Formality for algebroids II: Formality theorem for gerbes

arXiv:1308.3951v2 [math.QA] 29 Oct 2014 FORMALITY THEOREM FOR GERBES PAUL BRESSLER, ALEXANDER GOROKHOVSKY, RYSZARD NEST, AND BORIS TSYGAN Abstract. The main result of the present paper is an analogue of Kontsevich formality theorem in the context of the deformation theory of gerbes. We construct an L∞ deformation of the Schouten algebra of multi-vectors which controls the deformation theory of a gerbe. 1. Introduction The main result of the present paper is an analogue of Kontsevich formality theorem in the context of the deformation theory of gerbes. A differential graded Lie algebra (DGLA) controlling the deformation theory of gerbes was constructed in ([1, 2, 3]). As it turnes out it is not quite formal in the sense of D. Sullivan (see for example [7]). More precisely, it turns out to be L∞ quasi-isomorphic to the algebra of multi-vectors with the L∞ structure determined by the class of the gerbe. The argument uses a proof of the theorem of M. Kontsevich on the formality of the Gerstenhaber algebra of a regular commutative algebra over a field of characteristic zero. For simplicity, consider for the moment the case of a C ∞ -manifold X with the × -gerbe there structure sheaf OX of complex valued smooth functions. With an OX is a canonically associated “linear object,” the twisted form S of OX (Section 6). × ∼ ) = H 3 (X; Z). Twisted forms of OX are classified up to equivalence by H 2 (X; OX One can formulate the formal deformation theory of algebroid stacks ([21, 20]) which leads to the 2-groupoid valued functor Def(S) of commutative Artin Calgebras. We review this construction in Section 6. It is natural to expect that the deformation theory of algebroid stacks is “controlled” by a DGLA. For a nilpotent DGLA g which satisfies gi = 0 for i ≤ −2, P. Deligne [6] and, independently, E. Getzler [11] associated the (strict) 2-groupoid, denoted MC2 (g) (see [4] 3.3.2), which we refer to as the Deligne 2-goupoid. The DGLA gDR (JX )ω (see 4.1) is the de Rham complex of the Gerstenhaber algebra of the algebra JX of jets of functions twisted by a representative ω of the class of the gerbe S. The following theorem is proved in [2] (Theorem 1 of loc. cit.): 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. In Section 5 we show that with every closed 3-form one can associate a ternary operation on the algebra of multi-vector fields on X. The Schouten algebra of multi-vector fields with this additional ternary operation becomes an L∞ -algebra. A. Gorokhovsky was partially supported by NSF grant DMS-0900968, R. Nest was partially supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92), B. Tsygan was partially supported by NSF grant DMS-0906391. 1 2 P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN In particular, a choice of de Rham representative H of the class of S leads to an L∞ -algebra s(OX )H . The principal technical result of the present paper (Theorem 5.7) says in this context the following: Theorem 1.1. Let X be a C ∞ -manifold. Then the DGLA gDR (JX )ω is L∞ quasiisomorphic to the L∞ -algebra s(OX )H . In conjunction with our previous results (see [2, 3, 4]) we get the description of Def(S) in terms of the L∞ -algebra s(OX )H (see Theorem 6.5). Let us explain this passage in more detail. We would like to appeal to the invariance of Deligne 2-groupoid construction under quasi-isomorphisms. However, since one of the algebras appearing in Theorem 1.1 is not a DGLA but an L∞ -algebra, we need an extension of the notion of Deligne 2-groupoid to this context. In order to do this, we use the construction of Hinich [13], extended by Getzler in [11] to the case of nilpotent L∞ algebras, which associates a Kan simplicial set Σ(g) to any nilpotent L∞ -algebra g. Duskin’s work ([9]) associates with any Kan simplicial set K a bigroupoid Bic Π2 (K), see Section 6.3 for a brief description. Thus with a nilpotent DGLA g which satisfies gi = 0 for i ≤ −2 one can associate bigroupoids Bic Π2 (Σ(g)) and MC2 (g). In [4] we show that in this situation there is a natural equivalence MC2 (g) ∼ = Bic Π2 (Σ(g)) (see Theorem 3.7 or, alternatively, Theorem 6.6 of [4]). This statement, combined with Theorem 1.1 yields the following result: Theorem 1.2. For any Artin algebra R with maximal ideal mR there is an equivalence of 2-groupoids Def(S)(R) ∼ = Bic Π2 (Σ(s(OX )H ⊗ mR )) natural in R. Objects of the bigroupoid Bic Π2 (Σ(s(OX )H ⊗ mR )) are Maurer-Cartan elements of the L∞ -algebra sDR (OX )H ⊗ mR . These are twisted Poisson structures in the V terminology of P. Ševera and A. Weinstein, [27], i.e. elements π ∈ Γ(X; 2 TX )⊗mR , satisfying the equation [π, π] = Φ(H)(π, π, π), see Remark 5.6. A construction of an algebroid stack associated to a twisted formal Poisson structure (using the formality theorem) was proposed by P. Ševera in [26]. The proof of Theorem 1.1 is based on the approach of [8] to the theorem of M. Kontsevich conjectured in [15] and proven in [16] on formality of the Gerstenhaber algebra of a regular commutative algebra over a field of characteristic zero. To give a uniform treatment of the case of a plain C ∞ manifold discussed above as well as of the complex-analytic and other settings, we work in this paper in the natural generality of a C ∞ manifold X equipped with an integrable complex distribution (and, eventually, with two transverse integrable distributions). More precisely, suppose that X is a C ∞ manifold and P is an integrable complex distribution such that the P-Dolbault Lemma 3.2 holds. We assume that P admits an integrable complement with the same property. These assumptions are fulfilled when P is a complex structure or trivial. We denote by OX/P the sheaf of (complex valued) P-holomorphic functions and by F• Ω•X the Hodge filtration. FORMALITY THEOREM FOR GERBES 3 × -gerbe). The class of S Let S be a twisted form of OX/P (equivalently, a OX/P in de Rham cohomology can be represented by a form H ∈ Γ(X; F−1 Ω3X ). Such a form H determines an L∞ -algebra structure on the P-Dolbeault resolution of Pholomorphic multi-vector fields (see 5.4 for details). We denote this L∞ -algebra by s(OX/P ). In this setting the main theorem (Theorem 6.5) says: Theorem 1.3. Suppose that X is a C ∞ manifold equipped with a pair of complementary complex integrable distributions P and Q, and S is a twisted form of OX/P (6.2). Let H ∈ Γ(X; F−1 Ω3X ) be a representative of [S] (6.2). Then, for any Artin algebra R with maximal ideal mR there is an equivalence of bi-groupoids Bic Π2 (Σ(s(OX/P )H ⊗ mR )) ∼ = Def(S)(R), natural in R. In particular, when P = 0 we recover Theorem 1.1. The paper is organized as follows. Section 2 contains a short exposition of the proof of Kontsevich formality theorem given in [8]. Section 3 contains a short review of differential calculus and differential geometry of jets in the presence of an integrable distribution. In Section 4, the proof from [8] is modified to the twisted case. Section 5 is devoted to constructions of L∞ -structures on the algebra of multi-vectors and related algebras. In particular, we construct a morphism of DGLA from the shifted de Rham complex (equipped with the trivial bracket) to the deformation complex of the Schouten Lie algebra of multi-vectors (see Lemma 5.1) which may be of independent interest. Finally, in Section 6 we prove the main results on deformations of algebroid stacks. 2. Formality This section contains a synopsis of results of the paper [8] in the notation 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 operad e2 is Koszul, and we denote by e2 ∨ the dual cooperad (see (5.6) for an explicit description of Fe2 ∨ ). 2.1. Hochschild cochains. For an associative k-algebra A, p = 0, 1, 2, . . . the space C p (A) = C p (A; A) of Hochschild cochains of degree p with values in (the A ⊗k Aop module) A is defined by C 0 (A) = A and C p (A) = Homk (A⊗k p , A) for p ≥ 1. Let gp (A) = C p+1 (A). There is a canonical isomorphism of graded k-modules g(A) = Derk (coAss(A[1])), where coAss(V ) denotes the co-free co-associative co-algebra on the graded vector space V . In particular, g(A) = C(A)[1] has a canonical structure of a graded Lie algebra under the induced Gerstenhaber bracket. Let m ∈ C 2 (A) denote the multiplication on A and let δ denote the adjoint action of m with respect to the Gerstenhaber bracket. Thus, δ is an endomorphism of degree one and a derivation of the Gerstenhaber bracket. Associativity of m implies that δ ◦ δ = 0. The graded space C • (A) equipped with the Hochschild differential δ is called the Hochschild complex of A. The graded space g(A) = C • (A)[1] equipped with the Gerstenhaber bracket and the Hochschild differential is a DGLA which controls the deformation theory of A. 4 P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN 2.2. Outline of the proof of formality for cochains. For an associative kalgebra A the complex of Hochschild cochains C • (A) has a canonical structure of a brace algebra. By [22], [29], and [18], C • (A) has a structure of a homotopy algebra over the operad of chain complexes of the topological operad of little discs. By [30] and [17], the chain operad of little discs is formal, that is to say weakly equivalent to the Gerstenhaber operad e2 . Therefore, the Hochschild cochain complex has a structure of a homotopy e2 -algebra. The construction of [30] depends on a choice of a Drinfeld associator, while the construction of [17] does not. It has been shown in [28] that the latter construction is a particular case of the former, corresponding to a special choice of associator. The homotopy e2 -algebra described above is encoded in a differential (i.e. a coderivation of degree one and square zero) M : Fe2 ∨ (C • (A)) → Fe2 ∨ (C • (A))[1]. We assume from now on that A is a regular commutative algebra over a field k 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)). In the notation of [8], Theorem 1, Be2 ∨ (V • (A)) = (Fe2 ∨ (V • (A)), dV • (A) ). 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) with its image under ι. By [8], 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: (2.1) σ ι − (Ξ(A), dV • (A) ) − → Be2 ∨ (V • (A)) (Fe2 ∨ (C • (A)), M ) ← Applying the functor Ωe2 (adjoint to the functor Be2 ∨ , see [8], Theorem 1) to (2.1) we obtain the diagram Ωe (σ) 2 −− (2.2) Ωe2 (Fe2 ∨ (C • (A)), M ) ←−− Ωe (ι) 2 −→ Ωe2 (Be2 ∨ (V • (A))) Ωe2 (Ξ(A), dV • (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 (2.3) Ωe (σ) ν 2 −− Ωe2 (Ξ(A), dV • (A) ) − → V • (A) Ωe2 (Fe2 ∨ (C • (A)), M ) ←−− of differential graded e2 -algebras. Theorem 2.1 ([8], Theorem 4). Suppose that A is a regular commutative algebra over a field k of characteristic zero. Then, the maps Ωe2 (σ) and ν in the diagram (2.3) are quasi-isomorphisms. Additionally, concerning the DGLA structures relevant to applications to deformation theory, deduced from respective e2 -algebra structures we have the following result. Theorem 2.2 ([8], Theorem 2). The DGLA Ωe2 (Fe2 ∨ (C • (A)), M )[1] and C • (A)[1] are canonically L∞ -quasi-isomorphic. Corollary 2.3 (Formality). The DGLA C • (A)[1] and V • (A)[1] are L∞ -quasi-isomorphic. FORMALITY THEOREM FOR GERBES 5 2.3. Some (super-)symmetries. For applications to deformation theory of algebroid stacks we will need certain equivariance properties of the maps described in 2.2. 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 (−1)k D(a1 , . . . , ai , a, ak+1 , . . . , an ). ia D(a1 , . . . , 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 . 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) is given in terms of the cup product and the brace operations. Therefore, 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 of a 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 (2.1) commute with ia and, therefore, so do the maps in (2.2) and (2.3). 2.4. Extensions and generalizations. Constructions and results of [8] apply in a variety of situations. First of all, observe that constructions of all objects and morphisms involved can be carried out in any closed symmetric monoidal category such as, for example, the category of sheaves of k-modules, k a sheaf of commutative algebras (over a field of characteristic zero). As is pointed out in [8], Section 4, the proof of Theorem 2.1 is based on the flatness of the module Der(A) and the Hochschild-Kostant-Rosenberg theorem. The considerations above apply to a sheaf of k-algebras K yielding the sheaf of Hochschild cochains C p (K), the Hochschild complex (of sheaves) C • (K) and the (sheaf of) DGLA g(K). If X is a C ∞ manifold the sheaf C p (OX ) coincides with the sheaf of multilinear ×p differential operators OX → OX by the theorem of J. Peetre [23], [24]. 2.5. Deformations of O and Kontsevich formality. Suppose that X is a C ∞ manifold. Let OX (respectively, TX ) denote the structure sheaf (respectively, the sheaf of vector fields). The construction of 2.2 yield the diagram of sheaves of differential graded e2 -algebras (2.4) Ωe (σ) ν 2 → V • (OX ), −− Ωe2 (Ξ(OX ), dV • (OX ) ) − Ωe2 (Fe2 ∨ (C • (OX )), M ) ←−− 6 P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN where C • (OX ) denotes the sheaf of multi-differential operators and V • (OX ) := Sym•OX (TX [−1]) denotes the sheaf of multi-vector fields. Theorem 2.1 says that the morphisms Ωe2 (σ) and ν in (2.4) are quasi-isomorphisms of sheaves of differential graded e2 -algebras. 3. Calculus in the presence of distribution In this section we briefly review basic facts regarding differential calculus in the presence of an integrable complex distribution. We refer the reader to [19], [25] and [10] for details and proofs. For a C ∞ manifold X we denote by OX (respectively, ΩiX ) the sheaf of complex valued C ∞ functions (respectively, differential forms of degree i) on X. Throughout this section we denote by TXR the sheaf of real valued vector fields on X. Let TX := TXR ⊗R C. 3.1. Complex distributions. A (complex) distribution on X is a sub-bundle1 of TX . A distribution P is called involutive if it is closed under the Lie bracket, i.e. [P, P] ⊆ P. For a distribution P on X we denote by P ⊥ ⊆ Ω1X the annihilator of P (with respect to the canonical duality pairing). A distribution P of rank r on X is called integrable if, locally on X, there exist functions f1 , . . . , fr ∈ OX such that df1 , . . . , dfr form a local frame for P ⊥ . It is easy to see that an integrable distribution is involutive. The converse is true when P is real, i.e. P = P (Frobenius) and when P is a complex structure, i.e. P ∩ P = 0 and P ⊕ P = TX (Newlander-Nirenberg). More generally, according to Theorem 1 of [25], a sufficient condition for integrability of a complex distribution P is (3.1) P ∩ P is a sub-bundle and both P and P + P are involutive. 3.2. The Hodge filtration. Suppose that P is an involutive distribution on X. Let F• Ω•X denote the filtration by the powers of the differential ideal generated V j−i ⊆ ΩjX . Let ∂ denote the differential in Gr•F Ω•X . by P ⊥ , i.e. F−i ΩjX = i P ⊥ ∧ ΩX The wedge product of differential forms induces a structure of a commutative DGA on (Gr•F Ω•X , ∂). In particular, Gr0F OX = OX , Gr0F Ω1X = Ω1X /P ⊥ and ∂ : OX → Gr0F Ω1X is equal d ∂ to the composition OX − → Ω1X → Ω1X /P ⊥ . Let OX/P := ker(OX − → Gr0F Ω1X ); equivalently, OX/P = (OX )P ⊂ OX , the subsheaf of functions constant along P. Note that ∂ is OX/P -linear. Theorem 2 of [25] says that, if P satisfies the condition (3.1) higher ∂-cohomology of OX vanishes, i.e.  OX/P if i = 0 i F • (3.2) H (Gr0 ΩX , ∂) = 0 otherwise. In what follows we will assume that the complex distribution P under consideration is integrable and satisfies (3.2). This is implied by the condition (3.1). 1A sub-bundle is an O -submodule which is a direct summand locally on X X FORMALITY THEOREM FOR GERBES 7 3.3. ∂-operators. Suppose that E is a vector bundle on X, i.e. a locally free OX module of finite rank. A connection along P on E is, by definition, a map ∇P : E → Ω1X /P ⊥ ⊗OX E which satisfies the Leibniz rule ∇P (f e) = f ∇P (e) + ∂f · e. Equivalently, a connection along P is an OX -linear map ∇P (•) : P → EndC (E) which P P satisfies the Leibniz rule ∇P ξ (f e) = f ∇ξ (e)+∂f ·e. In particular, ∇ξ is OX/P -linear. P The two avatars of a connection along P are related by ∇P ξ (e) = ιξ ∇ (e). A connection along P on E is called flat if the corresponding map ∇P (•) : P → EndC (E) is a morphism of Lie algebras. We will refer to a flat connection along P on E as a ∂-operator on E. A connection on E along P extends uniquely to a derivation of the graded Gr0F Ω•X 2 module Gr0F Ω•X ⊗OX E. A ∂-operator ∂ E satisfies ∂ E = 0. The complex (Gr0F Ω•X ⊗OX E, ∂ E ) is referred to as the (corresponding) ∂-complex. Since ∂ E is OX/P -linear, the sheaves H i (Gr0F Ω•X ⊗OX E, ∂ E ) are OX/P -modules. The vanishing of higher ∂-cohomology of OX (3.2) generalizes easily to vector bundles. Lemma 3.1. Suppose that E is a vector bundle and ∂ E is a ∂-operator on E. Then, H i (Gr0F Ω•X ⊗OX E, ∂ E ) = 0 for i 6= 0, i.e. the ∂-complex is a resolution of ker(∂ E ). Moreover, ker(∂ E ) is locally free over OX/P of rank rkOX E and the map OX ⊗OX/P ker(∂ E ) → E (the OX -linear extension of the inclusion ker(∂ E ) ֒→ E) is an isomorphism. Remark 3.2. Suppose that F is a locally free OX/P -module of finite rank. Then, OX ⊗OX/P F is a locally free OX -module of rank rkOX/P F and is endowed in a canonical way with a ∂-operator, namely, ∂ ⊗ Id. The assignments F 7→ (OX ⊗OX/P F, ∂ ⊗ Id) and (E, ∂ E ) 7→ ker(∂ E ) are mutually inverse equivalences of suitably defined categories. 3.4. Calculus. The adjoint action of P on TX preserves P, hence descends to an action on TX /P. The latter action defines a connection along P, i.e. a canonical ∂-operator on TX /P which is easily seen to coincide with the one induced via the duality pairing between the latter and P ⊥ .2 Let TX/P := (TX /P)P (the subsheaf of P invariant section, equivalently, the kernel of the ∂-operator on TX /P. The Lie bracket on TX (respectively, the action of TX on OX ) induces a Lie bracket on TX/P (respectively, an action of TX/P on OX/P ). The bracket and the action on OX/P endow TX/P with a structure of an OX/P -Lie algebroid. The action of P on Ω1X by Lie derivative restricts to a flat connection along V P, i.e. a canonical ∂-operator on P ⊥ and, therefore, on i P ⊥ for all i. It is V F Ω• [i] is an isoeasy to see that the multiplication map Gr0F Ω• ⊗ i P ⊥ → Gr−i Vi ⊥ F Ω• [i]. Let Ωi morphism which identifies the ∂-complex of P with Gr−i X/P := V i ⊥ i F • i 0 i H (Gr−i ΩX , ∂) (so that OX/P := ΩX/P ). Then, ΩX/P ⊂ P ⊂ ΩX . The wedge product of differential forms induces a structure of a graded-commutative algebra on Ω•X/P := ⊕i ΩiX/P [−i] = H • (Gr F Ω•X , ∂). The multiplication induces an isomorV phism iOX/P Ω1X/P → ΩiX/P . The de Rham differential d restricts to the map • • d : ΩiX/P → Ωi+1 X/P and the complex ΩX/P := (ΩX/P , d) is a commutative DGA. 2In the case of a real polarization this connection is known as the Bott connection. 8 P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN The Hodge filtration F• Ω•X/P is defined by Fi Ω•X/P = ⊕j≥−i ΩjX/P , so that the inclusion Ω•X/P ֒→ Ω•X is filtered with respect to the Hodge filtration. It follows from Lemma 3.1 that it is, in fact, a filtered quasi-isomorphism. The duality pairing TX /P ⊗ P ⊥ → OX restricts to a non-degenerate pairing TX/P ⊗OX/P Ω1X/P → OX/P . The action of TX /P on OX/P the pairing and the de Rham differential are related by the usual formula ξ(f ) = ιξ df , for ξ ∈ TX/P and f ∈ OX/P . 3.5. Jets. Let pri : X × X → X, i = 1, 2, denote the projection on the ith factor. The restriction of the canonical map pr∗i : pr−1 i OX → OX×X to the subsheaf OX/P takes values in the subsheaf OX×X/P×P hence induces the map pr∗i : OX/P → (pri )∗ OX×X/P×P . Let ∆X : X → X × X denote the diagonal embedding. It follows from the Leibniz rule that the restriction of the canonical map ∆∗X : OX×X → (∆X )∗ OX to the subsheaf OX×X/P×P takes values in the subsheaf (∆X )∗ OX/P . Let IX/P := ker(∆∗X ) ∩ OX×X/P×P . The sheaf IX/P plays the role of the defining ideal of the “diagonal embedding X/P → X/P × X/P”: there is a short exact sequence of sheaves on X × X 0 → IX/P → OX×X/P×P → (∆X )∗ OX/P → 0 For a locally-free OX/P -module of finite rank E let   k+1 k E , ⊗pr−1 OX/P pr−1 JX/P (E) := (pr1 )∗ OX×X/P×P /IX/P 2 2 k JX/P k := JX/P (OX/P ) . k It is clear from the above definition that JX/P is, in a natural way, a commutative k k algebra and JX/P (E) is a JX/P -module. Let k 1(k) : OX/P → JX/P denote the composition pr∗ 1 k (pr1 )∗ OX×X/P×P → JX/P OX/P −−→ k (E) as a O In what follows, unless stated explicitly otherwise, we regard JX/P X/P - module via the map 1(k) . Let k j k : E → JX/P (E) denote the composition e7→1⊗e k E −−−−→ (pr1 )∗ OX×X/P×P ⊗C E → JX/P (E) FORMALITY THEOREM FOR GERBES 9 Note that the map j k is not OX/P -linear unless k = 0. k+1 l+1 induces the surjective map πl,k : → IX/P For 0 ≤ k ≤ l the inclusion IX/P l k (E). The sheaves J k (E), k = 0, 1, . . . together with the maps JX/P (E) → JX/P X/P ∞ (E) := limJ k (E). Thus, πl,k , k ≤ l form an inverse system. Let JX/P (E) = JX/P X/P ←− JX/P (E) carries a natural (adic) 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 := lim1(k) , ←− j ∞ := limj k . ←− Let d1 : OX×X/P×P ⊗pr−1 OX/P pr−1 2 E → 2 1 −1 → pr−1 1 ΩX/P ⊗pr−1 OX/P OX×X/P×P ⊗pr−1 OX/P pr2 E 1 2 denote the exterior derivative along the first factor. It satisfies k+1 ⊗pr−1 OX/P pr−1 d1 (IX/P 2 E) ⊂ 2 1 k −1 pr−1 1 ΩX/P ⊗pr−1 OX/P IX/P ⊗pr−1 OX/P pr2 E 1 2 for each k and, therefore, induces the map (k) k−1 k (E) d1 : JX/P (E) → Ω1X/P ⊗OX/P JX/P (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/P (E) → Ω1X/P ⊗OX/P JX/P (E) . E Let JX,P (E) := OX ⊗OX/P JX/P (E) JX,P := JX,P (OX/P ) J X,P := JX,P /1(OX ) Here and below by abuse of notation we write (•) ⊗OX/P JX/P (E) for lim(•) ⊗OX/P k (E). JX/P The canonical flat connection extends to the flat connection ←− ∇can : JX,P (E) → Ω1X ⊗OX JX,P (E) . E 3.6. De Rham complexes. Suppose that F is an OX -module and ∇ : F → Ω1X ⊗OX F is a flat connection. The flat connection ∇ extends uniquely to a differential ∇ on Ω•X ⊗OX F subject to the Leibniz rule with respect to the Ω•X -module structure. We will make use of the following notation: ∇ (ΩiX ⊗OX F)cl := ker(ΩiX ⊗OX F − → Ωi+1 X ⊗OX F) Suppose that (F • , d) is a complex of OX -modules with a flat connection ∇ = (∇i )i∈Z , i.e. for each i ∈ Z, ∇i is a flat connection on F i and [d, ∇] = 0. Then, (Ω•X ⊗OX F • , ∇, Id⊗d) is a double complex. We denote by DR(F) the total complex. 10 P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN 4. Formality for the algebroid Hochschild complex Until further notice we work with a fixed C ∞ manifold X equipped with a complex distribution P which satisfies the standing assumptions of 3.2 and denote JX,P (respectively, J X,P , JX,P (E)) by J (respectively, J , J (E)). 4.1. Hochschild cochains in formal geometry. In what follows we will be interested in the Hochschild complex in the context of formal geometry. To this end we define C p (J ) to be the sheaf of continuous (with respect to the adic topology) OX -multilinear Hochschild cochains on J . OX -modules equipped with flat connections form a closed monoidal category. In particular, the sheaf of OX -multilinear maps JX×p → JX is endowed with a canonical flat connection induced by ∇can . It follows directly from the definitions that ∇can preserves C p (JX ). The Gerstenhaber bracket endows g(J ) = C • (J )[1] with a structure of a graded Lie algebra. The product on J is a global section of C 2 (J ), hence the Hochschild differential preserves C • (J ). The complex Γ(X; DR(C • (J ))) = (Γ(X; Ω•X ⊗ C • (JX )), ∇can + δ) is a differential graded brace algebra in a canonical way. The abelian Lie algebra J = C 0 (J ) acts on the brace algebra C • (JX ) by derivations of degree −1 via the restriction of the adjoint action with respect to the Gerstenhaber bracket. The above action factors through an action of J . Therefore, the abelian Lie algebra Γ(X; Ω2X ⊗ J ) acts on the brace algebra Ω•X ⊗ C • (J ) by derivations of degree +1; the action of an element a is denoted by ia . Due to commutativity of J , for any ω ∈ Γ(X; Ω2X ⊗J ) the operation ιω commutes with the Hochschild differential δ. Moreover, if 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 • (J ))ω := (Γ(X; Ω•X ⊗ C • (J )), ∇can + δ + iω ) as the ω-twist of Γ(X; DR(C • (J )). Let gDR (J )ω := Γ(X; DR(C • (JX ))[1])ω regarded as a DGLA. 4.2. Formality for jets. Let V • (J ) = Sym•J (Dercont OX (J )[−1]). Working now in the category of graded OX -modules we have the diagram (4.1) Ωe (σ) ν 2 −− Ωe2 (Ξ(J ), dV • (J ) ) − → V • (J ) Ωe2 (Fe2 ∨ (C • (J )), M ) ←−− of sheaves of differential graded OX -e2 -algebras. According to the Theorem 2.1 the morphisms Ωe2 (σ) and ν in (4.1) are quasi-isomorphisms. The sheaves of DGLA Ωe2 (Fe2 ∨ (C • (J )), M )[1] and C • (J )[1] are canonically L∞ -quasi-isomorphic. The canonical flat connection ∇can on J induces a flat connection which we denote ∇can as well on each of the objects in the diagram (4.1). Moreover, the maps Ωe2 (σ) and ν are flat with respect to ∇can , hence induce the maps of respective de Rham complexes (4.2) DR(Ωe (σ)) DR(Ωe2 (Fe2 ∨ (C • (J )), M )) ←−−−−2−−− DR(ν) DR(Ωe2 (Ξ(J ), dV • (J ) )) −−−−→ DR(V • (J )). FORMALITY THEOREM FOR GERBES 11 All objects in the diagram (4.2) have canonical structures of differential graded e2 -algebras and the maps are morphisms of such. The DGLA Ωe2 (Fe2 ∨ (C • (J )), M )[1] and C • (J )[1] are canonically L∞ -quasi-isomorphic in a way compatible with ∇can . Hence, the DGLA DR(Ωe2 (Fe2 ∨ (C • (J )), M )[1]) and DR(C • (J )[1]) are canonically L∞ -quasi-isomorphic. 4.3. Formality for jets with a twist. Suppose that ω ∈ Γ(X; Ω2X ⊗ J ) satisfies ∇can ω = 0. For each of the objects in (4.2) we denote by iω the operation which is induced by the operation described in 2.3 and the wedge product on Ω•X . Thus, for each differential graded e2 -algebra (N • , d) in (4.2) 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 (4.2) commute with the respective operations iω , they give rise to morphisms of respective ω-twists (4.3) DR(Ωe (σ)) DR(Ωe2 (Fe2 ∨ (C • (J )), M ))ω ←−−−−2−−− DR(ν) DR(Ωe2 (Ξ(J ), dV • (J ) ))ω −−−−→ DR(V • (J ))ω . • Let G• Ω•X denote the filtration given by Gi Ω•X = Ω>−i X . The filtration G• ΩX induces a filtration denoted G• DR(K • , d)ω for each object (K • , d) of (4.3) defined by Gi DR(K • , d)ω = Gi Ω•X ⊗ K • . As is easy to see, the associated graded complex is given by (4.4) G Gr−p DR(K • , d)ω = (ΩpX [−p] ⊗ K • , Id ⊗ d). It is clear that the morphisms DR(Ωe2 (σ)) and DR(ν) are filtered with respect to G• . Theorem 4.1. The morphisms in (4.3) are filtered quasi-isomorphisms, i.e. the maps GriG DR(Ωe2 (σ)) and GriG DR(ν) are quasi-isomorphisms for all i ∈ Z. Proof. We consider the case of DR(Ωe2 (σ)) leaving GriG 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 (4.5) Id ⊗ Ωe2 (σ) : ΩpX ⊗ Ωe2 (Ξ(J ), dV • (J ) ) → ΩpX ⊗ Ωe2 (Fe2 ∨ (C • (J )), M ). The map σ is a quasi-isomorphism by Theorem 2.1, therefore so is Ωe2 (σ). Since ΩpX is flat over OX , the map (4.5) is a quasi-isomorphism.  Corollary 4.2. The maps DR(Ωe2 (σ)) and DR(ν) in (4.3) are quasi-isomorphisms of sheaves of differential graded e2 -algebras. Additionally, the DGLA DR(Ωe2 (Fe2 ∨ (C • (J )), M )[1]) and DR(C • (J )[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 • (J )), M )[1])ω and DR(C • (J )[1])ω are canonically L∞ -quasi-isomorphic. 12 P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN 5. L∞ -structures on multivectors 5.1. L∞ -deformation complex. For a graded vector space V we denote by coComm(V ) the co-free co-commutative co-algebra co-generated by V . Thus, a graded vector space W gives rise to the graded Lie algebra Der(coComm(W [1])). P∞ An element µ ∈ Der(coComm(W [1])) of degree one is of the form µ = i=0 µi with V µi : i W → W [2 − i]. If µ0 = 0 and [µ, µ] = 0, then µ defines a structure of an L∞ -algebra on W . (If µ0 is non-trivial, one obtains a “curved” L∞ -algebra.) An element µ as above determines a differential ∂µ := [µ, .] on Der(coComm(W [1])), such that (Der(coComm(W [1])), ∂µ ) is a DGLA. If g is a graded Lie algebra and µ is the element of Der(coComm(g[1])) corresponding to the bracket on g, then (Der(coComm(g[1])), ∂µ ) is equal to the shifted Chevalley-Eilenberg cochain complex C • (g; g)[1]. 5.2. L∞ -structures on multivectors. The canonical pairing h , i : Ω1X/P ⊗TX/P → OX extends to the pairing h , i : Ω1X/P ⊗ V • (OX/P ) → V • (OX/P )[−1] (5.1) For k ≥ 1, ω = α1 ∧ . . . ∧ αk , αi ∈ Ω1X/P , i = 1, . . . , k, let Φ(ω) : Symk V • (OX/P )[2] → V • (OX/P )[k] denote the map given by the formula Pk−1 Φ(ω)(π1 , . . . , πk ) = (−1) i=1 (k−i)(|πi |−1) X sgn(σ)hασ(1) , π1 i ∧ . . . ∧ hασ(k) , πk i, σ l (O ). X where |π| = l for π ∈ V For α ∈ OX let Φ(α) = α ∈ V 0 (OX ). We use the following explicit formula for the bracket on the Lie algebra complex: [Φ, Ψ] = Φ ◦ Ψ − (−1)|Φ||Ψ| Ψ ◦ Φ, where (Φ ◦ Ψ)(π1 , . . . , πk+l−1 ) = X 1 ǫ(σ, |π1 |, . . . , |πk+l−1 |)Φ(Ψ(πσ(1) , . . . , πσ(k) ), πσ(k+1) , . . . , πσ(k+l−1) ). k!(l − 1)! σ∈Sk+l−1 he sign ǫ(σ, |π1 |, . . . , |πn |) is defined by πσ(1) ∧ πσ(2) ∧ . . . ∧ πσ(n) = ǫ(σ, |π1 |, . . . , |πn |)π1 ∧ π2 ∧ . . . ∧ πn in V • (OX/P ). In particular, a transposition of πi and πj contributes a factor (−1)|πi ||πj | . The differential in the complex C • (V • (OX/P )[1]; V • (OX/P )[1])[1] is given by the formula ∂Φ = [m, Φ] where m(π, ρ) = (−1)|π| [π, ρ]. In what follows we consider the (shifted) de Rham complex Ω•X/P [2] as a differential graded Lie algebra with the trivial bracket. FORMALITY THEOREM FOR GERBES 13 Lemma 5.1. The map ω 7→ Φ(ω) defines a morphism of sheaves of differential graded Lie algebras Φ : Ω•X/P [2] → C • (V • (OX/P )[1]; V • (OX/P )[1])[1]. (5.2) Proof. First, we show that Φ is a morphism of graded Lie algebras. Since Ω•X/P [2] is Abelian, it suffices to show that for α, β ∈ Ω•X/P (5.3) [Φ(α), Φ(β)] = 0. Let α = α1 ∧ . . . ∧ αk and β = β1 ∧ . . . ∧ βl , with αi , βj ∈ Ω1X/P . Direct calculation shows that Φ(β) ◦ Φ(α) is the antisymmetrization with respect to αi , βj , πm of the expression k−1+ (−1) k+l−2 P (k+l−1−i)(|πi |−1) i=1 hβ1 α1 , π1 ihα2 , π2 i . . . hαk , πk ihβ2 , πk+1 i . . . hβl , πk+l−1 i, (k − 1)!(l − 1)! where hβα, πi = hβ, hα, πii. Interchanging α with β (and k with l) we obtain a similar expression for Φ(α) ◦ Φ(β). Direct comparison of signs, left to the reader, shows that Φ(α) ◦ Φ(β) = (−1)kl Φ(β) ◦ Φ(α) which implies (5.3). We now verify that Φ is a morphism of complexes. Recall the explicit formula for the Schouten bracket: for f, g ∈ OX/P , Xi , Yj ∈ TX/P X ci . . . Xk Y1 . . . Yl + (5.4) [f X1 . . . Xk , gY1 . . . Yl ] = (−1)1+i f Xi (g)X1 . . . X i X (−1)j Yj (f )gX1 . . . Xk Y1 . . . Ybj . . . Yl + j X ci . . . Xk Y1 . . . Ybj . . . Yl (−1)i+j f g[Xi , Yj ]X1 . . . X i,j Note that for a one-form ω ∈ and for vector fields X and Y (5.5) − hω, [X, Y ]i + [hω, Xi, Y ] + [X, hω, Y i] = Φ(dω)(X, Y ) Direct calculation using formulas (5.4) and (5.5) shows that for π, ρ ∈ V • (OX/P ) (−1)|π|−1 (−hω, [π, ρ]i + [hω, πi, ρ] + [π, hω, ρi]) = Φ(dω)(π, ρ). From the definition of the differential, we see that ∂Φ(α)(π1 , . . . , πk+1 ) is the antisymmetrizations with respect to αi and πj of the expression k P (−1)i=1 (k+1−i)(|πi |−1) (−hα1 , [π1 , π2 ]i + [hα1 , π1 i, π2 ] + [π1 , hα1 , π2 i]) hα2 , π3 i . . . hαk , πk+1 i 2 Computing Φ(dα) with the help of (5.5) we conclude that ∂Φ(α) = Φ(dα).  Remark 5.2. As we shall explain below, a closed three-form actually defines a deformation of the homotopy Gerstenhaber algebra of multi-vector fields, not just of the underlying L∞ algebra. Recall that, for a graded vector space W , (5.6) Fe2 ∨ (W ) = coComm(coLie(W [1])[1])[−2] 14 P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN and Maurer-Cartan elements of the graded Lie algebra Der(Fe2 ∨ (W )) (respectively, Der(coComm(W [1]))) are in bijective correspondence with the homotopy Gerstenhaber algebra structures (respectively, L∞ algebra structures) on W . There is a canonical morphism of graded Lie algebras (5.7) Der(Fe2 ∨ (W )) → Der(coComm(W [2])) such that the map of the respective sets of Maurer-Cartan elements MC(Der(Fe2 ∨ (W ))) → MC(Der(coComm(W [2]))) induced by (5.7) sends a homotopy Gerstenhaber algebra structure on W to the underlying homotopy Lie (i.e. L∞ ) algebra structure on W [1]. The canonical projection Fcolie (W [1]) → W [1] induces the map (5.8) Fe2 ∨ (W ) = coComm(coLie(W [1])[1])[−2] → coComm(W [2])[−2]. Under the map (5.7) the subspace of derivations which annihilate the kernel of (5.8) is mapped isomorphically onto Der(coComm(W [2])). Thus, the map (5.7) admits a canonical splitting as a morphism of graded vector spaces (not compatible with the respective Lie algebra structures). Suppose that A is a homotopy Gerstenhaber algebra, in particular, A is a homotopy commutative algebra and A[1] is an L∞ algebra. The structure of a homotopy Gerstenhaber algebra on A (respectively, of an L∞ algebra on A[1]) gives rise to a differential on Der(Fe2 ∨ (A)) (respectively, on Der(coComm(A[1]))) making the latter a DGLA. The canonical map Der(Fe2 ∨ (A)) → Der(coComm(A[2])) is a morphism of DGLA. Suppose that A is a differential graded Gerstenhaber algebra so that A[1] is a DGLA. Then, Der(coComm(A[2])) = C • (A[1], A[1])[1], the complex of ChevalleyEilenberg cochains of the DGLA A[1]. The subcomplex of C • (A[1], A[1])[1] of cochains which are derivations of the commutative product on A in each variable is isomorphic to Hom•A (SymA (Ω1A [2]), A)[2]. The complex of multi-derivations HomA (SymA (Ω1A [2]), A) is equipped with a natural structure of an e3 -algebra. First of all, it is equipped with an obvious commutative product. The Lie bracket on Hom•A (SymA (Ω1A [2]), A)[2] is completely determined by the Leibniz rule with respect to the commutative product and (1) [D, a] = D(a) for a ∈ A and D ∈ Der(A), (2) it coincides with the commutator bracket on Der(A). It is easy to verify that the bracket described above coincides with the one induced by the embedding of Hom•A (SymA (Ω1A [2]), A)[2] into Der(coComm(A[2])), i.e. the former is a sub-DGLA of the latter. The canonical splitting of (5.7) gives rise to the map of graded vector spaces (5.9) Hom•A (SymA (Ω1A [2]), A)[2] → Der(Fe2 ∨ (A)) Direct calculation shows that this is a map of DGLA. Needless to say, all of the above applies in the category of sheaves of vector spaces and, in particular, to the e2 -algebra A := V • (OX/P ). The adjoint of the pairing (5.1) is the map Ω1X/P → Der(A)[−1] FORMALITY THEOREM FOR GERBES 15 which extends to the map of commutative algebras Ω•X/P → SymA (Der(A)[−2]) = Hom•A (SymA (Ω1A [2]), A) such that the map (5.10) Ω•X/P [2] → SymA (Der(A)[−2])[2] = Hom•A (SymA (Ω1A [2]), A)[2] is a map of DGLA with Ω•X/P [2] Abelian. Therefore, the composition of (5.10) with (5.9) Φ : Ω•X/P [2] → Der(Fe2 ∨ (A)) is a morphism of DGLA and so is the composition of the latter with the canonical map (5.7), which is to say, the map which is the subject of Lemma 5.1. We conclude that every closed three-form defines a Maurer-Cartan element of Der(Fe2 ∨ (V • (OX/P ))), i.e. a structure of a homotopy Gerstenhaber algebra on V • (OX/P ). 5.3. L∞ -structures on multivectors via formal geometry. Let C • (V • (J )[1]; V • (J )[1]) denote the complex of continuous OX -multilinear Chevalley-Eilenberg cochains. k b bk b• Let Ω J /O := J (ΩX/P ). Let ddR denote the (OX -linear) differential in ΩJ /O bdR is horizontal with induced by the de Rham differential in Ω• . The differential d X/P ∇can b • , hence we have the double respect to the canonical flat connection on Ω J /O b • , ∇can , Id ⊗ b b • ). complex (Ω•X ⊗ Ω d ) whose total complex is denoted DR(Ω dR J /O J /O b• The Hodge filtration F• Ω is induced by that on Ω• , that is, we set J /O The map of DGLA (5.11) X/P • b• bj Fi Ω J /O := J (Fi ΩX/P ) = ⊕j≥−i ΩJ /O . b: Ω b • [2] → C • (V • (J )[1]; V • (J )[1])[1] Φ J /O defined in the same way as (5.2) is horizontal with respect to the canonical flat connection ∇can and induces the map (5.12) b : DR(Ω b • )[2] → DR((C • (V • (J )[1]; V • (J )[1])[1]) DR(Φ) J /O There is a canonical morphism of sheaves of differential graded Lie algebras (5.13) DR(C • (V • (J )[1]; V • (J )[1])[1]) → C • (DR(V • (J )[1]); DR(V • (J )[1]))[1] b • )) determines an L∞ -structure Therefore, a degree three cocycle in Γ(X; DR(F−1 Ω J /O b • )), determine on DR(V • (J )[1]). Two cocycles, cohomologous in Γ(X; DR(F−1 Ω J /O quasiisomorphic L∞ structures. Notation. For a section B ∈ Γ(X; Ω2X ⊗J ) we denote by B its image in Γ(X; Ω2X ⊗J ). Lemma 5.3. If B ∈ Γ(X; Ω2X ⊗ J ) satisfies ∇can B = 0, then b • )); (1) b ddR B is a (degree three) cocycle in Γ(X; DR(F−1 Ω J /O (2) The L∞ -structure induced by b ddR B is that of a differential graded Lie algebra • equal to DR(V (J )[1])B . 16 P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN Proof. For the first claim it suffices to show that ∇can b ddR B = 0. This follows from can b b1 the assumption that ∇ B = 0 and the fact that ddR : Ω•X ⊗ J → Ω•X ⊗ Ω J /O factors through Ω•X ⊗ J . The proof of the second claim is left to the reader.  Notation. For a 3-cocycle b • )) ω ∈ Γ(X; DR(F−1 Ω J /O we will denote by DR(V • (J )[1])ω the L∞ -algebra obtained from ω via (5.12) and (5.13). Let sDR (J )ω := Γ(X; DR(V • (J )[1]))ω . Remark 5.4. Lemma 5.3 shows that this notation is unambiguous with reference to the previously introduced notation for the twist. In the notation introduced above, b b1 ) ddR B is the image of B under the injective map Γ(X; Ω2X ⊗ J ) → Γ(X; Ω2X ⊗ Ω J /O bdR and allows us to identify B with b which factors d ddR B. 5.4. Dolbeault complexes. We shall assume that the manifold X admits two complementary integrable complex distributions P and Q both satisfying (3.2). In other words, P ∩ Q = 0 and P ⊕ Q =LTX . The latter decomposition Vq ⊥a Vp ⊥ induces p,q Q . P ⊗ bi-grading on differential forms: ΩnX = p+q=n Ωp,q with Ω = X X L p,n−p n The bi-grading splits the Hodge filtration: F−i Ω = p≥i ΩX . Two cases of particular interest in applications are • P=0 • P is a complex structure, and P = Q. The map (5.2) extends to the morphism of sheaves of DGLA (5.14) 0,• • • Φ : Ω•X [2] → C • (Ω0,• X ⊗OX/P V (OX/P )[1]; ΩX ⊗OX/P V (OX/P )[1])[1]. 0,• • Let F• (Ω0,• X ⊗OX/P V (OX/P )) denote the filtration defined by F−i (ΩX ⊗OX/P L p V • (OX/P )) = p≥i Ω0,• X ⊗OX/P V (OX/P ). The complex 0,• • • C • (Ω0,• X ⊗OX/P V (OX/P )[1]; ΩX ⊗OX/P V (OX/P )[1])[1] carries the induced filtration. We leave the verification of the following claim to the reader. Lemma 5.5. The map (5.14) is filtered. Thus, the image under (5.14) of a closed 3-form H ∈ Γ(X; F−1 Ω3X ), dH = 0, gives • rise to a structure of an L∞ -algebra on Ω0,• X ⊗OX/P V (OX/P )[1] (whereas general closed 3-forms give rise to curved L∞ -structures). Moreover, cohomologous closed 3forms give rise to gauge equivalent Maurer-Cartan elements, hence to L∞ -isomorphic L∞ -structures. Notation. For H as above we denote by s(OX/P )H the P-Dolbeault complex of the sheaf of multi-vector fields equipped with the corresponding L∞ -algebra structure: • s(OX/P )H = Γ(X; Ω0,• X ⊗OX/P V (OX/P ))[1]  FORMALITY THEOREM FOR GERBES 17 Remark 5.6. In the case when P = 0, in other words, X is a plain C ∞ manifold, the map (5.14) simplifies to Φ : Ω•X [2] → C • (V • (OX )[1]; V • (OX )[1])[1] and s(OX/P ) = s(OX ) = Γ(X; V • (OX ))[1], a DGLA with the Schouten bracket and the trivial differential. These are the unary and the binary operations in the L∞ -structure on s(OX )H , H a closed 3-form on X; the ternary operation is induced by H and all operations of higher valency are equal to zero. The L∞ -structure on multi-vector fields induced by a closed three-form appeared earlier in [26] and [27].  5.5. Formal geometry vs. Dolbeault. Compatibility of the two constructions, one using formal geometry, the other using Dolbeault resolutions, is the subject of the next theorem. Theorem 5.7. Suppose given B ∈ Γ(X; Ω2X ⊗ J ) and H ∈ Γ(X; F−1 Ω3X ) such b • )). Then, the that dH = 0 and j ∞ (H) is cohomologous to b ddR B in Γ(X; DR(F−1 Ω X L∞ -algebras gDR (J )B and s(OX/P )H are L∞ -quasi-isomorphic. Before embarking upon a proof of Theorem 5.7 we introduce some notations. L n b p,q = JX (Ωp,q ), Ω bn = b p,q Let Ω X p+q=n ΩX = JX (ΩX ). The differentials ∂ and ∂ X X b in Ω b •,• which are horizontal with induce, respectively, the differentials ∂b and ∂ X b is b p,• with the differential ∂ respect to the canonical flat connection. The complex Ω X b• b • . The filtration on Ω b • is defined by bp b 0,• = Ω and Ω ⊗J Ω a resolution of Ω J /O J /O X X X b • = JX (Fi Ω• ). With filtrations defined as above the map Fi Ω X X b •X ) j ∞ : Ω•X → DR(Ω is a filtered quasi-isomorphism. The map (5.11) extends to the map of DGLA b: Ω b • [2] → C • (Ω b 0,• ⊗J V • (J )[1]; Ω b 0,• ⊗J V • (J )[1])[1] Φ X X X which gives rise to the map of DGLA b : DR(Ω b • [2]) → C • (DR(Ω b 0,• ⊗J V • (J )[1]); DR(Ω b 0,• ⊗J V • (J )[1]))[1] (5.15) DR(Φ) X X X b • [2])) determines an L∞ -structure Therefore, a degree three cocycle in Γ(X; DR(F−1 Ω X 0,• • b ⊗J V (J )[1]) and cohomologous cocycles determine L∞ -quasi-isomorphic on DR(Ω X structures. b • [2])) we denote by DR(Ω b 0,• ⊗J Notation. For a degree three cocycle ω in Γ(X; DR(F−1 Ω X X • V (J )[1])ω the L∞ -algebra obtained via (5.15).  Proof of Theorem 5.7. The map (5.16) 0,• b 0,• ⊗J V • (J )[1]) j ∞ : ΩX ⊗OX/P V • (OX/P )[1] → DR(Ω X induces a quasi-isomorphism of sheaves of L∞ -algebras 0,• b 0,• ⊗J V • (J )[1])j ∞ (H) . (5.17) j ∞ : (ΩX ⊗OX/P V • (OX/P )[1])H → DR(Ω X b • )) the Since, by assumption, j ∞ (H) is cohomologous to b ddR B in Γ(X; DR(F−1 Ω X b 0,• ⊗J V • (J )[1])j ∞ (H) and DR(Ω b 0,• ⊗J V • (J )[1])b L∞ -algebras DR(Ω are L∞ X X ddR B quasi-isomorphic. 18 P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN b 0,• ⊗J V • (J )[1] induces the quasi-isomorphism The quasi-isomorphism V • (J )[1] → Ω X of sheaves of L∞ -algebras (5.18) DR(V • (J )[1])bd dR B b 0,• ⊗J V • (J )[1])b → DR(Ω X d dR B • (J )[1]) B by Lemma 5.3. The former is equal to the DGLA DR(V According to Corollary 4.2 the sheaf of DGLA DR(V • (J )[1])B is L∞ -quasiisomorphic to the DGLA deduced form the differential graded e2 -algebra DR(Ωe2 (Fe2 ∨ (C • (J )), M ))B . The latter DGLA is L∞ -quasi-isomorphic to DR(C • (J )[1])B . Passing to global sections we conclude that sDR (J )j ∞ (H) and gDR (J )B are L∞ quasi-isomorphic. Together with (5.17) this implies the claim.  6. Deformations of algebroid stacks 6.1. Algebroid stacks. Here we give a very brief overview of basic definitions and facts, referring the reader to [5, 14] for the details. Let k be a field of characteristic zero, and let R be a commutative k-algebra. Definition 6.1. 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. 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 stack then the substack of isomorphisms iC is 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 f+ denote the associated prestack in linear categories given by U 7→ K(U )+ . Let K f+ is an algebroid stack equivalent to the stack of locally free Kop stack. Then, K modules of rank one. f+ . By a twisted form of K we mean an algebroid stack locally equivalent to K The equivalence classes of twisted forms of K are in bijective correspondence with H 2 (X; Z(K)× ), where Z(K) denotes the center of K. To see this note that there ^+ → is a canonical monoidal equivalence of stacks in monoidal categories α : iZ(K) f+ ). Here, iZ(K) ^+ is the stack of locally free modules of rank one over the Aut(K commutative algebra Z(K) and isomorphisms thereof with the monoidal structure f+ ) is the stack of auto-equivalences of K f+ . The given by the tensor product; Aut(K f+ . The inverse ^+ and L ∈ K functor α is given by α(a)(L) = a ⊗Z(K) L for a ∈ Z(K) associates to an auto-equivalence F the Z(K)-module Hom(Id, F ). 6.2. Twisted forms of O. Twisted forms of OX/P are in bijective correspondence × -gerbes: if S is a twisted form of OX/P , the corresponding gerbe is the with OX/P substack iS of isomorphisms in S. We shall not make a distinction between the two notions. The equivalence classes of twisted forms of OX/P are in bijective × ). correspondence with H 2 (X; OX/P FORMALITY THEOREM FOR GERBES 19 The composition log j∞ × × /C× −−→ OX/P /C −−→ DR(J ) → OX/P OX/P × ) → H 2 (X; DR(J )) ∼ induces the map H 2 (X; OX/P = H 2 (Γ(X; Ω•X ⊗ J ), ∇can ). We denote by [S] the image in the latter space of the class of S. Let B ∈ Γ(X; Ω2X ⊗ J ) denote a representative of [S]. Since the map Γ(X; Ω2X ⊗ J ) → Γ(X; Ω2X ⊗ J ) is surjective, there exists a B ∈ Γ(X; Ω2X ⊗ J ) lifting B. b • ) induces the isomorphism The quasi-isomorphism j ∞ : F−1 Ω• → DR(F−1 Ω b • )[1]) ∼ H 2 (X; DR(F−1 Ω = H 2 (X; F−1 Ω•X [1]) = H 3 (Γ(X; F−1 Ω•X )). Let H ∈ Γ(X; F−1 Ω3X ) denote the closed form which represents the class of b ddR B. 6.3. 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 6.2 ([2], Lemma 4.13). Suppose that S is an R-algebra and C is an Ralgebroid 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 6.3 ([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. 6.4. 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. Recall the DGLA gDR (J )ω := Γ(X; DR(C • (J ))[1])ω introduced in 4.1 for arbitrary ω ∈ Γ(X; Ω2X ⊗J ). It satisfies the vanishing condition gDR (J )iω = 0 for i ≤ −2. In particular we obtain DGLA gDR (J )B associated with the form B ∈ Γ(X; Ω2X ⊗ J ) constructed in 6.2. 20 P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN For a nilpotent DGLA g which satisfies gi = 0 for i ≤ −2, P. Deligne [6] and, independently, E. Getzler [11] associated the (strict) 2-groupoid, denoted MC2 (g) (see [4] 3.3.2), which we refer to as the Deligne 2-goupoid. The following theorem follows from the results of [2]; cf. also [1]: Theorem 6.4. For any Artin algebra R with maximal ideal mR there is an equivalence of 2-groupoids MC2 (gDR (J )B ⊗ mR ) ∼ = Def(S)(R) natural in R. The main result of the present paper (Theorem 6.5 below) is a quasi-classical description of Def(S), that is to say, in terms of the L∞ -algebra s(OX/P )H defined in 5.4 in the situation when X is a C ∞ -manifold which admits a pair of complementary integrable complex distributions P and Q satisfying (3.1). The statement of the result, which is analogous to that of Theorem 6.4, requires a suitable replacement for the Deligne 2-groupoid as the latter is defined only for nilpotent DGLA and not for nilpotent L∞ -algebras satisfying the same vanishing condition. The requisite extension of the domain of the Deligne 2-groupoid functor is provided by the theory of J.W. Duskin ([9]). Namely, for a nilpotent L∞ -algebra g which satisfies gi = 0 for i ≤ −2, we consider the 2-groupoid Bic Π2 (Σ(g)). Here, Σ(g) is the Kan simplicial set defined for any nilpotent L∞ -algebra (see [4] 3.2 for the definition and properties) and Π2 is the projector on Kan simplicial sets of Duskin ([9]) which is supplied with a natural transformation Id → Π2 . The latter transformation induces isomorphisms on sets of connected components as well as homotopy groups in degrees one and two (component by component), while higher homotopy groups of a simplicial set in the image of Π2 vanish (component by component). In [9] the image of Π2 is characterized as the simplicial sets arising as simplicial nerves of bi-groupoids (see [4] 2.1.3 and 2.2) and Bic denotes the functor which “reads the bi-groupoid off” the combinatorics of its simplicial nerve. For example, in our situation the simplicial set Π2 (Σ(g)) is the simplicial nerve of Bic Π2 (Σ(g)). The fact that g 7→ Bic Π2 (Σ(g)) is indeed an extension of the Deligne 2-groupoid functor (up to natural equivalence) is the principal result of [4]. Theorem 3.7 (alternatively, Theorem 6.6) of loc. cit. states that, for a nilpotent DGLA g which satisfies gi = 0 for i ≤ −2, Σ(g) and the simplicial nerve of the Deligne 2-groupoid of g are canonically homotopy equivalent. This implies that the Deligne 2-groupoid of g is canonically equivalent to Bic Π2 (Σ(g)). Theorem 6.5. Suppose that X is a C ∞ -manifold equipped with a pair of complementary complex integrable distributions P and Q, and S is a twisted form of OX/P (6.2). Let H ∈ Γ(X; F−1 Ω3X ) be a representative of [S] (6.2). Then, for any Artin algebra R with maximal ideal mR there is an equivalence of bi-groupoids Bic Π2 (Σ(s(OX/P )H ⊗ mR )) ∼ = Def(S)(R), natural in R. Proof. We refer the reader to [4] for notations. By Theorem 5.7, s(OX/P )H is L∞ -quasi-isomorphic to gDR (J )B . Proposition 3.4 of [4] implies that Σ(s(OX/P )H ⊗ mR ) is weakly equivalent to Σ(gDR (J )B ⊗ mR ). In particular, Σ(s(OX/P )H ⊗ mR ) is a Kan simplicial set with homotopy groups FORMALITY THEOREM FOR GERBES 21 vanishing in dimensions larger then two. By Duskin (cf. [9]), the natural transformation Id → Π2 induces a homotopy equivalence between Σ(s(OX/P )H ⊗ mR ) and N Bic Π2 (Σ(s(OX/P )H ⊗mR )), the nerve of the two-groupoid Bic Π2 (Σ(s(OX/P )H ⊗ mR )). On the other hand, by Theorem 3.7 (alternatively, Theorem 6.6) of [4], Σ(gDR (J )B ⊗ mR ) is homotopy equivalent to N MC2 (gDR (J )B ⊗ mR ). Combining all of the above equivalences we obtain an equivalence of 2-groupoids Bic Π2 (Σ(s(OX/P )H ⊗ mR )) ∼ = MC2 (gDR (J )B ⊗ mR ) The result now follows from Theorem 6.4.  Remark 6.6. In the case when P = 0, i.e. X is a plain C ∞ -manifold isomorphism classes of formal deformations of S are in bijective correspondence with equivalence classes of Maurer-Cartan elements of the L∞ -algebra sDR (OX )H ⊗ mR . 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Kontsevich graph complex and the Grothendieck-Teichmüller Lie algebra. arXiv:1009.1654v4 Universidad de los Andes, Bogotá E-mail address: paul.bressler@gmail.com Department of Mathematics, UCB 395, University of Colorado, Boulder, CO 803090395, 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: b-tsygan@northwestern.edu