FORMALITY FOR ALGEBROID STACKS PAUL BRESSLER, ALEXANDER GOROKHOVSKY, RYSZARD NEST, AND BORIS TSYGAN Abstract. We extend the formality theorem of M. Kontsevich from deformations of the structure sheaf on a manifold to deformations of gerbes. 1. Introduction In the fundamental paper [11] M. Kontsevich showed that the set of equivalence classes of formal deformations the algebra of functions on a manifold is in one-to-one correspondence with the set of equivalence classes of formal Poisson structures on the manifold. This result was obtained as a corollary of the formality of the Hochschild complex of the algebra of functions on the manifold conjectured by M. Kontsevich (cf. [10]) and proven in [11]. Later proofs by a different method were given in [14] and in [5]. In this paper we extend the formality theorem of M. Kontsevich to deformations of gerbes on smooth manifolds, using the method of [5]. Let X be a smooth manifold; we denote by OX the sheaf of complex valued C ∞ functions on X. For a twisted form S of OX regarded 3 as an algebroid stack (see Section 2.5) we denote by [S]dR ∈ HdR (X) the de Rham class of S. The main result of this paper establishes an equivalence of 2-groupoid valued functors of Artin C-algebras between Def(S) (the formal deformation theory of S, see [2]) and the Deligne 2-groupoid of Maurer-Cartan elements of L∞ -algebra of multivector fields on X twisted by a closed three-form representing [S]dR : Theorem 6.1. Suppose that S is a twisted form of OX . Let H be a 3 closed 3-form on X which represents [S]dR ∈ HdR (X). For any Artin algebra R with maximal ideal mR there is an equivalence of 2-groupoids MC2 (s(OX )H ⊗ mR ) ∼ = Def(S)(R) natural in R. A. Gorokhovsky was partially supported by NSF grant DMS-0400342. B. Tsygan was partially supported by NSF grant DMS-0605030. 1 2 P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN Here, s(OX )H denotes the L∞ -algebra of multivector fields with the trivial differential, the binary operation given by Schouten bracket, the ternary operation given by H (see 5.3) and all other operations equal to zero. As a corollary of this result we obtain that the isomorphism classes of formal deformations of S are in a bijective correspondence with equivalence classes of the formal twisted Poisson structures defined by P. Severa and A. Weinstein in [13]. The proof of the Theorem proceeds along the following lines. As a starting point we use the construction of the Differential Graded Lie Algebra (DGLA) controlling the deformations of S. This construction was obtained in [1, 2]. Next we construct a chain of L∞ -quasiisomorphisms between this DGLA and s(OX )H , using the techniques of [5]. Since L∞ -quasi-isomorphisms induce equivalences of respective Deligne groupoids, the result follows. The paper is organized as follows. Section 2 contains the preliminary material on jets and deformations. Section 3 describes the results on the deformations of algebroid stacks. Section 4 is a short exposition of [5]. Section 5 contains the main technical result of the paper: the construction of the chain of quasi-isomorphisms mentioned above. Finally, in Section 6 the main theorem is deduced from the results of Section 5. The paper was written while the first author was visiting Max-PlankInstitut für Mathematik, Bonn. 2. Preliminaries 2.1. Notations. Throughout this paper, unless specified otherwise, X will denote a C ∞ manifold. By OX we denote the sheaf of complexvalued C ∞ functions on X. A•X denotes the sheaf of differential forms on X, and TX the sheaf of vector fields on X. For a ring K we denote by K × the group of invertible elements of K. 2.2. Jets. Let pri : X × X → X, i = 1, 2, denote the projection on the ith factor. Let ∆X : X → X × X denote the diagonal embedding. Let IX := ker(∆∗X ). For a locally-free OX -module of finite rank E let   k+1 −1 k pr2 E , JX (E) := (pr1 )∗ OX×X /IX ⊗pr−1 2 OX JXk := JXk (OX ) . It is clear from the above definition that JXk is, in a natural way, a commutative algebra and JXk (E) is a JXk -module. Let FORMALITY FOR ALGEBROID STACKS 3 1(k) : OX → JXk denote the composition pr∗ 1 OX −−→ (pr1 )∗ OX×X → JXk In what follows, unless stated explicitly otherwise, we regard JXk (E) as a OX -module via the map 1(k) . Let j k : E → JXk (E) denote the composition e7→1⊗e E −−−−→ (pr1 )∗ OX×X ⊗C E → JXk (E) The map j k is not OX -linear unless k = 0. k+1 l+1 induces the surjective map → IX For 0 ≤ k ≤ l the inclusion IX l k k πl,k : JX (E) → JX (E). The sheaves JX (E), k = 0, 1, . . . together with the maps πl,k , k ≤ l form an inverse system. Let JX (E) = JX∞ (E) := lim JXk (E). Thus, JX (E) carries a natural topology. ←− The maps 1(k) (respectively, j k ), k = 0, 1, 2, . . . are compatible with the projections πl,k , i.e. πl,k ◦ 1(l) = 1(k) (respectively, πl,k ◦ j l = j k ). Let 1 := lim 1(k) , j ∞ := lim j k . ←− ←− Let d1 : OX×X ⊗pr−1 pr−1 2 E −→ 2 OX 1 pr−1 OX×X ⊗pr−1 pr−1 1 AX ⊗pr−1 2 E 1 OX 2 OX denote the exterior derivative along the first factor. It satisfies k+1 −1 1 k ⊗pr−1 pr−1 ⊗pr−1 pr−1 IX d1 (IX 2 E 2 E) ⊂ pr1 AX ⊗pr−1 2 OX 1 OX 2 OX for each k and, therefore, induces the map (k) d1 : J k (E) → A1X ⊗OX J k−1 (E) (k) The maps d1 for different values of k are compatible with the maps πl,k giving rise to the canonical flat connection ∇can : JX (E) → A1X ⊗OX JX (E) . 4 P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN 2.3. Deligne groupoids. In [4] P. Deligne and, independently, E. Getzler in [8] associated to a nilpotent DGLA g concentrated in degrees grater than or equal to −1 the 2-groupoid, referred to as the Deligne 2-groupoid and denoted MC2 (g) in [1], [2] and below. The objects of MC2 (g) are the Maurer-Cartan elements of g. We refer the reader to [8] (as well as to [2]) for a detailed description. The above notion was extended and generalized by E. Getzler in [7] as follows. To a nilpotent L∞ -algebra g Getzler associates a (Kan) simplicial set γ• (g) which is functorial for L∞ morphisms. If g is concentrated in degrees greater than or equal to 1 − l, then the simplicial set γ• (g) is an l-dimensional hypergroupoid in the sense of J.W. Duskin (see [6]) by [7], Theorem 5.4. Suppose that g is a nilpotent L∞ -algebra concentrated in degrees grater than or equal to −1. Then, according to [6], Theorem 8.6 the simplicial set γ• (g) is the nerve of a bigroupoid, or, a 2-groupoid in our terminology. If g is a DGLA concentrated in degrees grater than or equal to −1 this 2-groupoid coincides with MC2 (g) of Deligne and Getzler alluded to earlier. We extend our notation to the more general setting of nilpotent L∞ -algebras as above and denote by MC2 (g) the 2-groupoid furnished by [6], Theorem 8.6. For an L∞ -algebra g and a nilpotent commutative algebra m the L∞ -algebra g ⊗ m is nilpotent, hence the simplicial set γ• (g ⊗ m) is defined and enjoys the following homotopy invariance property ([7], Proposition 4.9, Corollary 5.11): Theorem 2.1. Suppose that f : g → h is a quasi-isomorphism of L∞ algebras and let m be a nilpotent commutative algebra. Then the induced map γ• (f ⊗ Id) : γ• (g ⊗ m) → γ• (h ⊗ m) is a homotopy equivalence. 2.4. Algebroid stacks. Here we give a very brief overview, referring the reader to [3, 9] for the details. Let k be a field of characteristic zero, and let R be a commutative k-algebra. Definition 2.2. A stack in R-linear categories C on X is an R-algebroid stack if it is locally nonempty and locally connected, i.e. satisfies (1) any point x ∈ X has a neighborhood U such that C(U ) is nonempty; (2) for any U ⊆ X, x ∈ U , A, B ∈ C(U ) there exits a neighborhood V ⊆ U of x and an isomorphism A|V ∼ = B|V . For a prestack C we denote by Ce the associated stack. FORMALITY FOR ALGEBROID STACKS 5 For a category C denote by iC the subcategory of isomorphisms in C; equivalently, iC is the maximal subgroupoid in C. If C is an algebroid e is stack then the stack associated to the substack of isomorphisms iC a gerbe. For an algebra K we denote by K + the linear category with a single object whose endomorphism algebra is K. For a sheaf of algebras K on X we denote by K+ the prestack in linear categories given by f+ denote the associated stack. Then, K f+ is an U 7→ K(U )+ . Let K op algebroid stack equivalent to the stack of locally free K -modules of rank one. By a twisted form of K we mean an algebroid stack locally equivalent f+ . It is easy to see that the equivalence classes of twisted forms of to K K are bijective correspondence with H 2 (X; Z(K)× ), where Z(K) denotes the center of K. 2.5. Twisted forms of O. Twisted forms of OX are in bijective cor× -gerbes: if S is a twisted form of OX , the correrespondence with OX sponding gerbe is the substack iS of isomorphisms in S. We shall not make a distinction between the two notions. The equivalence classes of twisted forms of OX are in bijective cor× ). The composition respondence with H 2 (X; OX log j∞ × × /C× −→ OX /C −→ DR(JX /OX ) → OX OX × ) → H 2 (X; DR(JX /OX )) ∼ induces the map H 2 (X; OX = H 2 (Γ(X; A•X ⊗ can JX /OX ), ∇ ). We denote by [S] the image in the latter space of the class of S. The short exact sequence 1 0 → OX − → JX → JX /OX → 0 gives rise to the short exact sequence of complexes 0 → Γ(X; A•X ) → Γ(X; DR(JX )) → Γ(X; DR(JX /OX )) → 0, hence to the map (connecting homomorphism) H 2 (X; DR(JX /OX )) → 3 HdR (X). Namely, if B ∈ Γ(X; A2X ⊗ JX ) maps to B ∈ Γ(X; A2X ⊗ JX /OX ) which represents [S], then there exists a unique H ∈ Γ(X; A3 ) such that ∇can B = DR(1)(H). The form H is closed and represents the image of the class of B under the connecting homomorphism. Notation. We denote by [S]dR the image of [S] under the map 3 H 2 (X; DR(JX /OX )) → HdR (X). 6 P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN 3. Deformations of algebroid stacks 3.1. Deformations of linear stacks. Here we describe the notion of 2-groupoid of deformations of an algebroid stack. We follow [2] and refer the reader to that paper for all the proofs and additional details. For an R-linear category C and homomorphism of algebras R → S we denote by C ⊗R S the category with the same objects as C and morphisms defined by HomC⊗R S (A, B) = HomC (A, B) ⊗R S. For a prestack C in R-linear categories we denote by C ⊗R S the prestack associated to the fibered category U 7→ C(U ) ⊗R S. Lemma 3.1 ([2], Lemma 4.13). Suppose that A is a sheaf of R-algebras and C is an R-algebroid stack. Then C^ ⊗R S is an algebroid stack. Suppose now that C is a stack in k-linear categories on X and R is a commutative Artin k-algebra. We denote by Def(C)(R) the 2-category with • objects: pairs (B, ̟), where B is a stack in R-linear categories flat over R and ̟ : B^ ⊗R k → C is an equivalence of stacks in k-linear categories • 1-morphisms: a 1-morphism (B (1) , ̟(1) ) → (B (2) , ̟(2) ) is a pair (F, θ) where F : B (1) → B (2) is a R-linear functor and θ : ̟(2) ◦ (F ⊗R k) → ̟(1) is an isomorphism of functors • 2-morphisms: a 2-morphism (F ′ , θ′ ) → (F ′′ , θ′′ ) is a morphism of R-linear functors κ : F ′ → F ′′ such that θ′′ ◦ (Id̟(2) ⊗ (κ ⊗R k)) = θ′ The 2-category Def(C)(R) is a 2-groupoid. Let B be a prestack on X in R-linear categories. We say that B is flat if for any U ⊆ X, A, B ∈ B(U ) the sheaf HomB (A, B) is flat (as a sheaf of R-modules). Lemma 3.2 ([2], Lemma 6.2). Suppose that B is a flat R-linear stack on X such that B^ ⊗R k is an algebroid stack. Then B is an algebroid stack. 3.2. Deformations of twisted forms of O. Suppose that S is a twisted form of OX . We will now describe the DGLA controlling the deformations of S. The complex Γ(X; DR(C • (JX )) = (Γ(X; A•X ⊗ C • (JX )), ∇can + δ) is a differential graded brace algebra in a canonical way. The abelian Lie algebra JX = C 0 (JX ) acts on the brace algebra C • (JX ) by derivations of degree −1 by Gerstenhaber bracket. The above action factors through an action of JX /OX . Therefore, the abelian Lie algebra FORMALITY FOR ALGEBROID STACKS 7 Γ(X; A2X ⊗ JX /OX ) acts on the brace algebra A•X ⊗ C • (JX ) by derivations of degree +1. Following longstanding tradition, the action of an element a is denoted by ia . Due to commutativity of JX , for any ω ∈ Γ(X; A2X ⊗ JX /OX ) the operation ιω commutes with the Hochschild differential δ. If, moreover, ω satisfies ∇can ω = 0, then ∇can + δ + iω is a square-zero derivation of degree one of the brace structure. We refer to the complex Γ(X; DR(C • (JX ))ω := (Γ(X; A•X ⊗ C • (JX )), ∇can + δ + iω ) as the ω-twist of Γ(X; DR(C • (JX )). Let gDR (J )ω := Γ(X; DR(C • (JX ))[1])ω regarded as a DGLA. The following theorem is proved in [2] (Theorem 1 of loc. cit.): Theorem 3.3. For any Artin algebra R with maximal ideal mR there is an equivalence of 2-groupoids MC2 (gDR (JX )ω ⊗ mR ) ∼ = Def(S)(R) natural in R. 4. Formality We give a synopsis of the results of [5] in the notations of loc. cit. Let k be a field of characteristic zero. For a k-cooperad C and a complex of k-vector spaces V we denote by FC (V ) the cofree C-coalgebra on V . We denote by e2 the operad governing Gerstenhaber algebras. The latter is Koszul, and we denote by e2 ∨ the dual cooperad. For an associative k-algebra A the Hochschild complex C • (A) has a canonical structure of a brace algebra, hence a structure of homotopy e2 -algebra. The latter structure is encoded in a differential (i.e. a coderivation of degree one and square zero) M : Fe2 ∨ (C • (A)) → Fe2 ∨ (C • (A))[1]. Suppose from now on that A is regular commutative algebra over a field of characteristic zero (the regularity assumption is not needed for the constructions). Let V • (A) = Sym•A (Der(A)[−1]) viewed as a complex with trivial differential. In this capacity V • (A) has a canonical structure of an e2 -algebra which gives rise to the differential dV • (A) on Fe2 ∨ (V • (A)); we have: Be2 ∨ (V • (A)) = (Fe2 ∨ (V • (A)), dV • (A) ) (see [5], Theorem 1 for notations). In addition, the authors introduce a sub-e2 ∨ -coalgebra Ξ(A) of both Fe2 ∨ (C • (A)) and Fe2 ∨ (V • (A)). We denote by σ : Ξ(A) → Fe2 ∨ (C • (A)) and ι : Ξ(A) → Fe2 ∨ (V • (A)) respective inclusions and identify Ξ(A) 8 P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN with its image under the latter one. By [5], Proposition 7 the differential dV • (A) preserves Ξ(A); we denote by dV • (A) its restriction to Ξ(A). By Theorem 3, loc. cit. the inclusion σ is a morphism of complexes. Hence, we have the following diagram in the category of differential graded e2 ∨ -coalgebras: (4.0.1) σ ι (Fe2 ∨ (C • (A)), M ) ← − (Ξ(A), dV • (A) ) − → Be2 ∨ (V • (A)) Applying the functor Ωe2 (adjoint to the functor Be2 ∨ , see [5], Theorem 1) to (4.0.1) we obtain the diagram Ωe (σ) 2 (4.0.2) Ωe2 (Fe2 ∨ (C • (A)), M ) ←−− −− Ωe (ι) Ωe2 (Ξ(A), dV • (A) ) −−2−→ Ωe2 (Be2 ∨ (V • (A))) of differential graded e2 -algebras. Let ν = ηe2 ◦ Ωe2 (ι), where ηe2 : Ωe2 (Be2 ∨ (V • (A))) → V • (A) is the counit of adjunction. Thus, we have the diagram (4.0.3) Ωe (σ) ν 2 Ωe2 (Fe2 ∨ (C • (A)), M ) ←−− −− Ωe2 (Ξ(A), dV • (A) ) − → V • (A) of differential graded e2 -algebras. Theorem 4.1 ([5], Theorem 4). The maps Ωe2 (σ) and ν are quasiisomorphisms. Additionally, concerning the DGLA structures relevant to applications to deformation theory, deduced from respective e2 -algebra structures we have the following result. Theorem 4.2 ([5], Theorem 2). The DGLA Ωe2 (Fe2 ∨ (C • (A)), M )[1] and C • (A)[1] are canonically L∞ -quasi-isomorphic. Corollary 4.3 (Formality). The DGLA C • (A)[1] and V • (A)[1] are L∞ -quasi-isomorphic. 4.1. Some (super-)symmetries. For applications to deformation theory of algebroid stacks we will need certain equivariance properties of the maps described in 4. For a ∈ A let ia : C • (A) → C • (A)[−1] denote the adjoint action (in the sense of the Gerstenhaber bracket and the identification A = C 0 (A)). It is given by the formula n X ia D(a1 , . . . , an ) = (−1)k D(a1 , . . . , ai , a, ak+1 , . . . , an ). i=0 The operation ia extends uniquely to a coderivation of Fe2 ∨ (C • (A)); we denote this extension by ia as well. Furthermore, the subcoalgebra Ξ(A) is preserved by ia . FORMALITY FOR ALGEBROID STACKS 9 Since the operation ia is a derivation of the cup product as well as of all of the brace operations on C • (A) and the homotopy-e2 -algebra structure on C • (A) given in terms of the cup product and the brace operations it follows that ia anti-commutes with the differential M . Hence, the coderivation ia induces a derivation of the differential graded e2 -algebra Ωe2 (Fe2 ∨ (C • (A)), M ) which will be denoted by ia as well. For the same reason the DGLA Ωe2 (Fe2 ∨ (C • (A)), M )[1] and C • (A)[1] are quasi-isomorphic in a way which commutes with the respective operations ia . On the other hand, let ia : V • (A) → V • (A)[−1] denote the adjoint action in the sense of the Schouten bracket and the identification A = V 0 (A). The operation ia extends uniquely to a coderivation of Fe2 ∨ (V • (A)) which anticommutes with the differential dV • (A) because ia is a derivation of the e2 -algebra structure on V • (A). We denote this coderivation as well as its unique extension to a derivation of the differential graded e2 -algebra Ωe2 (Be2 ∨ (V • (A))) by ia . The counit map ηe2 : Ωe2 (Be2 ∨ (V • (A))) → V • (A) commutes with respective operations ia . The subcoalgebra Ξ(A) of Fe2 ∨ (C • (A)) and Fe2 ∨ (V • (A)) is preserved by the respective operations ia . Moreover, the restrictions of the two operations to Ξ(A) coincide, i.e. the maps in (4.0.1) commute with ia and, therefore, so do the maps in (4.0.2) and (4.0.3). 4.2. Deformations of O and Kontsevich formality. Suppose that X is a manifold. Let OX (respectively, TX ) denote the structure sheaf (respectively, the sheaf of vector fields). The construction of the diagram localizes on X yielding the diagram of sheaves of differential graded e2 -algebras (4.2.1) Ωe (σ) ν 2 Ωe2 (Fe2 ∨ (C • (OX )), M ) ←−− −− Ωe2 (Ξ(OX ), dV • (OX ) ) − → V • (OX ), where C • (OX ) denotes the sheaf of multidifferential operators and V • (OX ) := Sym•OX (TX [−1]) denotes the sheaf of multivector fields. Theorem 4.1 extends easily to this case stating that the morphisms Ωe2 (σ) and ν in (4.2.1) are quasi-isomorphisms of sheaves of differential graded e2 -algebras. 5. Formality for the algebroid Hochschild complex 5.1. A version of [5] for jets. Let C • (JX ) denote sheaf of continuous (with respect to the adic topology) OX -multilinear Hochschild cochains on JX . Let V • (JX ) = Sym•JX (Dercont OX (JX )[−1]). 10 P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN Working now in the category of graded OX -modules we have the diagram (5.1.1) Ωe (σ) ν 2 Ωe2 (Fe2 ∨ (C • (JX )), M ) ←−− −− Ωe2 (Ξ(JX ), dV • (JX ) ) − → V • (JX ) of sheaves of differential graded OX -e2 -algebras. Theorem 4.1 extends easily to this situation: the morphisms Ωe2 (σ) and ν in (5.1.1) are quasi-isomorphisms. The sheaves of DGLA Ωe2 (Fe2 ∨ (C • (JX )), M )[1] and C • (JX )[1] are canonically L∞ -quasi-isomorphic. The canonical flat connection ∇can on JX induces a flat connection which we denote ∇can as well on each of the objects in the diagram (5.1.1). Moreover, the maps Ωe2 (σ) and ν are flat with respect to ∇can hence induce the maps of respective de Rham complexes DR(Ωe (σ)) (5.1.2) DR(Ωe2 (Fe2 ∨ (C • (JX )), M )) ←−−−2−−− DR(ν) DR(Ωe2 (Ξ(JX ), dV • (JX ) )) −−−→ DR(V • (JX )) where, for (K • , d) one of the objects in (5.1.1) we denote by DR(K • , d) the total complex of the double complex (A•X ⊗K • , d, ∇can ). All objects in the diagram (5.1.2) have canonical structures of differential graded e2 -algebras and the maps are morphisms thereof. The DGLA Ωe2 (Fe2 ∨ (C • (JX )), M )[1] and C • (JX )[1] are canonically L∞ -quasi-isomorphic in a way compatible with ∇can . Hence, the DGLA DR(Ωe2 (Fe2 ∨ (C • (JX )), M )[1]) and DR(C • (JX )[1]) are canonically L∞ quasi-isomorphic. 5.2. A version of [5] for jets with a twist. Suppose that ω ∈ Γ(X; A2X ⊗ JX /OX ) satisfies ∇can ω = 0. For each of the objects in (5.1.2) we denote by iω the operation which is induced by the one described in 4.1 and the wedge product on A•X . Thus, for each differential graded e2 -algebra (N • , d) in (5.1.2) we have a derivation of degree one and square zero iω which anticommutes with d and we denote by (N • , d)ω the ω-twist of (N • , d), i.e. the differential graded e2 -algebra (N • , d + iω ). Since the morphisms in (5.1.2) commute with the respective operations iω , they give rise to morphisms of respective ω-twists DR(Ωe (σ)) (5.2.1) DR(Ωe2 (Fe2 ∨ (C • (JX )), M ))ω ←−−−2−−− DR(ν) DR(Ωe2 (Ξ(JX ), dV • (JX ) ))ω −−−→ DR(V • (JX ))ω . Let F• A•X denote the stupid filtration: Fi A•X = A≥−i X . The filtration • • F• AX induces a filtration denoted F• DR(K , d)ω for each object (K • , d) FORMALITY FOR ALGEBROID STACKS 11 of (5.1.1) defined by Fi DR(K • , d)ω = Fi A•X ⊗ K • . As is easy to see, the associated graded complex is given by (5.2.2) Gr−p DR(K • , d)ω = (ApX ⊗ K • , Id ⊗ d). It is clear that the morphisms DR(Ωe2 (σ)) and DR(ν) are filtered with respect to F• . Theorem 5.1. The morphisms in (5.2.1) are filtered quasi-isomorphisms, i.e. the maps Gri DR(Ωe2 (σ)) and Gri DR(ν) are quasi-isomorphisms for all i ∈ Z. Proof. We consider the case of DR(Ωe2 (σ)) leaving Gri DR(ν) to the reader. The map Gr−p DR(Ωe2 (σ)) induced by DR(Ωe2 (σ)) on the respective associated graded objects in degree −p is equal to the map of complexes (5.2.3) Id⊗Ωe2 (σ) : ApX ⊗Ωe2 (Ξ(JX ), dV • (JX ) ) → ApX ⊗Ωe2 (Fe2 ∨ (C • (JX )), M ). The map σ is a quasi-isomorphism by Theorem 4.1, therefore so is Ωe2 (σ). Since ApX is flat over OX , the map (5.2.3) is a quasi-isomorphism.  Corollary 5.2. The maps DR(Ωe2 (σ)) and DR(ν) in (5.2.1) are quasiisomorphisms of sheaves of differential graded e2 -algebras. Additionally, the DGLA DR(Ωe2 (Fe2 ∨ (C • (JX )), M )[1]) and DR(C • (JX )[1]) are canonically L∞ -quasi-isomorphic in a way which commutes with the respective operations iω which implies that the respective ω-twists DR(Ωe2 (Fe2 ∨ (C • (JX )), M )[1])ω and DR(C • (JX )[1])ω are canonically L∞ quasi-isomorphic. 5.3. L∞ -structures on multivectors. The canonical pairing h , i : A1X ⊗ TX → OX extends to the pairing h , i : A1X ⊗ V • (OX ) → V • (OX )[−1] For k ≥ 1, ω = α1 ∧ . . . ∧ αk , αi ∈ A1X , i = 1, . . . , k, let Φ(ω) : Symk V • (OX )[2] → V • (OX )[k] denote the map given by the formula Φ(ω)(π1 , . . . , πk ) = (−1)(k−1)(|π1 |−1)+...+2|(πk−3 |−1)+(|πk−2 |−1) × X sgn(σ)hα1 , πσ(1) i ∧ · · · ∧ hαk , πσ(k) i, σ where |π| = l for π ∈ V l (OX ). For α ∈ OX let Φ(α) = α ∈ V 0 (OX ). 12 P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN Recall that a graded vector space W gives rise to the graded Lie algebra Der(coComm(W [1])). An element γ ∈ Der(coComm(W [1])) of degree one which satisfies [γ, γ] = 0 defines a structure of an L∞ algebra on W . Such a γ determines a differential ∂γ := [γ, .] on Der(coComm(W [1])), such that (Der(coComm(W [1])), ∂γ ) is a differential graded Lie algebra. If g is a graded Lie algebra and γ is the element of Der(coComm(g[1])) corresponding to the bracket on g, then (Der(coComm(g[1])), ∂γ ) is equal to the shifted Chevalley cochain complex C • (g; g)[1]. In what follows we consider the (shifted) de Rham complex A•X [2] as a differential graded Lie algebra with the trivial bracket. Lemma 5.3. The map ω 7→ Φ(ω) defines a morphism of sheaves of differential graded Lie algebras (5.3.1) Φ : A•X [2] → C • (V • (OX )[1]; V • (OX )[1])[1]. Proof. Recall the explicit formulas for the Schouten bracket. If f and g are functions and Xi , Yj are vector fields, then X ci . . . Xk Y1 . . . Yl + [f X1 . . . Xk , gY1 . . . Yl ] = (−1)k−i f Xk (g)X1 . . . X X j i (−1)j Yj (f )gX1 . . . Xk Y1 . . . Ybj . . . Yl + X i,j ci . . . Xk Y1 . . . Ybj . . . Yl (−1)i+j f gX1 . . . X Note that for a one-form ω and for vector fields X and Y (5.3.2) hω, [X, Y ]i − h[ω, X], Y i − hX, [ω, Y ]i = Φ(dω)(X, Y ) From the two formulas above we deduce by an explicit computation that hω, [π, ρ]i − h[ω, π], ρi − (−1)|π|−1 hπ, [ω, ρ]i = (−1)|π|−1 Φ(dω)(π, ρ) Note that Lie algebra cochains are invariant under the symmetric group acting by permutations multiplied by signs that are computed by the following rule: a permutation of πi and πj contributes a factor (−1)|πi ||πj | . We use the explicit formula for the bracket on the Lie algebra complex. [Φ, Ψ] = Φ ◦ Ψ − (−1)|Φ||Ψ| Ψ ◦ Φ X (Φ ◦ Ψ)(π1 , . . . , πk+l−1 ) = ǫ(I, J)Φ(Ψ(πi1 , . . . , πik ), πj1 , . . . , πjl−1 ) I,J FORMALITY FOR ALGEBROID STACKS 13 Here ` I = {i1 , . . . , ik }; J = {j1 , . . . , jl−1 }; i1 < . . . < ik ; j1 < . . . < jl−1 ; I J = {1, . . . , k + l − 1}; the sign ǫ(I, J) is computed by the same sign rule as above. The differential is given by the formula ∂Φ = [m, Φ] where m(π, ρ) = (−1)|π|−1 [π, ρ]. Let α = α1 . . . αk and β = β1 . . . βl . We see from the above that both cochains Φ(α) ◦ Φ(β) and Φ(β) ◦ Φ(α) are antisymmetrizations with respect to αi and βj of the sums X ±hα1 β1 , πp ihα2 , πi1 i . . . hαk , πik−1 ihβ2 , πj1 i . . . hβl , πjl−1 i I,J,p ` ` over all partitions {1, . . . , k+l−1} = I J {p} where i1 < . . . < ik−1 and j1 < . . . < jl−1 ; here hαβ, πi = hα, hβ, πii. After checking the signs, we conclude that [Φ(α), Φ(β)] = 0. Also, from the definition of the differential, we see that ∂Φ(α)(π1 , . . . , πk+1 ) is the antisymmetrizations with respect to αi and βj of the sum X ±(hα1 , [πi , πj ]i − h[α1 , πi ], πj i − (−1)|πi |−1 [πi , hα1 , πj i])· i<j hα2 , π1 i . . . hαi , πi−1 ihαi+1 , πi+1 i . . . hαj−1 , πj−1 ihαj , πj+1 ihαk , πk+1 i We conclude from this and (5.3.2) that ∂Φ(α) = Φ(dα).  Thus, according to Lemma 5.3, a closed 3-form H on X gives rise to a Maurer-Cartan element Φ(H) in Γ(X; C • (V • (OX )[1]; V • (OX )[1])[1]), hence a structure of an L∞ -algebra on V • (OX )[1] which has the trivial differential (the unary operation), the binary operation equal to the Schouten-Nijenhuis bracket, the ternary operation given by Φ(H), and all higher operations equal to zero. Moreover, cohomologous closed 3forms give rise to gauge equivalent Maurer-Cartan elements, hence to L∞ -isomorphic L∞ -structures. Notation. For a closed 3-form H on X we denote the corresponding L∞ -algebra structure on V • (OX )[1] by V • (OX )[1]H . Let s(OX )H := Γ(X; V • (OX )[1])H . 5.4. L∞ -structures on multivectors via formal geometry. In order to relate the results of 5.2 with those of 5.3 we consider the analog of the latter for jets. k bk Let Ω J /O := JX (AX ), the sheaf of jets of differential k-forms on b• X. Let b ddR denote the (OX -linear) differential in Ω induced by the J /O de Rham differential in The differential b ddR is horizontal with can b • , hence we have respect to the canonical flat connection ∇ on Ω J /O A•X . 14 P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN b • , ∇can , Id ⊗ b the double complex (A•X ⊗ Ω ddR ) whose total complex J /O b • ). is denoted DR(Ω J /O Let 1 : OX → JX denote the unit map (not to be confused with the map j ∞ ); it is an isomorphism onto the kernel of b ddR : JX → 1 b b• ΩJ /O and therefore defines the morphism of complexes 1 : OX → Ω J /O which is a quasi-isomorphism. The map 1 is horizontal with respect b • ), to the canonical flat connections on OX and JX (respectively, Ω J /O therefore we have the induced map of respective de Rham complexes b • ), a quasiDR(1) : A•X → DR(JX ) (respectively, DR(1) : A•X → DR(Ω J /O isomorphism). Let C • (g(JX ); g(JX )) denote the complex of continuous OX -multilinear cochains. The map of differential graded Lie algebras b: Ω b •J /O [2] → C • (V • (JX )[1]; V • (JX )[1])[1] (5.4.1) Φ defined in the same way as (5.3.1) is horizontal with respect to the canonical flat connection ∇can and induces the map b : DR(Ω b •J /O )[2] → DR((C • (V • (JX )[1]; V • (JX )[1])[1]) (5.4.2) DR(Φ) There is a canonical morphism of sheaves of differential graded Lie algebras (5.4.3) DR(C • (V • (JX )[1]; V • (JX )[1])[1]) → C • (DR(V • (JX )[1]); DR(V • (JX )[1]))[1] b • )) determines an Therefore, a degree three cocycle in Γ(X; DR(Ω J /O L∞ -structure on DR(V • (JX )[1]) and cohomologous cocycles determine L∞ -isomorphic structures. Notation. For a section B ∈ Γ(X; A2X ⊗ JX ) we denote by B it’s image in Γ(X; A2X ⊗ JX /OX ). Lemma 5.4. If B ∈ Γ(X; A2X ⊗ JX ) satisfies ∇can B = 0, then b • )); (1) b ddR B is a (degree three) cocycle in Γ(X; DR(Ω J /O (2) there exist a unique H ∈ Γ(X; A3X ) such that dH = 0 and DR(1)(H) = ∇can B. Proof. For the first claim it suffices to show that ∇can B = 0. This follows from the assumption that ∇can B = 0 and the fact that b ddR : • • 1 • b AX ⊗ JX → AX ⊗ ΩJ /O factors through AX ⊗ JX /OX . We have: b ddR ∇can B = ∇can b ddR B = 0. Therefore, ∇can B is in the image of DR(1) : Γ(X; A3X ) → Γ(X; A3X ⊗ JX ) which is injective, whence the existence and uniqueness of H. Since DR(1) is a morphism of complexes it follows that H is closed.  FORMALITY FOR ALGEBROID STACKS 15 Suppose that B ∈ Γ(X; A2X ⊗ JX ) satisfies ∇can B = 0. Then, the differential graded Lie algebra DR(g(JX ))B (the B-twist of DR(g(JX ))) is defined. On the other hand, due to Lemma 5.4, (5.4.2) and (5.4.3), b ddR B gives rise to an L∞ -structure on DR(V • (JX )[1]). Lemma 5.5. The L∞ -structure induced by b ddR B is that of a differential • graded Lie algebra equal to DR(V (JX )[1])B . Proof. Left to the reader.  b • )) we will denote by DR(V • (JX )[1])ω Notation. For a 3-cocycle ω ∈ Γ(X; DR(Ω J /O the L∞ -algebra obtained from ω via (5.4.2) and (5.4.3). Let sDR (JX )ω := Γ(X; DR(V • (JX )[1]))ω . Remark 5.6. Lemma 5.5 shows that this notation is unambiguous with reference to the previously introduced notation for the twist. In the notations introduced above, b ddR B is the image of B under the injective b 1 ) which factors b ddR and map Γ(X; A2X ⊗ JX /OX ) → Γ(X; A2X ⊗ Ω J /O “allows” us to “identify” B with b ddR B. Theorem 5.7. Suppose that B ∈ Γ(X; A2X ⊗ JX ) satisfies ∇can B = 0. Let H ∈ Γ(X; A3X ) denote the unique 3-form such that DR(1)(H) = ∇can B (cf. Lemma 5.4). Then, the L∞ -algebras gDR (JX )B and s(OX )H are L∞ -quasi-isomorphic. Proof. The map j ∞ : V • (OX ) → V • (JX ) induces a quasi-isomorphism of sheaves of DGLA (5.4.4) j ∞ : V • (OX )[1] → DR(V • (JX )[1]). Suppose that H is a closed 3-form on X. Then, the map (5.4.4) is a quasi-isomorphism of sheaves of L∞ -algebras j ∞ : V • (OX )[1]H → DR(V • (JX )[1])DR(1)(H) . Passing to global section we obtain the quasi-isomorphism of L∞ algebras (5.4.5) j ∞ : s(OX )H → sDR (JX )DR(1)(H) . By assumption, B provides a homology between b ddR B and ∇can B = DR(1)(H). Therefore, we have the corresponding L∞ -quasi-isomorphism (5.4.6) L∞ DR(V • (JX )[1])DR(1)(H) ∼ = DR(V • (JX )[1])❜ = DR(V • (JX )[1]) ddR B (the second equality is due to Lemma 5.5). B 16 P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN According to Corollary 5.2 the sheaf of DGLA DR(V • (JX )[1])B is L∞ quasi-isomorphic to the DGLA deduced form the differential graded e2 algebra DR(Ωe2 (Fe2 ∨ (C • (JX )), M ))B . The latter DGLA is L∞ -quasiisomorphic to DR(C • (JX )[1])B . Passing to global sections we conclude that sDR (JX )DR(1)(H) and gDR (JX )B are L∞ -quasi-isomorphic. Together with (5.4.5) this implies the claim.  6. Application to deformation theory Theorem 6.1. Suppose that S is a twisted form of OX (2.5). Let H be 3 a closed 3-form on X which represents [S]dR ∈ HdR (X). For any Artin algebra R with maximal ideal mR there is an equivalence of 2-groupoids MC2 (s(OX )H ⊗ mR ) ∼ = Def(S)(R) natural in R. Proof. Since cohomologous 3-forms give rise to L∞ -quasi-isomorphic L∞ -algebras we may assume, possibly replacing H by another representative of [S]dR , that there exists B ∈ Γ(X; A2X ⊗ JX ) such that B represents [S] and ∇can B = DR(1)(H). By Theorem 5.7 s(OX )H is L∞ -quasi-isomorphic to gDR (JX )B . By the Theorem 2.1 we have a homotopy equivalence of nerves of 2-groupoids γ• (s(OX )H ⊗ mR ) ∼ = γ• (gDR (JX )B ⊗ mR ). Therefore, there are equivalences MC2 (s(OX )H ⊗ mR ) ∼ = MC2 (gDR (JX ) ⊗ mR ) ∼ = Def(S)(R), B the second one being that of Theorem 3.3.  Remark 6.2. In particular, the isomorphism classes of formal deformations of S are in a bijective correspondence with equivalence classes of b · C[[t]]. These Maurer-Cartan elements of the L∞ -algebra sDR (OX )H ⊗t are the formal twisted Poisson in theVterminology of [13], P∞ structures k i.e. the formal series π = k=1 t πk , πk ∈ Γ(X; 2 TX ), satisfying the equation [π, π] = Φ(H)(π, π, π). A construction of an algebroid stack associated to a twisted Poisson structure was proposed by P. Ševera in [12]. References [1] P. Bressler, A. Gorokhovsky, R. Nest, and B. Tsygan. Deformation quantization of gerbes. Adv. Math., 214(1):230–266, 2007. [2] P. Bressler, A. Gorokhovsky, R. Nest, and B. Tsygan. Deformations of gerbes on smooth manifolds. In K-theory and noncommutative geometry. European Mathematical Society, 2008. FORMALITY FOR ALGEBROID STACKS 17 [3] A. D’Agnolo and P. Polesello. Stacks of twisted modules and integral transforms. In Geometric aspects of Dwork theory. Vol. I, II, pages 463–507. Walter de Gruyter GmbH & Co. KG, Berlin, 2004. [4] P. Deligne. Letter to L. Breen, 1994. [5] V. Dolgushev, D. Tamarkin, and B. Tsygan. 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Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany E-mail address: paul.bressler@gmail.com Department of Mathematics, UCB 395, University of Colorado, Boulder, CO 80309-0395, USA E-mail address: Alexander.Gorokhovsky@colorado.edu Department of Mathematics, Copenhagen University, Universitetsparken 5, 2100 Copenhagen, Denmark E-mail address: rnest@math.ku.dk Department of Mathematics, Northwestern University, Evanston, IL 60208-2730, USA E-mail address: tsygan@math.northwestern.edu