An $L_infty$ algebra structure on polyvector fields

An L ∞ algebra structure on pol yvector fields Boris Shoikhet Abstract It is well-kno wn that the Kontsevich formality [K97] for Ho chschild co chains of the po lynomial algebra A = S ( V ∗ ) fails if the vector space V is inﬁnite-dimensional. In the present pap er, we study the corres p onding obstr uctions. W e construct an L ∞ structure on po lyv ector ﬁelds on V ha ving the even degree T aylor comp onen ts. The degree 2 comp onent is given by the Sc ho uten-Nijenh uis brack et, but all its hig her even deg ree comp onen ts are no n- zero. W e p rov e that this L ∞ algebra is quasi-isomor phic to the co rresp onding Ho chschild co c hain complex. W e prov e that o ur L ∞ algebra is L ∞ quasi-isomo rphic to the Lie alg ebra of p olyvector ﬁelds on V with the Schouten-Nijenh uis bra c k et, if V is ﬁnite-dimens io nal. Intr oduction 0.1 The formalit y th eo rem of Maxim K on tsevic h [K97] is one of th e most imp ortan t breakthroughs in the deformation theory of algebraic stru ctures. It sa ys that the dg Lie algebra of Ho c hschild co c hains (with the Gerstenhab er Lie br ac k et) is q u asi-iso morphic, as a dg Lie algebra, to its cohomology , for an arbitrary regular commutativ e algebra of ﬁ nite typ e o v er C . Th e statemen t w as ﬁ rstly p ro ven for the p olynomial algebra A = S ( V ∗ ), for a ﬁnite-dimensional v ect or sp ac e V o ver C , by making use a kind of F eynmann diagram expans ion in top ological quantum ﬁeld theory . Later on, another pro of was found b y Dmitri T amarkin [T], using some op eradic meth- o ds. F or b oth pro ofs, the assumption that V is ﬁnite-dimensional , is essen tial . 1 In App endix A, we pro vide a pro of of the general statemen t of the f ail ure of the Kon tsevic h formalit y for an inﬁnite-dimensional V . 2 This pap er grew up f r om the author’s attempt to single out the obs tructions to th e K on t- sevic h formalit y in the inﬁ nite-dimensional case. 1 See Section 1.6 b elo w for an explanation why th e assumption d im V < ∞ is essentia l for Kontsevich’s pro of, and App endix B for the failure of T amarkin’s pro of when dim V = ∞ . 2 In general, by the Kontsevic h formalit y for S ( V ∗ ) we alwa y s mean gl ( V )-equiva rian t formalit y . Note that our pro of in A ppend ix A needs only a weak er equiv ariance. 1 Recall that the K ontsevic h form alit y (for the case of p olynomial algebra S ( V ∗ )) is equ iv alen t to existence of an L ∞ quasi-isomorphism L : T poly ( V ) → Ho c h q ( S ( V ∗ )) When V is inﬁnite-dimensional, b oth sides can b e deﬁned in an appropriate wa y (see Section 1.1-1.3, they are d enote d by T ﬁn ( V ) and Ho c h q ﬁn ( S ( V ∗ )), corresp ondingly). W e sho w that, as complexes, the l.h.s. and the r.h.s. are quasi-isomorphic, in T heorem 1. 6 (that is, “the Ho c hschild-Kostan t-Rosenberg theorem holds”). Ho w ev er, any L ∞ quasi-isomorphisms fails to exist . In this pap er, w e construct a new L ∞ structure on T L ﬁn ( V ) whose un derlying graded v ector space is T ﬁn ( V ), called “the exotic L ∞ structure”. Its T aylo r comp onen ts L 2 , L 4 , L 6 , . . . are all nonzero, with L 2 equal to the Sc houten-Nijenh uis b r ac k et (the o dd degree comp on ents v anish b y a sym metry r easo n). Our Main Theorem 2. 9 states th at there is an L ∞ quasi-isomorphism L ﬁn : T L ﬁn ( V ) → Ho c h q ﬁn ( S ( V ∗ )) regardless of is V ﬁnite- or inﬁnite-dimensional. Then the Kont sevic h formalit y theorem implies that, for a ﬁn ite -dimensional V , the L ∞ algebras T poly ( V ) (with the S c houten-Nijenhuis brac k et and v an ish ing higher T a ylor comp onen ts) and T L ﬁn ( V ) are L ∞ quasi-isomorphic. 0.2 The p revious arc hiv e v ersion(s) of this pap er d at e(s) bac k to 2008. Recen tly , some other pap ers making use and furth er dev eloping the id ea s and results of this pap er h av e app eared, see e.g. [KMW], [MW1], [MW2]. T he author b eliev es that the constructions in tro duced in this pap er ma y ﬁnd m ore fr uitful applications in n ear future. This 2017 v ersion is essen tially impr o ve d and expanded . Th e most imp ortant change s are: (i) Section 1 is mostly r ewritten with sev eral new pro ofs and examples; (ii) we su pplied th e pap er with t w o App endices, A and B. App endix A provides a general pro of of the failure on the Kon tsevic h formalit y for S ( V ∗ ) for an inﬁnite-dimensional V , and App endix B sho ws why the T amarkin pro of [T] do es not w ork in the inﬁ n ite- dimensional case. As well , w e corrected English (which hop efully has impr o ve d s ince 2008). A cknowledgements I am indebted to Boris F eigin who shared with me, around ’98-’99, some his conjecture on inﬁnite-dimensional Duﬂo formula, w hic h stimulate d my work on inﬁ nite-dimensional formalit y in general. 2 Maxim Kontsevic h found a mistak e in the pr oof of Lemma 2.2.4 of the ﬁrst archiv e version, whic h failed Th eorem 2.2.4 therein. The p roblem had b een ﬁxed ﬁxed in a later arc hiv e version, b y an interpretation of our form er integ rals as the T a ylor comp onen ts of a new L ∞ structure on p olyv ector ﬁelds, s ee Section 2. As w ell, the pro of of Lemma A. 4 in App endix A was comm unicated to the author by Maxim Kont sevic h. I am thankfu l to Maxim f or his in terest in m y work an d for th e fr uitful corresp ondence. I am thankful to P a v el Etingof for h is interest and for his man y suggestions. The author is thankful to the anon ymous r eferee for his careful reading of the pap er, and for his r emarks and suggestions w hic h help ed to im p ro ve the exp osition. The biggest part of the pap er w as completed during the author’s 5-y ear app ointmen t at the Univ ersit y of Luxemb ourg, wh en he w as a mem b er of the researc h group of Prof. Martin Sc hlic h enmaier. I am thankful to Martin S c hilec henmaier for h is kindness and his p ers onal participation, as w ell as for v ery nice working atmosphere, which made this pap er p ossible to come int o existence. The w ork w as partially su p p orted by the researc h gran t R1F105L15 of the Un iv ersity of Luxem b ourg and by the researc h gran t nr. 65 25 “Kredieten aan Na vo rsers” of Flemish Researc h F oundation (FW O). 1 The set-up 1.1 It is k n o wn that the K on tsevic h f ormalit y theorem [K97] for the Ho c hsc hild co c hains of the p olynomial alge bra A = S ( V ∗ ) fails, wh en the dimension of V is inﬁnite (we sh o w in Section 1.6 that the original pro of in lo c.cit. f ails, and in App endix B that the p roof of T amarkin [T] fails as w ell; the f ail ure in general is pr o ve n in Ap p endix A). 3 In this pap er, we formulate and pro v e a statemen t closely related to the K ontsevic h f ormalit y of Ho c h s c hild coc hain on S ( V ∗ ), whic h holds f or an inﬁnite-dimensional V as w ell. In this pap er, we r estrict ourselv es by considering the invinite- dimensional ve ctor sp ace s V o v er C , whic h are graded: V = ⊕ i ≥ 0 V i , dim V i < ∞ (1.1) and such that all graded comp onen ts V i are ﬁnite-dimensional . W e assume the v ect or sp ac e V to ha v e the homological degree 0. The degree i of all elemen ts in V i is called the auxiliary de gr e e . F or such a v ector sp ace V , w e deﬁn e the algebra of p olynomial functions on V as S ( V ∗ ) := S ( ⊕ i ≥ 0 V ∗ i ) (1.2) 3 Here w e mean t h e failure of the gl ( V )-equiv ariant formalit y . As w ell, we mean the formalit y of the Hochshild complex Ho c h ﬁn ( S ( V ∗ )), introdu ced in Section 1.3 b elo w, as a dg Lie algebra. Note that the pro of in A ppendix A requires only a wea ker equiv ariance. 3 The algebra S ( V ∗ ) inh erits fr om V the grading, with d eg V ∗ a = − a . W e start w ith d eﬁning suitable v ersions of p olyv ector ﬁ elds on V and of the cohomologi cal Ho c hschild complex of S ( V ∗ ). Then w e pr o ve in our setting a direct analog of th e Ho c hsc h ild- Kostan t-Rosen b erg theorem. 1.2 The pol yvector fields T ﬁn ( V ) W e deﬁne a su itable analogue of the L ie algebra of p olyv ect or ﬁ elds T poly ( V ), whic h w e denote T ﬁn ( V ) . W e wa n t to allo w inﬁnite sums of monomials. Here is the precise deﬁnition. A k -p olyv ecto r ﬁeld of the gradin g (ak a the auxiliary degree) i is an element in T ﬁn ( V ) k ,i = { γ ∈ Y a 1 ,...,a k ≥ 0  S ( V ∗ ) ⊗ Λ a 1 ,...,a k V  | deg aux γ = i } (1.3) where Λ a 1 ,...,a k V = V a 1 ∧ V a 2 ∧ · · · ∧ V a k Note that Λ a 1 ,...,a k V is ﬁ nite-dimensional, b ecause all the graded comp onen ts V i are. The grading d eg aux is d eﬁ ned ju st b elow. Note that an y element in S ( V ∗ ) is a ﬁnite sum, by the deﬁn ition (1.6). The grading deg aux is d eﬁ ned for f ⊗ λ ∈ S ( V ∗ ) ⊗ Λ a 1 ,...,a k V as deg aux ( f ⊗ λ ) = deg f + a 1 + · · · + a k (note that th e grading of an arbitary elemen t in Λ a 1 ,...,a k V is a 1 + · · · + a k , and the grading of an arbitary elemen t in V ∗ a = − a ). Then we d eﬁne T k ﬁn ( V ) = M i T k ,i ﬁn ( V ) (1.4) W e d enote T q ﬁn ( V ) = M k T k ﬁn ( V )[ − k ] (1.5) Example 1. 1. Let g = ⊕ i g i b e a graded Lie algebra, w ith all comp onent s g i ﬁnite-dimensional (but g may b e inﬁnite-dimensional). Then the Kostan t-Kirillo v biv ector of g is an elemen t of T 2 ﬁn ( g ∗ ). Here g ∗ := ⊕ i g ∗ i . Th is Kostan t-Kirillo v bive ctor is a pro duct of monomials, eac h of whic h has the grading (the auxilary degree) 0. Recall the Sc h outen-Nijenh u is br ac k et of p olyv ector ﬁ elds on a (smo oth manifold, smooth algebraic v ariet y) M . It is a graded Lie br ac k et on T poly ( M )[ 1]. T he classical counte rpart of the Sc houten-Nijenh uis brack et w e consider here is the one on the p olynomial p olyv ector ﬁelds on 4 a ﬁn ite-dimensional v ect or space W , T poly ( W ). F or γ 1 , γ 2 ∈ T poly ( W ), the Sc houten-Nijenhuis brac k et is [ γ 1 , γ 2 ] = γ 1 ◦ γ 2 − ( − 1) ( p − 1)( q − 1) γ 2 ◦ γ 1 (1.6) (where γ 1 ∈ T p poly ( W ), γ 2 ∈ T q poly ( W )). The op eration γ 1 ◦ γ 2 is deﬁned as follo ws. Let γ 1 ∈ S a ( W ∗ ) ⊗ Λ p ( W ) , γ 2 ∈ S b ( W ∗ ) ⊗ Λ q ( W ). Then the op eration γ 1 ◦ γ 2 is an op eration − ◦ − :  S a ( W ∗ ) ⊗ Λ p ( W )  ⊗  S b ( W ∗ ) ⊗ Λ q ( W )  → S a + b − 1 ( W ∗ ) ⊗ Λ p + q − 1 W There is the canonical gl ( W )-in v arian t e ∗ ∈ ( W ⊗ W ∗ ) ∗ . Within the canonical isomorp hism W ⊗ W ∗ ∼ → En d( W ), the op erator e ∗ is corresp onded to the trace op erator T r : End( W ) → C . The assump tio n that dim W < ∞ is crucial for e ∗ to exist. The op erator e ∗ can b e extended in a un ique w a y to an gl ( W )-in v ariant op erator e ∗ pb : S b ( W ∗ ) ⊗ Λ p ( W ) → S b − 1 ( W ∗ ) ⊗ Λ p − 1 W No w γ 1 ◦ γ 2 is deﬁned in tw o steps. At the ﬁr st step, w e jus t take the tensor pr odu ct of all four factors S a ( W ∗ ) ⊗ Λ p ( W ) ⊗ S b ( W ∗ ) | {z } e ∗ pb ⊗ Λ q ( W ) → S a ( W ∗ ) ⊗  Λ p − 1 W ⊗ S b − 1 ( W ∗ )  ⊗ Λ q W and app ly the op erator e ∗ pb to the tw o f ac tors in the middle. After that, we tak e the pro duct of all factors, to consider it as an elemen t of S q ( W ∗ ) ⊗ Λ q ( W ): γ 1 ◦ γ 2 = ( f 1 ⊗ ℓ 1 ) ◦ ( f 2 ⊗ ℓ 2 ) = f 1 · e ∗ pb ( ℓ 1 , f 2 ) · ℓ 2 (1.7) where f i and ℓ i are the factors in S q ( W ∗ ) and in Λ q ( W ), corresp ondingly . This deﬁ n itio n of γ 1 ◦ γ 2 is not v alid wh en W is an inﬁnite-dimensional v ector space. The reason is that the elemen tary inv ariant e ∗ : W ⊗ W ∗ → C is not deﬁn ed wh en dim W = ∞ . Ho w ever, f or th e graded space T q ﬁn ( V ) the op eration γ 1 ◦ γ 2 is still w ell-deﬁned, d espite the corresp onding m ap T r : V ∗ ⊗ V → C do es not exist. Indeed, it follo ws from (1.3) that the co eﬃcie n t at eac h exterior algebra comp onen t Λ a 1 ,...,a k V is a (ﬁnite) p olynomial in S ( V ∗ ). Let γ 1 ∈ Y a 1 ,...,a k S ( V ∗ ) ⊗ Λ a 1 ,...,a k V γ a 1 ,...,a k 1 , γ 2 ∈ Y b 1 ,...,b ℓ S ( V ∗ ) ⊗ Λ b 1 ,...,b ℓ V γ b 1 ,...,b ℓ 2 (1.8) Denote by γ a 1 ,...,a k 1 , γ b 1 ,...,b ℓ 2 the corresp ondin g comp onen ts of γ 1 and γ 2 , corresp ondin gly . W e claim that the follo wing d eﬁnition mak es sense: γ 1 ◦ γ 2 = Y a 1 ,...,a k b 1 ,...,b ℓ γ a 1 ,...,a k 1 ◦ γ b 1 ,...,b ℓ 2 (1.9) 5 That is, w e claim that the coeﬃcient at any ﬁxe d comp onen t Λ c 1 ,...,c k + ℓ − 1 V is a ﬁnite sum of p olynomials. Assume that γ a 1 ,...,a k 1 ◦ γ b 1 ,...,b ℓ 2 con tributes to S ( V ∗ ) ⊗ Λ c 1 ,...,c k + ℓ − 1 . Then one has { b 1 , . . . , b ℓ } ⊂ { c 1 , . . . , c k + ℓ − 1 } , th e remaining p − 1 of { c 1 , . . . , c k + ℓ − 1 } are in { a 1 , . . . , a k } . That is, (for any particular su mmand) there is exactly 1 index i su c h that a i 6∈ { c 1 , . . . , c k + ℓ − 1 } . As the co eﬃcien t within Λ b 1 ,...,b ℓ V is a p olinomial f b 1 ,...,b ℓ ∈ S ( V ∗ ), w e get a cond ition on { a 1 , . . . , a k } : one has ∂ ∂ x a j f b 1 ,...,b ℓ 6 = 0 for some j and x a j ∈ V ∗ a j (1.10) This condition depicts a ﬁ nite set of p ossiblities for a j , for ﬁxed { b 1 , . . . , b ℓ } and { c 1 , . . . , c k + ℓ − 1 } . As we ll, there is a ﬁnite num b er of c hoices of { b 1 , . . . , b ℓ } ⊂ { c 1 , . . . , c k + ℓ − 1 } (for a ﬁxed { c 1 , . . . , c k + ℓ − 1 } ). It pr ov es that th e co eﬃcien ts in (1.9) are ﬁnite sum s, w hat m ak es sense of (1.9). Then w e deﬁn e the Schouten-Nijenh u is brac ket by (1.6). One easily sees that in this w a y one gets a graded L ie brack et on T q ﬁn ( V )[1]. W e ha v e: Lemma 1. 2. L et V b e as in (1.1) , and let T q ﬁn ( V ) b e as ab ove. Then the c onstruction of the cir cle op er ation γ 1 ◦ γ 2 , γ 1 , γ 2 ∈ T q ﬁn ( V ) deﬁnes, via (1.6) , a gr ade d Lie br acket on T ﬁn ( V )[1] . The graded Lie br ac k et on T q ﬁn ( V )[1], giv en in Lemma 1. 2 , is called the Schouten-Nijenhuis Lie br acket . Example 1. 3. Let us consider ve ctor ﬁelds v 1 , v 2 ∈ T q ﬁn ( V ), moreo v er, line ar vect or ﬁelds. Th at is, v 1 , v 2 ∈ Y a ( ⊕ b V ∗ b ) ⊗ V a In this case, we r ec o v er the pro duct of the generalized J ac obian matrices. Remark 1. 4. Note that all results of this S ubsection w ould remain tru e if w e deﬁn ed T q ﬁn ( V ) as ∼ T q ﬁn ( V ) = Y a 1 ,...,a ℓ S ( V ∗ ) ⊗ Λ a 1 ,...,a ℓ V (1.11) th us dropp ing the assump tio n on the auxilary degree in (1.3) and (1.5). W e adopt our previous deﬁnition, by the follo wing reason. Later on, w e deﬁn e the corr esp ond ing “ﬁnite” version of the Ho c hschild cohomologica l complex Ho ch q ﬁn ( S ( V ∗ )), and pro v e the analog ue of the Hoc hsc hild- Kostan t-Rosen b erg theorem. Th at is, one h as H q (Ho ch q ﬁn ( S ( V ∗ ))) = T q ﬁn ( V ) W e d o not kn o w any deﬁnition of ∼ Ho c h q ﬁn ( S ( V ∗ )) such that one h ad H q ( ∼ Ho c h q ﬁn ( S ( V ∗ ))) = ∼ T q ﬁn ( V ) 6 1.3 The Hochschild complex Ho ch q ﬁn ( S ( V ∗ )) Here w e deﬁn e a su ita ble version of the Ho c hsc hild cohomologica l complex of the algebra S ( V ∗ ) = S ( ⊕ i V ∗ i ). W e wan t its cohomology to b e equal to T q ﬁn ( V ). Deﬁne Ho c h k ﬁn ( S ( V ∗ )) = { Ψ ∈ Hom( S ( V ∗ ) ⊗ k , S ( V ∗ )) | ∃ N (Ψ) ∈ Z suc h that | deg Ψ( f 1 ⊗ · · · ⊗ f k ) − k X i =1 deg f k | ≤ N (Ψ) for all homogeneous f 1 , . . . , f k } (1.12) In other words, a co c hain from Ho c h k ﬁn ( S ( V ∗ )) preserves th e d egree , up to a ﬁ nite in tegral n um b er, dep ending on the co c hain. One easily sees that Ho c h k ﬁn ( S ( V ∗ )) is a v ector sp ace . It inh erits th e grading, so that Ho c h k ﬁn ( S ( V ∗ )) = ⊕ i Ho c h k ,i ﬁn ( S ( V ∗ )) where Ho c h k ,i ( S ( V ∗ )) = { Ψ ∈ Ho c h k ﬁn ( S ( V ∗ )) | deg Ψ( f 1 ⊗ · · · ⊗ f k ) = k X i =1 deg f k + i f or all homogeneous f 1 , . . . , f k } F or any ﬁxed i , Ho c h q ,i ﬁn ( S ( V ∗ )) = M k Ho c h k ,i ﬁn ( S ( V ∗ ))[ − k ] b ecomes a complex w ith the Ho c hschild diﬀerentia l. Therefore, Ho c h q ﬁn ( S ( V ∗ )) = M k ,i Ho c h k ,i ﬁn ( S ( V ∗ ))[ − k ] is a complex. One easily sees that Ho c h q ﬁn ( S ( V ∗ ))[1] is a dg L ie alg ebra, with the Gerstenh aber Lie brac k et. The cohomology of Ho c h q ﬁn ( S ( V ∗ )) can b e in terpreted as a der ived functor, as follo w s . Denote by A the categ ory whose ob jects are gr ade d S ( V ∗ )-bimo dules, and whose morp hisms are the grading preserving map s of them. Then A is an abelian catego ry . F or an ob ject X ∈ O b( A ) , denote by X h j i the ob ject of A whose inn er grading is shifted by j : ( X h j i ) i = X j + i . W e h a ve th e follo win g lemma: Lemma 1. 5. The c oho molo gy H ℓ (Ho ch q ﬁn ( S ( V ∗ ))) is e q ual to the ve ctor sp ac e H ℓ (Ho ch q ﬁn , tot ( S ( V ∗ ))) = Ext ℓ A ( S ( V ∗ ) , ⊕ j ∈ Z S ( V ∗ ) h j i ) 7 Pr o of. Th e bar-resolution of S ( V ∗ ) is clea rly a pro jective resolution in A of the tautologic al bimo dule S ( V ∗ ). W e compute th e E xt’s functors b y making use of this resolution. Th e complex Hom A (Bar q ( S ( V ∗ )) , ⊕ k ∈ Z S ( V ∗ ) h k i ) is exactly the complex Ho c h q ﬁn ( S ( V ∗ )). 1.4 The Hochschild-K ost ant-Rosenber g theore m Deﬁne the Ho c hschild-Kostan t-Rosenberg map ϕ H K R : T q ﬁn ( V ) → Ho c h q ﬁn ( V ) as ϕ H K R ( γ ) = 1 k !  f 1 ⊗ · · · ⊗ f k 7→ γ ( d f 1 ∧ · · · ∧ d f k )  (1.13) for γ ∈ T k ﬁn ( V ). W e h a ve : Theorem 1. 6. The map ϕ H K R : T q ﬁn ( V ) → Ho c h q ﬁn ( S ( V ∗ )) is a quasi-isomorp hism of c om- plexes. Pr o of. Con s ider the follo wing K oszul complex K q : . . . d − → K 3 d − → K 2 d − → K 1 d − → K 0 → 0 (1.14) where K k = M a 1 ,...,a k S ( V ∗ ⊕ V ∗ ) ⊗ Λ a 1 ,...,a k V ∗ (1.15) with the diﬀerent ial d (( ξ i 1 ∧ · · · ∧ ξ i k ) ⊗ f ) = k X j =1 ( − 1) j − 1 ( ξ i 1 ∧ · · · ∧ ˆ ξ i j ∧ · · · ∧ ξ i k ) ⊗ (( x j − y j ) f ) (1.16) where { x i } is a basis in V ∗ compatible with the decomp osition V = ⊕ i V i , { y i } is the same basis in the second cop y of V ∗ , and { ξ i } is the corresp onded basis in V [1]. This Koszul complex is clearly a resolution of the tautological S ( V ∗ )-bimo dule S ( V ∗ ) b y free bimo dules. It is as w ell a resolution in the category A , b ecause the diﬀerentia l d preserves the auxilary grad in g. W e compute: Hom A  K ℓ , M j ∈ Z S ( V ∗ ) h j i  = Hom C  M a 1 ,...,a ℓ Λ a 1 ,...,a ℓ V ∗ , M j ∈ Z S ( V ∗ ) h j i  = Y a 1 ,...,a ℓ Hom C  Λ a 1 ,...,a ℓ V ∗ , M j ∈ Z S ( V ∗ ) h j i  = Y a 1 ,...,a ℓ M j [ S ( V ∗ ) ⊗ Λ a 1 ,...,a ℓ V ] j (1.17) 8 where the subscript j d enotes the elements of the auxilary grading j . In the last equalit y we essen tially use th at all Λ a 1 ,...,a ℓ V ∗ are ﬁn ite -dimensional. The rightmost term in (1.17) is exactly T ℓ ﬁn ( V ). One easily c h ec ks that the induced diﬀeren tial v anish es. It completes the computation of the cohomology H q (Ho ch q ﬁn ( S ( V ∗ ))). It r emains to note th at the image of the Ho c hsc hild-Kostan t-Rose n b erg map ϕ H K R coincides with the cohomology classes in Ho c h q ﬁn ( V ) pro duced by the Koszul r esolution. 1.5 The pol ydifferential opera tors, as socia ted with grap hs Here we r eca ll the construction f rom [K97], assigning a Ho c hsc hild co c hain of the algebra S ( W ∗ ), to several p olyv ector ﬁelds on W , and to a com binatorial datum giv en by a Kontsevich admissible gr aph . W e r efer the reader to [K97, Section 6.1] for more d eta il. Throughout this Sub secti on, W denotes a ﬁnite-dimensional vecto r space o v er C . The goal is to constr u ct th e most general Hochsc h ild co c hains, asso ciated with an ordered sequence of p olyvec tor ﬁelds γ 1 , . . . , γ n on W , suc h that th e construction is gl ( W )-equiv ariant. The idea is to subsequentl y apply the elemen tary inv ariant op erato r e ∗ : ( W ⊗ W ∗ ) → C , asso ciated to all edges of the graph. In fact, th e construction is a generalization of the construction of γ 1 ◦ γ 2 (see S ec tion 1.2), whic h is corresp onded to the graph with t w o ve rtices “of the ﬁrst t yp e” (see b elo w ), and a single orien ted edge, see Figure 1. e * Figure 1: T o eac h oriente d edge, is asso ciated th e elemen tary in v arian t e ∗ ∈ ( W ⊗ W ∗ ) ∗ ∼ W ∗ ⊗ W F or a general graph Γ , we use th e notation V Γ for the set of vertice s of Γ, and E Γ for the set of its edges. Definition 1. 7. A Kon tsevic h admissib le graph Γ is an oriented graph with t w o t yp es of lab elled ve rtices, the ve rtices of the ﬁrs t t yp e, lab elled { 1 , . . . , n } , and the v ertices of the second t yp e, lab elled { ¯ 1 , . . . , ¯ m } , such that (i) ♯E Γ = 2 n + m − 2 ≥ 0, V Γ = { 1 , . . . , n } ⊔ { ¯ 1 , . . . , ¯ m } , (ii) every edge ( v 1 , v 2 ) ∈ E Γ starts at a verte x of the ﬁrst t yp e, v 1 ∈ { 1 , . . . , n } , (iii) there are no simple lo ops (ak a tadp oles), that is, edges of the form ( v , v ), Remark 1. 8. Our deﬁnition is sligh tly diﬀerent fr om the original one, as w e do not consider an ordering of all sets Star( v ), v a ve rtex of the ﬁr st typ e, as a part of the data. The reason is 9 that, as so on as the sets of all vertice s of the ﬁrst t yp e and of all v ertices of the second t yp e are ordered, there is an in duced ordering of th e edges. F or a Kontsevic h admissible graph Γ, we denote b y V I Γ the set of vertice s of th e ﬁrst type, and by V I I Γ the set of ve rtices of the second type, so that V Γ = V I Γ ⊔ V I I Γ . Let us recall the constru ctio n, asso ciating to a Kon tsevic h admissible graph Γ with n v ertices of the ﬁrst t yp e and m vertic es of the second typ e, n h omog eneous p olyve ctor ﬁelds γ 1 , . . . , γ n on W , and m f unctions f 1 , . . . , f m ∈ S ( W ∗ ), a p olyve ctor ﬁeld U Γ ( γ 1 , . . . , γ n ; f 1 , . . . , f m ) of the (cohomolog ical) degree deg U Γ ( γ 1 , . . . , γ n ; f 1 , . . . , f m ) = n X i =1 deg γ i − 2 n − m + 2 (1.18) When n X i =1 deg γ i = 2 n + m − 2 (1.19) the p olyv ecto r ﬁeld U Γ ( γ 1 , . . . , γ n ; f 1 , . . . , f m ) has (cohomologica l) degree 0, that is, is a fun c- tion. I n this case, we get a Ho chschild c o chain U Γ ( γ 1 , . . . , γ n ) ∈ Hom C ( S ( W ∗ ) ⊗ m , S ( W ∗ )). A p olynomial p olyv ect or ﬁ eld on W is an elemen t of the graded commutativ e algebra T poly ( W ) = S ( W ∗ ⊕ W [ − 1]). Let Γ b e a Kon tsevic h admissible graph , with ♯V I Γ = n , ♯V I I Γ = m . Consider the asso ciat iv e graded commuta tiv e algebra A Γ = Y v ∈ V I Γ ( S ( W ∗ ⊕ W [ − 1])) v ⊗ Y v ∈ V I I Γ ( S ( W ∗ )) v (1.20) Although the construction of Γ U Γ do es n ot dep end on the c hoice of b asis in W , w e c ho ose one a wr ite the construction “in co ordinates”, as it mak es it more r eadable. Let { x i } , i = 1 , . . . , N = d im W b e a basis in W ∗ , let { ξ ′ i } , i = 1 , . . . , N b e the dual basis in W , and let { ξ i } , i = 1 , . . . , N b e the corresp ond ing basis in W [1]. One assigns to the elementa ry inv ariant e ∗ ∈ ( W ⊗ W ∗ ) ∗ the op erator e ∗ = N X i =1 ∂ ∂ x i ⊗ ∂ ∂ ξ i (1.21) whic h do es not dep end on the choic e of the basis { x i } . Let ( v 1 , v 2 ) b e an orient ed edge of Γ. W e assign to it the op erator acting in A Γ : e ∗ ( v 1 ,v 2 ) = N X i =1 ( ∂ ∂ x i ) v 2 ⊗ ( ∂ ∂ ξ i ) v 1 (1.22) 10 where the sub -indices v 1 and v 2 indicate the factors in A Γ on whic h the corresp ondin g op erators act. The op erators e ∗ ( v 1 ,v 2 ) , acti ng on A Γ , comm u te up to a sign, for diﬀerent edges. The lab ellings whic h is a p art of the deﬁn itio n of an admissib le graph, ﬁx an ord er on all edges. In the form ula b elo w this order is assumed: D Γ = Y ( v 1 ,v 2 ) ∈ E Γ e ∗ ( v 1 ,v 2 ) : A Γ → A Γ (1.23) T ak e homogeneous p olyv ector ﬁelds γ 1 , . . . , γ n , and fun cti ons f 1 , . . . , f m . The ordering of the sets V I Γ and V I I Γ ﬁxes an ordering of them. In the form ula b elo w we assu me th is ordering: U Γ ( γ 1 , . . . , γ n )( f 1 , . . . , f m ) = ∆ ∗  D Γ ( γ 1 ⊗ · · · ⊗ γ n ⊗ f 1 ⊗ · · · ⊗ f m )  (1.24) where ∆ ∗ ( − ) is the “restriction of the function to the diagonal” , whic h is, b y deﬁ nition, the pro duct of the comp onents: ∆ ∗ : A Γ → S ( W ∗ ⊕ W [ − 1]) (1.25) In general, U Γ ( γ 1 , . . . , γ n )( f 1 , . . . , f m ) ∈ T poly ( W ) is a p olyvec tor ﬁeld. When (1.19 ) holds, it is a fun ction. In this case, U Γ ( γ 1 , . . . , γ n ) is a Ho c h sc h ild co c hain of S ( W ∗ ). 1.6 To w ards the formality f or an infinite- dimensional sp ace Here we explain why the Kontsevic h solution of the formalit y theorem fails for the the algebra of p olynomials on an inﬁnite-dimensional v ector space S ( V ∗ ), in th e sense of S ecti on 1.1. One of the main parts of th e construction in [K97] is an assignment asso ciating to eac h K on tsevic h admissible graph Γ, and n = ♯V I Γ of homogeneous p olyv ector ﬁ elds γ 1 , . . . , γ n on W , with (1.19), the Ho c hsc hild co c hain U Γ ( γ 1 , . . . , γ n ) ∈ Ho c h q ( S ( W ∗ )), see S ect ion 1.5 . Consider the inﬁnite-dimensional framew ork of Sections 1.1, 1.2. W e claim that, f or a graph Γ con taining a f ragmen t wh ic h is an oriente d cycle b etwe e n the vertic es of the ﬁrst typ e , the corresp onding p olydiﬀerentia l op erator U Γ is in general ill-deﬁned. An inf ormal explanation is that the p olydiﬀerentia l op erator corresp ondin g to an orien ted cycle “looks lik e a trace op erator on an inﬁ n ite- dimensional ve ctor sp ace ”, which is ill-deﬁn ed wh en dim V = ∞ . See Example 1. 10 b elo w, for an exp licit computation. In fact, the graphs with orien ted cycles b et ween the vertic es of the ﬁrs t t yp e are th e only “bad” graphs : Lemma 1. 9. L et Γ b e a Kontsevich admissible gr aph in the sense of [K97] with n vertic es of the ﬁrst typ e and m v e rtic es of the se c ond typ e. S upp ose that the gr ap h Γ do e s not c ontain any oriente d cycle b e twe en the ve rtic es of the ﬁrst typ e, as its sub- gr aph. L et V b e as ab ove, and let γ 1 , . . . , γ n ∈ T ﬁn ( V ) . Then the Kontsevich p olydiﬀer ential op er ator U Γ ( γ 1 ∧ · · · ∧ γ n ) is wel l-deﬁne d as an element of Ho c h q ﬁn ( S ( V ∗ )) . 11 Pr o of of L e mma: In our deﬁnitions (1.1 ) and (1.6 ), an elemen t of S ( V ∗ ) is a ﬁnite sum . In con trast, an elemen t of T ﬁn ( V ) is an inﬁnite pr o duct , see (1.3). Let Γ b e a Kontsevic h admissib le graph w ith oriente d cycles b et w een the v ertices of the ﬁr st t yp e. Then there is a ve rtex of the ﬁrst t yp e, sa y v 0 , suc h that all edges outgoing from v 0 ha v e as their targets vertic es of the second t yp e. Let γ 1 , . . . , γ n and f 1 , . . . , f m b e ﬁxed. Consider all op erators e ∗ t = ( ∂ ∂ ξ i ) v 0 ⊗ ( ∂ ∂ x i ) v t , as- so ciat ed with all edges t outgoing fr om v 0 . As the f unctions f 1 , . . . , f m are ﬁn ite s ums, th e op erator Y t =( v 0 ,v t ) e ∗ t ( γ v 0 ⊗ f 1 ⊗ · · · ⊗ f m ) is n on -zero only f or a ﬁn itely many comp onen ts γ ( a 1 ,...,a ℓ ) v 0 ∈ S ( V ∗ ) ⊗ Λ ( a 1 ,...,a ℓ ) V . Namelly , the n um b ers a 1 , . . . , a n should ob ey the condition: ∂ ∂ x a i ( f 1 · · · · · f m ) 6 = 0 f or i = 1 , . . . , ℓ (1.26) F or eac h suc h comp onen t, denote g ( γ v 0 ) ( a 1 ,...,a ℓ ) = γ v 0 ( dx a 1 ∧ · · · ∧ dx a ℓ ) Then we remov e the v ertex v 0 from Γ , as w ell as all in coming to and outgoing from v 0 edges. Denote th e obtained graph by Γ (1) . The graph Γ (1) fulﬁls the assump tions of the Lemma as w ell. W e ﬁnd a vertex v 1 of the ﬁ rst type of Γ (1) suc h that an y edge outgoi ng from v 1 targets at a v ertex of the second t yp e. Th e comp onents γ ( b 1 ,...,b s ) v 1 ∈ S ( V ∗ ) ⊗ Λ ( b 1 ,...,b s ) whic h contribute to U Γ ( γ 1 , . . . , γ n )( f 1 , . . . , f m ) by a non-zero s u mmand are those for whic h ∂ ∂ x b i ( g ( γ v 0 ) ( a 1 ,...,a ℓ ) · f 1 · · · · · f m ) 6 = 0 f or all i = 1 . . . s and for some ( a 1 , . . . , a ℓ ) ob eying (1.26) (1.27) The set of all p ossible ( a 1 , . . . , a ℓ ) and ( b 1 , . . . , b s ) wh ic h con tribute by a non-zero su m mand is th us ﬁnite. Then we remo v e the v ertex v 1 from Γ (1) , with all its incoming and outgoing edges, and so on. After successiv e iteration of this pro cedure, w e get only v ertices of the second t yp e, and a ﬁnite num b er of summ an d s whic h do ﬁnally contribute to U Γ ( γ 1 , . . . , γ n )( f 1 , . . . , f m ). Example 1. 10. Here w e consider an example of a graph Γ with orien ted cycles b et w een the v ertices of the ﬁr st t yp e, and of an inﬁn ite-dimensional space V , su c h that the Ho c h sc hild co c hain U Γ ( γ 1 , . . . , γ n ) is ill-deﬁn ed for some γ 1 , . . . , γ n ∈ T ﬁn ( V ). Consider the graph with ♯V I Γ = 2 , ♯V I I Γ = 2 sho wn in Figure 2. (The lab elling is not essen tial for this Examp le). 12 t f g a b s Figure 2: A graph Γ with an orien ted cycle b etw een the type I v ertices, w hic h leads to dive rgen t op erator U Γ Consider the v ector space V as in Section 1.1 su c h that V = ⊕ a ≥ 0 V a , dim V a = 1 f or all a W e put in the v ertice s of the second t yp e linear functions, for simplicit y . Sa y , f = x a inV ∗ a , g = x b ∈ V ∗ b . W e put in th e v ertices of the ﬁrst t yp e (the same) quadr atic p olyv ector ﬁ eld α in T ﬁn ( V ), α = X 0 ≤ ia,t>b s + a − t ≥ 0 ,t + b − s ≥ 0 x s + a − t x t + b − s − X sb s + a − t ≥ 0 ,t + b − s ≥ 0 x s + a − t x t + b − s − X s>a,t 0 , c ∈ C } is a 3-dimensional group of tr ansformations. In particular, if E = 0 we r eco gnize the Kontsevic h’s conﬁ guratio n space C n from [K97]. Dimension d im C n, Γ is equal to 2 n − 3 and do es not dep end on Γ. No w supp ose that ♯E (Γ) = 2 n − 3. If e = − − → z i z j is an edge of Γ, we asso ciate with it the diﬀeren tial 1-form φ e = d Arg ( z j − z i ) = 1 2 i d Log z j − z i z j − z i (2.3) A lab elling of the v ertices of Γ yields a lab elling of th e edges. T o deﬁ n e it, one ﬁr stly tak es the edges outgoing f rom the v ertex 1, in the order ﬁxed b y the lab elling of their targets, then we tak e the edges outgoing f r om th e v ertex 2, and so on. Deﬁne W Γ = 1 π 2 n − 3 Z C n, Γ ^ e ∈ E (Γ) φ e (2.4) Here in the wedge pro duct we use the order of th e 1-forms φ e corresp onded to th e ordering of the edges of Γ, d escrib ed just ab o v e. W e ﬁ rstly sho w that W Γ = 0 f or an o dd n . Lemma 2. 2. F or any admissible gr aph with an o dd numb er n of vertic es, the inte gr al W Γ = 0 . 15 Pr o of. Map a vertex p to the p oin t 0 + 0 i , using the actio n of the group G 3 , and let another p oin t q m o ve along the u n it circle around p . (These t w o are the only vertices with the restricted degrees of f reedom, in this w a y w e get r id of the action of G 3 ). Dra w the v ertical line ℓ th rough p and consider the symmetry σ with resp ect to ℓ . One has th e follo wing general formula: Z c σ ∗ ω = Z σ ∗ c ω (2.5) where c is an oriente d chain. In our case c = C n, Γ and ω = V e ∈ E (Γ) φ e . W e ha v e: σ ∗ φ e = − φ e (there are 2 n − 3 edges), and at eac h “mo v able” p oin t (there are n − 1 of suc h p oin ts) the orienta tion c hanges to the opp osite. Finally , if I is the in tegral, we hav e from (2.5): ( − 1) 2 n − 3 I = ( − 1) n − 1 I whic h imp lies that I = 0 for an o dd n . It is clear that when Γ is the graph with tw o v ertices and a single edge, W Γ = 1. F or an admissible graph Γ with n v ertices to ha v e a non-zero in tegral W Γ , it should ha ve 2 n − 3 = dim C n, Γ edges. C onsider several examples. Example 2. 3. Consider the graph Γ sho wn in Figure 4. 4 Let u s compute W Γ for this graph. 1 2 3 4 Figure 4: An adm issible graph Γ w ith n = 4 and nonzero W Γ Fix the ve rtex 2 to th e p oin t 0 + 0 · i by the action of group G 3 , and let the vertex 3 mo v e along the un it lo w er h alf-ci rcle around 2. Let x b e the angle of th e arr o w − − − → (2 , 3), − π ≤ x ≤ 0. Th en 4 An example similar to this one was considered by M.Kontsevic h, in his email to the auth or regarding th e ﬁrst archiv e versio n of the pap er 16 w e can integrat e o v er p ositions of the v ertice s 1 an d 4 sep arat ely . Let us ﬁrs tly integ rate (for a ﬁxed x ) ov er 1, denote it z . W e n eed to compute th e integ ral I 1 = Z Im z ≥ 0 d Arg( − z ) ∧ d Arg( − z + exp( ix )) (2.6) W e u se the Stok es formula: I 1 = Z ∂ (Im z ≥ 0) Arg( − z ) d Arg ( − z + exp( ix )) (2.7) The Stok es formula is applied to th e d omain D = { Im z ≥ 0 , ε ≤ | z | ≤ R } where ε → 0 and R → ∞ . T here are t w o b oun dary strata whic h cont ribute to the int egral: I 1 = − lim ε → 0 Z | z | = ε, Im z ≥ 0 Arg( − z ) d Arg ( − z +exp( ix ))+ lim R →∞ Z | z | = R, Im z ≥ 0 Arg( − z ) d Arg ( − z +exp( ix )) (2.8) The ﬁrs t in tegral in the limit ε → 0 is equal to π x , and the second one in the limit R → ∞ is equal to 1 2 π 2 . Th e total answe r is I 1 = − π x + 1 2 π 2 (2.9) Analogously w e compute the integral I 2 o v er p osition of the verte x 4. W e get the same answe r: I 2 = − π x + 1 2 π 2 (2.10) Finally , W Γ = 1 π 5 Z 0 − π I 1 ( x ) I 2 ( x ) dx = 13 12 (2.11) Example 2. 4. More generally , consider the graph Γ sho wn in Figure 5. An analogous compu- tation sh o ws th at W Γ = 1 π 2 m +2 n +1 Z 0 − π ( π x − 1 2 π 2 ) m + n dx = 1 π 2 m +2 n +1 ·  1 π 1 m + n + 1 ( π x − 1 2 π 2 ) m + n +1  | 0 − π = ( ( − 1) m + n 3 m + n +1 − 1 ( m + n +1)2 m + n +1 if m + n is ev en 0 if m + n is o dd (2.12) (The additional b oundary strata, coming from the comp onents when some ≥ 2 of the m up p er p oin ts, or some ≥ 2 of the n lo wer p oints, app roac h eac h other, clearly do not con tribute to the in tegral). It follo ws that W Γ 6 = 0 f or an y m, n such that m + n is ev en. 17 ... ... m n points points Figure 5: The grap h Γ f r om E x amp le 2 2.2 The exo tic L ∞ structure W e are going to deﬁne p olylinear op erators L n : Λ n T ﬁn ( V ) → T ﬁn ( V )[2 − n ] , n ≥ 2 whic h later are prov en to b e th e T a ylor comp onents of an L ∞ structure. Let γ 1 , . . . , γ n are homogeneous p olyv ector ﬁ elds. W e are going to deﬁne the v alue L n ( γ 1 ∧ · · · ∧ γ n ). First of all, w e d eﬁne with an admissible graph Γ with n v ertices and for an ordered set γ 1 , . . . , γ n of p olyv ecto r ﬁ elds a p olyve ctor ﬁeld L Γ ( γ 1 ⊗ · · · ⊗ γ n ) wh ich is  P n i =1 deg γ i + n − ♯E (Γ)  -v ecto r ﬁeld. (Here w e denote by deg γ the Lie al gebra d egree , that is, if V is concen trated in the cohomologic al degree zero, d eg γ = k − 1 for a k -v ector ﬁeld γ ). If the target v ector space V were ﬁnite-dimensional, the p olyvec tor ﬁeld L Γ ( γ 1 ⊗ · · · ⊗ γ n ) w ould b e the su m L Γ ( γ 1 ⊗ · · · ⊗ γ n ) = X I : E (Γ) →{ 1 , 2 ,..., dim V } L I Γ ( γ 1 ⊗ · · · ⊗ γ n ) (2.13) The p olyv ector ﬁeld L I Γ ( γ 1 ⊗ · · · ⊗ γ n ) is th e pr odu ct o v er the ve rtices of Γ: L I Γ ( γ 1 ⊗ · · · ⊗ γ n ) = ^ v ∈ V (Γ) Ψ I v (2.14) 18 where the v ertices are taken in the order corresp onded to the lab elling of th e vertice s. Eac h Ψ v is deﬁned as Ψ I v =   Y e ∈ I n ( v ) ∂ ∂ x I ( e )    γ n ( v ) , ∧ e ∈ S tar ( v ) dx I ( e )  (2.15) where n ( v ) is the lab el of the vertex v , and the order in the we dge-pro duct is ﬁxed b y the ordering of the set S tar ( v ), as ab o v e (see Remark 1. 8 ). It completes th e deﬁnition of L Γ ( γ 1 ⊗ · · · ⊗ γ n ) for a ﬁ nite-dimensional v ector space V . No w supp ose that V = L i ∈ Z ≥ 0 V i where all graded compon ents V i are ﬁnite-dimensional, as in Section 1.1. W e lea v e to the reader th e follo wing lemma: Lemma 2. 5. L et γ i ∈ T ﬁn ( V ) , and a gr aph Γ do es not c ontain any oriente d cycles (this as- sumption is f ulﬁl le d if Γ is an admissible g r aph). Then the p olyve ctor ﬁeld L Γ ( γ 1 ⊗ · · · ⊗ γ n ) is wel l-deﬁne d for the c ase of inﬁnite-dimensional V , as ab ove. Let γ 1 , . . . , γ n ∈ T ﬁn ( V ). Deﬁn e the p olyve ctor ﬁ eld in T ﬁn ( V ) L n ( γ 1 ∧ · · · ∧ γ n ) = Alt γ 1 ,...,γ n  X Γ ∈ G n, 2 n − 3 W Γ · L Γ ( γ 1 ⊗ · · · ⊗ γ n )  (2.16) Here Alt is the sum o ver all p er mutations of γ 1 , . . . , γ n with signs, such that when t wo p olyv ecto r ﬁelds γ i and γ j are p erm u ted, th e sign ( − 1) (deg γ i +1) · (deg γ j +1) app ears; G n, 2 n − 3 stands for the set of the connected ad m issible graphs with n v ertices and 2 n − 3 edges. Example 2. 6. L 2 ( γ 1 ∧ γ 2 ) = { γ 1 , γ 2 } is the Schouten-Nijenhuis br acket. Remark 2. 7. In what fol lows, we c onsider dg Lie algebr as, or, mor e gener al ly, L ∞ alge- br as. Ther e is an ambiguity with the sign c onventions. Inde e d, for a dg Lie algebr a ther e ar e two p ossible ways to deﬁne the skew- c ommutativity of the br acket: the ﬁrst is [ a, b ] 1 = ( − 1) deg a deg b +1 [ b, a ] 1 , and the se c ond is [ a, b ] 2 = ( − 1) (deg a +1)( deg b +1) [ b, a ] 2 . In what f ol lows we stick to the se c ond deﬁnition of skew-c ommutativity. Recall that an L ∞ algebra structure on a Z -graded v ector space g is a co deriv ation Q of degree +1 of th e free coalgebra S ( g [1]) such that Q 2 = 0. If g is a Lie algebra, suc h Q is giv en b y the d iﬀeren tial in the Chev alley-Eilen b erg chain complex of the Lie algebra g . In co ordinates, an L ∞ structure on g is giv en by a collection of maps (the ”T a ylor comp o- nen ts” of the L ∞ structure) L n : Λ n g → g [2 − n ] , n ≥ 1 (2.17) whic h satisfy , for eac h N ≥ 1, the follo wing quadratic equation: Alt g 1 ,...,g N X a + b = N +1 , a,b ≥ 1 ± 1 a ! b ! L b  ( L a ( g 1 ∧ · · · ∧ g a ) ∧ g a +1 ∧ · · · ∧ g N )  = 0 (2.18) 19 No w turn bac k to the p olyv ector ﬁeld L Γ ( γ 1 ⊗ · · · ⊗ γ n ), for an admissib le graph Γ. This p olyv ect or ﬁ eld h as degree P n i =1 deg γ i + n − ♯E (Γ) − 1. When ♯E (Γ) = 2 n − 3, this degree is P deg γ i − n + 2, wh at agrees with th e shift of degrees in (2.17). Theorem 2. 8. The maps L n : Λ n T ﬁn ( V ) → T ﬁn ( V )[2 − n ] ar e the T aylor c omp onents of an L ∞ algebr a structur e on T ﬁn ( V ) . Pr o of. Con s ider the relation (2.18) for some ﬁxed N . It can b e rewritten as the l.h.s. of (2 . 18) = X Γ ∈ G N, 2 N − 4 c Γ · Alt γ 1 ,...,γ N L Γ ( γ 1 ⊗ · · · ⊗ γ N ) = 0 (2.19) where the summation is tak en o ver all the connected admissib le graphs with N v ertices and 2 N − 4 edges (for 1 edge less than in (2.16)), and c Γ are s ome (real) num b ers. W e need to prov e that c Γ = 0 f or eac h Γ ∈ G N , 2 N − 4 . F or, consid er th e integ ral Z C N, Γ d  ^ e ∈ E (Γ) φ e  (2.20) This in tegral is clearly equal to 0, b ecause all 1-forms φ e are closed (moreov er, they are exact). No w w e wa n t to apply the Stok es’ form ula. F or this we need to constru ct the compactiﬁcati ons C n, Γ of the spaces C n, Γ whic h is a smo oth manifold with corn er s , and suc h that the forms φ e can b e extended to a smo oth form s on C n, Γ . It can b e d on e in the standard w a y , see [K97, S ect . 5]. Here w e describ e the b oun dary strata of co dimension 1 which are the only strata w hic h con tribute to the integ rals w e consider. Here is the list of th e b oundary strata of co dimension 1: T1) some S p oint s p i 1 , . . . , p i S among th e n p oin ts appr oa c h eac h other, suc h that 2 ≤ ♯S ≤ n − 1; in this case let Γ 1 b e the restriction of the graph Γ in to these S p oints, and let Γ 2 b e the graph obtained from con tracting of the S v ertice s int o a single new vertex. Thus, Γ 1 has S v ertices, and Γ 2 has n − S + 1 ve rtices. In this ca se the b ou n dary stratum is isomorphic to C S, Γ 1 × C n − S +1 , Γ 2 ; T2) a p oin t q connected by an edge − → pq w ith a p oin t p approac h es the h orizonta l line passing through the p oin t p . W e contin ue: 0 = Z C N, Γ d  ^ e ∈ E (Γ) φ e  = Z C N, Γ d  ^ e ∈ E (Γ) φ e  = Z ∂ C N, Γ ^ e ∈ E (Γ) φ e (2.21) Only the b oundary strata of co dimension 1 do cont ribute to the r.h.s. in teg ral. The strata of t yp e T2) do not contribute b ecause the form V e φ e v anishes ther e. W e can ther efore consider only the strata of t yp e T1). 20 F or these strata we ha v e the f oll o wing f actorization principle: it sa ys th at the integ ral o v er a stratum T of type T 1) is the p rod u ct: Z T ^ e ∈ Γ φ e =  Z C S, Γ 1 ^ e ∈ Γ 1 φ e  ×  Z C n − S +1 , Γ 2 ^ e ∈ Γ 2 φ e  (2.22) The same factorizatio n h olds therefore for th e weig h ts W Γ . W e get the follo wing identi t y: 0 = Z ∂ T 1 C N, Γ ^ e ∈ E (Γ) φ e = X T ∈ ∂ T 1  Z C S, Γ 1 ^ e ∈ Γ 1 φ e  ×  Z C n − S +1 , Γ 2 ^ e ∈ Γ 2 φ e  (2.23) where the str ata T come w ith its orienta tion. The summ ands in th e r.h.s. are in 1-1 corresp ond en ce with the summ an d s in (2.18) which con tribute to c Γ . Th erefore, all c Γ = 0. It f oll o ws from Lemma 2. 2 th at the L ∞ structure giv en by Theorem 2. 8 has only comp onen ts of even ≥ 2 degrees: L 2 , L 4 , L 6 , . . . . On the ot her hand, Example 2. 4 sho ws that the higher comp onen ts L 2 n , n ≥ 2, are nonzero. W e d enote the L ∞ algebra T ﬁn ( V ) with th e constru cted L ∞ structure b y T L ﬁn ( V ), and call it the exotic L ∞ structure on T ﬁn ( V ). 2.3 Infinite-dimensional fo rmality No w we are ready to state the main result of this pap er: Main Theorem 2. 9. L et V = L i ∈ Z ≥ 0 V i b e a non-ne gatively gr ade d ve ctor sp ac e with dim V i < ∞ . Then ther e is an L ∞ quasi-isomorphism fr om the exotic L ∞ algebr a T L ﬁn ( V ) c onstructe d in Se ction 2.2 to the Ho c hschild c omplex Hoch q ﬁn ( S ( V ∗ )) with the Gerstenhab er br acket. Its ﬁrst T aylor c omp onent is given by the H o chschild-Kostant-R osenb er g map. The pro of of a more precise statement is giv en in Section 3, see Theorem 3. 3 . Here we discuss some d irect consequences. There is an im m ediate corollary , obtained by comparison of Theorem 2. 9 with the Kont sevic h formalit y [K97]: Corollar y 2. 10. L et V b e ﬁnite- dimensiona l. Then the exotic L ∞ structur e T L ﬁn ( V ) on p olyve ctor ﬁelds is L ∞ quasi-isomorphic to the classic al gr ade d Lie algebr a structur e on it (given by the Schouten-Ni jenhuis br acket). Pr o of. By the lo c.cit., there is an L ∞ quasi-isomorphism U : T poly ( V ) → Ho c h q ( S ( V ∗ )). By our Main Theorem, there is an L ∞ quasi-isomorphism F : T L ﬁn ( V ) → Ho c h q ( S ( V ∗ )). It im p lies that the tw o L ∞ structures on p olyv ecto r ﬁelds are L ∞ quasi-isomorphic. 21 Ho w ever, for an inﬁnite-dimensional space V , the t w o structures ma y b e complete ly diﬀeren t (and, in fact, they are, as follo ws from the failure of Kon tsevic h formalit y in this case). F or example, supp ose we h a ve a bive ctor α ∈ T ﬁn ( V ) f or an inﬁn ite- dimensional V . The right concept what is that α is P oisson is giv en by ou r Main Theorem 2. 9 as follo ws: 1 2 L 2 ( α ∧ α ) + 1 24 L 4 ( α ∧ α ∧ α ∧ α ) + · · · = 0 (2.24) F or a ﬁxed α w hic h is p olynomial in coord inates the su m is actually ﬁnite. W e see, in particular, that if α s at isﬁes (2.24), the bivecto r ﬁ eld λ · α , λ ∈ C , m ay not satisfy . That is, the equatio n (2.24) is not homogeneous. W e call (2.24) the quasi- P oisso n e quation . In th e case w hen α is a lin ea r bivec tor ﬁ eld the quasi-Poi sson equ ation (2.24) coincides with the classical one: Lemma 2. 11. L et α b e a line ar b ive c tor on V . Then the higher c omp onents L 2 n ( α ∧ 2 n ) = 0 , n ≥ 2 . That is, (2.24) is e quivalent to the Poisson e quation { α, α } = 0 . Pr o of. Any admissib le graph with k vertices wh ic h con tributes to the L ∞ algebra L has 2 k − 3 edges. If k > 2 it implies that there is at least one v ertex with at least t wo incoming edges. Then the corresp onding op erator L Γ ( α ∧ k ) is zero b ecause α has linear co eﬃcien ts. W e exp ect that there are P oisson biv ectors which are of degree ≥ 2 on an inﬁ nite-dimensional v ector space which ar e imp ossible to q uantize in the sense of deformation quanti zation. As the condition (2.24) is non-homogeneous, our formalit y theorem giv es a d eformati on quanti zation only if all L 2 n ( α ∧ 2 n ) = 0, n ≥ 2, separately , so we ha v e a sequence of homo gene ous equ a- tions. On e can sa y that this series of equations giv es the higher obst ructions f or deformation quan tizatio n pr oblem in the in ﬁnite-dimensional case. Compute no w the ﬁ rst obstru ctio n L 4 ( α ∧ 4 ). The list of all 6 admissible graphs (up to the lab elling) with 4 vertices and 2 · 4 − 3 = 5 edges is s h o wn in Figure 6 . Their w eigh ts are 2 × 13 12 , 2 × 13 12 , 2 × 1 3 , 13 12 , 7 12 , 7 12 , resp ectiv ely . (Here 2 × . . . count s the t w o p ossible labelings). Clearly only Γ 1 , Γ 4 , Γ 5 con tribute to L 4 ( α ∧ 4 ), b eca use other graphs has a vertex with 3 outgoing edges. W e get the follo wing equation for the v anishing of the ﬁr st obstruction in the inﬁ n ite- dimensional case:  13 6 L Γ 1 + 13 12 L Γ 4 + 7 12 L Γ 5  ( α ∧ α ∧ α ∧ α ) = 0 When α is a quadr atic P oisson bive ctor, on ly L Γ 4 survive s b ecause other graphs h a ve a v ertex with 3 incoming ed ges. Then in th e quadr ati c case th e ﬁr st obstruction reads L Γ 4 ( α ∧ α ∧ α ∧ α ) = 0 (2.25) 22 Γ Γ Γ 4 5 6 Γ Γ Γ 1 2 3 Figure 6: The ad m issible graph s in G 4 , 5 3 A pr oof of the Main Theore m 2.9 3.1 The new pr op aga tor Recall th e conﬁgur atio n space C n,m in tro duced in [K97]. First of all, Conf n,m is the conﬁguration space of m + n pairwise distinct p oin ts among whic h n b elong to the op en u p p er half-plane { z ∈ C , Im z > 0 } , and the remaining m b elong to the real line, which is thou ght on as th e b oundary of the op en u pp er-half p lane. Then C n,m = Conf n,m /G 2 (3.1) where G 2 is th e tw o-dimensional group of symmetries of the upp er half-plane of the form { z 7→ az + b | , a ∈ R + , b ∈ R } . Here a p oint z of the upp er h alf-plane is considered as a complex n um b er with p ositiv e imaginary part. M.Kon tsevic h pro vided lo c.c it. a compactiﬁcation C n,m of these spaces. W e r efer the reader to [lo c.cit. , Sect. 5], for a detailed description of this compactiﬁcation. Then he constructed a top degree diﬀeren tial form on C n,m , asso ciated with a Kontsevic h admissible graph Γ with n v ertices of the ﬁrs t t yp e and m vertice s of th e second t yp e (see [lo c.c it., Sect. 6.1]). The top d eg ree diﬀerentia l form on C n,m is constructed as follo ws. One ﬁr stly constru cts a closed 1-form φ on the space C 2 , 0 (“the propagator”). Assu me th at a graph Γ with n + m 23 v ertices has exactly 2 n + m − 2 orien ted edges. F or eac h edge e of Γone has the forgetful map t e : C n,m → C 2 , 0 . No w if the 1-form φ is c hosen, one deﬁn es the top d eg ree form on C n,m asso cia ted with an admissible graph Γ as φ Γ = ^ e ∈ E (Γ) t ∗ e ( φ ) (3.2) (one should imp ose some order on the edges of Γ to deﬁn e the wedge-prod uct of 1-forms in (3.2); this order is a part of d ata for a Kontsevic h adm iss ible graph). W e will need a slightl y reﬁned conce pt of a Kon tsevic h admissible graph (see Section 1.5 or [K97 , Sect. 6.1]). W e call this concept a norma lize d Kontsevic h admissible grap h ; by the normaliza tion we mean the cond ition (3.3) b elo w. Definition 3. 1. A normalize d Kontsevich admissible gr aph is a Kontsevic h admissible graph (see Deﬁnition 1. 7 ) suc h that the follo wing cond ition holds (where for a v ertex v of th e ﬁrst t yp e n ( v ) denotes it lab elling): F or t w o v ertices of the ﬁrst t yp e connected by an edge − − → v 1 v 2 one has n ( v 1 ) < n ( v 2 ) (3.3) Note that (3.3) implies that ther e ar e no oriente d cycles b etwe en the vertic es of the ﬁrst typ e . W e do n ot consider a c hoice of the orderings of the sets S tar ( v ) of outgoing edges for the v ertices v of Γ as a p art of the d ata of an admissible graph, see Remark 1. 8 . Recall ﬁrstly w h at the space C 2 , 0 is. It lo oks like “an ey e” (see the left picture in Figure 7). Th e t w o b ound ary lines comes when one of the t w o p oints z 1 or z 2 approac hes the r eal line, whic h is the b oun dary of the upp er h alf-plane. The circle is the b oundary stratum co rresp onded to th e case wh en the p oin ts z 1 and z 2 approac h eac h other and are far from the real line. The role of the t w o p oints z 1 and z 2 here is completely symmetric. No w w e are going to br eak this symmetry down. Sub divide the space C 2 , 0 in to tw o s u bspaces, as follo ws. Let the p oint z 1 b e ﬁxed. Dra w the half-circle orthogonal to the real line (a geo desic in the P oincar ´ e mo del of hyp erb oli c geometry) suc h that z 1 is the top p oin t of th e h alf-ci rcle, see Figure 8. The half-circle is a geo desic, and the group G 2 in (3.1) is the group of symmetries for the Po incar’e mo del of hyperb olic geometry . It pro v es that a half-circle orthogonal to the real line is tr an s formed to an analogous half-circle, b y an y g ∈ G 2 . Th erefore, the image of th e half-circle is well-deﬁned on the “ey e ” C 2 , 0 . W e dr a w th is image in Figure 7, as the b ord er lin e b et ween the light and the dark parts. W e w an t the 1-form φ to v anish wh en the orien ted pair ( z 1 , z 2 ) is in the ligh t area of Figure 8 ( z 2 is outside th e half-circle). W e contrac t all ligh t area in th e left-hand side picture in Figure 7 to a p oin t, and get th e righ t-hand side picture therefrom. Here b oth v ertices of th e eye f r om the left-hand side picture, and the u p p er b oun dary comp onen t of it, as we ll as the upp er h alf of the circle, are con tracte d to a one p oin t. The external b oundary in the right -hand side picture is 24 formed b y the lo w b ound ary comp onen t in the left-hand side picture. W e call the sp ace dr a wn in the right-hand side p icture in Figure 7 the mo diﬁe d (contrac ted) C 2 , 0 , an d denote it by C m 2 , 0 . Definition 3. 2 . A mo diﬁe d angle function is any map f r om the mo diﬁe d (c ontr acte d) C 2 , 0 to the cir c le unit S 1 such that the internal cir cle i s mapp e d i som orphic al ly to S 1 , as the E uclide an angle , and the external b oundary c omp onent is also mapp e d to S 1 (in the homotopic al ly unique way, with the p erio d π ). Here the angle in the in ternal circle is the Euclidian angle wh en z 2 approac hes z 1 . The p erio d of this angle inside the half-circle is π , n ot 2 π . There is a mo diﬁed Kon tsevich’s harmonic angle function, w h ic h pr o vides an example of suc h a m ap, see (3.7) b elo w. Let us recall that the Kontsevich angle function , intro d uced in [K97, Sect. 6.2], is a map of the ey e C 2 , 0 (dra wn in the left picture of Figure 7) to S 1 suc h that the inn er circle is mapp ed isomorphically as the angle function, the u pp er b oundary is con tracted to a p oin t, and the lo w er b oundary is mapp ed with p erio d 2 π . Let us stress a d iﬀerence b et w een the t w o deﬁn itio ns: in our case, an y an gle fun ctio n θ is a function in a prop er sense, wh ence in Kon tsevic h’s deﬁnition it is a multi- v alued function deﬁned only up to 2 π . Therefore, in our case, the de Rham deriv ative φ = dθ is an exact 1-form, whence in Kontsevic h’s case it is only closed. What also mak es this diﬀerence essen tia l is the observ ation that in our case th e p ropagat or φ is n ot a smo oth function on the manifold with corners C 2 , 0 . It will b e th e main source of problems for the pro of of the L ∞ iden tities using Stok es’ formula in the n ext S u bsection. Deﬁne the weight of an admissible gr aph Γ as W Γ = 1 π 2 n + m − 2 Z C + n,m ^ e ∈ E (Γ) f ∗ e φ (3.4) where φ is a mo diﬁed an gle fun ctio n, and f e is the forgetful map f e : C n,m → C 2 , 0 asso cia ted with an edge e ∈ E (Γ). This deﬁ nition although b eing correct h as a small dr awbac k, d ue to the non -sm oothness of the 1-form φ on C 2 , 0 . Pote n tially it m ay mak e troub les in pro ofs using the Stoke s form ula on manifold with corners. W e encourage to the reader not b e confus ed by this drawbac k no w, and to w ait u ntill Section 3.2.2, when we pro vide a b etter reformulation of the deﬁnition of w eigh ts, in (3.8). Note that, although the 1-form φ is exact, the integrals W Γ are in general nonzero, b ecause the sp ac es C n,m are manifolds (with corners) with b oun dary . 3.2 The proo f 3.2.1 St a tement of the re sul t Let V b e a Z ≥ 0 -graded v ector sp ac e ov er C with ﬁn ite- dimensional comp onents V i . 25 Figure 7: The K on tsevic h’s conﬁguration sp ace C 2 , 0 (left) and our mo diﬁed space C m 2 , 0 (righ t) z z 1 2 Figure 8: The height ordering of tw o p oints, z 2 ≤ z 1 Recall our concept of normalized K on tsevic h admissib le graphs, see Deﬁnition 3. 1 . Let G n,m, 2 n + m − 2 b e the set of all connected adm iss ible graphs with n v ertices of the ﬁ rst t y p e, m v ertices of the second t yp e, and 2 n + m − 2 edges. Recall the p olydiﬀeren tial op erators U Γ ( γ 1 ∧ · · · ∧ γ n ), Γ ∈ G n,m , asso ciate d with a (general) Kon tsevic h admissible graph Γ and with p olyv ector ﬁelds γ 1 , . . . , γ n , see [K97, Sect. 6.3]. Deﬁne F n ( γ 1 ∧ · · · ∧ γ n ) = X Γ ∈ G n,m W Γ × U Γ ( γ 1 ∧ · · · ∧ γ n ) (3.5) where th e w eigh t W Γ is deﬁn ed in (3.4) via the mo diﬁed angle function. Consider the follo w in g t w o cases: either Γ con tains an orien ted cycle (and in th is case clearly our W Γ = 0), or it do es not con tain any orien ted cycle (and in the latter case U Γ is w ell- deﬁned by Lemma 1. 9 ). Therefore, the co c hain F n ( γ 1 ∧ · · · ∧ γ n ) is well-deﬁned. W e p ro ve 26 Theorem 3. 3. L et V b e a Z ≥ 0 -gr ade d ve ctor sp ac e over C with ﬁnite-dimensional gr ade d c om- p onents V i . Then the maps F n ar e wel l- deﬁne d and ar e the T aylor c omp onents of an L ∞ quasi-isomorphism F : T L ﬁn ( V ) → Ho c h q ﬁn ( S ( V ∗ )) . (Her e in the l.h.s. T ﬁn ( V ) L is the exotic L ∞ algebr a intr o duc e d in Se c tion 2). The ﬁrst T aylor c omp onent F 1 of the L ∞ quasi-isomorphism F is the Ho c hsch ild-Kostant-R osenb er map (1.13). Theorem 3. 3 p ro vides an explicit form of the L ∞ quasi-isomorphism which existence is s tated in Main Theorem 2. 9 , and, therefore, it implies th e latter Th eo rem. Belo w w e p ro ve T h eorem 3. 3 . 3.2.2 Configura tion sp aces T o pro v e T h eorem 3. 3 , we wo uld lik e to app ly the w ell-kno wn argument with Stok es’ form ula f or manifolds with corners, see e.g. [K97, Sect. 6.4]. Th e main troub le, whic h mak es it imp ossible to apply this argum en t str aig h tforw ardly , is that the de Rham deriv ativ e of the m o diﬁed angle function is not a sm ooth diﬀerent ial form on the compactiﬁed sp ace C 2 , 0 . Th ere could b e (at least) tw o diﬀeren t wa ys to o v ercome this p roblem. Th e ﬁrst one is to sub d ivid e the conﬁ guratio n spaces C n,m suc h that the we dge-pro duct of the angle 1-forms would b e a smo oth d iﬀeren tial form on the sub division. In particular, it is clear h o w to sub divid e C 2 , 0 : the su b division is sho wn in Figur e 9. T his approac h, ho w ev er, wouldn’t wo rk, b ecause the sub divided spaces are not manifold with corners an ymore, and one can n ot apply the Stok es formula for them. Figure 9: A p ossible su b division C 2 , 0 (whic h is not a manifold with corners) The second w a y is to deﬁne another conﬁguration spaces, cutting the area z 1 ≤ z 2 oﬀ if there is an ed ge from z 1 to z 2 (see Figure 8) . This solution would complicate the constru cti ons, as the c onﬁgur ation sp ac e s would dep end on a gr aph Γ, that is, for eac h Γ w e would ha v e its o wn conﬁguration sp ac e. Ho wev er, w e will show in the r est of the p aper that this solution wo rks. Let Γ b e an normalized Kon tsevic h admissible graph with n vertice s of th e ﬁ rst type and m v ertices of the second type. Recall that all ve rtices of Γ of th e ﬁ rst t yp e are lab eled as { 1 , 2 , . . . , n } , and all vertices of the second t y p e are ord ered and lab eled as { 1 , 2 , . . . , m } . Recall 27 also that (3.3 ) is assumed . Deﬁne the conﬁguration sp ace C n,m, Γ as follo w s: C n,m, Γ = { z 1 , . . . , z n ∈ H ; t 1 , . . . , t n ∈ R | z i 6 = z j for i 6 = j , t 1 < · · · < t m , z j ≤ z i if − − → ( i, j ) is an edge in Γ and t j ≤ z i if − − → ( i, j ) is an edge in Γ } /G 2 (3.6) Here H = { z ∈ C , Im z > 0 } is the u pp er half-plane, and the relations z j ≤ z i and t j ≤ z i in the r.h.s. are u ndersto od in the sense of the ord ering with half-circle, sho wn in Figure 8. Th e group G 2 in the r.h.s. is G 2 = { z 7→ az + b, a ∈ R > 0 , b ∈ R } . Note that th e graph Γ in the deﬁnition of C n,m, Γ ma y ha ve an arbitrary (not necessarily 2 n + m − 2) num b er of edges. If Γ has no edges at all, w e recognize the Kon tsevic h ’s original space C n,m . It is easy to construct a Kontsevic h-t yp e compactiﬁcation C n,m, Γ whic h is a manifold with corners, with pro jections p Γ , Γ ′ : C n,m, Γ → C n ′ ,m ′ , Γ ′ deﬁned when Γ ′ is a subgraph of Γ. The space C 2 , 0 , Γ 0 , where Γ 0 is ju st one orien ted edge connecting p oin t 1 with p oint 2, is sho w n in Figure 10. Figure 10: The ”Eye” C 2 , 0 , Γ 0 Here the lo w er comp onen t of the b oun dary comes when the p oin t z 2 (see Figure 8) ap- proac hes the real line, th e left (corresp., the righ t) u pp er b oun dary comp onent is corresp onded to the conﬁgurations wh en the p oin t z 2 approac hes the left (corresp., the righ t) part of the geod esic h alf-c ircle in Figure 8, and the third upp er b oundary comp onent (the half-circle) comes when z 2 approac hes z 1 inside the geo desic half-circle. W e can u pgrade the deﬁn ition of the mo diﬁed angle function from this new p oint of view: Definition 3. 4. A mo diﬁe d angle function is a c ontinuous map θ : C 2 , 0 , Γ 0 → S 1 such that θ is given by the Euclide an angle varying fr om − π to 0 on the upp er half-cir cle, and θ c ontr acts the two other u pp er b oundary c omp onents to a p oint 0 ∈ S 1 . An example of the mo diﬁed angle fun ction is the d oubled Kon tsevic h ’s harmon ic an gle: 28 f ( z 1 , z 2 ) = 1 i Log ( z 1 − z 2 )( z 1 − z 2 ) ( z 1 − z 2 )( z 1 − z 2 ) where z 1 ≥ z 2 in the sense of Figure 8 (3.7) It is clear that the function f ( z 1 , z 2 ) d eﬁned in this wa y is w ell-deﬁned wh en z 1 ≥ z 2 and is equal to 0 on th e ”b order” circle, see Figure 8. No w w e deﬁn e f or an edge e of an admissible graph Γ the 1-form φ e on C n,m, Γ as p ∗ Γ , Γ 0 ( dθ ), where Γ 0 is the graph w ith t w o ve rtices and one edge e . Finally , w e give a rigorous deﬁn itio n of the weigh t: W Γ = 1 π 2 n + m − 2 Z C n,m, Γ ^ e ∈ E (Γ) φ e (3.8) Let us describ e the b oun dary strata of co dimension 1 of C n,m, Γ . The b oun dary strata will b e expressed in terms of sp ace s C n ′ ,m ′ , Γ ′ , as we ll as of spaces C n, Γ , in tro duced in S ec tion 2. Belo w is a complete list of the t yp es of b oun dary strata of co dimension 1 in C n,m, Γ : S1) some p oin ts p 1 , . . . , p S ∈ H , S ≥ 2, appr oac h eac h other and remain far from the geo desic half-circles of an y p oint q 6 = p 1 , . . . , p S suc h that there is an edge − → q p i , f or some 1 ≤ i ≤ S . In this case w e get the b oundary str at um of codimension 1 isomorphic to C n − S +1 ,m, Γ 1 × C S, Γ 2 where Γ 2 is the subgraph of Γ of the edges connecting the p oin t p 1 , . . . , p S with eac h other, and Γ 1 is obtained from Γ by collapsing the graph Γ 2 in to a (new) single verte x; S2) some p oin ts p 1 , . . . , p S ∈ H and some p oin ts q 1 , . . . , q R ∈ R , 2 S + R ≥ 2, 2 S + R ≤ 2 m + n − 1, ap p roac h eac h other and a p oin t of th e real line, but are far from the geo desic half-circle of any other p oint connected with these p oints by an edge. I n this case we get a b oun dary stratum of cod imension 1 isomorph ic to C S,R, Γ 1 × C n − S,m − R +1 , Γ 2 where Γ 1 is the graph formed from the edges b et w een th e app roac hing p oin ts, and Γ 2 is obtained from Γ by collapsing the sub graph Γ 1 in to a n ew vertex of the second type; S3) some p oin t p is placed on th e geo desic half-circle of exactly one p oin t q 6 = p wh ich is far from p , such th ere is an edge − → q p . These b ound ary s trata will b e irrelev ant for u s b ecause they d o not contribute to the integral s we consid er. W e are r eady to prov e Theorem 3. 3 . 29 3.2.3 Applica tion of the Stokes’ f ormula and the bo und ar y stra t a W e n eed to prov e the L ∞ morphism quadr ati c relations on th e maps F n . Recall, th at for eac h k ≥ 1 and f or any p olyv ector ﬁelds γ 1 , . . . , γ k ∈ T L ﬁn ( V ) it is the relation: d Hoch ( F k ( γ 1 ∧ · · · ∧ γ k ))+ X 2 ≤ N ≤ k 1 k !( N − k )! X σ ∈ S k ±F k − N  L N ( γ σ (1) ∧ · · · ∧ γ σ ( N ) ) ∧ γ σ ( k − N +1) ∧ · · · ∧ γ σ ( k )  + 1 2 X a,b ≥ 1 ,a + b = k 1 a ! b ! X σ ∈ S k ± [ F a ( γ σ (1) ∧ · · · ∧ γ σ ( a ) ) , F b ( γ σ ( a +1) ∧ · · · ∧ γ σ ( k ) )] = 0 (3.9) The l.h.s. of (3.9) is a sum o v er the admissible graphs Γ ′ ∈ G n,m, 2 n + m − 3 with n ve rtices of the ﬁrst t yp e, m vertic es of the second type, and 2 n + m − 3 egdes (that is, having for 1 edge less than the graphs con tributing to F n ). This sum is of the form: l.h.s = X Γ ′ ∈ G n,m, 2 n + m − 3 α Γ ′ U Γ ′ (3.10) where α Γ ′ are some complex n um b ers, and U Γ ′ are the Kon tsevic h ’s p olydiﬀeren tial op erators from [K97]. It is cle ar that al l U Γ ′ do not c onta in any oriente d cycle . Our goal is to pro v e that all num b ers α Γ ′ = 0. Eac h α Γ ′ is a qu adratic- linear com bination of our weig h ts W Γ for Γ ∈ G n,m, 2 n + m − 2 . There is a general construction pro ducing identitie s on quadratic-linear com binations of W Γ , as follo ws: Consider some Γ ′ ∈ G n,m, 2 n + m − 3 . With Γ is asso ciat ed a diﬀerent ial form V e ∈ E (Γ ′ ) φ e on C n,m, Γ ′ , as ab o v e. Consider Z C n,m, Γ ′ d  ^ e ∈ E (Γ ′ ) φ e  (3.11) This expression is 0 b ecause the form V e ∈ E (Γ ′ ) φ e is closed (it is, moreov er, exact). Next, b y the Stokes theorem, we ha v e 0 = Z C n,m, Γ ′ d  ^ e ∈ E (Γ ′ ) φ e  = Z ∂ C n,m, Γ ′  ^ e ∈ E (Γ ′ ) φ e  (3.12) The only str ata of co dimension 1 in ∂ C n,m, Γ ′ do con tribute to the r .h.s. They are giv en by the list S1)-S3) in Section 3.2.2. The strata of t yp e S3) clearly do not con tribute b ecause of our b oundary conditions on the mo diﬁed angle fu nction. F or the strata of t yp es S 1) and S2) one has th e follo wing factorization pr op e rty : the integral o v er the pr odu ct is equal to the pro duct of inte grals. It follo ws from the factorization of diﬀeren tial forms on the p rod uct of corresp onding spaces. It makes p ossible to prov e the f oll o wing ke y-lemma: 30 Key-lemma 3. 5. F or any normalize d Kontsevich admissible gr aph Γ ′ ∈ G n,m, 2 n + m − 3 , the c o e f- ﬁcient α Γ ′ in (3.10) is e qual to R ∂ C n,m, Γ ′  V e ∈ E (Γ ′ ) φ e  (which is zer o by the Stokes’ formula). Theorem 3. 3 f ollo ws directly from this Key-Lemma 3. 5 . T o pr o ve K ey-Lemma 3. 5 , we expr ess Z ∂ C n,m, Γ ′  ^ e ∈ E (Γ ′ ) φ e  = Z ∂ S 1 C n,m, Γ ′  ^ e ∈ E (Γ ′ ) φ e  + Z ∂ S 2 C n,m, Γ ′  ^ e ∈ E (Γ ′ ) φ e  + Z ∂ S 3 C n,m, Γ ′  ^ e ∈ E (Γ ′ ) φ e  (3.13) the in tegral ov er the b oundary ∂ C n,m, Γ ′ as the sum of the integral s ov er th e three t yp es of b oundary strata S 1)-S3) of co dimension 1. As w e ha v e already noticed, the integ ral R ∂ S 3 C n,m, Γ ′  V e ∈ E (Γ ′ ) φ e  = 0, due to the b oun dary conditions for the p ropaga tor. The summand R ∂ S 2 C n,m, Γ ′  V e ∈ E (Γ ′ ) φ e  corresp onds exactly to the ﬁr s t and to the third summands of the l.h.s. of (3.9) contai ning the Ho c hsc h ild diﬀeren tial and the Gerstenh aber brac k et, by the factorization prop ert y . It remains to asso ciate a b ound ary stratum of typ e S 1) with a summan d of the second summand of (3.9), cont aining op erations L N . It is clear th at the part of the second summ and of (3.9) for a ﬁxed N is in 1-to-1 corresp ond en ce with that summands in R ∂ S 1 C n,m, Γ ′  V e ∈ E (Γ ′ ) φ e  where S = N p oin ts in the u pp er h alf-plane app r oac h eac h other. Theorems 3. 3 an d Theorem 2. 9 are p ro ven. Remark 3. 6. In [K97], w here computations of this typ e are originated from, the strata analo- gous to our strata of t yp e S 1), con tribute zero inte grals, f or N > 2. This is why M.Kon tsevic h gets his “pur e” form ality theorem there. This v anishing is pro v en b y a quite non -trivial com- putation, see lo c.cit., S ect. 6.6. W e saw in Examples 2. 3 and 2. 4 that the analogous v anishin g fails in the fr amew ork of our construction. A The f ailure of the Kontsevich forma lity f or H o c h q ﬁn ( S ( V ∗ )) f or an infinite-dimensional V In this App endix, w e pro vide an argumen t whic h sho ws that, for a general inﬁnite-dimensional V , the formalit y of th e dg Lie algebra Ho c h q ﬁn ( S ( V ∗ )) fails. More precisely , we pro v e the follo wing statement : Theorem A. 1. L et W b e a ﬁnite- dimensiona l ve ctor sp ac e over a ﬁeld k of char acteristic 0, and L = Lie( W ) b e the fr e e Lie algebr a gener ate d by W . Denote by V the underlying gr ade d sp ac e of L , with the inherite d gr ading: V = ⊕ n ≥ 1 V n , V n = Lie n ( W ) 31 wher e Lie n ( W ) is the subsp ac e of Lie( W ) gener ate d by the length n Lie monomials. Then ther e do e s not e xi st any gl ( W ) -e quivariant L ∞ morphism U : T q ﬁn ( V ∗ )[1] → Ho c h q ﬁn ( S ( V ))[1] such that the ﬁrst T aylor c omp onent U 1 is the Ho chschild-Kostant-R osenb er g map. The pr o of go es as follo ws. Step 1. Assu me that such an L ∞ morphism U exists. T ake the linear P oisson b iv ector α in T ﬁn ( V ∗ )[1] corresp onded to the fr ee Lie alge bra structure on V = Lie ( W ) (the Kostant - Kirillo v biv ector). Consider U ∗ ( ~ α ), wh ic h is a Maurer-Cartan elemen t in Ho c h ﬁn ( S ( V ))[[ ~ ]][1]. Explicitly , one has: U ∗ ( ~ α )( f , g ) = f · g + ~ U 1 ( α )( f ⊗ g ) + 1 2 ~ 2 U 2 ( α, α ) + . . . (A.1) W e p ro ve the follo w ing Pr oposition A. 2. The star-algebr a S ( V ) ∗ [[ ~ ]] is isomorphic to the u ni v ersal en- veloping algebr a U ( ~ Lie( W ))[[ ~ ]] ( to the unive rsa l enveloping algebr a of the Lie algebr a whose br acket is ~ [ − , − ] , wher e [ − , − ] is the Lie br acke t of Lie( W )) . Note th at the statemen t analogous to this one fails for a ge neral L ∞ map T poly ( g ∗ )[1] → Ho c h q ( S ( g ))[1] and general Lie algebra g , ev en for th e case of a ﬁnite-dimensional Lie algebra g . In particular, M.Kon tsevic h p r o vided a rather sp eciﬁc argu m en t in [K97, Sect. 8.3.1], to sho w the claim for the L ∞ morphism constructed in lo c.cit. Pr o of. W e mak e use the obstru cti on theory in tro duced in [GM, Sect. 2.6]. T he idea is to show that any t w o Maurer-Cartan elements in T ﬁn ( V ∗ )[[ ~ ]][1] of the form ~ α + ~ 2 α 1 + ~ 3 α 2 + . . . (A.2) (where α is th e ﬁxed Kostan t-Kirillo v biv ecto r) are “connected” by a gauge transformation exp( v ) corresp onded to a ve ctor ﬁ eld of the form v = ~ v 1 + ~ 2 v 2 + . . . (A.3) W e u se th e we ll-kno wn fact that any L ∞ quasi-isomorphism of dg Lie algebras in duces an isomorphism on π 0 ( − ) of the corresp onding Deligne groupp oids. In particular, the star-pro duct on S ( V )[[ ~ ]], corresp onded to U ( ~ Lie( W ))[[ ~ ]] by the PBW theorem, giv es a Maurer-Cartan elemen t of the form (A.2), while the star-pro duct (A.1) is corresp onded to the simp lest elemen t of th e form (A.2), the one with α 1 = α 2 = · · · = 0. It is enough to p ro ve that any (and, in particular, these t w o) Maurer-Cartan elemen ts are gauge equiv alent. 32 In fact, the p roblem can b e reformula ted in terms of the localized dg Lie algebra T lo c ( V ∗ )[[ ~ ]][1] = ( T ﬁn ( V ∗ )[[ ~ ]][1] , d = ad α ) (in this lo calize d dg Lie algebra, w e consider the Maurer-Cartan elements of the f orm ~ α 1 + ~ 2 α 2 + . . . , with 0 co eﬃcien t at ~ 0 ). One easily sees that there is an isomorp h ism of sets:  ~ α + ~ 2 α 1 + ~ 3 α 2 + · · · ∈ MC( T ﬁn ( V ∗ )[[ ~ ]][1])  /  gauge actions exp ( ~ v 1 + ~ 2 v 2 + . . . )  ≃  ~ 1 α 1 + ~ 2 α 2 + · · · ∈ MC( T lo c ( V ∗ )[[ ~ ]][1])  /  gauge actions exp ( ~ v 1 + ~ 2 v 2 + . . . )  (A.4) W e app ly th e obstruction theory for extension of a gauge equ iv alence f r om [GM, Pr op . 2.6(2)] for the scalars extension from C [ ~ ] / ~ n to C [ ~ ] /h n +1 , to the d g Lie algebra T lo c ( V ∗ )[[ ~ ]][1] (cor- resp onded to the r .h.s. of (A.4)). W e need to show that the obstr u ctio ns v anish for any n . Accordinding to lo c.cit., the obstructions for extension of the gauge transf ormatio n, b elong to H 1 ( g ⊗ J ), where in our case g = T lo c ( V ∗ )[[ ~ ]][1], and J = ~ n C [ ~ ] / ~ n +1 . T herefore, the obstructions b elong to H 2 ( T lo c ( V ∗ )[[ ~ ]] , − ). W e h a ve : T lo c ( V ∗ )[[ ~ ]] = C q CE (Lie( W ) , S (Lie W )[[ ~ ]]) (A.5) The Ch ev alley-Eilen b erg cohomology of a f r ee Lie algebra v anish in all d egree s exce pt for degrees 0 and 1 (for any coeﬃcients); in particular, H 2 CE (Lie( W ) , − ) = 0. It giv es th e result. Step 2. One can sp ecialize to ~ = 1. The r esult of Prop osition A. 2 giv es, u nder the assumption con trary to the one of the statemen t of T h eorem A. 1 , an existence of a g l ( W )- equiv arian t L ∞ morphism U lo c : T lo c ( V ∗ )[1] → Ho c h ﬁn ( U (Lie ( W ))[1] (A.6) where T lo c ( V ∗ ) = C CE (Lie( W ) , S (Lie( W ))). (T o lo ca lize an L ∞ -morohism by a Maurer-Cartan equation, we use the s tand ard well-kno wn form ula). The cohomol ogy of b oth sides of (A.6) v anish except for the d egree s 0 and 1, b ecause Lie( W ) is a f ree Lie algebra. Recall that the univ ersal en v eloping algebra U (Lie( W )) = T ( W ) is the free asso ciativ e algebra generated by W . The cohomology in d egree 0 is equal to C and is not in teresting; the cohomology in d egree 1 is “v ery b ig” and “v ery in teresting” . W e consider the degree 1 cohomolo gy of b oth sides of (A.6) as Lie algebras. The existence of an L ∞ morphism (A.6) implies that these degree 1 cohomology are isomor- phic as Li e algebr as . W e sho w that the latter statemen t is false. Consequ ently , the statemem t of Th eo rem A. 1 is true. 33 The degree 1 cohomology is easy to ﬁn d. F or the r.h.s., it is g 2 = Der Asso c ( T ( W )) / Inn the Lie algebra of th e outer d eriv ations of the free asso ciativ e algebra T ( W ). F or the l.h.s., it is g 1 = Der Po is ( S (Lie( W ))) / Inn the Lie algebra of the outer P oisson d eriv ations of the fr ee Poisson algebra Pois( W ) = S (Lie( W )). Assuming the con trary to the statemen t of Theorem A. 1 , there exists a gl ( W )-equiv arian t isomorphism φ : g 1 → g 2 (A.7) One has: Pr oposition A. 3. Ther e do es not exist any gl ( W ) -e quivariant isomorphism φ of Lie algebr as, as in (A.7) . Pr o of. It is r ather easy to see, in fact. The idea is to consider the L ie subalgebra Der( S ( W )) of g 1 , as follo ws. An y Po isson der iv atio n D of S (Lie( W )) is u niquely d ete rmined by its restriction to the generators W ⊂ Poi s( W ). I n general, it is a linear map D : V → S (Lie( W )) The vecto r sp ac e of all su c h deriv atio ns con tains a su bspace of the deriv ations, giv en b y a linear map D : W → S ( W ) (A.8) These d eriv ations form a Lie sub alg ebra in Der Po is ( S (Lie( W ))), isomorphic to the Lie alge bra V ect( W ∗ ) = Der Comm ( S ( W )) of p olynomial v ector ﬁelds on W ∗ . Consider the restriction Der Comm ( S ( W )) → Der Po is ( S (Lie( W ))) / Inn φ − → Der Asso c ( T ( W )) / Inn (A.9) It is clearly an im b edding. Lemma A. 4. Ther e do es not exist any gl ( W ) -e quivariant imb e dding of Lie algebr as φ 0 : Der Comm ( S ( W )) → Der Asso c ( T ( W )) / Inn . Pr o of. 5 W e ca n (n ot canonically) im b ed Der Asso c ( T ( W )) / Inn ֒ → Der Asso c ( T ( W )), as gl ( W )- mo dules. Th en any im b edding φ 0 giv es an imb edding of gl ( W )-mod ules ˆ φ 0 : Der Comm ( S ( W )) → Der Asso c ( T ( W )) (A.10 ) 5 The author is indeb ted to Maxim Kontsevich for the pro of of Lemma A. 4 given b elo w. 34 An y deriv ation in Der Comm ( S ( W )) is a linear com bination of the homogeneous maps W → S N ( W ), and an y deriv ation in Der Asso c ( T ( W )) is a linear com b ination of the homogeneous maps W → T N ( W ). The m ap ˆ φ 0 , b eing gl ( W )-equiv arian t, preserv es the homegeneit y degree N . Thus, it is giv en by its gl ( W )-equiv atia n t comp onen ts ˆ φ ( N ) 0 : W ∗ ⊗ S N ( W ) → W ∗ ⊗ T N ( W ) (A.11) T ak e N ≪ dim W so that the inv arian ts can b e describ ed b y th e H.W eyl theorem, without any relations on them. As the ﬁrst step, we describ e all gl ( W )-in v arian t maps ˆ φ ( N ) 0 , by the H.W eyl inv arian t theo- rem. One easily sees that the vect or sp ace of such inv ariants has dimension N + 1, and it can b e describ ed as follo ws . Th e ﬁrst basic inv ariant t 1 is obtained as the p ost-composition with the symmetrization m ap W → S N ( W ) PBW − − − → T N ( W ) (A.12) and the r emai ning basic inv ariants t 2 , . . . , t N +1 are obtained as the comp osition(s): W ∗ ⊗ S N ( W ) div − − → S N − 1 ( W ) PBW − − − → T N − 1 ( W ) e ∈ W ∗ ⊗ W − − − − − − → W ∗ ⊗ T N ( W ) (A.13) where the rightmost map inserts the “new” factor W after ℓ ﬁrs t fact ors, on the ( ℓ + 1)-p osition, ℓ = 0 , . . . , N − 1, and e ∈ W ∗ ⊗ W is the canonical inv ariant . T urn back to the statement of Lemma. If an imb edding φ 0 existed, the comp onent s of the corresp onding im b edding ˆ φ ( N ) 0 w ere linear com binations of the describ ed N + 1 basic inv ariants, ˆ φ ( N ) 0 = a 1 N t 1 + a 2 N t 2 + · · · + a N +1 ,N t N +1 , a ij ∈ C (A.14) and these comp onents wo uld deﬁne a map of Lie algebras, m odu lo the inn er deriv ations of T ( W ). T o get a con tradiction, it is easier computing with div ergence zero vecto r ﬁelds, so that t 2 = · · · = t N +1 = 0 on suc h vec tor ﬁelds. That is, for any tw o div ergence 0 (homogeneous) ve ctor ﬁ elds v 1 , v 2 of degrees M , N m uc h smaller than dim W , one wo uld ha v e [ t 1 ( v 1 ) , t 1 ( v 2 )] = c M N t 1 ( { v 1 , v 2 } ) mo dulo the inner derivations , c M N ∈ C (A.15) where t 1 is the symmetrization m ap , see (A.12). W e p resen t t w o quadr atic div ergence 0 vecto r ﬁelds v 1 , v 2 for which (A.15 ) f ails. Condider dim W ≥ 3, let x, y , z b e the ﬁ r st th ree co ordinates. T ak e v 1 = y 4 ∂ ∂ x , v 2 = z 2 ∂ ∂ y 35 The comm utator { v 1 , v 2 } = 4 z 2 y 3 ∂ ∂ x On the other hand , [ t 1 ( v 1 ) , t ( v 2 )] = ( z 2 y 3 + y z 2 y 2 + y 2 z 2 y + y 3 z 2 ) ∂ ∂ x and it is not equal to c · t 1 (4 z 2 y 3 ∂ ∂ x ) ev en mo dulo th e inner deriv ations. (The latter statemen t is true b ecause all inner d er iv atio ns in Der Asso c T ( W ) should con tain x as a factor in the co eﬃcien t at ∂ ∂ x ). Prop osition A. 3 is prov en. Theorem A. 1 is p r o ve n. Remark A. 5. If there existed a gl ( W )-equiv ariant isomorphism φ : g 1 → g 2 of Lie alg ebras, the Kont sevic h formalit y theorem w ould follo w from it. It can b e sho wn by taking the free Koszul resolution of the p olynomial algebra S ( W ). Let R → S ( W ) b e this r esol ution. Th en R is a free asso ciat iv e dg algebra w h ose und erlying graded v ector space is S (Lie( S ( W [1])[ − 1])) . The diﬀerentia l in R is tangen tial to Lie( S ( W [1])[ − 1]). T ake W 1 = S ( W [1])[ − 1]. It is easy to see th at ( g 1 ( W 1 ) , d R ) ≃ T poly ( W ∗ ), ( g 2 ( W 1 ) , d R ) ≃ Ho c h( S ( W )) as d g Lie algebras. This idea, suggested by Boris F eigin around ’98-’99, w as a sour ce of inspiration for the author’s w ork on this pap er. Remark A. 6. It wo uld b e v ery interesting to compu te the Chev alley-Eilen b erg cohomology H q CE ( g 1 ( W ) , gl ( W ); g 1 ( W )) with the adjoin t co eﬃcien ts, at least in the inductiv e limit dim W → ∞ . Th e argumen t from the previous Remark sh ows that the latter stable cohomology is closely related with the stable cohomology H q CE ( T poly ( W ) , gl ( W ); T poly ( W )), whic h has b een activ ely studied in the recen t y ears, see e.g. [W]. B Where does T amarkin’s pr oof f ail f or an infin ite- dimensional V Here w e sho w where T amarkin’s pro of of th e Kon tsevic h formalit y for S ( V ∗ ) d oes not work, where dim V = ∞ . F or the deﬁnitions of S ( V ∗ ), T ﬁn ( V ), Ho ch q ﬁn ( S ( V ∗ )) see S ect ions 1.1, 1.2, and 1.3. The answer to the question in the title of this App end ix is the follo w ing. F or a ﬁ nite- dimensional vecto r space W , the graded commutat iv e algebra T poly ( W ∗ ) of p olyv ector ﬁelds is 36 equal to S ( W ⊕ W ∗ [ − 1]) and therefore is smo oth graded commuta tiv e algebra. Its smo othness is crucial in the pro of of one of th e key steps in the T amarkin p r oof, see Prop osition B. 1 b elo w. Its smo othness implies the v anish ing of higher An dre-Quillen cohomology . On the other side, for an inﬁnite-dimensional V , the graded commutati v e alge bra T ﬁn ( V ∗ ) is not of the form S ( L ), for a grad ed vect or space L , as follo ws immediately from its deﬁn itio n. In fact, it fails to b e smo oth. It r esults in presence of non-trivial higher (Andre-Quillen-lik e) cohomology , wh ic h breaks the pro of of Prop osition B. 1 . W e p ro vide more d eta il b elo w. The ke y-p oin t of T amarkin’s p roof of the Kontsvic h form ali t y is the follo wing f act : Pr oposition B. 1. L et W b e a ﬁnite-dimensional ve ctor sp ac e over a ﬁeld of char acteris- tic 0. Con sider the p olynomials p olyve ctor ﬁelds T poly ( W ) as a Gerstenhab e r algebr a (aka 2-algebr a). Then the c ohomolo gy of the Aﬀ ( W ) -invariant p art of the deformation c omplex Def hoe 2 ( T poly ( W )) of T poly ( W ) as a homotopy 2-algebr a, vanishes in al l de gr e es. W e r ecall th e main steps in the pro of, b ecause the failure of T amarkin’s pro of for an in ﬁnite- dimensional vecto r sp ace V comes fr om th e failure of th is Pr oposition. Let W b e a ﬁnite-dimensional vect or space. Th e d eformatio n complex Def hoe 2 ( T poly ( W )) = Def ( e 2 → End Op ( T poly ( W ))) is computed as the cod eriv ations Co der(Cobar e 2 ( T poly ( W ))). The Koszul d u al co op erad to e 2 is equal to e ∗ 2 {− 2 } , and Cobar e 2 ( X ) = ( e ∗ 2 {− 2 } ◦ X , d ), where the diﬀeren tial d comes from the e 2 -algebra stru cture on X . The diﬀeren tial d = d Comm + d Lie has t w o comp onen ts, expressed via the graded comm utativ e p rod uct, and via the shifted by 1 Lie brac k et, corresp ondingly . See ??? for more detail. F or any graded ve ctor s p ace T , one has e 2 ◦ T = S ∗ (Lie ∗ ( T [ − 1])[1]) (where S ∗ ( − ) and Lie ∗ ( − ) stand for the cofree commutat iv e (corresp., Lie) coalgebra), and th us e ∗ 2 {− 2 } ( T poly ( W )) = ( e ∗ 2 ◦ T poly ( W ) { 2 } ) {− 2 } = S ∗ (Lie ∗ ( T poly ( W )[1])[1])[ − 2] The corresp onding complex of co deriv ations is equal Def hoe 2 ( T poly ( W )) = Hom k  S ∗ (Lie ∗ ( T poly ( W )[1])[1])[ − 2] , T poly ( W )  = Hom k  S ∗ (Lie ∗ ( T poly ( W )[1])[1]) , T poly ( W )  [2] The diﬀeren tial has t w o comp onents, d = d Comm + d Lie . One uses the sp ectral sequence of the bicomplex, and computes the cohomology of d Comm at the ﬁr st s tep. One has:  Def hoe 2 ( T poly ( W )) , d Comm  = Y k ≥ 1  Hom k ( S k (Lie ∗ ( T poly ( W )[1])[1]) , T poly )[2] , d Comm  (B.1) 37 (the comp onen t with k = 0 do es not ﬁ gu r e out in th e d eform at ion complex). Eac h comp onen t, corresp onded to an y k ≥ 1, is a s u b-complex with resp ect to th e diﬀeren tial d Comm . The simplest one among them is corresp onded to the case k = 1, it is C Harr ( T poly ( W )) =  Hom k (Lie( T poly ( W )[1] , T poly ( W ))[1] , d Comm  (B.2) It is the Harrison complex, computing the And re-Quillen cohomolog y . It is well-kno wn that for any smo oth over k (graded) commuta tiv e algebra A , the complex C Harr ( A ) has vanishing c ohomolo gy exc ept for the de gr e e 0 , where its cohomology is equal to Der Comm ( A ): H q ( C Harr ( A )) = Der Comm ( A )[0] (B.3) Analogously , for higher k , one has H q (Hom k ( S k (Lie( A [1])[1]) , A [2]) , d Comm )) = S k A (Der Comm ( A )[ − 2])[2 ] (B.4) The rest of the computation is not relev an t for ou r goal in this App endix, and we r efer th e in terested reader to [T] f or m ore detail on the T amarkin computation. The matter is that the smo othness of A is essenti al for v anish ing of the h igher cohomolog y in (B.3) an d (B.4). T urn bac k to our case of an in vin ite -dimensional vec tor space V , as in Section 1.1. Then the p olyv ector ﬁelds T ﬁn ( V ) is not smo oth as a c ommuta tive algebr a . Indeed, to establish the smo othness of T poly ( W ) f or a ﬁ nite-dimensional W , w e use T poly ( W ) = S ( W ∗ ⊕ W [ − 1]) (B.5) whic h is smo oth as a p olynomial algebra. F or T ﬁn ( V ) (see Section 1.2 for the deﬁn itio n), one d o es not ha v e an equalit y similar to (B.5): T ﬁn ( V ) 6 = S ( − ) In fact, th e graded comm utativ e algebra T ﬁn ( V ) is n ot smo oth. It implies that the h igher homology of the complexes Hom k (( S k (Lie( A [1])[1]) , A [2]) , d Comm ), k ≥ 1, ma y not v anish . It results in the failure of th e ov erall argument. Reference s [D] V. Drinfeld, On quasitriangular quasi-Hopf algebras and a group closely connected with Gal( ¯ Q / Q ), Leningrad Math. J., vo l. 2 (1991), no. 4, [Du] M. Duﬂo, Caract ´ eres des alg ` ebres d e Lie r ´ esolubles, C.R. A c ad. Sci . , 269 (1969), s´ erie a, pp. 437-438, 38 [GM] W.Goldman, J.Millson, The deformation theory of represen tatio ns of fundamental groups of compact Kahler manifolds, Pub lications mathematiques de lI.H.E.S., tome 67 (1988), p. 43-96 [K97] M. Kon tsevic h, Deformation quanti zation of Poisson manifolds I, preprint q-alg/97090 40 , [K99] M. Kon tsevic h , Op erads and Motiv es in d eformatio n quan tizati on, preprin t q-alg/9 90405 5 , [Ka] V. Kathotia, Kontsevic h’s Un iversal F orm ula for Deformation Quantiza tion and the Campb ell-Bak er-Hausd orﬀ F ormula, I, arxiv:98111 74, [Kel] B. K elle r, Deriv ed in v ariance of higher s tructures on the Ho c hsc hild complex, av ailable at www.math.jussieu.f r / ˜ k eller, [KMW] A.Khoroshkin, S.Merkulo v, T.Willwa c her, O n quantiz able o dd Lie bialgebras, archiv e preprint 1512.04710 [L] J.-L. Lo da y , Cyclic Homolo gy , Sp r inger-V erlag Series of Comprehens ive Stud ies in Math- ematics, V ol. 301, 2nd Edition, Sprin ge r, 1998, [MW1] S.Merkulo v, T.Wilw ac her, Deformation theory of Lie bialgebra prop erads, arc hiv e preprint 1605.01282 [MW2] S.Merkulo v, T .Wil lw ac her , An explicit t w o step quan tizatio n of P oisson structures and Lie bialgebras, arc hiv e pr ep rin t 1612.00368 [PT] M. P evzner, Ch. T orossian, Isomorphisme de Duﬂo et la cohomologie tangen tielle , arXiv: 03101 28, Journal Ge ometry and Physics , v ol. 51(4), 2004, p p . 486- 505, [Sh1] B. Shoikhet, V anishin g of the Kontsevic h integ rals of the wheels, arXiv:000708 0, L ett. Math. Phys. , 56(200 1) 2, 141-149 , [Sh2] B. Shoikhet, The PBW p r operty for associativ e algebras as an in tegrabilit y condition, Mathematic al R e se ar ch L etters 21:6 (2014), 1407-1 434 [Sh3] B. Shoikhet, Koszul dualit y in deformation quantiz ation and T amarkin ’s app roac h to the Kon tsevic h formalit y , A dvanc es in M ath . , 224 (2010), 731-771 [T] D. T amarkin , Another pro of of M. Kont sevic h formalit y theorem, preprint math 9803 025. [W] T. Willw ac her, M. Kon tsevic h’s graph complex and th e Grothendiec k-T eic h mller Lie al- gebra, Invent. Math. 200 (2015), n o. 3, 671-760 39 Universiteit Antwerpen, Campus Middelheim, Wiskunde en Informa tica, Gebouw G Middelheimlaan 1, 2020 Antwerpen, Belgi ¨ e e-mail : Boris.Sh oikhet@uantwe rpen.be 40

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