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Gerbes of chiral differential operators. II

Results of our previous note,"Gerbes of chiral differential operators"(Math. Res. Letters, 7(2000), 55-66), are discussed in the algebraic category.


Introdution
The aim of this paper is a study of certain class of chiral (vertex) algebras which we call chiral algebras of differential operators.
I.0. To explain this notion, let us take up a "naive" approach to vertex algebras. Let us fix (in this Introduction) a ground field k of characteristic 0. Recall (cf. [K]) that a Z ≥0 -graded vertex algebra is a Z ≥0 -graded k-vector space V = ⊕ i≥0 V i equipped with a distinguished vacuum vector 1 ∈ V 0 and a family of bilinear operations (n ∈ Z), the operation (n) having the degree −n − 1, that is, V i(n) V j ⊂ V i+j−n−1 where by definition V i = 0 for i < 0. These operations must satisfy a family of quadratic Borcherds identities. Throughout this paper we shall deal only with Z ≥0graded vertex algebras, so we shall omit the words "Z ≥0 -graded" when speaking about them.
Note that in particular the operation (−1) has degree 0. It is neither commutative nor associative in general. 1 We are interested in a question: what structure on the subspace V ≤1 := V 0 ⊕ V 1 is induced by the structure of a vertex algebra on V ?
The claims in italics below are easily proved using the Borcherds identities, see the main body of the paper.
First of all, V 0 is a commutative associative k-algebra with unit 1 with respect to (−1) . Let us denote this algebra by A, and the operation a (−1) b on it by ab. Note that for each i the operation (−1) gives a mapping to be denoted ax, but this mapping does not make V i an A-module since the associativity abx = (ab)x is not satisfied in general.
We have a mapping ∂ : V i −→ V i+1 defined by ∂x = x (−2) 1. Let us denote by Ω ⊂ V 1 a k-subspace generated by all elements a∂b, a, b ∈ A. The operation (I1) makes Ω an A-module (i.e. the associativity holds true on this subspace) and ∂ : A −→ Ω is a derivation.
Let T be the quotient space V 1 /Ω. The operation (I1) induces the structure of an A-module on T and operation (0) induces a structure of a Lie algebra on T and an action of T on A by derivations. These structures are compatible: they make T a Lie algebroid (see 0.2) over A.
The operation (0) induces an action of T on Ω which makes Ω a module over the Lie algebra T , and ∂ is a morphism of T -modules.
The operation (1) induces an A-bilinear pairing T ⊗Ω −→ A which is a morphism of T -modules.
Thus, we have assigned to a vertex algebra V (with a splitting (I2)) a collection of data A = (A, T, Ω, ∂, γ, , , c) as above satisfying properties (A1) -(A5). We call such collection of data a vertex algebroid which is a central hero of our story.
The above procedure gives a functor form the category of vertex algebras to the (appropriately defined) category of vertex algebroids. This functor admits a left adjoint U , called vertex envelope. A vertex algebra of the form U A is called a chiral algebra of differential operators. Such vertex algebroids (with suitably defined morphisms) form a groupoid (i.e. a category where all morphisms are invertible) Alg(A). This category is nonempty if there exists an A-base of T consisting of commuting vector fields. This groupoid has a remarkable additional structure. To describe this structure, let us define another groupoid Ω [2,3 (A) as follows. By definition, its objects will be all closed 3-forms on A. If ω, ω ′ are two such 3-forms, the set of morphisms between them is defined as is an equivalence of categories.
I.3. Sheafifying the previous construction one gets a sheaf of groupoids, i.e. gerbe Alg X over X, with the lien Ω [2,3 X := Ω 2 X −→ Ω 3,closed (a short complex of sheaves over X, with Ω 2 X sitting in degree 0). By a simple general homological construction (see 7.3), this gerbe gives rise to a characteristic class c(Alg X ) ∈ H 2 (X; Ω [2,3 X ) with the following property.
The groupoid of global sections Alg X (X) is nonempty iff c(Alg X ) = 0. If this is the case then its set of connected components is a nonempty H 1 (X; Ω [2,3 X )-torseur and the group of automorphisms of its object is isomorphic to H 0 (X; Ω [2,3 X ). Our next aim is to calculate the class c(Alg X ). One shows (see Theorem 7.10) that it essentially coincides with the second component of the Chern character of the tangent bundle T X . Let us describe an "explicit formula" for it.
Let E be an arbitrary finite dimensional vector bundle over X, given by a Cech cocycle (g ij ) ∈ Z 1 (U; GL r (O X )) on a Zariski open covering U = {U i } of X. Define a Cech two-cocycle ch 2 (E) ∈ Z 2 (U; Ω [2,3 X ) by ch 2 (E) = 1 2 tr(g −1 ij g −1 jk dg jk dg ij ), This class may be called the "Atiyah-Chern-Simons class" of E. Its first component, α(E), is an element of Z 2 (U; Ω 2 X ) which is the usual "Atiyah" representative of the degree 2 part of the Chern character of E "style Hodge" living in H 2 (X; Ω 2 X ), while the second component, β(E) ("Chern-Simons"), is a Cech 1cochain with coefficients in Ω 3,closed X . The de Rham differential of α(E) is equal to the Cech coboundary of β(E). One checks that the cohomology class of (ACS) does not depend on the choice of trivialization of E.
Our second main result (Theorem 7.5) is Theorem I.2. The class c(Alg) is equal to (the cohomology class of ) 2ch 2 (T X ).
I.4. Our third main topic is an explicit construction of the enveloping algebra U A of a vertex algebroid A and "Poincaré-Birkhoff-Witt" type theorem for it. Let us formulate the last theorem (see Theorem 9.18).
Theorem I.3. Each sheaf of algebras of chiral differential operators D = U A, A ∈ Alg X (X) admits a canonical filtration F , compatible with the conformal grading and finite on each component of fixed conformal weight, such that the corresponding graded algebra (which is a sheaf of Z ≥0 -graded commutative vertex algebras) is where T (i) (resp. Ω (i) ) is a copy of T X (resp. of Ω 1 X ) living in conformal weight i. As a preparation to this theorem, we study in Section 8 the vertex envelope U C of an arbitrary conformal algebra C. Although its construction is more or less contained in Kac's book [K], we present here in a sense more direct construction of C, maybe of independent interest. Note in particular an amusing Lie bracket (8.21.1) which is defined on an arbitrary conformal algebra.
In the hope of possible arithmetical applications, we adopted in this paper a "characteristic free" approach to vertex algebras. Contrary to this Introduction (and to the tradition), in the main body of this work the ground ring will be an arbitrary commutative ring containing 1/2 (with one exception: the PBW theorem 9.18). This generalization is achieved without great effort. Note that the original Borcherds' definition of vertex algebras was characteristic-free.
I.4. This paper may be regarded as an "algebraic" version of our last note [GMS] where we worked in the analytical category. However, the approach adopted here is quite different from op. cit. (cf. also [MSV] and [MS]).
During the preparation of this work we have enourmously benefited from the discussions and correspondence with Sasha Beilinson. He was the first to suggest (by analogy with the classical picture, [BB]) that in the algebraic category the situation is more subtle than in the analytical one. He has sent us a note [BD1] where similar questions are treated from a different point of view. In a sense, a great part of this work is a result of the attempts to understand this note (which still remains mysterious for us). For the details on Beilinson-Drinfeld approach, see [BD2], 3.8.
Another special gratitude goes to Hélène Esnault. She has found a mistake in the previous formulation of our main result whose correction lead to the discovery of the class (ACS), and greatly helped in some computations. Theorem 7.10 was obtained in collaboraion with her.
This work was mostly done while V.S. visited IHES. He is grateful to this institute for the support and excellent working conditions. § 0. Recollections and Notation 0.0. Throughout this paper, a commutative ring (algebra) will mean a commutative associative ring (algebra) with unit.
k will denote a fixed ground commutative ring. We will often assume that 1/2 ∈ k; we will indicate this assumption when necessary.
Algebra will mean a k-algebra; ⊗ will mean the tensor product over k.
In a nonassociatve algebra, abc . . . will mean a(b(c(. . . )) . . . ). Z ≥0 will denote the set of nonnegative integers. The binomial coefficients are defined by The de Rham differential d : A −→ Ω 1 A commutes with the T A -action. We have the canonical A-bilinear pairing (Of course the last formula is a particular case of (0.1.2).)

A Lie
A-algebroid is a Lie algebra T acting by derivations on A and equipped with a structure of an A-module, such that for all τ, ν ∈ T ; a, b ∈ A.
0.2.1. Direct image (pushout). Let B be a commutative A-algebra, i : A −→ B be the structure morphism and T be an A-algebroid Lie. Assume that T , as a Lie algebra, acts on B by derivations in such a way that i is a morphism of T -modules and Then the B-module T B := B ⊗ A T admits a canonical structure of a B-algebroid Lie. Namely, the Lie bracket on T B is given by and the action of T B on B is defined by Vertex algebras 0.3. In his original paper, [B], Borcherds defined a vertex algebra over an arbitrary commutative ring. However, later most people preferred to work over the complex numbers. It is fairly obvious that all the general theorems of Kac's book [K] are true over an arbitrary field of characteristic 0. A little less obvious, but true, is that with a minor modification of the definitions, they remain true over an arbitrary commutative ring. Below we recall the definitions and results to be used in the sequel, and explain these modifications.
Throughout this work, we will deal only with Z ≥0 -graded vertex algebras.

Definition.
A Z ≥0 -graded conformal algebra is a Z ≥0 -graded k-module C = ⊕ C i , together with a family of endomorphisms and a family of bilinear operations for all a, b, c ∈ C; m, n, i ∈ Z ≥0 .
Cf. [K], Definition 2.7b. In each conformal algebra we have the following identities: and We leave their direct proof to the reader. The proof of (0.4.5) uses only the axiom (0.4.4) (and (0.4.1)), and the proof of (0.4.6) uses the axioms (0.4.2) and (0.4.3).
We also have This is proven by induction on j.
0.5. First Definition of a Vertex Algebra. A Z ≥0 -graded vertex algebra is a Z ≥0 -graded k-module V = ⊕ V i , equipped with a distinguished vector 1 ∈ V 0 (vacuum vector) and a family of bilinear operations for all a, b, c ∈ V, m, n, l ∈ Z.
Cf. [B], Section 4, [K], Prop. 4.8. (b). The important particular case of (0.5.3) corresponds to m = 0: This way we get endomorphisms ∂ (j) of V of degree j. It follows from (0.5.2) that ∂ (j) 1 = δ j,0 1 (0.5.6) and and (0.5.4) applied to b = c = 1 gives We have the commutativity formula for all a, b ∈ V, n ∈ Z. The proof will be given below, see the last paragraph of the next subsection.
One deduces from (0.5.4) that and for all n ∈ Z. We have the Operator Product Expansion (OPE) formula, cf. [K], (4.6.7) and the end of the next Subsection.
0.6. Following [K], let us give an equivalent definition of a vertex algebra. If M is a k-module, M [[z, z −1 ]] will denote as usually the module of formal power series ∞ n=−∞ a n z n , a n ∈ M . Let us define endomorphisms ∂ (j) , j ∈ Z ≥0 , of M [[z, z −1 ]] by ∂ (j) ( n a n z n ) = n n j a n z n−j (0.6.1) These data should satisfy the following axioms.
This is a modification of the definition given in [K], 4.1, which works over an arbitrary base commutative ring. Proposition 4.1 of op. cit. remains true if we understand by e z∂ the expression (we cannot use Lemma 4.1 of op.cit. anymore!). The argument of op. cit. 4.2 shows that the commutativity formula (0.5.9) is true for the vertex algebras in the second definition. Finally, the proof of op. cit., Proposition 4.8 (b) shows that the first and the second definitions are equivalent. The proof from [K], 4.6 works to give the proof of the OPE formula (0.5.12). 0.7. Theorem. Let V = ⊕V i be a Z ≥0 -graded k-module equipped with a distinguished vector 1 ∈ V 0 and a family of endomorphisms ∂ (j) of degree j, j ∈ Z ≥0 such that ∂ (0) = Id, ∂ (i) · ∂ (j) = i+j i ∂ (i+j) and ∂ (j) 1 = δ j,0 1. Assume that we are given a family of homogeneous vectors {a α } ⊂ V and a family of formal power series ("distributions") satisfying the following conditions (v) a α (n) 1 = 0 for n ≥ 0; a α −1 1 = a α ; (l) The distributions a α (z) are mutually local, i.e. for all α, β, (z−w) N [a α (z), a β (w)] = 0 for sufficiently large N . Let S denote the set of all vectors of the form Assume that we are given a map having the following property: (P) There exists a subset S ′ ⊂ S which generates V as a k-module such that for each a α1...αN j1...jN ∈ S ′ , we have a α1...αN j1...jN (z) =: ∂ j1 α α1 (z) . . . ∂ jN a αN (z) : (0.7.2) Then the mapping (*) defines the structure of a Z ≥0 -graded vertex algebra on V , and (0.7.2) holds true for all a α1...αN j1...jN ∈ S. This is a version of "Existence Theorem", [K], Theorem 4.5 and Corollary 4.5, whose proof goes through. 0.8. Let Vert, Conf denote the categories of Z ≥0 -graded vertex and conformal algebras respectively. We have a functor c : Vert −→ Conf (0.8.1) which assigns to a vertex algebra V the same space V , with the operations (n) , n < 0, forgotten. The axioms of a conformal algebra are satisfied due to (0.5.10), (0.5.9) and (0.5.12). This functor admits a left adjoint, to be constructed in Section 8 below.
In the sequel we will call Z ≥0 -graded vertex (conformal) algebras simply vertex (conformal) algebras. 0.9. A vertex algebra V is called commutative if a (n) b = 0 for all a, b ∈ V, n ≥ 0 (Kac uses the term "holomorphic").
Let V be a commutative vertex algebra. Then, with respect to the operation ab := a (−1) b, V becomes a commutative associative algebra with the unity 1. The operations with negative indices are recovered from the formula This way we get an equivalence of categories Here VertAb denotes the category of commutative vertex algebras and ∂ − Alg denotes the category whose objects are Z ≥0 -graded vector spaces V = ⊕V i equipped with a structure of a commutative algebra such that V i · V j ⊂ V i+j and 1 ∈ V 0 , and with a family of endomorphisms ∂ (j) of degree j, j ∈ Z ≥0 , such that and Cf. [B]. Objects of ∂ − Alg will be called ∂-algebras. § 1. Vertex Algebroids 1.1. Let us define an extended Lie algebroid to be a quintuple T = (A, T, Ω, ∂, , ) where A is a commutative k-algebra, T is a Lie A-algebroid, Ω is an A-module equipped with a structure of a module over the Lie algebra T , ∂ : A −→ Ω is an A-derivation and a morphism of T -modules, , : T × Ω −→ A is an A-bilinear pairing.
Let define a morphism between two extended Lie algebroids T = (A, T, . . . ) and T ′ = (A ′ , T ′ , . . . ) to be a triple g = (g A , g T , g Ω ) where g A : A −→ A ′ is a morphism of k-algebras, g T : T −→ T ′ is a morphism of Lie algebras and Amodules, g Ω : Ω −→ Ω ′ is a morphism of A-modules. We require that the following properties should hold: Composition of morphisms is defined in the obvious way. This way we get a category LieAlg of extended Lie algebroids. The correspondence T → T T provides an equivalence of the category of Lie algebroids with the full subcategory LieAlg perf ⊂ LieAlg of perfect extended Lie algebroids.
For example, if T is as in the previous example, then Ω i = Hom A (Λ i A T, A).
Let us define the maps d DR = d i DR : Ω i −→ Ω i+1 as follows. For i = 0 we set d DR a = −∂a. For i ≥ 1 we set for ω ∈ Ω 1 = Ω; and Let us introduce the action of the Lie algebra T on the modules Ω i by Let us define the convolution operators τ, · : Ω i −→ Ω i−1 by τ, h (τ 1 , . . . , τ i−2 = h(τ, τ 1 , . . . , τ i−2 ) (1.3.6) The maps {d i DR } may be characterized as a unique collection of maps such that d 0 DR = −∂ and the Cartan formula holds true.
The maps d DR commute with the action of T . One checks that d 2 DR = 0, so we get a complex (Ω • (T ), d DR ) called the de Rham-Chevalley complex of T .

1.4.
In the definition below we assume that 1/2 ∈ k.
A vertex algebroid is a septuple A = (A, T, Ω, ∂, γ, , , c) where A is a commutative k-algebra, T is a Lie A-algebroid, Ω is an A-module equipped with an action of the Lie algebra T , ∂ : A −→ Ω is a derivation commuting with the T -action, is a symmetric k-bilinear pairing equal to zero on Ω × Ω and such that T A = (A, T, Ω, ∂, , | T ×Ω ) is an extended Lie A-algebroid; c : T × T −→ Ω is a skew symmetric k-bilinear pairing and γ : A × T −→ Ω is a k-bilinear map.
The following axioms must hold (a, b ∈ A; τ, τ i ∈ T ): where d Lie is defined by (1.3.2).
To stress the dependence on A, we shall sometimes say that A is a vertex Aalgebroid.
1.8. Let A be a commutative k-algebra and T = (A, T, Ω, ∂, , ) be an extended Lie A-algebroid. Let T 0 ⊂ T be a k-submodule which generates T as an A-module, and assume that we are given k-bilinear mappings It is clear that there exists not more than one extension of the mappings to mappings satisfying (A1), (A2), (A3). This extension, if it exists, must be given by the formulas (a, b ∈ A, τ, τ i ∈ T 0 ): When deducing the last formula, one should take into account (1.1.3).
Then the axioms (A1) -(A5) hold true for all τ, Proof. Let us check the axiom (A1). It is enough to show that if (A1) holds true for some τ and all a, b then it holds true for cτ (c ∈ A) and all a, b. Thus, we have to check the identity The left hand side of it equal to On the other hand, the first two terms in the right hand side are equal to Comparing, we get the desired identity (?).
The other axioms are checked in a similar way. This is a tiresome but straightforward calculation. When checking (A5) one should use the identity (A4) bis Assume that the Lie algebra T acts on B by derivations in such a way that τ (i(a)) = i(τ (a)) and (aτ ) Then there exists a unique extension of γ to a k-bilinear mapping there exists a unique extension of the pairing , : there exists a unique extension of the pairing c : Proof. Apply 1.9 to T 0 := Im(T −→ T B ). △ 1.11. Example. Let A be a commutative k-algebra, Ω = Ω 1 k -the A-module of Kählerian 1-differentials, ∂ : A −→ Ω the canonical A-derivation. Let g be a Lie k-algebra acting on A by derivations and equipped with an invariant bilinear form , : g × g −→ k.
Due to the morphism g −→ Der k (A), Ω is equipped with a canonical structure of a g-module such that ∂ is a morphism of g-modules and there is a canonical pairing g × Ω −→ A, cf. 0.1.
Then T is canonically a Lie A-algebroid, there exists a unique extension of the zero map A×g −→ A (resp., of , and of the zero map g×g −→ Ω) to the map γ : A × T −→ Ω (resp., to the pairing , : T × T −→ A and to the map c : T × T −→ Ω) which satisfies (A1) (resp., (A2) and (A3)).
For the proof, apply Theorem 1.9 to T 0 = g ⊂ T .
1.12. Example. Let A be a commutative k algebra, g ⊂ T = Der k (A) a Lie k-subalgebra such that T = A ⊗ g, Ω = Ω 1 k . For example, A is the coordinate ring of an algebraic group, g is the Lie algebra of left invariant vector fields.
Assume that g is equipped with an invariant bilinear form , . Then we are in the situation of 1.11 and get a vertex A-algebroid A = A A,g, , .
1.13. Example. Let A be a commutative k-algebra equipped with anétale Such an f is called anétale coordinate system on A; it exists Zariski locally for every A smooth over k.
The commuting vector fields ∂/∂x i ∈ Der k (A 0 ) admit unique liftings to vector fields τ i ∈ T = Der k (A). The k-submodule g ⊂ T spanned by τ i is an abelian Lie subalgebra; let us equip it with the trivial bilinear form. Then we are in the situation of 1.11 and we get a vertex A-algebroid A = A A;f . §2. From Vertex Algebras to Vertex Algebroids 2.1. Let V be a vertex algebra over k, cf. 0.5. In this section we assume that 1/2 ∈ k; recall that we always deal with Z ≥0 -graded vertex algebras. We denote ∂ := ∂ (1) .
The identity (0.5.4) at n = 0 gives for all n, i.e. the operation (0) is a derivation with respect to all operations (n) .
We shall denote the operation a (−1) b by ab or a · b.
Set A = V 0 . In the sequel a, b, c will denote elements of A. Since a (n) b = 0 for all n ≥ 0, (0.5.4) and (0.5.9) imply that A is a commutative associative k-algebra with the unity 1 with respect to the operation ab.
Elements of V 1 will be denoted x, y, z. It follows from (0.5.4), taking into account that −1 The identity (0.5.9) implies Let Ω ⊂ V 1 be the k-submodule generated by all elements a∂b; ω will denote an element of Ω.
Let T denote the quotient module V 1 /Ω; let π : V 1 −→ T be the canonical projection. Elements of T will be denoted τ . The operation ax induces an operation i.e. the operation aτ provides T with a structure of an A-module.
Note that by (0.5.10) and (2.1.7) On the other hand, , and Therefore the operation (0) induces an operation T × T −→ T , to be denoted [, ]. By (0.5.9) is skew symmetric. (2.1.1) implies that [, ] satisfies the Jacobi identity, hence it provides a structure of a Lie algebra (over k) on T .
It follows that the operation . Thus, T is equipped with a structure of a Lie A-algebroid.

2.2.
Let us assume that the projection π : V 1 −→ T admits a splitting, i.e. there exists a morphism of k-modules s : T −→ V 1 such that π • s = Id T . Let us fix such a splitting. A vertex algebra with a chosen s will be called split.
Define a skew symmetric operation [, ] : Let us use the notation x, y for x (1) y. It follows form (0.5.9) that Proof. We have to check the axioms (A1) -(A5) from 1.4.
Check of (A5). We shall use the formulas and (2.3.4). We get Interchanging τ 1 with τ 2 we get where we have used the formula which is a consequence of (0.5.9). The axiom (A5) follows. This completes the proof of the theorem. △ 2.4. Thus, to a pair (V, s), where V is a vertex algebra and s : T −→ V 1 is a splitting on V , we have assigned a vertex algebroid A(V, s) = (A, T, Ω, ∂, γ, , , c). Note that by definition A(V, s) has the property (Sur) The module Ω is generated as an A-module by the subspace ∂A. Equivalently, the morphism Ω 1 The following axioms must be satisfied: whenever the both sides are defined.
This way we get a category Vert ≤1 of 1-truncated vertex algebras. We have an obvious truncation functor which assignes to a vertex algebra V = ⊕V i the truncated algebra tV := (V 0 , V 1 , . . . ).
In Section 9 below we shall construct a left adjoint to t.

Let
Let us call v splittable if π admits a k-linear splitting s : T −→ V 1 , cf. 2.2. This is of course a weak condition; it holds true for example if T is a projective k-module. Let Vert ′ ≤1 ⊂ Vert ≤1 denote the full subcategory of splittable 1-truncated vertex algebras.
A 1-truncated vertex algebra with a chosen splitting s will be called split. The argument of Section 2 assigns a vertex algebroid A(v; s) to every split 1-truncated vertex algebra (v; s).

We have by definition
for every split vertex algebra (V ; s).
3.3. Conversely, let A = (A, T, Ω, . . . ) be a vertex algebroid. We want to assign to it a split 1-truncated vertex algebra. To this end one needs simply to invert the construction of the previous Section.
Let us define the operations (i) as follows: By inverting the argument of the previous Section, one sees easily that we get a split 1-truncated vertex algebra (V 0 , V 1 , . . . ), to be denoted by uA. Let For two vertex algebroids A, A ′ define the set of morphisms between A and A ′ to be the subset of Hom Vert ≤1 (uA, uA ′ ) which consists of all morphisms g : uA −→ uA ′ such that g(Ω) ⊂ Ω ′ . This way we get a category of vertex algebroids, to be denoted Alg. The mapping u induces a functor which is in fact an equivalence of categories, due to (3.3.5).

Let
where h : T −→ Ω ′ is some k-linear mapping.
The similar claims connected with the equations (3.5 , ) and (3.5c) are proved analogously, and we leave them to the reader. △ §4. Cofibered Structure 4.1. Let us introduce objects which lie "in between" extended Lie algebroids and vertex algebroid. Let us define a vertex prealgebroid to be a sextuple B = (A, T, Ω, ∂, γ, , ) where A, T, Ω, ∂, γ, , are as in 1.4.
Let us define a morphism between two vertex prealgebroids B = (A, T, . . . ) and B ′ = (A ′ , T ′ , . . . ) to be a quadruple g = (g A , g T , g Ω , h) as in Theorem 3.5, satisfying the properties (3.5A) -(3.5 , ). The composition of morphisms and the identity morphisms are defined by the rules (3.7.1), (3.7.2). This way we get a category PreAlg of vertex prealgebroids.
We have obvious forgetful functors Note that P is injective on Hom's. The composition Q • P : Alg −→ LieAlg will be denoted by Θ.
We will use the standard notation for fibers. For example, if B ∈ PreAlg then Alg B will denote the category whose objects are A ∈ Alg such that P (A) = B and morphisms are morphisms in A which go to Id B after applying P , and similarly with the other fibers. Note that all such g are invertible, namely g −1 = (Id A , Id T , Id Ω , −h), cf. (3.7.1), (3.7.2). In other words, Alg B is a groupoid.

the set Hom Alg B
(A, A · + α) may be canonically identified with the set of h ∈ Ω 2 such that d DR h = −α.
Consequently, the set of isomorphism classes of objects of the groupoid Alg B is equal to the third de Rham cohomology H 3 (Ω · (T )).
If g ′ : T ′′ −→ T ′ is another morphism of extended Lie algebroids, we have a composition map It is a morphism of A ′ -modules. We have ν g,g ′ (Ω 2,cl g × Ω 2,cl g ′ ) ⊂ Ω 2,cl gg ′ (cf. Remark 4.4 below). The maps (4.3.4) are associative in the obvious sense (with respect to triples of composable morphisms in LieAlg).
4.4. Remark. Generalizing 1.3, one can define the de Rham-Chevalley complexes Ω · g such that the modules Ω 2 g and Ω 2,cl g from the previous no. become really the module of two-forms and closed ones respectively. These complexes come equipped with the composition maps Ω · g × Ω · g ′ −→ Ω · gg ′ which are compatible with the de Rham differentials and satisfy associativity.
We leave the necessary definitions as an exercise to the reader. 4.5. Theorem. Let g = (g A , g T , g Ω , h) : B −→ B ′ be a morphism between vertex prealgebroids such that g T is an isomorphism. Let A ∈ Alg B . There exists a unique pair (g * A,g) where g * A ∈ Alg B ′ andg : A −→ g * A is a morphism of vertex algebroids such that P (g) =g.
The other two axioms, (A3) and (A5), are checked in a similar manner, and we leave them to the reader. It is convenient to check (A5) in its equivalent form (A5) ter , see 1.6. △ §5. Chern-Simons 5.1. Let us define a frame of a Lie A-algebroid T to be a k-submodule g ⊂ T such that A ⊗ k g = T . For example, if T is a free A-module and {τ i } is some A-base of T then g {τi} := kτ i is a frame in T .
A frame of a vertex A-algebroid (resp. prealgebroid, extended Lie algebroid) is by definition a frame of the underlying Lie algebroid T . A vertex algebroid (resp. prealgebroid, ...) equipped with a frame will be called framed.
We will call an extended Lie algebroid T quasiregular if it is perfect (see 1.2) and admits a frame.

5.2.
Let us consider the situation of 4.3, and assume that T ′ is equipped with a frame g ′ and T is perfect. Let us assume that g T is an isomorphism. Then g := g T (g ′ ) is a frame in T . Let us define a k-linear map h g ′ : g ′ −→ Ω by the condition Since g T is an isomorphism, (5.2.1) defines h g ′ uniquely. Then the condition (3.5 , ) is obviously satisfied for all τ i ∈ g ′ . There is a unique extension of h g ′ to a map h g ′ : T ′ −→ Ω satisfying (3.5γ) for all τ ∈ g ′ .
If g is a Lie subalgebra of T then A T ;g may be defined as follows. Take the vertex k-algebroid A g = (k, g, g * , 0, 0, , , 0) where g acts on g * in the coadjoint way, , | g×g * is the obvious pairing, the other components of , being zero. Then apply to A g the pushout with respect to the structure morphism k −→ A, where we set γ(a, τ ) = 0 for τ ∈ g and the Lie algebra acts on A due to the embedding g ⊂ T , cf. 1.10. We get a vertex A-algebroid which is equal to A T ;g . We set B T ;g := P (A T ;g ).
Let us call the frame g abelian if g is an abelian Lie subalgebra of T , i.e. [τ, τ ′ ] = 0 for all τ, τ ′ ∈ g. We call an extended Lie algebroid T regular if it is perfect and admits an abelian frame.

5.4.
Let T = (A, T, Ω, . . . ) be a regular extended Lie algebroid. Let g, g ′ be two abelian frames of T . Let us assume that g and g ′ are free k-modules of finite rank n. In the sequel we will need some formulas pertaining to this situation.

Define the matrices
for all i, j, q. Applying τ r to (5.4.6) we get for all i, j, q, r. Setting here r = q and summing up by q we get for all i, j. Applying τ r we get It follows from (5.4.5) that for all r. Multiplying (5.4.6) by ρ ia and summing up by i, we get Thus, τ a (ρ bc ) = τ c (ρ ba ) (5.4.13) 5.5. In the situation 5.4, consider the vertex prealgebroids B T ;g = (A, T, Ω, ∂, γ, , ) and B T ;g ′ = (A, T, Ω, ∂, γ ′ , , ′ ). According to 5.2, we have an isomorphism which is (5.2.1) in our situation. Now consider the vertex algebroids A T ;g = (. . . , c), A T ;g ′ = (. . . , c ′ ). We have g * A T ;g ′ , A T ;g ∈ Alg BT ;g . Therefore, by 4.2.1 A T ;g = g * A T ;g ′ · + β for some closed 3-form β.
Let us define a form β = β g,g ′ ∈ Ω 3 (T ) by 5.6. Magic Lemma. The form β g,g ′ is closed and we have A T ;g = g * A T ;g ′ · + β g,g ′ .

5.7.
Proof. Let us write down the things explicitly. The following formulas hold true in the algebroid A T ;g (they are (1.8.3) in our situation): The left hand side of (5.5.2) is equal to φ ip τ p , h jq ω q = φ ip h jp By (5.7.1) , , the right hand side is equal to Thus, the equation (5.5.2) takes the form wherefrom, applying (5.4.5), We have to prove that where g * c ′ is defined by by (4.5.1). Thus, we have to prove that By definition, c ′ (τ ′ i , τ ′ j ) = 0. By (5.7.1) c and (5.4.8) we have On the other hand, by (5.4.1) and (5.4.3), where we have used (5.4.1). Thus, (5.7.3) takes the form We have to prove that the matrix (h ij ) defined by (5.7.2) satisfies the differential equation (5.7.5). Using (5.4.4), rewrite (5.7.5) in the form We have to prove that A + B + C = β ijr ρ rl ω l where h ij is given by (5.7.2) and β ijr is given by (5.5.3). Thus, we have Let us denote the n-th summand in an expression X by Xn (where we open the brackets). We have We see that B221 = −A1, B222 = −B213 and B223 = −B212. Similarly, A2 = −C221. We compute B23 using (5.4.10): Finally, B211 + C211 = β ijr ρ rl ω l . Everything except these last terms cancels out, and this proves the Lemma. △ §6. Atiyah 6.1. Let T = (A, T, Ω, ∂) be a perfect extended Lie algebroid. Let g, g ′ , g ′′ be three frames in T . According to 5.2, we have the morphisms of the corresponding vertex prealgebroids as well as the morphism g g;g ′′ : B T ;g ′′ −→ B T ;g , all of them over Id T . Recall that Hom PreAlgT (B T ;g ′′ , B T ;g ′ ) is an Ω 2 (T )-torseur, cf. 4.3.1. We are aiming to compute the discrepancy α g,g ′ ,g ′′ := g g;g ′ • g g ′ ,g ′′ − g g,g ′′ ∈ Ω 2 (T ) (6.1.2) We have the functions h g,g ′ , etc., acting from T to Ω (not A-linear!), as in the previous Section, which define our morphisms. By (3.7.1) the composition g g;g ′ • g g ′ ,g ′′ is defined by the function h g;g ′ + h g ′ ;g ′′ ; therefore the discrepancy (6.1.2) is defined by the A-linear function which by definition coincides with (6.1.2).

Choose some bases {τ
The maps h g,g ′ , etc., are defined by the matrices (h ij g,g ′ ), etc., where h g,g ′ (τ ′ i ) = h ij g,g ′ ω j , etc. By (3.5γ) and (5.7.1) γ we have In the computation we shall use the same convention for the notation of various terms in A, B, C, . . . as in 5.7.

So, we have
We see first of all the terms of the second order cancel out, as they should: A11 = −C12 and B11 = −C11. Most of the other terms also cancel out, and in A + B + C we are left only with Rewriting the right hand side in the base {ω ′′ i } we get So we see that α ir 1 = −α ri 2 , i.e. the matrix α ir is skew symmetric, that is, α ∈ Ω 2 (T ) as it should be.
6.3. Let us rewrite the expression (6.2.3) in terms of vector fields τ ′′ i : Using (5.4.13), we have Skew symmetrizing, we arrive at the first part of 6.4. Theorem. (a) The cocycle α g,g ′ ,g ′′ , (6.1.3) is given in coordinates by the expression Part (b) is proven by the same argument as in 6.3, and we leave it to the reader.
From the expression (6.4.2) we see directly that the form β(φ) is closed, and from (6.4.1) one checks that the form α(φ, ψ) satisfies the group (or Cech) cocycle condition for all φ, ψ, χ ∈ GL n (A), which is the same as (6.1.4), and where Ω 2 (T ) sits in degree 0 (the action of GL n (A) on Ω [2,3 being trivial).
The cocycle α is classical, and essentially goes back to Atiyah; it is written down explicitly by Harris, [H], p. 280. The class β resembles "Chern-Simons" form.
The whole cocycle c(T ) may be thought of as an integration of a cocyclec from [MSV], (5.16), (5.17). §7. Gerbes of Vertex Algebroids 7.1. Let us reformulate the results of the last three Sections in language of Torseurs. This reformulation was inspired by [BD1].
Similarly, let Ω 2,cl T denote a groupoid with the unique object and the set of morphisms equal to Ω 2,cl (T ). It is also an abelian group in categories. We have a fully faithful monoidal functor sending the unique object in Ω 2,cl T to 0 ∈ Ω 3,cl (T ).
Consider the groupoid Alg T . According to 5.3, it is nonempty.
We can define an Action of Ω as follows. For A ∈ Alg, ω ∈ Ω 3,cl (T ), a couple (A, ω) goes to the vertex algebroid Let us fix A and consider the functor induced by (7.1.2). By 4.2.2 this functor is fully faithful.
Let A ′ ∈ Alg T be another object. By 5.2, there exists a morphism of vertex prealgebroids g : P (A) −→ P (A ′ ) lying over Id T ; it is necessarily an isomorphism. Consider the vertex algebroid g * A constructed in Theorem 4.5. By definition, g is lifted to an isomorphismg : A ∼ −→ g * A. Since g * A ∈ Alg P (A ′ ) , by 4.2.1 A ′ = g * A · + ω for some ω ∈ Ω 3,cl (T ). Thereforeg induces an isomorphismg · + ω : In other words, we have checked that (7.1.3) is surjective on isomorphism classes of objects, hence it is an equivalence of categories. This proves 7.2. Theorem. Let T be a quasiregular extended Lie algebroid. Then the Action (7.1.2) makes the groupoid Alg T a nonempty Ω For example, if T is a Lie O X -algebroid which is locally free as an O X -module then the corresponding extended algebroid T T is quasiregular. If X is a smooth k-scheme of finite type and T = T X/k is the sheaf of vector fields then T T is regular.
We can sheafify the constructions of the previous Subsections and obtain the sheaves (champs) of groupoids (i.e. gerbes) Alg T , etc.
Let T be quasiregular. Consider the gerbe Alg T . According to 7.1 and 7.2, it is locally nonempty but not locally connected in general; its sheaf of connected components is an H 3 DR (T )-torseur. By the general procedure this gerbe defines a characteristic class c(T ) := c(Alg T ) ∈ H 2 (X; Ω [2,3 (T )) (7.3.1) Here in the right hand side we consider the hypercohomology with coefficients in the complex Ω [2,3 (T ).
Let us explain how to define the class (7.3.1). Choose an open covering U = {U i } of X such that all groupoids Alg T (Ui) are nonempty. Choose an object A i in each Alg T (Ui) . Over double intersections U ij := U i ∩ U j we get two objects A i | Uij , A j | Uij ∈ Alg T (Uij ) . Choose 3-forms ω ij ∈ Ω 3,cl (T (U ij ) such that there exist isomorphisms Choose some isomorphisms (7.3.2). Then on triple intersections we get isomorphisms The isomorphisms (7.5.3) must be given by the 2-forms η ijk ∈ Ω 2 (T ) such that d DR (η ijk ) = c ij − c ik + c jk . Then (ω ij , η ijk ) is a 2-cocycle in the Cech complex C · (U; Ω [2,3 (T )) representing the class (7.3.1).
7.4. Now let us assume that the groupoid T is regular. Theorem 6.4 calculates the class (7.3.1).

△
It is instructive to compare the previous discussion with [BB], 2.1. In the chiral situation the degree of cohomology goes one step up. 7.6. Let us identify the class c(T ) when X is a smooth k-scheme and T = T X is the tangent bundle. If E is an arbitrary vector bundle over X given by a Cech 1-cocycle g = (g ij ) ∈ Z 1 (U; GL r (O X )) (7.6.1) on some open covering U then (7.4.1) -(7.4.3) define a cocycle c(g) = (α(g), β(g)) ∈ Z 2 (U; Ω [2,3 X ) (7.6.2) then one checks by a direct computation that where e = (η, γ) ∈ C 1 (U; Ω [2,3 X ) (7.6.8) △ Therefore (7.6.2) gives rise to a well defined characteristic class c(E) ∈ H 2 (X, Ω [2,3 X ). The following Lemma is obvious.
It is obvious that the natural map . . is the stupidly truncated (and shifted, so that Ω 2 X sits in degree 0) de Rham complex. The proof of the following lemma was provided to us by H. Esnault.

Lemma. The inverse image map
In fact, more is true. One can consider the cohomology theory which assigns to a smooth X a collection of cohomology groups {H i (X; Ω [i )}. This theory has the standard Grothendieck's properties needed to define the Chern classes, cf. [Gr]. This fact was communicated to us by A. Beilinson.
Anyway, the inverse image map H 2 (X; Ω 7.11. Corollary. In the situation of 7.6 the class described in Theorem 7.5 is equal to 2ch 2 (T X ). △ §8. Vertex Envelope of a Conformal Algebra 8.1. Our aim in this Section will be to construct a left adjoint U to the forgetful functor (0.8.1) and to prove the "Poincaré-Birkhoff-Witt" theorem for algebras U C, C ∈ Conf .
Let V be a vertex algebra. We have the following two particular cases of the OPE formula (0.5.12). The first one corresponds to m = n = −1: where we have used (0.5.10). The second one corresponds to m ≥ 0, n = −1: 8.2. Let C = ⊕ C i be a conformal algebra. Let T C = j∈Z ≥0 T j C, T j C := C ⊗j be the tensor algebra of C over k. The multiplication in T C will be denoted x · y or xy. The Z ≥0 -grading of C induces the Z ≥0 -grading of T C such that T C becomes a Z ≥0 -graded associative algebra with the unit 1 ∈ T 0 C = k. We have a canonical embedding of k-modules C = T 1 C ⊂ T C.
There is a unique extension of the operators ∂ (j) on C to the whole space T C satisfying cf. (0.4.6). These operators will satisfy (0.4.1).
Let R ⊂ T C be a two-sided ideal generated by all elements cf. (8.1.1). Set U C = T C/R. We have a canonical morphism i : C −→ U C equal to the restriction of the projection p : T C −→ U C to C.
Since the relations (8.2.1) are homogeneous, the algebra U C inherits a Z ≥0grading from T C. Using (0.4.6) and an obvious identity ∂ (i) ∂ (j) = ∂ (j) ∂ (i) one sees easily that the operators ∂ (i) respect the ideal R. Hence they induce the operators ∂ (i) on U C of degree i which satisfy (0.4.1).

Theorem.
There is a unique structure of a vertex algebra on the k-module U C such that for all x ∈ C, z ∈ T C, The corresponcence C → U C defines a functor U : Conf −→ Vert left adjoint to the forgetful functor.
The algebra U C will be called a vertex envelope of a conformal algebra C.
This Theorem will be proven in 8.20, after some preparation.
8.4. Let us define k-linear operators x (j) , x ∈ C, j ∈ Z of degree −j − 1 acting on the module T C. If j < 0, j = −n − 1, we set In particular, x (−1) z = xz.
Each element of T C is a linear combination of monomials z = z 1 z 2 . . . z n , z i ∈ C. We define x (j) z, j ∈ Z ≥0 by induction on the length n of the monomial. If n = 1, i.e. z ∈ C, then we already have x (j) z due to the structure of a conformal algebra on C.
If z = yu, y ∈ C, we set We leave to the reader an easy proof of the lemma below.
8.4.1. Lemma. For all i ∈ Z ≥0 , n ∈ Z, The operators x (j) introduced above respect the ideal R.
The proof will be given in 8.6 -8.8 below.
8.6. Since R is a left ideal, it is respected by all operators x (n) with n < 0.
It follows from the commutation formula (8.4.2) that if all the operators x (n) , n ≥ 0, respect a subset S ⊂ T C then they respect the left ideal generated by S. Therefore we need to prove that for all u, x, y ∈ C, z ∈ T C, n ≥ 0. Let us denote the summands in (8.6.1) by A, B and C. We have We shall use the same agreement as in 5.7: in an expression X, the n-th summand will be denoted by Xn. We have Similarly, (8.6.9) Next, u (p) x) (n+j−p) y z (8.6.10) and u (q) x) (j+n−p−q) y z (8.6.11) We have A21+B21+C2 = r(x, y)u (n) z ∈ R. Next, A23 = −B32 and A32 = −B23. Next, Similarly, 8.7. Below we shall often use the identity n p p r = n r n − r n − p (8.7.1) We claim that Using (8.7.1), we have n p n − p q = n n − p n − p q = n q n − q p Consider the part of (8.7.4) at a fixed q, 0 ≤ q ≤ n − 1: It is easy to see that the part of (8.7.5) corresponding to j = p − 1 is equal to −A31 q and the part corresponding to j ≥ p equals −C113 q . This proves (8.7.2).
After bookkeeping, the computations of 8.6 -8.8 show that the left hand side of (8.6.1) belongs to the ideal R, which finishes the proof of Lemma 8.5. △ 8.9. By Lemma 8.5, the operators x (j) induce operators x (j) : U C −→ U C, x ∈ C, j ∈ Z (8.9.1) 8.10. Lemma. The operators (8.9.1) satisfy the OPE formula (0.5.12).
This lemma will be proven in 8.11 -8.19 below.
8.11. We have to prove the identity between the operators acting on U C, for all m, n ∈ Z.
Let us discuss separately three cases.
8.12. Case A. m < 0 and n < 0. First of all, note that for m = n = −1 the relation (8.11.1) is nothing but (8.2.1) and holds true by definition, since R is a left ideal.
In general, set m = −1 − a, n = −1 − b for a, b ≥ 0. We have by definition (8.4.1) [ 8.14. Case B. m ≥ 0 and n < 0. If n = −1 then (8.11.1) is the same as the definition (8.4.2). Now let n = −1 − a, a ≥ 0. The lhs of (8.11.1) is equal to The rhs of (8.11.1) is equal to Consider the first sum in the rhs of (8.14.1). It is equal to By Lemma 8.13, this is equal to the first sum in the rhs of (8.14.2).
The second term in the rhs of (8.14.1) is equal to which is the same as the second sum in (8.14.2).
8.15. Case C. m < 0 and n ≥ 0. First let us treat the case m = −1. We need to prove that The lhs of (8.15.1) is equal to On the other hand, the rhs of (8.15.1) equals (we use (0.4.3)) − j≥0 q≥0 Therefore, (8.15.1) follows from the identity below, which is easily checked by induction on n: for all n, p ≥ 0. Now assume that m = −1 − a, a ≥ 0. We need to prove that The lhs is equal to First of all, A1 + B1 = R1 by induction hypothesis. Next, A2 = −B4 and A4 = −B2. Further, A5 + B5 = R2 by the axiom of a conformal algebra. Next, Similarly, We have Similarly, with a = m − j, which is obvious from the combinatorial definition of the binomial coefficients, shows that E + F + G = R3.
This completes the proof of Lemma. △ 8.18. Remark. Lemmas 8.17 and 8.4.1 mean that T C is canonically a module over conformal algebra C.
8.19. Lemma 8.17 implies Case D, which completes the proof of Lemma 8.10. △ 8.20. Now we can finish the proof of Theorem 8.3. According to Lemmas 8.5 and 8.10, we have the collection of mutually local fields a(z) = n∈Z a (n) z −n−1 , a ∈ C, acting on the space U C. These fields satisfy the conditions of Theorem 0.7, which gives the desired structure of a vertex algebra on U C. The space C with the Lie algebra structure given by (8.21.1) will be denoted C Lie .
The proof is given in 8.23 -8.26 below.
8.23. Skew symmetry. Using (0.4.3), we have and we again conclude after an easy rearrangement of the indices and using Lemma 8.13.
This completes the proof of lemma and of the theorem. △ 8.27. Corollary. The vertex enevlope U C of a conformal algebra C is equal to the associative enveloping algebra U C Lie of the Lie algebra C Lie .
If i : C −→ U C denotes the canonical morphism, for all x, y ∈ C.
This follows immediately from the definition in 8.2.
Note that in an arbitrary vertex algebra, although the operation x (−1) y is not associative, the commutator x (−1) y − y (−1) x is a Lie bracket, according to a remark by A.Radul cited in [K], 3.1, page 82.
This corollary supports the point of view advocated by Beilinson and Drinfeld, namely, that conformal algebras (Lie * -algebras in the language of BD) are analogs of Lie algebras, while vertex algebras (chiral algebras) are analogs of associative algebras.
8.28. Let us define a canonical increasing filtration F · U C on U C by induction: Obviously, the operation x (−1) y induces on the associated graded space gr We have an evident surjective morphism of commutative algebras Sym k (C) −→ gr F (U C) (8.28.1) 8.29. Theorem. If C is a free k-module then (8.28.1) is an isomorphism.
This is immediate consequence of Corollary 8.27 and the usual Poincaré-Birkhoff-Witt theorem for Lie algebras, cf. [Se], Part I, Chapter III, Theorem 4.3. §9. Enveloping Algebra of a Vertex Algebroid Abelian case 9.1. Let us define a ∂-module (over k) to be a Z ≥0 -graded k-module C = ⊕ i≥0 C i equipped with a family of endomorphisms ∂ (j) : C −→ C, j ≥ 0, where ∂ (j) has degree j, ∂ (0) = Id and Obviously, a ∂-module is the same as an abelian conformal algebra, i.e. a conformal algebra in which all operations (n) are trivial. These objects, with obvious morphisms, form a category ∂ − Mod.
Let us define a 1-truncated ∂-module to be a triple X = (C 0 , C 1 , ∂) where C i are k-modules and ∂ : C 0 −→ C 1 is a map of k-modules. These objects form a category ∂ − Mod ≤1 . We have an obvious truncation functor It is easy to construct the left adjoint to (9.1.1). Namely, given X = (C 0 , C 1 , ∂) ∈ ∂ − Mod ≥1 , consider the direct sum where ∂ (i) X denotes a copy of a k-module X, whose elements are denoted ∂ (i) x, x ∈ X. Define a ∂-module X ∂ = ⊕ i≥0 C i as the quotient of M modulo the submodule generated by the following elements: The grading and operators ∂ (j) : C i −→ C i+j are defined in the obvious manner.
Note that if the ground ring k contains the field of rational numbers Q then X is equal simply to Recall that in 0.9 one introduced the category ∂ − Alg of ∂-algebras which are identified with commutative vertex algebras. A ∂-ideal J in a ∂-algebra B is a Z ≥0 -graded ideal stable under all endomorphisms ∂ (i) ; this is the same as a vertex ideal if B is understood as a vertex algebra.

This functor admits a left adjoint
Sym : ∂ − Mod −→ ∂ − Alg (9.2.2) that assigns to a ∂-module C the symmetric algebra (over k) Sym(C). It is nothing but the restriction of the functor defined in 8.3 to the full subcategory of abelian conformal algebras.
Note that the grading on S(A) is defined in such a way that A ⊂ S(A) has grading 0 and Ω ⊂ S(A) has grading 1.
Let R ⊂ S(A) be a ∂-ideal generated by all elements the second arrow being the evident canonical map.
9.5. Theorem. Assume that k ⊃ Q and Ω is a free A-module. Then (i) the maps Ω −→ Ω (i) sending ω to (the image of ) ∂ (i−1) ω, i ≥ 1, are isomorphisms of A-modules; (ii) both maps in (9.4.3) are isomorphisms of A-algebras.
Indeed, choose an A-base {ω r } of Ω. LetΩ (i) denote a free A-module with the base {∂ (i−1) ω r }. LetJ be the symmetric algebra Sym A {A ⊕ (⊕ i≥1Ω (i) )}. It has an obvious structure of a ∂-algebra, and satisfies the universal property 9.3.1. Therefore, we have canonical isomorphism J(A) ∼ −→J (9.5.1) We can apply the construction of 9.4 to the algebraJ as well, and for it the claims of Theorem 9.5 are clear. On the other hand, the maps Ω (i) −→Ω (i) induced by (9.5.1) are isomorphisms. This implies the theorem. △ Conformal algebra associated with a vertex algebroid 9.6. Following the pattern of 3.1, let us define a 1-truncated conformal algebra to be a quintuple c = (C 0 , C 1 , ∂, (0) , (1) ) where C i are k-modules, ∂ : C 0 −→ C 1 is a k-module map and (i) : (C 0 ⊕ C 1 ) ⊗2 −→ (C 0 ⊕ C 1 ) are bilinear operations of degree −i − 1.
Elements of C 0 (resp. of C 1 ) will be denoted a, b, c (resp. x, y, z).
These data must satisfy the following axioms.
We have an evident forgetful functor Vert ≤1 −→ Conf ≤1 (9.6.1) 9.7. We have an obvious truncation functor Conf −→ Conf ≤1 (9.7.1) which assigns to a conformal algebra C its part C ≤1 of degree ≤ 1. Let us construct a left adjoint to this functor.
Given c = (C 0 , C 1 , . . . ) ∈ Conf ≤1 , consider c as a 1-truncated ∂-module (forgetting the operations), and consider the corresponding ∂-module C := c ∂ , cf. 9.1. 9.8. Theorem. There is a unique structure of a conformal algebra on C such that operations (i) and ∂ on the subspace C ≤1 coincide with the ones given by the structure of a 1-truncated conformal algebra on c.
The assignement c → C gives a functor Conf ≤1 −→ Conf (9.8.1) 9.13. LetR ⊂ U CA denote the image of R under the canonnical projection T CA −→ U CA. The previous lemma means thatR is a vertex ideal in U CA. Let us denote by U A the vertex algebra U CA/R. It is equipped with an obvious splitting T ֒→ T ⊕ Ω = U A 1 .
Conversely, given a splittable vertex algebra V , choose a splitting s : T −→ V 1 , and consider the vertex algebroid A(V ; s), cf. §2. Let Vert ′ ⊂ Vert denote the full subcategory of splittable vertex algebras. We get a functor A : Vert ′ −→ Alg (9.13.1) which is in fact the composition of the truncation functor (3.1.1) (restricted to Vert ′ ) and of the functor quasiinverse to (3.3.6).
Indeed, this is evident from the construction.
Poincaré-Birkhoff-Witt 9.15. Let A = (A, T, Ω, . . . ) be a vertex algebroid. The enveloping algebra U A is generated as a k-module by the monomials of the form x p · x p−1 · . . . · x 1 (9.15.1) where x i has the form ∂ (j) y, y = a, ω or τ . Here we have denoted for brevity by x · x ′ the operation x (−1) x ′ .
Let us introduce a canonical increasing exhaustive filtration F 0 U A ⊂ F 1 U A ⊂ . . . U A by setting F i U A to be equal the k-submodule of U A generated by all monomials (9.15.1) where there are ≤ i letters x j of the form ∂ (a) τ .
Obviously all submodules F i U A are stable under ∂ (j) and F i U A · F j U A ⊂ F i+j U A. Consider the associated graded module gr F U A = ⊕ i≥0 F i U A/F i−1 U A.
It is easy to see that the operation x · y induces a commutative and associative multiplication on gr F U A, i.e. gr F U A becomes a ∂-algebra (an abelian vertex algebra).
Let A ab denote an abelian vertex algebroid (A, Ω ⊕ T, ∂). We have A(gr F U A) = A ab (9.15.2) hence by adjunction we get a canonical map of ∂-algebras J(A ab ) −→ gr F U A (9.15.3) It is clear the this map is surjective.
Recall that by 9.4 we have a canonical filtration on J(A ab ) (let us denote it by G here) and a canonical surjective map of ∂-algebras Sym A (⊕ i≥1 T (i) ) ⊕ (⊕ i≥1 Ω (i) ) −→ gr G J(A ab ) (9.15.4) where X (i) (X = T or Ω) denotes a copy of X sitting in weight i.
By refining the filtration F using the filtration G, we get a canonical filtration H on U A together with a surjective map (of ∂-algebras over A) The multiplication in the right hand side is induced by the operation (−1) on U A.
9.16. Theorem. Assume that k ⊃ Q and both Ω and T are free A-modules. Then the maps (9.15.3) and (9.15.5) are isomorphisms.
Filtration H is compatible with the conformal grading, finite on each component with the fixed conformal weight and is canonical in following sense: for any morphism of vertex algebroids A −→ A ′ , the corresponding morphism of vertex envelopes U A −→ U A ′ respects these filtrations.
To prove this, we use the strategy of Serre's proof of the usual PBW theorem, cf. [Se], pp. 14-16. Let us choose well ordered A-bases {τ s }, {ω r } of T and Ω. They give rise to a well ordered A-base {x u } u∈I = {∂ (a) τ s , ∂ (b) ω r } of (⊕ i≥1 T (i) ) ⊕ (⊕ i≥1 Ω (i) ). For a sequence M = (u 1 , . . . , u m ), u j ∈ I, u 1 ≤ u 2 ≤ . . . ≤ u m define a monomial x M ∈ U A by Similarly, M defines monomialsx M ∈ gr H U A and x M ∈ S := Sym A (⊕ i≥1 T (i) ) ⊕ (⊕ i≥1 Ω (i) ) Obviously, the monomials {x M } form an A-base of S. On the other hand, it is easy to see that each element of U A may by written as a M · x M with some a M ∈ A. To prove our claim it is enough to show that a relation a M · x M = 0 (9.16.1) implies that all a M = 0 (cf. loc. cit., Lemma 4.5).
Proceeding in a manner similar to 8.4, we define an action of the Lie algebra CA Lie , and hence of its envelope U CA Lie = U CA, on S. Next one checks the relations (9.11.1) and (9.11.2) and therefore gets an action of U A on S.
One sees immediately from the definitions that (a · x M ) · 1 S = ax M . Hence (9.16.1) implies the relation a MxM = 0 in S and therefore a M = 0, which proves the theorem. We leave the details to the reader. △ 9.17. Let X be a smooth k-scheme, U ⊂ X a Zariski open in X. A vertex algebra of the form U A where A ∈ Alg TX (U ) is a section of the gerbe Alg TX discussed in Section 7, is called an algebra of chiral differential operators on U .
These algebras form a gerbe Dif f X , by definition isomorphic to the gerbe Alg X .
Note that all isomorphisms of algebras of chiral differential operators respect the canonical filtrations on them, and there is no obstruction to the gluing of associated graded algebras.
In particular, we have 9.18. Theorem. Assume that k ⊃ Q. Each algebra of chiral do D X ∈ Dif f X (X) admits a canonical filtration whose graded algebra is isomorphic to where T (i) X (resp. Ω 1(i) X ) denotes a copy of the tangent bundle (resp. of the bundle of 1-forms) sitting in conformal degree i. △