Orbifold cup products and ring structures on Hochschild cohomologies

ORBIF OLD CUP PR ODUCTS AND R ING STR UCTURES ON HOCHSCHILD COHOM OLOGIES M.J. PFLAUM, H.B. POSTHUMA, X. T ANG, AND H.-H . TSENG Abstract. In this pap er we study the Hochsc hil d cohomology ring of conv o- lution algebras asso ciated to orbifolds, as w ell as their deform ation quant iza- tions. In the ﬁrst case the ring structure is giv en in terms of a wed ge pro duct on t wisted polyvecto rﬁelds on the inertia or bi fold. After deformation quanti- zation, the ring structure deﬁnes a product on the cohomology of the inertia orbifold. W e study the relation b etw een t his pro duct and an S 1 -equiv ariant v ersion of the Chen–Ruan pr oduct. In particular, we give a de Rham mo del for this equiv ari an t orbif old coho mology . Contents 1. Int ro duction 2 2. Outline 4 3. Cup pro duct on the Ho chschild co homology of the convolution a lgebra 11 3.1. Lo c a lization metho ds 11 3.2. The g lobal quotient case 13 4. Cup pro duct on the Ho chsc hild cohomo logy of the deformed conv olutio n algebra 18 4.1. Reduction to th e ˇ Cech complex 20 4.2. Twisted c o cycles on the for mal W eyl alg ebra 20 4.3. The F edosov–W einstein–Xu resolutio n ov er B 0 22 4.4. Lo c a l computatio ns 25 4.5. The g eneral cas e 27 4.6. F rob enius algebr as from Ho chsc hild cohomolo gy 29 5. Chen-Ruan o rbifold co homology 30 5.1. S 1 -Equiv ar ia nt Chen-Ruan orbifold cohomolo gy 30 5.2. Equiv ariant de Rha m mo del 32 5.3. T op ologica l and algebr a ic Hochschild cohomolo gy 35 Appendix A. H omologica l algebra of b or nological algebra s and mo dules 37 A.1. Bornolog ies on vector s paces 37 A.2. Bornolog ical algebras and mo dules 38 A.3. Resolutions and homo logy 39 A.4. Hochschild homology a nd B ar reso lution 41 A.5. The cup pro duct on Ho chschild c o homology 42 A.6. Bornolog ical structures on co nv olution a lgebras and their mo dules 42 A.7. Morita equiv alence for bo rnologica l a lgebras 44 References 49 Date : May 29th, 2007. 1 2 M.J. PFLA UM, H.B. POSTHUMA, X. T ANG, AND H.-H. TSENG 1. Introduction In this pap er, we study orbifolds within the languag e of noncommutativ e ge- ometry . According to [ Mo ], an orbifold X can b e represe nted by a prop er ´ etale Lie groupo id G , and diﬀerent repres ent ations o f the sa me o rbifold X are Morita equiv alent. A paradig m from noncommutativ e g eometry tells that o ne s hould view the group oid algebra A ⋊ G of a prop er ´ etale g roup oid G r epresenting the orbifold X as the “ a lgebra of functions” on X , where A is the sheaf of functions on the unit space G 0 of G . Though it is noncommutativ e , the algebra A ⋊ G con tains m uch impo rtant information o f X . W e provide in this pap er a complete descr iptio n of the r ing structures on the Ho ch schild coho mology of the group oid a lgebra A ⋊ G and its deformation q uanti- zation A ~ ⋊ G when X is a symplec tic orbifold. W e thus complete pr o jects initiated in [ T a ] and [ N ePfPoT a ]. By o ur re s ults one obta ins a cup product on the space of m ultivector ﬁelds on the inertia orbifold e X as so ciated to X , and a F rob enius structure on the de Rham cohomology of the ine r tia or bifold e X . This F rob enius structure is clos ely re lated to the Chen-Ruan o rbifold cohomolo gy [ ChRu04 ], and inspires us to intro duce a de Rham mo del for so me S 1 -equiv aria n t Chen-Ruan orbifold cohomology . W e prov e that the algebra of the Ho chsc hild cohomology of the deformation quan tiza tion A ~ ⋊ G is isomorphic to the gr a ded alge br a of the Chen-Ruan o rbifold co homology with resp ect to a natural ﬁltration. In this pap er, we view the alg ebra A ⋊ G as a b ornolog ical alg ebra with the canon- ical b orno logy inherited from the F r echet top olo gy and co mpute its Hochschild cohomolog y respe c t to this borno logy . In [ NePfPoT a ], we constructed a vector space isomo rphism H • ( A ⋊ G , A ⋊ G ) = Γ ∞  ∧ •− ℓ T B 0 ⊗ ∧ ℓ N  G , where B 0 is the space o f lo o ps of G deﬁned as { g ∈ G | s ( g ) = t ( g ) } , ℓ is a lo cally constant function on B 0 , namely the co dimension of the germ of B 0 inside G 1 , and where N is the norma l bundle o f B 0 in G 1 . In this article, w e determine the cup pro duct on the Ho chsc hild co chain H • ( A ⋊ G , A ⋊ G ). T o do so, we need to understand the maps rea lizing the above is omorphisms of vector spaces. In [ NePfPoT a ], the ring str uctures got lost a t the end of the ﬁna l equality , s inc e there we w ere de a ling with a clumsy c hain of quasi-iso mo rphisms. The ﬁr st goa l of t his work is to present a sequence of explicit quasi-isomorphisms o f diﬀ erential graded algebr a s pr e serving cup pro ducts. Some par ts of these quasi-isomorphisms hav e alrea dy app eared in [ NePfPoT a ] and [ HaT a ], but in this work we succeeded to put all ing rediants to gether in the right w ay and th us determine the cup pro ducts we were loo king for. The new input consists in the following. Firstly , w e in tro duce a complex of ﬁne pr esheav es H • on X whic h has a na tural cup pro duct a nd the global sections of which form a complex qua si-isomor phic to the Ho chschild cohomolog y complex. Seco ndly , we use ˇ Cech cohomolo gy metho ds to loca lize the computation of the coho mology ring of H • ( X ). Thirdly , we use the twisted co cycle construction and the lo cal qua si-inv er se map T from [ HaT a ] to c o mpute the cup pro duct lo ca lly . By gluing together the lo cal cup pro ducts to a g lobal one we ﬁnally arr ive at a transpare nt computation of the cup pro duct on H • ( A ⋊ G , A ⋊ G ). W e would like to mention tha t in [ A n ], some s imilar but incomplete res ults in the lo cal situatio n were o btained. ORBIF OLD C UP PRODUCTS 3 The ab ove sequence of explicit qua si-isomor phisms opens a way to compute the Ho ch schild co ho mology of the deformation quantization A ~ ⋊ G , which originally has b een constr uc ted in [ T a ]. In the case of a globa l quotient , the Ho chsc hild cohomolog y of this a lg ebra has been computed by Dolg ushev and Etingo f [ Do Et ] as a vector space using v an den Berg duality . Our method is co mpletely diﬀer e n t from [ DoEt ] and allows ev en to determine the ring structure on H • ( A ~ ⋊ G , A ~ ⋊ G ) in full g enerality . The crucial step in our approa ch is that we generalize the ab ov e complex of preshev es H • on X and the assso ciated ˇ Cech double complex to the deformed case. With the quasi-isomorphisms for th e undeformed a lgebra, o ne can chec k that there are natura l morphis ms of diﬀerential gr aded algebr as from the Ho ch schild co chain complex of A ~ ⋊ G to the presheaf co mplex H • loc , ~ and the asso ciated ˇ Cech double complex. W e prove these maps to be quasi- isomorphisms by lo ok ing at the E 1 terms of the spectr al seque nce asso ciated to the ~ -ﬁltration, which a grees with the undefor med c o mplexes. T o p erfor m the lo cal computations, we generalize the F edosov–W einstein– Xu re s olution in [ Do ] for the computation of the Ho chsc hild cohomo lo gy o f a deformation quantization to the G -twisted situation using ideas of F edosov [ Fe ]. W e use esse ntially an ex plic it map from the Koszul resolution of the W eyl algebr a to the corresp onding Bar resolution by Pinczon [ Pi ]. Our main theor em is that we hav e a natural isomorphism of algebras ov er C (( ~ )) H •  A ~ ⋊ G , A ~ ⋊ G  ∼ = H •− ℓ  ˜ X , C (( ~ ))  , where the pr o duct structure on the rig ht hand side is deﬁned (cf. Section 4) by [ α ] ∪ [ β ] = Z m ℓ pr ∗ 1 α ∧ pr ∗ 2 β . (1.1) This genera lizes Alv arez’s res ult [ Al ] on the Ho chsc hild co homology ring of the crossed pr o duct alg ebra of a ﬁnite gro up with the W eyl algebra . The cup product (1.1) together with integration with r esp e ct to the symplectic volume form deﬁnes a F rob enius structure on the de Rham co homology of the iner- tia orbifold e X . O ne notices that there is similarity b etw een (1.1) a nd the de Rham mo del deﬁned by Chen a nd Hu [ ChHu ]. How ever, Chen a nd Hu’s mo del was only deﬁned for abe lia n or bifolds a nd w orks in a for mal level. T o connect the Ho ch schild cohomolog y of A ~ ⋊ G to the Chen-Ruan orbifold cohomolo gy , we extend the de Rham mo del to an ar bitrary almost c o mplex orbifold us ing metho ds fro m equi- v ariant cohomolog y theor y and [ JaKaKi ]’s result on obstruction bundles. More precisely , we pr ove that the algebra ( H • ( A ~ ⋊ G , A ~ ⋊ G ) , ∪ ) is isomorphic to the graded algebr a of the S 1 -equiv aria n t Chen- Ruan orbifold co homology with resp ect to a na tural ﬁltr ation. In g eneral, the Ho chsc hild cohomo logy and the Chen- Rua n orbifold co homology are not isomor phic as algebras. By co nstruction, the Chen- Ruan or bifold co homology dep ends on the c ho ice of an almost complex structur e, but the Hochsc hild cohomo logy is indep endent of the choice of an almost co mplex or symplectic structure. Ther efore, o ne n aturally ex p ects that infor mation o n the almost complex structure s hould b e contained in the ﬁltration on the de Rham mo del. It is a v er y interesting question whether one ca n detect diﬀeren t almost complex structures through the ﬁltr ation on the de Rha m model. O ur de Rham mo del and the computation o f the Ho chsc hild cohomo logy ring of the deformed con- volution alg e br a give more insig h t to the Ginzburg- K aledin conjecture [ GiKa ] f or hyper-K¨ ahler or bifolds. Our computations wit hin the diﬀer e n tial categor y sug gest 4 M.J. PFLAUM, H. B. POSTHUMA, X. T ANG, AND H.-H. TSENG that it is cr ucial to work in the holomor phic categor y of deformation quantiza- tion, otherwise conjecture from [ G iKa ] that there is an is omorphism bet ween the Ho ch schild cohomology ring of a deforma tio n quantization and the Chen- Rua n or b- ifold cohomology will in gener al not b e true. Concerning the de Rham model for orbifold cohomolo g y let us a lso men tio n that recently , a similar mo del has b een obtained indep endantly b y R. Ka ufmann [ Ka ]. Our pap er is or ganized as follows. In Section 2, we outline the str ategy and the main results of this pap er , in Sectio n 3, we provide a detailed computation of the Hochsc hild cohomology and its ring structure of the a lgebra A ⋊ G . Next, in Section 4 , w e c o mpute the Ho chsch ild coho mo logy and its r ing str ucture of the deformed algebr a A ~ ⋊ G . Then w e switch in Section 5 to orbifold cohomo logy theory . W e in tro duce a de Rha m model for some S 1 -equiv aria n t Chen- Ruan orb- ifold cohomo logy and connect this mo del to the ring struc tur e of the Ho chsc hild cohomolog y of the deformed co n volution alge br a. In the Appendix, we provide a full in tro duction to bor nological algebras , their modules and their Mor ita theor y . W e wan t to emphas ize that the App endix contains some orig inal res ults on Morita equiv alence of b ornolog ical a lgebras, which to o ur knowledge ha s not b een cov ered in the literature before. W e hav e chosen to k eep thes e r esults in the Appendix to av oid to o technical ar guments in the main par t of our pap er. Ac knowledgemen ts: H.P . and X.T. would like to thank Goethe- Univ ersit¨ at F rank - furt/Main for hosting their visits. M.P ., X.T., and H.-H.T. tha nk the o rganizer s of the trimester “Gro up oids and Stacks in Physics and Geometry” for hosting their visits of the Institut de Henri Poincar´ e, Paris. X.T. would like to thank G. Pinczo n and G. Halb out for helpful discussions, M.P . a nd H.P . thank M. Cr ainic for fr uitful discussions. M.P . ac knowledges supp ort b y the DF G. H.P . is suppor ted by NWO. X.T. a ckno wledg es suppo r t by the NSF. 2. Outline As is mentioned ab ove, the main goa l of this ar ticle is to deter mine the ring structure of the Ho chsch ild cohomolo gy of a defo rmation quantization on a pr op er ´ etale Lie gr oup oid. In this section, w e outline the strateg y to achiev e that goa l and beg in with a br ief ov er v iew ov er the ba s ic no tation and results needed from the theory of gro upo ids. F or further details on the latter we refer the interested reader to the mono g raph [ MoMr ] and also to our previous article [ NePfPoT a ]. Recall that a gr oup oid is a small category G with set o f o b jects denoted by G 0 and set of morphis ms by G 1 such that all morphisms ar e inv ertible. The structure maps o f a gr oup oid are depicted in the diagra m G 1 × G 0 G 1 m → G 1 i → G 1 s ⇒ t G 0 u → G 1 , where s and t are the sour ce and ta r get map, m is the multiplication resp. comp osi- tion, i denotes the inv er se and ﬁnally u is the inclusion of ob jects by identit y mor- phisms. In the mos t interesting c ases, the gr o upo id carr ies an additional structure, like a top ologic a l or diﬀer en tiable str uc tur e. If G 1 and G 0 are both top olo gical space s and if all structure maps are contin uo us , then G is called a top olog ical group oid. Such a top olo gical g roup oid is called pr op er , if the map ( s, t ) : G 1 → G 0 × G 0 is a prop er map, and ´ etale , if s and t are bo th lo cal homeo morphisms. In case G 1 and ORBIF OLD C UP PRODUCTS 5 G 0 carry the structure of a C ∞ -manifold such that s , t, m, i and u a re s mo o th and s, t submersions, then G is sa id to b e a Lie g r oup oid . The s ituation s tudied in this article consists o f an orbifold represented by a prop er ´ etale Lie group o id G . As a top olo gical s pace, the orbifold coincides with the or bit space X = G 0 / G . In the following w e intro duce several sheaves o n G and X . By A we alwa ys denote the s heaf of smo oth functions o n G 0 , and by A ⋊ G the conv olutio n a lgebra, i.e. the space C ∞ cpt ( G 1 ) tog ether with the con volution pro duct ∗ which is deﬁned by the formula a 1 ∗ a 2 ( g ) = X g 1 · g 2 = g a 1 ( g 1 ) a 2 ( g 2 ) for all a 1 , a 2 ∈ C ∞ cpt ( G 1 ) a nd g ∈ G 1 . (2.1) Next, let ω b e a G -in v ariant s ymplectic form on G 0 and choo se a G -inv ar iant (loc a l) star pro duct ⋆ o n G 0 . The res ulting sheaf of defor med algebras of smoo th functions will b e denoted b y A ~ . The cro ssed pro duct algebra A ~ ⋊ G has the underlying vector space C ∞ cpt ( G 1 )[[ ~ ]] = Γ ∞ cpt ( G 1 , s ∗ A ~ ) a nd carries the pro duct ⋆ c given by [ a 1 ⋆ c a 2 ] g = X g 1 · g 2 = g [ a 1 ] g 1 g 2 ⋆ [ a 2 ] g 2 for a ll a 1 , a 2 ∈ C ∞ cpt ( G 1 ) a nd g ∈ G 1 . (2.2) Hereby , [ a ] g denotes an element of the stalk ( s ∗ A ~ ) g ∼ = A ~ s ( g ) , and it has b een used that G a cts fro m the right on the sheaf A ~ (see [ NePfPoT a , Sec. 2] for details). F or every op en subset U ⊂ X deﬁne the space e A ( U ) by e A ( U ) := ( π s ) ∗ s ∗ A ( U ) = C ∞ (( π s ) − 1 ( U )) , where π : G 0 → X is the cano nical pr o jection. Denote by e A fc ( U ) the subs pace  f ∈ C ∞ (( π s ) − 1 ( U )) | supp f ∩ ( πs ) − 1 ( K ) is compact for all compact K ⊂ U  of all smo o th functions o n ( π s ) − 1 ( U ) with ﬁb erwise compact supp ort. Obser ve that the conv olution pro duct ∗ can b e extended naturally b y Eq. (2.1) to ea ch of the spaces e A fc ( U ). Indeed, since for K i := supp a i with a i ∈ e A fc ( U ), i = 1 , 2 the set m  ( K 1 × K 2 ) ∩ ( G 1 × G 0 G 1 ) ∩ ( πs ) − 1 ( K )  = m  ( K 1 ∩ ( π s ) − 1 ( K )) × ( K 2 ∩ ( π s ) − 1 ( K ))  ∩ ( G 1 × G 0 G 1 )  is compact by as s umption o n the a i , the pro duct a 1 ∗ a 2 is w ell-deﬁned and lies again in e A fc ( U ). Hence, the spaces e A fc ( U ) all c a rry the structure of an algebra a nd form the sectio na l spaces of a sheaf e A fc on X . Lik ewise, one constructs the sheaf e A ~ fc . Finally note that the natura l maps A ⋊ G ֒ → e A fc ( X ) a nd A ~ ⋊ G ֒ → e A ~ fc ( X ) are b oth alg e bra homomo r phisms. F rom Appe ndix A.6 one can derive the following result. Theorem O . The algebr as A ⋊ G and A ~ ⋊ G c arry in a natur al way t he structur e of a b ornolo gic al algebr a and ar e b oth quasi-unital. Likewise, the she aves e A fc and e A ~ fc ar e she aves of quasi-unital b ornolo gic al algebr as. Mor e over, the natur al homomorphisms A ⋊ G ֒ → e A fc ( X ) and A ~ ⋊ G ֒ → e A ~ fc ( X ) ar e b ounde d. Pr o of. P rop. A.6 and P rop. A.8 in the app endix show that A ⋊ G a nd A ~ ⋊ G ar e quasi-unital b orno logical algebra s. By exa ctly the sa me metho ds a s in ther e one shows that e A fc and e A ~ fc are sheaves of quasi-unital b or nological alg e bras. That the homomorphisms in T he o rem O. ar e b ounded is straig h tforward.  6 M.J. PFLAUM, H. B. POSTHUMA, X. T ANG, AND H.-H. TSENG According to Appendix A.4, Theor em O implies in particular that eac h one of the alg ebras in the c la im is H-unital and that the Bar complex provides a pro jective resolution. This will b e the star ting p oint for proving that several of the chain maps constructed in the following s teps are indeed qua s i-isomorphis ms . T o formulate the next step, consider the Ho chschild co chain complex (see A.4 ) C • ( A ⋊ G , A ⋊ G ) := Hom ( A ⋊ G ) e (Bar • ( A ⋊ G ) , A ⋊ G ) , where ( A ⋊ G ) e is th e en veloping algebra (see Sec. A.4), and deﬁne for each op en U ⊂ X the b or nological space H k G ( U ) by H k G ( U ) := Hom  ( A | U ⋊ G | U ) ˆ ⊗ k , e A fc ( U )  , where G | U is the gr oup oid with ob ject s e t G | U 0 = π − 1 ( U ) and mor phism set G | U 1 = ( π s ) − 1 ( U ) a nd where A | U is the sheaf o f smo oth functions on π − 1 ( U ). Obviously , the spaces H k G ( U ) form the sectiona l spaces of a pr esheaf H k G on X which w e denote by H k if no co nfusion can ar ise. The Ho chsc hild co bo undary map β := b ∗ on C • ( A | U ⋊ G | U , A | U ⋊ G | U ) extends to a cobounda ry map β on H • ( U ) b y the fo llowing deﬁnition: β F ( a 1 ⊗ . . . ⊗ a k ) := a 1 F ( a 2 ⊗ . . . ⊗ a k )+ + k − 1 X i =2 ( − 1) i +1 F ( a 1 ⊗ . . . ⊗ a i a i +1 ⊗ . . . ⊗ a k ) + F ( a 1 ⊗ . . . ⊗ a k − 1 ) a k , for F ∈ H k ( U ) a nd a 1 , . . . , a k ∈ C ∞ cpt (( π s ) − 1 ( U )). Moreov er, there is a pr o duct ∪ : H • ( U ) × H • ( U ) → H • ( U ), which is called the cup pr o duct on H • ( U ) a nd which is given as follows: ∪ : H k ( U ) × H l ( U ) → H k + l ( U ) , ( F, G ) 7→ F ∪ G, F ∪ G ( a 1 ⊗ · · · ⊗ a k + l ) := F ( a 1 ⊗ · · · ⊗ a k ) G ( a k +1 ⊗ · · · ⊗ a k + l ) for a 1 , · · · , a k + l ∈ C ∞ cpt (( π s ) − 1 ( U )) . It is stra ightf orward to chec k tha t the cup product is asso cia tive a nd passes down to the coho mology of H • ( U ). The compatibilit y betw een the Ho chschild cohomolo g y ring of A ⋊ G and the ring structur e on the c o homology of H • is express e d by the following s tep and will be pr ov ed in Sec tio n 3.1. Theorem I . The c anonic al emb e dding ι : C • ( A ⋊ G , A ⋊ G ) → H • ( X ) is a quasi-isomorp hism which pr eserves cup pr o ducts. Since H • is a co mplex of preshea ves o n X , o ne c an use localiza tio n techniques for the computation of its coho mology ring. Thus, metho ds from ˇ Cech cohomolog y theory come into play . T o mak e t hese ideas prec is e let U be an open cov er of the orbit space X , and d enote b y ˇ H • , • U := ˇ H • , • G , U := ˇ C • U ( H • G ) the ˇ Cech double complex asso ciated to the preshea f complex H • G . This means that ˇ H p,q U := ˇ C q U ( H p ) := Y ( U 0 , ··· , U q ) ∈U q +1 H p ( U 0 ∩ · · · ∩ U q ) . ORBIF OLD C UP PRODUCTS 7 The cob ounda ries on ˇ H • , • U are given in p -dir ection by the Ho chsc hild co bo undary β : ˇ H p,q U → ˇ H p +1 ,q U and, in q -direction, by the ˇ Cech cobo undary δ : ˇ H p,q U → ˇ H p,q +1 U ,  H ( U 0 , ··· , U q )  ( U 0 , ··· , U q ) ∈U q +1 7→  X 0 ≤ i ≤ q +1 ( − 1) i H ( U 0 , ··· , c U i , ··· , U q +1 ) | U 0 ∩···∩ U q +1  ( U 0 , ··· ,U q +1 ) ∈U q +2 . The cohomolo gy of the double co mplex ˇ H • , • U , i.e. the co homology o f the total com- plex T ot • ⊕ ( ˇ H • , • U ), w ill b e deno ted by ˇ H • U ( H • ). The inductive limit ˇ H • ( H • ) := lim ← − U ˇ H • U ( H • ) , where U runs throug h the set of op en cov ers of X , then is the ˇ Ce ch c ohomolo gy of the pr esheaf complex H • . The crucial claim, which will be prov ed in Section 3.1 as well, no w is the following. Theorem I I . The pr eshe aves H p ar e al l ﬁne, henc e the ˇ Ce ch c ohomolo gy of the pr eshe af c omplex H • is c onc entr ate d in de gr e e q = 0 , i.e. ˇ H • U ( H • ) is c anonic al ly isomorphi c t o the c ohomolo gy of the c o chain c omplex H • , 0 U . Mor e over, t he ˇ Ce ch c ohomolo gy ˇ H • ( H • ) is given by the glob al se ctions of a c ohomolo gy she af on X . Fi- nal ly, for e ach suﬃciently ﬁne and lo c al ly ﬁnite op en c overing U of X the c anonic al chain map H p ( X ) → ˇ Z p, 0 U ( H • ) , H 7→ ( H | U ) U ∈U is a quasi-isomorp hism, wher e ˇ Z p, 0 U ( H • ) :=  H = ( H U ) U ∈U ∈ ˇ H p, 0 U | δ H = 0  . By Theo rem I I one o nly needs to compute the co homology of the complexes H • ( U ) for all elemen ts U of a suﬃciently ﬁne op en covering of X . This is the purp ose of the following steps. Let us consider now a weak equiv alence of prop er ´ etale gro upo ids ϕ : H ֒ → G . Assume further that ϕ is an op en e m bedding and denote b y H • H and H • G the complexes of pres heav es as deﬁned ab ov e. Then ϕ indu ces a bounded linear map ϕ ∗ : C ∞ cpt ( H 1 ) → C ∞ cpt ( G 1 ) by putting for a ∈ C ∞ cpt ( H 1 ), g ∈ G 1 ϕ ∗ ( a )( g ) = ( a ◦ ϕ − 1 ( g ) , if g ∈ im ϕ, 0 , else . Moreov er, one o btains b ounded chain maps ϕ ∗ : C • ( A ⋊ G , A ⋊ G ) → C • ( A ⋊ H , A ⋊ H ) , F 7→ ϕ ∗ ( F ) and ϕ ∗ : H • G → H • H , F 7→ ϕ ∗ ( F ) , where in b oth ca ses ϕ ∗ ( F ) is deﬁned by ϕ ∗ ( F )( a 1 ⊗ · · · ⊗ a k ) = F  ϕ ∗ a 1 ⊗ · · · ϕ ∗ a k  ◦ ϕ for a 1 , · · · , a k ∈ C ∞ cpt ( H 1 ) . By Theorem I, the c hain map ι : C • ( A ⋊ G , A ⋊ G ) → H • G ( X ) is a quasi- isomorphism. Hence by Theorems A.11 and A.1 4 from the Appendix the following result ho lds true. Theorem II I . Under the assumptions on G , H and ϕ fro m ab ov e, the co nv olution 8 M.J. PFLAUM, H. B. POSTHUMA, X. T ANG, AND H.-H. TSENG algebras A ⋊ G and A ⋊ H ar e Morita equiv alent as b ornolo gical algebr as. Mor eov er , there is a co mm utative diagr am co nsisting of quas i- isomorphisms C • ( A ⋊ G , A ⋊ G ) ϕ ∗ / / ι   C • ( A ⋊ H , A ⋊ H ) ι   H • G ( X ) ϕ ∗ / / H • H ( X ) such that the upp er horizont al chain map coincides with the natural isomorphism betw een the Ho chschild cohomolo g ies H • ( A ⋊ G , A ⋊ G ) → H • ( A ⋊ H , A ⋊ H ) induced b y the Mor ita co n text b etw een A ⋊ G and A ⋊ H . As a n applicatio n of this r e sult, we consider a p oint x ∈ G 0 in the ob ject set o f a prop er ´ etale Lie group oid G , and denote by G x the isotr opy group o f x tha t means the gro up of all ar rows starting and ending a t x . Cho o se for each g ∈ G x an op en connected neighbo rho o d W g ⊂ G 1 (whic h c an b e c ho sen to be suﬃcien tly sma ll) such that bo th s | W g : W g → G 0 and t | W g : W g → G 0 are diﬀeomor phis ms onto their images. Let M x be the connected comp onent of x in T g ∈ G x s ( W g ) ∩ t ( W g ) a nd put M g := W g ∩ s − 1 ( M x ) fo r all g ∈ G x . Deﬁne an action of G x on M x by G x × M x → M x , ( g , y ) 7→ t  s − 1 | W g ( y )  . It is now stra ightf orward to chec k that the cano nical embedding G x ⋉ M x ֒ → G | π ( M x ) , ( y , g ) 7→ s − 1 | W g ( y ) , is op en and a weak equiv alence of Lie gro upo ids. In this article, w e will call a manifold M x together with a G x -action on M x and a G x -equiv aria n t embedding ι x : M x ֒ → G 0 a slic e around x , if the induced embedding G x ⋉ M x ֒ → G | π ι ( M x ) is op en a nd a weak equiv alence of Lie group oids. The argument ab ov e shows that for every point x ∈ G 0 there ex ists a s lice. As a corollar y to the a bove o ne obtains Theorem II Ib . L et x ∈ G 0 b e a p oint and ι x : M x ֒ → G 0 a slic e ar oun d x . L et ϕ x : G x ⋉ M x ֒ → G | U x with U x := π ι x ( M x ) b e the c orr esp onding we ak e quivalenc e. Then the c onvolution algebr as C ∞ ( M x ) ⋊ G x and A | U x ⋊ G | U x ar e Morita e qu ivalent. Mor e over, the c anonic al chai n map ϕ ∗ x : H • ( U x ) → C •  C ∞ ( M x ) ⋊ G x , C ∞ ( M x ) ⋊ G x  is a quasi-isomorp hism which implements the quasi-isomorphism induc e d in Ho ch- schild c ohomolo gy by the Morita c ontext b etwe en C ∞ ( M x ) ⋊ G x and A | U x ⋊ G | U x . Theorems I I and I II ena ble us to lo ca lize the computation of the Ho chsc hild cohomolog y rings. Lo cally , we hav e the following result, also shown in Section 3 . Theorem IV . L et M b e a smo oth manifold, and Γ a ﬁnite gr oup acting on M . Then the Ho chschild c ohomolo gy ring H • ( C ∞ cpt ( M ) ⋊ Γ , C ∞ cpt ( M ) ⋊ Γ) is given as fol lows. As a ve ctor sp ac e, one has H • ( C ∞ cpt ( M ) ⋊ Γ , C ∞ cpt ( M ) ⋊ Γ) = M γ ∈ Γ Γ ∞  Λ •− ℓ ( γ ) T M γ ⊗ Λ ℓ ( γ ) N γ  Γ , ORBIF OLD C UP PRODUCTS 9 wher e ℓ ( γ ) is the c o dimension of M γ in M , and N γ is the normal bund le to M γ in M . F or elements ξ =  ξ α  α ∈ Γ , η =  η β  β ∈ Γ ∈ M γ ∈ Γ Γ ∞  Λ •− ℓ ( γ ) T M γ ⊗ Λ ℓ ( γ ) N γ  Γ their cup pr o duct i s gi ven by ( ξ ∪ η ) γ = X α · β = γ , ℓ ( α )+ ℓ ( β )= ℓ ( γ ) ξ α ∧ η β . By g lobalization o f Theore m IV, one obtains the following result. Theorem V . L et G b e a pr op er ´ e tale gr oup oid. Denote by B 0 := { g ∈ G 1 | s ( g ) = t ( g ) } its sp ac e of lo ops, and by N → B 0 the normal bund le to T B 0 in T | B 0 G 1 . Then the Ho chschild c ohomolo gy ring of the c onvolution algebr a i s g iven by H • ( A ⋊ G , A ⋊ G ) = Γ ∞  Λ •− ℓ T B 0 ⊗ Λ ℓ N  G , wher e ℓ is the lo c al ly c onstant function on B 0 the value of which at g ∈ B 0 c oincides with the c o dimension of the germ of B 0 at g within G 1 . The cup-pr o duct is given by the formula ξ ∪ η = Z m pr ∗ 1 ξ ∧ pr ∗ 2 η , for “mu ltive ctorﬁelds” ξ , η ∈ Γ ∞  Λ •− ℓ T B 0 ⊗ Λ ℓ N  G . In the formula ab ove, t he maps m, pr 1 , pr 2 : S → B 0 ar e the multiplic ation and the pr oje ction onto the ﬁrst and se c ond c omp onent, wh er e S := { ( g 1 , g 2 ) ∈ B 0 × B 0 | s ( g 1 ) = t ( g 2 ) } . Final ly, the inte gr al over m simply me ans sum m ation over the discr ete ﬁb er of m . Remark 2.1. W e remar k that one can also compute the Gerstenhab er bra ck et on H • ( A ⋊ G , A ⋊ G ) by tracing down the quasi-isomo r phisms constr ucted in Theorem I-I II. Tho ugh, there is no na tural Gerstenhab er brack e t deﬁned on the co mplex H • , the brac ket is well deﬁned on the subc omplex of lo ca l cochains whic h take v alues in co mpactly suppo rted functions. And this sub complex is quas i-isomorphic to the whole complex by T eleman’s lo ca lization as is explained in Section 3. Therefor e, the Gerstenha ber brack et is well deﬁned o n the Ho chsc hild coho mology . Using the presheaf H • and the loca l computation in [ HaT a ], one can g e neralize the compu- tation o f the Ger stenhab er bracket from [ H aT a ] to gener a l or bifolds. Let us now consider the deformed case. The strategy in computing the Hochschild cohomolog y of the defor med algebra A ~ ⋊ G is basically the same as in the unde- formed alg ebra. W e deﬁne the defor med ana lo gue of the complex o f pre sheav es H • G in the obvious way and denote it by H • G , ~ . The asso ciated ˇ Cech complex is denoted by ˇ H • , • U , ~ . With this, the defor med versions of Theor ems I– I I I a r e str aightforw ard to prov e : the maps in these theor ems generalize trivially to the sheaf A ~ . Using t he ~ -adic ﬁltratio ns on the complexes, Theor ems I–I I I imply that in the z ero’th order approximation these maps are quasi- is omorphisms. By an easy spectral sequence argument, cf. Section 4 one then shows this m ust b e quasi-iso morphisms in general. 10 M.J. PFLAUM, H. B. POSTHUMA, X. T ANG, AND H.-H. TSENG Again, this enables us to lo calize the computation of the Ho chschild co homology rings. F or a g lobal quotient orbifold, we have the following r esult, prov ed in Section 4.4. Theorem VI . L et M b e a smo oth s ymple ct ic manifold, and Γ a ﬁnite gr oup acting on M pr eserving the symple ctic structu r e. Then the Ho chschild c ohomolo gy ring H •  A (( ~ )) cpt ( M ) ⋊ Γ , A (( ~ )) cpt ⋊ Γ  is given as fol lows. A s a ve ctor sp ac e, one has H •  A (( ~ )) cpt ( M ) ⋊ Γ , A (( ~ )) cpt ⋊ Γ  ∼ = M ( γ ) ⊆ Γ H •− ℓ ( γ ) Z ( γ )  M γ , C (( ~ ))  , wher e Z ( γ ) is the c entr alizer of γ in Γ , and ( γ ) st ands for the c onju gacy class o f γ inside Γ . F or elements α =  ξ γ  γ ∈ Γ , β =  β γ  γ ∈ Γ ∈ M γ ∈ Γ  H •− l ( γ )  M γ , C (( ~ ))   Γ their cup pr o duct i s gi ven by α ∪ β = X γ 1 γ 2 = γ , ℓ ( γ 1 )+ ℓ ( γ 2 )= ℓ ( γ 1 γ 2 ) ι ∗ γ 1 α γ 1 ∧ ι ∗ γ 2 α γ 2 . Given this result, one migh t hope for a quas i-isomorphism H • G , ~ → Ω •− ℓ ˜ X to exist, which implements the isomorphism of the theorem ab ov e. The situation how ever is more co mplicated than that, a nd this is where the defor med case nota bly diﬀers from the undefor med case . First of all, it turns out o ne has to consider a sub-complex o f presheaves H • G , lo c , ~ ⊂ H • G , ~ , of co chains that a re lo ca l in a se ns e expla ined in the b eginning of Section 4. Second, instea d of one quasi- isomorphism, ther e is a chain H • G , lo c , ~ ֒ → C • , • ˜ X ← ֓ Ω •− ℓ ˜ X , where the intermediate double co mplex o f shea ves C • , • ˜ X is a twisted v er sion o f the F edosov–W einstein–Xu resolution used in [ Do ]. With this, w e ﬁnally obtain the following r esult: Theorem VI I . L et G b e a pr op er ´ etale gr oup oid with an invariant symple ctic struc- tur e, mo deling a symple ctic orbifo ld X . F or any invariant deformation quant ization A ~ of G , we have a natur al i somorphism H •  A (( ~ )) ⋊ G , A (( ~ )) ⋊ G  ∼ = H •− ℓ  ˜ X , C (( ~ ))  . With t his isomorphism, the cup pr o du ct is given by α ∪ β = Z m ℓ pr ∗ 1 α ∧ pr ∗ 2 β , for α, β ∈ H •− ℓ  ˜ X , C (( ~ ))  and wher e m ℓ is the r estriction of m , cf. The or em V, to those c onne cte d c omp onent s of S that satisfy ℓ ( g 1 g 2 ) = ℓ ( g 1 ) + ℓ ( g 2 ) , ( g 1 , g 2 ) ∈ S . Mor e over, this cup pr o duct and s ymple ct ic vo lume form t o gether deﬁne a F r ob enius algebr a structur e on H •− ℓ ( ˜ X , C (( ~ ))) . On the other hand, on H •− ℓ ( ˜ X , C (( t ))), there is the famous Chen-Ruan o rbifold pro duct [ ChR u04 ]. In Section 5, we study the c o nnection b etw een the cup pro duct ORBIF OLD C UP PRODUCTS 11 deﬁned in Theo rem VI a nd the Chen-Rua n orbifold pr o duct. W e intro duce a de Rham mo del for some pa rticular S 1 -equiv aria n t Chen-Ruan orbifold cohomology and relate this de Rha m mo del to the ab ove computation of Ho chsc hild cohomolo gy of A ~ ⋊ G . Given an arbitr ary almo s t complex orbifold X , we intro duce a tr iv ial S 1 -action on X , but a non trivial S 1 -action on the bundle T X → X by ro tating each ﬁb er. This S 1 -action is compatible with a ll the o rbifold str uc tur es on X and the inertia orbifold ˜ X . Therefore, we have the S 1 -equiv aria n t Chen-Ruan o rbifold cohomolo gy ( H • C R ( X )(( t )) , ⋆ t ) a s intro duced in Section 5.1 with ⋆ t the equiv ariant Chen-Ruan orbifold pro duct. The de Rham mo del ( H T • ( X )(( t )) , ∧ t ) for the ab ov e S 1 -equiv aria n t Chen-Ruan orbifold cohomo logy is deﬁned as a v ec tor space equal to H • ( ˜ X )(( t ))[ ℓ ] with the pro duct deﬁned by putting ( ξ ∧ t η ) γ := X γ = γ 1 γ 2 ι ∗ γ  ι γ 1 ∗ ( ξ γ 1 ) ∧ ι γ 2 ∗ ( η γ 2 )  , ξ , η ∈ H • ( ˜ X )(( t )) , where ι γ i is the embedding o f X γ i int o X . The following theorem is prov ed in Section 5 . Theorem VI I I . The two algebr as ( H • C R ( X )(( t )) , ⋆ t ) and ( H T • ( X )(( t )) , ∧ t ) ar e isomorphi c. T o connect ( H T • ( X )(( t )) , ∧ t ) to the ab ove Ho chsc hild cohomo logy ring, we de- ﬁne a decr easing ﬁltra tion F ∗ on H T • ( X )(( t )) by F ∗ = { α ∈ H • ( X γ )(( t )) | deg( α ) − ℓ ( γ ) ≥ ∗ } . W e prov e in Sectio n 5 the following r esult and thus ﬁnish our article. Theorem IX The gr ade d algebr a g r ( H T • ( X )(( t ))) asso ciate d to ( H T • ( X )(( t )) , ∧ t ) with r esp e ct to the ﬁlt r ation F ∗ is isomorphic to t he Ho chschild c ohomolo gy algebr a ( H • ( A ~ ⋊ G, A ~ ⋊ G ) , ∪ ) by identifying t with ~ . 3. Cup product on the Hochschild cohomology of the convolution algebra 3.1. Lo calization m etho ds. W e s tart with the pro of of Theor em I b y using a lo calization metho d going bac k to T eleman [ Te ]. Recall that the o rbifold X = G 0 / G represented by a proper ´ etale Lie gro up oid G car ries in a natural way a sheaf C ∞ X of smoo th functions. More precise ly , for every open U ⊂ X the alg ebra C ∞ ( U ) coincides naturally with the a lg ebra  C ∞ ( π − 1 ( U ))  G of smo oth functions on G 0 inv aria n t under the a ction o f G . Clea rly , C k ( A ⋊ G , A ⋊ G ) is a mo dule ov er C ∞ ( X ), and H k is a mo dule preshe a f ov er the C ∞ X for every k ∈ N . Since C ∞ X is a ﬁne sheaf, this implies in par ticular that H k has to b e a ﬁne presheaf. Next recall f rom [ NePfPoT a , Sec. 3, Step I] that there is a canonical isomo r- phism ˆ : C k ( A ⋊ G , A ⋊ G ) → Hom( A ⋊ G ˆ ⊗ k , C ∞ ( G 1 )) = C k red ( A ⋊ G , C ∞ ( G 1 )) , F 7→ ˆ F . (3.1) 12 M.J. PFLAUM, H. B. POSTHUMA, X. T ANG, AND H.-H. TSENG Hereby , the map ˆ F : A ⋊ G ˆ ⊗ k → C ∞ ( G 1 ) is uniquely determined by the requirement that for e very compa ct K ⊂ G 1 and all a 1 , . . . , a k ∈ A ⋊ G the rela tion ˆ F ( a 1 ⊗ . . . ⊗ a k ) | K = F ( ϕ K δ u ⊗ a 1 ⊗ . . . ⊗ a k ⊗ ϕ K δ u ) holds true, where ϕ K : G 0 → [0 , 1] is a smo oth function with compact supp or t such that ϕ K ( x ) = 1 for all x in a neig h b o rho o d of s ( K ) ∪ t ( K ), and wher e δ u : G 1 → R is the lo cally constant function whic h coincides with 1 on G 0 and which v anishes elsewhere. Now let us ﬁx a smo o th function  : R → [0 , 1 ] which has suppo r t in ( −∞ , 3 4 ] and which s atisﬁes  ( r ) = 1 for r ≤ 1 2 . F o r ε > 0 we de no te by  ε the resc aled function r 7→  ( s ε ). Next c ho ose a G -inv ar iant complete r iemannian metric on G 0 , and denote by d the corr esp onding geo desic distance on G 0 (where we put d ( x, y ) = ∞ , if x and y are not in the same connected comp onent of G 0 ). T he n d 2 is smo oth o n the set of pairs o f points of G 0 having ﬁnite distance. Put for ev ery k ∈ N ∪ {− 1 } , i = 1 , · · · , k + 1 and ε > 0: Ψ k,i,ε ( g 0 , g 1 , · · · , g k ) = i − 1 Y j =0  ε  d 2 ( s ( g j ) , t ( g j +1 ))  , where g j ∈ G 1 and g k +1 := g 0 . Moreov er, put Ψ k,ε := Ψ k,k +1 ,ε . Using the ab ove identiﬁcation (3.1) we then deﬁne for F ∈ C k := C k ( A ⋊ G, A ⋊ G ) a Ho chsc hild co chain Ψ k,ε F as follows: Ψ k,ε F ( a 1 ⊗ · · · ⊗ a k ) ( g 0 ) := F  Ψ k,ε ( g − 1 0 , − , · · · , − ) · ( a 1 ⊗ · · · ⊗ a k )  ( g 0 ) , for g 0 ∈ G 1 and a 1 , · · · , a k ∈ C ∞ cpt ( G 1 ) . One immediately c he cks that Ψ • ,ε forms a c hain map on the Ho chschild cochain complex. L ikewise, one deﬁnes a c hain map Ψ • ,ε acting on the shea f of co chain complexes H • . In [ NePfPoT a , Sec. 3, Step 2] it has bee n sho wn that there exist homotopy oper ators H k,ε : C k → C k − 1 such that ( β H k,ε + H k +1 ,ε β ) F = F − Ψ k,ε F (3.2) for all F ∈ C k . By a s imilar arg ument lik e in [ N ePfPoT a ] one sho ws that this algebraic homotopy holds a lso for F ∈ H k ( X ). By completeness of the metric d , the co chain Ψ • ,ε F is an element of C k red ( A ⋊ G, A ⋊ G ) for F ∈ C k or F ∈ H k ( X ). Hence Ψ • ,ε is a quasi-inv ers e to the canonical embedding C • red ( A ⋊ G , A ⋊ G ) ֒ → C • ( A ⋊ G , A ⋊ G ) resp. to ι : C • red ( A ⋊ G , A ⋊ G ) → H • ( X ). This prov es Theorem I. Next, we study the proper ties of the ˇ Cech double complex ˇ H • , • U asso ciated to an op en c overing U of X and prove Theo rem II. W e already have shown ab ov e that eac h pr esheaf H k is ﬁne. Denote by ˆ H k the sheaf associa ted to the presheaf H k . Then the ˇ Cech co ho mology o f H • coincides with the ˇ Cech cohomolo gy o f ˆ H • , a nd the latter is given b y the global sections of the cohomology sheaf o f ˆ H • (see for example [ Sp , Sec. 6.8]). T o prov e the last part of Theorem II cho ose a lo cally ﬁnite o pen covering U o f X such that e a ch elemen t U ∈ U is r elatively compact and let ( ϕ U ) U ∈U be a sub ordinate pa rtition o f unity b y smo o th functions on X . Then the ˇ Cech double co mplex ˇ C • U ( H • ) c ollapses at the E 1 term, hence its cohomolog y can b e computed by the cohomolog y of ˇ Z • , 0 U ( H • ). By a ssumption on U there exists for every U ∈ U a ε U > 0 such that for every H U ∈ H p ( U ) the co chain ϕ U Ψ p,ε U H U ∈ H p ( U ) can be extended b y zero to an element of H p ( X ) which w e ORBIF OLD C UP PRODUCTS 13 also denote by ϕ U Ψ p,ε U H U . Then it is eas ily chec ked that the restr iction map H p ( X ) → ˇ Z p, 0 U ( H • ) ⊂ Y U ∈U H p ( U ) , H 7→ ( H | U ) U ∈U is a quasi- isomorphism with qua si-inv er s e g iven b y ˇ Z p, 0 U ( H • ) ∋ ( H U ) U ∈U 7→ X U ∈U ϕ U Ψ p,ε U H U ∈ H p ( X ) . (3.3) This ﬁnishes the pr o of of Theorem I I. 3.2. The gl obal quotien t case. In this par t, w e provide a complete pro of of Theorem IV. Let Γ b e a ﬁnite group acting o n a smooth or ient able manifold M . This deﬁnes a transformatio n group oid G := (Γ ⋉ M ⇒ M ). In this case, the group oid a lg ebra of G is equal to the cr ossed pro duct a lgebra C ∞ cpt ( M ) ⋊ Γ . In [ NePfPoT a , Thm. 3], w e prov ed that as a v ector space the Ho chsc hild co- homology of the alg ebra C ∞ cpt ( M ) ⋊ Γ is equal to H • ( C ∞ cpt ( M ) ⋊ Γ , C ∞ cpt ( M ) ⋊ Γ) =  M γ ∈ Γ Γ ∞ (Λ •− ℓ ( γ ) T M γ ⊗ Λ ℓ ( γ ) N γ )  Γ , where ℓ ( γ ) is the co dimension of M γ in M , and N γ is the normal bun dle to M γ in M . The ma in goal of this section is to compute the cup pr o duct betw een multi- vector ﬁelds on the iner tia orbifold, that means b etw een elemen ts o f the right hand side of the preceding equation, f rom the cup pro duct on the Ho chsc hild cohomol- ogy of the left hand side of that equation. W e hereb y restrict our considerations to the par ticula r ca se, where M ca r ries a Γ-in v a riant riemannian metric s uch that M is g eo desically convex. This condition is in particular satisﬁed for a linear Γ- representation space carrying a Γ-inv ar iant scaler pro duct. Theorem IV can there- fore be immediately reduced to the ca se considered in the following by Theorems II a nd I I I and the slice theorem. In the ﬁrst part of our co ns truction, we o utline how to determine the Ho chsc hild cohomolog y as a v ecto r space. T o this end w e construct two co chain maps L and T be t ween the Ho chsch ild co chain complex and the spa c e of sections o f multi- vector ﬁelds on the inertia o r bifold. These t wo co chain maps ar e actually q uasi- isomorphisms. The map L has already b een co nstructed in [ NePfPoT a ], the map T in [ H aT a ]. In the second par t of our construc tio n, we will use the co chain maps T and L to co mpute the cup pr o duct. 3.2.1. The c o chain map L . F ollowing [ NePfPoT a , Theorem 3.1] we construct L : C • ( C ∞ cpt ( M ) ⋊ Γ , C ∞ cpt ( M ) ⋊ Γ) − →  M γ ∈ Γ Γ ∞ (Λ •− ℓ ( γ ) T M γ ⊗ Λ ℓ ( γ ) N γ )  Γ . The map L is the co mpo sition of three co chain maps L 1 , L 2 and L 3 deﬁned in the following. T o deﬁne the ﬁrst map L 1 recall th at Γ ac ts on F ∈ C k ( C ∞ cpt ( M ) , C ∞ cpt ( M ) ⋊ Γ) by γ F :=  C ∞ cpt ( M ) ˆ ⊗ k ∋ f 1 ⊗ · · · ⊗ f k 7→ δ γ · F  γ − 1 ( f 1 ) ⊗ · · · ⊗ γ − 1 ( f k )  · δ γ − 1  . Given f ∈ C ∞ cpt ( M ) and γ ∈ Γ w e hereby (and in the following) use the no tation f δ γ for the function in C ∞ cpt ( M ) ⋊ Γ which maps ( σ , p ) ∈ Γ × M to f ( γ p ), if σ = γ , 14 M.J. PFLAUM, H. B. POSTHUMA, X. T ANG, AND H.-H. TSENG and to 0 else. Now we put L 1 : C k ( C ∞ cpt ( M ) ⋊ Γ , C ∞ cpt ( M ) ⋊ Γ) − → C k ( C ∞ cpt ( M ) , C ∞ cpt ( M ) ⋊ Γ) Γ , F 7→ L 1 F :=  C ∞ cpt ( M ) ˆ ⊗ k ∋ f 1 ⊗ · · · ⊗ f k 7→ F ( f 1 δ e ⊗ · · · ⊗ f k δ e )  . Next we e xplain how the map L 2 is co nstructed. It has the following form: L 2 : C • ( C ∞ cpt ( M ) , C ∞ cpt ( M ) ⋊ Γ ) Γ − →  M γ ∈ Γ Γ ∞ (Λ • T | M γ M )  Γ , where T | M γ M is the restriction o f the vector bundle T M to M γ , and where the diﬀerential on the complex L γ ∈ Γ Γ ∞ (Λ • T | M γ M ) is given b y the ∧ -pro duct with a no wher e v a nishing v ector ﬁeld κ on M which we deﬁne later. Actually , we will deﬁne a Γ-equiv ar iant chain map L 2 slight mo re genera l than wha t is stated ab ov e, namely a map L 2 : C • ( C ∞ cpt ( M ) , C ∞ cpt ( M ) ⋊ Γ) − → M γ ∈ Γ Γ ∞ (Λ • T | M γ M ) . As a C ∞ cpt ( M )- C ∞ cpt ( M ) bimodule, C ∞ cpt ( M ) ⋊ Γ has a natural splitting in to a direct sum of submo dules L γ ∈ Γ C ∞ cpt ( M ) γ . According ly , the H o chsc hild co chain complex C •  C ∞ cpt ( M ) , C ∞ cpt ( M ) ⋊ Γ  naturally splits a s a direct sum M γ ∈ Γ C •  C ∞ cpt ( M ) , C ∞ cpt ( M γ )  . Therefore, to deﬁne the ma p L 2 , it is enoug h to consider each single map L γ 2 : C •  C ∞ cpt ( M ) , C ∞ cpt ( M γ )  − → Γ ∞ (Λ • T | M γ M ) . In the following we use ideas from the pap er [ Co ] to construct L γ 2 . T o this end let pr 2 : M × M → M b e the pro jection onto the s econd facto r of M × M , and ξ the vector ﬁeld o n M × M whic h maps ( x 1 , x 2 ) to exp − 1 x 2 ( x 1 ). By our a ssumptions on the riemannian metric on M the vector ﬁeld ξ is w ell-deﬁned and Γ-in v a riant. According to [ Co , Lemma 44], the complex K • =  Γ ∞ cpt (pr ∗ 2 (Λ • T ∗ M )) , ξ x  de- ﬁnes a pro jectiv e res olution of C ∞ cpt ( M ). Essentially , it is a Koszul r esolution fo r C ∞ cpt ( M ). F ollowing App endix A.4, we use the reso lution K • to de ter mine the Ho ch schild cohomology H •  C ∞ cpt ( M ) , C ∞ cpt ( M γ )  as the cohomology of the cochain complex Ho m C ∞ cpt ( M 2 )  K • , C ∞ cpt ( M γ )  . By [ Co ] the following chain map is a quasi- isomorphism b etw een the r esolution K • and the Bar r esolution Bar • ( C ∞ cpt ( M )  : Φ : K k → Bar k ( C ∞ cpt ( M )) = C ∞ cpt ( M k +2 ) , ω 7→  M k +2 ∋ ( a, b, x 1 , · · · , x k ) 7→ h ξ ( x 1 , b ) ∧ · · · ∧ ξ ( x k , b ) , ω ( a, b ) i  . Hence the dual of the chain map Ψ deﬁnes a quasi-isomo rphism Φ ∗ : Hom C ∞ cpt ( M 2 ) ( C ∞ cpt ( M k +2 ) , C ∞ cpt ) → Hom C ∞ cpt ( M 2 ) ( K k , C ∞ cpt ( M )) . Now consider the em be dding ∆ γ : M → M × M given by ∆ γ ( x ) = ( γ ( x ) , x ). According to [ N ePfPoT a , Sec. 3 , Step 4 ], the map η : Γ ∞ (Λ k T M ) → Hom C ∞ cpt ( M 2 ) ( K k , C ∞ cpt ( M )) , τ 7→ η ( τ ) , ORBIF OLD C UP PRODUCTS 15 deﬁned by η ( τ )( ω ) = h ∆ ∗ γ ω , τ i for ω ∈ Γ ∞ cpt pr ∗ 2 (Λ • T ∗ M )) is an is omorphism. So ﬁ- nally w e can deﬁne for F ∈ C k ( C ∞ cpt ( M ) , C ∞ cpt ( M ) γ ) an elemen t L γ 2 ( F ) ∈ Γ ∞ (Λ k T M ) by L γ 2 ( F ) = η − 1 Φ ∗ Ψ k,ε ( F ) , where w e hav e used the cut-oﬀ cochain map Ψ k,ε deﬁned ab ov e. Th us w e obtain an iso morphism of complex e s L γ 2 :  C k ( C ∞ cpt ( M ) , C ∞ cpt ( M )) , b  → (Λ • T M , κ γ ∧ ) , where κ γ is the re s triction o f the vector ﬁeld ξ on M × M to the γ -dia gonal ∆ γ . In the case, wher e M is a (ﬁnite dimensional) v ector space V with a linear Γ- action, w e can write do wn L γ 2 explicitly . Cho ose coo rdinates x i , i = 1 , . . . , dim V on V . Then the vector ﬁeld ξ on V × V ca n b e written as ξ ( x 1 , x 2 ) = X i ( x 1 − x 2 ) i ∂ ∂ x i 2 . (3.4) Moreov er, L γ 2 ( F ) is g iven as follows: L γ 2 ( F ) ( x ) = X i 1 , ··· , i k F  Ψ k,ε ( − , x 1 , · · · , x k ) · · h ξ ( x 1 , x ) ∧ · · · ∧ ξ ( x k , x ) , pr ∗ 2 ( dx i 1 ∧ · · · ∧ dx i k ) i  ( x ) ∂ ∂ x i 1 ∧ · · · ∧ ∂ ∂ x i k = X i 1 , ··· i k  X σ ∈ S k ( − 1) σ F  ( x σ (1) − x ) i 1 · · · ( x σ ( k ) − x ) i k   ( x ) ∂ ∂ x i 1 ∧ · · · ∧ ∂ ∂ x i k , (3.5) where x ∈ V , and where we have identiﬁed F w ith a b ounded linea r map from C ∞ cpt ( V k ) to C ∞ ( V ). T o deﬁne L 3 we construct for ea ch γ ∈ Γ a localiza tion map to the ﬁxed point sub m anifold M γ . Recall that w e ha ve chosen a Γ-inv a riant complete riemannian metric on M , and consider the no rmal bundle N γ to the embedding ι γ : M γ ֒ → M . The riemannian metr ic allows us to r egard N γ as a s ubbundle of the restricted tangent bundle T | M γ M . No w w e denote b y pr γ the or tho gonal pro jectio n from Λ • T | M γ M to Λ •− ℓ ( γ ) T M γ ⊗ Λ ℓ ( γ ) N γ . The chain map L 3 :  M γ ∈ Γ Γ ∞ (Λ • T M ) , κ ∧ −  − → M γ ∈ Γ  Γ ∞ (Λ •− ℓ ( γ ) T M γ ⊗ Λ ℓ ( γ ) N γ ) , 0  . is then cons tructed as the sum of the maps L γ 3 deﬁned b y L γ 3 ( X ) = pr γ ( X | M γ ) for X ∈ Γ ∞ (Λ • T M ) . In [ NePfPo T a , Sec. 3] we pr ov ed that L = L 3 ◦ L 2 ◦ L 1 is a quas i-isomorphism of co chain complex e s. 3.2.2. The chain map T . Under the assumption that M is a linear Γ-re presentation space V we construct in this section a quasi-inv erse T :  M γ ∈ Γ Γ ∞ (Λ •− ℓ ( γ ) T M γ ⊗ Λ ℓ ( γ ) N γ )  Γ − → C • ( C ∞ cpt ( M ) ⋊ Γ , C ∞ cpt ( M ) ⋊ Γ) . 16 M.J. PFLAUM, H. B. POSTHUMA, X. T ANG, AND H.-H. TSENG to the abov e co chain ma p L . T o this end w e ﬁrst recall the construction of the normal twiste d c o cycle Ω γ ∈ C l ( γ ) ( C ∞ cpt ( V ) , C ∞ cpt ( V ) γ ) from [ HaT a ]. Since Γ acts linearly on V , V γ is a linear subspace of U and has a normal s pace V ⊥ . Let x i , i = 1 , · · · , n − ℓ ( γ ), b e co ordinates on V γ , and y j , j = 1 , · · · , ℓ ( γ ), coordinates on V ⊥ . W e write ˜ y = γ y , and for ev ery σ ∈ S ℓ ( γ ) we int ro duce the fo llowing vectors in V ⊥ : z 0 = ( y 1 , · · · , y ℓ ( γ ) ) , z 1 = ( y 1 , · · · , ˜ y σ (1) , · · · , y ℓ ( γ ) ) , z 2 = ( y 1 , · · · , ˜ y σ (1) , · · · , ˜ y σ (2) , · · · , y ℓ ( γ ) ) , · · · z ℓ ( γ ) − 1 = ( ˜ y 1 , · · · , y σ ( ℓ ( γ )) , · · · , ˜ y ℓ ( γ ) ) , z ℓ ( γ ) = ( ˜ y 1 , · · · , ˜ y ℓ ( γ ) ) . Then we deﬁne a co chain Ω γ ∈ C ℓ ( γ ) ( C ∞ cpt ( V ) , C ∞ cpt ( V ) γ ) a s follows: Ω γ ( f 1 , · · · , f ℓ ( γ ) )( x, y ) := 1 ℓ ( γ )! · · X σ ∈ S ℓ ( γ ) ( f 1 ( x, z 0 ) − f 1 ( x, z 1 ))( f 2 ( x, z 1 ) − f 2 ( x, z 2 )) · · · ( f ℓ ( γ ) ( x, z n − 1 ) − f ℓ ( γ ) ( x, z n )) ( y 1 − ˜ y 1 ) · . . . · ( y ℓ ( γ ) − ˜ y ℓ ( γ ) ) , where f 1 , · · · , f ℓ ( γ ) ∈ C ∞ cpt ( V ), x ∈ V γ and y ∈ V ⊥ . It is straightf orward to c heck that Ω γ is a co c y cle in C ℓ ( γ ) ( C ∞ cpt ( V ) , C ∞ cpt ( V ) γ ) indeed. Now deﬁne the co chain map T 1 :  M γ ∈ Γ Γ ∞ (Λ •− ℓ ( γ ) T V γ ⊗ Λ ℓ ( γ ) N γ )  Γ − → C • ( C ∞ cpt ( V ) , C ∞ cpt ( V ) ⋊ Γ) Γ as the s um of maps T γ 1 : Γ ∞ (Λ k − ℓ ( γ ) T V γ ⊗ Λ ℓ ( γ ) N γ ) − → C k ( C ∞ cpt ( V ) , C ∞ cpt ( V ) γ ) , deﬁned b y T γ 1 ( X ⊗ Y γ ) = Y γ ( y 1 , · · · , y ℓ ( γ ) ) X ♯ Ω γ , where X ∈ Γ ∞ (Λ k − ℓ ( γ ) T V γ ), Y γ ∈ Γ ∞ (Λ ℓ ( γ ) N γ ), and where X ♯ Ω γ ( f 1 , · · · , f k ) is equal to X ( f 1 , · · · , f k − ℓ ( γ ) ) Ω γ ( f k − ℓ ( γ )+1 , · · · , f ℓ ( γ ) ) . Observe hereby that X to act on f 1 , · · · , f k − ℓ ( γ ) , we need to use a Γ-inv ar iant connection, i.e. the Levi-Civita connection of the in v ar iant metric, on the normal bundle of V γ in V to lift X to a vector ﬁeld o n V . The map T :  M γ ∈ Γ Γ ∞ (Λ •− ℓ ( γ ) T V γ ⊗ Λ ℓ ( γ ) N γ )  Γ − → C • ( C ∞ cpt ( V ) ⋊ Γ , C ∞ cpt ( V ) ⋊ Γ) is now written a s the compos ition of T 1 and T 2 , where T 2 is the standard co chain map fro m the Eilenberg -Zilb er theore m: T 2 : C • ( C ∞ cpt ( V ) , C ∞ cpt ( V ) ⋊ Γ) Γ − → C • ( C ∞ cpt ( V ) ⋊ Γ , C ∞ cpt ( V ) ⋊ Γ) . More prec is ely , for F ∈ C k ( C ∞ cpt ( V ) , C ∞ cpt ( V ) ⋊ Γ) one has T 2 ( F )( f 1 δ γ 1 , · · · , f k δ γ k ) = F ( f 1 , γ 1 ( f 2 ) , · · · , γ 1 · · · γ k − 1 ( f k )) δ γ 1 ··· γ k . The following r esult then holds for the c ompo sition T = T 2 ◦ T 1 . Its pro of is per formed by a stra ig ht forward check (cf. [ Ha T a , Sec. 2 ] for some more details). ORBIF OLD C UP PRODUCTS 17 Theorem 3.1. L et V b e a ﬁnite dimensional r e al line ar Γ -r epr esentation sp ac e. Then the c o chain maps L and T deﬁne d ab ove satisfy L ◦ T = id . In p articular, T is a quasi-inverse t o L . By the ab ove consider ations one concludes that there is an iso morphism of vector spaces b etw een the Ho chsc hild cohomolog y o f C ∞ cpt ( M ) ⋊ Γ a nd the spa ce of smo o th sections of alter nating multi-vector ﬁelds o n the cor resp onding inertia or bifold. 3.2.3. The cup pr o duct. In this part, we use the a bove constructed maps to co mpute the cup pro duct on the Hochschild coho mology of the algebra C ∞ cpt ( M ) ⋊ Γ. By proving the fo llowing pro po sition, we will complete pro of of Theorem IV. Prop ositio n 3. 2. F or every smo oth Γ - manifold M the cup pr o duct on the Ho chschild c ohomolo gy H • ( C ∞ cpt ( M ) ⋊ Γ , C ∞ cpt ( M ) ⋊ Γ) ∼ =  L γ ∈ Γ Γ ∞ (Λ •− ℓ ( γ ) T M γ ⊗ Λ ℓ ( γ ) N γ )  Γ is given for two c o chains ξ = ( ξ α ) α ∈ Γ , η = ( η β ) β ∈ Γ ∈  M γ ∈ Γ Γ ∞ (Λ •− ℓ ( γ ) T M γ ⊗ Λ ℓ ( γ ) N γ )  Γ as t he c o chain ξ ∪ η with c omp onents ( ξ ∪ η ) γ = X αβ = γ ,ℓ ( α )+ ℓ ( β )= ℓ ( γ ) ξ α ∧ η γ . Pr o of. It suﬃces to prov e the claim under the assumption that M is a linear Γ- representation space V . Then w e hav e the abov e deﬁned quasi-inv erse T to the co chain ma p L at our disp osa l. T o c o mpute ξ ∪ η w e thus have to determine the m ultivector ﬁeld L ( T ( ξ ) ∪ T ( η )). Since L is the comp ositio n o f L 1 , L 2 , and L 3 , w e compute L 1 ( T ( ξ ) ∪ T ( η )) ﬁr s t. Reca ll t hat the co chain L 1 ( T ( ξ ) ∪ T ( η )) ∈ C p + q ( C ∞ cpt ( V ) , C ∞ cpt ( V ) γ ) is deﬁned by L 1 ( T ( ξ ) ∪ T ( η ))( f 1 , · · · , f p + q ) = X αβ = γ T α 1 ( ξ α )( f 1 , · · · , f p ) α ( T β 1 ( η β )( f p +1 , · · · , f p + q )) , (3.6) where f 1 , · · · , f p + q ∈ C ∞ cpt ( V ), Recall also that the co chain map L 2 : C k ( C ∞ cpt ( V ) , C ∞ cpt ( V ) ⋊ Γ) → M γ ∈ Γ Γ ∞  Λ k T V γ V  essentially is the an ti-symmetrization o f the linear ter ms of a co chain. Hence L 2 ( L 1 ( T ( ξ ) ∪ T ( η ))) is equal to X αβ = γ L α 2 ( T α 1 ( ξ α )) ∧ α ( L β 2 ( T β 2 ( η β ))) . T o compute L 2 ( L 1 ( T ( ξ ) ∪ T ( η ))), it thus suﬃces to determine L α 2 ( T α 1 ( ξ α )) ∧ α ( L β 2 ( T β 2 ( η β ))) , which deﬁnes a ( p + q )-multiv ector ﬁeld Z supp orted in a neig h b o rho o d o f V α ∩ V β in V . By Equation (3 .5), one observes that when the restrictions of the norma l bundles N α and N β to V α ∩ V β hav e a nontrivial in tersection, for instance a lo ng a coo rdinate x 0 , then in Equatio n (3.6), the deriv ative ∂ ∂ x 0 shows up in b oth L α 2 ( T α 1 ( ξ α )) and α ( L β 2 ( T β x ( η β ))). Ther efore their wedge pro duct then has to v anish. T his argumen t shows that the nontrivial contribution of the cup pro duct ξ ∪ η comes from t hose 18 M.J. PFLAUM, H. B. POSTHUMA, X. T ANG, AND H.-H. TSENG comp onents, where N α and N β do not hav e a no n trivial intersection. By the following L e mma (3.3) this implies that a t a p oint x ∈ V α ∩ V β with N α x ∩ N β x = { 0 } one has T x V α + T x V β = T x V and ther efore V αβ = V α ∩ V β . This la st condition by Lemma 5.6 is equiv alent to ℓ ( α ) + ℓ ( β ) = ℓ ( γ ). When V α ∩ V β = V αβ and N α ∩ N β = { 0 } , one co mputes L 3 ( Z ) using the deﬁnition of L 3 and obta ins L ( T ( ξ ) ◦ T ( η )) = X αβ = γ ℓ ( α )+ ℓ ( β )= ℓ ( β ) L 3  L α 2 ( T α 1 ( ξ α )) ∧ α ( L β 2 ( T β 2 ( η β )))  = X αβ = γ ℓ ( α )+ ℓ ( β )= ℓ ( β ) ξ α ∪ α ( η β ) . Note t hat on V α ∩ V β = V αβ one has α ( η β ) = η β . This ﬁnishes t he pro o f o f the claim.  T o end this section we ﬁna lly show a lemma which already has b een used in the pro of o f the preceding result. Lemma 3. 3. L et α, β b e two line ar automorphisms on t he r e al ve ct or sp ac e V . L et h− , −i b e a sc alar pr o du ct pr eserve d by α and β and let V α , V β b e the c orr esp onding ﬁxe d p oint s u bsp ac es. If V α + V β = V , then V α ∩ V β = V αβ . Pr o of. O b viously , V α ∩ V β ⊂ V αβ . It is enough to show that if v ∈ V αβ , then v ∈ V α ∩ V β . Since v ∈ V αβ , one has αβ ( v ) = v , hence β ( v ) = α − 1 ( v ). Deﬁne w = β ( v ) − v = α − 1 ( v ) − v . W e prove that w is or thogonal to b oth V α and V β . F o r every u ∈ V α one ha s h w, u i = h α − 1 ( v ) − v , u i = h α − 1 ( v ) , u i − h v , u i = h v , α ( u ) i − h v , u i = h v , u i − h v, u i = 0 , where in the ﬁrst equality of the second line we have used the fact that α preserves the metric h− , −i , and in the second equality of the second line we have used that u is α -inv ar iant. Therefor e one co ncludes that w is orthog onal to V α . Likewise one shows tha t w is orthogona l to V β . Therefor e , w is orthogona l to V α + V β = V , hence w has to b e 0. This implies tha t v is in v ariant under bo th α a nd β .  4. Cup product on the Hochschild cohomology of the def ormed convolution algebra In this section w e compute the Hochschild cohomo logy together with the cup pro duct of a formal deformation of the conv olution a lg ebra of a prop er ´ etale group o id G . F or this we a ssume that the o r bifold X is symplectic o r in o ther words that G 0 carries a G -in v ariant s ymplectic form ω , i.e., satisfying s ∗ ω = t ∗ ω . W e let A ~ be a G -inv ar iant for ma l deformation quan tizatio n o f A = C ∞ G 0 , whe r e the def ormation parameter is denoted by ~ . This means that A ~ is a G -sheaf over G 0 and the asso- ciated cross ed pro duct A ~ ⋊ G is a formal deforma tio n of the conv olutio n alge br a, cf. [ T a ]. As a formal deforma tion the algebra A ~ ⋊ G is ﬁltered by p ow e r s of ~ , i.e., F k ( A ~ ⋊ G ) := ~ k ( A ~ ⋊ G ) and we hav e F k  A ~ ⋊ G  F k − 1  A ~ ⋊ G  ∼ = A ⋊ G . (4.1) ORBIF OLD C UP PRODUCTS 19 As usual, the Ho chschild co chain complex is deﬁned by C •  A ~ ⋊ G , A ~ ⋊ G  := Hom C [[ t ]]  ( A ~ ⋊ G ) ˆ ⊗• , A ~ ⋊ G  , with diﬀere ntial β deﬁned with resp ect to the defo r med con volution algebra . The justiﬁcation for t his deﬁnition co mes from P rop osition A.8, whic h als o shows that the cup-pro duct, de ﬁned by (A.6), extends this complex to a diﬀeren tial graded algebra (DGA). The ~ -adic ﬁltratio n of A ~ ⋊ G a bove induces a c o mplete and ex- haustive ﬁltration of the Hochschild complex. Since the pro duct in A ~ ⋊ G is a formal deformatio n of the conv olution pro duct, cf. eq uation (4.1), the asso ciated sp ectral sequence has E 0 -term just the undeformed Ho chsc hild complex of the co n- volution a lgebra. This has the following use ful co nsequence that we will use several times in the course of the argument: Supp ose that A ~ 1 and A ~ 2 are formal deformations of the algebras A 1 and A 2 , and f : C •  A ~ 1 , A ~ 1  → C •  A ~ 2 , A ~ 2  is a morphis m of ﬁltered complexes. Then f is a quasi-is omorphism, if it induces an isomorphism a t level E 1 . The pro of of this sta tement is a direct application of the Eilenberg– Mo ore s pectr al se q uence compa rison theor em, cf. [ We , Thm. 5.5.11 ]. Let us apply this to the following situa tion: consider the following subspace of the space of Ho chsc hild co chains on A ~ ⋊ G : C k loc  A ~ ⋊ G , A ~ ⋊ G  := n Ψ ∈ C k ( A ~ ⋊ G , A ~ ⋊ G ) | π s  supp Ψ( a 1 , . . . , a k )  ⊂ k \ i =1 π s (supp a i ) o . Here s upp( a ) denotes the supp ort of a function. These are the lo c al co chains with resp ect to the underlying orbifold X . Notice that b ecause of the conv o lution nature of the alg ebra A ~ ⋊ G , which inv olves the action of G , it is unreaso na ble to req uire lo cality with respect to G 0 or G 1 . The imp ortant p oint now is: Prop ositio n 4.1. The c omplex of lo c al Ho chschild c o chains C • loc  A ~ ⋊ G , A ~ ⋊ G  is a su b c omplex of C •  A ~ ⋊ G , A ~ ⋊ G  , and the c anonic al incl usion map is a quasi- isomorphi sm pr eserving c up-pr o ducts. Pr o of. The o rbifold X can b e identiﬁed with the quotient space G 0 / G 1 . The de- formed conv olution pro duct o n A ~ ⋊ G inv olves t he lo cal pro duct o n the G -sheaf A ~ on G 0 and the group oid action, and the pro duct in turn deﬁnes the Ho chsc hild complex as well as the cup-pro duct. With this, it is eas y to check that the lo cality condition o n X is compatible w ith b oth the diﬀerential and the pro duct. T o sho w th at the cano nical inclusion is a qua si-isomor phism, ﬁrst observe that the map clea r ly r esp ects the ~ -a dic ﬁltratio n. It follows from Theo rem IV that for the undeformed co nv olution algebra the local Ho chsc hild co chain complex computes the same co ho mology , since the vector ﬁelds clearly satisfy the locality condition. Therefore the inclus ion map is a q uasi-isomor phism a t the E 0 -level, and by the ab ov e, a qua s i-isomorphism in gener al.  Remark 4.2. In the following we will o ften co nsider the ring extension A (( ~ )) := A ~ ˆ ⊗ C [[ ~ ]] C (( ~ ) , 20 M.J. PFLAUM, H. B. POSTHUMA, X. T ANG, AND H.-H. TSENG where C (( ~ )) denotes the ﬁeld o f forma l Laurent ser ies in ~ , and will then r e- gard A (( ~ )) as an algebra ov er the gro und ﬁeld C (( ~ )) . By standard results from Ho ch schild (co)homology theory one k nows that H •  A (( ~ )) ⋊ G , A (( ~ )) ⋊ G  = H •  A ~ ⋊ G , A ~ ⋊ G  ˆ ⊗ C [[ ~ ]] C (( ~ ) . (4.2) In the remainder o f this article we will tacitly make use of this fact. 4.1. Reduction to the ˇ Cec h comple x. As in the undeformed c ase, the idea is to use a ˇ Cech complex to co mpute the cohomo logy . F or U ⊂ X , introduce H k G , ~ ( U ) := Hom C [[ ~ ]]  Γ ∞ cpt ( U, ˜ A ~ fc ) ˆ ⊗ k , ˜ A ~ fc ( U )  . This is clea rly a deformation of the sheaf H • G . The sheaf H k G , lo c , ~ is similarly deﬁned. As in the undeformed ca se, we now hav e an o b vious map I ~ loc : C • loc  A ~ ⋊ G , A ~ ⋊ G  → H • G , lo c , ~ ( X ) . Prop ositio n 4 . 3. The ma p I ~ loc is a quasi-isomorp hism of DGA ’s. Pr o of. By the Eilenberg- Mo ore sp ectr a l sequence, this follows fro m Theorem I.  4.2. Twisted co cycles on the formal W eyl alg ebra. Our aim is to r educe the co mputation of Hochschild to sheaf cohomolog y . The pr esent section ca n b e viewed as a stalkwise c omputation. Let V = R 2 n equipp e d with the standard symplectic for m ω , and supp ose that Γ ⊂ S p ( V , ω ) is a ﬁnite gr oup acting o n V by linear symplectic transformatio ns. The action of an elemen t γ ∈ Γ induces a decomp osition V = V γ ⊕ V ⊥ int o s y mplectic subspaces. Put ℓ ( γ ) := dim( V ⊥ ) = dim( V ) − dim( V γ ) . Let W 2 n be the fo r mal W eyl a lgebra, i.e., W 2 n = C [[ y 1 , . . . , y n ]][[ ~ ]] equipp ed with the Moy al pr o duct f ⋆ g = ∞ X k =0 X 1 ≤ i 1 ,...,i k ≤ n 1 ≤ j 1 ,...,j k ≤ n Π i 1 j 1 · · · Π i k j k ~ k k ! ∂ k f ∂ y 1 . . . ∂ y k ∂ k g ∂ y 1 . . . ∂ y k , where Π := ω − 1 is the Poisson tensor a sso ciated to ω . With this pr o duct, the formal W eyl alg ebra W 2 n is a unital a lg ebra ov e r C [[ ~ ]]. It is a for mal deformation of the commutativ e a lgebra C [[ y 1 , . . . , y 2 n ]]. With an auto mo rphism γ ∈ Γ, we can consider the W eyl alg e bra bimodule W 2 n,γ which equals W 2 n except for the fact that the rig ht action of W 2 n is twisted by γ . With this we hav e: Prop ositio n 4 . 4 (cf. [ Pi ]) . The twiste d Ho chschi ld c ohomolo gy is giv en by H k ( W 2 n , W 2 n,γ ) = ( C [[ ~ ]] , for k = ℓ ( γ ) , 0 , else. Ther e exist s a gener ator Ψ γ in t he r e duc e d Ho chschild c omplex satisfy ing Ψ γ | Λ V ∗ =  Π ⊥ γ  ℓ ( γ ) / 2 . In fact, Ψ is, up to a c ob oundary, uniquely determine d by this p r op erty. ORBIF OLD C UP PRODUCTS 21 Pr o of. The ﬁr st part of the Pro po sition is essen tially well-known, cf. [ AlF aLaSo , Al ]. It is conv enie ntly proved by the Kosz ul reso lution of the W eyl algebra 0 ← − W 2 n ⋆ ← − W 2 n ⊗ W op 2 n ∂ ← − K 1 ∂ ← − K 2 ∂ ← − . . . where K p := W 2 n ⊗ Λ p V ∗ ⊗ W op 2 n , and wher e the diﬀerential ∂ : K p → K p − 1 is deﬁned by ∂ ( a 1 ⊗ a 2 ⊗ dy i 1 ∧ . . . ∧ dy i p ) := p X j =1 ( − 1) j  ( y i j ⋆ a 1 ) ⊗ a 2 − a 1 ⊗ ( a 2 ⋆ y i j )  dy i 1 ∧ . . . ∧ c dy i j ∧ . . . ∧ dy i p , with r esp ect to a Darb oux basis of V , i.e., ω ( y i , y i + n ) = 1 a nd zero other wise. T o compute the H o chsc hild cohomolog y , w e take Hom W 2 n ( − , W 2 n,γ ) to o btain the complex K p γ := Λ p V ⊗ W 2 n , (4.3) with diﬀer en tial d γ : K p → K p +1 given by d γ  a ⊗ y i 1 ∧ . . . ∧ y i p +1  = 2 n X j =1 ( − 1) j ( y j ⋆ a − a ⋆ γ y j ) y j ∧ y i 1 ∧ . . . ∧ y i p . The cohomology of this complex can easily b e computed using the sp ectra l sequence of the ~ -adic ﬁltration. In degree z e r o o ne ﬁnds the ordinary , i.e. c o mm utative Koszul co mplex and there fo re we ﬁnd E p,q 1 = Λ p + q V ⊥ . The diﬀer ent ial d 1 : E p,q 1 → E p +1 ,q 1 is given by the Poisson cohomo logy diﬀerential, which ha s triv ia l coho mology ex cept in maximal degre e and therefore E p,q 2 = ( Λ ℓ ( γ ) V ⊥ , for p + q = ℓ ( γ ), 0 , else . The spectral sequence degenerates at this point and the ﬁrs t part of the Prop osi- tion is prov ed. The second part is a s in [ Pi ]: the Koszul complex is naturally a sub c omplex o f the r e duced Bar complex ( K • , ∂ ) ⊂ ( B red • , b ), where B red k = W 2 n ⊗ ( W 2 n / C [[ ~ ]]) ⊗ k ⊗ W 2 n , and the em b edding is induced by the natur al inclusion V ∗ ֒ → W 2 n as degree one homogeneous p olynomia ls. This leads to a natural pr o jection R V : ( C • red ( W 2 n , W 2 n ) , β γ ) → ( K • , d γ ) given by restricting c o chains to Λ V ∗ . It is ea sily chec ked tha t  Π ⊥ γ  ℓ ( γ ) / 2 deﬁnes a co cycle of degr ee ℓ ( γ ) in the complex ( K • , d γ ), a nd the statement follows.  Let µ γ : K • γ → C [[ ~ ]] b e t he morphism deﬁned by µ γ ( a ⊗ v i 1 ∧ . . . ∧ v i k ) := a (0)  ω ⊥ γ  ℓ ( γ / 2) ( v i 1 , . . . , v i k ) . Clearly , this map is only non trivia l in de g ree ℓ ( γ ) and ma ps t he diﬀerential d γ on K • γ to zer o. Deﬁne P γ : C • red ( W , W γ ) → C [[ ~ ]] 22 M.J. PFLAUM, H. B. POSTHUMA, X. T ANG, AND H.-H. TSENG to b e P γ := µ ◦ R V . On the other hand, c ho osing Ψ γ as in the prop os ition deﬁnes a morphism I Ψ γ : C [[ ~ ]][ ℓ ( γ )] → C • red ( W , W γ ). The arg umen t in the pro o f of the prop osition then shows: Corollary 4. 5. The inclus ion I Ψ γ and the pr oje ction P γ ar e quasi-isomorphisms satisfying P γ ◦ I Ψ γ = id . Finally , we come to the full crossed pro duct W 2 n ⋊ Γ. As usual, we hav e H • ( W 2 n ⋊ Γ , W 2 n ⋊ Γ) ∼ =   M γ ∈ Γ H • ( W 2 n , W 2 n,γ )   Γ , where the Γ- action is as explained in Section 3.2. Corollary 4.6 . The gener ators Ψ γ , γ ∈ Γ satisfy γ 1 · Ψ γ 2 − Ψ γ 1 γ 2 γ − 1 1 = e x act . Ther efor e they deﬁne a c anonic al i somorphism H • ( W 2 n ⋊ Γ , W 2 n ⋊ Γ) = M h γ i⊂ Γ C [[ ~ ]][ ℓ ( h γ i )] . Pr o of. Res tr icting to Λ • V ∗ , we ﬁnd  γ 1 · Ψ γ 2 − Ψ γ 1 γ 2 γ − 1 1  | Λ • V ∗ = γ 1 ·  Π ⊥ γ 2  ℓ ( γ 2 ) / 2 −  Π ⊥ γ 1 γ 2 γ − 1 1  ℓ ( γ 1 γ 2 γ − 1 1 ) / 2 = 0 in K ℓ ( γ 2 ) γ 1 γ 2 γ − 1 1 . By the ar g ument ab ov e, the co cycles γ 1 · Ψ γ 2 and Ψ γ 1 γ 2 γ − 1 1 therefore diﬀer by a cob ounda ry , a nd the r esult follows.  4.3. The F edosov–W einstein–Xu res olution o v er B 0 . Let B 0 be the s pace of lo ops in G : B 0 := { g ∈ G 1 | s ( g ) = t ( g ) } . W e recall from [ NePfPoT a ] that the canonical inclusion ι : B 0 ֒ → G 1 gives B 0 a symplectic for m by pull-back. Denote by A ~ B 0 := ι − 1 A ~ the pull-back o f the defo r - mation quantization of G ; this is no t quite a deformation quantization of ( B 0 , ι ∗ ω ), bec ause it in volv e s the g erm of B 0 inside G 1 . Recall from [ P fPoT a ] that the shea f A ~ B 0 has a canonical lo c a l auto mo rphism, denoted θ , co ming from the fa ct that B 0 has a cyclic structure [ Cr ]. This enables us to deﬁne t he f ollowing co mplex of sheav es on B 0 : C 0 β θ − → C 1 β θ − → . . . where C k := Hom C [[ ~ ]]   A ~ B 0  ˆ ⊗ k , A ~ B 0 ,θ  is the sheaf o f Ho chsc hild k -co chains, and β θ : C k → C k +1 is the twisted Ho chsc hild cob oundary . W e will no w write do wn a resolution of this complex of sheav es. F or this, let W G be the bundle of W eyl algebra s ov er G 0 . This is just the bundle F Sp , G 0 × Sp W asso ciated to the symplectic fra me bundle ov er G 0 with typical ﬁb er W . This construction shows tha t W G carries a canonica l action of the gr oup oid G . In the following we will denote its sheaf of sections by the s a me symbo l W G . ORBIF OLD C UP PRODUCTS 23 Prop ositio n 4 . 7 (cf. [ Fe ]) . On B 0 ther e exists a r esolution 0 − → A ~ B 0 − → Ω 0 B 0 ⊗ W G D − → Ω 1 B 0 ⊗ W G D − → . . . , (4.4) wher e D is a F e dosov c onne ction on W G . Pr o of. Let us ﬁrst construct the F edosov diﬀer e n tial D . The sequence of ma ps T B 0 ˜ ω 0 − → T ∗ B 0 → T ∗ G → W G determines a n element A 0 ∈ Ω 1 ( B 0 , W G ). O n eas ily veriﬁes that [ A 0 , A 0 ] = ω 0 ∈ Ω 2 B 0 ⊗ C [[ ~ ]] , which is central in W G . Therefore, δ = ad( A 0 ) deﬁnes a diﬀerential on Ω • ( B 0 , W G ), and we hav e H k  Ω • B 0 ⊗ W G , ad( A 0 )  = ( Ω 0 B 0 ⊗ W N , for k = 0 , 0 , for k 6 = 0 , bec ause ad( A 0 ) is simply a Koszul diﬀerential in the tangential dir ections along B 0 . Cho ose a symplectic connectio n ∇ B 0 on B 0 and a symplectic co nnection ∇ N on the normal bundle N → B 0 . W e will co nsider connectio ns on W G of the form D = δ + ∇ + a d( A • ) , where ∇ = ( ∇ B 0 ⊗ 1 + 1 ⊗ ∇ N ) and A ∈ Ω 1 ( B 0 , W G ) ha s deg( A ) ≥ 2. Notice that deg( δ ) = 0 and deg ( ∇ ) = 1. Such a connection has W eyl cur v a ture given by Ω = ω 0 + ˜ R + ∇ A + 1 2 [ A, A ] . By the usual F edosov metho d, w e can ﬁnd A such that Ω is central, i.e ., Ω ∈ Ω 2 B 0 ⊗ C [[ ~ ]]. Since the diﬀerent ial D is a defo rmation of the K oszul complex ab ove, acyclicity of the se q uence (4.4) follows. This shows the e x istence of the r esolution 0 − → Ω 0 B 0 ⊗ W N − → Ω 0 B 0 ⊗ W G D − → Ω 1 B 0 ⊗ W G D − → . . . , so it remains to co nstruct an isomorphism Ω 0 B 0 ⊗ W N ∼ = ι − 1 A ~ G . This is done by minimal coupling in [ Fe ].  Consider now the following do uble complex ( C • , • , β θ , D ) of sheaves: . . . . . . . . . Ω 0 B 0 ⊗ C 1 β θ O O D / / Ω 1 B 0 ⊗ C 1 β θ O O D / / Ω 2 B 0 ⊗ C 1 β θ O O D / / . . . Ω 0 B 0 ⊗ C 0 β θ O O D / / Ω 1 B 0 ⊗ C 0 β θ O O D / / Ω 2 B 0 ⊗ C 0 β θ O O D / / . . . where C • is the sheaf of formal p ow er series of poly diﬀerential oper ators on W G . This complex is a twisted version o f the F edosov–W einstein– Xu resolution consider ed in [ Do ]. Prop ositio n 4 . 8. The total c ohomolo gy of t his c omplex is g iven by H k (T ot ( C • , • ) , β θ + D ) = H k ( C • , β θ ) . 24 M.J. PFLAUM, H. B. POSTHUMA, X. T ANG, AND H.-H. TSENG Pr o of. Co nsider the sp ectral sequence by ﬁltering the total complex by rows. This yields E p,q 1 = H p v ( C • ,q ) = ( Ω 0 B 0 ⊗ C q , for p = 0 , 0 , for p 6 = 0 , since the r e solution (4 .4) is acyclic. A t the sec ond stag e E 0 ,q 2 = H q ( C • , β θ ) ⇒ H q (T ot ( C • , • )) . Since the sp ectra l sequence colla ps es at this p oint, this proves the statement.  Of course the cohomology sheaf of the vertical complex is computed b y Propo - sition 4.4. If we repla ce the vertical complex b y its reduced coun ter part, we get a natural pro jection R : Γ  B 0 , C ℓ  → Γ  B 0 , Ω ℓ G  , where ℓ is the lo ca lly cons ta n t function on B 0 deﬁned in Section 2. The transversal Poisson s tructure induced by ω ∈ Ω 2 ( G 0 ) induce s a s e c tion  Π ⊥ θ  ℓ/ 2 ∈ Γ  B 0 , Ω ℓ G  , and w e ch o ose Ψ ∈ Γ  B 0 , C ℓ  which gener ates the ﬁberwis e v ertical c o homology and pro jects o n to the section ab ove. Prop ositio n 4 . 9. The ma p I Ψ extends to a morphi sm I Ψ :  Ω • B 0 ⊗ C [[ ~ ]][ ℓ ] , d  → (T ot • ( C • , • ) , D + β θ ) of c o chain c omplexes o f she aves by t he formula I Ψ ( α ) := α ⊗ Ψ . In fact, this is a quasi-isomorphism. Pr o of. Let us ﬁrst prove that I Ψ is a ma p of co chain complexes of sheaves. Consider the pro jection map P : Ω • B 0 ⊗ C • r ed → Ω • B 0 ⊗ C (( t )). It satisﬁes P ◦ I Ψ = id. W e therefore have to show that P ◦ D ◦ I Ψ = d. Since the statement of the pr op osition is a lo ca l statement we can write the F edos ov connection as D = d + ad( A ) , with A ∈ Ω 1 B 0 ⊗ W G = Ω 1 B 0 ⊗ C 0 . It then follows ea sily that D : Ω k B 0 ⊗ C • → Ω k +1 B 0 ⊗ C • is given by D = d + [ β A, ] , where [ , ] is the Gerstenha ber brack et o n the Hochschild cochain complex. W e now compute P DI Ψ ( α ) − dα = α ⊗ P ([ β A, Ψ]) = α ⊗ P ( β ([ A, Ψ ]) − [ A, β Ψ ]) = 0 , where we ha ve used that Ψ is a co cy cle, i.e., β Ψ = 0, and the fact that P maps cob oundaries to zero. This proves that I Ψ is a map o f co chain complexes of sheav es.  Using Prop os ition 4.8 we now ﬁnd: ORBIF OLD C UP PRODUCTS 25 Corollary 4.1 0. Ther e is a natur al i somorphism H • ( C • B 0 , β θ ) ∼ = H •− ℓ ( B 0 , C [[ ~ ]]) . Finally , notice that C • , • B 0 is in a natura l w ay a double complex of Λ( G )-sheaves on B 0 , b ecause the F edosov resolution of Prop osition 4.7 ca rries a natural G -a ction. Therefore, we can deﬁne the following sheaf on ˜ X : C • , • ˜ X ( ˜ U ) := C • , • B 0 ( ˜ π − 1 ( ˜ U )) Λ( G ) | ˜ U . Because G , and therefore a lso Λ( G ), is prop er, it follo ws fro m Coro llary 4.1 0 that its hyper c o homology is g iven by H • ( C • , • ˜ X , β θ ) ∼ = H •− ℓ  ˜ X , C [[ ~ ]]  . (4.5) 4.4. Lo cal computations. In this section we will p erform several explicit co mpu- tations in s ome op en or bifold charts. This suﬃce s to prove the result in the case of a g lobal quotient or bifold. The genera l case is tr e ated in the next sectio n. Let U ⊂ R 2 n be an open orbifold chart with a ﬁnite group Γ U acting b y linear symplectic tr a nsformations, so tha t we have U / Γ U ⊂ X . Prop ositio n 4 . 11. Ther e ex ist a n atur al quasi-isomorphism H • M ⋊ Γ , l oc , ~ ( M / Γ) → C • , • ] M / Γ  ] M / Γ  . Pr o of. W e ﬁr st use the natural map H • G , lo c , ~ ( U / Γ U ) → C • loc  A ~ ( U ) ⋊ Γ U , A ~ ( U ) ⋊ Γ U  of the “ deformed version” of Theorem I I Ib. As in the undefor med case, there is a quasi-isomo rphism L ~ 1 : C • loc  A ~ ( U ) ⋊ Γ U , A ~ ( U ) ⋊ Γ U  → C • loc  A ~ cpt ( U ) , A ~ cpt ( U ) ⋊ Γ U  Γ U , given by the same for m ula a s for L 1 . The right hand side is the space of Γ U - inv aria n ts of a complex which decomp oses int o M γ ∈ Γ U C • loc  A ~ ( U ) , A ~ ( U ) γ  . There is a natural morphism C • loc  A ~ ( U ) , A ~ ( U ) γ  → C • loc  ι − 1 A ~ ( U γ ) , ι − 1 A ~ ( U γ ) γ  , where ι − 1 A ~ ( U γ ) := Γ  U γ , ι − 1 A ~  is by deﬁnition the algebra given b y the jets along the em bedding ι : U γ ֒ → U . Indeed the loc a lity condition for co chains Ψ ∈ C k loc states in this case that the v a lue Ψ( f 1 , . . . , f k )( x ) can only dep end o n the germs of f 1 , . . . , f k at the p oints o f the γ -or bit o f x ∈ U . Restricted to U γ ⊂ U , s uch co chains ther efore pres erve the subalgebr a ι − 1 A ~ ( U γ ) ⊂ A ~ ( U ), and the r estriction map ab ov e is well-deﬁned. E ven strong er, it is a quasi-iso morphism beca use on E 0 - level of the spectral sequence assoc ia ted to the ﬁltration by pow ers o f ~ this map is simply given b y lo ca lization, which we already k now to b e a q uasi-isomor phism, cf. Se c tion 3.2. As r emarked ab ov e, lo cal co chains, in the sense deﬁned a bove, are truly lo ca l with resp ect to U γ , b ecause p oints in U γ by deﬁnition hav e a trivial γ -or bit. F rom this we se e that there is a canonica l isomorphism C • loc  ι − 1 A ~ cpt ( U γ ) , ι − 1 A ~ cpt ( U γ ) γ  ∼ = C • ( U γ ) , 26 M.J. PFLAUM, H. B. POSTHUMA, X. T ANG, AND H.-H. TSENG compatible with diﬀer en tials. T aking the sum ov er all γ ∈ Γ U , and taking Γ U - inv aria n ts, deﬁnes the map of the prop osition. As the a rgument shows, it is a quasi-isomo rphism.  Using the F edosov–W einstein–Xu resolutio n, this result suﬃces to compute the Ho ch schild cohomolog y for a globa l quotient symplectic or bifold: Corollary 4.12. F or a glob al quotient X = M / Γ of a ﬁnite gro up Γ acting on a symple ctic manifold M , ther e is a natur al isomorphism H • ( A (( ~ )) cpt ( M ) ⋊ Γ , A (( ~ )) cpt ⋊ Γ) ∼ = M ( γ ) ⊂ Γ H •− ℓ γ Z ( γ ) ( M γ , C (( ~ ))) . Pr o of. This follows from the isomor phism (4.5).  Next, we c o nsider the cup-pro duct. An eas y computation sho ws that the map L ~ 1 induces the following pr o duct on the complex C • loc  A ~ cpt ( U ) , A ~ cpt ( U ) ⋊ Γ U  Γ U : ( ψ • φ ) γ := X γ 1 γ 2 = γ ψ γ 1 ∪ tw φ γ 2 where the ma p ∪ tw : C k ( A ~ ( U ) , A ~ ( U ) γ 1 ) × C l ( A ~ ( U ) , A ~ ( U ) γ 2 ) → C k + l ( A ~ ( U ) , A ~ ( U ) γ 1 γ 2 ) is deﬁned as ( ψ γ 1 ∪ tw φ γ 2 ) ( a 1 , . . . , a k + l ) := ψ γ 1 ( a 1 , . . . , a k ) γ 1 φ γ 2 ( a k +1 , . . . , a k + l ) . Indeed one ea sily chec k s that β γ 1 γ 2 ( ψ γ 1 ∪ tw φ γ 2 ) = ( β γ 1 ψ γ 1 ) ∪ tw φ γ 2 + ( − 1 ) deg ( ψ ) ψ γ 1 ∪ tw ( β γ 2 φ γ 2 ) . Restricting to ˜ U = F γ U γ , this induces the following pro duct on the F edosov– W einstein–Xu reso lution C • , • ˜ U / Γ U ( ˜ U / Γ U ): (( α ⊗ ψ ) • ( β ⊗ φ )) γ := X γ 1 γ 2 = γ  ι ∗ γ ( α γ 1 ) ∧ ι ∗ γ ( β γ 2 )  ⊗  ι ∗ γ ( ψ γ 1 ) ∪ tw ( ι ∗ γ ( φ γ 2 )  , where an element of C • , • ˜ U / Γ U ( ˜ U / Γ U ) is written as α ⊗ ψ = P γ α γ ⊗ ψ γ , with α γ ∈ Ω • ( U γ ) and ψ γ is a lo cal s e ction the sheaf o f Ho chsc hild c o cycle on W G ov er U γ ⊂ U . W e therefore hav e: Prop ositio n 4.13. The map of Pr op osition 4.11 is c omp atible with pr o ducts that me ans deﬁnes a quasi-isomorphism of she aves of DGA’s on U / Γ U . F or a global quotient orbifold, this leads immediately to: Corollary 4 .14. Under the isomorp hism of Cor ol lary 4.12, the cup-pr o duct is given by α • β = X γ 1 γ 2 = γ ℓ ( γ 1 ) + ℓ ( γ 2 ) = ℓ ( γ 1 γ 2 ) ι ∗ γ α γ 1 ∧ ι ∗ γ 2 α γ 2 . Pr o of. The isomor phism of Coro llary 4.1 2 is induced b y the quasi-iso morphism I Ψ :  Ω • ˜ X ⊗ C [[ ~ ]] , d  →  C • , • ˜ X , D + β tw  ORBIF OLD C UP PRODUCTS 27 of Pr op osition 4.9. W e therefore ﬁnd I Ψ ( α ) • I Ψ ( β ) = X γ 1 γ 2 = γ  ι ∗ γ ( α γ 1 ) ∧ ι ∗ γ ( β γ 2 )  ⊗  ι ∗ γ (Ψ γ 1 ) ∪ tw ( ι ∗ γ (Ψ γ 2 )  . W e will now consider the second compone nt ι ∗ γ (Ψ γ 1 ) ∪ tw ( ι ∗ γ (Ψ γ 2 ) for a momen t. An ea sy calcula tio n shows that in the Koszul complex (4.3), the pr o duct ∪ tw : K p γ 1 × K q γ → K p + q γ 1 γ 2 is given by  a 1 ⊗ y I q  ∪ tw  a 2 ⊗ y I q  = ( a 1 γ 1 a 2 ) ⊗ y I p ∧ γ 1 y I q , where I p and I q are mult i-indices of length p r esp. q . Therefo re,  Π ⊥ γ 1  ℓ ( γ 1 ) / 2 ∪ tw  Π ⊥ γ 2  ℓ ( γ 2 ) / 2 = (  Π ⊥ γ 1 γ 2  ℓ ( γ 1 γ 2 ) / 2 , if ℓ ( γ 1 ) + ℓ ( γ 2 ) = ℓ ( γ 1 γ 2 ) , 0 , else. In the reduced Ho chschild co mplex, this gives Ψ γ 1 ∪ tw Ψ γ 2 = ( Ψ γ 1 γ 2 + exa ct , if ℓ ( γ 1 ) + ℓ ( γ 2 ) = ℓ ( γ 1 γ 2 ) , exact , else. T aking cohomolog y we ﬁnd the pr o duct as stated ab ov e.  4.5. The general case. Reca ll that we hav e spaces and morphisms a s in the f ol- lowing dia gram: B 0 / / ˜ π   G 0 π   ˜ X ψ / / X As in [ ChHu ], deﬁne the spa ce S 1 G := { ( g 1 , g 2 ) ∈ G 1 × G 1 | s ( g 1 ) = t ( g 1 ) = s ( g 2 ) = t ( g 2 ) } . It co mes equipped with three maps pr 1 , m, pr 2 : S 1 G → B 0 , where pr 1 ( g 1 , g 2 ) = g 1 , m ( g 1 , g 2 ) = g 1 g 2 and pr 2 ( g 1 , g 2 ) = g 2 . F o r diﬀerential forms α, β ∈ Ω • ( B 0 ), deﬁne the following pr o duct: α • β = Z m ℓ pr ∗ 1 α ∧ pr ∗ 2 β , (4.6) where m ℓ : S 1 G ,ℓ → B 0 is the restriction o f the multiplication map a nd S 1 G ,ℓ := { ( g 1 , g 2 ) ∈ S 1 G | ℓ ( g 1 ) + ℓ ( g 2 ) = ℓ ( g 1 g 2 ) } . Both S 1 G ,ℓ and B 0 carry a natural action of G by conjugating lo ops and the quotient B 0 / G = ˜ X . W e hav e Ω • ( ˜ X ) = Ω • ( B 0 ) G , and the product (4.6 ) deﬁnes an asso ciative gr aded pro duct on Ω • ( ˜ X ). T o gether with the de Rham diﬀer e ntial, it turns Ω • ( ˜ X ) into a diﬀerential gra ded algebra . Of course, Ω • ( ˜ X ) is the global sections of a sheaf on ˜ X , but it is impo r tant to notice that the pro duct (4.6) is not lo ca l. Ho w ever, if we consider ψ ∗ Ω • ˜ X , the push-for ward to X , we hav e Γ( X, ψ ∗ Ω • ˜ X ) = Ω • ( ˜ X ) and now the pro duct is lo c al, i.e., ( ψ ∗ Ω • ˜ X , d, • ) do es deﬁne a shea f of DGA’s on X . 28 M.J. PFLAUM, H. B. POSTHUMA, X. T ANG, AND H.-H. TSENG The same can b e do ne fo r the FWX-resolution on ˜ X . This time we in tro duce the pro duct ( α ⊗ ψ ) • ( β ⊗ φ ) := Z m ℓ ( pr ∗ 1 α ∧ pr ∗ 2 β ) ⊗ (pr ∗ 1 ψ ∪ tw pr ∗ 2 φ ) . (4.7) Again, b ecause of the integration ov er the ﬁb e r, this deﬁnes a lo cal pro duct on ψ ∗ C • , • ˜ X , so tha t the tota l sheaf complex is a sheaf of DGA’s. Lemma 4.15. The emb e dding of Pr op osition 4. 9 de ﬁnes a quasi-isomorp hism I Ψ : ψ ∗ Ω •− ℓ ˜ X ⊗ C [[ ~ ]] → ψ ∗ C • , • ˜ X c omp atible with pr o ducts up to h omotopy. Pr o of. By Cor ollary 4.6, if we choo se Ψ ∈ Γ  B 0 , C ℓ  to b e G -inv a riant, it descends to a morphism I Ψ : ψ ∗ Ω •− ℓ ˜ X ⊗ C [[ ~ ]] → ψ ∗ C • , • ˜ X By a ssumption, the gr oup oid G is prop er, so we hav e H  ˜ X , (Ω • ˜ X , d )  ∼ = H  B 0 , (Ω • B 0 , d )  G , and similarly for C • , • ˜ X . Ther efore the morphism I Ψ is a qua si-isomor phism be- cause it is a q ua si-isomor phism o n B 0 . The fact that it pres erves pro ducts up to a cob oundary , is a simple calcula tio n as in the pro of of Co rollar y 4.14.  Prop ositio n 4 . 16. Ther e is a quasi-isomorphism H • G , lo c , ~ → ψ ∗ C • , • ˜ X which maps the cup-pr o duct to t he pr o duct (4.7) Pr o of. F or a n y x ∈ X , choose a lo cal s lice to o btain a morphism H • G , lo c , ~ ( U x ) → C •  A ~ ( M x ) ⋊ G x , A ~ ( M x ) ⋊ G x  , as in Theo rem II Ib. By this very same Theo r em I I Ib, one knows that on E 0 of the sp ectral sequences asso cia ted to the ~ -ﬁltr ation, the above chain mo rphism is a quasi-isomo r phism, and therefo r e it is a quasi-is omorphism on the or iginal complexes. W e now co mpo se with the morphis m of Prop ositio n 4.11 to g et a ma p H • G , lo c , ~ → ψ ∗ C • , • ˜ M x / G x ( U x ) ∼ = ψ ∗ C • , • ˜ X ( U x ) . Because the sheav es ar e ﬁne, this in fact deﬁnes a g lobal quasi-isomo rphism o ver the orbifold X .  Finally , co mbining Lemma 4.15 w ith Prop ositio n 4.16, we ha ve arrived at the main co nc lus ion: Theorem 4.17. L et G b e a pr op er ´ etale gr oup oid with an invariant symple ctic structur e, mo deling a symple ctic orbifold X . F or any invaria nt deformatio n quan- tization A ~ of G , we have a natur al i somorphism H •  A (( ~ )) ⋊ G , A (( ~ )) ⋊ G  ∼ = H •− ℓ  ˜ X , C (( t ))  . With t his isomorphism, the cup pr o du ct is given by (4.6) . ORBIF OLD C UP PRODUCTS 29 Remark 4 .18. W e ex pla in the pro duct (4.6) using the o rbifold la ng uage. Let X be repr e sented by the g roup oid G such that X = G 0 / G , a nd ˜ X b e the corres po nding inertia orbifold repr esented by B 0 / G . Lo ca lly , a n op en chart of X is like U / Γ with Γ a ﬁnite g roup acting linearly on an op en subset U of R n . Accordingly ˜ X is lo cally represented b y  ` γ ∈ Γ U γ  / Γ ∼ = ` ( γ ) ⊂ Γ U γ / Z ( γ ), where ( γ ) is the c onjugacy clas s of γ in Γ. W e usually use X γ to sta nd for the comp onent of ˜ X containing U γ / Z ( γ ). With these no tations, the a bove S 1 G and S 1 G ,ℓ are lo ca lly repr esented as S 1 G = a γ 1 ,γ 2 U γ 1 ∩ U γ 2 , S 1 G ,ℓ = a γ 1 , γ 2 , ℓ ( γ 1 ) + ℓ ( γ 2 ) = ℓ ( γ 1 γ 2 ) U γ 1 ∩ U γ 2 . As w e are considering G -in v ariant diﬀerential for ms on B 0 , their pull-ba cks through pro jections pr 1 , pr 2 to S 1 G and S 1 G ,ℓ are inv ar iant under the following G - action on S 1 G and S 1 G ,ℓ , which is deﬁned as ( g 1 , g 2 ) g = ( g − 1 g 1 g , g − 1 g 2 g ) , ( g 1 , g 2 ) ∈ S 1 G ,ℓ , s ( g ) = t ( g 2 ) = t ( g 2 ) . Lo cally this actio n can b e written as a Γ -action on ` γ 1 ,γ 2 U γ 1 ∩ U γ 2 , ( x, γ 1 , γ 2 ) γ = ( γ − 1 ( x ) , γ − 1 γ 1 γ , γ − 1 γ 2 γ ) , γ 1 ( x ) = γ 2 ( x ) = x, γ ∈ Γ . The cor resp onding quo tien t space S 1 G / G is usua lly denoted by X 3 = { ( x, ( g 1 , g 2 , g 3 )) | g 1 , g 2 ∈ Stab( x ) , g 1 g 2 g 3 = id , x ∈ X } . (4.8) One can see that lo cally X 3 = X g 1 ∩ X g 2 . The pullbacks of G -inv ar ia nt diﬀerent ial forms on B 0 are diﬀerential forms o n X 3 . Therefore, the formula (4.6) can be int erpreted as follows. F or α 1 , α 2 ∈ Ω • ( ˜ X ), α 1 • α 2 | γ = X γ 1 , γ 2 , ℓ ( γ 1 ) + ℓ ( γ 2 ) = ℓ ( γ 1 γ 2 ) ι ∗ γ 1 ( α 1 | γ 1 ) ∧ ι ∗ γ 2 ( α 2 | γ 2 ) , where ι ∗ γ i is the embedding o f X 3 in X γ i , i = 1 , 2. 4.6. F rob enius algebras from Ho c hsc hi ld cohomology . The pro duct struc- ture of Theore m 4.17 is par t o f a natural gra ded F rob enius algebra asso cia ted to A (( ~ )) ⋊ G . Recall that a F ro b enius algebr a is a commutativ e unital alge bra equipp e d with an inv a riant trace. The co nstruction of this F r ob enius alge bra on the Ho chschild cohomolo gy us es one additional piece of data, namely the trace o n the algebra A ~ ⋊ G constr ucted in [ PfPoT a ]. Let A be a unital algebra o ver a ﬁeld k equipped with a trace tr : A → k . As we hav e seen, the cup-pro duct (A.6) gives the Ho chsc hild cohomology H • ( A, A ) the structure of a g r aded algebr a. The Ho chschild homology H H • ( A ) is a natura l mo dule over this algebra if we let a cochain ψ ∈ C k ( A, A ) act as ι ψ : C p ( A ) → C p − k ( A ) given by ι ψ ( a 0 ⊗ . . . ⊗ a p ) = ( − 1) deg ( ψ ) a 0 ψ ( a 1 , . . . , a k ) ⊗ a k +1 ⊗ . . . ⊗ a p . With this mo dule str ucture, the tra ce induces a pair ing h , i : H • ( A, A ) × H H • ( A ) → k which is giv en by h ψ , a 0 ⊗ . . . ⊗ a k i = tr ( ι ψ ( a 0 ⊗ . . . ⊗ a k )) . 30 M.J. PFLAUM, H. B. POSTHUMA, X. T ANG, AND H.-H. TSENG Let us assume, as in our ca se, that the Ho chschild cohomolog y a nd homology a re ﬁnite dimens io nal a nd that this pairing is p erfect. Prop ositio n 4.19. Under these assumptions, the ring structur e on H • ( A, A ) is p art o f a natur al gr ade d F r ob enius algebr a structur e. Pr o of. The pairing gives us a canonica l isomo rphism H • ( A, A ) ∼ = H • ( A ) ∗ . The trace deﬁnes a canonical element [tr] ∈ H 0 ( A ) ∗ which deﬁnes the unit. F urthermore, the unit 1 ∈ H 0 ( A ) ∼ = H 0 ( A, A ) ∗ deﬁnes an inv ariant trace.  In the case at hand, the deforma tio n of the co n volution a lgebra on a symplectic orbifold, the Ho chschild homology was computed in [ NePfPo T a ] to b e H •  A (( ~ )) ⋊ G  ∼ = H 2 n −• cpt  ˜ X , C (( ~ ))  . With the trac e of [ PfPo T a ], one checks that the pairing b etw een Ho chschild ho- mology a nd cohomo logy ab ove is nothing but Poincar´ e duality o n ˜ X . 5. Chen-R uan orbif old cohomology In this section, we study S 1 -equiv aria n t Chen-Ruan orbifo ld cohomolog ie s o n an almost complex orbifold. In a special case, w e apply the idea from [ ChHu ] to int ro duce a de Rham model (top olo gical Ho chsc hild c ohomology ) to compute this equiv ariant co homology . In the last subsectio n, we compare this de Rham model with the pre vious computation of Hochschild cohomology of the qua nt ized gro upo id algebra. The Ho chsc hild cohomolog y of the quantized group oid a lgebra is identiﬁed as a gr aded algebr a of the de Rham mo del with respect to so me ﬁltratio n. 5.1. S 1 -Equiv arian t Chen-R uan orbifold cohomo logy. In this subsection, we brieﬂy in tro duce the idea of S 1 -equiv aria n t Chen-Ruan orbifold cohomolo g y . Let X be an orbifold with an S 1 action. This means that there is a mor phism f : S 1 × X → X of orbifolds. One ca n think of a morphism b etw een tw o orbifolds as a c ollection o f morphisms be tw een charts and group homomorphisms b etw een lo cal gro ups such that the morphisms b etw een charts ar e equiv a riant with r esp e ct to lo cal group a ctions, and are co mpatible with ov er laps of charts. (See [ ChRu04 ] and [ AdLeRu ] for mor e details.) The ac tion is assumed to be a sso ciative, whic h is a somewhat delicate prop erty since the categ o ry of orbifo lds is not a categor y but a 2- c ate gory . This mea ns that the sta ndard asso ciativity diagr am for a group a c tio n on an orbifold is only required to be commutativ e up to 2-morphisms. Generalities on group a c tio ns on ca tegories can b e found in [ Ro ]. The S 1 -equiv aria n t coho mo logy is deﬁned via the standar d Bor el constructio n: H • S 1 ( X ) := H • ( X × S 1 E S 1 ) . W e identify H • S 1 (pt) = C [ t ]. With this, H • S 1 ( X ) is considere d a s a C [ t ]-mo dule. As usual, we cons ider the fra ction ﬁeld C (( t )) and put H • S 1 ( X )(( t )) := H • S 1 ( X ) ⊗ C [ t ] C (( t )) . In the following, we shall use the Carta n mo del for equiv a r iant co homology to represent cohomology cla s ses by equiv a riant diﬀerential forms. As b efore, ˜ X is the iner tia or bifold, and p : ˜ X → X the natural pro jection. It is easy to chec k that the S 1 -action lifts to ˜ X . Indeed, for any s ∈ S 1 the action mo rphism f s : s × X → X deﬁnes for ea ch x ∈ X a group homo morphism ORBIF OLD C UP PRODUCTS 31 ρ s : Sta b( x ) → Stab( f s ( x )). This induces the S 1 -action on ˜ X , who se p oints are pairs ( x, ( γ )) , x ∈ X , γ ∈ Stab( x ). Mo r e pre c isely , the actio n is given by S 1 × ˜ X → ˜ X , ( s, ( x, ( γ ))) 7→ ( f s ( x ) , ( ρ s ( γ ))) . As a C [ t ]-mo dule, the S 1 -Chen-Ruan or bifold cohomolo gy can be deﬁned exactly in the same fashion a s its non-equiv ar iant version [ ChRu04 ], that is H • S 1 ( ˜ X ) := H • ( ˜ X × S 1 E S 1 ) . There is a natura l inv olution I : ˜ X → ˜ X which maps a p oint ( x, ( γ )) to ( x, ( γ − 1 )). The or bifold Poincar´ e pairing h , i , which is deﬁned by h a, b i := Z ˜ X a ∧ I ∗ b, naturally extends t o a non-degenerate pairing on H • S 1 ( ˜ X ). The a dditio na l structures one deﬁnes o n Chen-Ruan or bifold cohomolo gy require a choice of an almost co mplex structure on the tangent bundle T X , which we now make. W e also assume that the S 1 -action on T X is compatible with this almost complex s tructure. W e will as sume an S 1 -action on the tangent bundle T X which commutes with the S 1 -action on the base X . It should be noted that we do n ot nece s sarily w or k with the ca nonical action o n T X induced fro m that on X . This will be impo r tant in what follows. Therefore the pull-back bundle p ∗ T X admits an S 1 -action co vering that on ˜ X . L e t X γ be a comp onent o f ˜ X . The bundle p ∗ T X | X γ splits into a direct sum of γ -eig enb undles. This allows one to deﬁne the age function, deno ted by ι ( γ ) (c.f. [ ChRu02 ]). This is a lo cally cons ta n t function on ˜ X . W e consider the shifted S 1 -equiv aria n t cohomolog y of ˜ X , H • S 1 ( ˜ X )(( t ))[ − 2 ι ( γ )] . Here t is ass igned degree 2. The S 1 -action on p ∗ T X | X γ restricts to a n S 1 -action on each eig en bundle. Now consider the tri-cyclic sector (4.8), i.e., the quotient S G / G . There are three ev alu- ation maps e i : X 3 → ˜ X , e i (( x, ( γ 1 , γ 2 , γ 3 )) = ( x, ( γ i )). The S 1 -action also lifts to tri-cyclic sec tor: S 1 × X 3 → X 3 , ( s, ( x, ( γ 1 , γ 2 , γ 3 ))) 7→ ( f s ( x ) , ( ρ s ( γ 1 ) , ρ s ( γ 2 ) , ρ s ( γ 3 ))) . The e v aluation maps a re clea rly S 1 -equiv aria n t. It follows from the a bove discussion that the obstruction bundle Θ ov er the tri-cyclic secto r X 3 is an S 1 -equiv aria n t orbifold bundle on ˜ X . Therefor e, we can deﬁne S 1 -equiv aria n t orbifold cup pr o duct ⋆ t by h α 1 ⋆ t α 2 , α 3 i = Z X 3 e ∗ 1 ( α 1 ) ∧ e ∗ 2 ( α 2 ) ∧ e ∗ 3 ( I ∗ ( α )) ∧ eu S 1 (Θ) , where eu S 1 (Θ) is the equiv ariant Euler class of the o bs truction bundle. Man y prop erties o f the Chen-Rua n or bifo ld cohomo logy algebra holds for the algebr a ( H • S 1 ( ˜ X )(( t ))[ − 2 ι ( γ )] , ⋆ t ) , with the s ame pro ofs. F or example, the a s so ciativity of ⋆ t is reduced to the ratio nal equiv alence b etw een tw o p oints in the mo duli space M 0 , 4 of genus zero stable curves with four marked p oints. (Note that M 0 , 4 ≃ CP 1 .) See [ ChRu04 ] for mor e details. 32 M.J. PFLAUM, H. B. POSTHUMA, X. T ANG, AND H.-H. TSENG 5.2. Equiv arian t de R ham mo del. In this s ubsection, we deﬁne an equiv aria nt de Rham mo del for a specia l ca se of the ab ov e in tro duced S 1 -equiv aria n t Chen- Ruan orbifold cohomology . W e introduce our deﬁnition of topolo gical Hochsc hild cohomolog y algebra with the following steps. Step I: W e start with an arbitrar y almost complex orbifold X lo cally like M / Γ, and in tro duce a trivial S 1 action on X a nd therefore also on ˜ X w hich is lo ca lly like ( ` γ ∈ Γ M γ ) / Γ. Acco rdingly , the S 1 -equiv aria n t cohomolog y of ˜ X is equal to H • ( ˜ X )(( t )). Step I I: W e in tro duce a “nontrivial” S 1 -action on the tangent bundle T X of X which commutes with the tr iv ial S 1 -action on X . Since T X is a n almost co mplex bundle, S 1 ident iﬁed with U (1) acts on T X as the center of the principal group GL(dim C ( T X ) , C ). Geometrically , this action is simply rotation b y an angle. W e remark that since S 1 is identiﬁed as the center of the principal gr oup, the abov e S 1 -action commutes with all the orbifold structure. And we ha ve made T X into an S 1 -equiv aria n t orbifold bundle on X , and the same is for p ∗ T X on ˜ X . Step I I I: W e consider the normal bundle N γ of the embedding of X γ int o X . Since the S 1 -action on T X commutes with the γ -actio n, N γ inherits an S 1 -action, and becomes an S 1 -equiv aria n t vector bundle on X γ . W e decompos e N γ int o a direct sum of S 1 -equiv aria n t line bundles ⊕ i N γ i , with resp ect to the eigen v a lue of γ -action, i.e. exp(2 π iθ i ) a nd 0 ≤ θ i < 1. Let t i be the eq uiv ariant Thom form for N γ i , and the equiv a riant Thom cla s s T γ of N γ be de ﬁned by T γ := Y i t i . F or the following, it is important to r emark that T γ is inv ertible in Ω • S 1 ( N γ ). Deﬁnition 5.1. Deﬁne the topo logical Ho chschild coho mology H T • ( X )(( t )) o f a n orbifold X to be M H • ( X γ )(( t ))[ − ℓ ( γ )] , where ℓ ( γ ) is, a s b efore, the co dimension of X γ ∈ X . On H T • ( X )(( t )), w e deﬁne a cup pr o duct ∧ t as follows. First of all, the cup pro duct is C (( t )) linea r. F or α i ∈ Ω •− ℓ ( γ i ) ( X γ i )(( t )), i = 1 , 2, α 1 ∧ α 2 is deﬁned by the following integral, h α 1 ∧ t α 2 , α 3 i = Z X γ 1 γ 2 ι ∗ ( α 1 ∧ T γ 1 ∧ α 2 ∧ T γ 2 ) ι ∗ ( T γ 1 γ 2 ) ∧ I ∗ ( α 3 ) , for a ny α 3 ∈ Ω •− ℓ ( γ 1 γ 2 ) ( X γ 1 γ 2 )(( t )), Remark 5.2. More e xplicitly , if ι ∗ is the pushforward of Ω ∗ ( ˜ X ) into Ω ∗ ( X ) we hav e that α 1 ∧ t α 2 = ι ∗ ( ι ∗ ( α 1 ) ∧ ι ∗ ( α 2 )). A mor e glo bal wa y to write the pro duct, in the style o f Section 4.5, is as follows: α 1 ∧ t α 2 = Z m pr ∗ 1 ( α 1 ∧ T ) ∧ pr ∗ 2 ( α 2 ∧ T ) m ∗ T , where, as before m : S → B 0 is the multiplication and R m means in tegration o ver the discrete ﬁb er . ORBIF OLD C UP PRODUCTS 33 W e remark that the a sso ciativity of ∧ t is an easy co rollary of th e asso ciativity of the w e dg e pro duct on diﬀerential forms of X . The following are a few simple observ atio ns o f H T • ( X )(( t )), which we sta te without pro ofs. (They a re coro llaries of Theor em 5.2.) (1) the pro duct ∧ t is C (( t ))-linear; (2) ( H T • ( X )(( t )) , ∧ t ) is a graded a lgebra. In summary , with the S 1 -action int ro duced in Step I and II, we can in tro duce t wo co homology algebra structures on Ω • ( ˜ X )(( t )). (1) S 1 -equiv aria n t Chen-Ruan orbifold cohomo logy algebra as in Section 5 .1; (2) top ologica l Ho chsc hild cohomo logy algebra as in Deﬁnition 5.1. In the rest of this subsection, we relate the top ologica l Ho c hschild co homol- ogy ( H T • ( ˜ X )(( t ))[ − l ] , ∧ t ) with the S 1 -equiv aria n t Chen-Ruan orbifold cohomo logy ( H • C R ( ˜ X )(( t ))[ − 2 ι ] , ⋆ t ). The key ingredien t connecting these tw o a lgebra structures is a certa in equiv ar iant Euler cla ss naturally as s o ciated to the orbifold. W e consider a stringy K-gr oup class [ JaKaKi ] s γ asso ciated to the norma l bundle N γ of an orbifold X , i.e., s γ := M i θ i N γ i . (5.1) The equiv a riant Euler class t γ of s γ is deﬁned to b e t γ := ι ∗ ( Y i t θ i i ) ∈ H 2 ι ( γ ) ( X γ )(( t )) , where t i is the S 1 -equiv aria n t Thom class of N γ i , ι ∗ is the pullback of the Thom form to X γ which is embedded as the zer o section. W e remark that t θ i i and t γ are well deﬁned in the S 1 -equiv aria n t coho mology H • ( X γ )(( t )) by using the T a ylor expansion o f t θ i for the θ -p ower. W e deﬁne the following is o morphism J γ : H • ( X γ )(( t ))[ − 2 ι ( γ )] → H •− l ( γ ) ( X γ )(( t )) of vector spaces , J γ ( α ) = α/t γ − 1 , for all α ∈ H •− 2 ι ( γ ) ( X γ )(( t )) . Remark 5.3. W e observe that t γ is in vertible in H • ( X γ )(( t )), b e cause it has a nonzero constant term. Accordingly , J γ is a linear iso morphism of the vector spaces. The colle c tion of all J γ deﬁnes an is omorphism J = ⊕ γ J γ on H • ( ˜ X ). The map J preser ves g rading. The deg ree of t γ − 1 is equa l to ℓ ( γ ) − 2 ι ( γ ). If α is an element in H • ( X γ )(( t ))[ − 2 ι ( γ )] = H •− 2 ι ( γ ) ( X γ )(( t )), J ( α ) is of degre e • − 2 ι ( γ ) − ( ℓ ( γ ) − 2 ι ( γ )) = • − ℓ ( γ ) . The following theorem is a gene r alization o f the result in [ ChHu ]. Theorem VI. The map J is an isomorphism of C (( t )) -a lgebr as fr om the algebr a ( H • C R ( X )(( t )) , ⋆ t ) t o ( H T • ( X )(( t )) , ∧ t ) . Pr o of. As w e hav e rema rked, J is an isomorphis m of vector spaces pre serving the degrees. It is suﬃcient to show that J is compatible with the a lg ebra str uctur es. 34 M.J. PFLAUM, H. B. POSTHUMA, X. T ANG, AND H.-H. TSENG F or α i ∈ H • ( X γ i )(( t ))[ − 2 ι ( γ i )], i = 1 , 2, and α 3 ∈ H • ( X γ 1 γ 2 )(( t ))[ − 2 ι ( γ 1 γ 2 )] we ha ve h J ( α 1 ) ∧ J ( α 2 ) , α 3 i = Z X γ 1 γ 2 ι ∗ ( α 1 t γ 1 − 1 ∧ T γ 1 ∧ α 2 t γ 2 − 1 ∧ T γ 2 ) ι ∗ ( T γ 1 γ 2 ) ∧ I ∗ ( α 3 ) = Z X γ 1 ,γ 2 ι ∗ ( α 1 t γ 1 − 1 ∧ T γ 1 ∧ α 2 t γ 2 − 1 ∧ T γ 2 ) ι ∗ ( T γ 1 γ 2 ) ∧ I ∗ ( α 3 ) ∧ ι ∗ ( T γ 1 γ 2 ) ι ∗ ( T γ 1 ,γ 2 ) = Z X γ 1 ,γ 2 ι ∗ ( α 1 ) ∧ ι ∗ ( α 2 ) ∧ ι ∗ ( I ∗ ( α 3 )) ∧ ι ∗ ( T γ 1 ) ∧ ι ∗ ( T γ 2 ) ι ∗ ( t γ 1 − 1 ) ∧ ι ∗ ( t γ 2 − 1 ) ∧ ι ∗ ( T γ 1 ,γ 2 ) , where X γ 1 ,γ 2 := X γ 1 T X γ 2 , and T γ 1 ,γ 2 is the equiv ariant Thom for m for the norma l bundle of X γ 1 ,γ 2 in X , and ι ∗ is the pullbac k of t he f orms to X γ 1 ,γ 2 . And w e can summarize the ab ove computation in the following equation, for γ 3 = ( γ 1 γ 2 ) − 1 , h J ( α 1 ) ∧ J ( α 2 ) , α 3 i = Z X γ 1 ,γ 2 ι ∗ ( α 1 ) ∧ ι ∗ ( α 2 ) ∧ ι ∗ ( I ∗ ( α 3 )) ∧ R γ 1 ,γ 2 ,γ 3 , (5.2) with R γ 1 ,γ 2 ,γ 3 = ι ∗ ( T γ 1 ) ∧ ι ∗ ( T γ 2 ) ι ∗ ( t γ 1 − 1 ) ∧ ι ∗ ( t γ 2 − 1 ) ∧ ι ∗ ( T γ 1 ,γ 2 ) W e no w apply the result of [ JaKaKi ] to better understand the term R γ 1 ,γ 2 ,γ 3 . By [ JaKaKi ][Thm. 1.2], for γ 1 γ 2 γ 3 = i d , when re s tricted to X γ 1 ,γ 2 := X γ 1 T X γ 2 , the obstruction bundle Θ γ 1 ,γ 2 as a str ing y K -gr oup class has a natural splitting Θ γ 1 ,γ 2 = T ( X γ 1 ,γ 2 ) ⊖ T X | X γ 1 ,γ 2 ⊕ s γ 1 | X γ 1 ,γ 2 ⊕ s γ 2 | X γ 1 ,γ 2 ⊕ s γ 3 | X γ 1 ,γ 2 , (5 .3) where we remind that s γ i is a n element in the s tringy K -gr oup [ JaKaKi ] as deﬁned in Eq. (5.1). W e remark that the ab ov e isomorphism for Θ γ 1 ,γ 2 again ho lds a s S 1 − equiv aria n t bundles b ecause the S 1 actions on the re spe c tiv e bundles are deﬁned by the a lmost complex structures a nd the equation (5.3) preser ves almost complex structur es. Now taking the equiv a r iant Euler classes of the bundles in Eq. (5.3) on X γ 1 ,γ 2 , we hav e that o n X γ 1 ,γ 2 eu S 1 (Θ γ 1 ,γ 2 ) = ι ∗ ( t γ 1 ) ∧ ι ∗ ( t γ 2 ) ∧ ι ∗ ( t γ 3 ) ι ∗ ( T γ 1 ,γ 2 ) = ι ∗ ( T γ 1 ) ∧ ι ∗ ( T γ 2 ) ∧ ι ∗ ( T γ 3 ) ι ∗ ( t γ 1 − 1 ) ∧ ι ∗ ( t γ 2 − 1 ) ∧ ι ∗ ( t γ 3 − 1 ) ∧ ι ∗ ( T γ 1 ,γ 2 ) , where in the s e cond equality , we hav e used the fact that on X γ 1 ,γ 2 , ι ∗ ( t γ i ) = ι ∗ ( T γ i ) ι ∗ ( t γ − 1 i ) for i = 1 , 2 , 3 . ORBIF OLD C UP PRODUCTS 35 W e use the a bove ex pression for eu S 1 (Θ γ 1 ,γ 2 ) to compute h J ( α 1 ⋆ t α 2 ) , α 3 i . h J ( α 1 ⋆ t α 2 ) , α 3 i = Z X γ 1 γ 2 α 1 ⋆ t α 2 t ( γ 1 γ 2 ) − 1 ∧ I ∗ ( α 3 ) = Z X γ 1 ,γ 2 α 1 ∧ α 2 ∧ ι ∗ ( I ∗ ( α 3 )) ι ∗ ( t ( γ 1 γ 2 ) − 1 ) ∧ eu S 1 (Θ γ 1 ,γ 2 ) = Z X γ 1 ,γ 2 α 1 ∧ α 2 ∧ ι ∗ ( I ∗ ( α 3 )) ι ∗ ( t ( γ 1 γ 2 ) − 1 ) ∧ ι ∗ ( T γ 1 ) ∧ ι ∗ ( T γ 2 ) ∧ ι ∗ ( T γ 3 ) ι ∗ ( t γ 1 − 1 ) ∧ ι ∗ ( t γ 2 − 1 ) ∧ ι ∗ ( t γ 3 − 1 ) ∧ ι ∗ ( T γ 1 ,γ 2 ) . Using the equa lit y ι ∗ ( t ( γ 1 γ 2 ) − 1 ) = ι ∗ ( t γ 3 ) = ι ∗ ( T γ 3 ) ι ∗ ( t γ − 1 3 ) , we conclude that h J ( α 1 ⋆ t α 2 ) , α 3 i = Z X γ 1 ,γ 2 ι ∗ ( α 1 ) ∧ ι ∗ ( α 2 ) ∧ ι ∗ ( I ∗ ( α 3 )) ∧ ι ∗ ( T γ 1 ) ∧ ι ∗ ( T γ 2 ) ι ∗ ( t γ 1 − 1 ) ∧ ι ∗ ( t γ 2 − 1 ) ∧ ι ∗ ( T γ 1 ,γ 2 ) = h J ( α 1 ) ∧ t J ( α 2 ) , α 3 i . The las t eq uation, co m bining with Poincar´ e dua lit y , implies that J − 1 ( J ( α 1 ) ∧ J ( α 2 )) = α 1 ⋆ t α 2 . This co mpletes the pro o f.  Remark 5. 4. Note tha t when t is equal to 0, the map J is not inv ertible genera lly . How ever, one can solve this pr oblem by working in the for mal framework as in [ ChHu ]. In this ca se our mo del extends Chen-Hu’s mo del to an ar bitrary almost complex o rbifold. 5.3. T op ologi cal and algebraic Ho c hschild cohomology . In the case of a s ym- plectic orbifold ( X, ω ), we ha ve t wo cohomology algebra str uctures from diﬀeren t approaches. One is the H o chsc hild cohomology algebr a o f the quan tized groupoid algebra computed in Theorem 4.17, the other is the topo logical Hochschild coho- mology H T • ( X )(( t )) deﬁned in Deﬁnition 5 .1 using e ssentially a unique (up to homotopy) compatible almo st complex structure to the sy mplectic str ucture on X . W e observe that the alg ebra structur e on the Hochsc hild cohomo logy of the quan- tized gr oup oid algebra is co mpletely top olo gical, which do es not dep end on the symplectic structures o r the almo st complex structures a t all. On the other ha nd, the topolo gical Ho chschild cohomolog y H T • ( X )(( t )) does dep end on the choices of almo st complex structures. Therefore, it is natural to exp ect that these tw o algebras ar e not isomorphic. In this subsectio n, w e w ould like to study the con- nections b etw e e n these t wo alge br a structures. W e sho w in t he f ollowing that the graded alg ebra of the t op ologica l Ho chschild co homology a lgebra is isomorphic to the Ho chschild cohomology of the co r resp onding quantized group oid algebr a. W e introduce a decreasing ﬁltration on the top ologica l H o chsc hild cohomolog y H T • ( X )(( t )) as follows F ∗ = { α ∈ H T • ( X γ )(( t )) | deg( α ) − ℓ ( γ ) ≥ ∗ } . Lemma 5.5. ( H T • ( X )(( t )) , ∧ t , F ∗ ) is a ﬁlter e d algebr a. 36 M.J. PFLAUM, H. B. POSTHUMA, X. T ANG, AND H.-H. TSENG Pr o of. O ne needs to prove that F k ∧ t F l ⊂ F k + l . T o this end let α 1 ∈ F k and α 2 ∈ F l and consider α 1 ∧ t α 2 . Without lo s s of ge ner ality , let us assume that α 1 ∈ H T k 1 ( X γ 1 )(( t )) and α 2 ∈ H T k 2 ( X γ 2 )(( t )) with k 1 ≥ k + ℓ ( γ 1 ) and k 2 ≥ l + ℓ ( γ 2 ). Since ( H T • ( X )(( t )) , ∧ t ) is a graded a lgebra resp ect to • , w e hav e that deg( α 1 ∧ α 2 ) = k 1 + k 2 ≥ k + ℓ ( γ 1 ) + l + ℓ ( γ 2 ). Since ℓ ( γ 1 ) + ℓ ( γ 2 ) ≥ ℓ ( γ 1 γ 2 ), w e hav e that deg( α 1 ∧ t α 2 ) − ℓ ( γ 1 γ 2 ) ≥ k + l + ℓ ( γ 1 ) + ℓ ( γ 2 ) − ℓ ( γ 1 γ 2 ) ≥ k + l. Therefore, α 1 ∧ t α 2 belo ngs to F k + l .  Lemma 5.6. L et Γ b e a ﬁnite gr oup acting a ve ct or sp ac e V . Then for every γ 1 , γ 2 ∈ Γ one has ℓ ( γ 1 ) + ℓ ( γ 2 ) = ℓ ( γ 1 γ 2 ) if and only if V γ 1 + V γ 2 = V and V γ 1 γ 2 = V γ 1 ∩ V γ 2 . Pr o of. By linear alg ebra one knows that dim( V γ 1 ) + dim( V γ 2 ) = dim( V γ 1 + V γ 2 ) + dim( V γ 1 ∩ V γ 2 ) . Moreov er, one ha s ℓ ( γ 1 ) + ℓ ( γ 2 ) = 2 dim( V ) − (dim( V γ 1 ) + dim( V γ 2 )) = 2 dim( V ) − dim ( V γ 1 + V γ 2 ) − dim( V γ 1 ∩ V γ 2 ) = dim( V ) − dim( V γ 1 + V γ 2 ) + dim( V ) − dim( V γ 1 ∩ V γ 2 ) . Since V γ 1 + V γ 2 ⊂ V and V γ 1 ∩ V γ 2 ⊂ V γ 1 γ 2 , we ha ve dim( V ) − dim( V γ 1 + V γ 2 ) ≥ 0 , dim( V ) − dim( V γ 1 ∩ V γ 2 ) ≥ dim( V ) − dim( V γ 1 γ 2 ) . Therefore ℓ ( γ 1 ) + ℓ ( γ 2 ) ≥ ℓ ( γ 1 γ 2 ) , and equality holds, if and only if dim( V ) = dim( V γ 1 + V γ 2 ) and dim( V γ 1 ∩ V γ 2 ) = dim( V γ 1 γ 2 ).  Theorem VI I. The gr ade d algebr a gr( H T • ( X )(( t ))) of ( H T • ( X )(( t )) , ∧ t ) with r esp e ct to the ﬁltra tion F ∗ is isomorph ic to the H o chschild c ohomolo gy algebr a ( H • ( A (( ~ )) ⋊ G ; A (( ~ )) ⋊ G ) , ∪ ) by identifying t wi th ~ . Pr o of. O b viously , the tw o vector spaces ov er C (( t )) a re is omorphic. It is suﬃcient to prove that the tw o pro duct structure s ag ree. According to the pro of o f Lemma 5.5, we hav e that for α 1 ∈ F k and α 2 ∈ F l , the graded pro duct g r ( α 1 ∧ t α 2 ) is not equal to zero only when ℓ ( γ 1 ) + ℓ ( γ 2 ) = ℓ ( γ 1 γ 2 ). In the case of ℓ ( γ 1 ) + ℓ ( γ 2 ) = ℓ ( γ 1 γ 2 ), b y Lemma 5.6, w e hav e that V γ 1 + V γ 2 = V and V γ 1 γ 2 = V γ 1 ∩ V γ 2 . This implies that N γ 1 ⊕ N γ 2 = N γ 1 γ 2 on X γ 1 γ 2 . Therefore, the following identit y of equiv a riant Thom cla s ses ho lds true: ι ∗ ( T γ 1 ∧ T γ 2 ) = ι ∗ ( T γ 1 γ 2 ) . Hence, by Deﬁnition 5 .1, one obtains h α 1 ∧ t α 2 , α i = Z X γ 1 ,γ 2 ι ∗ γ 1 ( α 1 | γ 1 ) ∧ ι ∗ γ 2 ( α 2 | γ 2 ) ∧ I ∗ ( α ) | γ 1 γ 2 , where the ∧ on the right hand side is the w edge pro duct on diﬀerential for ms. One concludes that g r( α 1 ∧ t α 2 ) agrees w ith the cup pro duct on the Hochsc hild cohomolog y algebra.  ORBIF OLD C UP PRODUCTS 37 F rom Theo rem VI I, we can view that the top ologica l Ho chsc hild cohomolog y ( H T • ( X )(( t )) , ∧ t ) a s a deforma tion of the alg ebraic Ho chsc hild coho mology ( H • ( A (( ~ )) ⋊ G ; A (( ~ )) ⋊ G ) , ∪ ) . It is very interesting to s tudy this defor mation using the Ho chsc hild co homology metho d a g ain, which will illustrate the role of the almost complex str ucture chosen to deﬁne ∧ t . W e leave this topic for future re s earch. Appendix A. Homol ogical algebra of bornological algebras and modules A.1. Bornologi es on v e ctor spaces. In this app endix we r ecollect the basic deﬁnitions and constructions in the theory of b or nological vector spaces. F or further details o n this see [ Bo ] and [ Ho ]. Let k b e the ground ﬁeld R or C , and V b e a v ector space o ver k . A set B of subsets of V is ca lled a ( c onvex line ar ) b ornolo gy on V and ( V , B ) a ( c onvex line ar ) b ornolo gic al ve ct or sp ac e , if the following a x ioms hold true: (BOR1) Every subset of an e le men t of B b elongs to B . (BOR2) Every ﬁnite union of elements of B b elong s to B . (BOR3) The set B is c overing for V that means ev er y element of V is contained in some s et b elonging to B . (BOR4) F or every B ∈ B , the absolutely c onvex hul l B ✸ := { λ 1 v 1 + λ 2 v 2 | v 1 , v 2 ∈ V , λ 1 , λ 2 ∈ k , | λ 1 | + | λ 2 | ≤ 1 } is ag ain B . The elemen ts of a b ornolog y B are called its b ounde d sets or sometimes it s small sets . Given an a bs olutely co nvex set S ⊂ V , we denote its linear span by V S and b y k · k B the seminorm o n V S with unit ball S := T λ> 1 λS . If k · k S is a norm on V S , then S is said to be norming , a nd c ompletant , if ( V S , k · k S ) is even a Banach space. A b ornolo gical vector space ( V , B ) is ca lled s ep ar ate d (resp. c omplete ), if every bounded a bsolutely c o nv ex se t B ⊂ V is no rming resp. c ompletant. Prop ositio n A. 1. L et V b e a b ornolo gic al ve ctor sp ac e. Then ther e exists a c om- plete b ornolo gic al ve ctor sp ac e ˆ V to gether with a b ounde d line ar map ι : V → ˆ V such that the fol lowing universal pr op erty is fulﬁl le d: • F or every c omplete b ornolo gic al ve ctor sp ac e W and every b ounde d line ar map f : V → W ther e exists a unique b oun de d line ar map ˆ f : ˆ V → W such that t he diagr am V f / / ι   W ˆ V ˆ f 8 8 q q q q q q q q q q q q q (A.1) c ommutes. Pr o of. F or the pro of of this see [ Me99 ].  F or ( V , B ) a nd ( W, D ) tw o b or nological vector spa ces, a linear map f : V → W is called b oun de d , if for every S ∈ B the image f ( S ) is in D . The s pa ce of b ounded linear maps V → W will be denoted b y Hom( V , W ). It carr ie s itself a cano nical 38 M.J. PFLAUM, H. B. POSTHUMA, X. T ANG, AND H.-H. TSENG bo rnology , na mely the b orno logy of e quib ounde d sets of linear maps, i.e. of subsets E ⊂ Hom( V , W ) such that for eac h S ∈ B the set E ( S ) is b ounded in ( W, D ). Obviously , the bo rnologica l v e ctor spaces together with the b ounded linear maps then form a categor y . Since the direct s um V ⊕ W of tw o bo rnologic a l vector spaces obviously inherits a canonica l b ornolo gical structure from its co mp onents, the categ ory of b orno logical vector spaces is even an additive catego ry . Moreov er, it carries the s tr ucture of a tensor category , since the a lgebraic tenso r pro duct V ⊗ W of t wo bo rnologica l vector spaces ( V , B ) and ( W , D ) carr ies a natural borno logy which is generated by the sets S ⊗ T , wher e S ∈ B , T ∈ D . In c a se V and W a re b oth complete b or nological vector space s, t he direct s um V ⊕ W is obviously a complete b ornolog ical v ector space as well. F or the tensor pro duct V ⊗ W , though, with its canonical bor no logical structure, completeness need not nece ssarily hold. Ther e fore, one intro duces the completed tensor pr o duct V ˆ ⊗ W := ( V ⊗ W ) ˆ for any pair of b ornolo gical v ector spaces V , W . Note that the catego ry of co mplete b ornolog ical vector spaces with ⊕ and ˆ ⊗ as dir ect sum resp. tensor functor a lso satisﬁes the axioms of an additive tenso r catego ry . W e denote the categ ory of co mplete b or nological vector spaces and bo unded linear maps by Bor . Example A.2. Let V b e a lo ca lly co nvex top olo gical vector space. Then Bnd ( V ) := { S ⊂ V | p ( S ) < ∞ for every seminorm p on V } and Cpt ( V ) := { S ⊂ V | S is precompa ct in V } are tw o, in g eneral diﬀerent, bor nologies o n V , w hich o ne calls, res p ectively , the von Neu mann and the pr e c omp act bo rnology . A.2. Bornologi cal algebras and mo dule s. By a b ornolo gic al algebr a one under- stands a k -algebra A together with a complete c onv ex b or nology B suc h that the pro duct map m : A ⊗ A → A is b ounded. B y the univ er sal proper t y of the com- pleted b ornolo gical tensor pr o duct one knows that for such an A the m ultiplication m lifts uniquely to a b ounded map A ˆ ⊗ A → A . F or any (real o r complex) algebra A we denote by A + the unital alg ebra A ⊕ k , and by A u the smallest unital algebr a cont aining A , which means that A u coincides with A , if A is unital, and w ith A + otherwise. Obviously , A + and A u are a gain bo rnologic a l alg ebras, if that is the ca se already for A . F or every bo rnologica l algebra A we denote by A e its enveloping algebr a which is deﬁned as the b or no logical tensor pro duct a lgebra A u ˆ ⊗ ( A u ) op . By a ( left ) A - mo dule o ver a b orno logical a lgebra A one understands a complete bo rnologic a l vector space M together with a b ounded linear map A u ˆ ⊗ M → M such that the following axio ms ar e satisﬁed: (MOD1) One has ( a 1 · a 2 ) · m = a 1 · ( a 2 · m ) for all a 1 , a 2 ∈ A a nd m ∈ M . (MOD2) The relation 1 · m = m holds for a ll m ∈ M . Example A.3. F or every complete bo rnologic al vector space V the tensor pro duct A u ˆ ⊗ V carries in a natural way the structure of a left A -mo dule. Mo dules of this form a re called fr e e left A -mo dules; likewise o ne deﬁnes free rig h t A -mo dules. Given left A -mo dules M and N w e write Hom A ( M , N ) for the space o f b ounded A -mo dule homomorphisms with the equib ounded b or nology . Obviously , the left A -mo dules together with these morphisms form a categ o ry , which w e will denote by Mod( A ). Note that every morphism f : M → N in Mo d( A ) has a kernel a nd a ORBIF OLD C UP PRODUCTS 39 cokernel. The kernel simply coincides with the vector space k er nel equipped with the subs pace bo rnology , where the cokernel is the quotient N /f ( M ) ˆ together with the quotient b o rnology . Simila r ly , one deﬁnes right A -mo dules ov er a bornolo g ical algebra A and writes Mo d( A op ) (resp. Hom A op ( M , N )) for the c a tegory of right A -mo dules (res p. the s et of rig h t A -mo dule morphis ms from M to N ). Finally , a n ob ject in the ca tegory Mo d( A e ) w ill b e ca lled an A - bimo dule . F or any right A -mo dule M and a ny left A -mo dule N we denote by M ˆ ⊗ A N the A -b alanc e d tensor pr o duct that means the co kernel o f the b ounded linear map M ˆ ⊗ A ˆ ⊗ N → M ˆ ⊗ N , m ⊗ a ⊗ n 7→ m · a ⊗ n − m ⊗ a · n. A bo rnologic a l algebra A is said to hav e a n appr oximate identity , if for every bo unded subset S ⊂ A ther e is a bo unded sequence ( u S,k ) k ∈ N and an abs olutely conv ex bo unded T S ⊂ A such that the following prop erties hold tr ue: (AID1) F or every a ∈ A S one ha s u S,k · a ∈ A T S and a · u S,k ∈ A T S . (AID2) F or all a ∈ S , the sequences u S,k · a and a · u S,k conv er ge to a in the B anach space A T S , and the co nvergence is unifor m in a . (AID3) F or b ounded subsets S 1 , S 2 ⊂ A such that S 1 ⊂ S 2 one ha s k u S 2 ,k · a − a k T S 2 ≤ k u S 1 ,k · a − a k T S 2 for a ll a ∈ A T S 1 and k ∈ N . In other words, a n appr oximate identit y  u S,k  S ∈B ,k ∈ N is es sentially a net in A s uch that each of the nets  u S,k a  and  a u S,k  conv er ges to a . A b or nological a lgebra A whic h p ossesses an approximate identit y and which, additionally , is pro jective bo th as a left and a right A -mo dule, is called qu asi-unital . Note t hat under the as sumption tha t A has an approximate ident ity , pr o jectivity of A is equiv alent to the existence of a bo unded left A -mo dule ma p l : A → A u ˆ ⊗ A and a bounded right A -mo dule map r : A → A ˆ ⊗ A u which are b oth sections of the m ultiplication ma p (cf. [ Me04 ]). Given a quasi-unital bo rnologica l algebra A , a left A -mo dule M (resp. a right A - mo dule N ) is called ess en t ial , if the ca nonical map A ˆ ⊗ A M → M (r esp. N ˆ ⊗ A A → N ) is an is omorphism. If the left A -module M (resp. the right A -module N ) has the prop erty that the canonical map M → Hom A ( A, M ) (resp. N → Ho m A op ( A, N )) is an isomorphism, one ca lls M (re s p. N ) a r ough mo dule. The categor y of essen- tial left A -mo dules (resp. right A -mo dules) will be denoted by Mod e ( A ) (resp. by Mo d e ( A op )). Since A is assumed to be qua si-unital, one concludes tha t for every A -mo dule M , the tensor pro duct A ˆ ⊗ A M is an essential mo dule. A.3. Resol utions and homolo gy. In this a rticle we conside r homology theories in the a dditiv e but in general not a belia n category of modules over a b o r nologica l algebra A . This implies that we have to use metho ds from relative ho mological algebra. Essent ially this means that only so- c alled allowable chain complexes and allow a ble pro jective resolutions are use d to determine homologies and coho mo logies. T o deﬁne the notion o f a llow ability precisely recall that a b ounded epimor phis m of le ft A -modules f : M − → N or in other w ords a short exact se q uence o f left A -mo dules and b ounded ma ps 0 − → K − → M f − → N − → 0 is called line arly split , if there ex ists a b ounded linear ma p N → M which is a section o f f . A left A -mo dule P is now called pr oje ctive , if the functor Hom A ( P, − ) is exact on linea rly s plit short exact sequences in Mo d ( A ). Moreov er, a chain complex 40 M.J. PFLAUM, H. B. POSTHUMA, X. T ANG, AND H.-H. TSENG ( C • , ∂ ) of A -mo dules a nd b ounded maps ∂ k : C k → C k − 1 is called al lowable , if for every k the image o f ∂ k is in Mo d ( A ), i.e. is a complete bor no logical subspace of C k − 1 , and if the b ounded epimorphism ∂ k : C k → im ∂ k induced by ∂ k is linearly split. Likewise one deﬁnes allow able co chain complexes. F o r homolog y theories in categorie s o f mo dules of b orno logical algebr as the following result now is crucial. Prop ositio n A.4. Le t A b e a quasi-un ital b ornolo gic al algebr a A . Then every fr e e left A -mo dule is pr oje ctive. Mor e over, the c ate gory Mo d ( A ) has en ou gh pr oje ctives, that me ans for every left A -mo dule M ther e exists a pr oje ctive left A - mo dule P to gether wi th a split epimorphism of A -m o dules P → M . Her eby, P c an b e chosen to b e fr e e. Final ly, the f unctor Mo d ( A ) → Mo d e ( A ) , M 7→ A ˆ ⊗ M pr eserves pr oje ctive mo dules, and t he c ate gory M o d e ( A ) of essent ial left A -m o dules has enough pr oje ct ives as wel l. Pr o of. See [ Me04 , Sec. 4]).  The pr op osition implies tha t for ev er y left A -mo dule there exists an al lowable pr oje ctive r esolution of M , i.e. an allow able acyclic complex ( P • , ∂ ) of pro jective left A -modules P k , k ≥ 0 together with a quasi-isomorphism ε : P • → M • in the category o f left A -mo dules, wher e M • denotes the c o mplex which is co nc e n trated in degree 0 a nd coincides there w ith the A -mo dule A . Thes e conditions ar e equiv alent to the req uirement that ε is a s plit b ounded A -linea r sur jection ε : P 0 → M which satisﬁes ε ◦ ∂ 1 = 0 and that there exists an A -linea r splitting h : M → P 0 and a fa mily ( h k ) k ∈ N of bo unded linear maps h k : P k → P k +1 such that ∂ 1 h 0 = id P 0 − hε and ∂ k +1 h k − h k − 1 ∂ k = id P k for a ll k ≥ 1 . The pro o f of the following result is standard in (relative) homolo g ical algebra . Theorem A.5 (Comparison Theorem) . ( cf. [ Me99 , Thm. A.9] ) Assume that M and N ar e two left A -mo dules over a b ornolo gic al algebr a A . L et P • → M • and Q • → N • b e al lowable r esolutions of M r esp. N . If P • is pr oje ctive, then ther e exists for every morphism f : M → N of left A -mo dules a lifting of f , i. e. a chai n map F : P • → Q • in t he c ate gory o f left A -mo dules su ch that the diagr am P • / / F   M • f   Q • / / N • c ommutes. Any two su ch liftings of f ar e homotopic. In p articular, any two al low- able pr oje ctive r esolutions of M ar e homotopy e quivalent. The c omparison theor em allows the construction of derived functor s in the cate- gory of A -mo dules. In particula r, the functors E xt and T or can now be deﬁned for A -mo dules as usual. ORBIF OLD C UP PRODUCTS 41 A.4. Ho chsc hild ho mology a nd Ba r resolution. Given a b orno logical alg ebra A and an A -bimo dule M , the Ho chsc hild homology H • ( A, M ) and cohomology H • ( A, M ) are deﬁned as de r ived functors in the categ ory Mo d  A e  of A -bimo dules as fo llows: H • ( A, M ) := T or A e • ( A, M ) , H • ( A, M ) := Ext • A e ( A, M ) . (A.2) A particularly useful resolutio n of the A -bimodule A is given by the Bar c omplex (Bar • ( A ) , b ′ ) together with the mu ltiplication map inducing the qua si-isomor phis m Bar • ( A ) → A . Hereby , Bar k ( A ) = A ˆ ⊗ A ˆ ⊗ k ˆ ⊗ A, and b ′ is the standa rd b oundar y map on the Bar complex: b ′ 0 ( a 0 ⊗ a 1 ) = 0 , b ′ k ( a 0 ⊗ a 1 . . . a k ⊗ a k +1 ) = k X i =0 ( − 1) i a 0 ⊗ . . . ⊗ a i a i +1 ⊗ . . . ⊗ a k +1 . Obviously , if A is q uasi-unital, then the Ba r complex of A provides an a llow able pro jective resolution of A in the categ ory of A - bimo dules . In other words, this means that a qua s i-unital b orno logical alg ebra A is H- unital in the sens e of W o dzicki (see [ Wo , Lo ]). Thus, for quas i-unital A , the Ho chsc hild homology and cohomolog y groups H • ( A, M ), H • ( A, M ) ar e computed as the homology resp. cohomolo gy of the Ho chschild complexes ( C • ( A, M ) , b ′ ∗ ) w ith C • ( A, M ) := Bar • ( A ) ˆ ⊗ A e M , and ( C • ( A, M ) , b ′ ∗ ) w ith C • ( A, M ) := Hom A e (Bar • ( A ) , M ) . (A.3) F or some applica tions, in particular to deﬁne the cup pro duct on the Ho chsc hild co chain complex o f a quasi-unital b orno lo gical algebra which do es not have a unit, the left and right r educed Bar co mplexes  Bar l-red • ( A ) , b ′  and  Bar r-red • ( A ) , b ′  are quite useful. They carry the same bo undary as the Ba r complex, and hav e compo - nent s Bar l-red k ( A ) := A u ˆ ⊗ A ˆ ⊗ k ˆ ⊗ A a nd Bar r-red k ( A ) := A ˆ ⊗ A ˆ ⊗ k ˆ ⊗ A u . Obviously , under the assumption that A is quasi- unital, the left and right reduced Bar co mplexes a re bo th a llow able pro jective resolutions o f A . Moreov er, the canon- ical em bedding s Bar • ( A ) ֒ → Bar l-red • ( A ) and Bar • ( A ) ֒ → Bar r-red • ( A ) hav e the fol- lowing q ua si-inv er ses: r k : Bar l-red k ( A ) → Bar k ( A ) , a 0 ⊗ . . . ⊗ a k +1 7→ a 0 r ( a 1 ) · . . . · r ( a k +1 ) , l k : Bar r-red k ( A ) → Bar k ( A ) , a 0 ⊗ . . . ⊗ a k +1 7→ l ( a 0 ) · . . . · l ( a k ) a k +1 , (A.4) where l : A → A ˆ ⊗ A res p. r : A → A ˆ ⊗ A is an A -left resp. A -r ight linear sec tion of the m ultiplication ma p o n A . Finally in this section we consider the r e duc e d Ho chschild chain and r e duc e d Ho chschild c o chain c omplexes . These ar e deﬁned by C red k ( A, M ) := ( A ˆ ⊗ A M ˆ ⊗ A A, if k = 0 , M ˆ ⊗  A u / k  ˆ ⊗ k if k ≥ 1 , C k red ( A, N ) := ( Hom A e  A ˆ ⊗ A, N  , if k = 0 , Hom k  A u / k  ˆ ⊗ k , N  , if k ≥ 1 , (A.5) 42 M.J. PFLAUM, H. B. POSTHUMA, X. T ANG, AND H.-H. TSENG and carry the same b ounda ry resp. cob oundary maps a s the unreduced complexes. By co nstruction the ca nonical maps C • ( A, M ) → C red • ( A, M ) and C • red ( A, N ) → C • ( A, N ) are then chain maps . If M is an essent ial A -bimodule (resp. N a rough A -bimodule), then the ﬁrst (re s p. the second) o f these chain maps is a q ua si-isomor phism. Since A is assumed to be quasi-unital, the ﬁr st chain map is alwa ys a quasi-isomo rphism for M = A . In ma n y applications, a nd in pa rticular those app ear ing in this article, the second c hain ma p is also a quasi-is omorphism for N = A . In c ase A is unital, bo th chain maps ar e always quasi- is omorphisms. A.5. The cup pro duct on Ho c hsc hi ld cohomo logy . Under the g eneral as- sumption from ab ove that A is a (p ossibly no nunital) bo rnolog ic a l alge bra, we will now explain the co nstruction o f the cup pro duct ∪ : H • ( A, A ) × H • ( A, A ) → H • ( A, A ) . One way to deﬁne ∪ is via the Y oneda pro duct on (b ounded) extensio ns 0 → A → E 1 → · · · → E k → A → 0 and the in ter pretation of Ext k A e ( A, A ) as the space of e q uiv alence cla sses of suc h extensions. Alternatively , and that is the appro ach we will follow here, one can use the qua si-isomor phisms fr om Eq. (A.4) to directly deﬁne a cup pro duct ∪ : C • ( A, A ) × C • ( A, A ) → C • ( A, A ) on the Ho chsc hild co chain complex, which on cohomolog y co incide s with the Y oneda pro duct. Mo re pre cisely , w e deﬁne for f ∈ C k ( A, A ) and g ∈ C l ( A, A ) the pro duct f ∪ g ∈ C k + l ( A, A ) by f ∪ g ( a 0 ⊗ · · · ⊗ a k + l +1 ) := f  l k ( a 0 ⊗ · · · ⊗ a k ⊗ 1)  g  r l (1 ⊗ a k +1 ⊗ · · · ⊗ a k + l +1 )  , (A.6) where a 0 , · · · , a k + l +1 ∈ A . It is s traightforw ard to chec k tha t the thu s deﬁned map ∪ is a chain ma p and asso ciative up to homotopy . The cup-pr o duct induced on the reduced Hochsc hild cochain co mplex by th e em b edding C • red ( A, A ) → C • ( A, A ) is given by f ∪ g ( a 1 ⊗ · · · ⊗ a k + l ) := f ( a 1 ⊗ · · · ⊗ a k ) g ( a k +1 ⊗ · · · ⊗ a k + l ) (A.7) for f ∈ C k red ( A, A ), g ∈ C l red ( A, A ) and a 1 , · · · , a k + l ∈ A . A.6. Bornologi cal structures on conv olution algebras and thei r mo dules . Consider a pro per ´ etale Lie group oid G and let A ⋊ G denote its conv o lution algebr a (see Sec. 2). A subset S ⊂ A ⋊ G is said to b e b ounde d , if there is a co mpact subset in K ⊂ G 1 such that supp a ⊂ K for every a ∈ S , and if for each diﬀerential op erator D on G 1 one ha s sup a ∈ S k D a k K < ∞ . The b ounded subset of A ⋊ G form a b or nology which co incides b oth with the von Neumann and t he precompact bornolo gy deﬁned in Example A.2. In this article, we always assume that A ⋊ G car ries this bo rnology . By an immediate argument o ne chec ks that the co n volution pro duct is b ounded and that t he b orno logy on A ⋊ G is complete. Th us A ⋊ G beco mes a bornolo gical algebra. Let us chec k that it is quasi-unital. T o this end choo se a s e quence of smoo th maps ϕ k : G 0 → [0 , 1 ] suc h that the suppor t of each ϕ k is compac t and such that  ϕ 2 k  k ∈ N is a lo ca lly ﬁnite partition of unity on G 0 . Obviously , o ne can even achieve that (supp ϕ k ) ◦ = supp ϕ k (A.8) ORBIF OLD C UP PRODUCTS 43 holds for every k ∈ N ; this is a prop er t y we will need la ter . No w e x tend ea ch ϕ k by zero to a smo oth fun ction on G 1 and denote the re sulting element of A ⋊ G ag ain by ϕ k . Then put u k := P l ≤ k ϕ l ∗ ϕ k and chec k that  u k  k ∈ N is an appr oximate ident it y . Moreov er, the maps l : A ⋊ G → A ⋊ G ˆ ⊗ A ⋊ G , a 7→ a ∗ ϕ k ⊗ ϕ k and r : A ⋊ G → A ⋊ G ˆ ⊗ A ⋊ G , a 7→ X k ∈ N ϕ k ⊗ ϕ k ∗ a are b oth sections of the convolution pr o duct. This prov es Prop ositio n A. 6 . The c onvolution algebr a A ⋊ G of a pr op er ´ etale Lie gr oup oid G to gether with the von Neumann b ornolo gy is a quasi-unital b ornolo gic al algebr a. Next let us consider the case where G 0 carries a G -inv ar iant symplectic form ω and where a G -inv ar iant lo cal star pro duct ⋆ on G 0 has b een c hosen. Under thes e assumptions consider the cro ssed pro duct algebr a A ~ ⋊ G , where A ~ denotes the sheaf C ∞ G 0 [[ ~ ]] with pro duct ⋆ . A subset B ⊂ A ~ ⋊ G is said to b e b ounde d , if there is a compact subset in K ⊂ G 1 such th at supp a ⊂ K for ev ery a ∈ B , a nd such that for e a ch diﬀerential o per ator o n G 1 and k ∈ N one ha s sup a ∈ B k D a k k,K < ∞ . Hereby , k · k k,K is the semino r m on A ~ ⋊ G deﬁned by k a k k,K := sup g ∈ K | a k ( g ) | , a ∈ A ~ ⋊ G , where the a l ∈ C ∞ cpt ( G 1 ), l ∈ N are the unique co eﬃcients in the formal p ow er series expansio n a = P l ∈ N a l ~ l . The b ounded subsets of A ~ ⋊ G deﬁne a co mplete bo rnology which we call the c anonic al b ornolo gy on A ~ ⋊ G . One immediately chec ks that the conv olution pro duct ⋆ c deﬁned by E q . (2.2) is b ounded, hence A ~ ⋊ G is a bo rnologica l a lgebra. Obviously , the family  u k  k ∈ N from a bove for ms an approximate unit also for A ~ ⋊ G . By the assumption (A.8) it is clear that each of the functions ϕ k has o nly zeros of inﬁnite or der. No w chec k the following lemma by using standa rd ar guments fro m the theory of deforma tion quantization. Lemma A.7. L et ϕ : G 0 → [0 , 1 ] b e a smo oth function which has only zer os of inﬁnite or der, and put u = ϕ 2 . Then u has a st ar pr o duct r o ot, that me ans ther e exists an element Φ = P l ∈ N Φ l ~ l ∈ C ∞ [[ ~ ]] such that Φ ⋆ Φ = u , Φ 0 = ϕ, and supp Φ ⊂ supp ϕ. Using this result choose Φ k ∈ A ~ ⋊ G with supp ort in G 0 such that Φ k ⋆ c Φ k = ϕ 2 k and Φ k − ϕ k ∈ ~ A ~ ⋊ G . The maps l : A ~ ⋊ G → A ~ ⋊ G ˆ ⊗ A ~ ⋊ G , a 7→ X k ∈ N a ⋆ c Φ k ⊗ Φ k and r : A ~ ⋊ G → A ~ ⋊ G ˆ ⊗ A ~ ⋊ G , a 7→ X k ∈ N Φ k ⊗ Φ k ⋆ c a then a re b oth sections of the conv olution pro duct. Hence we obtain Prop ositio n A. 8. The cr osse d pr o duct a lgebr a A ~ ⋊ G asso ciate d to an invariant lo c al deformation quantization on the sp ac e of obje cts of a pr op er ´ etale Lie gr oup oid G with an invariant symp le ctic form is a quasi-unital b ornolo gic al algebr a. 44 M.J. PFLAUM, H. B. POSTHUMA, X. T ANG, AND H.-H. TSENG A.7. Morita equiv alence for b o rnological alg ebras. Assume that A and B are tw o bor nological alge br as. Reca ll that by an A - B -bimo dule one understands an element of the categ ory Mo d  A u ˆ ⊗ ( B u ) op  . Under the condition that the bo rnolog- ical algebr as A a nd B are bo th quasi-unita l, o ne calls A and B Mo rita e quivalent , if there e x ist bimo dules P ∈ Mod  A u ˆ ⊗ ( B u ) op  and Q ∈ Mo d  B u ˆ ⊗ ( A u ) op  such that the following ax ioms ho ld true: (MOR1) P is esse n tial b oth as an A -left mo dule and a s a B -r ight mo dule. (MOR2) Q is essential both as a B - left mo dule a nd as an A -right mo dule. (MOR3) There exist b ounded bimo dule isomor phisms u : P ˆ ⊗ B Q → A and v : Q ˆ ⊗ A P → B . (MOR4) P is pro jective as a B -right mo dule, a nd Q is pr o jective as an A -rig ht mo dule. W e sometimes say in this situation that ( A , B , P , Q , u, v ) is a Morita c ontext . The following r esult follows ea sily fr o m the deﬁnition of a Morita context. Prop ositio n A. 9. L et ( A, B , P , Q , u, v ) b e a Mo rita c ontext. Then the funct ors Mo d( A ) → Mo d( B ) , M 7→ Q ˆ ⊗ A M Mo d( B ) → Mo d( A ) , N 7→ P ˆ ⊗ B N ar e b oth exact and quasi-inverse to e ach other. In p articular this me ans that Mo d( A ) and Mo d( B ) ar e e quivalent c ate gories. Example A. 10. Le t ϕ : H → G b e a weak equiv a lence of prop er ´ etale Lie group oids. By [ Mr , Cor . 3.2] it follows that the conv olution alg ebras A := A ⋊ G and B := A ⋊ H are Morita equiv alent. A Mo rita context is given by the bimo dules P = C ∞ cpt ( h ϕ i ) and Q = C ∞ cpt ( h ϕ i − ), where h ϕ i := G 1 × ( s,ϕ ) H 0 and h ϕ i − := H 0 × ( ϕ,t ) G 1 . Let us pr ovide the details for the case, where ϕ is even a n op en embedding. Then, h ϕ i is the op en subset s − 1 ( ϕ ( H 0 )) ⊂ G 1 , a nd h ϕ i − = t − 1 ( ϕ ( H 0 )) ⊂ G 1 . Moreov er, the A - B -bimo dule structure on P is giv en by a ∗ p ∗ b ( g ) = X g 1 g 2 ϕ ( h )= g g 1 ∈ G 1 ,g 2 ∈h ϕ i ,h ∈ H 1 a ( g 1 ) p ( g 2 ) b ( h ) , where a ∈ C ∞ cpt ( G 1 ), b ∈ C ∞ cpt ( H 1 ) a nd p ∈ C ∞ cpt ( s − 1 ( ϕ ( H 1 )). The bimo dule structure for Q is deﬁned analog ously . Theorem A.11. Under the assu mption that the we ak e quivalenc e ϕ : H ֒ → G is an op en e mb e dding, the fol lowing holds t rue: (1) The bimo dules P = C ∞ cpt ( s − 1 ( ϕ ( H 0 )) and Q = C ∞ cpt ( t − 1 ( ϕ ( H 0 )) satisfy axioms (MOR1) a nd ( MOR2). (2) P r esp. Q is pr oje ct ive b oth as an A - as a B -m o dule. (3) The ma ps u : C ∞ cpt ( s − 1 ( ϕ ( H 0 )) ˆ ⊗ C ∞ cpt ( H 1 ) C ∞ cpt ( t − 1 ( ϕ ( H 0 )) → C ∞ cpt ( G 1 ) , a ⊗ ˜ a 7→ a ∗ ˜ a, v : C ∞ cpt ( t − 1 ( ϕ ( H 0 )) ˆ ⊗ C ∞ cpt ( G 1 ) C ∞ cpt ( s − 1 ( ϕ ( H 0 )) → C ∞ cpt ( H 1 ) , ˜ a ⊗ a 7→ ϕ ∗ (˜ a ∗ a ) . ar e b ounde d bimo dule isomorphi sms. This me ans in p articular that the tuple ( A ⋊ G , A ⋊ H , P, Q, u, v ) is a Morita c ontext b etwe en b ornolo gic al algebr as. ORBIF OLD C UP PRODUCTS 45 Pr o of. Co nsider the family ( ϕ k ) k ∈ N of elemen ts of A = A ⋊ G from ab ov e. Then the map l P : P → A ˆ ⊗ P , p 7→ X k p ∗ ϕ k ⊗ ϕ k resp. r Q : Q → Q ˆ ⊗ A, q 7→ X k ϕ k ⊗ ϕ k ∗ q is a section of the left (resp. right) A -a ction on P (resp. Q ). Since A is qua si-unital, this implies that P (resp. Q ) is essential and pro jective as a left (resp. r ig ht ) A - mo dule. Likewise, one pr ov es the ex istence of a section l Q (resp. r P ) of the left (resp. right) B -a c tion on Q (r e s p. P ). Hence, Q (r esp. P ) is es sential and pr o jective as a left (resp. right) B -mo dule. Thus, we have pr ov ed (1) a nd (2 ). Next we s how that ther e exists a b ounded sectio n of the map ˆ v : C ∞ cpt ( t − 1 ( ϕ ( H 0 )) ˆ ⊗C ∞ cpt ( s − 1 ( ϕ ( H 0 )) → C ∞ cpt ( H 1 ) , ˜ a ⊗ a 7→ ϕ ∗ (˜ a ∗ a ) . T o th is end choo se a family ( ψ k ) k ∈ N in B = A ⋊ H whic h has supp ort in H 0 , and such that ( ψ 2 k ) k ∈ N is a lo cally ﬁnite partition of unity on H 0 . F or ea ch element b ∈ B deﬁne elements ϕ ∗ b in P and Q b y extension by zero. Then the map ˆ ν : C ∞ cpt ( H 1 ) → C ∞ cpt ( t − 1 ( ϕ ( H 0 )) ˆ ⊗C ∞ cpt ( s − 1 ( ϕ ( H 0 )) , b 7→ X k ∈ N ϕ ∗ b ∗ ψ k ⊗ ϕ ∗ ψ k is a bo unded section of ˆ v and a mor phism of le ft B -mo dules. Hence ν := π ˆ ν , where π : Q ˆ ⊗ P → Q ˆ ⊗ A P denotes the canonical pr o jection, is a b ounded section of v . Note that b y construction the image of ν lies in the algebr aic tensor pro duct Q ⊗ A P , and that the image of ν is a complete bo rnologic a l subspace of Q ⊗ A P which has to be is omorphic to B . Since b y [ Mr , Thm. 4 ] the restriction to the algebraic tensor pro duct v | Q ⊗ A P : Q ⊗ A P → B is a n (algebr aic) isomor phism of B - bimo dules, one then concludes that Q ⊗ A P = Q ˆ ⊗ A P and that v is a (b or nological) isomor phism of B -bimo dules. The pro of that u is a (b or nological) isomorphism o f A -bimo dules is more compli- cated. W e sho w this claim under the additional assumption that H 0 is connected. The genera l cas e is slightly mo re in volv ed, but c a n b e proved a long the sa me lines. Denote by G α the connected co mp onents of G 0 , and by G α 0 the image ϕ ( H 0 ). Let G α,β = s − 1 ( G α ) ∩ t − 1 ( G β ). Then G 1 is the disjoint union o f the G α,β . Since ϕ is a weak eq uiv alence, o ne derives the following: (i) The image ϕ ( H 1 ) c oincides with G α 0 ,α 0 . (ii) The bibundle h ϕ i = s − 1 ( ϕ ( H 0 )) co incides with the disjoint union of the com- po nent s G α 0 ,α , and h ϕ i − = t − 1 ( ϕ ( H 0 )) with the disjoin t union of the comp o- nent s G α,α 0 . (iii) Ther e exist unique op e n em b eddings σ α,β : G α,β ֒ → G α,α 0 and τ α,β : G α,β ֒ → G α 0 ,β such that s ◦ σ α,β = s | G α,β resp. t ◦ τ α,β = t | G α,β . Next choo se for every β functions ψ β ,k ∈ C ∞ cpt ( G β ,β ∩ G 0 ) such that ea ch family ( ψ 2 β ,k ) k ∈ N is a lo cally ﬁnite partition of unit y . Ex tend ψ β ,k by 0 to a smo oth function with compact support in G β ,β . Deﬁne ψ (1) β ,k ∈ C ∞ cpt ( H 1 ) and ψ (2) β ,k ∈ C ∞ cpt ( t − 1 ϕ ( H 0 )) by ψ (1) β ,k = ( ϕ ∗  ( τ α 0 ,β σ β ,β ) ∗ ( ψ β ,k )  ov er H 0 0 else . 46 M.J. PFLAUM, H. B. POSTHUMA, X. T ANG, AND H.-H. TSENG and ψ (2) β ,k = ( ( τ β ,β ) ∗ ( ψ β ,k ) o ver G α 0 ,β 0 else . Now w e have the means to constr uct a b ounded section of the map ˆ u : C ∞ cpt ( s − 1 ( ϕ ( H 0 )) ˆ ⊗C ∞ cpt ( t − 1 ( ϕ ( H 0 )) → C ∞ cpt ( G 1 ) , a ⊗ ˜ a 7→ a ∗ ˜ a. Put ˆ µ : C ∞ cpt ( G 1 ) → C ∞ cpt ( s − 1 ( ϕ ( H 0 )) ˆ ⊗C ∞ cpt ( t − 1 ( ϕ ( H 0 )) , C ∞ cpt ( G α,β ) ∋ a 7→ X k ∈ N ( σ α,β ) ∗ ( a ) ∗ ψ (1) β ,k ⊗ ψ (2) β ,k . and c heck that ˆ µ is a bounded section of ˆ u indeed. One pro ceeds now exactly as for v to show that P ⊗ B Q = P ˆ ⊗ B Q and that u is a b or nological isomo rphism of A -bimo dules.  Ho ch schild (co)homology of b orno lo gical algebra s and their bimo dules is inv a riant under a Mor ita context as the following res ult shows. Theorem A. 1 2 (cf. [ Lo , Thm. 1.2.7]) . Assu me that A and B ar e quasi-unital b ornolo gic al algebr as and assume that ther e is a Morita c ontext ( A, B , P , Q, u, v ) with the addi tional pr op erty that q u ( p ⊗ q ′ ) = v ( q ⊗ p ) q ′ and pv ( q ⊗ p ′ ) = u ( p ⊗ q ) p ′ for a l l p, p ′ ∈ P and q, q ′ ∈ Q . (A.9) L et M b e an essential A -bimo dule. Then ther e ar e n atur al iso morphisms H • ( A, M ) ∼ = H • ( B , Q ˆ ⊗ A M ˆ ⊗ A P ) and H • ( A, M ) ∼ = H • ( B , Q ˆ ⊗ A M ˆ ⊗ A P ) (A.10) Pr o of. W e only prov e the ho mo logy cas e . The cohomolo gy case is proven similar ly . T o pr ov e the claim ﬁrst choo se approximate ident ities ( u S,k ) S ∈B ,k ∈ N for A and ( v T ,l ) T ∈ D , l ∈ N for B , where B a nd D ar e the bo rnologies of A and B resp ectively . Since P and Q are esse ntial modules ov er A and B , there exis ts a b ounded s ection ˆ µ : A → P ˆ ⊗ Q (resp. ˆ ν : B → Q ˆ ⊗ P ) of the comp os ition of u : P ˆ ⊗ B Q → A with the cano nical pr o jection P ˆ ⊗ Q → P ˆ ⊗ B Q (resp. of V : Q ˆ ⊗ A P → B with Q ˆ ⊗ P → Q ˆ ⊗ A P ). The section ˆ µ (re sp. ˆ ν ) ca n even be c hos en to be a morphism of left A -mo dules (resp. o f left B -mo dules). F or ev ery b ounded S ⊂ A and k ∈ N (resp. b ounded T ⊂ B and l ∈ N ) let X i ∈ N p S,k,i ⊗ q S,k,i := ˆ µ ( u S,k ) a nd X j ∈ N q ′ T ,l,j ⊗ p ′ T ,l,j := ˆ ν ( v T ,l ) . Then deﬁne for every n ∈ N a bounded map  n : M ˆ ⊗ A ˆ ⊗ n → ( Q ˆ ⊗ A M ˆ ⊗ A P ) ˆ ⊗ B ˆ ⊗ k by  n ( m ⊗ a 1 ⊗ . . . ⊗ a n ) = lim ( S,k ) ∈B × N X i 0 ,i 1 , ··· , i n ( q S,k,i 0 ⊗ m ⊗ p S,k,i 1 ) ⊗ v ( q S,k,i 1 ⊗ a 1 p S,k,i 2 ) ⊗ . . . . . . ⊗ v ( q S,k,i n ⊗ a n p S,k,i 0 ) . ORBIF OLD C UP PRODUCTS 47 Note that this map is well-deﬁned since ( u S,k ) is a n approximate iden tit y , ˆ µ is a bo unded mor phism of left A -modules , and since M is essential. Analogo usly we deﬁne maps θ n : ( Q ˆ ⊗ A M ˆ ⊗ A P ) ˆ ⊗ B ˆ ⊗ k → M ˆ ⊗ A ˆ ⊗ n by θ n ( q ⊗ m ⊗ p ⊗ b 1 ⊗ . . . ⊗ b n ) = lim ( T ,l ) ∈D × N X j 0 ,j 1 , ··· , j n u ( p ′ T ,l,j 0 ⊗ q ) m u ( p ⊗ q ′ T ,l,j 1 ) ⊗ u ( p ′ T ,l,j 1 ⊗ b 1 q ′ T ,l,j 2 ) ⊗ . . . . . . ⊗ u ( p ′ T ,l,j n ⊗ b n q ′ T ,l,j 0 ) . Again, con vergence is guara nt eed by the fact that A a nd B are quasi-unital and that M is essential. By assumption (A.9) on u and v the maps  and θ are both chain maps. The comp osites θ ◦ ψ and  ◦ θ are homotopic to the identit y . A simplicial ho motopy b etw e e n θ ◦  and the identit y is given by h i ( m ⊗ a 1 ⊗ . . . ⊗ a n ) = lim ( S,k ) ∈B× N ( T ,l ) ∈D× N X i 0 ,i 1 , ··· ,i i j 0 ,j 1 ,...,j i m u ( p S,k,i 0 ⊗ q ′ T ,l,j 0 ) ⊗ u ( p ′ T ,l,j 0 ⊗ q S,k,i 0 ) a 1 u ( p S,k,i 1 ⊗ q ′ T ,l,j 1 ) ⊗ . . . ⊗ u ( p ′ T ,l,j i − 2 ⊗ q S,k,i i − 2 ) a i − 1 u ( p S,k,i i − 1 ⊗ q ′ T ,l,j i − 1 ) ⊗ ⊗ u ( p ′ T ,l,j i − 1 ⊗ q S,k,i i − 1 ) ⊗ a i ⊗ a i +1 ⊗ . . . ⊗ a n . By a s tr aightforw ard though somewhat length y a rgument (cf. [ Ha , Chap. 5]) one chec ks that the thu s deﬁned h i are well-deﬁned and form a b ounded s implicial homotopy indeed. Similarly , one constructs a bounded homoto p y betw een ψ ◦ θ and the identit y . Since [  ] = [ θ ] − 1 , whe r e [  ] and [ θ ] denote the induced maps on the Hochschild homology of A ⋊ G resp. A ⋊ H , and since θ neither dep ends on the pa r ticular choice of an approximate identit y on A ⋊ G nor on the choice of the elements p S,k,i and q S,k,i , o ne concludes that [  ] is indep endent of the particular choice of these data. Likewise on s hows that [ θ ] do e s not dep end on the choice of an appr oximate ident it y on A ⋊ H and of the elemen ts p ′ T ,l,j and q ′ T ,l,j . O ne c o ncludes fro m this that [ θ ] and [  ] are na tural is omorphisms. This ﬁnishes the pro of of the claim.  Remark A.13. If one repre sents the Ho chschild cohomology groups H n ( A, A ) as equiv alence classes o f bo unded e x tensions 0 → A → E 1 → · · · → E k → A → 0 , (A.11) the isomor phism H • ( A, A ) → H • ( B , B ) b etw e en the Hochsc hild cohomolo gies of t wo Morita equiv alent bounded alg ebras is giv en b y tensoring (A.11) with P and Q , i.e. by mapping it to the b ounded extension 0 → B → Q ˆ ⊗ A E 1 ˆ ⊗ A P → · · · → Q ˆ ⊗ A E k ˆ ⊗ A P → B → 0 , Let us now apply the preceding theorem to the situation of Ex ample A.10 . Then we can prove the following result. Theorem A.14 . Assu m e that ϕ : H ֒ → G is a we ak e quivalenc e of pr op er ´ etale Lie gr oup oids whic h also is an op en emb e dding, and let ( A ⋊ G , A ⋊ H , P, Q, u, v ) denote the Morita c ontext fr om The or em A .11. The r esult ing natur al isomorphisms in Ho chschild ho molo gy and c ohomolo gy ar e implemente d by the fol lowing natur al 48 M.J. PFLAUM, H. B. POSTHUMA, X. T ANG, AND H.-H. TSENG chain maps ϕ ∗ : C • ( A ⋊ H , A ⋊ H ) → C • ( A ⋊ G , A ⋊ G ) , a 0 ⊗ a 1 ⊗ . . . ⊗ a n 7→ ϕ ∗ ( a 0 ) ⊗ ϕ ∗ ( a 1 ) ⊗ . . . ⊗ ϕ ∗ ( a n ) and ϕ ∗ : C • ( A ⋊ G , A ⋊ G ) → C • ( A ⋊ H , A ⋊ H ) , F 7→  ( A ⋊ H ) ˆ ⊗ n ∋ a 1 ⊗ . . . ⊗ a n 7→ F  ϕ ∗ ( a 1 ) ⊗ . . . ⊗ ϕ ∗ ( a n )  ◦ ϕ  . Pr o of. W e only show the claim in the ho mo logy case. The cla im for Ho chschild cohomolog y can b e proved by a dual argument. As ab ov e let A = A ⋊ G and B = A ⋊ H . Cho ose a lo ca lly ﬁnite family ( ψ k ) k ∈ N ∈ C ∞ cpt ( H 1 ) of smo o th functions ψ k : H 1 → [0 , 1] with compact and co nnec ted supp ort on H 0 such that ( ψ 2 k ) is a pa rtition o f unity on H 0 . Put for m ∈ N : ψ [ m ] ,k := ( p P m l =0 ψ 2 l for k = 0 , ψ k + m for k ≥ 1 . (A.12) and u [ m ] ,k = k X l =0  ψ [ m ] ,l  2 . (A.13) Then ( u [ m ] ,k ) k ∈ N is an approximate unit in A ⋊ H for e very m ∈ N . Denote by K [ m ] the compact set  u − 1 [ m ] , 0 (1)  ◦ , i.e. the closur e of the s et o f all points x ∈ H 0 on a neighbo rho o d of which u [ m ] , 0 (and thus ψ [ m ] , 0 also) ha s v alue 1. Then for ev ery compact K ⊂ H 0 there exists a n m K ∈ N such that K ⊂ K ◦ [ m ] for all m ≥ m K . Hence one ha s ψ [ m ] , 0 | K = u [ m ] , 0 | K = 1 fo r a ll m ≥ m K . (A.14) Let K [ m ] = S i m i =0 K [ m ] ,i be the decomp osition of K [ m ] in connected components, i.e. each K [ m ] ,i is compact and connected and K [ m ] ,i 6 = K [ m ] ,i ′ for i 6 = i ′ . By the pro of o f Theor e m A.11 one can c o nstruct for ev er y ﬁxed m ∈ N a se c tion ˆ ν [ m ] : C ∞ cpt ( H 1 ) → C ∞ cpt ( t − 1 ( ϕ ( H 0 )) ˆ ⊗C ∞ cpt ( s − 1 ( ϕ ( H 0 )) and elements p ′ [ m ] , 0 ,j ∈ C ∞ cpt ( s − 1 ( ϕ ( H 0 )), q ′ [ m ] , 0 ,j ∈ C ∞ cpt ( t − 1 ( ϕ ( H 0 )), j = 0 , . . . , j m such that every p ′ [ m ] , 0 ,j and every q ′ [ m ] , 0 ,j has connected supp or t in ϕ ( H 0 ), such that ˆ ν [ m ] ( u [ m ] , 0 ) = j m X j =0 q ′ [ m ] , 0 ,j ⊗ p ′ [ m ] , 0 ,j (A.15) and ﬁnally such that  p ′ [ m ] , 0 ,j  | ϕ ( K [ m ] ,j ′ ) = ( 1 if j = j ′ , 0 else ,  q ′ [ m ] , 0 ,i  | ϕ ( K [ m ] ,j ′ ) = ( 1 if j = j ′ , 0 else . (A.16) Put p ′ [ m ] , 0 ,j = 0 and q ′ [ m ] , 0 ,j = 0 for j > j m and let X j ∈ N q ′ [ m ] ,k,j ⊗ p ′ [ m ] ,k,j := ˆ ν [ m ] ( u [ m ] ,k ) for k ≥ 1 . (A.17) ORBIF OLD C UP PRODUCTS 49 F or ev ery ﬁxed m ∈ N we no w ha ve the data to construct the quasi-isomorphism θ [ m ] : C • ( A ⋊ H , A ⋊ H ) → C • ( A ⋊ G , A ⋊ G ) as in the pro of of the preceding theorem. Note that the induced maps [ θ [ m ] ] b etw een the Ho chsc hild ho mology groups a ll coincide. Now let K ⊂ H 0 be c ompact a nd co nsider a 0 , a 1 , . . . , a n ∈ C ∞ cpt ( s − 1 ( K ) ∩ t − 1 ( K )) ⊂ A ⋊ H . Then one for m ≥ m k by construction: θ [ m ] ( a 0 ⊗ a 1 ⊗ . . . ⊗ a n ) = = X j 0 ,...,j n u ( p ′ [ m ] , 0 ,j 0 ⊗ a 0 q ′ [ m ] , 0 ,j 1 ) ⊗ u ( p ′ [ m ] , 0 ,j 1 ⊗ a 1 q ′ [ m ] , 0 ,j 2 ) ⊗ . . . . . . ⊗ u ( p ′ [ m ] , 0 ,j n ⊗ a n q ′ [ m ] , 0 ,j 0 ) = ϕ ∗ ( a 0 ) ⊗ ϕ ∗ ( a 1 ) ⊗ . . . ⊗ ϕ ∗ ( a n ) . Since A ⋊ H is the inductiv e limit of the s ubspaces C ∞ cpt ( s − 1 ( K ) ∩ t − 1 ( K )) the claim follows.  References [ AdLeR u] Adem, A., J. Leida, and Y. Ruan: Orbifolds and stringy top olo gy , bo ok to appear. 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