check b ( ( ω ) 0 ) = b ( 1 ) = 0. Then observe that for k ≥ 1 b ( ω ) k = b ( 1 ⊗ ω ⊗ · · · ⊗ ω ) = ω ⊗ · · · ⊗ ω + + k − 1 ∑ j = 1 ( − 1 ) j 1 ⊗ ω ⊗ · · · ⊗ ω 2 ⊗ · · · ⊗ ω + ( − 1 ) k ( − 1 ) k − 1 ω ⊗ · · · ⊗ ω = k − 1 ∑ j = 1 ( − 1 ) j 1 ⊗ ω ⊗ · · · ⊗ ¯ h ( Ω − ˜ R − ∇ ω ) ⊗ · · · ⊗ ω = ¯ h ∇ ( ( ω ) k − 1 ) + ¯ h k − 1 ∑ j = 1 ( − 1 ) j 1 ⊗ ω ⊗ · · · ⊗ ( Ω − ˜ R ) ⊗ · · · ⊗ ω .  Lemma 3.3. For ω as in Lemma 3.1 and every a ∈ C l ( A ) one has B ( a × ( ω ) k ) = Ba × ( ω ) k . Proof. The claim follows by a straightforward computation: B ( a × ( ω ) k ) = = B  ( − 1 ) deg ( b 0 ) ( ∑ j deg ( a j ) ) Sh l , k ( a 0 ⊗ a 1 ⊗ · · · ⊗ a l ⊗ ω ⊗ · · · ⊗ ω )  = k + l ∑ i = l + 1 ( − 1 ) i ( k + l ) 1 ⊗ Sh l , k ( ω ⊗ · · · ⊗ ω ⊗ a 0 ⊗ a 1 ⊗ · · · ⊗ a l ⊗ ω ⊗ · · · ⊗ ω ) + l ∑ i = 1 ( − 1 ) i ( k + l ) 1 ⊗ Sh l , k ( a i ⊗ · · · ⊗ a l ⊗ ω ⊗ · · · ⊗ ω ⊗ a 0 ⊗ a 1 ⊗ · · · ⊗ a i − 1 ) = l ∑ i = 0 ( − 1 ) i l Sh l + 1, k ( 1 ⊗ a i ⊗ · · · ⊗ a l ⊗ a 0 ⊗ · · · ⊗ a i + 1 ⊗ ω ⊗ · · · ⊗ ω ) = B a × ( ω ) k .  3.3. Cyclic cocycles on deformation quantiza tions of symplectic mani folds. In this section, we study the cyclic cohomolog y of the quantum algebra A ( ( ¯ h ) ) D : = A ¯ h D [ ¯ h − 1 ] , which is the kernel of a Fedosov connection D = d + 1 ¯ h [ A , − ] on W [ ¯ h − 1 ] . Note that since D is a local opera tor we in fact obtain a sheaf of quantum algebr a s on CYCLIC COCYCL E S O N DE FORMA TIO N QUANTIZA TIO NS 13 M , which we also denote by A ( ( ¯ h )) D . Let A ( ( ¯ h )) cpt be its space of sections with com- pact support. W e will deﬁne in this section a n S- morphism Q be tween the mixed complexes  Ω • ( M ) , d , 0  and  C •  A ( ( ¯ h )) cpt  , b , B  . In the construction of Q we will use the mixed sheaf complex  C •  A ( ( ¯ h ) )  , b , B  de- ﬁned in Appendix A.2 and Theorem A.3 which tells that the complex of its global section spaces is quasi-isomorphi c to the mixed complex  C •  A ( ( ¯ h ) ) cpt  , b , B  . In the followin g deﬁnitions, we consider the shuf ﬂe product on the Hochschild chains of the gra ded algebra W ⊗ C ∞ ( M ) Ω • ( M ) ( ( ¯ h )) with a degree 1 deriva tion ∇ , the symplectic connection, and a generalized Ma urer-Cartan element A , the Fedosov connection. Remark 3 . 4. The c yclic c ocycle ( τ 0 , . . . , τ 2 n ) ∈ T ot 2 n B C • ( W poly 2 n ) deﬁned in Def. 2. 2 extends uniq uely to a continuous cyclic cocycle on the algebra W with the sa me properties as Prop. 2.10. Deﬁnition 3.5. Deﬁne Ψ i 2 k ∈ Ω i ( M ) ⊗ C ∞ ( M )  W ⊗ ( 2 k − i + 1 )  ∗ ( M ) by putting Ψ i 2 k  a 0 ⊗ · · · ⊗ a 2 k − i  : =  1 ¯ h  i τ 2 k  ( a 0 ⊗ · · · ⊗ a 2 k − i ) × ( A ) i  . T o explain this deﬁnition a bit more: for a given point x ∈ M , we have used the natural identiﬁcation of the ﬁber of W [ ¯ h − 1 ] over x with the W eyl a lgebra W 2 n in the formula above. The cochain τ 2 k is deﬁned as in Deﬁnition 2.2, ( − ) ∗ denotes the dual bundle functor , and a 0 , · · · , a 2 k − i are ger ms of smooth sections of W at x . It is important to remark that the de ﬁnition above d oes not depend on the decomposition D = ∇ + A of the Fedosov connection: a different choice amounts to ad ding a s p 2 n valued one-form to A . By Proposition 2 .10, this yields the same result. Proposition 3.6. For every chain a 0 ⊗ · · · ⊗ a 2 k − i ∈ C 2 k + 1 − i  A ( ( ¯ h ) ) cpt  the above deﬁned Ψ i 2 k satisﬁes the following equality: ( − 1 ) i d Ψ i 2 n − 2 k ( a 0 ⊗ · · · ⊗ a 2 k + 1 − i ) = Ψ i + 1 2 n − 2 k ( b ( a 0 ⊗ · · · ⊗ a 2 k + 1 − i ) ) + Ψ i + 1 2 n − 2 k + 2 ( B ( a 0 ⊗ · · · ⊗ a 2 k + 1 − i ) ) . (3.2) Proof. T o prove E q. ( 3.2), apply τ 2 k to Eq. (3.1) with ω = A and check that  1 ¯ h  i τ 2 k ( b ( a × ( A ) i ) ) = =  1 ¯ h  i τ 2 k ( b ( a ) × ( A ) i ) + ( − 1 ) 2 k − i + 1  1 ¯ h  i τ 2 k ( a × b ( A ) i ) − ( − 1 ) 2 k − i + 1 2 k + 1 − i ∑ j = 0  1 ¯ h  i − 1 τ 2 k  ( a 0 ⊗ · · · ⊗ 1 ¯ h [ A , a j ] ⊗ · · · · · · ⊗ a 2 k + 1 − i ) × ( A ) i − 1  (3.3) 14 M.J. PFLAUM, H. POSTHUMA, AND X. T ANG for ever y a = a 0 ⊗ · · · ⊗ a 2 k + 1 − i ∈ C 2 k + 1 − i  A ( ( ¯ h ) ) cpt  . Recall that by deﬁnition, every a j ∈ A ( ( ¯ h ) ) satisﬁes the equality ∇ a j + 1 ¯ h [ A , a j ] = 0. Therefore, we have τ 2 k ( ( a 0 ⊗ · · · ⊗ 1 ¯ h [ A , a j ] ⊗ · · · ⊗ a 2 k + 1 − i ) × ( A ) i − 1 ) = = − τ 2 k ( ( a 0 ⊗ · · · ⊗ ∇ a j ⊗ · · · ⊗ a 2 k + 1 − i ) × ( A ) i − 1 ) . By Lemma 3.2, we obtain 1 ¯ h a × b ( ( A ) i ) = = a × ∇ ( ( A ) i − 1 ) + i − 1 ∑ j = 1 ( − 1 ) j a × ( 1 ⊗ A ⊗ · · · ⊗ ( Ω − ˜ R ) ⊗ · · · ⊗ A ) . (3.4) Recall that Ω ∈ Ω 2 ( M , C [ [ ¯ h ]] ) is in the center of W and ˜ R is in the image of sp 2 n in W . Therefore, since the τ 2 k are r educed sp 2 n basic cochains by Prop. 2.10, τ 2 k ( a × ( 1 ⊗ A ⊗ · · · ⊗ ( Ω − ˜ R ) ⊗ · · · ⊗ A ) ) = 0. Applying τ 2 k to Eq. (3.4), one gets 1 ¯ h τ 2 k ( a × b ( A ) i ) = τ 2 k ( a × ∇ ( ( A ) i − 1 ) ) . Therefore, we have that ( − 1 ) 2 k + 1 − i  1 ¯ h  i τ 2 k ( a × b ( A ) i ) − ( − 1 ) 2 k + 1 − i 2 k + 1 − i ∑ j = 0  1 ¯ h  i − 1 τ 2 k ( ( a 0 ⊗ · · · ⊗ 1 ¯ h [ A , a j ] ⊗ · · · · · · ⊗ a 2 k + 1 − i ) × ( A ) i − 1 ) = ( − 1 ) 2 k + 1 − i  1 ¯ h  i − 1 τ 2 k ( a × ∇ ( ( A ) i − 1 ) ) + ( − 1 ) 2 k + 1 − i 2 k + 1 − i ∑ j = 0  1 ¯ h  i − 1 τ 2 k ( ( a 0 ⊗ · · · ⊗ ∇ a j ⊗ · · · ⊗ a 2 k + 1 − l ) × ( A ) i − 1 ) = ( − 1 ) 2 k + 1 − i  1 ¯ h  i − 1 d τ 2 k ( a × ( A ) i − 1 ) . Applying Corollary 2 .8, we have that b τ 2 k ( a ⊗ ( A ) i ) = − B τ 2 k + 2 ( a × ( A ) i ) = − τ 2 k + 2 ( B ( a × ( A ) i ) ) = − τ 2 k + 2 ( B ( a ) × ( A ) i ) ( by Le mma 3.3 ) . (3.5) Eq. (3.5) entails  1 ¯ h  i τ 2 k ( b ( a × ( A ) i ) ) = −  1 ¯ h  i τ 2 k + 2 ( B ( a ) × ( A ) i ) . CYCLIC COCYCL E S O N DE FORMA TIO N QUANTIZA TIO NS 15 Now going back to Eq. (3.3), we obtain −  1 ¯ h  i τ 2 k + 2 ( B ( a ) × ( A ) i ) = =  1 ¯ h  i τ 2 k ( b ( a ) × ( A ) i ) + ( − 1 ) i − 1  1 ¯ h  i − 1 d τ 2 k ( a × ( A ) i − 1 ) . (3.6) But this is equivalent to ( − 1 ) i − 1  1 ¯ h  i d τ 2 k ( a × ( A ) i ) = =  1 ¯ h  i + 1 τ 2 k + 2 ( B ( a ) × ( A ) i + 1 ) +  1 ¯ h  i + 1 τ 2 k ( b ( a ) × ( A ) i + 1 ) , which by the deﬁnition of Ψ i 2 k entails Eq. (3.2).  Deﬁnition 3.7. For e very i , r with 2 r ≤ i a nd every open U ⊂ M d e ﬁne a mor- phism χ i − 2 r i , U : Ω i ( U ) ( ( ¯ h )) → C i − 2 r  A ( ( ¯ h ) ) cpt  ( U ) by χ i − 2 r i , U ( α ) ( a 0 ⊗ · · · ⊗ a i − 2 r ) = Z U α ∧ Ψ 2 n − i 2 n − 2 r ( a 0 ⊗ · · · ⊗ a i − 2 r ) , where α ∈ Ω i ( U ) ( ( ¯ h )) a nd a 0 , · · · , a i − 2 r ∈ A ( ( ¯ h )) cpt ( U ) . The integra l converges because a 0 , . . . , a i − 2 r have compact support. Obviously , the χ i − 2 r i , U form the local components of sheaf morphisms χ i − 2 r i : Ω i M ( ( ¯ h ) ) → C i − 2 r  A ( ( ¯ h ))  . Using these, deﬁne further sheaf morphisms χ i : Ω i M ( ( ¯ h ) ) → T ot i B C • ( A ( ( ¯ h )) ) by (3.7) χ i = ∑ 2 r ≤ i χ i − 2 r i . The χ i have the following crucial property . Proposition 3.8. For ev ery α ∈ Ω • ( U ) ( ( ¯ h )) with U ⊂ M op en one has ( b + B ) χ • ( α ) = χ • ( d α ) . Proof. W riting out the deﬁnition of χ , we have to show that Z M d α ∧ ∑ 2 r ≤ i + 1 Ψ 2 n − i − 1 2 n − 2 r ( a 0 ⊗ · · · ⊗ a i + 1 − 2 r ) = = Z M α ∧ ∑ 2 r ≤ i + 1 Ψ 2 n − i 2 n − 2 r ( b ( a 0 ⊗ · · · ⊗ a i + 1 − 2 r ) ) + + Z M α ∧ ∑ 2 r ≤ i + 1 Ψ 2 n − i 2 n − 2 r + 2 ( B ( a 0 ⊗ · · · ⊗ a i + 1 − 2 r ) ) 16 M.J. PFLAUM, H. POSTHUMA, AND X. T ANG holds true f or all chains a 0 ⊗ · · · ⊗ a i − 2 r + 1 ∈ C 2 k + 1 − i  A ( ( ¯ h ) ) cpt  . Since M is a closed manifold, by integration by parts, this equality is equivalent to ( − 1 ) i Z M α ∧ ∑ 2 r ≤ i + 1 d Ψ 2 n − i − 1 2 n − 2 r ( a 0 ⊗ · · · ⊗ a i + 1 − 2 r ) = = Z M α ∧ ∑ 2 r ≤ i + 1 Ψ 2 n − i 2 n − 2 r ( b ( a 0 ⊗ · · · ⊗ a i + 1 − 2 r ) ) + + Z M α ∧ ∑ 2 r ≤ i + 1 Ψ 2 n − i 2 n − 2 r + 2 ( B ( a 0 ⊗ · · · ⊗ a i + 1 − 2 r ) ) . This is a coroll ary of Prop. 3 .6  As a corol lary of Proposition 3.8, we obtain for every i a sheaf morphism Q i : T ot i B Ω • M ( ( ¯ h ) ) : = M 2 r ≤ i Ω i − 2 r M ( ( ¯ h ) ) → T ot i B C •  A ( ( ¯ h ) )  which over U ⊂ M open evaluated on forms α i − 2 r ∈ Ω i − 2 r ( U ) ( ( ¯ h )) gives (3.8) Q i U  ∑ 2 r ≤ i α i − 2 r  = 1 ( 2 π √ − 1 ) n ∑ 2 r ≤ i χ i − 2 r , U ( α i − 2 r ) , where we have viewed χ i − 2 r , U ( α i − 2 r ) as an element in T ot i B C •  A ( ( ¯ h ))  ( U ) via the embedding T ot i − 2 r B C •  A ( ( ¯ h ) )  ֒ → T ot i B C •  A ( ( ¯ h ) )  . Theorem 3.9. The above deﬁned sheaf morphism Q :  T ot • B Ω • M ( ( ¯ h ) ) , d  →  T ot • B C • ( A ( ( ¯ h ) ) ) , b + B  is an S-morphism between mixed coch ain complex es of sheaves and a qu asi-isomorphism of the sheaves of cyclic cochains. Proof. By Propositio n 3. 8, Q is a morphism of sheaf complexes. T ogether with Eqs. (3.7) and (3 .8) this entails that Q is an S -morphism. T o p rove the second claim it therefore sufﬁces by [ L O , Prop. 2.5. 15] that the  χ i i  i ∈ N form a quasi- isomorphism of sheaf complexes χ : Ω • M ( ( ¯ h ) ) → C •  A ( ( ¯ h ))  . This follows f rom a spectral sequence argument provided in the following. W e remark that χ does not preserve the ¯ h -ﬁltration on both complexes. Therefore, we need to modify χ i i by 1 ¯ h i − n without changing the ﬁnal conclusion. Under this change, we will have a cochain map χ : ( Ω • M ( ( ¯ h ) ) , ¯ hd ) → ( C •  A ( ( ¯ h ) )  , b ) compat- ible with the ¯ h -ﬁltration. Then we consider the induced morphi sm on the c orre- sponding spectral sequences. The E 0 terms of C •  A ( ( ¯ h ) )  is equal to the localized Hochschild cochain sheaf complex C •  C ∞ ( M ) ( ( ¯ h ))  , which is quasi-isomorphic to the sheaf of de Rham currents on M , cf. [ C O ] . The induced differential on E 0 under this quasi-isomorphism is dual to the Poisson differential on the sheaf of differential forms on M As all the a bove sheaves are ﬁne, it sufﬁcient to prove the claim over e a ch ele- ment of an open cover of M where each of its open sets is symplectic diffeomor- phic to an open contractible subset of R 2 n equipped with the standard symplectic form: a Darboux chart. W e check that the induced χ i i on E 0 over such open set U is CYCLIC COCYCL E S O N DE FORMA TIO N QUANTIZA TIO NS 17 a quasi-isomorphism. Over U , the E 0 component ˜ χ i i of χ i i in Def . 3.7 is computed to be (3.9) ˜ χ i i ( α ) ( a 0 ⊗ a 1 ⊗ · · · ⊗ a i ) = Z U α ∧ ∗ ( a 0 d a 1 ∧ · · · ∧ d a i ) , where ∗ : Ω i M → Ω 2 n − i M is the symplectic Hodge sta r operator on M introduced by Brylinski [B R ]. By the identity d π = ( − 1 ) i ∗ d ∗ for the Poisson homology differential d π on Ω i + 1 M : Z U d α ∧ ∗ ( a 0 d a 1 ∧ · · · ∧ d a i + 1 ) = = ( − 1 ) i Z U α ∧ d ∗ ( a 0 d a 1 ∧ · · · ∧ d a i + 1 ) = Z U α ∧ ∗ ( d π ( a 0 d a 1 ∧ · · · ∧ d a i + 1 ) ) . (3.10) Combining Eq. (3.9)-(3.10), we see that ˜ χ i i maps the d e Rham differential on Ω • M ( ( ¯ h ) ) to a differ ential on the cohomology of C •  C ∞ ( M ) ( ( ¯ h ))  , which is dual to the Pois - son differential d π . Ther efore, we conclude that on each U , the chain map ( ˜ χ i i ) i ∈ N is a quasi-isomorphism at the E 0 level. This proves that ( χ i i ) i ∈ N is a quasi-isomorphism.  Corollary 3.10. Over global sections, Q induces an S-quasi-isomorphism Q :  T ot • B Ω • ( M ) ( ( ¯ h )) , d  →  T ot • B C •  A ( ( ¯ h )) cpt  , b + B  . 4. A L G E B R A I C I N D E X T H E O R E M S In this section we study Connes’ pairing betwee n the K - theory of A ( ( ¯ h ) ) cpt and a cocycle Q ( c ) ∈ T ot • B C • ( A ( ( ¯ h ) ) cpt ( M ) ) , where c is an element in T ot • B Ω • ( M ) = L 2 l ≤ • Ω • − 2 l ( ( ¯ h ) ) . This results in an algebra ic index theorem which computes this pairing in terms of topological data of the underlying manifold M . 4.1. The pairing bet ween cycli c cohomology and K -theory. W e start with br ieﬂy reviewing the general theory [ L O , S ec. 8.3] of a pairing between cyclic cohomology and K 0 group of a unital algebra. Let A be a unital a lgebra over a ﬁeld k a nd let e be an idempotent of A . The Chern character Ch k ( e ) is a cocycle in B 2 k ( A ) = A ⊗ A ⊗ ( 2 k ) ⊕ A ⊗ A ⊗ ( 2 k − 2 ) ⊕ · · · ⊕ A deﬁned by the following formulas Ch k ( e ) = ( c k , c k − 1 , · · · , c 0 ) ∈ B 2 k ( A ) , where c i = ( − 1 ) i ( 2 i ) ! i ! ( e − 1 2 ) ⊗ e ⊗ ( 2 i ) ∈ A ⊗ A 2 i for i = 1, . . . , k c 0 = e ∈ A . (4.1) It is e a sy to check that Ch k ( e ) is b + B closed. One then deﬁnes a pairing between a ( b + B ) -cocycle φ = ( φ 2 k , · · · , φ 0 ) and a projection e ∈ A by the ca nonical pairing 18 M.J. PFLAUM, H. POS THUMA, AND X. TAN G between C k ( A ) and C k ( A ) , h φ , e i : = h φ , Ch k ( e ) i = k ∑ l = 0 ( − 1 ) l ( 2 l ) ! l ! φ 2 l  ( e − 1 2 ) ⊗ e ⊗ · · · ⊗ e  . This constructio n descends to cohomology and yields the desired pairing H C k ( A ) × K 0 ( A ) → k . Now let M be a symplectic manifold, and A ( ( ¯ h )) ( M ) be a Fedosov deformation quantization of M as constructed in the previous section. W e apply the above to obtain a pa iring between the cyclic cohomology H C • ( A ( ( ¯ h ) ) cpt ) a nd the K 0 group of A ( ( ¯ h )) cpt ( M ) . ( T o deﬁne the Chern charac te r like (4.1), we usually adjoin a unit to the algebra A ( ( ¯ h ) ) cpt ( M ) . ) Recall from [F E 9 5 , 6.1 ] that an element in K 0  A ( ( ¯ h ) ) cpt  can be represented by a pairing of projections P 0 , P 1 in M k  A ( ( ¯ h ))  for some k ≥ 0 such that P 0 − P 1 is compactly supported. (B y M k  A ( ( ¯ h ))  we mean the a lgebr a of k × k -matr ices with coefﬁcient in A ( ( ¯ h ) ) .) The set of all such pa irs of projections forms a semi-group. It is pro ved in [ F E 9 5 , 6.1] that mo dulo stabilization this semi-gr oup is isomorphic to the K -group of M . Now let φ be a ( b + B ) -cocycle of A ( ( ¯ h )) which has degree 2 k . Then the pairing b e tween φ = ( φ 0 , · · · , φ 2 k ) and e = ( P 1 , P 2 ) a representative o f a K -group element of A ( ( ¯ h ) ) is deﬁned as h φ , e i : = h φ , P 1 i − h φ , P 2 i . 4.2. Quantiz ation twisted by a vector bu ndle. In the following, we explain how to reduce the computation of the above pa iring to the trivial case that e = 1 in A ( ( ¯ h )) . Deﬁne p 1 = P 1 | ¯ h = 0 and p 2 = P 2 | ¯ h = 0 . Since P 1 ⋆ P 1 = P 1 and P 2 ⋆ P 2 = P 2 , the matrice s p 1 and p 2 are p rojections in M n ( C ∞ ( M ) ) a nd therefore d eﬁne vector bundles V 1 and V 2 on M . Furthermore, V 1 and V 2 are isomo rphic outside a compact of M . Following [ F E 9 5], we can twist the quantum algebra A ¯ h by the bundles V 1 and V 2 . W e consider the twisted W e yl algebra bundles W V 1 = W ⊗ End ( V 1 ) and W V 2 = W ⊗ End ( V 2 ) . Fixing connections ∇ 1 and ∇ 2 on V 1 resp. V 2 , we obtain connections ∇ V 1 = ∇ ⊗ 1 + 1 ⊗ ∇ 1 and ∇ V 2 = ∇ ⊗ 1 + 1 ⊗ ∇ 2 on W V 1 resp. W V 2 . F E D O S O V proved in [ F E 9 5 ] that there a re ﬂat connections D V 1 = ∇ V 1 + 1 ¯ h [ A V 1 , − ] and D V 2 = ∇ V 2 + 1 ¯ h [ A V 2 , − ] on W V 1 resp. W V 2 such that the algebra of ﬂat sec- tions f orms a def ormation quantization twisted by V 1 resp. V 2 . The corresponding deformation quantization sheaf is denoted by A ( ( ¯ h ) ) V 1 resp. A ( ( ¯ h )) V 2 . Observe that the cocycle ( τ 0 , · · · , τ 2 n ) on W poly 2 n can be extended to the algebra W poly , V 2 n : = W poly 2 n ⊗ End ( V ) for any ﬁnite dimensional vector space V by putting τ V 2 k  ( a 0 ⊗ M 0 ) ⊗ · · · ⊗ ( a 2 k ⊗ M 2 k )  : = τ ( a 0 ⊗ · · · ⊗ a 2 k ) tr ( M 0 M 1 · · · M 2 k ) . CYCLIC COCYCL E S O N DE FORMA TIO N QUANTIZA TIO NS 19 As in the proof of Cor ollary 2.8 one checks that ( τ V 0 , · · · , τ V 2 n ) is a ( b + B ) -cocycle on W poly , V 2 n . Hence we can extend Def. 3.5 to deﬁne twisted Ψ i V j ,2 k for j = 1 , 2 by Ψ i V j ,2 k  a 0 ⊗ · · · ⊗ a 2 k − i  = =  1 ¯ h  i τ V j 2 k  ( a 0 ⊗ · · · ⊗ a 2 k − i ) × ( A V j ) i  , where a 0 , · · · , a 2 k − i are germs of smooth sections of W V j at x . Mor eover , we deﬁne sheaf morphisms χ i − 2 l V j , i : Ω i M ( ( ¯ h ) ) → C i − 2 l  A ( ( ¯ h ) ) V j  by setting over U ⊂ M open χ i − 2 l V j , i , U ( α ) ( a 0 ⊗ · · · ⊗ a i − 2 l ) : = Z M α ∧ Ψ 2 n − i V j ,2 n − 2 l  a 0 ( x ) · · · ⊗ a i − 2 l ( x )  , where α ∈ Ω i ( U ) ( ( ¯ h )) and where a 0 , · · · , a 2 k − i ∈ A ( ( ¯ h )) V j , cpt ( U ) are sections of the twisted de formation quantization sheaf with compact support in U . Like in Sec- tion 3.3 we then obtain S-quasi-isomorphisms of mixed sheaf complexes Q V j :  T ot • B Ω • M ( ( ¯ h ) ) , d  →  T ot • B C • ( A ( ( ¯ h )) V j ) , b + B  , j = 1, 2. Over global sections, Q V j then induces an S-quasi-isomorphism Q V j :  T ot • B Ω • ( M ) ( ( ¯ h )) , d  →  T ot • B C •  A ( ( ¯ h )) V j , cpt  , b + B  . Generalizing [C H D O , Thm. 3] , we have the follow ing proposition: Proposition 4.1 . For any a c losed d ifferen tial α ∈ T ot • B Ω • ( M ) ( ( ¯ h )) and projections P 1 and P 2 of A ( ( ¯ h ) ) with P 1 − P 2 compactly supported, one has h Q ( α ) , P 1 − P 2 i = h Q V 1 ( α ) , 1 i − h Q V 2 ( α ) , 1 i . Proof. The proof of [C H D O , Thm. 3] applies verbatim.  4.3. Lie al ge bra cohomology . In the f ollowing para graphs, we use L ie algebra cohomolog y to determine the pairing h Q V ( α ) , 1 i locally for a vector bundle V on M . B y deﬁnition, the pairing h Q V ( α ) , 1 i for an element α = ( α 0 , · · · , α 2 k ) ∈ T ot 2 k B Ω • ( M ) ( ( ¯ h )) is equal to D 1 ( 2 π √ − 1 ) n ∑ l ≤ k χ V ,2 k − 2 l ( α 2 k − 2 l ) , 1 E = D 1 ( 2 π √ − 1 ) n ∑ l ≤ k , j ≤ k − l χ 2 k − 2 l − 2 j V ,2 k − 2 l ( α 2 k − 2 l ) , 1 E = 1 ( 2 π √ − 1 ) n ∑ l ≤ k , j ≤ k − l ( − 1 ) k − l − j ( 2 k − 2 l − 2 j ) ! ( k − l − j ) ! Z M α 2 k − 2 l ∧ Ψ 2 n − 2 k + 2 l V ,2 n − 2 j ( 1 ⊗ · · · ⊗ 1 ) . Now observe that Ψ 2 n − 2 k + 2 l V ,2 n − 2 j ( 1 ⊗ · · · ⊗ 1 ) vanishes when j < k − l since τ 2 n − 2 j is a normalized cochain. Hence h Q V ( α ) , 1 i = ∑ l ≤ k 1 ( 2 π √ − 1 ) n Z M α 2 k − 2 l ∧ Ψ 2 n − 2 k + 2 l V ,2 n − 2 k + 2 l ( 1 ) = = k ∑ l = 0 1 ( 2 π √ − 1 ) n Z M α 2 l ∧ Ψ 2 n − 2 l V ,2 n − 2 l ( 1 ) . 20 M.J. PFLAUM, H. POS THUMA, AND X. TAN G These considerations show that f or the computation of the pairing betwee n an el- ement α ∈ T ot • B Ω • ( M ) ( ( ¯ h )) a nd a class in K 0  A ( ( ¯ h ) )  it is sufﬁcient to de te rmine Ψ 2 n − 2 l V ,2 n − 2 l ( 1 ) for all l ≤ n . T o achieve this goal we will a pply methods from Lie algebra cohomology , namely the Chern–W eil homomorphism. T o this end let us ﬁrst review the standard map from the Hochschild cochain complex to the corresponding Lie algebra cochain complex, which can be found in [ L O ] . Let A be a unital algebra . Cons ider Lie algebra gl N ( A ) of N × N -matrices with coefﬁcients in A . There is a chain map φ N from the Hochschild cochain complex C • ( A ) to the Lie algebra cochain complex C •  gl N ( A ) ; gl N ( A ) ∗  : φ N ( c )  ( M 1 ⊗ a 1 ) ⊗ · · · ⊗ ( M k ⊗ a k )  ( M 1 ⊗ a 1 ) = ∑ σ ∈ S k sgn ( σ ) c ( a 0 ⊗ a σ ( 1 ) ⊗ · · · ⊗ a σ ( k ) ) tr ( M 0 M σ ( 1 ) · · · M σ ( k ) ) . (4.2) W e deﬁne Θ V , N ,2 k to be φ N ( τ V 2 k ) ∈ C 2 k  gl N ( W V 2 n ) ; gl N ( W V 2 n ) ∗  . It is easy to check that Ψ 2 n − 2 k V ,2 n − 2 k ( 1 ) =  1 ¯ h  2 n − 2 k 1 ( 2 n − 2 k ) ! Θ V ,2 n − 2 k ( A ∧ · · · ∧ A ) ( 1 ) . Proposition 4.2 . For any k ≤ n, Θ V , N ,2 k ( 1 ) is a cocy cle in the relative Lie algebra cohomology complex C 2 k  gl N ( W V 2 n ) , gl N ⊕ gl V ⊕ sp 2 n  and satisﬁes Θ N V ,2 n ( p 1 ∧ q 1 ∧ · · · ∧ p n ∧ q n ) = N dim ( V ) . Proof. Since 1 is in the center of W V 2 n , we have the following equation ∂ Lie ( ( Θ V , N ,2 k ) ( 1 ) ) = ∂ Lie ( Θ V , N ,2 k ) ( 1 ) . On the right hand side of the above equation, Θ V , N ,2 k is viewed as a Lie algebra cochain in C 2 k  gl N ( W V 2 n ) ; gl N ( W V 2 n ) ∗  . Furthermore, since φ N is a morphism of cochain complexes, we have that ∂ Lie Θ V , N ,2 k ( 1 ) = ∂ Lie φ N ( τ V 2 k ) = φ N ( b ( τ V 2 k ) ) . Since ( τ V 0 , · · · , τ V 2 n ) is a ( b + B ) -cocycle, we have b ( τ V 2 k ) = − B ( τ V 2 k + 2 ) a nd ∂ Lie Θ V , N ,2 k ( 1 ) = − φ N ( B ( τ V 2 k + 2 ) ) ( 1 ) . Now we compute φ N ( B ( τ V 2 k + 2 ) ) ( 1 )  ( a 1 ⊗ M 1 ) ⊗ · · · ⊗ ( a 2 k + 1 ⊗ M 2 k + 1 )  = ∑ σ ∈ S 2 k + 1 sgn ( σ ) B ( τ V 2 k + 2 ) ( 1 ⊗ a σ ( 1 ) ⊗ · · · ⊗ a σ ( 2 k + 1 ) ) · tr ( M σ ( 1 ) · · · M σ ( 2 k + 1 ) ) = ∑ σ ∈ S 2 k + 1 ∑ i sgn ( σ ) τ V 2 k + 2 ( 1 ⊗ a σ ( i ) ⊗ · · · ⊗ a σ ( 2 k + 1 ) ⊗ 1 ⊗ a σ ( 1 ) ⊗ · · · ⊗ a σ ( i − 1 ) ) · · tr  M σ ( 1 ) · · · M σ ( 2 k + 1 )  = 0. One concludes that Θ V , N ,2 k ( 1 ) is a closed 2 k -cocycle in C 2 k  gl N ( W V 2 n ) ; C ( ( ¯ h ))  . Since τ 2 k is a normalized cochain, one can easily check that Θ V , N ,2 k is in fact a cocycle relative to the Lie subalgebra gl N ⊕ gl V of gl N ( W V 2 n ) . The fac t that Θ V , N ,2 k is a cocycle relative to sp 2 n is a corollary of Proposition 2.10. Thus the claim is proven.  4.4. Local Rie m ann-Roch t heorem. In this subsection, we use Chern-W eil the- ory to compute the Lie algebra cocycle Θ V , N ,2 k , usin g the strategy in the p roof of [ F E F E S H , Thm. 5.1] . CYCLIC COCYCL E S O N DE FORMA TIO N QUANTIZA TIO NS 21 W e star t with recalling the constructio n of the Chern-W eil homomorphism. Let g be a Lie algebra and h a L ie subalgebra with a n h -inva riant pr ojection pr : g → h . The curvature C ∈ Hom ( ∧ 2 g , h ) of pr is deﬁned by C ( u ∧ v ) : = [ pr ( u ) , pr ( v ) ] − pr ( [ u , v ] ) . Let ( S • h ∗ ) h be the algebra of h -invariant polynomials on h gra ded by polynomial degree. Deﬁne the homomorphism ρ : ( S • h ∗ ) h → C 2 • ( g , h ) by ρ ( P ) ( v 1 ∧ · · · ∧ v 2 q ) = 1 q ! ∑ σ ∈ S 2 q , σ ( 2 i − 1 ) < σ ( 2 i ) ( − 1 ) σ P  C ( v σ ( 1 ) , v σ ( 2 ) ) , · · · , C ( v σ ( 2 q − 1 ) , v σ ( 2 q ) )  . The right hand side of this equation d eﬁnes a cocycle, and the induced map in cohomolog y ρ : ( S • h ∗ ) h → H 2 • ( g , h ) is independent of th e choice of the projection pr. This is the Chern–W eil homomorphism . In our case, we consider g = gl N ( W V 2 n ) and h = gl N ⊕ gl V ⊕ sp 2 n . The projection pr : g → h is deﬁned by pr ( M 1 ⊗ M 2 ⊗ a ) : = : = 1 N a 0 tr ( M 2 ) M 1 + 1 dim ( V ) a 0 tr ( M 1 ) M 2 + 1 N dim ( V ) tr ( M 1 ⊗ M 2 ) a 2 , where a j is the component of a homogeneous of d egree j in y , M 1 ∈ gl N , and M 2 ∈ gl V . The essential point about the Chern-W eil homom orphism in this case is contained in the following result. Proposition 4.3. For N ≫ n and q ≤ 2 k , th e Chern-W eil homomorph ism ρ : ( S q h ∗ ) h → H 2 q ( g , h ) is an isomorphism. Proof. The proof of this result goes along the same lines as the pro of of Pr oposition 4.2 in [F E F E S H ].  Recall the f ollowing invariant polynomials on the Lie algebras gl N and sp 2 n : First on gl N we have the Chern character Ch ( X ) : = tr ( exp X ) , f or X ∈ gl N . On sp 2 n , we have the ˆ A -genus: ˆ A ( Y ) : = det  Y / 2 sinh ( Y /2 )  1/ 2 , for Y ∈ sp 2 n . W e will need the rescaled version ˆ A ¯ h ( Y ) : = ˆ A ( ¯ hY ) . W ith this, we can now state: Theorem 4.4. In H 2 k  gl N ( W V 2 n ) , gl N ⊕ gl V ⊕ sp 2 n  we have the identity [ Θ V , N ,2 k ] = ρ ( ( ˆ A ¯ h Ch V Ch ) k ) for k ≤ n and N ≫ 0 . Proof. When k = n the equality is proved in [ F E F E S H , Thm. 5.1]. Actually , one can literally repeat the constructions and arguments in the p roof of [ F E F E S H , Thm. 5.1 ] for all k ≤ n . W e remark that we have different sign convent ion with respect to [ F E F E S H , Thm. 5.1] due to the change of sign in the cocycle τ 2 n , cf. Remark 2.3 .  22 M.J. PFLAUM, H. POS THUMA, AND X. TAN G 4.5. Higher a lgebraic i ndex theorem. In this section, we use Theorem 4 .4 to com- pute the pairing h Q ( α ) , P 1 − P 2 i . Theorem 4.5. For a sequence of closed forms α = ( α 0 , · · · , α 2 k ) ∈ T ot 2 k B Ω • ( M ) ( ( ¯ h )) and two projectors P 1 , P 2 in A ( ( ¯ h ) ) with P 1 − P 2 compactly supported, one has h Q ( α ) , P 1 − P 2 i = k ∑ l = 0 1 ( 2 π √ − 1 ) l Z M α 2 l ∧ ˆ A ( M ) Ch ( V 1 − V 2 ) exp  − Ω 2 π √ − 1 ¯ h  , where V 1 and V 2 are v ector bundles on M determined by the zer o-th order terms of P 1 and P 2 . Proof. According to Proposition 4.1, h Q ( α ) , P 1 − P 2 i = h Q V 1 ( α ) , 1 i − h Q V 2 ( α ) , 1 i . Furthermore, by the argum ents at the beginning of S e ction 4.3, h Q V i ( α ) , 1 i , i = 1, 2 is given by (4.3) ∑ l ≤ k 1 ( 2 π √ − 1 ) n Z M α 2 l ∧ Ψ 2 n − 2 l V ,2 n − 2 l ( 1 ) . Moreover , recall that Ψ 2 n − 2 l V ,2 n − 2 l ( 1 ) is equal to 1 ( 2 n − 2 l ) ! ¯ h n − l Θ V ,2 n − 2 l ( A ∧ · · · ∧ A ) ( 1 ) . Note that the dir ect sum wi th trivia l bundles does not change the value of the pa ir- ing. Therefor e, we ca n add a large enough trivial bundle to both V 1 and V 2 so that we can apply Theorem 4.4 to compute Θ V ,2 n − 2 l . For vector ﬁelds ξ 1 , · · · , ξ 2 n − 2 l on M we have Θ V , N ,2 n − 2 l ( 1 ) ( A ∧ · · · A ) ( ξ 1 , · · · , ξ 2 n − 2 l ) = = ( 2 n − 2 l ) ! ρ ( ( ˆ A ¯ h Ch V Ch ) 2 n − 2 l ) ( A ( ξ 1 ) ∧ · · · ∧ A ( ξ 2 n − 2 l ) ) = ( 2 n − 2 l ) ! ( n − l ) ! ∑ σ ( 2 j − 1 ) < σ ( 2 j ) sgn ( σ ) P 2 n − 2 l  C ( A ( ξ σ ( 1 ) ) , A ( ξ σ ( 2 ) ) ) , · · · , C ( A ( ξ σ ( 2 n − 2 l − 1 ) ) , A ( ξ σ ( 2 n − 2 l ) ) )  , where P 2 n − 2 l = ( ˆ A ¯ h Ch V Ch ) n − l ∈ ( S n − l h ) ∗ h . By [ F E F E S H , Thm. 5. 2], for two any vector ﬁelds ξ , η on M , C ( A ( ξ ) , A ( η ) ) is equal to ˜ R V ( ξ , η ) + ˜ R ( ξ , η ) − Ω ( ξ , η ) , where ˜ R (and ˜ R V ) is the lifting of the curva ture of the bundle T M (and V ) and Ω is the curvature for the Fe d osov connection. Therefor e, we have Θ V , N ,2 n − 2 l ( 1 ) ( A ∧ · · · A ) ( ξ 1 , · · · , ξ 2 n − 2 l ) =( 2 n − 2 l ) ! ρ ( P 2 n − 2 l ) ( ( R V + R − Ω ) 2 n − 2 l ) . Replacing Ψ 2 n − 2 l V ,2 n − 2 l ( 1 ) by 1 ¯ h n − l ( 2 n − 2 l ) ! Θ V , N ,2 k in Equation 4.3, we obtain  Q V 1 ( α ) − Q V 2 ( α ) , 1  = ∑ l ≤ k 1 ( 2 π √ − 1 ) l Z M α 2 l ∧ ˆ A ( M ) C h ( V 1 − V 2 ) exp  − Ω 2 π √ − 1 ¯ h  . This completes the proof.  CYCLIC COCYCL E S O N DE FORMA TIO N QUANTIZA TIO NS 23 5. G E N E R A L I Z AT I O N T O O R B I F O L D S In this se c tion we show how the previous constructions can be generalize d to orbifolds. The result is an a lgebr aic index theorem for ( b + B ) -cocycles on certain formal deformations of proper ´ etale groupoids, which in turns generalize s the in- dex formula for traces in [ P F P O T A ] . 5.1. Prelimin a ries. Let ( M , ω ) be a symplectic orbifold, i.e., a paracompact Haus- dorff space locally modeled on a quotient of an open subset of R 2 n , equipped with the standard symplectic form, by a ﬁnite subgroup Γ ⊂ S p ( 2 n , R ) . As an abstract notion of an atlas, we ﬁx a proper ´ etale groupoid G 1 ⇒ G 0 with the property that G 0 / G 1 ∼ = M , a nd G 0 is equipped with a G -invariant symplectic form ω . Denot- ing the two structure maps of the groupoid by s , t : G 1 → G 0 , this means that s ∗ ω = t ∗ ω . Remark that for any symplectic orbifold, such a groupoid always ex- ists and is unique up to Morita equivalence. Associated to the groupoid G is its convolution algebra A ⋊ G : = C ∞ cpt ( G 1 ) with product given by convolution: ( f 1 ∗ f 2 ) ( g ) : = ∑ g 1 g 2 = g f 1 ( g 1 ) f 2 ( g 2 ) , where f 1 , f 2 ∈ C ∞ cpt ( G 1 ) and g ∈ G 1 . The symplectic structure on G equips A ⋊ G with a noncommutative Poisson struc- ture, that is, a degree 2 Hochschild cocycle whose Gerstenhaber brac ke t with itself is a coboundary . Let A ¯ h be a G -invariant deformation quantization of ( G 0 , ω ) , for example given by Fedosov’s method, using an invariant connection as is explained in [ F E 0 2 ]. This mea ns that A ¯ h forms a G -sheaf of algebras over G 0 , and we can take the crossed pr oduct A ¯ h ⋊ G : = Γ cpt  G 1 , s − 1 A ¯ h  with algebra structure [ a 1 ⋆ c a 2 ] g = ∑ g 1 g 2 = g ( [ a 1 ] g 1 g 2 ) [ a 2 ] g 2 , for a 1 , a 2 ∈ s − 1 A ¯ h ( G 1 ) and g ∈ G 1 . This is a noncommutative algebra def orming the convolution algebra of the un- derlying groupoid. In [ N E P F P O T A ], the cyclic cohomology of A ( ( ¯ h )) ⋊ G was computed to be given by (5.1) H C • ( A ( ( ¯ h ) ) ⋊ G ) = M r ≥ 0 H • − 2 r  e M , C ( ( ¯ h ))  , where e M is the so-called inertia orbifold which we will now describe. Intr oduce the “space of loops” B ( 0 ) given by B ( 0 ) : = { g ∈ G 1 | s ( g ) = t ( g ) } . In the sequel, we denote by σ 0 the local embedding obtained as the composition of the canonical embedding B ( 0 ) ֒ → G 1 with the source map s . If no confusion can arise, we also denote the embedding B ( 0 ) ֒ → G 1 by σ 0 . The groupoid G acts on B ( 0 ) and the associate action groupoid Λ G : = B ( 0 ) ⋊ G turns out to be proper and ´ etale as well. It therefore models a nother orbifold e M : = B ( 0 ) / G called the inertia orbifold. As done in the previous sections for smooth manifolds, we will lift the isomor- phism (5. 1) to a morphism of cochain complexes where on one side we have a complex of differential forms and on the other side Conn es’ ( b , B ) -complex. There 24 M.J. PFLAUM, H. POS THUMA, AND X. TAN G are two natural choices for a de Rham-type of complex that computes the coho- mology of e M . One is to use a simplicial resolution of B ( 0 ) given by the so-called “higher Burghelea spaces” B k and the a ssociated simplicial d e Rham complex. The other one, and this is the complex we use, is to use G -invariant differential forms on B ( 0 ) . The fact that the two models compute the same cohomology is true be- cause G is a proper gr oupoid. It was observed in [ C R ] that Λ G is a so-called cyclic groupoid, that is, comes equipped with a canonical nontrivial section θ : B ( 0 ) → Λ G 1 of both sour ce a nd target map. In this case θ is given by θ ( g ) = g , g ∈ B ( 0 ) . A s a consequence of this, when we pull-ba ck the shea f A ¯ h to B ( 0 ) , it comes equipped with a ca nonical section θ ∈ Aut ( ι − 1 ( A ¯ h G ) ) . As we have seen, for a smooth symplectic manifold, the local model for a def or- mation quantization was given by the W eyl algebra. In this case, it is given by the W eyl algebr a together with an a utomorphism. 5.2. Adding an a utomorphism to the W eyl alge bra. A s remarked in § 2 . 1, the symplectic group Sp ( 2 n , R ) acts on the W eyl a lgebra W poly 2 n by a utomorphisms. Let us ﬁx an e lement γ ∈ Sp ( 2 n , R ) of ﬁnite order . It induces a dec omposition of V = R 2 n into two components, V = V ⊥ ⊕ V γ , where V γ is the subspac e of ﬁxed points. Since γ is a linear symplectic transformation, this d e composition is symplectic, and we put l : = dim ( V ⊥ ) /2. Adding the a utomorphism γ to the deﬁnition of cyclic cohomolo gy has quite an effect in th e sense that we now have Proposition 5.1. T he twisted cyclic cohomology of the Weyl algebra is given b y H C k γ  W poly 2 n  = ( C [ ¯ h , ¯ h − 1 ] , if k = 2 n − 2 l + 2 p , p ≥ 0 0, else . W e will now give an explicit generator for the nonzero class in cyclic cohomol- ogy . Le t A and ˜ A be algebr a s over a ﬁeld k , possibly equipped with a utomor- phisms γ ∈ Aut ( A ) and ˜ γ ∈ Aut ( ˜ A ) . The Alexa nder–Whitney map deﬁnes a cochain map # : C • ( A ) ⊗ C • ( ˜ A ) → C • ( A ⊗ ˜ A ) , where the Hochschild differentials are twisted b y resp. γ , ˜ γ and γ ⊗ ˜ γ . According to the Eilenberg–Zilber theorem, this is in fact a quasi-isomorphism. The c yclic version of this theorem, cf. [ L O , § 4.3], states that the map above c a n be completed to a quasi-isomorphis m of the cochain complexes T ot • B C • . Below we will only be interested in the ca se wh ere one of the two cochains is of degree 0, that means a twisted trace. Recall that a ˜ γ -twisted trace on ˜ A is a linear functional tr ˜ γ : ˜ A → k satisfying (5.2) tr ˜ γ ( ˜ a 1 ˜ a 2 ) = tr ˜ γ ( ˜ γ ( ˜ a 2 ) ˜ a 1 ) for all ˜ a 1 , ˜ a 2 ∈ ˜ A . Lemma 5.2 . Let ψ = ( ψ 0 , . . . , ψ 2 k ) ∈ T ot 2 k B C • ( A ) be a γ - t wisted ( b + B ) -cocycle on A and tr a ˜ γ -twisted trace on ˜ A. Then the cochain ψ # tr = ( ψ 0 # tr, . . . , ψ 2 k # tr ) is a γ ⊗ ˜ γ -twisted cocycle of degree 2 k in T ot • B C • ( A ⊗ ˜ A ) . CYCLIC COCYCL E S O N DE FORMA TIO N QUANTIZA TIO NS 25 Proof. Explicit computation.  In our case, we have W poly 2 n = W poly 2 l ⊗ W poly 2 n − 2 l according to the decomposition V = V ⊥ ⊕ V γ of the underlying symplectic vector space. Notice that b y deﬁnition, the automorphism γ ∈ Sp 2 n is trivial on W poly 2 n − 2 l . Therefore we can simply use the cyclic cocycle ( τ 0 , . . . , τ 2 n − 2 l ) of d egree 2 n − 2 l on this part of the tensor product. On the transversal part, i.e., associated to V ⊥ = R 2 l we use the twisted trace tr γ : W poly 2 l → C [ ¯ h , ¯ h − 1 ] constructed by Fedosov in [ F E 0 2 ]: For this, we choose a γ -invaria nt complex structure on V ⊥ , identifying V ⊥ ∼ = C l so that γ ∈ U ( l ) . The inverse Caley transform c ( γ ) = 1 − γ 1 + γ is an anti-hermitian matrix, i.e., c ( γ ) ∗ = − c ( γ ) . W ith this, d e ﬁne tr γ ( a ) : = µ 2 l  det − 1 ( 1 − γ − 1 ) exp  ¯ h c ( γ − 1 ) i j ∂ ∂ z i ∂ ∂ ¯ z j  a  , where c ( γ − 1 ) i j is the inverse matrix of c ( γ − 1 ) and where we sum over the repeated indices i , j = 1, . . . , l . It is pr oved in [ F E 0 2 , Thm. 1.1], that this functional is a γ - twisted trace density , i.e., satisﬁes equation ( 5 .2). Clearly , tr γ ( 1 ) = det − 1 ( 1 − γ − 1 ) , so the cohomology class of tr γ is independent o f the chosen polariza tion. W ith this we have: Proposition 5.3. Let γ ∈ Sp ( 2 n , R ) . Then th e # -product ( τ 0 # tr γ , . . . , τ 2 n − 2 l # tr γ ) ∈ T ot 2 n − 2 l B C • ( W poly 2 n ) deﬁnes a nontrivial γ -twisted cocycle of degree 2 n − 2 l on th e W eyl alg ebra. 5.3. Cyclic cocycles on formal deformat ions of proper ´ etale groupoids. In this section we will show how to use the twisted ( b + B ) -cocycle of the previous sec- tion to construct a rbitrary ( b + B ) - cocycles on formal deformations of pr oper ´ etale groupoids. Consider aga in the Burgh elea spac e B ( 0 ) . Generically , this space wil l not be connected, and has component s of different dimens ions. Introduce the lo- cally constant function ℓ : B ( 0 ) → N by putting ℓ ( g ) equal to half the codimension of the ﬁxed point set of g in a local orbifold char t. Deﬁnition 5.4. Deﬁne Ψ i 2 k ∈ Ω i ( B ( 0 ) ) ⊗ C ∞ ( B ( 0 ) )  ( σ ∗ 0 W ) ⊗ ( 2 k − 2 ℓ − i + 1 )  ∗ by Ψ i 2 k  a 0 ⊗ . . . ⊗ a 2 k − 2 ℓ − i  : = : =  1 ¯ h  i τ θ 2 k − 2 ℓ  ( a 0 ⊗ . . . ⊗ a 2 k − 2 ℓ − i ) × ( σ ∗ 0 A ) i  . Hereby , W is the W eyl algebr a bundle on G 0 for the G -invariant Fed osov defor- mation quantization A ( ( ¯ h ) ) , A is the corresponding connection 1-form on G 0 , and a 0 , . . . , a 2 k − 2 ℓ − i are ger ms of smooth sections of σ ∗ 0 W at a point g ∈ B ( 0 ) . Notice that as a cochain on σ ∗ 0 W , the de gree of Ψ i 2 k varies over the connected compo nents of B ( 0 ) according to the function ℓ introduced above. Proposition 5.5. T he Ψ i 2 k are G -equ ivariant and satisfy the equalities ( − 1 ) i d Ψ i − 1 2 k = Ψ i 2 k ◦ b θ + Ψ i 2 k + 2 ◦ B θ . 26 M.J. PFLAUM, H. POS THUMA, AND X. TAN G Proof. Since the Fedosov connection on G 0 is a ssumed to be G -invaria nt, Ψ i 2 k is easily checked to be G -equivariant. W e observe that b θ ( σ ∗ 0 A ) k = b ( σ ∗ 0 A ) k and B θ ( σ ∗ 0 A ) k = B ( σ ∗ 0 A ) k on G 0 . The proof of the equality follows the same lines as the proof of its untwisted version Propos ition 3.6.  Remark 5.6. In particular , for g ∈ B ( 0 ) , i = 2 n − 2 ℓ ( g ) and k = n , we ﬁnd that over each connected neighbor hood of g ∈ B ( 0 ) d Ψ 2 n − 2 ℓ ( g ) − 1 2 n = Ψ 2 n − 2 ℓ ( g ) 2 n ◦ b θ . Thus the form Ψ 2 n − 2 ℓ 2 n is a “twisted tr a ce de nsity” in the notation of [ P F P O T A , Def. 2.1]. In fact unraveling the d e ﬁnitions, the ide ntity above is exac tly [ P F P O T A , Prop. 4.2]. Deﬁnition 5.7. For 2 r ≤ i , deﬁne sheaf morphisms χ i − 2 r i : Ω i B ( 0 ) ( ( ¯ h ) ) → C i − 2 r ( σ ∗ 0 A ( ( ¯ h ) ) ) , by the formula χ i − 2 r i , U ( α ) ( a 0 , . . . , a i − 2 r ) : = Z B ( 0 ) α ∧ Ψ 2 n − 2 ℓ − i 2 n − 2 r ( σ − 1 0 a 0 , . . . , σ − 1 0 a i − 2 r ) , where U ⊂ B ( 0 ) is open, α ∈ Ω i ( U ) ( ( ¯ h )) , and a 0 , . . . , a 2 k − 2 r ∈ Γ cpt  σ ∗ A ( ( ¯ h ))  . T ogether these morphisms deﬁne sheaf morphisms χ i : Ω i B ( 0 ) ( ( ¯ h ) ) → T ot i B C •  σ ∗ 0 A ( ( ¯ h ) )  , χ i : = ∑ 2 r ≤ i χ i − 2 r i . By an argument similar to the untwisted case we obtain Theorem 5.8. The morphism χ • is a morphism of sheaves of cochain complexes, i.e., ( b + B ) χ • ( α ) = χ • ( d α ) , for all α ∈ Ω • ( U ) and U ⊂ B • open. W ith this we can now deﬁne an S -morphism of mixed G -sheaf complexes over the inertia orbifold ˜ M as follows: Q i : T ot i B Ω • B ( 0 ) ( ( ¯ h ) ) = M 2 r ≤ i Ω i − 2 r B ( 0 ) ( ( ¯ h ) ) → T ot i B C •  A ( ( ¯ h ) )  by Q i  ∑ 2 r ≤ i α i − 2 r  : = 1 ( 2 π √ − 1 ) ℓ ∑ 2 r ≤ i χ i − 2 r ( α i − 2 r ) . Forming global invariant sections we ﬁnally obtain the S-morphism Q : T ot i B Ω • ( ˜ M ) ( ( ¯ h )) = M 2 r ≤ i Ω i − 2 r ( ˜ M ) ( ( ¯ h )) → T ot i B C •  A ( ( ¯ h ) ) ⋊ G  . Proposition 5.9. The map Q is an S-quasi-isomorphism establishing th e isomorphism (5.1) . CYCLIC COCYCL E S O N DE FORMA TIO N QUANTIZA TIO NS 27 5.4. T wisting b y vector bun dles. It is our aim to compute the pairing of the co- cycles in Connes’ ( b + B ) -complex obtained by the ma p Q above with K -theory classes on A ( ( ¯ h ) ) ⋊ G . Let us ﬁrst explain how orbifold ve c tor bundles de ﬁne el- ements in K 0  A ( ( ¯ h ) ) ⋊ G ) . Recall that an orbifold vector bundle is a vector bun- dle V → G 0 together with an action of G . T aking formal differences of isomor- phism classes, these deﬁne the orbifold K - group K 0 orb ( M ) . An orbifold vector bundle deﬁnes a projective A ⋊ G -module Γ cpt ( G 0 , V ) , where f ∈ C ∞ cpt ( G 1 ) acts on ξ ∈ Γ cpt ( G 0 , V ) by ( f · ξ ) ( x ) = ∑ t ( g ) = x f ( g ) ξ ( s ( g ) ) , for x ∈ G 0 . On the other hand, K -theory is stable under formal def ormations, which mea ns that K 0 ( A ¯ h ⋊ G ) ∼ = K 0 ( A ⋊ G ) , where the isomorphism is induced by taking the ze ro’th or der term of a projector in a matrix algebra over A ¯ h ⋊ G . Altogether , we have deﬁned a map K 0 orb ( M ) → K 0 ( A ¯ h ⋊ G ) . It therefore makes sense to pair our cyclic cocycles with formal differences of isomorphism classes of vector bundles. T o compute this pairing we a gain use quantization with values in the vector bundle to extend our c yclic c ocycles. For this, notice that when we pull-bac k an orbifold v e ctor b undle V → G 0 to B ( 0 ) , the c yclic structure θ acts on σ ∗ 0 V . W e therefore consider the algebra W poly , V 2 n = W poly 2 n ⊗ End ( V ) equipped with the a utomorphism γ acting both via Sp ( 2 n ) on W poly 2 n and on the second factor by an element in E nd ( V ) , also denoted by γ . This leads to cochains τ V , γ 2 k  ( a 0 ⊗ M 0 ) ⊗ . . . ⊗ ( a 2 k ⊗ M 2 k )  : = τ γ 2 k ( a 0 ⊗ . . . ⊗ a 2 k ) tr V ( γ M 0 · · · M 2 k ) . for 0 ≤ k ≤ 2 n − 2 l . T ogether these cochains constitute a γ -twisted ( b + B ) -cocycle ( τ V , γ 0 , τ V , γ 2 , . . . , τ V , γ 2 n − 2 l ) ∈ T ot 2 n − 2 l B C • ( W V 2 n ) . W ith this, one genera lizes the deﬁ- nition of Ψ i 2 k , χ • and Q • in the obvious manner to Ψ i V ,2 k , χ V , i and Q i V . Proposition 5.10 . Let α = ( α 0 , . . . , α 2 k ) ∈ T ot 2 k B Ω • ( e M ) be a c lo sed differential form, and P 1 and P 2 projection in the matrix algebras over A ( ( ¯ h ) ) ⋊ G with P 1 − P 2 compactly supported on e M. Then we have  Q ( α ) , P V 1 − P V 2  =  Q V 1 ( α ) − Q V 2 ( α ) , 1  = k ∑ i = 0 Z e M 1 ( 2 π √ − 1 ) ℓ m α 2 i ∧  Ψ 2 n − 2 ℓ − 2 i V 1 ,2 n − 2 i ( 1 ) − Ψ 2 n − 2 ℓ − 2 i V 2 ,2 n − 2 i ( 1 )  . Here t h e function m : e M → N is the locally constant function which coincides for each sector O ⊂ B ( 0 ) with m O , t he order of th e isotopy group of t he principal stratum of O / G ⊂ e M. Proof. The ﬁrst equality is just as in Prop. 4.1. For the second, again observe that the twisted cyclic cocycles are normalized, so we can throw away a ll terms that contain more than one 1 . Finally , the reduction to an integral over e M is as in [ P F P O T A , Prop. 4.4].  28 M.J. PFLAUM, H. POS THUMA, AND X. TAN G 5.5. A twisted Riemann–Roch th eorem. By the previous proposition, it remains to evaluate Ψ 2 n − 2 ℓ − 2 i V ,2 n − 2 i ( 1 ) , wh ich is of course done by interpreting it as a cocycle in Lie algebra cohomology . Deﬁne the inclus ion of Lie algebr a s h ⊂ g by setting g : = gl N  W V , γ 2 n  , h : = gl N ⊕ gl V ⊕ sp γ 2 n , where the superscript γ means taking γ - invariants. W e will now construct Lie a l- gebra cocycles of g relative to h in C • ( g ; h ) as follows. First the standard morphism from Hochschild cochains to Lie a lgebra cochains, cf. E q. (4.2), is still a morphism of cochain complexes when we twist the differentials: φ N :  C • ( M N ( W 2 n ) , M N ( W 2 n ) ∗ ) , b γ  →  C • ( gl N ( W 2 n ) , M N ( W 2 n ) ∗ ) , ∂ Lie , γ  . Here the twisted Lie algebra cochain c omplex is as deﬁned in [ P F P O T A , § 4 .1.]. Second, evaluation at 1 ∈ M n ( W 2 n ) induces a morphism ev 1 :  C • ( gl N ( W γ 2 n ) , M N ( W 2 n ) ∗ ) , ∂ Lie , γ  →  C • ( g , C ( ( ¯ h )) , ∂ Lie  . Notice that this is onl y a morphism of cochain complexes when r estricted to the γ - invariant par t o f gl N ( W V 2 n ) , because the evaluation morphi sm above only respects the module structure of this sub-Lie algebra . W ith this we now have: Proposition 5.11. For k ≤ n the cochain Θ N , γ V ,2 k : = 1 ¯ h k ev 1  φ N ( τ γ 2 k )  ∈ C 2 k ( g ; h , C ( ( ¯ h ) ) ) , is a Lie algebra cocycle relative to h , which means ∂ Lie Θ N , γ V ,2 k = 0. W ith this we have Ψ 2 n − 2 ℓ − 2 r V ,2 n − 2 r ( 1 ) =  1 ¯ h  n − ℓ − r 1 ( 2 n − 2 ℓ − 2 r ) ! Θ N , θ V ,2 n − 2 ℓ − 2 r ( A ∧ . . . ∧ A ) ( 1 ) T o explicitly compute the class [ Θ N , γ V ,2 k ] ∈ H 2 k ( g ; h , C ( ( ¯ h )) ) , we use the Chern–W e il homomorphi sm ρ :  S k h ∗  h → H 2 k ( g ; h , C ( ( ¯ h )) ) , which, by [ P F P O T A , Prop. 5.1], is again an isomorphism for k ≤ n − l and N ≫ n as in the untwisted c a se, cf. Prop. 4 .3. Le t us now describe the ingredients of the unique polynom ial in S k h ∗ that is deﬁned by Θ N , γ V ,2 k . For this we split h = sp 2 n − 2 ℓ ⊕ sp γ 2 ℓ ⊕ gl γ V ⊕ gl N , and write X = ( X 1 , X 2 , X 3 , X 4 ) for an element in h . Deﬁne ( ˆ A ¯ h J γ Ch V , γ Ch ) ( X ) : = ˆ A ¯ h ( X 1 ) J γ ( X 2 ) Ch V , γ ( ¯ hX 3 ) Ch ( X 4 ) , where Ch and ˆ A ¯ h are as before, Ch V , γ is the Chern character twisted by γ . Con- cretely , th is means Ch V , γ ( X 3 ) = tr V ( γ exp ( X 3 ) ) . Finally , J γ is deﬁned by J γ ( X 2 ) : = ∞ ∑ i = 0 1 i ! tr γ ( X 2 ⋆ · · · ⋆ X 2 | {z } i ) , where we use the embedding of sp γ 2 ℓ ⊂ sp 2 n as degree two polynomials in the W eyl algebra. Strictly speaking, this is no t an element of S • ( h ∗ ) h , but we will only need a ﬁnite number of terms in the expansion in the theor em below . In fact, in CYCLIC COCYCL E S O N DE FORMA TIO N QUANTIZA TIO NS 29 the a pplication to the higher index theor em, the speciﬁc element X 2 turns out to be pro-nilpotent . Theorem 5.12. I n H 2 k ( g ; h ) we h ave the equality [ Θ N , γ V ,2 k ] = ρ  ( ˆ A ¯ h J γ Ch V , γ Ch ) k  . Proof. Given Theorem 4.4, this follows as in [P F P O T A , Thm. 5 . 3].  5.6. The higher index theorem for proper ´ etale groupoids. W e ﬁnally arrive at our ma in result. T o state it properly , we nee d to introduce a few character istic classes. Let V be an orbifold vector bundle. Using the cyclic structure θ , we can twist the Chern character of the pull-back σ − 1 0 V to deﬁne Ch θ ( ι − 1 V ) by Ch θ ( ι − 1 V ) : = tr  θ exp  R V 2 π √ − 1  ∈ H e v ( e M ) where R V denotes the curvature of a connection on V . Denote by N , the normal bundle over B ( 0 ) coming from the embedding into G 1 . It is easy to see that the element Ch θ ( λ − 1 N ) : = 2 ℓ ∑ i = 0 ( − 1 ) i Ch θ ( Λ i N ) ∈ H e v ( e M ) , is invertible. If we use R ⊥ to denote the curvature on N , then 2 ℓ ∑ i = 0 ( − 1 ) i Ch θ ( Λ i N ) = de t ( 1 − θ − 1 exp ( − R ⊥ 2 π √ − 1 ) ) . W ith this observation, we can now state: Theorem 5.13. Let α = ( α 2 k , · · · , α 0 ) ∈ T ot 2 k B Ω • ( e M ) ( ( ¯ h )) be a sequen ce of closed forms on the inertia orbifold, and P 1 , P 2 be two pr ojectors in the m a t rix algebra over A ( ( ¯ h ) ) with P 1 − P 2 compactly supported. Then we have h Q ( α ) , P 1 − P 2 i = k ∑ j = 0 Z e M 1 ( 2 π √ − 1 ) j m α 2 j ∧ ˆ A ( e M ) Ch θ ( ι ∗ V 1 − ι ∗ V 2 ) exp ( − ι ∗ Ω 2 π √ − 1 ¯ h ) Ch θ ( λ − 1 N ) , where V 1 and V 2 are the orbifold vector bundles on M determined by t he zero-th order terms of P 1 and P 2 , and m is a loca l constant funct ion deﬁned by t he order of the isotopy group of the principal stratum of a sector O / G ⊂ e M. 6. T H E H I G H E R A N A LY T I C I N D E X T H E O R E M O N M A N I F O L D S The higher a lgebraic index theorems proved in S ection 4 gives us the mea ns to der ive Connes–Moscovici’s higher index theorem in a de f ormation theoretic framework. T o this e nd we ﬁrst recall Alexa nder–Spanier cohomology which is needed to deﬁne a higher a nalytic index for elliptic operators on manifolds and then de termine the cyclic Alexander–Spa nier cohomol ogy . An ¯ h -dependent sym- bol calculus for pseudodifferential o perators gives rise to a deformation quantiza- tion on the cotangent bundle. This together with the computation of the cyclic Alexander–Spanier cohomology enable us to relate the analytic with the alge- braic higher index. The higher algebraic index theo rems c an then be de rived from Thm. 4.5. 30 M.J. PFLAUM, H. POS THUMA, AND X. TAN G 6.1. Alexa nder–Spanie r cohomology. Assume to be given a smooth manifold M . Like in App. A.2 denote by k one of the commutative rings R , R [ [ ¯ h ] ] and R ( ( ¯ h )) , and let O M , k be one of the sheaves C ∞ M , C ∞ M [[ ¯ h ] ] and C ∞ M ( ( ¯ h ) ) , respectively . In other words, O M , k ( U ) : = C ∞ ( U ) ˆ ⊗ k with U ⊂ M open consists of all smooth functions on U with values in k . If no confusion ca n arise, we shortly write O instead of O M , k . For k ∈ N denote by O ˆ ⊠ k the completed exterior te nsor product sheaf which is a sheaf on M k and which is deﬁned by the property O ˆ ⊠ k ( U 1 × · · · × U k ) ∼ = O ( U 1 ) ˆ ⊗ · · · ˆ ⊗O ( U k ) for all U 1 , · · · , U k ⊂ M open, where ˆ ⊗ means the completed bornological tensor product. Put now C k AS ( O ) : = ∆ ∗ k + 1  O ˆ ⊠ k + 1  and deﬁne sheaf map s δ : C k − 1 AS ( O ) → C k AS ( O ) as follows. First observe that C k AS ( O ) ( U ) ∼ = O ˆ ⊠ k + 1  U k + 1  / J  ∆ k + 1 ( U ) , U k + 1  , where J  ∆ k + 1 ( U ) , U k + 1  denotes the ideal of sections of O ˆ ⊠ k + 1 over U k + 1 which vanish on the diagonal ∆ k + 1 ( U ) . Then deﬁne δ f ∈ O ˆ ⊠ k + 1  U k + 1  for f ∈ O ˆ ⊠ k  U k  by the formula δ f = k ∑ i = 0 ( − 1 ) i δ i f , where δ i f ( x 0 , . . . , x k + 1 ) = f ( x 0 , . . . , x i − 1 , x i + 1 , . . . , x k + 1 ) , x 0 , . . . , x k + 1 ∈ U . Additionally , put δ ′ f = k − 1 ∑ i = 0 ( − 1 ) i δ i f . By construction, δ f a nd δ ′ f lie in J  ∆ k + 2 ( U ) , U k + 2  , if f ∈ J  ∆ k + 1 ( U ) , U k + 1  . Hence one can pass to the quotients and obtains maps δ : C k − 1 AS ( O ) ( U ) → C k AS ( O ) ( U ) and δ ′ : C k − 1 AS ( O ) ( U ) → C k AS ( O ) ( U ) which are the components of sheaf maps. Since δ 2 = ( δ ′ ) 2 = 0, we have two sh eaf cochain complexes  C • AS ( O ) , δ  and  C • AS ( O ) , δ ′  . Denote by C • AS ( O ) : = Γ ( M , C • AS ( O ) ) the complex of global sections with differ ential given by δ . This is the Alexander– Spanier co c hain com plex of O . Its cohomology is de noted by H • AS ( O ) and called the Alexander–Spanier cohomolog y of O . In the particular case, where k = R and O = C ∞ M , one recovers the Alexander–Spanier cohomology H • AS ( M ) of M . Proposition 6.1. Let ι : k → C 0 AS ( O ) be the canonical embedding of the locally constant sheaf k into C 0 AS ( O ) . Then k ι → C 0 AS ( O ) δ → C 1 AS ( O ) δ → . . . δ → C k AS ( O ) δ → is a ﬁne re solution of the locally constant sheaf k . More over ,  C • AS ( O ) , δ ′  is contractible. Proof. Obviously , each of the sheaves C k AS ( O ) is ﬁne. So it remains to show that for each x ∈ M the sequence of stalks 0 ֒ → k ι → C 0 AS ( O ) x δ → . . . δ → C k AS ( O ) x δ → . . . CYCLIC COCYCL E S O N DE FORMA TIO N QUANTIZA TIO NS 31 is exact. T o this end note ﬁrst that the sta lk C k AS ( O ) x is given as the inductive limit of quotients O ˆ ⊠ ( k + 1 )  U k + 1  / J  ∆ k + 1 ( U ) , U k + 1  , where U runs through the open neighborho ods of x . Deﬁne now for k ∈ N so-called extra degeneracy maps s k x : O ˆ ⊠ ( k + 2 )  U k + 2  → O ˆ ⊠ ( k + 1 )  U k + 1  , f 7 → f ( x , − ) , and s k + 1, k : O ˆ ⊠ ( k + 2 )  U k + 2  → O ˆ ⊠ ( k + 1 )  U k + 1  , f 0 ⊗ . . . ⊗ f k + 1 7→ f 1 ⊗ . . . ⊗ f k ⊗ f k + 1 f 0 . Additionally put ε x : O ( U ) → k , f 7→ f ( x ) . Then one checks immediately that (6.1) s k + 1 x δ + δ s k x = id for all k ∈ N and s 0 x δ + ι ε = id . This proves the ﬁrst claim. For the pr oof of the second it sufﬁces to verify that (6.2) s k + 1, k δ 0 = id and s k + 1, k δ i = δ i − 1 s k , k − 1 , since then s k + 1, k δ ′ + δ ′ s k , k − 1 = id for all k ∈ N ∗ and s 1,0 δ ′ = id . But Eq. (6.2) is obtained by straightforward computation, and the propositi on fol- lows.  Remark 6. 2 . B y the preced ing result the A lexander–Spanier cohomolo gy has to coincide both with the ˇ Cech coho mology of the locally constant sheaf k and the de Rham cohomology of M with va lues in k (cf. [ S P , C O M O ]) . Let us sketch the con- struction of the corresponding quasi-isomorphisms. T o this end choose an open covering U of M and a subordinate smooth partition of unity ( ϕ U ) U ∈ U . Consider a ˇ Cech cochain c = ( c U 0 ,. . ., U k ) ( U 0 ,. . ., U k ) ∈ N k ( U ) with values in the ring k , where N k ( U ) : = { ( U 0 , . . . , U k ) ∈ U k + 1 | U 0 ∩ . . . ∩ U k 6 = ∅ } is the nerve of the covering. A ssociate to c the Alexander–Spanier cochain ρ U ( c ) ( x 0 , . . . , x k ) = ∑ U 0 . . . U k c U 0 . . . U k ϕ U 0 ( x 0 ) · . . . · ϕ U k ( x k ) . One checks easily that the resulting map ρ U : ˇ C • U ( M , k ) → C • AS ( O ) is a chain map. M oreover , if U is a good covering, i.e. if it is locally ﬁnite and if the in- tersection of each ﬁnite f amily of elements of U is contractible, then ρ U is even a quasi-isomorphism. T o deﬁne a quasi-isomorphism λ : C • AS ( O ) → Ω • ( M , k ) ﬁrst choose a com plete riemannian metric on M , and denote by exp the corresponding exponential function. For f ∈ O ˆ ⊠ ( k + 1 ) ( M k + 1 ) , x ∈ M a nd v 1 , . . . , v k ∈ T x M then put λ ( f ) x ( v 1 , . . . , v k ) : = = 1 k ! ∑ σ ∈ S k sgn ( σ ) ∂ ∂ s 1 . . . ∂ ∂ s k f  x , exp x ( s 1 v σ ( 1 ) ) , . . . , exp x ( s k v σ ( k ) )  | s i = 0 . Clearly , this de ﬁnes a k -va lued smooth k -form λ ( f ) , which vanishes, if one has f ∈ J  ∆ k + 1 ( M ) , M k + 1  . Moreover , one checks easily that λ δ ( f ) = d λ ( f ) . By passing to the quotient C k AS ( M ) = O ˆ ⊠ ( k + 1 ) ( M k + 1 ) / J  ∆ k + 1 ( M ) , M k + 1  we thus 32 M.J. PFLAUM, H. POS THUMA, AND X. TAN G obtain the d esired chain map which is denoted by λ . By [ C O M O ] , λ is a quasi- isomorphism. Remark 6. 3. For later purposes let us present here another representation of Ale- xander-Spanier cochains in ca se O is the sheaf of smooth functions on M . This representation allows a lso for a dualization, i.e. the construction of Alexa nder- Spanier homology groups. T o this end consider an open covering U of M , a nd denote by U k the neighborho od S U ∈ U U k of the diagonal ∆ k ( M ) in M k . Then put (6.3) C k AS ( M , U ) : = C ∞  U k  Obviously ,  C • AS ( M , U ) , δ  then f orms a complex where, in degree k , δ denotes here the Alexander-Spanier differential restricted to C ∞  U k  . Moreover , for every re- ﬁnement V ֒ → U of open coverings, one has a canonical chain map C • AS ( M , U ) → C • AS ( M , V ) . The d irect limit of these chain complexes with respect to U r unning through th e directed set Cov ( M ) of open coverings of M coincides natur a lly with the Alexander-Spanier coch ain complex over M : (6.4) lim − → U ∈ Cov ( M ) C • AS ( M , U ) ∼ = C • AS ( C ∞ ) ( M ) . Hence the direct limit of the cochain complexes C • AS ( M , U ) computes the Alexan- der-Spanier cohomo logy of M . Note that since homology functors commute with direct limits , Alex ander-Spanier cohomol ogy also coincides naturally with the di- rect limit lim − → U ∈ Cov ( M ) H • AS ( M , U ) . Now let C AS k ( M , U ) be the topological dual of C k AS ( M , U ) , i.e. the spa c e of com- pactly supported distributions on M k + 1 . T ra nsposing δ gives r ise to a chain com- plex  C AS • ( M , U ) , δ ∗  , the homology of which is denoted by H AS • ( M , U ) . The in- verse limit (6.5) H AS • ( M ) : = lim ← − U ∈ Cov ( M ) H AS • ( M , U ) is called the Alexander-Spanier homology of M . B y [M O W U , Prop. 1.2] one has for every open covering U of M a natural isomorphis m be tween the Alexander- Spanier homol ogy and ˇ Cech homolo gy (6.6) H AS • ( M , U ) ∼ = ˇ H • ( M , U ) . This implies in particular , that Alexander-Spa nier homology c oincides natura lly with ˇ Cech homology . Moreover , for a good open cover U of M , i.e an open cover such that a ll ﬁnite nonempty intersections of elements of U a re contracible, the homology H AS • ( M , U ) of the cover U then has to coindide with the Alexander- Spanier homol ogy H AS • ( M ) of the total space (cf. [ B O T U , § 15]). By duality of the deﬁning complexes, Alexander-Spanier homolog y and coho- mology pair naturally , which means that in each degree k one has a natural map (6.7) h − , − i : H AS k ( M ) × H k AS ( M ) → R . Let us describe this pairing in some more detail, since we will later need it. L et [ f ] be an Alexander-Spanier cohomology class represented by some cochain f ∈ CYCLIC COCYCL E S O N DE FORMA TIO N QUANTIZA TIO NS 33 C k AS ( M , U ) . Let µ =  [ µ V ]  V ∈ Cov ( M ) be an Alexa nder-Spanier homology c la ss, where the µ V are appropriate cycles in C AS k ( M , V ) . Then, one puts (6.8) h µ , [ f ] i : = µ U ( f ) . It is stra ightforward to check that this deﬁnition of the pairing h µ , [ f ] i does not depend on the choice of representatives for the hom ology classes [ µ V ] r espectively for the cohomolo gy cla ss [ f ] . Besides the above deﬁned shea f complex  C • AS ( O ) , δ  , one ca n d eﬁne the sheaf complex  C • aAS ( O ) , δ  of antisymmetric Alexander–Spanier cochains and the sheaf complex  C • λ AS ( O ) , δ  of cyclic Alexa nder–Spanier cochains. A section of C k AS ( O ) over U ⊂ M open which is represented by some f ∈ O ˆ ⊠ k + 1  U k + 1  is ca lled ant i- symmetric resp. c y clic , if f ( x σ ( 0 ) , . . . , x σ ( k + 1 ) = sgn ( σ ) f ( x 0 , . . . , x k ) for all ( x 0 , . . . , x k ) close to the diagonal and every permutation resp. ever y cyclic permutation σ in k + 1 var ia bles. In the following we show how to determine the cohomolog y of these sheaf complexes. T o this end we ﬁrst deﬁne d egeneracy map s s i , k for 0 ≤ i ≤ k as follows: s i , k : O ˆ ⊠ ( k + 2 )  U k + 2  → O ˆ ⊠ ( k + 1 )  U k + 1  , f 0 ⊗ . . . ⊗ f k + 1 7→ f 0 ⊗ . . . ⊗ f i f i + 1 ⊗ . . . ⊗ f k + 1 . Obviously , these maps s i , k induce sheaf morphisms s i , k : C k + 1 AS ( O ) → C k AS ( O ) . Moreover , one chec ks immediately that the following cosimplicial identities are satisﬁed: δ j δ i = δ i δ j − 1 , if i < j (6.9) s j , k − 1 s i , k = s i , k − 1 s j + 1, k , if i ≤ j (6.10) s j , k δ i =      δ i s j − 1, k − 1 for i < j , id for i = j or i = j + 1, δ i − 1 s j , k − 1 for i > j + 1 . (6.11) Next we introduce the cyclic operators (6.12) t k x : C k AS ( O ) x → C k AS ( O ) x , [ f 0 ⊗ . . . ⊗ f k ] x 7→ ( − 1 ) k [ f 1 ⊗ . . . ⊗ f k ⊗ f 0 ] x Note that the cyclic operator t k x is induced by a globally deﬁned sheaf morphism t k : C k AS ( O ) → C k AS ( O ) . One ea sily checks that the t k satisﬁes the follow ing cyclic identities: t k δ i = δ i − 1 t k − 1 , if 1 ≤ i ≤ k (6.13) t k s i , k = s i − 1, k t k + 1 , if 1 ≤ i ≤ k (6.14)  t k  k + 1 = id . (6.15) This means that the tuple  C k AS ( O ) , δ i , s i , k , t k  is a cyclic cosimplicial sheaf over M . Its cyclic cohomology can be computed as the cohomology of either one of the following complexes: 34 M.J. PFLAUM, H. POS THUMA, AND X. TAN G (1) the total complex of the associated cyclic bicomplex with vertical differen- tials given by δ in even d e gree resp. by − δ ′ in odd d egree, and horizontal differentials given by id − t k in even degree resp. by N k : = ∑ k l = 0  t k  l in odd degree; (2) the complex obtained a s the 0-th cohomolog y of the horizontal differentials in the cyclic bicomplex; in other words this is the cyclic A lexander–Spanier complex C • λ AS ( O ) with differential δ ; (3) the total complex of the associated mixed cochain complex with d ifferen- tials δ and B AS , where B k AS : = N k s 0, k  id − t k − 1  with s 0, k the extra degener- acy deﬁned above. By Propositio n 6.1 the Hochschild cohomology of the mixed complex (3) is given by k in degree 0 and by 0 in all other degrees. Hence th e cyclic cohom ology of th is mixed compl ex coincides with k in even degree and with 0 else. Since the cyclic cohomolog y is also computed by C • λ AS ( O ) x one obtains the cla im about the cyclic Alexander–Spanier coh omology in the following result. Proposition 6. 4. In t he derived category of sheaves on M , both sheaf com p lexes C • AS ( O ) and C • aAS ( O ) are isomorphic to k , whereas C • λ AS ( O ) is isomorph ic to the cyclic sh eaf com - plex k → 0 → k → . . . → 0 → k → 0 → . . . . Moreover , the antisymm et rizat ion ε • : C • AS ( O ) → C • aAS ( O ) , ε k  [ f 0 ⊗ . . . ⊗ f k ]  = ∑ σ ∈ S k + 1 sgn σ ( k + 1 ) ! [ f σ ( 0 ) ⊗ . . . ⊗ f σ ( k ) ] is a quasi-isomorphism. Proof. By the previous considerations it remains only to p rove that ε • is a quasi- isomorphism. T o this end one checks along the lines of the proof of Proposition 6.1 and by using the maps s k x that C • aAS ( O ) is a ﬁne resolution of k , and that ε • is a sheaf morphism between these ﬁne resoluti ons over the identity of k .  Remark 6.5. Abstractly , a cyclic object in a category C is a contravening functor from Connes’ cyclic category ∆ C to C , cf. [ L O , § 6.1]. The cyclic category ∆ C has the remarkable property of being isomo rphic to its opposite ∆ C o p via an e xplicit f unc- tor a s in Prop. 6 .1.11 . in [L O ]. Therefore, out of any cyclic object, one constr ucts a co cyclic object -that is, a covariant functor ∆ C → C - by precomposing with this isomorphism, called the dual. W ith this, one recognizes the cocyclic sheaf C • AS ( O ) as the dual of the cyclic sheaf O ♮ • associated to O a s a sheaf of algebras. Next we construct a quasi-isomorphism fr om the sheaf complex  C k λ AS ( O ) , δ  to the total complex of the mixed sheaf complex  Ω • ( − , k ) , d , 0  . T o this end d eﬁne for 2 r ≤ k and U ⊂ M open a morphism λ k − 2 r k , U : Γ  U , C k AS ( O )  → Ω k − 2 r ( U , k ) CYCLIC COCYCL E S O N DE FORMA TIO N QUANTIZA TIO NS 35 as follows. First let f ∈ O ˆ ⊠ k + 1 ( U k + 1 ) be a representative of a section of C k AS ( O ) over U , let x ∈ U and v 1 , . . . , v k − 2 r ∈ T x M . Then put λ k − 2 r k , U ( f ) x ( v 1 , . . . , v k − 2 r ) : = ( k − 2 r ) ! ( k + 1 ) ! ∑ ν ∈ S 2 r + 1, k − 2 r ∑ σ ∈ S k − 2 r sgn ( ν ) sgn ( σ ) ∂ ∂ s 1 . . . ∂ ∂ s k − 2 r ( ν f )  x , x , . . . , x , exp x ( s 1 v σ ( 1 ) ) , . . . , exp x ( s k − 2 r v σ ( k − 2 r ) )  | s i = 0 Hereby , S p , q denotes the set of ( p , q ) -shufﬂes of the set { 0, . . . , p + q } , and ν f for ν ∈ S k + 1 is deﬁned by ν f ( x 0 , x 1 , . . . , x k ) : = f ( x ν ( 0 ) , x ν ( 1 ) , . . . , x ν ( k ) ) . Obviously , λ k − 2 r k , U ( f ) vanishes, if f vanishes around the diagonal of U k + 1 . Hence one can deﬁne λ k − 2 r k , U  f + J ( ∆ k + 1 , U k + 1 )  : = λ k − 2 r k , U ( f ) which provides us with the desired morphism. B y an immediate computation one checks that for f 0 , . . . , f k ∈ O ( U ) λ k − 2 r k , U ( f 0 ⊗ . . . ⊗ f k ) = = ( k − 2 r ) ! ( k + 1 ) ! ∑ ν ∈ S 2 r + 1, k − 2 r sgn ( ν ) f ν ( 0 ) · . . . · f ν ( 2 r ) d f ν ( 2 r + 1 ) ∧ . . . ∧ d f ν ( k ) . (6.16) Proposition 6.6. Let λ k : C k λ AS ( O ) → T ot k B Ω • ( − , k ) be the sheaf morphism deﬁned by λ k : = ∑ 2 r ≤ k λ k − 2 r k . Then the following r elat io n is satisﬁed: λ k + 1 δ = d λ k . Proof. First check that for 0 ≤ i ≤ k λ k − 2 r k , U  δ i ( f 0 ⊗ . . . ⊗ f k − 1 )  = = ( − 1 ) i λ k − 2 r k , U  1 ⊗ f 0 ⊗ . . . ⊗ f k − 1  = ( − 1 ) i ( k − 2 r ) ! ( k + 1 ) ! ∑ ν ∈ S 2 r , k − 2 r sgn ( ν ) f ν ( 0 ) ⊗ . . . ⊗ f ν ( 2 r − 1 ) d f ν ( 2 r ) ∧ . . . ∧ d f ν ( k − 1 ) , and then that d λ k − 1 − 2 r k − 1, U ( f 0 ⊗ . . . ⊗ f k − 1 ) = = ( k − 2 r − 1 ) ! k ! ∑ ν ∈ S 2 r + 1, k − 2 r − 1 sgn ( ν ) d ( f ν ( 0 ) ⊗ . . . ⊗ f ν ( 2 r ) ) ∧ d f ν ( 2 r + 1 ) ∧ . . . ∧ d f ν ( k − 1 ) = ( k − 2 r ) ! k ! ∑ ν ∈ S 2 r , k − 2 r sgn ( ν ) f ν ( 0 ) ⊗ . . . ⊗ f ν ( 2 r − 1 ) d f ν ( 2 r + 1 ) ∧ . . . ∧ d f ν ( k − 1 ) . By the deﬁnition of δ , these two equations entail the claimed equality .  36 M.J. PFLAUM, H. POS THUMA, AND X. TAN G 6.2. Higher indi c e s. Alexander–S p a nier cohomology has been used by C O N N E S – M O S C O V I C I [ C O M O ] to deﬁne higher (analytic) indices of an elliptic operator act- ing on the space of smoot h sections of a ( hermitian) vector bundle over a closed (riemannian) manifold. M ore precisely , the Connes–Moscovici higher indices c a n be understood as a pairing of the Chern charac ter of a K-theory class deﬁned by an elliptic operator with the cyclic cohomology class de ﬁned by a n Alexa nder– Spanier cohomology class (cf . [ M O W U ] ). Unlike for the K-theoretic formulation of the Atiyah–Singer index formula, where the K-theory of the algebra of smooth sec- tions over the cosphere b undle of the underlying manifold is considered, it turns out that f or the K-theoretic formulation of higher index theorems the appropri- ate algebra is the algebra of trace class operators acting on the Hilbert spa ce of square integrable sections of the given vector bundle. This point of view and the fact that the pseudo-differential calculus on the unde r lying manifold gives rise to a deformation quantization enable us to c ompare the higher analytic index with the higher algebr a ic index a nd then der ive the Connes–Moscovici higher index formula. In the follow ing we provide the details and proceed in sev e ral steps. Step 1. A ssume that Ψ ∈ Ω 2 n ( M ) ⊗ C ∞ ( M ) W ∗ is a trac e density for the star product algebra A ( ( ¯ h ) ) cpt on M . In other wor ds this means that T r : A ( ( ¯ h ) ) cpt → k , a 7 → Z M Ψ ( a ) is a trace functional on A ( ( ¯ h ) ) cpt . Then we deﬁne a chain map X T r : C • AS  C ∞ M ( ( ¯ h ) )  → C •  A ( ( ¯ h ) )  as follows. For f 0 , f 1 , . . . , f k ∈ C ∞ ( U ) ( ( ¯ h )) with U ⊂ M open and a 0 , . . . , a k ∈ A ( ( ¯ h )) cpt ( U ) put (6.17) X T r ( f 0 ⊗ f 1 ⊗ . . . ⊗ f k ) ( a 0 ⊗ . . . ⊗ a k ) : = T r k  f 0 ⋆ a 0 , . . . , f k ⋆ a k  , where (6.18) T r k  a 0 , . . . , a k ) : = T r  a 0 ⋆ . . . ⋆ a k  . Since the star product is local and the trace functional T r is given as an integral over the trace density , which also is loca l in its argument, one concludes that the cochain X T r ( f ) vanishes, if f ∈ C ∞ ( U k + 1 ) ( ( ¯ h ) ) vanishes around the diagonal ∆ k + 1 ( U ) . By passing to the quotient we obtain the d esired maps X T r : C k AS  C ∞ M ( ( ¯ h ) )  ( U ) → C k  A ( ( ¯ h ))  ( U ) . B y straightforward computation on e checks that b X T r = X T r δ and B X T r ( f ) = 0, if f ∈ C k λ AS  C ∞ M ( ( ¯ h ) )  ( U ) . Hence X T r provides a chain map from the cyclic Alex ander–Spanier complex to the cyclic complex of the deformed algebra. Remark 6. 7 . Let A b e a shea f of k -a lgebras. Assume that on O a local product denoted by · is deﬁned, and that A ca rries an O -module structure. Finally let τ : A ( M ) → k be a trace. Then E qs. (6.17) and (6.18) deﬁne a map X τ : C k λ AS  O  → C k λ  A ( M )  . Later in this section we will make use of this observation. CYCLIC COCYCL E S O N DE FORMA TIO N QUANTIZA TIO NS 37 W e now want to compare the morphism X T r with Q ◦ λ . T o this end, let Ψ denote the trace density Ψ 2 n 2 n deﬁned in 3.5. Note that by Proposition 3.6, Ψ 2 n 2 n is a trac e density , indeed . Furthermore, let U ⊂ M be a contrac tible open Dar boux domain. By Theorem 3.9 on e knows that Q 2 k U ( 1 ) = T r is a generator of the cyclic cohomology group H 2 k  T ot • B C • ( A ( ( ¯ h )) ) ( U )  for k ≥ 0, and that a ll other cyclic cohom ology gro ups H l  T ot • B C • ( A ( ( ¯ h )) ) ( U )  , with l odd. Moreover , observe that for all k ∈ N X T r ( 1 2 k + 1 ) = T r 2 k and Q 2 k U λ ( 1 2 k + 1 ) = Q 2 k U ( 1 ) = T r . But since 1 2 k ( b + B )  T r 1 , − T r 3 , . . . , ( − 1 ) k − 1 T r 2 k − 1  = T r 0 +( − 1 ) k − 1 T r 2 k for k > 0, both X T r ( 1 2 k + 1 ) and Q 2 k U λ ( 1 2 k + 1 ) are generators of the cyclic cohomology groups H 2 k  T ot • B C • ( A ( ( ¯ h ) ) ) ( U )  for k ≥ 0. Hence one concludes Proposition 6. 8. Th e sheaf morphisms X T r : C • λ AS  C ∞ M ( ( ¯ h ) )  → T ot • B C •  A ( ( ¯ h ))  and Q ◦ λ : C • λ AS  C ∞ M ( ( ¯ h ) )  → T ot • B C •  A ( ( ¯ h ) )  ֒ → T ot • B C •  A ( ( ¯ h ) )  coincide in the d e- rived ca t egory of sheaves on M. In particular , X T r : C • λ AS  C ∞ M ( ( ¯ h ) )  → T ot • B C •  A ( ( ¯ h ) )  is a quasi-isomorphism. Step 2. Next we explain how a global symbol c alculus for pseudodifferential operators on a riema nnian manifold Q gives rise to a deforma tion qu antization on the cotangent bundle T ∗ Q . Given an open subset U ⊂ Q denote by S ym m ( U ) , m ∈ Z , the space of symbols of or d er m on U , that means the space of sm ooth functions a on T ∗ U such that in each local coor dinate system of U and each compact set K in the domain of the local coordinate system there is an e stimate of the form   ∂ α x ∂ β ξ a ( x , ξ )   ≤ C K , α , β ( 1 + | ξ | 2 ) m −| β | 2 , x ∈ K , ξ ∈ T ∗ x Q , α , β ∈ N n , for some C K , α , β > 0 . Mor eover , put Sym ∞ ( U ) : = [ m ∈ Z Sym m ( U ) , Sym − ∞ ( U ) : = \ m ∈ Z Sym m ( U ) . Obviously , the spa ces Sym m ( U ) with m ∈ Z ∪ { ± ∞ } form the section spa c es of a sheaf Sym m on Q . Similarly , one constru cts the presheaves Ψ DO m of pseud odif- ferential operators of order m ∈ Z ∪ {± ∞ } on Q . Nex t let us recall the deﬁnition of the symbol map σ and its quasi-inverse, the quantization map Op. The sym- bol map associates to every operator A ∈ Ψ DO m ( U ) a symbol a ∈ Sym m ( U ) by setting (6.19) a ( x , ξ ) : = A  χ ( · , x ) e i h ξ ,Exp − 1 x ( · ) i  ( x ) , where Exp − 1 x is the inverse map of the exponential map on T x Q , and (6.20) χ : Q × Q → [ 0 , 1 ] is a smooth cut-off function such that χ = 1 on a neighborhood of the dia gonal, χ ( x , y ) = χ ( y , x ) f or all x , y ∈ Q , supp χ ( · , x ) is compact for each x ∈ Q , a nd ﬁnally 38 M.J. PFLAUM, H. POS THUMA, AND X. TAN G such that the restriction of Exp x to an open neighbor hood of Exp − 1 x  supp χ ( · , x )  is a diffeomorphism on to its image. The quantization map (6.21) Op : S ym m ( U ) → Ψ DO m ( U ) ⊂ Hom  C ∞ cpt ( U ) , C ∞ ( U )  , is then given by (6.22)  Op ( a ) f  ( x ) : = Z T ∗ x Q Z Q e − i h ξ ,Exp − 1 x ( y ) i χ ( x , y ) a ( x , ξ ) f ( y ) d y d ξ , f ∈ C ∞ cpt ( U ) . The maps σ and Op are n ow inverse to each other up to elements Ψ DO − ∞ respec- tively Sym − ∞ . Note that by deﬁnition of the operator map, the Schwartz kernel K Op ( a ) of Op ( a ) is given by (6.23) K Op ( a ) ( x , y ) = Z T ∗ x Q e i h ξ ,exp − 1 x ( y ) i χ ( x , y ) a ( ξ ) d ξ . By the space ASym m ( U ) , m ∈ Z of asymptot ic symbols over an open U ⊂ Q one understands the space of all q ∈ C ∞ ( T ∗ U × [ 0, ∞ ) ) such that for each ¯ h ∈ [ 0 , ∞ ) the f unction q ( − , ¯ h ) is in Sym m ( U ) a nd such that q has an asymptotic expansion of the form q ∼ ∑ k ∈ N ¯ h k a m − k , where ea c h a m − k is a symbol in Sym m − k ( U ) . More precisely , this mea ns that one has for all N ∈ N lim ¯ h ց 0  q ( − , ¯ h ) − ¯ h − N N ∑ k = 0 ¯ h k a m − k  = 0 in Sym m − N ( U ) . Like above one then obtains sheaves ASym m for m ∈ Z ∪ {± ∞ } . Now consider the subsheav e s JSym m ⊂ A S ym m consisting of all asymptotic symbol s which van- ish to inﬁnite order at ¯ h = 0. The quotient sheaves A m : = AS ym m / JSym m can then be identiﬁed with the formal power series sheaves Sym m [[ ¯ h ] ] . The operator product on Ψ DO ∞ induces an a symptotically associative product on ASym ∞ ( Q ) by deﬁning for q , p ∈ ASym ∞ ( Q ) (6.24) q ⊛ p : = ( σ ¯ h  Op ¯ h ( q ) ◦ Op ¯ h ( p )  if ¯ h > 0, q ( − , ¯ h ) · p ( − , ¯ h ) if ¯ h = 0. Hereby , Op ¯ h = Op ◦ ι ¯ h and σ ¯ h = ι ¯ h − 1 ◦ σ , where ι ¯ h : Sym ∞ ( Q ) → Sym ∞ ( Q ) is the map which maps a symbol a to the symbol ( x , ξ ) 7 → a ( x , ¯ h ξ ) . By standard techniques of p seudodifferential calculus (cf. [ P F 9 8]), one checks that ⊛ has an asymptotic expansion of the following form: (6.25) q ⊛ p ∼ q · p + ∞ ∑ k = 1 c k ( q , p ) ¯ h k , where the c k are bidifferential oper a tors on T ∗ Q such that c 1 ( a , b ) − c 1 ( b , a ) = − i { a , b } for all symbols a , b ∈ Sym ∞ ( Q ) . Hence, ⊛ is a star product on the quotient sheaf A ∞ , which gives rise to a de- formation quantization for the sheaf A T ∗ Q of smooth functions on the cotangent CYCLIC COCYCL E S O N DE FORMA TIO N QUANTIZA TIO NS 39 bundle T ∗ Q . By deﬁnition of the pro duct ⊛ it is clear that for the Schwartz kernels of two operators Op ¯ h ( q ) and Op ¯ h ( q ) one has the following relation: (6.26) K Op ¯ h  q ⊛ p  ( x , y ) = Z Q K Op ¯ h ( q ) ( x , z ) K Op ¯ h ( p ) ( z , y ) d z . Even though ⊛ is not obtained by a Fedosov c onstructio n, it is equivalent to a Fedosov star product ⋆ on T ∗ Q by [ N E T S 95]. In the following, we ﬁx ⋆ to be such a Fedosov star product, and assume that it is obtained by a Fedosov conn ection A constant along the ﬁbe r s of T ∗ Q . Note that by the equivalence of ⊛ a nd ⋆ , each trace functional for ⊛ is one for ⋆ a nd vice versa. Using the riemannian metric on Q on e even obtains a trace function al T r on A ∞ by the follo wing construction . Pseudodifferential operators Ψ DO − dim Q cpt ( Q ) act as trace class operators on the Hilbert space L 2 ( Q ) . Thus there is a map T r : A − ∞ cpt ( Q ) → C [ ¯ h − 1 , ¯ h ] ] , q 7 → tr  Op ¯ h ( q )  , where tr is the oper ator tra ce. By construction, T r has to be a tra ce with respect to ⊛ and is ad ( A ∞ ) -invar ia nt. U sing the global symbol calculus for pseudodifferential operators [W I D , P F 9 8] the following formula can be derived: (6.27) T r ( q ) = 1 ( 2 π √ − 1 ¯ h ) dim Q Z T ∗ Q q ( − , ¯ h ) ω dim Q ( dim Q ) ! , where ω is the canonical symplectic form on T ∗ Q . Moreover , by the remarks above, T r is also a tr a ce with respect to the Fedosov star product ⋆ . Finally note that for all operators A ∈ A Ψ DO ∞ ( Q ) (6.28) tr  Op ¯ h σ ¯ h ( A ) − A  = 0. Step 3. The ﬁnal step reduces the c omputation of the high er indices to the alge- braic higher indices using the global symbol calculus of step 2. W e begin by deﬁn- ing the a na lytic higher index using the localized K -theory of M O S C O V I C I – W U [ M O W U ]. Let Q be a compact riemannian manifold and consider the smoothin g operators Ψ D O − ∞ ( Q ) acting on L 2 ( Q ) . These operators have a smooth Schwartz kernel, and therefore Ψ D O − ∞ ( Q ) ∼ = C ∞ cpt ( Q × Q ) . Note that by a ssumptions on Q , every element K ∈ Ψ DO − ∞ ( Q ) is trace-c la ss, and Ψ D O − ∞ ( Q ) is dense in the space of trace class opera tors on L 2 ( Q ) . For any ﬁnite open covering U of Q , we deﬁne Ψ DO − ∞ ( Q , U ) : = { K ∈ Ψ D O − ∞ ( Q ) | supp ( K ) ⊂ U 2 } , where U k : = S U ∈ U U k for k ∈ N ∗ . Now let M ∞  Ψ DO − ∞ ( Q , U )  be the induc- tive limit of all N × N -matrices with entries in Ψ DO − ∞ ( Q , U ) . Likewise, d eﬁne M ∞  Ψ DO − ∞ ( Q , U ) ∼  and M ∞ ( C ) , where Ψ D O − ∞ ( Q , U ) ∼ : = Ψ D O − ∞ ( Q , U ) ⊕ C . W ith these preparations, one deﬁnes K 0 ( Q , U ) : = K 0  Ψ DO − ∞ ( Q , U )  : = : =  ( P , e ) ∈ M ∞  Ψ DO − ∞ ( Q , U ) ∼  × M ∞ ( C )  | P 2 = P , P ∗ = P , e 2 = e , e ∗ = e and P − e ∈ M ∞  Ψ DO − ∞ ( Q , U )  / ∼ , (6.29) where ( P , e ) ∼ ( P ′ , e ′ ) for projections P , P ′ ∈ M ∞  Ψ DO − ∞ ( Q , U ) ∼  and e , e ′ ∈ M ∞ ( C ) , if the elements P and P ′ can be joined by a continuous and p ie c ewise C 1 path of pro jections in som e M N  Ψ DO − ∞ ( Q , U )  with N ≫ 0 and likewise for e 40 M.J. PFLAUM, H. POS THUMA, AND X. TAN G and e ′ (see [ M O W U , Sec. 1.2 ] for further d e tails). Elements of K 0 ( Q , U ) are repre- sented as e quivalence classes of differences R : = P − e , where P is an idempotent in M ∞  Ψ DO − ∞ ( Q , U ) ∼  , e is a projection in M ∞ ( C ) , and the difference P − e lies in M ∞  Ψ DO − ∞ ( Q , U )  . A (ﬁnite) reﬁnement V ⊂ U obviously leads to an inclusion Ψ D O − ∞ ( Q , V ) ֒ → Ψ DO − ∞ ( Q , U ) which induces a map K 0 ( Q , V ) → K 0 ( Q , U ) . W ith these maps, the localized K -theory of Q is deﬁned a s (6.30) K 0 loc ( Q ) : = lim ← − U ∈ Cov ﬁn ( Q ) K 0 ( Q , U ) . Concretely , this means that elements of K 0 loc ( Q ) are given by f amilies (6.31)  [ P U − e U ]  U ∈ Cov ﬁn ( Q ) of equi valence classes of pairs of pr ojectors in matrix spaces over Ψ DO − ∞ ( Q , U ) ∼ such that e U ∈ M ∞ ( C ) for eve r y ﬁnite covering U and ( P U , e U ) ∼ ( P V , e V ) in M ∞  Ψ − ∞ ( Q , U ) ∼  whenever V ⊂ U . Following [M O W U ], we now cons truct the so-called (even) Alex ander-Spanier- Chern character map Ch AS 2 • : K 0 loc ( Q ) → H AS 2 • ( Q ) . As a preparation f or the constructio n we set for every subset W ⊂ Q , k ∈ N and every ﬁnite covering V of Q st k ( W , V ) : = [ ( V 1 ,. . ., V k ) ∈ cha in k ( W , V ) V 1 ∪ . . . ∪ V k , where chain k ( W , V ) : = : = { ( V 1 , . . . , V k ) ∈ V k | W ∩ V 1 6 = ∅ , V 1 ∩ V 2 6 = ∅ , . . . , V k − 1 ∩ V k 6 = ∅ } . Then we deﬁne st k ( V ) as the open covering of Q with elements st k ( V , V ) where V runs thr ough the elements of V . Obviously , one then has Ψ DO − ∞ ( Q , V ) · . . . · Ψ DO − ∞ ( Q , V ) | {z } k − times ⊂ Ψ D O − ∞ ( Q , st k ( V ) ) . Next let us ﬁx an even homology d egree 2 k and a ﬁnite open covering U of Q . Then choose a ﬁnite open covering U 0 of Q such tha t st k ( U 0 ) is a reﬁnement of U . Now let R U 0 : = P U 0 − e U 0 ∈ Ψ D O − ∞ ( Q , U 0 ) represent an element of K 0 ( Q , U 0 ) as deﬁned above, and put for f 0 , . . . , f 2 k ∈ C ∞ ( Q )  Ch AS 2 k ( R U 0 )  ( f 0 ⊗ . . . ⊗ f 2 k ) : = : = ( − 2 π i ) k ( 2 k ) ! k ! ε 2 k tr  ( f 0 P U 0 f 1 . . . f 2 k P U 0 ) − ( f 0 e U 0 f 1 . . . f 2 k e U 0 )  (6.32) It has been shown in [ M O W U , Sec. 1.4 ] that the right hand side even deﬁnes a cycle in C AS 2 k ( M , U ) , hence one obtains a homology class Ch AS 2 k ( R U 0 , U ) ∈ H AS 2 k ( M , U ) . Moreover , a fa mily R = ( R V ) V ∈ Cov ﬁn ( Q ) deﬁning a local K-theory class gives r ise to a family of compatible homology classes Ch AS 2 k ( R U 0 , U ) , U ∈ Cov ﬁn ( Q ) , hence by the universal properties of in verse limits one ﬁnally obtains a character map Ch AS 2 k : K 0 loc ( Q ) → H AS 2 k ( Q ) inde e d. Le t us now reformulate the pairing  [ f ] , Ch AS 2 k ( [ R ] )  , where [ f ] denotes an Alexa nde r-Spanier cohomolo gy class of degree 2 k . W ithout CYCLIC COCYCL E S O N DE FORMA TIO N QUANTIZA TIO NS 41 loss of generality , we can assume that [ f ] has the form [ f 0 ⊗ . . . ⊗ f 2 k ] with the f i being smooth functions on Q . Then note that the opera tor trace tr on L 2 ( Q ) induces a trace on Ψ − ∞ ( Q ) . W ith this trace, equation (6.17) deﬁnes a morphism X U tr : C 2 k A S ( Q , U ) → C 2 k λ  Ψ − ∞ ( Q , U 0 )  which is uniquely determined by the requir ement X U tr ( f 0 ⊗ . . . ⊗ f k ) ( R 0 ⊗ . . . ⊗ R k ) : = tr ( f 0 R 0 · · · f k R k ) , where the f i on the right hand side are viewed as bounded multiplication oper- ators on L 2 ( Q ) , a nd the R i are elements of Ψ − ∞ ( Q , U 0 ) . Note that on the right hand side C 2 k λ ( Ψ − ∞ ( Q , U 0 ) ) is the restriction of the space of cyclic 2 k -c ochains C 2 k λ ( Ψ − ∞ ( Q ) ) ) to elements of Ψ − ∞ ( Q , U 0 ) . By the construction of the pairing in Alexander-Spanier homo logy in Remark 6.3 and the deﬁnition of Ch AS 2 k above, the pairing between localized K -theory and Alexander–S p a nier cohomology can be rewritten as (6.33)  [ f ] , Ch AS 2 k ( [ R ] )  = D X U tr ( ε 2 k f ) , Ch ( R U 0 ) E , where R = ( R V ) V ∈ Cov ﬁn ( Q ) is as above, Ch is the no ncommutative Chern charac - ter (on the chain level) as deﬁned by Eq. (4.1), and where U is a sufﬁciently ﬁne covering such that in pa rticular U 2 k + 1 is contained in the domain of the f unction f deﬁning the Alexander-Spanier cohomology cla ss [ f ] . Let us now come to the deﬁnition of the localized index, or in other words, the higher index which originally was deﬁned by C O N N E S – M O S C O V I C I in [ C O M O , § 2.]. T o this end a ssume ﬁrst that E → Q is a n He rmitian vector bundle over Q and tha t D is an elliptic pseudodifferential operator acting on the spa ce of smooth sections Γ ∞ ( E ) . The operator D gives rise to an invertible pr inciple sym- bol σ pr ( D ) ∈ C 0 ( T ∗ Q \ Q ) . Its restriction to the cosphere bundle wil l be denoted by σ res ( D ) : = σ pr ( D ) | S ∗ Q . The restricted principa l symbol σ res ( D ) deﬁnes an element in the odd K-group K 1  C ∞ ( S ∗ Q )  . Morever , as explained in [ C O M O , p. 3 5 3], one can associate to σ res ( D ) and each ﬁnite covering U of Q an element R U = P U − e U ∈ Ψ − ∞ ( Q , U ) which is constructed as a d ifference of a certain pseudodifferential projection P of order − ∞ on Q a nd a projection in the matrix algebra over C and which f ulﬁlls the crucial relation ind ( D ) = tr R U . Note that R U is homotopic to the gra ph projection of D (cf. [ E L N A N E ] ), and that the induced class [ R ] ∈ K 0 loc ( Q ) of the family R = ( R U ) U depends only on the class of σ res ( D ) in K 1  C ∞ ( S ∗ Q )  . One thus obtains a map ∂ : K 1  C ∞ ( S ∗ Q )  → K 0 loc ( Q ) which we call the local ind ex map . Next let [ f ] be an even Alexa nder–Spanier co- homology class of degree 2 k which is represented by the function f ∈ C ∞ ( Q 2 k + 1 ) . Then one deﬁnes the localized index or higher index of D at [ f ] as the pa iring (6.34) ind [ f ] ( D ) : =  [ f ] , Ch AS 2 k  ∂ [ σ res ( D ) ]  . 42 M.J. PFLAUM, H. POS THUMA, AND X. TAN G Note that according to the work of [ M O W U ], this localize d index can be trans- formed into th e original deﬁnition of the localized in dex by C O N N E S – M O S C O V I C I : ind [ f ] ( D ) : = ( − 1 ) k Z Q 2 k + 1 tr  R V ( x 0 , x 1 ) · . . . · R V ( x 2 k − 1 , x 2 k )  f ( x 0 , . . . , x 2 k ) d µ 2 k + 1 = X U tr ( f )  R V ⊗ . . . ⊗ R V  , (6.35) where here µ is the volume form on Q , R : = ( R U ) U : = ∂ σ res ( D ) , a nd V is a ﬁnite covering sufﬁciently ﬁne such that V 2 k + 1 is contained in U 2 k + 1 , the domain of the function f deﬁning the Alexander-Spa nier cohomology class [ f ] . Now let a i = σ ¯ h ( A i ) , i = 0, . . . , k be the asymptotic symbols of pseudodifferen- tial operators A i ∈ A Ψ DO − ∞ . For all Alexander Sp a nier cochains f 0 ⊗ . . . ⊗ f k ∈ C ∞ ( Q k + 1 ) the following relation then holds true asymptotically in ¯ h : X tr  f 0 ⊗ . . . ⊗ f k  A 0 ⊗ . . . ⊗ A k  : = tr  f 0 A 0 · . . . · f k A k  = = tr  f 0 Op ¯ h ( a 0 ) · . . . · f k Op ¯ h ( a k )  = tr  Op ¯ h ( f 0 a 0 ) · . . . · Op ¯ h ( f k a k )  = tr  Op ¯ h ( f 0 a 0 ⊛ . . . ⊛ f k a k )  = T r  f 0 ⊛ a 0 ⊛ . . . ⊛ f k ⊛ a k  = T r  f 0 ⋆ a 0 ⋆ . . . ⋆ f k ⋆ a k  = X T r  f 0 ⊗ . . . ⊗ f k  a 0 ⊗ . . . ⊗ a k  . (6.36) Hereby , we have used that f i Op ¯ h ( a i ) = Op ¯ h ( f i a i ) , and that by Eq. (6 .26) tr  Op ¯ h ( a i ⊛ a i + 1 )  = tr  Op ¯ h ( a i ) Op ¯ h ( a i + 1 )  . Using the results from Step I together with Eqns. (6.33) a nd (6. 36) one now obtains with r V = : σ ¯ h R V the asymptotic symbol of R V , and f = f 0 ⊗ . . . ⊗ f 2 k ind [ f ] ( D ) =  [ f ] , Ch AS 2 k  ∂ [ σ res ( D ) ]  ( 6. 33 ) = D X U tr ε 2 k ( f ) , Ch  R V  E = ( 6. 36 ) = D X T r ε 2 k ( f ) , Ch r V E = D X T r ε 2 k ( [ f ] ) , Ch r E Prop. 6.8 = D Q λε 2 k ( [ f ] ) , Ch r E = = 1 ( 2 π √ − 1 ) k Z T ∗ Q f 0 d f 1 ∧ . . . ∧ d f 2 k ∧ ˆ A ( T ∗ Q ) Ch ( V 1 − V 2 ) . Hereby , V 1 − V 2 is the virtual vector bundle obtained by the a symptotic limit ¯ h ց 0 of r , and r is the symbol of R Q with Q denoting here the trivial covering of Q . W e have thus repro ved the followin g result from [C O M O ] . Theorem 6.9. For an elliptic differential op erator D on a riemannian manifold Q and an Alexander–Spanier cohom ology class [ f ] of degree 2 k with compact support t h e localized index is given by ind [ f ] ( D ) = 1 ( 2 π √ − 1 ) k Z T ∗ Q f 0 d f 1 ∧ . . . ∧ d f 2 k ∧ ˆ A ( T ∗ Q ) Ch ( V 1 − V 2 ) . CYCLIC COCYCL E S O N DE FORMA TIO N QUANTIZA TIO NS 43 7. A H I G H E R A N A LY T I C I N D E X T H E O R E M F O R O R B I F O L D S In this section, which comprises the ﬁnal part of this work, we prove the higher index theorem for elliptic differential operators on orbifolds as an application of the higher algebr a ic index theorem for proper ´ etale groupoids of Section 5. This generalizes the higher index theorem by C O N N E S – M O S C O V I C I to the orbifold set- ting. Although our stra tegy f or the proof is the same a s in Section 6, the general- ization is by no means straightforward: we start in Section 7.1 with deﬁning the Alexander–Spanier cochain complex for proper ´ etale groupoids G . This cochain complex depends on the groupoid structure, and instead of being localized to the diagonal, the cochains are localize d to the so-called “higher Burghelea spaces”. In Section 7.2, we e xplain how cohomology classes are represented by functions in a sufﬁciently small neighbourhood of these Burghelea spaces so that we c a n have cocycles a cting on L 2 ( G 0 ) by convo lution. This is important in Section 7 .4 f or the pairing with orbifold localized K -theory . In Section 7.3, we relate orbifold Alexander–Spanier cohomology to the cyclic cohomolog y of a deformation quantization of the convolution algebra . In Section 7.4, the orbifold version of localized K -theory is introduced in terms of a ﬁltra- tion in which smoothing operators on G 0 are localized to the diagonal and invari- ance is impos ed. V ia a Cher n character , such K - theory cla sses p a ir with localized Alexander–Spanier cocycles. The link between the two pairings of Alexa nder–Spanier cohomology , namely on the one side the pairing with loca liz ed K -theory and on the other side with the cyclic cohomology of a def ormation quantization, is given by a global ¯ h -dependent symbol c alculus for pseudodifferential opera tors on orbif olds as constructed in [ P F P O T A ]. This induces a deformation quantization over the cotangent bundle of the underlying orbifolds with which we ca n compare the two pairings. The high er index theor e m ﬁnally follow s by application of this idea to the canonical localized K -theory class induced by the elliptic operator . 7.1. Orbifold Al exande r-Spanier cohomology. As before, we de note by M a n orbifold given as a quotient space of a proper ´ eta le Lie groupoid G 1 ⇒ G 0 . The orbifold version of the Alexa nder–Spanier sheaf complex is constructed a s f ollows: again we consider the space of loops B ( 0 ) ⊂ G 1 . On this space deﬁne the following sheaves: C k AS,tw ( O ) : = s − 1 O ⊠ ( k + 1 ) G 0 , where O G 0 is a shea f of unital algebras a s be fore. W e introduce a cosimplicial structure with coface operators ¯ δ i : C k AS,tw ( O ) → C k + 1 AS,tw ( O ) , i = 0, . . . , k + 1 given by ¯ δ i ( f 0 ⊗ . . . ⊗ f k ) : = f 0 ⊗ . . . ⊗ f i − 1 ⊗ 1 ⊗ f i + 1 ⊗ . . . ⊗ f k , and degeneracies s i : C k AS,tw ( O ) → C k − 1 AS,tw ( O ) , i = 0, . . . , k − 1 deﬁned by ¯ s i ( f 0 ⊗ . . . ⊗ f k ) : = f 0 ⊗ . . . ⊗ f i f i + 1 ⊗ . . . ⊗ f k . So fa r , nothing new , but this time the cyclic structure ¯ t k : C k AS,tw ( O ) → C k AS,tw ( O ) is given by ¯ t ( f 0 ⊗ . . . ⊗ f k ) : = f 1 ⊗ . . . ⊗ f k ⊗ θ − 1 ( f 0 ) , 44 M.J. PFLAUM, H. POS THUMA, AND X. TAN G where θ : B ( 0 ) → G 1 is the cyclic structure of the groupoid Λ ( G ) . Reca ll, cf. [ C R , Def. 3.3.1 ], that θ , and also θ − 1 , equips Λ ( G ) with the structure of a cyc lic groupoid . Using that notion, it is not difﬁcult to ve rify that with these structure maps C • AS,tw ( O ) is a cocyclic sheaf on the cyclic groupoid ( Λ ( G ) , θ − 1 ) and gives rise to a n ∞ - cocyclic object ( C • AS,tw ( O ) , ¯ δ , ¯ s , ¯ t ) in the category of G -sheaves over B ( 0 ) such that ¯ t k + 1 = θ − 1 in each degree k . Remark 7. 1. Pulling back the standard cyclic sheaf of a lgebras O ♮ G 0 on G 0 to B ( 0 ) , there is a way to twist the structur e maps by the cyclic structure θ , c f. [C R ]. T he cyclic sheaf ab ove is simply the cyclic dua l of this one. Notice that there is no twist in the degeneracies because ex a ctly the fa ce operator containing the twist in s − 1 O ♮ G 0 is not used in the deﬁnition of the dual, cf. [ L O , § 6.1]. Associated to the underlying simplicial complex is the Hochschild sheaf com- plex ( C • AS,tw ( O ) , ¯ δ ) with differential ¯ δ = ∑ k i = 0 ( − 1 ) i ¯ δ i . Deﬁnition 7.2. The orbifold A lexander–Spanier cohomo logy H • AS,orb ( M , O ) of M with values in O is deﬁned to be the groupoid sheaf cohomology of the complex ( C • AS,tw ( O ) , ¯ δ ) . As alluded to in the notation, orbifold Alexander–Spa nier cohomology is inde- pendent of the particular groupo id G representing its Morita equivalence class. In fact we have: Proposition 7.3. T here is a natural isomorphism H • AS,orb ( M , O ) ∼ = H • ( ˜ M , k ) . Proof. As for manifolds, cf. Propositio n 6.1, the inclusion k ֒ → C • AS,tw is a quasi- isomorphism in Sh ( Λ ( G ) ) since it is clearly compatible with the G -action on both sheaves. But for the locally constant sheaf k we have the natural isomorphism H • ( Λ ( G ) , k ) ∼ = H • ( ˜ M , k ) .  As groupoid cohomology , orbifold Alexander–Spa nier cohomolo gy can be com- puted using the Ba r complex of Λ ( G ) . However , instead of using the ner v e of Λ ( G ) , we shall use the iso morphic Bur ghelea spaces associated to G to write down such a Bar complex. Intro duce B ( k ) : =  ( g 0 , . . . , g k ) ∈ G k + 1 | s ( g 0 ) = t ( g 1 ) , . . . , s ( g k − 1 ) = t ( g k ) , s ( g k ) = t ( g 0 )  . These Burghelea spaces B ( k ) form a simplicial manifold with face maps (7.1) d i ( g 0 , . . . , g k ) = ( ( g 0 , . . . , g i g i + 1 , . . . , g k ) , 0 ≤ i ≤ k − 1, ( g k g 0 , . . . , g k − 1 ) , i = k . Consider now the map ¯ σ k : B ( k ) → G ( k + 1 ) 0 given by ¯ σ k ( g 0 , . . . , g k ) = ( s ( g 0 ) , . . . , s ( g k ) ) . W ith this we deﬁne for ea ch k ∈ N the shea f S k : = ¯ σ ∗ k  O ˆ ⊠ ( k + 1 )  , the pullback sheaf of O ˆ ⊠ ( k + 1 ) to B ( k ) . W e write A S k ( G , O ) : = Γ  B ( k ) , S k  and observe that a (bornolo gically) dense subspace of the space of sections Γ  B ( k ) , S k  is given by sums of sections of the form f = f 0 ⊗ · · · ⊗ f k : ( g 0 , · · · , g k ) 7→ ( f 0 ) [ g 0 ] ⊗ . . . ⊗ ( f k ) [ g k ] , CYCLIC COCYCL E S O N DE FORMA TIO N QUANTIZA TIO NS 45 where ( f i ) [ g i ] ∈ O s ( g i ) and ( g 0 , . . . , g k ) ∈ B ( k ) . W ith this in mind, we introduce a simplicial structure on AS • ( G , O ) by means of the c oface maps δ i : AS k − 1 ( G , O ) → AS k ( G , O ) deﬁned as δ i ( f 0 ⊗ . . . ⊗ f k − 1 ) [ g 0 ,. . ., g k ] = ( 1 s ( g 0 ) ⊗ ( f 0 ) [ g 0 ] ⊗ ( f 1 ) [ g 2 ] ⊗ . . . ⊗ ( f k − 1 ) [ g k g 0 ] i = 0 ( f 0 ) [ g 0 ] ⊗ . . . ⊗ ( f i ) [ g i g i + 1 ] g − 1 i ⊗ 1 s ( g i ) ⊗ . . . ⊗ ( f k − 1 ) [ g k ] for 1 ≤ i ≤ k . Codegeneracies are given by s i ( f 0 ⊗ . . . ⊗ f k + 1 ) [ g 0 , · · · , g k ] = ( f 0 ) [ g 0 ] ⊗ . . . ⊗ ( f i − 1 ) [ g i − 1 ] ⊗ ( f i ) [ s ( g i − 1 ] · ( f i + 1 ) [ g i ] ⊗ ( f i + 2 ) [ g i + 1 ] ⊗ . . . ⊗ ( f k + 1 ) [ g k ] . Finally , we can deﬁne a compatible cyclic str ucture t k : A S k ( G , O ) → A S k ( G , O ) by t k ( f 0 ⊗ · · · ⊗ f k ) [ g 0 , · · · , g k ] = ( − 1 ) k ( f 0 ) [ g k ] ⊗ ( f 1 ) [ g 0 ] ⊗ . . . ⊗ ( f k ) [ g k − 1 ] . It is straightforward to show that with these structure maps A S ♮ ( G , O ) is a cyclic cosimplicial vector space indeed. T o relate the above introduced cosimplicial complex with the Bar complex of the sheaf cohomology of C • AS,tw ( O ) on Λ G in Deﬁni tion 7 .2 we identify B ( k ) with Λ G ( k ) by the map ν ν ( g 0 , · · · , g k ) = ( g 1 · · · g k g 0 , g 1 , · · · , g k ) . The induced isomorphism ν ∗ on A S ♮ is computed to be ν ∗ ( f 0 ⊗ · · · ⊗ f k ) ( g 1 · · · g k g 0 , g 1 , · · · , g k ) = =  ( f 0 ( g 0 ) ) g 1 · · · g k g 0 , ( f 1 ( g 1 ) ) g 2 · · · g k g 0 , · · · , ( f k ( g k ) ) g 0  . The image ν ∗ ( S k ) then is a sheaf on Λ G ( k ) . Observe that the sheaf S k can be un- derstood as the pullback of a sheaf O k tw on B ( 0 ) through the map ( g 1 · · · g k g 0 , . . . , g k ) 7→ g 1 · · · g k g 0 . It is e a sy to check that ν ∗ deﬁnes an iso morphism be tween the c omplex AS ♮ ( G , O ) and the Bar c omplex on Λ G of C ♮ AS,tw ( O ) . Since Λ ( G ) is proper , the Bar complex is quasi-isomorphic to the complex of invariant sections on B ( 0 ) . Denote by β : B ( k ) → B ( 0 ) the map β ( g 0 , . . . , g k ) = g 0 · · · g k . Puttin g all this together , we have Proposition 7.4. For ev ery proper ´ etale Lie gr oupoid G and sheaf O as above β ∗ : AS ♮ ( G , O ) → Γ inv  B ( 0 ) , C ♮ AS,tw ( O )  . is a quasi-isomorphism of cochain complexes. 7.2. Explici t reali z ation of Al exande r–Spanier cocycles. Reca ll that in the case of manifolds the Alexa nde r–Spanier cochain complex c ould be written as a direct limit of a cochain complex of functions deﬁned on a neighbourhoo d of the diag- onal ∆ k + 1 : M → M k + 1 given in terms of the c hoice of a n open covering of M . This realization of Alexa nde r–Spanier cocycles wa s crucial in the deﬁnition of the pairing with localized K -theory . In this section we will generalize this cons truction 46 M.J. PFLAUM, H. POS THUMA, AND X. TAN G to proper ´ eta le groupoids G , where this time the role of the diagonal is played by the Burghelea spa ce B ( k ) ֒ → G k + 1 1 . Let G 1 ⇒ G 0 be a proper ´ etale groupoid modeling an orbifold M , a nd U ⊂ G 0 an open set. A local bisection , cf. [ M O M R , § 5. 1], on U is a local section σ : U → G 1 of the source map s : G 1 → G 0 such that t ◦ σ : U → G 0 is an open embedd ing. This second p roperty of local bisections shows that they deﬁne local diffeomorphisms of G 0 and as such the product (7.2) ( σ 1 σ 2 ) ( x ) : = σ 1 ( t ( σ 2 ( x ) ) ) σ 2 ( x ) is deﬁned if the domain of σ 1 contains the image of t ◦ σ 1 . Deﬁnition 7.5. A covering U = { U i } i ∈ I of G 0 is said to be G -trivializing if it satis- ﬁes the followin g two conditions: i ) U is the pull-ba ck of a covering of M along the projection π : G 0 → M . By this we mean that there exists a covering U of M such that U consists of the connected components of π − 1 ( U ) , U ∈ U . ii ) For all g ∈ G 1 , ther e exists an i ∈ I and a bisection σ i deﬁned on U i with σ i ( s ( g ) ) = g . In particular , s ( g ) ∈ U i . Remark th at such a covering always exists and is completely determined by the induced covering of the quotient M . W e will therefore denote the set of coverings of M satisfying pr operty ii ) a b ove by Cov G ( M ) . C le a rly , Cov G ( M ) is directed by the notion of reﬁnement. Remark that the property of being G -trivializing very much depends on the groupoid G , and not the quotient. A s an easy example, consider a manifold M : when represented as a groupoid with only identity arrows, any covering satisﬁes the properties above. However , wh en represented as a ˇ Cech- groupoid associated to a ﬁxed covering U , only those coverings that reﬁne the covering U ′ = { U i ∩ U j } i , j ∈ I are trivializing. Also, the σ i in ii ) are uniquely determined by g because G is ´ eta le . Because of this, we shall write σ g i for this local bisection. Furthermore, since s ◦ π = t ◦ π , we have π ( t ( σ g i ( U i ) ) ) = π ( U i ) so there exists a j ∈ I such that ( t ◦ σ g i ) ( U i ) = U j . Associated to the covering are the subsets G i j ⊂ G 1 , i , j ∈ I deﬁned by G i j : = { g ∈ G 1 | s ( g ) ∈ U j , σ g j ( U j ) = U i } . The conditions on the covering ensures that S i , j ∈ I G i j = G . W ith this notation, we introduce B ( k ) U : = [ i 0 ,. . ., i k ∈ I G i 0 i 1 × . . . × G i k i 0 ֒ → G k + 1 1 . Remark that there is a ca nonical embedding B ( k ) ⊂ B ( k ) U . Lemma 7.6 . T he family of spac es B ( k ) U , k ∈ N carries a c a nonical cyc lic manifold struc- ture wh ich extends the cyclic structure on B ( • ) . Proof. Let ( g 0 , . . . , g k ) ∈ B ( k ) U . B y deﬁnition, ther e are i 0 , . . . , i k ∈ I such that g j ∈ G i j i j + 1 . Let σ j be the unique local bisection σ j : U i j → G 1 corresponding to g j . By construction , σ j + 1 ( U i j + 1 ) = U i j , and we can deﬁne the product of g j and g j + 1 as g j ⊙ g j + 1 : = ( σ j σ j + 1 ) ( s ( g j + 1 ) ) , CYCLIC COCYCL E S O N DE FORMA TIO N QUANTIZA TIO NS 47 with the product of the local bisection s as in (7.2). It is not difﬁcult to check that this deﬁnition of the p roduct is indep e ndent of the choice of i 0 , . . . , i k . T o see that it is a ssociative it is best to think of the local bisections σ j as elements in the p se u- dogroup of local diffeomorphisms of G 0 . Finally , with this composition, we can deﬁne the cyclic structure on B ( k ) U by the same formulae as in (7. 1). The proof that this indeed deﬁne a cyclic manifold is then routine. Clearly , it induces the canoni- cal cyclic structure on the Burghelea spaces.  Remark 7. 7. Note that the product g 1 ⊙ g 2 coincides with the groupoid composi- tion, if s ( g 1 ) = t ( g 2 ) . The product ⊙ can thus be understood as a n extension of the groupoid pr oduct a round the “orbif old diagonal” meaning around the set of composable arrows. Let us now introduce the follo wing complex: C k A S ( G , U ) : = C ∞ ( B ( k ) U ) . The differential δ : C k A S ( G , U ) → C k + 1 A S ( G , U ) is deﬁned by the formula ( δ f ) ( g 0 , . . . , g k + 1 ) : = : = k ∑ i = 0 ( − 1 ) i f ( g 0 , . . . , g i ⊙ g i + 1 , . . . , g k ) + ( − 1 ) k + 1 f ( g k + 1 ⊙ g 0 , . . . , g k ) . Since the differen tial is deﬁned in te rms of the underlying simplicial structure on B ( • ) U , we automatically have δ 2 = 0. Example 7.8. Consider the transformation groupoid Γ × X ⇒ X a ssociated to a group action of a discrete group Γ on a manifold X . By deﬁnition, s ( γ , x ) = x , t ( γ , x ) = γ ( x ) for x ∈ X , γ ∈ Γ , and a bisection on X is given by an element γ ∈ Γ . In this case, the trivial covering of X / Γ obviously satisﬁes the conditions i ) and ii ) above. Unravelling the deﬁnition this lead s to the followin g complex associated to the trivial covering of X : C k A S ( Γ × X , X ) : = C ∞  ( Γ × X ) k + 1  and the differential is given by ( δ f ) ( γ 0 , x 0 , . . . , γ k + 1 , x k + 1 ) : = k ∑ i = 0 ( − 1 ) i ( γ 0 , x 0 , . . . , x i − 1 , γ i γ i + 1 , x i + 1 , . . . , γ k + 1 , x k + 1 ) + ( − 1 ) k + 1 f ( γ k + 1 γ 0 , x 0 , . . . , γ k , x k ) . (7.3) In particular , for Γ the trivial group and any covering U , we ﬁnd e x actly the com- plex (6.3). Clearly , if a covering U satisﬁes condition i ) and ii ) above, a r eﬁnement V ֒ → U also satisﬁes these conditions and therefore induces a canonical map C • A S ( G , U ) → C • A S ( G , V ) . W ith this , we can take the direct limit over the set of coverings of the orbifold M . Proposition 7.9. I n the limit, there is a canonical isomorphism lim − → U ∈ Cov G ( M ) C • AS ( G , U ) ∼ = AS • ( G , O ) . 48 M.J. PFLAUM, H. POS THUMA, AND X. TAN G Proof. As the cover gets ﬁner , the set B ( k ) U ֒ → G k + 1 shrinks to the Burghelea space B ( k ) . Therefore, the restriction of a function f ∈ C ∞ ( B ( k ) U ) to the ge r m f | B ( k ) of B ( k ) induces the linear isomorphism a s in the sta tement of the Propositio n. It is straightforward to show that this map is compatible with the differentials.  By Propo sition 7.3, we therefore have that the c ohomology of the limit complex lim − → U ∈ Cov G ( M ) ( C • AS ( G , U ) , δ ) equals H • ( ˜ M , k ) . In fact, unravelling all the isomo rphisms involved, we have : Corollary 7.10. The cohomology c lass in H k ( ˜ M , k ) induced by a cocycle f = f 0 ⊗ . . . ⊗ f 2 k ∈ C ∞ ( B ( k ) U ) is repres ented by the closed invariant differential form on B ( 0 ) given by ν ∗  λ k k ( f ) | B ( k )  g = ∑ g 0 ,. . ., g k ∈ B ( k ) g 0 ·· · g k = g ∑ σ ∈ S k + 1 f σ ( 0 ) ( g 0 ) d f σ ( 1 ) ( g 1 ) ∧ . . . ∧ d f σ ( k ) ( g k ) . In particular , if f has com pact support, the re sulting differ ential form is comp actly sup- ported. This gives us an explicit way of representing cohomolo gy classes in H • ( ˜ M , k ) by cocycles deﬁned on a sufﬁciently small neighbourhood of the ” orbifold diago- nal” B ( k ) ֒ → G k + 1 . For example, for a transformation groupoid as in Example 7.8, we have e M = ∐ h γ i ∈ Conj ( Γ ) X h γ i , Z h γ i . As we have seen above, there are enough global bisections in th is ca se, a nd we can use the trivial covering. W e write f = ∑ γ ∈ Γ f γ U γ for a n element in f ∈ C ∞ ( Γ × X ) . The function f h γ i = ∑ ν ∈ Γ 1 U ν γ ν − 1 , is closed under the Alexander–Spanier differential, δ f h γ i = 0 , as an easy argument shows. By th e canonical pro jection onto th e direct limit complex, it induces a cocy- cle of degree z ero in the A lexander–Spanier complex. It is not difﬁcult to see that this is a generator of H 0 ( X h γ i / Z h γ i , k ) ⊂ H 0 ( ˜ M , k ) . 7.3. Relati ng orbifold A lexan der–Spanie r cohomology with cycli c cohomology . W e assume in this step G 0 is equipped with an invaria nt symplectic form ω . Let A ( ( ¯ h )) be a (loca l) deformation quantization on G 0 . According to [ T A ] , A ( ( ¯ h ) ) ⋊ G is a deformation quantization over th e groupoid G , whi ch by deﬁnition is a defor- mation of the convolution a lgebra on G . In [ P F P O T A ], we constructed a universa l trace T r on A ( ( ¯ h )) ⋊ G . The trace functional is deﬁned by (7.4) T r ( a ) : = Z B ( 0 ) Ψ 2 n − ℓ ( g ) 2 n ( a ) , a ∈ A ( ( ¯ h ) ) ( G 0 ) , where Ψ 2 n − ℓ 2 n is deﬁned in Section 5 (cf. Remark 5 .6). In this step, we will use T r to associate to each groupoid Alexander-Spanier cocycle on G a cyclic cocycle on A ( ( ¯ h )) ⋊ G . CYCLIC COCYCL E S O N DE FORMA TIO N QUANTIZA TIO NS 49 Note that there are two natural products on A ( ( ¯ h ) ) ⋊ G . Recall ﬁrst that linearly A ( ( ¯ h )) ⋊ G ∼ = Γ cpt  G 1 , s ∗ A ( ( ¯ h ) )  , and tha t th is space carries the convolution pr oduct ⋆ c deﬁned by (7.5) [ f 1 ⋆ c f 2 ] g = ∑ g 1 g 2 = g  [ f 1 ] g 1 g 2  [ f 2 ] g 2 , f 1 , f 2 ∈ Γ cpt  G 1 , s ∗ A ( ( ¯ h ) )  , g ∈ G 1 , where [ f ] g denotes the germ of a section ∈ Γ cpt  G 1 , s ∗ A ( ( ¯ h ) )  at the point g ∈ G 1 . Secondly , the star product ⋆ can be canonically extended to A ( ( ¯ h ) ) ⋊ G by putting (7.6) [ f 1 ⋆ f 2 ] g = s ∗ [ f 1 ] g ⋆ s ∗ [ f 2 ] g , f 1 , f 2 ∈ Γ cpt  G 1 , s ∗ A ( ( ¯ h ))  , g ∈ G 1 . Now we can deﬁne X G T r : AS •  G , C ∞ G 0 ( ( ¯ h ) )  → C •  A ( ( ¯ h )) ⋊ G  by X G T r ( f 0 ⊗ · · · ⊗ f k ) ( a 0 ⊗ · · · ⊗ a k ) = T r k  f 0 ⋆ a 0 , · · · , f k ⋆ a k  : = T r  ( f 0 ⋆ a 0 ) ⋆ c · · · ⋆ c ( f k ⋆ a k )  . (7.7) Since in the de ﬁnition of T r ( A ) by Eq. ( 7.4) only the germ of a ∈ A ( ( ¯ h ) ) ⋊ G at B ( 0 ) enters, X G T r ( f ) ( a ) with f = f 0 ⊗ · · · ⊗ f n and a = a 0 ⊗ · · · ⊗ a k depends only on the ger m of ( f 0 ⋆ a 0 ) ⋆ c · · · ⋆ c ( f k ⋆ a k ) at B ( 0 ) . By deﬁnition of the products ⋆ c and ⋆ on A ( ( ¯ h )) ⋊ G , the value X G T r ( f ) ( a ) then depends only on the germs of f 0 ⊗ · · · ⊗ f k and a 0 ⊗ · · · ⊗ a k at B ( k ) . In particular , if f vanishes around B ( k ) , then X G T r ( f ) = 0. This shows that for ea ch k , X G T r is well deﬁned a s a map from A S  G , C ∞ G 0 ( ( ¯ h ) )  k to C k  A ( ( ¯ h )) ⋊ G  . Mor eover , one checks immediately that b X G T r = X G T r δ and B X G T r ( f ) = 0, if f ∈ AS  G , C ∞ G 0 ( ( ¯ h ) )  k . W e conclude that X G T r deﬁnes a cochain map from the the groupoid Alexander– Spanier cochain complex of G to the cyclic cochain complex of A ( ( ¯ h ) ) ⋊ G . T o relate our cons truction to higher indices of elliptic opera tors on an orbif old M , we constr uct a cochain map X M T r from groupoid Alexander–Spannier cochain complex of G to the cyclic cochain complex of the algebr a A ( ( ¯ h ) ) M , which can be identiﬁed as the algebr a of G -invariant smo oth functions on G 0 equipped with a G -invariant star product. Let c be a smooth cut-off function on G 0 as is introduced in [ T U , Sec. 1]. Deﬁne e a smooth function on G by e ( g ) : = c ( s ( g ) ) 1 2 c ( t ( g ) ) 1 2 . It is ea sy to check that ∑ g = g 1 g 2 e ( g 1 ) e ( g 2 ) = e ( g ) . Let E be the corresponding projection in A ( ( ¯ h ) ) ⋊ G with E ¯ h = 0 = e . (W e point out that e and E may not be compactly supported but they ca n be chosen to be inside a proper completion of A ⋊ G and A ( ( ¯ h )) ⋊ G on which the convolution products are still well deﬁned.) It is easy to check that e comm utes with all G -invar iant functions on G 0 and similarly E commutes with all elements of A ( ( ¯ h ) ) M . W e will use E to deﬁne a cochain map X M T r from groupoid Alexa nde r-Spannier cochain complex of G to the cyclic cochain complex of A ( ( ¯ h ) ) M . 50 M.J. PFLAUM, H. POS THUMA, AND X. TAN G Deﬁne X M T r : AS •  G , C ∞ G 0 ( ( ¯ h ) )  → C •  A ( ( ¯ h )) M  by X M T r ( f 0 ⊗ · · · ⊗ f k ) ( a 0 , · · · , a k ) : = T r   f 0 ⋆ ( a 0 ⋆ c E )  ⋆ c · · · ⋆ c  f k ⋆ ( a k ⋆ c E )   = T r   ( f 0 ⋆ E ) ⋆ c a 0  ⋆ c · · · ⋆ c  ( f k ⋆ E ) ⋆ c a k   (7.8) where a 0 , · · · , a k are elements of A ( ( ¯ h ) ) M identiﬁed as G -invaria nt functions on G 0 . W e point out that since a 0 , · · · , a k and f 0 , · · · , f k are compactly supported,  ( f 0 ⋆ E ) ⋆ c a 0  ⋆ c · · · ⋆ c  ( f k ⋆ E ) ⋆ c a k  is a lso compactly supported. U sing the fact that E commutes wit h a i , we can quickly check the equality between the two expressio ns in the deﬁnition. Hence, the pairing X M T r ( f 0 ⊗ · · · ⊗ f k ) ( · · · ) is well de ﬁned. Sim- ilar to X G T r , one can easily check tha t X M T r is compatible with the differ entials and therefore deﬁnes a cochain map. Both X G T r and X M T r are morphisms of sheaves of complexes. Now we explain how to related X G T r and X M T r when M is reduced. As shown in [ N E P F P O T A , Prop. 5 . 5], the algebra A ( ( ¯ h ) ) ⋊ G is Morita equivalent to the invariant algebra  A ( ( ¯ h ) ) ( G 0 )  G , if M is reduced. The Morita equivalence bimodules are given by P : = A ( ( ¯ h )) ⋊ G ⋆ c E and Q : = E ⋆ c A ( ( ¯ h ) ) ⋊ G , where P (a nd Q ) is a lef t (right) A ( ( ¯ h )) ⋊ G and right (left)  A ( ( ¯ h ) ) ( G 0 )  G bimodule. In particular , the map ι :  A ( ( ¯ h ) ) ( G 0 )  G → A ( ( ¯ h )) ⋊ G deﬁned by ι ( a ) = E ⋆ c a ⋆ c E = a ⋆ c E is an algebra homomorphism between the two algebras, and one can easily check the following diagram to commute: AS • ( G , O ) C • ( A ( ( ¯ h )) ⋊ G ) AS • ( G , O ) C •  ( A ( ( ¯ h )) ( G 0 ) ) G  ✲ X G T r ❄ I d ❄ ι ✲ X M T r . W e point out that when G is a tra nsformation gr oupoid of a ﬁni te group Γ acting on a symplectic manifold X , then one can choose e = E to be the element e = 1 | Γ | ∑ γ ∈ Γ δ γ , where δ γ is the function on ˜ U × Γ such that δ γ ( x , γ ) = 1 for every x ∈ ˜ U , and which is 0 otherwise. Note that the Morita equ ivalence between the cr ossed prod- uct algebra A ( ( ¯ h ) ) ⋊ Γ and the invaria nt algebra A ( ( ¯ h ) ) ( X ) Γ ∼ = A ( ( ¯ h )) M with M = X / Γ was proved by D O L G U S H E V and E T I N G O F [ D O E T ]. After the above discussion, we end this subsection with comparing the c on- structions a bove with the quasi-isomorphism Q from Section 5 . 3. S ince all the cochain maps involved are sheaf morphisms, the same local computations as in the proof of Proposition 6.8 entail the following result. Proposition 7.11. The sheaf mo rphisms X M T r : C • λ AS  C ∞ G 0 ( ( ¯ h ) )  → T ot • B C •  A ( ( ¯ h ) ) M  and Q ◦ e λ : C • λ AS  C ∞ G 0 ( ( ¯ h ) )  → T ot • B C •  A ( ( ¯ h ) ) M  ֒ → T ot • B C •  A ( ( ¯ h ) ) M  coincide in th e CYCLIC COCYCL E S O N DE FORMA TIO N QUANTIZA TIO NS 51 derived ca t egory o f sheaves on M . I n particular , th e morph ism X T r : C • λ AS  C ∞ G 0 ( ( ¯ h ) )  → T ot • B C • ( A ( ( ¯ h )) M ) is a quasi-isomorphism. 7.4. Pairing with locali zed K -the ory. In this section we will deﬁne localized K - theory for orbifolds and its pairing with the Alexander–Spa nier cohomology de- ﬁned in Se ction 7.1. Let Q be a n orbifold modeled by a proper ´ etale groupoid G . Pseudodifferenti al operators on orbifolds were intr oduce d in [ G N 1 , GN 2 ] and [ B U ] a s operators on C ∞ ( Q ) that in any local orbifold char t can be lifted to invari- ant pseudodifferential operators on open subsets of R n . Here we are interested in the a lgebra of smoothing operators that a re lifts of such smoothing operators on Q . However , the notion of invariance is not straightforward, except f or global quotient orbifolds. First let us remark that C ∞ ( Q ) embeds into C ∞ ( G 0 ) as functions invariant un- der G via pull-ba ck a long the projection π : G 0 → Q . Con sider the algebra Ψ DO − ∞ ( G 0 ) of smoothing oper a tors on G 0 . Let U = { U i } i ∈ I be a G -trivializing covering of G 0 and denote by A i the restriction of A ∈ Ψ − ∞ ( G 0 ) to U i ∈ U . Deﬁne Ψ DO − ∞ inv ( G , U ) : = : = { A ∈ Ψ − ∞ ( G 0 ) | supp ( A ) ⊂ U 2 , A i ( g x , g y ) = A j ( x , y ) for all i , j ∈ I , g ∈ G i j } . Note that this deﬁnition really makes sense, since G is ´ etale, hence a ny arrow g ∈ G 1 induces, by the existence of a local bisection, a local diffeomorphism with support on a sufﬁciently small neighbourhood of s ( g ) ∈ G 0 . Ther efore, we ﬁnd: Proposition 7.12 . For a sufﬁciently ﬁne covering U of G 0 , any element A ∈ Ψ DO − ∞ inv ( G , U ) deﬁnes a smoothing operator on C ∞ cpt ( Q ) . Observe that Ψ DO − ∞ inv ( G , U ) is not a subalgebra of Ψ DO − ∞ ( G 0 ) because of both the support condition and the invariance condition . However , we shall consider the space C • λ ( Ψ DO − ∞ inv ( G , U ) ) of cyclic cochains nonetheless. Let tr be the densely deﬁned trace on Ψ DO − ∞ ( G 0 ) coming f rom the repr esentation on L 2 ( G 0 ) . L et ∗ be canonical commutative product on C ∞ ( G ) deﬁned by f 1 ∗ f 2 ( g ) : = f 1 ( g ) f 2 ( g ) for f 1 , f 2 ∈ C ∞ ( G ) . For f = f 0 ⊗ . . . ⊗ f 2 k an element in C ∞ cpt ( B ( 2 k ) U ) deﬁne, as before, X U tr ( f ) ( A 0 ⊗ . . . ⊗ A 2 k ) = tr k  ( f 0 ∗ e ) A 0 , . . . , ( f 2 k ∗ e ) A 2 k  , with A 0 , . . . , A 2 k ∈ Ψ DO − ∞ inv ( G , U 0 ) , where U 0 is a G-trivializing cover such that st 2 k ( U 0 ) reﬁnes U , and e is the projection in A ⋊ G introduced in Section 7.3 . Proposition 7.13. The following identities hold true: X U tr ( δ ( f ) ) ( A 0 ⊗ . . . ⊗ A 2 k ) = X U tr ( f ) ( b ( A 0 ⊗ . . . ⊗ A 2 k ) ) X U tr ( t ( f ) ) ( A 0 ⊗ . . . ⊗ A 2 k ) = X U tr ( f ) ( A 2 k ⊗ A 0 ⊗ . . . ⊗ A 2 k − 1 ) . Proof. This is a d irect computation: ﬁrst observe that for f ∈ C ∞ cpt ( B ( 2 k ) U ) and smooth- ing operators A 0 , . . . , A 2 k ∈ Ψ DO − ∞ inv ( G , U 0 ) , the pairing X U tr ( f ) ( A 0 ⊗ . . . ⊗ A 2 k ) can 52 M.J. PFLAUM, H. POS THUMA, AND X. TAN G be written as ∑ t ( g i )= x i , i = 0,. . .,2 k Z G 2 k + 1 0 f ( g 0 , . . . , g 2 k ) e ( g 0 ) · · · e ( g 2 k ) A 0 ( g 0 ( x 0 ) , x 1 ) · · · A 2 k ( g 2 k ( x 2 k ) , x 0 ) d x 0 · · · d x 2 k = ∑ s ( g 0 )= x i i = 0,. . .,2 k Z G 2 k + 1 0 f ( g 0 , . . . , g 2 k ) e ( g 0 ) · · · e ( g 2 k ) A 0 ( g 0 ⊙ . . . ⊙ g 2 k ( x 0 ) , x 1 ) · · · A 2 k ( x 2 k , x 0 ) d x 0 · · · d x 2 k , where, to pass to th e second line, we use inva r iance of the kernels A i , i = 0, . . . , 2 k , and the composition g 0 ⊙ . . . ⊙ g 2 k is as de ﬁned in Lemma 7. 6. From this expres- sion and the property that e is a projection, the identities of the Propositio n ea sily follow .  W e can therefore morally think of X U tr as being a morphism of c ochain com- plexes from the compactly supported Alexander–Spanier complex ( C • AS,c ( G , U ) , δ ) to the cyclic complex ( C • λ ( Ψ DO − ∞ inv ( G , U 0 ) ) , b ) . 7.4.1. Localized K -theory. After these prepara tions, we can give a deﬁnition of lo- calized K -theory for orbifolds. Let U be a G -trivializing covering of G 0 and con- sider the associated subset Ψ DO − ∞ inv ( G , U ) of smoothing operators. As before, uni- talization is denoted by a ∼ . W ith this, let us deﬁne K 0  Ψ DO − ∞ inv ( G , U )  : =  ( P , e ) ∈ M ∞  Ψ DO − ∞ inv ( G , U ) ∼  × M ∞ ( C ) | P 2 = P , P ∗ = P , e 2 = e , e ∗ = e and P − e ∈ M ∞  Ψ DO − ∞ inv ( G , U )   / ∼ , (7.9) where ( P , e ) ∼ ( P ′ , e ′ ) for projections P , P ′ ∈ M ∞  Ψ DO − ∞ inv ( G , U ) ∼  and e , e ′ ∈ M ∞ ( C ) , if the elements P and P ′ can be joined by a continuous and p ie c ewise C 1 path of pro jections in som e M N  Ψ DO − ∞ inv ( G , U )  with N ≫ 0 a nd likewise for e and e ′ . E lements of K 0  Ψ DO − ∞ inv ( G , U )  are represented as equivalence classes of differences R : = P − e , wher e P is a n idempotent in M ∞  Ψ DO − ∞ inv ( G , U ) ∼  , e is a projection in M ∞ ( C ) , and the difference P − e lies in M ∞  Ψ DO − ∞ inv ( G , U )  . A (ﬁnite) r eﬁnement V ⊂ U obviously lea d s to an inclus ion Ψ DO − ∞ inv ( G , V ) ֒ → Ψ DO − ∞ inv ( G , U ) which induces a map K 0  Ψ DO − ∞ inv ( G , V )  → K 0  Ψ DO − ∞ inv ( G , U )  . W ith these maps, the orbifold localized K -theory of Q is d eﬁned as (7.10) K 0 loc ( Q ) : = lim ← − U ∈ Cov G ( M ) K 0  Ψ DO − ∞ inv ( G , U )  . More precisely , this means that elements of K 0 loc ( Q ) are given by families (7.11)  [ P U − e U ]  U ∈ Cov G ( M ) of equivalence classes of pairs of projectors in matrix spaces over Ψ DO − ∞ inv ( G , U ) ∼ such that e U ∈ M ∞ ( C ) for every G -trivializing covering U and ( P U , e U ) ∼ ( P V , e V ) in M ∞  Ψ DO − ∞ inv ( G , U ) ∼  whenever V ⊂ U . CYCLIC COCYCL E S O N DE FORMA TIO N QUANTIZA TIO NS 53 7.4.2. Pairing with Alexander–Spanier cohom ology. Finally , le t us describe the pair- ing of the thus deﬁned localized K -theory with orbifold Alexa nder–Spanier co- homology . Let U be a G -trivia lizing covering of G 0 , a nd f = f 0 ⊗ . . . ⊗ f 2 k ∈ C ∞ cpt ( B ( 2 k ) U ) a cocycle, i.e, δ f = 0. Choose a G -trivializing covering U 0 of G 0 such that st 2 k ( U 0 ) reﬁnes U . Now let R U 0 : = P U 0 − Q U 0 ∈ Ψ DO − ∞ inv ( G , U 0 ) represent an element of K 0  Ψ DO − ∞ inv ( G , U 0 )  as deﬁned above. Deﬁne  Ch AS 2 k ( R U 0 )  ( f ) : = ( − 2 π i ) k ( 2 k ) ! k ! ε 2 k  tr  ( f 0 ∗ e ) P U 0 ( f 1 ∗ e ) . . . ( f 2 k ∗ e ) P U 0  − tr  ( f 0 ∗ e ) Q U 0 ( f 1 ∗ e ) . . . ( f 2 k ∗ e ) Q U 0   , (7.12) where tr is the ca nonical opera tor tr a ce on Ψ DO − ∞ ( G 0 ) and e is the projection introduced in Section 7.3. W e remark that the ( f 0 ∗ e ) P U 0 ( f 1 ∗ e ) . . . ( f 2 k ∗ e ) P U 0 and ( f 0 ∗ e ) Q U 0 ( f 1 ∗ e ) . . . ( f 2 k ∗ e ) Q U 0 are well d eﬁned trace class operators on L 2 ( G 0 ) , because st 2 k ( U 0 ) is ﬁner than U . The same arguments as in [M O W U , Sec. 2] now prove the f ollowing result. Proposition 7.14. In the limit when the covering gets ﬁner , th e pairing deﬁned by Eq. (7.12) is independent of all choices and induces a map H e v cpt ( ˜ Q , C ) × K 0 l o c ( Q ) → C . 7.5. Operator-Symbol calculu s on orbifolds and the higher ana lytic index . In this ﬁnal subsection we will deﬁne the higher analytic index of an elliptic differ- ential operator on a reduced orbifold a nd, using the algebraic index theorem, de- rive a topological expressio n computing this num ber . Througho ut this section, we denote by Q a reduced compact riemannian orbifold modeled by a proper ´ etale groupoid G . The groupoid T ∗ G therefore models the cotangent bundle T ∗ Q . 7.5.1. Orbifold pseudodiffer ential op erators and the symbol calculus. Here we recall the symbol calculus on proper ´ etale groupoids of [ P F P O T A ] and relate it to the the- ory of pseudodifferential operators on orbifolds by imposing invaria nce. As f or the smoothing operators in Section 7.4, invariance only makes sense when the operators are localize d to a sufﬁciently small neighbourhood of the diagonal in G 0 × G 0 . Let U be a G -trivializing cover of G 0 , and choose a cut-off function χ : G 0 × G 0 → [ 0, 1 ] a s in (6.20) with supp ( χ ) ⊂ U 2 which is invariant: χ ( g x , g y ) = χ ( x , y ) , for all g ∈ G i j , x , y ∈ U i × U i . These choices deﬁne a quantization map as in (6.21). Observe that the groupoid G acts on the sheaf S ym m of symbols on G 0 , since they are just functions on T ∗ G 0 . It therefore makes sense to consider the subspace S ym m inv of invariant global symbols of order m . W ith this, we see from the explicit f ormula (6. 22) that the quantiza tion provides a map Op : S ym m inv → Ψ DO m inv ( G , U ) , where, as f or the smoothi ng opera tors, Ψ DO m inv ( G , U ) : = : = { A ∈ Ψ DO m ( G 0 ) , | supp ( A ) ⊂ U 2 , g A i g − 1 = A j , f or all i , j ∈ I , g ∈ G i j } . 54 M.J. PFLAUM, H. POS THUMA, AND X. TAN G Indeed, both the support and the invariance properties follow from the corre- sponding properties of the c ut- off function χ . In the opposite d irection, the symbol map σ d eﬁned in equation (6.1 9) maps σ : Ψ DO m inv ( G , U ) → Sym m inv and we there- fore have an isomorphis m Sym ∞ inv / Sym − ∞ inv ∼ = Ψ DO ∞ inv ( G , U ) / Ψ DO − ∞ inv ( G , U ) , induced by Op and σ . Now , since p seud odifferential operators hav e smo oth ker- nels off the dia gonal, one observes that the r ight hand side inherits an algebra structure from the product in Ψ DO ∞ ( G 0 ) even though Ψ DO ∞ inv ( G , U ) is not closed under operator composition. Therefore, going over to asymptotic families of sym- bols, one obtains a deformation quantization of T ∗ Q by de ﬁning the product on in- variant a symptotic families of symbols as in equation (6.24). W e refer to [ P F P O T A , Appendix 2] for more details about this opera tor-symbol calculus. M oreover , the operator tr a ce on L 2 ( Q ) d eﬁnes a trace tr on this deformation q uantization. W e ob- served in [ P F P O T A ] that as in the manifold ca se, this canonical deformation quan- tization using the a symptotic symbol calculus is isomorphic to one constructed by a Fedosov connection. Under the corresponding isomorphism, the operator tr is identiﬁed with the trace T r deﬁned by Eq. (7.4). Beside s this, we can use now a similar a rgument a s in Sec. 7.3 for the construction of X T r , to show that asymptotically in ¯ h the locally deﬁned maps X tr ( U ) glue together to a sheaf mor- phism X tr : C • AS  C ∞ T ∗ G 0 ( ( ¯ h ) )  → C •  A ( ( ¯ h ))  . Furthermor e, we can pull back func- tions on Q to T ∗ Q , hence we obta in a quasi-isomorphism from C • AS  C ∞ Q ( ( ¯ h ) )  to C • AS  C ∞ T ∗ Q ( ( ¯ h ) )  . Using the same arguments as for the proof of Eq. (6 .36), we can show now that the induced cochain map X tr : C • AS  C ∞ T ∗ Q  → C •  A ( ( ¯ h ) ) T ∗ Q  agrees with the map X T r . 7.5.2. The orbifold higher analytic index. Le t D be an elliptic differential operator on the reduced orbifold Q . W e denote by the same symbol D its lift to a G -invariant elliptic operator on G 0 . W ith the symbol calculus developed in the previous section we can now prove the f ollowing: Proposition 7.15. The elliptic operator D deﬁnes a canonical element [ D ] ∈ K 0 loc ( Q ) . Proof. By the deﬁnitio n of localized K - theory , cf. (7.10), we ﬁrst ha ve to constr uct an element in K 0  Ψ DO − ∞ inv ( G , U )  for any G -trivializing cover U , a nd second for any r eﬁnement V ⊂ U a homotopy between the corresponding K-theory elements localized in V respectively U . T o achieve the ﬁrst we use the opera tor-symbol cal- culus de veloped in Section 7.5 .1 and follow the standard procedure (cf . [ E G S C , Sec. 3 .2.2]) to ﬁnd a symbol f unction e ∈ Sym ∞ inv ( Q ) such that σ ( D ) e − 1 and e σ ( D ) − 1 are in S ym − ∞ inv . Choose a G -trivializing covering U ′ such that st 2 ( U ′ ) reﬁnes U , and a correspon ding invariant c ut- off function χ , we deﬁne the quan- tization ma p as in ( 6.21). It follows that both D Op ( e ) − I and Op ( e ) D − I a re elements in Ψ DO − ∞ inv ( G , U ′ ) , since D is an invariant differential operator . W rite E = Op ( e ) , and deﬁne S 0 : = I − D E , and S 1 : = I − E D , and L =  S 0 − E − S 0 B D S 1  . CYCLIC COCYCL E S O N DE FORMA TIO N QUANTIZA TIO NS 55 Then the matrix R deﬁned by R = L  I 0 0 0  L − 1 −  0 0 0 I  is a formal difference of projectors in M 2  Ψ DO − ∞ inv ( G , U )  which deﬁnes an ele- ment in K 0  Ψ DO − ∞ inv ( G , U )  . Second, f or a reﬁnement V ⊂ U we have two element R U ∈ K 0  Ψ DO − ∞ inv ( G , U )  and R V ∈ K 0  Ψ DO − ∞ inv ( G , V )  deﬁned by using cut-of f functions χ U and χ V . But then the family of pro jectors R t , t ∈ [ 0, 1 ] deﬁned us- ing the cut-off function χ t = t χ U + ( 1 − t ) χ V gives the desir ed ho motopy provin g that both pr ojectors deﬁne the same element in K 0  Ψ DO − ∞ inv ( G , U )  . In total, this deﬁnes the element [ D ] ∈ K 0 l o c ( Q ) . It is indepe ndent of any choices made.  W e a re now ready to deﬁne the higher a na lytic index of D . Let [ f ] ∈ H 2 k cpt ( ˜ Q , C ) be a c ompac tly supported cohomology class of degree 2 k , represented by an Ale- xander–Spanier cocycle f ∈ C ∞ cpt ( B ( 2 k ) U ) satisfying δ ( f ) = 0 for some G -trivia lizing cover U . Choose a G -trivializing cover V such that st 2 k ( V ) reﬁnes U . Then by the above d iscussion, R V deﬁnes an element in K 0  Ψ − ∞ inv ( G , V )  , which ca n be paired with f . Hence we deﬁne the [ f ] - localized index of D to be ind [ f ] = : Ch AS 2 k ( R V ) ( f ) , which is indep endent of the choices of the representative f in its cohomology class and the coverings U , V . Using the previously obtained results from this section one proves exa c tly like for Eq. ( 6.36) that by comparing Eq. (7.8) and E q. (7.1 2) the higher analytic index of D on Q can be computed using the corresponding higher algebra index of r D , where r D is the a symptotic symbol of R D . Therefore, we can apply Thm. 5.13 to compute ind [ f ] ( D ) . This proves our last result. Theorem 7 .16. Let D be an elliptic pseudod ifferen tial operators on a reduced orbifold Q, and [ f ] a c ompactly supported orbifold cyclic Alexander–Spanier cohom ology class of degree 2 j. Then ind [ f ] ( D ) = j ∑ r = 0 Z g T ∗ Q 1 ( 2 π √ − 1 ) j − r m e λ 2 j − 2 r ( f ) ∧ ˆ A ( ] T ∗ M ) Ch θ ( σ pr ( D ) ) Ch θ ( λ − 1 N ) , where ℓ , Ch θ , λ − 1 N , and m are as in Theorem 5.13. W e end this section with two remarks a bout the above Theorem 7.16. (1) When we ta ke the Alex a nder–Spanier c ohomolo gy class 1 ∈ H 0 ( ˜ Q ) , the localized index ind [ 1 ] ( D ) is the classical index of the elliptic operator D on Q . Theorem 7 .16 in this case reduces to the Kawasa ki’s index theor em [ K A ] , and our proof is identical to the one given in [ P F P O T A ]. (2) In the ca se that Q is a global quotient orbifold represented by a transfor- mation groupoid as in E xample 7 .8, if we take the cocycle f h γ i introduced at the end of Section 7.2, the localized index ind f h γ i ( D ) can be computed using a theorem by Atiyah and Segal in [A T S E ]. 56 M.J. PFLAUM, H. POS THUMA, AND X. TAN G A P P E N D I X A. C Y C L I C C O H O M O L O G Y A.1. The cycl i c bicomple x. Here we brie ﬂy recall the deﬁnition of Connes’ ( b , B ) - complex computing cyclic cohomo logy . L et A be a unital a lgebr a over a ﬁeld k . The Hochschild chain complex  C • ( A ) , b  resp. the normalize d Hochschild chain complex  C • ( A ) , b  is given by C k ( A ) : = A ⊗ k A ⊗ k resp. C k ( A ) : = A ⊗ k ( A / k ) ⊗ k equipped with the differential b : C k ( A ) → C k − 1 ( A ) , b ( a 0 ⊗ . . . ⊗ a k ) : = k − 1 ∑ i = 0 ( − 1 ) i a 0 ⊗ . . . ⊗ a i a i + 1 ⊗ . . . ⊗ a k + ( − 1 ) k a k a 0 ⊗ . . . ⊗ a k − 1 . Note that b passes down to C • ( A ) . The homology of  C • ( A ) , b  is called the Hochschild homology of A and is d enoted by H H • ( A ) . It naturally coincides with the homology of the normalized Hochschild chain complex. Introduce the operator B : C k ( A ) → C k + 1 ( A ) by the formula B ( a 0 ⊗ . . . ⊗ a k ) : = k ∑ i = 0 ( − 1 ) i k 1 ⊗ a i ⊗ . . . ⊗ a k ⊗ a 0 ⊗ . . . ⊗ a i − 1 . This deﬁnes a differential, i.e., B 2 = 0, and we ha ve [ B , b ] = 0, so we ca n form the ( b , B ) -bicomplex . . . b   . . . b   . . . b   C 2 ( A ) b   C 1 ( A ) b   B o o C 0 ( A ) B o o C 1 ( A ) b   C 0 ( A ) B o o C 0 ( A ) The total complex associated to this (normalized) mixed complex B k ( A ) = [ k / 2 ] M i = 0 C k − 2 i ( A ) , equipped with the differ ential b + B , is the fundamental complex computing the cyclic homolog y H C • ( A ) . The dual theory is obtained by taking the Hom k ( − , k ) of this complex with the induced differentials, also denoted b and B . For exa mple the normalized Hochschild cochain complex is given by C • ( A ) : = Hom k ( C • ( A ) , k ) and this leads to the normalize d mixed cyclic cochain c omplex  B • ( A ) , b , B  . This is the mixed complex that we will mainly use thr oughout this pa per . For further information on Hochschild and cyclic hom ology theory and in particular for the deﬁnition of B and B • ( A ) in the general, not normalized, case see [ L O ]. CYCLIC COCYCL E S O N DE FORMA TIO N QUANTIZA TIO NS 57 Remark A.1 . Note that for A a sheaf of algebr a s over a topolo gical spa ce M , the assignments U 7 → C k  Γ ( U , A )  and U 7 → C k  Γ cpt ( U , A )  , wh ere U runs through the open subsets of M are presheaves on M . Remark A.2. Throughout this paper we consider only algebra s resp. sheaves of a l- gebras which additionally carry a bornology compatible with the algebr aic struc- ture. It is understood that the Hochschild and cyclic (co)homologies considered have to be compatible with the bornology mea ning that as tensor product functor we take the completed bornological tensor product and as Hom-spac e s we choose the space of bounded linear maps between two bornological linear spaces. See [ M E , P F P O T A T S ] for de ta ils on bornologies. A.2. Localiza tion. Let k denote one of the ground rings R , R [ [ ¯ h ] ] or R ( ( ¯ h ) ) , a nd let M be a smooth manifold. Le t O M , k , or just O if no confusion can arise, be the sheaf of smoo th functions C ∞ M , if k = R , the sheaf C ∞ M [[ ¯ h ] ] , if k = R [ [ ¯ h ] ] , and ﬁnally the sheaf C ∞ M ( ( ¯ h ) ) , if k = R ( ( ¯ h ) ) . Assume that O carries a n associative local product · , which can be either given by the standard poin twise product of smooth functions or by a formal deformation thereof. Note that in each case, O carries the structure of a sheaf of bornological a lgebras and that O M k , k = O ˆ ⊠ k M , k , where ˆ ⊠ denotes the completed bornological exterior tensor product. Now let X ⊂ M be a (locally) closed subset. Then put for each open U ⊂ M J X , M , k ( U ) : = { F ∈ O ( U ) | ( D F ) | X ∩ U = 0 for a ll differential operators D on M } . Obviously , these space s form the section spaces of a n ideal sheaf J X , M , k in O ; we denote it brieﬂy by J X if no confusion ca n arise. The pullba ck of the quotient sheaf O / J X , M , k by the canoni cal embedding ι : X ֒ → M gives rise to a sheaf of Whitney ﬁelds on X (cf . [M A , B R P F ]). The r esulting sh eaf ι ∗  O / J X , M , k  will be d enoted by E X , M , k or E X for short. Next let ∆ k : M → M k be the diagonal embedding. The constructions ab ove then give rise to sheaf complexes C • ( O ) a nd C • ( O ) de ﬁned as follows. For k ∈ N and U ⊂ M open put C k ( O ) ( U ) : = Γ  ∆ k + 1 ( U ) , E ∆ k + 1 ( M ) , M k + 1 , k  and (A.1) C k ( O ) ( U ) : = Hom  Γ cpt  ∆ k + 1 ( U ) , E ∆ k + 1 ( M ) , M k + 1 , k  , k  . (A.2) Clearly , the C k ( O ) ( U ) resp. C k ( O ) ( U ) are the sectional spa ces of a ﬁne sheaf on M . Since b a nd B map the id eal J ∆ k + 1 , M k + 1 , k ( U ) to J ∆ k , M k , k ( U ) resp. J ∆ k + 2 , M k + 2 , k ( U ) , the differentials b and B desce nd to C • ( O ) and C • ( O ) . Thus we obtain mixed sheaf complexes  C • ( O ) , b , B  and  C • ( O ) , b , B  . Obviously , there are no rmalized versions of these mixed sheaf complexes which we will a lso use in this article. Finally , for each open U ⊂ M we have natural maps ρ k : C k  Γ ( U , O )  → C k ( O ) ( U ) , (A.3) a 0 ⊗ . . . ⊗ a k 7→ a 0 ⊗ . . . ⊗ a k + J ∆ k + 1 ( U ) , U , k and ρ k : C k ( O ) ( U ) → C k  Γ cpt ( U , O ( U )  , (A.4) F 7→  a 0 ⊗ . . . ⊗ a k 7→ F  a 0 ⊗ . . . ⊗ a k + J ∆ k + 1 ( U ) , U , k   . 58 M.J. PFLAUM, H. POS THUMA, AND X. TAN G Clearly , these maps are even mo rphisms of presheaves preserving the mixed com- plex structur es. Theorem A.3 . The morphisms of m ix ed sheaf complexes ρ • and ρ • are quasi-isomorphisms. Proof. For k = R and O the sheaf of smooth functions on M the cla im has been proven in [B R P F ]. For k = R [ [ ¯ h ] ] the c la im follows by a spectra l sequence argu- ment. Note that O is ﬁltered by powers of ¯ h in that case which induces a ﬁltration on C •  Γ ( U , O ) and C k ( O ) ( U ) . Consider the associated spec tr al sequences. 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P FL A U M , markus.pflaum@c olorado.edu Department of Mathematics, University of Colorado, Boulder , USA H E S S E L P O S T H U M A , posthuma@math. uu.nl Mathematical Institute, Ut recht University , Utrecht, The Netherlands X I A N G T A N G , xtang@math.wustl .edu Department of Mathematics, W ashington University , St . L ouis, USA

Cyclic cocycles on deformation quantizations and higher index theorems

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