On quasi-classical limits of DQ-algebroids

arXiv:1510.01361v4 [math.QA] 18 Oct 2016 On quasi-classical limits of DQ-algebroids Paul Bressler, Alexander Gorokhovsky, Ryszard Nest and Boris Tsygan Abstract We determine the additional structure which arises on the classical limit of a DQalgebroid. 1. Introduction As is well-known, a one-parameter formal deformation of a commutative algebra gives rise to a Poisson bracket on the latter. The resulting Poisson algebra is usually referred to as the quasi-classical limit of the deformation. This circumstance is responsible for the link between deformation quantization and Poisson geometry. Although deformation quantization had been developed in the context of (sheaves of) algebras and deformations thereof, it became apparent from the work of M. Kontsevich ([Kon01]) and M. Kashiwara ([Kas96]) that the broader context of algebroid stacks provides a more natural setting for the theory leading to the notion of a DQ-algebroid due to M. Kashiwara and P. Schapira ([KS12]). Unlike the more traditional setting, the classical limit of a DQ-algebroid on a manifold X is not the structure sheaf OX 1 but, rather, a twisted form thereof which is, in essence, a linear × -gerbe. The purpose of the present note is to determine the additional structure version of a OX which arises on the classical limit of a DQ-algebroid and reflects the non-commutative aspect of the latter in the way in which the Poisson bracket reflects non-commutativity of the deformation quantization of OX . We shall presently describe our answer. A DQ-algebroid C gives rise to the associated Poisson bi-vector π C on X, hence to a Lie algebroid structure on the cotangent sheaf Ω1X which we denote ΠC . The notion of a connec× -gerbe readily generalizes to the notion of a ΠC -connective tive structure with curving on a OX structure with curving (and, in fact, to the notion of a B-connective structure with curving for any Lie algebroid B). The principal result of this note is the construction of a canonical ΠC -connective structure with flat curving on the classical limit of the DQ-algebroid C. As an × -gerbe S admits a deformation with the associapplication we show (Theorem 7.5) that an OX × ) lifts to Deligne–Poisson cohomology ated Poisson bi-vector π only if the class [S] ∈ H 2 (X; OX 2,cl × 1 2 H (X; OX → ΩΠ → ΩΠ ), where Π is the Lie algebroid (structure on Ω1X ) determined by π. Similar ideas have been used in [BGKP16] to give a necessary and sufficient condition for a line bundle on a lagrangian subvariety of a symplectic manifold to admit a quantization. In [Bur02] (see also [BW04]), the canonical flat contravariant connection on the classical limit of a bi-module over a pair of star-product algebras is constructed. Our variant of this construction 2010 Mathematics Subject Classification 53D55 Keywords: algebroid, deformation quantization, classical limit A. Gorokhovsky was partially supported by NSF grant DMS-0900968. B. Tsygan was partially supported by NSF grant DMS-0906391. R. Nest was supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92). 1 Here and below we consider complex valued functions. Paul Bressler, Alexander Gorokhovsky, Ryszard Nest and Boris Tsygan (see 6.8) is a key component of the present work. We work in the natural generality of a C ∞ manifold X equipped with an integrable complex distribution P satisfying additional technical conditions. Subsumed thereby are the case of plain C ∞ manifold X (P trivial) as well as X a complex analytic manifold (P a complex structure). The paper is organized as follows. Relevant facts about calculus in presence of an integrable distribution are recalled in Section 2. Section 3 contains basic facts about torsors and gerbes and establishes notation. In Section 4 we review the theory of Lie algebroids and modules. In particular, we describe the cohomological classification of invertible modules over a Lie algebroid which may be of independent interest. Section 5 is devoted to generalization of the notions of connective structures and curving to the Lie algebroid setting. In Section 6 we review basic facts on deformation quantization algebras and bi-modules and define the subprincipal curvature. Section 7 contains the necessary background material on DQ-algebroids and the principal result of the present note – the construction of the quasi-classical limit of a DQ-algebroid. In Section 8 we discuss the relationship between the construction of the quasi-classical limit and the quasiclassical description of [BGNT15] of the category of DQ-algebroids provided by the formality theorem. Section 8 concludes with a conjecture on the precise nature of the aforementioned relationship. It is the authors’ pleasure to express their gratitude to the referee for thorough reading of the original manuscript and thoughtful comments and suggestions. 2. Calculus in presence of an integrable 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 [Kos70], [Raw77] and [FW79] 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. 2.1 Complex distributions A (complex) distribution on X is a sub-bundle2 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 [Raw77], a sufficient condition for integrability of a complex distribution P is P ∩ P is a sub-bundle and both P and P + P are involutive. 2 A sub-bundle is an OX -submodule which is a direct summand locally on X. 2 (1) On quasi-classical limits of DQ-algebroids 2.2 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 by P ⊥ , i.e. V j−i ⊆ ΩjX . Let ∂ denote the differential in Gr•F Ω•X . The wedge product of F−i ΩjX = i P ⊥ ∧ ΩX 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 to the d ∂ composition OX − → Ω1X → Ω1X /P ⊥ . Let OX/P := ker(OX − → Gr0F Ω1X ). Equivalently, OX/P = (OX )P ⊂ OX , the subsheaf of functions locally constant along P. Note that ∂ is OX/P -linear. Theorem 2 of [Raw77] says that, if P satisfies the condition (1), the higher ∂-cohomology of OX vanishes, i.e.  OX/P if i = 0 i F • (2) H (Gr0 ΩX , ∂) = 0 otherwise. In what follows we will assume that the complex distribution P under consideration is integrable and satisfies (2). The latter is implied by the condition (1). 2.3 ∂-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 for e ∈ E and f ∈ OX . Equivalently, a connection along P is an P P OX -linear map ∇P (•) : P → EndC (E) which satisfies the Leibniz rule ∇ξ (f e) = f ∇ξ (e) + ∂f · e for e ∈ E and f ∈ OX . In particular, ∇P ξ is OX/P -linear. The two avatars of a connection along P (e). (e) = ι ∇ P are related by ∇P ξ ξ 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 ∂ E of the graded Gr0F Ω•X 2 module Gr0F Ω•X ⊗OX E which is a ∂-operator if and only if ∂ 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 (2) generalizes easily to vector bundles. Lemma 2.1. 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 2.2. 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. 2.4 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 3 Paul Bressler, Alexander Gorokhovsky, Ryszard Nest and Boris Tsygan coincide with the one induced via the duality pairing between the latter and P ⊥ .3 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 Vi ⊥ restricts to a flat connection along P, i.e. a canonical ⊥ P for all i. It is easy to see that the multiplication ∂-operatorVon P and, therefore, on Vi ⊥ map F Ω• [i] is an isomorphism which identifies the ∂-complex of Gr0F Ω• ⊗ i P ⊥ → Gr−i P with Vi ⊥ F • i i F • 0 i Gr−i Ω [i]. Let ΩX/P := H (Gr−i ΩX , ∂) (so that OX/P := ΩX/P ). Then, ΩX/P ⊂ P ⊂ ΩiX . 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 isomorphism Vi i+1 1 i i OX/P ΩX/P → ΩX/P . The de Rham differential d restricts to the map d : ΩX/P → ΩX/P and the complex Ω•X/P := (Ω•X/P , d) is a commutative DGA. The Hodge filtration F• Ω•X/P is defined by Fi Ω•X/P = ⊕j>−iΩjX/P , so that the inclusion Ω•X/P ֒→ Ω•X is filtered with respect to the Hodge filtration. It follows from Lemma 2.1 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. Algebroid stacks In what follows we will be considering gerbes with abelian lien. Below we briefly recall some relevant notions and constructions with the purpose of establishing notations. We refer the reader to [Del73] and [Mil03] for detailed treatment of Picard stacks and gerbes respectively. Suppose that X is a topological space. 3.1 Picard stacks We recall the definitions from 1.4.Champs de Picard strictement commutatifs of [Del73]. A (strictly commutative) Picard groupoid P is a non-empty groupoid equipped with a functor + : P × P → P and functorial isomorphisms – σx,y,z : (x + y) + z → x + (y + z) – τx,y : x + y → y + x rendering + associative and strictly commutative, and such that for each object x ∈ P the functor y 7→ x + y is an equivalence. A Picard stack on X is a stack in groupoids P equipped with a functor + : P × P → P and functorial isomorphisms σ and τ as above, which, for each open subset U ⊆ X, endow the category P(U ) a structure of a Picard groupoid. 3 In the case of a real polarization this connection is known as the Bott connection. 4 On quasi-classical limits of DQ-algebroids 3.2 Torsors Suppose that A is a sheaf of abelian groups on X. The stack of A-torsors will be denoted by A[1]; it is a gerbe since all A-torsors are locally trivial. Suppose that φ : A → B is a morphism of sheaves of abelian groups. The assignment A[1] ∋ T 7→ φ(T ) := T ×A B ∈ B[1] extends to a morphism φ : A[1] → B[1] of stacks. There is a canonical map of sheaves of torsors φ = φT : T → φ(T ) compatible with the map φ of abelian groups and respective actions. Suppose that A and B are sheaves of abelian groups. The assignment A[1] × B[1] ∋ (S, T ) 7→ S × T ∈ (A × B)[1] extends to a morphism of stacks × : A[1] × B[1] → (A × B)[1]. Suppose that A is a sheaf of abelian groups with the group structure + : A×A → A. The latter is a morphism of sheaves of groups since A is abelian. The assignment A[1] × A[1] ∋ (S, T ) 7→ S + T := +(S × T ) defines a structure of a Picard stack on A[1]. If φ : A → B is a morphism of sheaves of abelian groups the corresponding morphism φ : A[1] → B[1] is a morphism of Picard stacks. As a consequence, the set π0 A[1](X) of isomorphism classes of A-torsors is endowed with a canonical structure of an abelian group. There is a canonical isomorphism of groups π0 A[1](X) ∼ = H 1 (X; A). 3.3 Gerbes An A-gerbe is a stack in groupoids which is a twisted form of (i.e. locally equivalent to) A[1]. The (2-)stack of A-gerbes will be denoted A[2]. Since all A-gerbes are locally equivalent the (2-)stack A[2] is a (2-)gerbe. Every A-gerbe S (is equivalent to one which) admits a canonical action of the Picard stack A[1] by autoequivalences denoted + : A[1] × S → S, (T, L) 7→ T + L endowing S with a structure of a (2-)torsor under A[1]. We shall not make distinction between A-gerbes and (2-)torsors under A[1] and use the notation A[2] for both. Suppose that φ : A → B is a morphism of sheaves of abelian groups and S is an A-gerbe. In particular, for any two (locally defined) objects s1 , s2 ∈ S the sheaf HomS (s1 , s2 ) is an A-torsor. The stack φS is defined as the stack associated to the prestack with the same objects as S and HomφS (s1 , s2 ) := φ(HomS (s1 , s2 )). Then, φS is a B-gerbe and the assignment S 7→ φS extends to a morphism φ : A[2] → B[2]. There is a canonical morphism of stacks φ = φS : S → φS which induces the map φ : A → B on groups of automorphisms. The Picard structure on A[1] gives rise to one on A[2] defined in analogous fashion. As a consequence, the set π0 A[2](X) of equivalence classes of A[1]-torsors is endowed with a canonical structure of an abelian group. There is a canoncial isomorphism of groups π0 A[2](X) ∼ = H 2 (X; A). d → A1 be a complex of sheaves of abelian groups on X concentrated in degrees Let A0 − d → A1 )-torsor is a pair (T, τ ), where T is a A0 -torsor and τ is a zero and one. Recall that a (A0 − d → A1 )-torsors trivialization (i.e. a section) of the A1 -torsor d(T ) = T ×A0 A1 . A morphism of (A0 − 0 φ : (S, σ) → (T, τ ) is a morphism of A -torsors φ : S → T such that the induced morphism of A1 -torsors d(φ) : d(S) → d(T ) which commutes with respective trivializations, i.e. d(φ)(σ) = τ d → A1 )-torsors is defined as follows. Suppose The monoidal structure on the category of (A0 − d → A1 )-torsors. The sum (S, σ) + (T, τ ) is represented by (S + that (S, σ) and (T, τ ) are (A0 − T, σ + τ ), where σ + τ is the trivialization of d(S) + d(T ) = d(S + T ) induced by σ and τ . d d → → A1 )-torsors form a Picard stack on X which we will denote by (A0 − Locally defined (A0 − 1 A )[1]. By a result of P. Deligne ([Del73], Proposition 1.4.15) all Picard stacks arise in this way. 5 Paul Bressler, Alexander Gorokhovsky, Ryszard Nest and Boris Tsygan The group (under the operation induced by the monoidal structure) of isomorphism classes of d d d → A1 ). → A1 )[1](X) is canonically isomorphic to H 1 (X; A0 − → A1 )-torsors on X, i.e. π0 (A0 − (A0 − 3.4 2-torsors d → A1 )-gerbe is equivalent to the data (S, σ), where S is an A0 -gerbe and τ is a trivialization A (A0 − d τ of the A1 -gerbe dS, i.e. an equivalence τ : dS → A1 [1]. The composition S − → dS − → A1 [1] is a 1 functorial assignment of an A -torsor τ (s) to a (locally defined) object s ∈ S. The 2-stack d d d → A1 )[1] → A1 )[2]. The Picard structure on (A0 − → A1 )-gerbes will be denoted (A0 − of (A0 − d → A1 )[2] defined in an analogous fashion. As a consequence, the set gives rise to one on (A0 − d d → A1 )-gerbes is endowed with a canonical → A1 )[2](X) of equivalence classes of (A0 − π0 (A0 − d → A1 )[2](X) ∼ structure of an abelian group. There is a canonical isomorphism of groups π0 (A0 − = d → A1 ). H 2 (X; A0 − 3.5 Algebroids Recall that a k-algebroid, k a commutative ring with unit, is a stack in k-linear categories C such that the substack of isomorphisms iC (which is a stack in groupoids) is a gerbe. Suppose that A is a sheaf of k-algebras on X. For U open in X let A+ (U ) denote the category with one object whose endomorphism algebra is Aop . The assignment U 7→ A+ (U ) defines a prestack on X, which we denote A+ . Let g A+ denote the associated stack. Note that g g + + (X) is A is equivalent to the stack of A-modules locally isomorphic to A. The category A non-empty. Conversely, let C be a k-algebroid such that the category C(X) is non-empty. Let L ∈ C(X), and let A := EndC (L)op . The morphism A+ → C which sends the unique object to L induces g + → C. and equivalence A For a sheaf of k-algebras A on X a twisted form of A is a k-algebroid locally k-linearly equivalent to g A+ . Let X be a C ∞ manifold equipped with an integrable complex distribution P. Twisted × -gerbes via forms of the sheaf of C-algebras OX/P are in one-to-one correspondence with OX/P S 7→ iS. Equivalence classes of twisted forms of OX/P form an group canonically isomorphic to × ). H 2 (X; OX/P 4. Lie algebroids In this section we review basic definitions and facts concerning Lie algebroids, modules over Lie algebroids and cohomological classification of invertible modules. General references for this material are [Mac05] Throughout the section X is a C ∞ manifold equipped with an integrable complex distribution P, see Section 2 for definitions and notations. 4.1 OX/P -Lie algebroids An OX/P -Lie algebroids or, simply, a Lie algebroid is the datum of – an OX/P -module B, – an OX/P -linear map σ : B → TX/P (the anchor ), 6 On quasi-classical limits of DQ-algebroids – a structure of a C-Lie algebra on B given by [ , ] : B ⊗C B → B such that the Leibniz rule [b1 , f b2 ] = σ(b1 )(f )b2 + f [b1 , b2 ], b1 , b2 ∈ B, f ∈ OX/P , holds. As a consequence, the anchor map is a morphism of Lie algebras. A morphism of Lie algebroids is an OX/P -linear map φ : B1 → B2 of Lie algebras which is commutes with respective anchors. The sheaf TX/P is a Lie algebroid under the Lie bracket of vector fields with the anchor the identity map. It is immediate from the definitions that TX/P is the terminal object in the category of Lie algebroids. Further examples of Lie algebroids are given below. 4.1.1 Atiyah algebras For a locally free OX/P -module of finite rank E the Atiyah algebra of E, denoted AE is defined by the pull-back diagram AE   y σ −−−−→ TX/P  1⊗Id y σ Diff 61 (E, E) −−−−→ EndOX/P (E) ⊗ TX/P where the bottom horizontal arrow is the order one principal symbol map. It is a Lie algebroid under (the restriction of) the commutator bracket, the anchor given by the top horizontal map (the restriction of the principal symbol map). There is an exact sequence σ 0 → EndOX/P (E) → AE − → TX/P → 0 . V e : Ω1X/P → 4.1.2 Poisson structures Let π ∈ Γ(X; 2 TX/P ) be a bi-vector; we denote by π TX/P the adjoint map α 7→ π(α, ·). The bi-vector π is Poisson if the operation { , } : OX/P ⊗C OX/P → OX/P defined by {f, g} = π(df, dg) satisfies the Jacobi identity, i.e. endows OX/P with a structure of a C-Lie algebra. Let [ , ]π : Ω1X/P ⊗C Ω1X/P → Ω1X/P denote the map given by [α, β]π = Lπe(α) β − Lπe(β) α − dπ(α, β). (3) The bi-vector π is Poisson if and only if the bracket (3) on Ω1X/P satisfies the Jacobi identity; the map π e is compatible with respective brackets. Thus, a Poisson bi-vector π gives rise to a structure, denoted Π in what follows, of a Lie algebroid on Ω1X/P with bracket (3) and the anchor π e. 4.1.3 The de Rham complex We shall assume for simplicity that the Lie algebroid B in question is locally free of finite rank over OX/P . Let Ω0B = OX/P , Ω1B = B ∨ , where, for an OX/P -module E we put E ∨ := HomOX/P (E, OX/P ). L V Let ΩiB = i Ω1B . Then, Ω•B := i ΩiB [−i] is a graded commutative algebra. The composition σ∨ d OX/P − → Ω1X/P −−→ B ∨ = Ω1B extends uniquely to a square-zero derivation dB of degree one of the algebra Ω•B . Let Ωi,cl B := V d B i i TX/P and the de Rham differential ker(ΩiB −→ Ωi+1 B ). In the case B = Π of example 4.1.2 ΩΠ = 7 Paul Bressler, Alexander Gorokhovsky, Ryszard Nest and Boris Tsygan is given by dΠ = [π, ·], i.e the Schouten bracket with the bi-vector π. 4.2 Modules over Lie algebroids Suppose that B is a Lie algebroid. 4.2.1 B-modules Suppose that M is a OX/P -module. A B-module structure on M is given by an OX/P -linear map B → EndC (M), b 7→ (m 7→ b.m), b ∈ B, m ∈ M, which is a map of Lie algebras (with respect to the commutator bracket of endomorphisms) and satisfies the Leibniz rule b.f m = σ(b)(f )m + f · b.m. Suppose that E is a locally free OX/P -module of finite rank. It transpires from the above definition that a B-module structure on E amounts to a morphism of algebroids B → AE . Equivalently, it is a morphism of Lie algebroids B → B ×TX/P AE splitting the exact sequence pr B 0 → EndOX/P (E) → B ×TX/P AE −−→ B → 0. A morphism of B-modules is an OX/P -linear map which commutes with respective actions of B. B-modules form an Abelian category. The structure sheaf OX/P is a TX/P -module hence a B-module in a canonical way for any Lie algebroid B. 4.2.2 Tensor product of B-modules The category of B-modules is endowed with a monoidal structure: for any two modules M and N their tensor product M ⊗OX/P N is equipped with the canonical B-module structure given by the Leibniz rule. Namely, for m ∈ M, n ∈ M, b ∈ B, b.(m ⊗ n) = (b.m) ⊗ n + m ⊗ (b.n). The sheaf HomOX/P (M, N ) is equipped with the canonical structure of a B-module by (b.φ)(m) = b.(φ(m)) − φ(b.m). 4.2.3 Invertible B-modules For the purposes of this note an invertible B-module is a line bundle (i.e. a locally free OX/P -module of rank one) equipped with a structure of a B-module. Invertible B-modules form a Picard category, denoted PicB (X), under the tensor product over OX/P . Let PicB denote the Picard stack of invertible B-modules, i.e. PicB (U ) = PicB (U ) for U open in X. × -torsor (the Suppose that L is a line bundle; we shall denote by L× the corresponding OX/P subsheaf of nowhere vanishing sections). Recall that a B-module structure on L is a morphism of Lie algebroids ∇ : B → B ×TX/P AL splitting the short exact sequence pr i B 0 → OX/P − → B ×TX/P AL −−→ B → 0. (4) For any two such, ∇1 , ∇2 : B → B ×TX/P AL , their difference ∇2 − ∇1 satisfies prB ◦(∇2 − ∇1 ) = 0 and therefore factors through (the inclusion of) OX/P . Moreover, the resulting section ∇2 − ∇1 ∈ HomOX/P (B, OX/P ) = Ω1B satisfies dB (∇2 − ∇1 ) = 0, where dB is the de Rham differential introduced in 4.1.3. The sheaf of (locally defined) splittings of (4) by Lie algebroid morphisms is a Ω1,cl B -torsor ♭ and will be denoted ConnB (L). A structure of a B-module on L is a trivialization of Conn♭B (L). The morphism of sheaves of groups × → Ω1,cl dB log : OX/P B 8 On quasi-classical limits of DQ-algebroids is defined by f 7→ dB f . There is a canonical morphism of sheaves f dB log : L× → Conn♭B (L) (5) defined as follows. Let s ∈ L× be a (locally defined) section. The section s gives rise to a (locally ∼ = defined) isomorphism OX/P − → L. Via this isomorphism the canonical B-module structure on OX/P gives rise to a B-module structure on L which corresponds to a (locally defined) section denoted dB log(s) ∈ Conn♭B (L). We leave it to the reader to check that the morphism (5) is compatible with the map of sheaves of groups dB log and respective actions. Hence, (5) induces a ♭ × × ∼ canonical isomorphism of Ω1,cl Ω1,cl B . In particular, B -torsors ConnB (L) = dB log(L ) := L ×O × X/P a structure of a B-module on L amounts to a trivialization of dB log(L× ). 4.2.4 Classification of invertible B-modules Assigning to an invertible B-module L the pair (L× , ∇), where ∇ is the trivialization of dB log(L× ) corresponding to the B-module structure we obtain a morphism of Picard stacks d log B × −− −→ Ω1,cl PicB → (OX/P B )[1] (6) Lemma 4.1. The morphism (6) is an equivalence. × -torsor T determines Proof. A quasi-inverse to (6) is given by the following construction. A OX/P the line bundle L := T ×O× OX/P such that there is a canonical isomorphism L× ∼ = T , hence X/P dB log(T ) ∼ = dB log(L× ) ∼ = Conn♭B (L). Thus, a trivialization of dB log(T ) gives rise to a B-module structure on L. dB log × −− −→ Ω1,cl Corollary 4.2. π0 PicB (X) ∼ = H 1 (X; OX/P B ). The above result reduces to the well-known classification of line bundles equipped with flat connections in the case B = TX/P (see, for example, [Bry93]). In the case B = Π (see 4.1.2) Lemma 4.1 is a particular case of Proposition 2.8 of [Bur01]. 4.3 O-extensions e c, σ) which consists Suppose that B is a OX/P -Lie algebroid. An O-extension of B is a triple (B, of e – a Lie algebroid B, e – a central element c of the Lie algebra of sections Γ(X; B), – a morphism of Lie algebroids σ : Be → B such that these data give rise to the associated short exact sequence f 7→f ·c σ 0 → OX/P −−−−→ Be − →B→0 (7) Locally defined O-extensions of B form a Picard stack under the operation of Baer sum of extensions which we denote OEX T (B). e [b, f · c] = b(f ) · c, Since c is central, it follows from the Leibniz rule that, for f ∈ OX/P , b ∈ B, where b ∈ TX/P denotes the image of b under the anchor map. That is to say, the (adjoint) action of Be on OX/P · c ∼ = OX/P factors through TX/P and the latter action coincides with the Lie derivative action of vector fields on functions. 9 Paul Bressler, Alexander Gorokhovsky, Ryszard Nest and Boris Tsygan Suppose that B is locally free of finite rank over OX/P . Then there is a canonical equivalence of Picard stacks ∼ = OEX T (B) − → (Ω1B → Ω2,cl B )[1]. (8) e c, σ) the Ω1 -torsor ConnB (B) e of (locally defined) splittings of The functor (8) associates to (B, B 2,cl e ∋ ∇ 7→ c(∇) ∈ Ω2,cl ⊂ e → Ω , ConnB (B) σ : Be → B. The “curvature” map c : ConnB (B) B B V2 Hom( B, OX/P ) is determined by c(∇)(b1 , b2 ) · c = [∇(b1 ), ∇(b2 )] − ∇([b1 , b2 ]). We leave it to the reader to check that the standard calculation shows that the above formula defines a closed 2-form. 5. Connective structures Throughout this section X is a C ∞ manifold equipped with an integrable complex distribution P (see Section 2 for definitions and notations), B is a OX/P -Lie algebroid (Section 4). This section × -gerbes in terms of the formalism of is devoted to generalities on connective structures on OX/P Section 3. The exposition follows closely the terminology of [Bry93], Chapter V. 5.1 B-connective structures × -gerbe. A B-connective structure on S is a trivialization of the Ω1B -gerbe dB log S, Let S be a OX/P i.e. a morphism, thus an equivalence, of Ω1B -gerbes ∇ : dB log S → Ω1B [1]. The composition ∇ S → dB log S − → Ω1B [1] is a functorial assignment of a Ω1B -torsor ∇(L) to a (locally defined) object × → Ω1B on respective sheaves of groups of automorL ∈ S which induces the map dB log : OX/P phisms. For locally defined objects L1 , L2 ∈ S the connective structure induces the isomorphism ∇ : ConnB (HomS (L1 , L2 )) = HomdB log S (dB log(L1 ), dB log(L2 )) → HomΩ1 [1] (∇(L1 ), ∇(L2 )) ∼ = B ∇(L2 ) − ∇(L1 ). × -gerbes with B-connective It transpires from the above description that locally defined OX/P d log B × −− −→ Ω1B )[2]. As structure, i.e. pairs (S, ∇) as above are objects of the Picard 2-stack (OX/P × -gerbes with B-connective structure is a consequence, the group of equivalence classes of OX/P dB log × −−−→ Ω1B ). canonically isomorphic to H 2 (X; OX/P × 1 1 → Ω1B factors through the inclusion Ω1,cl Since the map dB log : OX/P B ֒→ ΩB the ΩB -gerbe ♭ dB log S is induced from the Ω1,cl B -gerbe which will be denoted dB log S . A connective structure 1 ∇ : dB log S → ΩB [1] is called flat if it is induced from the trivialization of dB log S ♭ , i.e. from an equivalence ∇ : dB log S ♭ → Ω1,cl B [1]. × -gerbes with flat B-connective It transpires from the above description that locally defined OX/P d log B × −− −→ Ω1,cl structure are objects of (OX/P B )[2]. As a consequence, the group of equivalence classes dB log × × −−−→ -gerbes with flat B-connective structure is canonically isomorphic to H 2 (X; OX/P of OX/P Ω1,cl B ). 10 On quasi-classical limits of DQ-algebroids 5.2 Curving × -gerbe S is a lift of the functor A curving κ on the a B-connective structure ∇ on a OX/P ∇ S− → Ω1B [1] to (Ω1B → Ω2B )[1], i.e. a factorization of ∇ as κ S− → (Ω1B → Ω2B )[1] → Ω1B [1]. κ We refer to the functor S − → (Ω1B → Ω2B )[1] as a connective structure with curving (given by κ). × -gerbes with B-connective structure with curving, i.e. pairs (S, κ) as Locally defined OX/P d log d B B × −− −→ Ω1B −→ Ω2B )[2]. The above are objects of the Picard 2-stack which we denote (OX/P × -gerbes with connective structure with curving is canonically group of equivalence classes of OX/P d log d B B × −− −→ Ω1B −→ Ω2B ). isomorphic to H 2 (X; OX/P The curving S → (Ω1B → Ω2B )[1] is flat if it factors through (Ω1B → Ω2,cl B )[1]. In this case 2,cl 1 we refer to the functor S → (ΩB → ΩB )[1] as a B-connective structure with a flat curv× -gerbes with B-connective structure with flat curving are objects of ing. Locally defined OX/P dB log d B × −−−→ Ω1B −→ Ω2,cl the Picard 2-stack which we denote (OX/P B )[2]. The group of equivalence × classes of OX/P -gerbes with connective structure with flat curving is canonically isomorphic to dB log d B × −−−→ Ω1B −→ Ω2,cl H 2 (X; OX/P B ). 2,cl 1 The morphism of complexes Ω1,cl B → (ΩB → ΩB ) induces a canonical flat curving on a flat B-connective structure. 6. DQ-algebras Throughout this section X a C ∞ manifold equipped with an integrable complex distribution P, see Section 2 for definitions and notations. In the context of complex manifolds the notion of a DQ-algebra was introduced in [KS12]. 6.1 Star-products A star-product on OX/P is a map OX/P ⊗C OX/P → OX/P [[t]] of the form f ⊗ g 7→ f ⋆ g = f g + ∞ X Pi (f, g)ti , (9) i=1 where Pi are bi-differential operators. Such a map admits a unique C[[t]]-bilinear extension OX/P [[t]] ⊗C[[t]] OX/P [[t]] → OX/P [[t]] and the latter is required to define a structure of an associative unital C[[t]]-algebra on OX/P [[t]]. Proposition 6.1 ([KS12], Proposition 2.2.3). Let ⋆ and ⋆′ be star-products and let ϕ : (OX/P [[t]], ⋆) → (OX/P [[t]], ⋆′ ) be a morphism of C[[t]]-algebras. Then, there exists a unique sequence of differ∞ P ential operators {Ri }i>0 on X such that R0 = 1 and ϕ(f ) = Ri (f )ti for any f ∈ OX/P . In i=0 particular, ϕ is an isomorphism. 11 Paul Bressler, Alexander Gorokhovsky, Ryszard Nest and Boris Tsygan Remark 6.2. The paper [KS12] and, in particular, Proposition 2.2.3 of loc. cit. pertain to the holomorphic context, i.e. the case when P is a complex structure. However, it is easy to see that the proof of Proposition 2.2.3 as well as the results it is based upon carry over to the case of a general integrable distribution. For a star-product given by (9) the operation f ⊗ g 7→ P1 (f, g) − P1 (g, f ) (10) is a Poisson bracket on OX/P which we refer to as the associated Poisson bracket. It follows from Lemma 6.6 below that isomorphic star-products give rise to the same associated Poisson bracket. Definition 6.3. A star-product given by (9) is said to be special if P1 is skew-symmetric. 1 Remark 6.4. If the star-product is special, then P1 (f, g) = {f, g}, where the latter is the 2 associated Poisson bracket. The following lemma is well-known. Lemma 6.5. Any star-product is locally isomorphic to a special one. Proof. Let P1+ (respectively, P1− ) denote the skew-symmetrization (respectively, the symmetrization) of P1 . Associativity of the star-product implies that P1 , P1+ and P1− are Hochschild cocycles of degree two on OX/P . The Hochschild-Kostant-Rosenberg theorem says that, locally on X, P1+ is a bi-vector and P1− is a Hochschild coboundary, i.e., locally on X, there exists a differential operator Q (acting on OX/P ) such that P1− (f, g) = Q(f g) − Q(f )g − f Q(g). The star-product ⋆′ defined by f ⋆′ g = exp(−tQ)(exp(tQ)(f ) ⋆ exp(tQ)(g)) = f g + P1+ (f, g)t + · · · is special. 6.2 DQ-algebras A DQ-algebra is a sheaf of C[[t]]-algebras locally isomorphic to a star-product. For a DQ-algebra A there is a canonical isomorphism A/t·A ∼ = OX/P . Therefore, there is a canonical map (reduction σ modulo t) A − → OX/P of C[[t]]-algebras. A morphism of DQ-algebras is a morphism of sheaves of C[[t]]-algebras. For an open subset U ⊆ X let DQ-algX/P (U ) denote the category of DQ-algebras on U , where U is equipped with the restriction of P. The assignment U 7→ DQ-algX/P (U ) defines a stack on X denoted DQ-algX/P . In what follows, we shall denote HomDQ-algX/P simply by HomDQ-alg . 6.3 The associated Poisson structure Suppose that A is a DQ-algebra. The composition [.,.] σ A ⊗ A −−→ A − → OX/P [.,.] is trivial. Therefore, the commutator A ⊗ A −−→ A takes values in tA. The composition [.,.] t−1 σ A ⊗ A −−→ tA −−→ A − → OX/P factors uniquely as {.,.} σ⊗σ A ⊗ A −−−→ OX/P ⊗ OX/P −−→ OX/P 12 On quasi-classical limits of DQ-algebroids The latter map, {., .} : OX/P ⊗ OX/P → OX/P is a Poisson bracket on OX/P , hence corresponds V to a bi-vector π ∈ Γ(X; 2 TX/P ). If A is a star-product we recover the (10). Lemma 6.6. Locally isomorphic DQ-algebras give rise to the same associated Poisson bracket. Proof. Suppose that Ai , i = 1, 2 is a DQ-algebras with associated Poisson bracket {., .}i and Φ : A1 → A2 is a morphism of such. Let f, g ∈ OX/P , fe, ge ∈ A1 with f = fe + tA1 , g = ge + tA1 . Since Φ induces the identity map modulo t, t · {f, g}2 + t2 A2 = Φ(fe)Φ(e g ) − Φ(e g )Φ(fe) + t2 A2 = Φ(fee g−e gfe + t2 A1 ) = t · {f, g}1 + t2 A2 Thus, if Ai , i = 1, 2 are locally isomorphic, then the associated Poisson brackets are locally equal hence equal. 6.4 Standard sections Let A be a DQ-algebra. Recall ([KS12], Definition 2.2.6) that a C-linear section φ : OX/P → A σ of the map A − → OX/P is called standard if there exist bi-differential operators Pi , i = 0, 1, . . ., such that for any f, g ∈ OX/P X φ(f )φ(g) = φ(f g) + φ(Pi (f, g))ti , (11) i>1 where the left-hand side product is computed in A. In this case X f ⊗ g 7→ f ⋆φ g := f g + Pi (f, g)ti i>1 defines a star-product. A standard section φ : OX/P → A extends by t-linearity to a morphism of DQ-algebras e φ : (OX/P [[t]], ⋆φ ) → A. Conversely, a morphism of DQ-algebras ϕ : (OX/P [[t]], ⋆) → A restricts to a standard section φ := ϕ|OX/P such that ⋆φ = ⋆ and φe = ϕ. We will call a standard section φ special if the corresponding star-product ⋆φ is special. Notation. Let Σ(A) denote the sheaf of locally defined special standard sections. For U ⊆ X and open subset, φ1 , φ2 ∈ Σ(A)(U ) and k = 1, 2, . . . we say that φ1 and φ2 φi are equivalent modulo tk+1 if the compositions OX/P (U ) −→ A(U ) → A(U )/tk+1 A(U ), i = 1, 2, coincide. Clearly, “equivalence modulo tk+1 ” is an equivalence relation on Σ(A) which we denote ∼k . We denote by Σk (A) the sheaf associated to the presheaf U 7→ Σ(A)(U )/ ∼k . Proposition 6.7. Let A be a DQ-algebra. (i) The sheaf Σ(A) is locally non-empty. (ii) The quotient map Σ(A) → Σk (A) is locally surjective on sets of sections. (iii) For a morphism Φ : A0 → A1 of DQ-algebras and φ ∈ Σ(A0 ) the composition Φ ◦ φ is a special standard section, i.e. Φ ◦ φ ∈ Σ(A1 ). Proof. Since the question is local we may assume that A = (OX/P [[t]], ⋆) is a star product given by (9). (i) It follows from Lemma 6.5 that locally there exists an isomorphism ϕ : (OX/P [[t]], ⋆) → A with ⋆ a special star-product. Then ϕ|OX/P is a special standard section. 13 Paul Bressler, Alexander Gorokhovsky, Ryszard Nest and Boris Tsygan (ii) Since the question is local we may assume that the map q : A → A/tk+1 A admits a splitting s : A/tk+1 A → A, q ◦ s = Id. The composition s ◦ q : A → A preserves equivalence modulo tk+1 , hence induces the map s ◦ q : Σk (A) → Σ(A). Since the composition Σk (A) → Σ(A) → Σk (A) is the identity map the claim follows. (iii) Since Φ induces the identity map modulo t the composition Φ◦φ is a section. By Proposition 6.1 the terms of X (Φ ◦ φ)(f )(Φ ◦ φ)(g) = (Φ ◦ φ)(f g) + (Φ ◦ φ)(Pi (f, g))ti , i>1 are bi-differential operators and therefore the section Φ ◦ φ is standard if φ is. The formula shows that, if φ is special, then so is Φ ◦ φ. Φ Thus, the assignment A 7→ Σ(A), (A0 − → A1 ) 7→ (Σ(Φ) : φ 7→ Φ ◦ φ) defines a functor, denoted Σ, on the category of DQ-algebras. It is clear that the map Σ(Φ) : Σ(A0 ) → Σ(A1 ) induced by the morphism of DQ-algebras Φ : A0 → A1 preserves equivalence modulo tk . Thus, the functor Σ gives rise to functors Σk for all k = 1, 2, . . .. 6.5 The functor Σ1 Suppose that A is a DQ-algebra with the associated Poisson tensor π and the corresponding Lie algebroid (structure on Ω1X/P ) denoted Π. Lemma 6.8. Suppose that φ0 and φ1 are standard sections of A with the corresponding starproducts ⋆(j) := ⋆φi given by (j) (j) f ⋆(j) g = f g + P1 (f, g)t + P2 (f, g)t2 + · · · for j = 0, 1. Let R = 1 + ∞ P i=1 f1 = φ f0 ◦ R Ri ti , Ri differential operators, denote the solution of φ (0) uniquely determined by Proposition 6.1. Then, P1 a section of TX/P . (1) = P1 if and only if R1 is a derivation, i.e. Proof. The operator R defines a morphism of algebras (OX/P [[t]], ⋆(1) ) → (OX/P [[t]], ⋆(0) ), i.e. R(f ) ⋆(0) R(g) = R(f ⋆(1) g). Comparing the calculations (0) R(f ) ⋆(0) R(g) = f g + (f R1 (g) + R1 (f )g + P1 (f, g))t + . . . . and (1) R(f ⋆(1) g) = f g + (R1 (f g) + P1 (f, g))t + . . . . one concludes that (0) (1) P1 (f, g) = P1 (f, g) if and only if f R1 (g) + R1 (f )g = R1 (f g), i.e. R1 is a derivation. Corollary 6.9. In the notation of Lemma 6.8, suppose in addition that φ0 is special. Then, φ1 is special if and only if R1 is a derivation, i.e. a section of TX/P . 14 On quasi-classical limits of DQ-algebroids For ξ ∈ TX/P = Ω1Π let Rξ := 1 + ξt. Corollary 6.9 implies that for φ ∈ Σ(A) the section φe ◦ Rξ |OX/P is special, i.e. φ 7→ φe ◦ Rξ |OX/P gives rise to a map Σ(A) → Σ(A). Suppose that φ0 , φ1 ∈ Σ(A) are equivalent modulo t2 , which is to say φ0 (f ) − φ1 (f ) ∈ t2 A f0 (f ) − φ f1 (f ) ∈ t2 A for any f ∈ OX/P [[t]]. Then, for for any f ∈ OX/P which implies that φ f0 (Rξ |O f1 (Rξ |O f0 ◦ Rξ |O f1 ◦ Rξ |O (f ) = φ (f )) − φ (f )) ∈ t2 A. any f ∈ OX/P , φ (f ) − φ X/P X/P X/P X/P Therefore, the map φ 7→ φe ◦ Rξ |O preserves equivalence modulo t2 , hence descends to a map X/P Σ1 (A) → Σ1 (A) denoted φ 7→ φ + ξ. Lemma 6.10. In the notation introduced above (i) The map Σ1 (A) × Ω1Π → Σ1 (A) given by (φ, ξ) 7→ φ + ξ defines a free action of the group Ω1Π on the sheaf Σ1 (A). Moreover, Σ1 (A) is a torsor under (the above action of) Ω1Π . (ii) For a morphism of DQ-algebras Φ : A0 → A1 the induced map Σ1 (Φ) : Σ1 (A0 ) → Σ1 (A1 ) is a morphism of Ω1Π -torsors. (iii) For a ∈ A× let Ad a denote the inner automorphism defined by (Ad a)(f ) = af a−1 . Then, for φ ∈ Σ1 (A) Σ1 (Ad a)(φ) = φ + dΠ log(σ(a)). Proof. (i) The sheaf Σ1 (A) is locally non-empty by Proposition 6.7. Note that, for ξ, η ∈ TX/P the equality Rξ ◦Rη = Rξ+η +t2 ξ ◦η holds. Therefore, the formula e ξ (f )) − φ(f e ) = (φ, ξ) 7→ φ + ξ does indeed define an action of the group Ω1Π . Since φ(R 2 tφ(ξ(f )) + t A, it follows that the action is free. Corollary 6.9 implies that the action is in fact transitive. (ii) By associativity of composition ^ Φ(φe ◦ Rξ (f )) = (Φ ◦ φ) ◦ Rξ (f ) for all φ ∈ Σ(A0 ), ξ ∈ TX/P and f ∈ OX/P . Therefore, Σ1 (φ + ξ) = Σ1 (φ) + ξ. (iii) Since the statement is local, we can assume that a = exp α, α ∈ A. Note that, for b ∈ A, Ad(exp α)(b) = ∞ X (ad α)i (b) i=0 i! = b + t{σ(α), σ(b)} + t2 A, where (ad α)(b) := [α, b]. Therefore, for f ∈ OX/P , since σ ◦ φ = Id, ((Ad a) ◦ φ)(f ) = φ(f ) + t{σ(α), σ(φ(f ))} + t2 A = φ(f ) + tdΠ log(σ(a))(f ) + t2 A = φe ◦ RdΠ log(σ(a)) (f ) + t2 A In view of Lemma 6.10, the assignment A 7→ Σ1 (A), Φ 7→ Σ1 (Φ), defines a morphism of stacks Σ1 : DQ-algX/P → Ω1Π [1]. 15 Paul Bressler, Alexander Gorokhovsky, Ryszard Nest and Boris Tsygan 6.6 Subprincipal symbols The construction presented below can be traced back to [Vey75]. Subprincipal curvature defined below appears in the context of classification of star-products in [BCG97] and in [Bur02], where it is called ”semiclassical curvature”. ∼ = Multiplication by tn induces the isomorphism OX/P − → tn A/tn+1 A. In particular, there is a short exact sequence ·t 0 → OX/P − → A/t2 A → OX/P → 0 (12) 1 , } endows A/t2 A with a Let {] , } denote operation on A/t2 A induced by [ , ]. The operation {] t structure of a Lie algebra so that the exact sequence (12) exhibits A/t2 A as an abelian extension of (the Lie algebra) OX/P equipped with the associated Poisson bracket. Lemma 6.11. For φ ∈ Σ(A) and f, g ∈ OX/P {φ(f^ ), φ(g)} − φ({f, g}) ∈ tA. Proof. Since (in the notations of (11)) [φ(f ), φ(g)] − (tφ({f, g}) + t2 φ(P2 (f, g) − P2 (g, f ))) ∈ t3 A, it follows that {φ(f^ ), φ(g)} − φ({f, g}) = tφ(P2 (f, g) − P2 (g, f )) + t2 A ∈ tA By Lemma 6.11 ·t {φ(f^ ), φ(g)} − φ({f, g}) + t2 A ∈ tA/A/t2 A ← − OX/P , For φ ∈ Σ(A) we define the map c(φ) : OX/P ⊗C OX/P → OX/P (13) by tc(φ)(f, g) = {φ(f^ ), φ(g)} − φ({f, g}) + t2 A ∈ tA/t2 A. Lemma 6.12. Let Φ : A0 → A1 be a morphism of DQ-algebras. Then c(Φ ◦ φ) = c(φ) Proof. We have tc(Φ ◦ φ)(f, g) = {Φ ◦ φ(f^ ), Φ ◦ φ(g)} − Φ ◦ φ({f, g}) + t2 A1 =   Φ {φ(f^ ), φ(g)} − φ({f, g}) + t2 A0 = Φ(tc(φ)(f, g). The statement follows from the commutativity of the diagram ·t OX/P −−−−→ tA0 /t2 A0   yΦ ·t OX/P −−−−→ tA1 /t2 A1 16 On quasi-classical limits of DQ-algebroids Proposition 6.13. (i) The map (13) is a skew-symmetric bi-derivation, hence c(φ) ∈ (ii) The bi-vector c(φ) satisfies dΠ c(φ) = 0. V2 TX/P . (iii) The map c(φ) depends only on the equivalence class of φ modulo t2 . (iv) For ξ ∈ TX/P c(φ + ξ) = c(φ) + dΠ ξ. Proof. e −1 to the identity (i) The skew-symmetry of c(φ) is clear. Applying (φ) [φ(f ), φ(g)φ(h)] = φ(g)[φ(f ), φ(h)] + [φ(f ), φ(g)]φ(h), expanding the result in powers of t and comparing the coefficients of t2 we obtain 1 1 1 c(φ)(f, gh) + {f, {g, h}} = gc(φ)(f, h) + {g, {f, h}} + c(φ)(f, g)h + {{f, g}, h}, 2 2 2 which implies that c(φ)(f, gh) = gc(φ)(f, h) + c(φ)(f, g)h i.e. c(φ) is a derivation in the second variable. Skew-symmetry implies that it is a biderivation. e −1 to the identity (ii) Applying (φ) [φ(f ), [φ(g), φ(h)]] = [φ(g), [φ(f ), φ(h)]] + [[φ(f ), φ(g)], φ(h)], expanding the result in powers of t and comparing the coefficients of t3 we obtain c(φ)(f, {g, h})+{f, c(φ)(g, h)} = c(φ)(g, {f, h})+{g, c(φ)(f, h)}+c(φ)({f, g}, h)+{c(φ)(f, g), h} which is equivalent to dΠ (φ) = 0. P i (iii) If φ, φ′ ∈ Σ(A), then there exists R = 1 + ∞ i=1 Ri t , where Ri are differential operators and ′ R1 = ξ is a vector field such that φ = φ ◦ R. Direct calculation shows that [φ′ (f ), φ′ (g)] = φ({f, g}t + ({ξ(f ), g} + {f, ξ(g)} + c(φ)(f, g))t2 + . . . ). (14) Hence, 1 ′ [φ (f ), φ′ (g)] − φ′ ({f, g}) t = φ({f, g}) + φ({ξ(f ), g} + {f, ξ(g)} + c(φ)(f, g))t − φ′ ({f, g}) = φ({ξ(f ), g} + {f, ξ(g)} − ξ({f, g}) + c(φ)(f, g))t + t2 A and c(φ′ )(f, g) = c(φ)(f, g) + {ξ(f ), g} + {f, ξ(g)} − ξ({f, g}). The sections φ and implies c(φ′ ) = c(φ). φ′ are equivalent modulo t2 (15) if and only if ξ = 0 in which case (15) (iv) Note that (15) is equivalent to c(φ′ )(f, g) = c(φ)(f, g) + dΠ ξ(f, g). According to parts (i), (ii) and (iii) of Proposition 6.13 the assignment c : φ 7→ c(φ) descends to a map c : Σ1 (A) → Ω2,cl Π . 17 (16) Paul Bressler, Alexander Gorokhovsky, Ryszard Nest and Boris Tsygan which, according to (iv) of Proposition 6.13 is a morphism of Ω1Π -torsors compatible with the map of sheaves of groups dΠ : Ω1Π → Ω2,cl Π . In other words, the map (16) endows Σ1 (A) with a d Π canonical structure of a (Ω1Π −→ Ω2,cl Π )-torsor. Definition 6.14. We refer to (16) as the subprincipal curvature map. Lemma 6.12 and Lemma 6.12 imply that the assignment A 7→ Σ1 (A), Φ 7→ Σ1 (Φ) defines a morphism of stacks d Π Σ1 : DQ-algX/P → (Ω1Π −→ Ω2,cl Π )[1]. 6.7 Bi-invertible bi-modules Suppose that A0 and A1 are DQ-algebras. op Definition 6.15 [KS12], Definition 2.1.6. A1 ⊗C[[t]] A0 -module A01 is called bi-invertible if locally on X there exists a section b ∈ A01 such that the map A1 ∋ a 7→ (a ⊗ 1)b ∈ A01 (respectively, op A0 ∋ a 7→ (1 ⊗ a)b ∈ A01 ) is an isomorphism of A1 - (respectively, A0 -) modules. op Lemma 6.16. Suppose that there exists a bi-invertible A1 ⊗C[[t]] A0 -module. Then, the associated Poisson bi-vectors of A0 and A1 are equal. Proof. Existence of a bi-invertible module implies that the algebras are locally isomorphic. It follows from Lemma 6.6 that locally isomorphic algebras give rise to the same Poisson bi-vector. op In what follows we suppose that A01 is a bi-invertible A1 ⊗C[[t]] A0 -module. Let gr A01 = A01 /tA01 and let gr : A01 → gr A01 denote the canonical projection. Let π denote the common Poisson bi-vector of A0 and A1 and let Π denote the corresponding Lie algebroid (structure on Ω1X/P , see 4.1.2). Let A× 01 denote the subsheaf of A01 consisting of sections b ∈ A01 such that the maps A1 ∋ op a 7→ (a ⊗ 1)b ∈ A01 and A0 ∋ a 7→ (1 ⊗ a)b ∈ A01 ) are isomorphisms of A1 - (respectively, A0 -) × modules. For b ∈ A01 let Φb : A0 → A1 denote the morphism of DQ-algebras uniquely determined by (Φb (a) ⊗ 1) · b = (1 ⊗ a)b. × The map gr : A01 → gr A01 restricts to gr : A× 01 → (gr A01 ) . The assignment b 7→ Φb defines a map A× 01 → HomDQ-alg (A0 , A1 ), (17) while the functor Σ1 give rise to the map Σ1 : HomDQ-alg (A0 , A1 ) → Hom dΠ (Ω1Π −→Ω2,cl Π )[1] (Σ1 (A0 ), Σ1 (A1 )). Lemma 6.17. (i) The composition (17) Σ 1 A× 01 −−→ HomDQ-alg (A0 , A1 ) −→ Hom factors through (gr A01 )× . 18 dΠ (Ω1Π −→Ω2,cl Π )[1] (Σ1 (A0 ), Σ1 (A1 )) On quasi-classical limits of DQ-algebroids (ii) The induced map (gr A01 )× → Hom dΠ (Ω1Π −→Ω2,cl Π )[1] (Σ1 (A0 ), Σ1 (A1 )) (18) × → Ω1Π . is a morphism of torsors compatible with the morphism of groups dΠ log : OX/P Proof. A section b ∈ A× 01 determines the isomorphism Φb : A0 → A1 and a compatible isomorop phism of A01 with A1 viewed as A1 ⊗C[[t]] A1 -module. Functoriality of Σ1 shows that it is sufficient to prove the statement in the case when A0 = A1 = A, and A01 = A as an A-bi-module. In this case the morphism A× → HomDQ-alg (A, A) is given by a 7→ Ad a, and both parts of the lemma follow from the formula for action of inner automorphisms on Σ1 (A) (Lemma 6.10, part (iii)): Σ1 (Ad a)(φ) = φ + dΠ log(σ(a)), φ ∈ Σ1 (A). . Notation. The map (18) will be denoted Σ1 (A01 ). The proof of the following lemma is left to the reader. (1) (0) op Lemma 6.18. Suppose that Ψ : A01 → A01 is an isomorphism of bi-invertible A1 ⊗C[[t]] A0 (1) (0) modules. Then Σ1 (A01 ) = (gr Ψ)× ◦ Σ1 (A01 ) 6.8 Quasi-classical limits of bi-modules The canonical contravariant connection on the classical limit of a bi-module deformation was constructed in [Bur02] (see also [BW04]) in the setting of star-products. The construction presented below is a generalization of that of loc. cit. to the case of DQ-algebras. op Suppose that A01 is a bi-invertible A1 ⊗C[[t]] A0 -module. By Lemma 6.16 A0 and A1 give rise to the same Poisson bi-vector π. Let Π denote the corresponding Lie algebroid (see 4.1.2). Then, gr A01 has a canonical structure of a OX/P -bi-module which is central, i.e. the two OX/P -module structures coincide. Moreover, gr A01 is locally free of rank one over OX/P . Therefore, the composition gr A1 ×OX/P A0 ⊗ A01 → A01 −→ gr A01 defined by (a1 , a0 ) ⊗ b 7→ (a1 b − ba0 ) + tA01 is trivial. Thus, the first map factors through the inclusion tA01 ֒→ A01 . The composition t−1 A1 ×OX/P A0 ⊗ A01 → tA01 −−→ A01 → gr A01 factors through a unique map A1 /t2 A1 ×OX/P A0 /t2 A0 ⊗ gr A01 → gr A01 which gives rise to the map A1 /t2 A1 ×OX/P A0 /t2 A0 → EndC (gr A01 ) 19 (19) Paul Bressler, Alexander Gorokhovsky, Ryszard Nest and Boris Tsygan The sheaf A1 /t2 A1 ⊖ A0 /t2 A0 (the Baer difference of extensions of OX/P by OX/P ) is defined by the push-out square (·t)×(·t) OX/P × OX/P −−−−−→ A1 /t2 A1 ×OX/P A0 /t2 A0     y y OX/P A1 /t2 A1 ⊖ A0 /t2 A0 −−−−→ where the left vertical map is defined by (f1 , f0 ) 7→ f1 − f0 . Here, for a DQ-algebra A we use the identification OX/P ∼ = tA/t2 A given by the multiplication by t. The sheaf A1 /t2 A1 ×OX/P A0 /t2 A0 has a canonical structure of a Lie algebra as a Lie subalgebra of A1 /t2 A1 × A0 /t2 A0 with the bracket {] , } (see 6.6) on each factor. The bracket on A1 /t2 A1 ×OX/P A0 /t2 A0 descends to a Lie bracket on A1 /t2 A1 ⊖ A0 /t2 A0 . Proposition 6.19. (i) The map (19) is a morphism of Lie algebras. (ii) The map (19) factors through A1 /t2 A1 ⊖ A0 /t2 A0 . (iii) The induced map A1 /t2 A1 ⊖ A0 /t2 A0 → EndC (gr A01 ) takes values in the Atiyah algebra Agr A01 of the line bundle gr A01 (see 4.1.1). (iv) The diagram OX/P   f 7→f ·idy −−−−→ A1 /t2 A1 ⊖ A0 /t2 A0 −−−−→ OX/P   d  yΠ y EndOX/P (gr A01 ) −−−−→ Agr A01 −−−−→ TX/P is commutative. Proof. For (a, a′ ) ∈ A1 ×OX/P A0 and b ∈ A01 let [(a, a′ ), b] := ab − ba′ . (i) For (ai , a′i ) ∈ A1 ×OX/P A0 , i = 1, 2 and b ∈ A01 [(a1 , a′1 ), [(a2 , a′2 ), b]] − [(a2 , a′2 ), [(a1 , a′1 ), b]] = [(a1 , a′1 ), a2 b − ba′2 ] − [(a2 , a′2 ), a1 b − ba′1 ] = a1 a2 b − a2 ba′1 − a1 ba′2 + ba′2 a′1 − a2 a1 b + a1 ba′2 + a2 ba′1 − ba′1 a′2 = [a1 , a2 ]b − b[a′1 , a′2 ] = [([a1 , a2 ], [a′1 , a′2 ]), b], which implies the claim. (ii) The composition
OX/P × OX/P ⊗ gr A01 → A1 /t2 A1 ×OX/P A0 /t2 A0 ⊗ gr A01 → gr A01 is given by (f1 , f0 ) ⊗ b 7→ (f1 − f0 ) · b. This means that the map (19) factors through A1 /t2 A1 ⊖ A0 /t2 A0 . (iii) [(a, a′ ), f b] − f [(a, a′ ), f b] = [a, f ]b = t{σ(a), f }b + t2 A01 which shows that A1 /t2 A1 ⊖ A0 /t2 A0 acts by differential operators of order one. 20 On quasi-classical limits of DQ-algebroids (iv) The same calculation shows that the principal symbol of the operator b 7→ [(a, a′ ), b] is given by dΠ σ(a). e gr A01 = Π ×T Let Π Agr A01 denote the OX/P -extension of Π obtained by pull-back by the X/P anchor map π e. Thus, we have the commutative diagram OX/P −−−−→ A1 /t2 A1 ⊖ A0 /t2 A0 −−−−→ OX/P     yd y OX/P −−−−→ For φi ∈ Σ1 (Ai ), i = 0, 1, the composition e gr A Π 01 −−−−→ Π (φ1 ,φ0 ) e gr A OX/P −−−−→ A1 /t2 A1 ⊖ A0 /t2 A0 → Π 01 is a derivation, hence factors uniquely as ∇ φ1 ,φ0 d e gr A OX/P − → Π −−− −→ Π 01 e gr A ). with ∇φ1 ,φ0 ∈ ConnΠ (Π 01 The assignment (φ1 , φ0 ) 7→ ∇φ1 ,φ0 defines the map e gr A ). ∇(A01 ) : Σ1 (A1 ) ⊖ Σ1 (A0 ) → ConnΠ (Π 01 (20) The proof of the following lemma is left to the reader. (1) (0) op Lemma 6.20. Suppose that Ψ : A01 → A01 is an isomorphism of bi-invertible A1 ⊗C[[t]] A0 (1) (0) modules. Then ∇(A01 ) = ∇(A01 ) ◦ ∇Π (Ad(Ψ)) op Proposition 6.21. Suppose that A01 is a bi-invertible A1 ⊗C[[t]] A0 -module. d Π (i) The map ∇(A01 ) is a morphism of (Ω1Π −→ Ω2,cl Π )-torsors. (ii) The diagram Hom d log e gr A ) −−Π−−→ ConnΠ (Π 01 x ∇(A )  01 (gr A01 )×   Σ1 (A01 )y dΠ (Ω1Π −→Ω2,cl Π )[1] (Σ1 (A0 ), Σ1 (A1 )) −−−−→ Σ1 (A1 ) ⊖ Σ1 (A0 ) is commutative. Proof. (i) Since the curvature is OX/P -bilinear it is sufficient to check that c(∇φ1 ,φ0 ) = c(φ1 ) − c(φ0 ) on exact forms. The latter identity follows from Proposition 6.19. (ii) Recall ((5), see 4.2.3) that the upper horizontal map dΠ log is characterized as follows. A × × → (gr A01 )× which -torsors τb : OX/P section b ∈ (gr A01 )× establishes a morphism of OX/P e gr A ) given by α 7→ ∇b + α, where ∇b induces the morphism of Ω1Π -torsors Ω1Π → ConnΠ (Π 01 × × . The map is the connection induced on (gr A01 ) by the canonical flat connection on OX/P 21 Paul Bressler, Alexander Gorokhovsky, Ryszard Nest and Boris Tsygan dΠ log is the unique map making the diagram × OX/P   τb y dΠ log Ω1Π  α7→∇ +α y b −−−−→ d log e gr A ) (gr A01 )× −−Π−−→ ConnΠ (Π 01 commutative. Therefore, it suffices to show that the composition ∇(A01 ) ◦ Σ1 (A01 ) has the same property. A section b ∈ A× 01 determines the isomorphism Φb : A0 → A1 and a compatible isomorphism op of A01 with A1 viewed as A1 ⊗C[[t]] A1 -module. Functoriality of Σ1 , Lemma 6.18 and Lemma 6.20 show that it is sufficient to prove the statement in the case when A0 = A1 = A, and A01 = A is the “diagonal” A-bi-module corresponding to b = 1 ∈ A. (Σ1 (A), Σ1 (A)) = Suppose that A0 = A1 = A and A01 = A. In this case Hom dΠ (Ω1Π −→Ω2,cl Π )[1] 1,cl 1,cl × 1 Ω1,cl Π , Σ1 (A) = dΠ log : OX/P → ΩΠ while the map ∇(A) : ΩΠ ֒→ ΩΠ is the inclusion, which proves the claim. Suppose that Ai , i = 0, 1, 2, is a DQ-algebra and Ai,i+1 , i = 0, 1 is a bi-invertible Ai+1 ⊗C[[t]] op Ai -module. Thus, all DQ-algebras Ai give rise to the same associated Poisson bi-vector π, hence the same Lie algebroid Π. op Let A02 := A01 ⊗A1 A12 . Then, A02 is a bi-invertible A2 ⊗C[[t]] A0 -module with gr A02 = gr A01 ⊗OX/P gr A12 . The bi-linear paring gr A01 × gr A12 → gr A02 gives rise to the map (gr A01 )× × (gr A12 )× → (gr A02 )× . Proposition 6.22. The diagram Σ1 (A01 )×Σ1 (A12 ) (gr A01 )× × (gr A12 )× −−−−−−−−−−−→ Hom(Σ1 (A0 ), Σ1 (A1 )) × Hom(Σ1 (A1 ), Σ1 (A2 ))   ◦  ⊗y y (gr A02 )× Σ1 (A02 ) −−−−−→ Hom(Σ1 (A0 ), Σ1 (A2 )) is commutative. Proof. The commutativity of the diagram (gr A01 )× × (gr A12 )× −−−−→ HomDQ-alg (A0 , A1 ) × HomDQ-alg (A1 , A2 )   ◦  ⊗y y (gr A02 )× HomDQ-alg (A0 , A2 ) −−−−→ with the horizontal maps as in (17) is left to the reader. The rest follows from functorial properties of Σ1 . 7. DQ-algebroids 7.1 DQ-algebroids Recall the definition of DQ-algebroids from [KS12]. 22 On quasi-classical limits of DQ-algebroids Definition 7.1 [KS12], Definition 2.3.1. A DQ-algebroid C is a C[[t]]-algebroid such that for each open set U ⊆ X with C(U ) 6= ∅ and any L ∈ C(U ) the C[[t]]-algebra EndC (L) is a DQ-algebra on U . In other words, a DQ-algebroid is a C[[t]]-algebroid locally equivalent to a star-product. Lemma 7.2. Suppose that C is a DQ-algebroid, U ⊆ X is an open subset such that C(U ) 6= ∅. For L, L′ ∈ C(U ) the EndC (L′ ) ⊗ EndC (L)op -module HomC (L, L′ ) is bi-invertible. Moreover, HomC (L, L′ )× coincides with the sub-sheaf of isomorphisms, i.e. HomiC (L, L′ ) = HomC (L, L′ )× . Proof. Left to the reader. For a DQ-algebroid C we denote by C/t · C the separated prestack with the same objects as C and HomC/t·C (L1 , L2 ) = Γ(U ; gr HomC (L1 , L2 )) for L1 , L2 ∈ C(U ) and by gr C the associated stack. We denote by gr : C → gr C the “principal symbol” functor. For U ⊆ X let DQX/P (U ) denote the 2-category of DQ-algebroids.The assignment U 7→ DQX/P (U ) extends to a 2-stack which we denote DQX/P . The assignment C 7→ gr× C := i gr C gives rise to the morphism of 2-stacks × [2]. gr× : DQX/P → OX/P (21) 7.2 The associated Poisson structure The following proposition is an immediate consequence of Lemma 7.2 and Lemma 6.16. Proposition 7.3. There exists a unique Poisson bracket {., .}C : OX/P ⊗ OX/P → OX/P such that for any U ⊂ X with C(U ) 6= ∅ and any L ∈ C(U ) the restriction of {., .}C to U coincides with the Poisson bracket associated to the DQ-algebra EndC (L). V Notation. We denote by π C ∈ Γ(X; 2 TX/P ) the Poisson bi-vector which corresponds to {., .}C . The Poisson bi-vector π C gives rise to a structure of a Lie algebroid on Ω1X/P as described in 4.1.2. We denote this Lie algebroid by ΠC . The assignment C 7→ π C give rise to the canonical morphism DQX/P → Λ2 TX/P (22) For π ∈ Γ(X; Λ2 TX/P ) we denote by DQπX/P the fiber of (22) over π. 7.3 The associated ΠC -connective structure Suppose that C is a DQ-algebroid. For U ⊆ X an open subset such that C(U ) 6= ∅, L, L′ ∈ C(U ) the EndC (L′ ) ⊗ EndC (L)op module HomC (L, L′ ) is bi-invertible by Lemma 7.2. An isomorphism φ : L → L′ induces the morphism of DQ-algebras Ad(φ) : EndC (L) → EndC (L′ ) defined by ψ 7→ φ ◦ ψ ◦ φ−1 . The proof of the following lemma is left to the reader. Lemma 7.4. The map Ad : HomC (L, L′ )× → HomDQ-alg (EndC (L), EndC (L′ )) coincides with the map (17). 23 Paul Bressler, Alexander Gorokhovsky, Ryszard Nest and Boris Tsygan For U ⊆ X an open subset such that C(U ) 6= ∅, L ∈ C(U ) the C[[t]]-algebra EndC (L) is a DQ-algebra and we set d Π ∇C (L) := Σ1 (EndC (L)) ∈ (Ω1ΠC −→ )[1]. Ω2,cl ΠC For U ⊆ X an open subset such that C(U ) 6= ∅, L, L′ ∈ C(U ), let ∇C (L, L′ ) := Σ1 (HomC (L, L′ )) : HomO× X/P × ′ × [1] (gr(L) , gr(L ) ) = (gr HomC (L, L′ ))× → Hom (Ω1 C Π dΠ −→ Ω2,cl )[1] ΠC (∇C (L), ∇C (L′ )). By Proposition 6.22 the above assignments define a functor, denoted d Π ∇C : i(C/tC) → (Ω1ΠC −→ )[1]. Ω2,cl ΠC As the target is a stack, this functor induces the functor d Π ∇C : i gr C → (Ω1ΠC −→ Ω2,cl )[1], ΠC i.e. a ΠC -connective structure with flat curving on i gr C. The assignment C 7→ (gr C, ∇C ) defines the morphism of 2-stacks 2,cl × 1 g × : DQπ gr X/P → (OX/P → ΩΠ → ΩΠ )[2] (23) (see 7.2 and 5.2 for notation) lifting the morphism (21). We refer to (23) as the morphism of quasi-classical limit 7.4 Obstruction to quantization The construction of the canonical Poisson connective structure with flat curving on the classical limit of a DQ-algebroid imposes restrictions on the class of the classical limit. We will V say that a twisted form S of OX/P admits a deformation along a Poisson bi-vector π ∈ Γ(X; 2 TX/P ) if there exists a DQ-algebroid C with π C = π and gr C ∼ = S. Theorem 7.5. A twisted form S of OX/P admits a deformation along a Poisson bi-vector π ∈ V × ) is in the image of the map Γ(X; 2 TX/P ) only if the class of S in H 2 (X; OX/P × × 2 → Ω1Π → Ω2,cl H 2 (X; OX/P Π ) → H (X; OX/P ), (24) where Π denotes the Lie algebroid associated with π. Proof. The map (24) is induced by the map of complexes of sheaves × × → Ω1Π → Ω2,cl (OX/P Π ) → OX/P . The induced morphism of 2-stacks × × → Ω1Π → Ω2,cl (OX/P Π )[2] → OX/P [2] is the functor of forgetting the Π-connective structure given by the (S, κ) 7→ S. If there exists a DQ-algebroid C with the associated Poisson bi-vector π and gr× C equivalent 2,cl 1 g × C in H 2 (X; O × to S, then the class of gr X/P → ΩΠ → ΩΠ ) is a lift of the class of S in × ). H 2 (X; OX/P The following corollary of Theorem 7.5 is well-known (see for example [BGNT07]). 24 On quasi-classical limits of DQ-algebroids Corollary 7.6. A twisted form S of OX/P admits a symplectic deformation only if the class × × ). ) is in the image of the map H 2 (X; C× ) → H 2 (X; OX/P of S in H 2 (X; OX/P Proof. If the associated Poisson bi-vector is non-degenerate, then the anchor map π e is an isomor× × 1 → Ω1Π → Ω2,cl is isomorphic to the the complex O phism and the complex OX/P Π X/P → ΩX/P → Ω2,cl X/P . The Poincaré Lemma holds for X/P, in other words, the map (inclusion of locally constant 2,cl × 1 functions) C× X → (OX/P → ΩX/P → ΩX/P ) is a quasi-isomorphism. Remark 7.7. In fact, as shown in [BGNT07], the converse of Corollary 7.6 holds. 8. The quasi-classical limit and formality for algebroids In this section we formulate a conjecture relating the quasi-classical limit with the results of [BGNT15] which describe the formal deformation theory of gerbes in terms of quasi-classical data. For simplicity we will assume that the distribution P is trivial, i.e. we are dealing with a plain C ∞ manifold. × -gerbe 8.1 The de Rham class of an OX × -gerbe. We recall the construction of a closed differential 3-form representing the class of an OX See [Bry93] for further details. d log Suppose that S is a twisted form of OX . Under the map H 2 (X; O× ) −−−→ H 2 (X; Ω1,cl ) ∼ = X X 3 3 (X) the equivalence class [S] ∈ H 2 (X; O × ) is mapped to the class [S] HdR dR ∈ HdR (X). We X 3,cl briefly recall the construction of a representative H ∈ Γ(X; ΩX ) of the class [S]dR . × × × ) is surjective, in other words, every OX → Ω1X → Ω2X ) → H 2 (X; OX The map H 2 (X; OX gerbe admits a connective structure with curving (a.k.a. a classical B-field). The de Rham dif× → Ω1X → Ω2X )[2] → Ω3,cl ferential gives rise to the map of complexes of sheaves d : (OX X . The × → Ω1X → Ω2X ) → Γ(X; Ω3,cl ) corresponds to the map induced map on cohomology d : H 2 (X; OX X which associates to a gerbe with connective structure with curving (S, κ) a closed 3-form c(κ) 3 (X) depends only on the equivacalled the curvature of the curving κ. The class of c(κ) in HdR lence class of S (and does not depend on the choice of the connective structure) and is denoted 3 (X). [S]dR ∈ HdR × 3 (X) given by [S] 7→ [S] ) → HdR The map H 2 (X; OX dR coincides with the composition × × 2 1,cl 3 2 ∼ → Ω1,cl H (X; OX ) → H (X; Ω)X ) = HdR (X) induced by the map of sheaves d log : OX X . 8.2 Deformations of twisted forms of OX ([BGNT15]) Suppose that S is a twisted form of OX . Theorem 6.5 in conjunction with Remark 6.6 of [BGNT15] imply that equivalence classes of DQ-algebroids with associated Poisson bi-vector π and classical limit (equivalent to) S are in bijective correspondence with equivalence classes of pairs (πt , H), where V b – πt = 0 + π1 t + π2 t2 + · · · ∈ Γ(X; 2 TX ⊗tC[[t]]) – H ∈ Γ(X; Ω3,cl X ) satisfying (i) π1 = π 3 (X) (ii) H is a representative of [S]dR ∈ HdR 25 Paul Bressler, Alexander Gorokhovsky, Ryszard Nest and Boris Tsygan (iii) [πt , πt ] = πet ∧3 (H), b where πet : Ω1X → TX ⊗tC[[t]] is the adjoint of πt . 8.3 Π-connective structures from quasi-classical data Suppose that π is a Poisson bi-vector on X with associated Lie algebroid denoted Π and (S, ∇) is gerbe equipped with a connective structure ∇ with curving whose curvature is equal to H ∈ Γ(X; Ω3,cl X ). V b such that the pair (πt , H) satisfies the conditions in 8.2 one Given πt ∈ Γ(X; 2 TX ⊗tC[[t]]) obtains a canonical Π-connective structure with flat curving on S as follows. To a locally defined object L ∈ S the connective structure ∇ associates a Ω1X -torsor ∇(L) equipped with the curvature map cL : ∇(L) → Ω2X . As H is the curvature of the curving of ∇, c d L the composition ∇(L) −→ Ω2X − → Ω3,cl X is constant and equal to H. 1 The ΩX -torsor ∇(L) gives rise to the Ω1Π -torsor π e∇(L). The map (e π (cL ) − π2 ) : π e∇(L) → Ω2Π (where π2 ∈ Ω2Π is viewed as a constant map) is a morphism of torsors compatible with the map of groups dΠ : Ω1Π → Ω2Π . The condition [πt , πt ] = πet ∧3 (H) implies that [π, π2 ] = π e∧3 (H). Therefore, the composition π e(cL )−π2 d Π π e∇(L) −−−−−−→ Ω2Π −→ Ω3Π is equal to zero. Consequently, the map π e(cL ) − π2 takes values in 2,cl ΩΠ and, hence, the assignment S ∋ L 7→ (e π ∇(L), π e(cL ) − π2 ) defines a Π-connective structure with flat curving on S. 8.4 Quasi-classical limit and formality Suppose that C is a DQ-algebroid. 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