REMARKS ON FORMAL DEFORMATIONS

AND BATALIN-VILKOVISKY ALGEBRAS

Vadim Schechtman

Introduction

This note consists of two parts. Part I is an exposition of (a part of) the V.Drinfeld’s letter, [D]. The responsibility for the style lies entirely on the author of the present note.

The sheaf of algebras of polyvector fields Λ^{•}T_{X} on a Calabi-Yau manifold X, equipped with the Schouten bracket, admits a structure of a Batalin-Vilkovisky algebra. This fact was probably first noticed by Z.Ran, [R]. Part II is devoted to some generalizations of this remark.

To get the above BV structure, one uses a trivialization of the canonical bundle K_{X}. We note that, in fact, it is sufficient to have an integrable connection on K_{X}. Moreover, one has a canonical bijection between the set of integrable connections on K_{X}, or, what is the same, the set of right D_{X}-module structures on O_{X}, and the set of BV structures on Λ^{•}T_{X}, cf. Theorem 4.3. We give some generalizations of this correspondence, cf. §4.

An interesting example of these data is a pair (a non compact CY manifold X, a function f ∈ Γ(X, O_{X})). If we fix a trivialization K_{X} ≅ O_{X}, we get the connection on K_{X} defined by the exact form df, see §2.

This note is an extended version of the talks given at Max-Planck-Institut für Mathematik in January, 1998. I am very grateful to MPI for the hospitality and the excellent working conditions. I am grateful to Vladimir Hinich for the enlightening correspondence. I am especially obliged to Yu.I. Manin, whose question triggered this work. The contents of Sections 1 and 2 of Part II was influenced by the M. Kontsevich’s talk at MPI in January, 1997.

TABLE OF CONTENTS

Intoduction 1 Part I. Maurer-Cartan and formal deformations §1. Standard functors and Maurer-Cartan equations 3 §2. Koszul complex and Maurer-Cartan scheme 8 §3. Deformations 11 Part II. Milnor ring and Batalin-Vilkovisky algebras §1. Koszul complex and Milnor ring 14 §2. Batalin-Vilkovisky algebras on Calabi-Yau manifolds 17 §3. Recollections on D-modules 21 §4. Connections and Batalin-Vilkovisky structures 24 References 29 PART I. MAURER-CARTAN AND FORMAL DEFORMATIONS

§1. Standard functors and Maurer-Cartan equations

1.1. In this Part, we fix once and for all a ground field k of characteristic 0. A ”vector space” will mean a vector space over k; ⊗ will mean ⊗_{k}.

V ect will denote the category of vector spaces; it is a symmetric monoidal category, with respect to the tensor product. Ass will denote the category of associative algebras in V ect (without unit). The forgetful functor Ass ⟶ V ect admits a left adjoint functor of a free associative algebra, to be denoted F_{Ass}. For V ∈ V ect, F_{Ass}(V) is equal to the tensor algebra

T^{≥1}(V) = ⊕_{n≥1} V^{⊗n},  (1.1)

with the obvious multiplication.

Let Com denote the category of commutative algebras in V ect (assotiative, without unit). The forgetful functor Com ⟶ V ect admits a left adjoint, F_{Com}. We have F_{Com}(V) = S^{≥1}(V) where

S^{≥1}(V) = ⊕_{n≥1} S^{n}(V)  (1.2)

Here S^{n}(V) is the n-th symmetric power of V, i.e. the quotient of V^{⊗n} over the obvious action of the symmetric group Σ_{n}.

The canonical projection V^{⊗n} ⟶ S^{n}(V) admits a canonical splitting i : S^{n}(V) ↪ V^{⊗n}, given by

i(x_{1} · . . . · x_{n}) = \frac{1}{n!} ∑_{σ∈Σ_{n}} x_{σ(1)} ⊗ … ⊗ x_{σ(n)}  (1.3)

We denote by Lie the category of Lie algebras in V ect, and by F_{Lie} the functor of a free Lie algebra, the left adjoint to the forgetful functor Lie ⟶ V ect.

1.2. Let C be a coassociative coalgebra in V ect, with comultiplication Δ : C ⟶ C ⊗ C. For n ≥ 2, define a map

Δ^{(n)} : C ⟶ C^{⊗n}  (1.4)

by induction. We set Δ^{(2)} = Δ, and

Δ^{(n+1)} = (id_{A^{⊗(n−1)}} ⊗ Δ) ∘ Δ^{(n)}  (1.5)

Let us consider the following condition

(F) For each x ∈ C, there exists n such that Δ^{(n)}(x) = 0.

We denote by Coass the category of coassociative coalgebras in V ect (without counit) satisfying (F). The forgetful functor Coass ⟶ V ect admits a right adjoint, the functor of a cofree coalgebra, to be denoted F_{Coass}. For V ∈ V ect, F_{Coass}(V) = T^{≥1}(V) as a vector space. The comultiplication is defined by

Δ(v_{1} ⊗ … ⊗ v_{n}) = ∑_{i=1}^{n−1} (v_{1} ⊗ … ⊗ v_{i}) ⊗ (v_{i+1} ⊗ … ⊗ v_{n})  (1.6)

We denote by Cocom the category of cocommutative coalgebras in V ect (coassociative, without counit). The forgetful functor Cocom ⟶ V ect admits a right adjoint, the functor of a cofree cocommutative coalgebra, to be denoted by F_{Cocom}. For V ∈ V ect, F_{Cocom}(V) = S^{≥1}(V) as a vector space. The comultiplication is defined as the composition

S^{≥1}(V) ↪ T^{≥1}(V) ⟶ T^{≥1}(V) ⊗ T^{≥1}(V) ⟶ S^{≥1}(V) ⊗ S^{≥1}(V)  (1.7)

Here the first arrow is the canonical injection (1.3), the second one is the comultiplication in T^{≥1}(V), and the third one is the canonical projection.

1.3. By a graded vector space, we mean a collection of vector spaces V^{•} = {V^{i}}_{i∈ℤ}, indexed by integers. Graded vector spaces form the category GV ect, with obvious morphisms. It is a monoidal tensor category. For V^{•}, W^{•} ∈ GV ect, their tensor product V^{•} ⊗ W^{•} is defined by

(V^{•} ⊗ W^{•})^{i} = ⊕_{p+q=i} V^{p} ⊗ W^{q}  (1.8)

The associativity constraint is the obvious one. The commutativity isomorphisms

s_{V^{•},W^{•}} : V^{•} ⊗ W^{•} ⟶^{∼} W^{•} ⊗ V^{•} are given by

s_{V^{•},W^{•}}(a ⊗ b) = (−1)^{pq}b ⊗ a,  (1.9)

for a ∈ V^{p}, b ∈ W^{q}. We often use the notation |a| = p.

We have the shift functors [i] : GV ect ⟶ GV ect (i ∈ ℤ), given by (V^{•}[i])^{j} = V^{i+j}. By a morphism of degree i between V^{•} and W^{•} we mean a morphism V^{•} ⟶ W^{•}[i] in GV ect. We define Hom(V^{•}, W^{•}) ∈ GV ect by

Hom(V^{•}, W^{•})^{i} = Hom_{GV ect}(V^{•}, W^{•})  (1.10)

We define the categories GAss, GCoass, GCom, GCocom and GLie as the categories of associative algebras, coassociative coalgebras, commutative algebras, cocommutative coalgebras (all without unit or counit) and Lie algebras in GV ect. The coalgebras are assumed to satisfy the condition (F). We regard V ect, etc. as a full subcategory of GV ect, etc., consisting of objects living in degree 0.

We have the functors of (co)free (co)algebras F_{GAss}, F_{GCoass}, F_{GCom}, F_{GCocom} and F_{GLie}, defined as in the case of non-graded algebras.

If A^{•} ∈ GAss and B^{•} ∈ GCoass then Hom(B^{•}, A^{•}) admits a canonnical structure of a coassociative coalgebra. For two maps f ∈ Hom(B^{•}, A^{•})^{i}, g ∈ Hom(B^{•}, A^{•})^{j}, their product f · g is defined as a composition

f · g : B^{•} ⟶ B^{•} ⊗ B^{•} ⟶^{f⊗g} A^{•}[i] ⊗ A^{•}[j] ⟶ A^{•}[i + j]  (1.11)

Here the first arrow is the comultiplication in B^{•}, and the third one is the multiplication in A^{•}. The second one is defined by

(f ⊗ g)(b ⊗ c) = (−1)^{j|b|}f(b) ⊗ g(c)  (1.12)

Similarly, if 𝔤^{•} ∈ GLie and B^{•} ∈ GCocom, Hom(B^{•}, 𝔤^{•}) admits a canonical structure of a graded Lie algebra.

If A^{•} ∈ GCom then A^{•} ⊗𝔤^{•} is canonically a graded Lie algebra, with the bracket defined by

[a ⊗ x, b ⊗ y] = (−1)^{|x||a|}ab ⊗ [x, y]  (1.13)

A differential graded (dg) vector space, or a complex, is a graded vector space V^{•} equipped with a differential of degree 1, d : V^{•} ⟶ V^{•}[1], d^{2} = 0. We can repeat all the discussion of this Subsection, adding the word ”differential” to ”graded”. The corresponding categories are denoted by DGV ect, DGAss, etc.

1.4. Let A be an associative algebra. Consider the cofree graded coalgebra F_{GCoass}(A[1]). There exists a unique differential d on this graded space, making it a dg coassociative coalgebra, whose degree −2 component

d^{−2} : F_{GCoass}(A[1])^{−2} = A ⊗ A ⟶ F_{GCoass}(A[1])^{−1} = A

coincides with the multiplication μ in A. Note that the condition d^{2} = 0 is equivalent to the associativity of μ.

Let us denote this dg coalgebra by C_{Coass}(A). This way we get a functor C_{Coass} : Ass ⟶ DGCoass which extends naturally to a functor

C_{Coass} : DGAss ⟶ DGCoass  (1.14)

Conversely, let B be a coassociative coalgebra. Consider the free graded algebra F_{GAss}(B[−1]). There exists a unique differential d on this space, making it a dg associative algebra, whose degree 1 component

d^{1} : F_{GAss}(B[−1])^{1} = B ⟶ F_{GAss}(B[−1])^{2} = B ⊗ B  (1.15)

coincides with the comultiplication ν in B. The condition d^{2} = 0 is equivalent to the coassociativity of ν.

Let us denote this dg algebra by C_{Ass}(B). This way we get a functor C_{Ass} : Coass ⟶ DGAss, which extends naturally to a functor

C_{Ass} : DGCoass ⟶ DGAss  (1.16)

1.5. Let A^{•} be a dg associative algebra. The Maurer-Cartan equation in A^{•} is the equation

da + a · a = 0,  (1.17)

a ∈ A^{1}. The set of all a ∈ A^{1} satisfying (1.7) will be denoted MC(A^{•}).

1.7. Let { \atop V ect} denote the category whose objects are topological vector spaces V, complete in a linear topology. Thus, the topology admits a base of neighbourhoods of 0 consisting of subspaces V^{(i)}, and V is the inverse limit of the spaces with discrete topology,

V = lim_{⟵} V/V^{(i)}  (1.19)

Morphisms are continuous linear maps. This is a monoidal tensor category, with the tensor product ˆ⊗ given by

V ˆ⊗W = lim_{⟵} V/V^{(i)} ⊗ W/W^{(j)}  (1.20)

We regard the tensor category V ect as embedded in { \atop V ect}, as a full subcategory of spaces with the discrete topology.

Let { \atop Coass}, { \atop Cocom}, etc. denote the category of coassociative coalgebras, cocommutative coalgebras, etc., in { \atop V ect}. We do not impose on the coalgebras the finiteness condition (F).

The forgetful functor { \atop Coass} ⟶ { \atop V ect}, resp., { \atop Cocom} ⟶ { \atop V ect}, admits a right adjoint

F{ \atop Coass}(V) = T̂^{≥1}(V) := ∏_{n≥1} V^{⊗n},  (1.21)

resp.,

F{ \atop Cocom}(V) = Ŝ^{≥1}(V) := ∏_{n≥1} S^{n}(V)  (1.22)

We define the category G{ \atop Coass}, DG{ \atop Coass}, etc., in the obvious way.

1.9. We define the functor

Ĉ_{Coass} : DGÂss ⟶ DG{ \atop Coass}  (1.23)

in the same way as (1.14).

Given A^{•} ∈ DGÂss, a solution a of (1.17) gives rise to a 0-cocycle of the dg coalgebra Ĉ_{Coass}(A^{•}), given by

∑_{n=1}^{∞} a^{⊗n} ∈ T̂^{≥1}(A^{1}) ⊂ Ĉ_{Coass}(A^{•})^{0}  (1.24)

1.10. Theorem. Let A^{•} ∈ DGAss, B^{•} ∈ DGCoass. We have canonically

Hom_{DGAss}(C_{Ass}(B^{•}), A^{•}) = MC(Hom(B^{•}, A^{•})) =
= Hom_{DGCoass}(B^{•}, C_{Coass}(A^{•}))  (1.25)

In particular, the functor C_{Ass} is left adjoint to C_{Coass}. △

1.11. Let 𝔤 be a Lie algebra. Consider the cofree coalgebra F_{GCocom}(𝔤[1]) on the graded space 𝔤[1]. There exists a unique differential

d : F_{GCocom}(𝔤[1]) ⟶ F_{GCocom}(𝔤[1])[1]

making F_{GCocom}𝔤[1] a dg cocommutative coalgebra, and whose degree −2 component

d^{−2} : F_{Cocom}(𝔤[1])^{2} = Λ^{2}(𝔤) ⟶ F_{Cocom}(𝔤[1])^{1} = 𝔤

coincides with the bracket in 𝔤. Let us denote this dg coalgebra C_{Cocom}(𝔤). This way we get a functor C_{Cocom} : Lie ⟶ DGCocom, which extends naturally to a functor

C_{Cocom} : DGLie ⟶ DGCocom  (1.26)

Conversely, let B be a cocommutative coalgebra. Consider the free graded Lie algebra F_{Lie}(B[−1]). There exists a unique differential on this space, making it a dg Lie algebra, whose degree 1 component

d^{1} : F_{Lie}(B[−1])^{1} = B ⟶ F_{Lie}(B[−1])^{2} = S^{2}(B)

coincides with the comultiplication in B. Let us denote this dg Lie algebra by C_{Lie}(B). This way we get a functor C_{Lie} : Cocom ⟶ DGLie which extends naturally to a functor

C_{Lie} : DGCocom ⟶ DGLie  (1.27)

In a similar manner, one defines the functor

Ĉ_{Cocom} : DGL̂ie ⟶ DG{ \atop Cocom}  (1.28)

1.12. Let 𝔤^{•} be a dg Lie algebra. The Maurer-Cartan equation in 𝔤^{•} is the equation

da + \frac{1}{2}[a, a] = 0,  (1.30)

a ∈ 𝔤^{1}. The set of all a ∈ 𝔤^{1} satisfying (1.3) will be denoted by MC(𝔤^{•}).

A solution a of (1.30) gives rise to a 0-cocycle of the dg coalgebra Ĉ_{Cocom}(𝔤^{•}), given by

∑_{n=1}^{∞} \frac{a^{n}}{n!} ∈ Ŝ^{≥1}(𝔤^{1}) ⊂ Ĉ_{Cocom}(𝔤^{•})  (1.31)

1.13. Theorem. Let 𝔤^{•} ∈ DGLie, B^{•} ∈ DGCocom. We have canonically

Hom_{DGLie}(C_{Lie}(B^{•}), 𝔤^{•}) = MC(Hom(B^{•}, 𝔤^{•})) =
= Hom_{DGCocom}(B^{•}, C_{Cocom}(𝔤^{•}))  (1.32)

In particular, the functor C_{Lie} is left adjoint to C_{Cocom}.

Assume for simplicity that B is concentrated in degree 0. By definition, a map of graded Lie algebras α : C_{Lie}(B) ⟶ 𝔤^{•} is the same as a map of vector spaces α^{1} : B ⟶ 𝔤^{1}; the map α is compatible with the differentials iff the map α^{1} satisfies the Maurer-Cartan equation. This proves the first equality in this case. The rest is proved similarly. △

§2. Koszul complex and Maurer-Cartan scheme

A. KOSZUL

2.0. In this section we will use (dg) algebras and coalgebras with unit (resp., counit). We will use the notation (·)^{+} for the objects obtained by formally adjoining a unit or a counit. We denote by Com^{+}, Cocom^{+} the categories of unital algebras, resp., coalgebras.

For a (dg) vector space V, we denote

S^{•}(V) = ⊕_{n≥0} S^{n}(V)  (2.0)

It is the underlying vector space for F_{Com}(V)^{+} and F_{Cocom}(V)^{+}.

2.1. Let 𝔤^{•} be a graded Lie algebra. Assume that 𝔤^{1} is finite dimensional. Then the Maurer-Cartan equation (1.30) defines a closed subscheme of 𝔤^{1}, considered as an affine space, to be called the Maurer-Cartan scheme of 𝔤^{•}, and denoted also by MC(𝔤^{•}). It represents the functor Com^{+} ⟶ Sets which assigns to a k-algebra B the set MC(𝔤^{•} ⊗ B), cf. (1.3).

More explicitely, set

A = S^{•}(𝔤^{1∗})  (2.1)

𝔤^{1∗} denotes the dual space. We have 𝔤^{1} = Spec(A).

Let us choose a base {e_{i}} of 𝔤^{2∗}. Define the elements

f_{i} = (e_{i} ∘ d, e_{i} ∘ [ , ]) ∈ 𝔤^{1∗} ⊕ S^{2}(𝔤^{1∗}) ⊂ A  (2.2)

Here d : 𝔤^{1} ⟶ 𝔤^{2} is the component of the differential, and [ , ] : S^{2}𝔤^{1} ⟶ 𝔤^{2} is the component of the bracket.

We have

MC(𝔤^{•}) = Spec(A/(f_{i}))  (2.3)

Here we have denoted by (f_{i}) the ideal generated by all f_{i}.

2.2. Let us assume that 𝔤^{i} = 0 for i ≠ 1, 2. Such a dg Lie algebra is the same as the following set of data:

two vector spaces V = 𝔤^{1}, W = 𝔤^{2}, a linear map d : V ⟶ W, and a symmetric map b : S^{2}(V) ⟶ W.

We have

C_{Cocom}(𝔤^{•})^{+} = S^{•}(V) ⊗ Λ^{•}(W)  (2.4)

as a graded coalgebra. Here Λ^{•}(W) = ⊕_{n≥0} Λ^{n}(W) is the exterior algebra. In (2.4), V has grading 0, and W has grading 1.

The differential acts as

0 ⟶ S^{•}(V) ⟶ S^{•}(V) ⊗ W ⟶ S^{•}(V) ⊗ Λ^{2}(W) ⟶ …  (2.5)

This complex is dual to a Koszul complex. To make a precise statement, assume that V, W are finite dimensional. Set A = S^{•}(V^{∗}), as in (2.1).

Then the (restricted) dual of (2.5) is

… ⟶ Λ^{2}(W) ⊗ A ⟶ W ⊗ A ⟶ A ⟶ 0  (2.6)

If we choose a basis {e_{i}} of the linear space W^{∗}, we get the elements f_{i} ∈ 𝔪_{A}, as in (2.2). The complex (2.5) is nothing but the Koszul complex K(A; (f_{i})), cf. II, 1.1.

2.3. Corollary. In the assumptions of 2.2, one has canonical isomorphism of commutative algebras

Ô_{MC(𝔤^{•});0} = (H^{0}(C_{Cocom}(𝔤^{•})^{+}))^{∗}  (2.7)

Here in the left hand side stays the completion of the local ring of the origin of the scheme MC(𝔤^{•}).

2.4. More generally,

C_{Cocom}(𝔤^{•})^{+} = S^{•}(𝔤^{odd}) ⊗ Λ^{•}(𝔤^{ev})  (2.8)

for an arbitrary dg Lie algebra 𝔤^{•}. Here 𝔤^{odd} = ⊕_{i∈ℤ} 𝔤^{2i+1}, 𝔤^{ev} = ⊕_{i∈ℤ} 𝔤^{2i}.

Assume that 𝔤^{i} = 0 for i ≤ 0. Then

C_{Cocom}(𝔤^{•})^{+} = ⊕_{p=1}^{∞} ⊕_{i_{1},... ,i_{p}≥0} (S^{i_{1}}(𝔤^{1}) ⊗ Λ^{i_{2}}(𝔤^{2}) ⊗ … ⊗ F^{i_{p}}(𝔤^{p})) (2.9)

where F = S if p is odd, and F = Λ if p is even. In this space the component 𝔤^{p} has degree p − 1. With the differential, this complex looks as

0 ⟶ S^{•}(𝔤^{1}) ⟶ S^{•}(𝔤^{1}) ⊗ 𝔤^{2} ⟶ S^{•}(𝔤^{1}) ⊗ (Λ^{2}(𝔤^{2}) ⊕ 𝔤^{3}) ⟶ …  (2.10)

In other words, the very beginning depends only on 𝔤^{≤2}. Therefore, we get

2.5. Theorem. Let 𝔤^{•} be a dg Lie algebra sitting in degrees ≥ 1. Assume that 𝔤^{1} is finite dimensional. We have a canonical isomorphism of commutative algebras

Ô_{MC(𝔤^{•});0} = (H^{0}(C_{Cocom}(𝔤^{•})^{+}))^{∗}  (2.11)

B. CHEVALLEY-EILENBERG

2.6. A dg Lie algebra 𝔤^{•} such that 𝔤^{i} = 0 for i ≠ 0, 1, is the same as the following set of data:

a Lie algebra U = 𝔤^{0}, a U-module V = 𝔤^{1}, and a linear map d : U ⟶ V such that d([u, u′]) = u · du′ − u′ · du, for all u, u′ ∈ U.

Here we have denoted by u · v the result of the action of an element u ∈ U on v ∈ V.

We want to describe the dg coalgebra C_{Cocom}(𝔤^{•})^{+}. We have

C_{Cocom}(𝔤^{•})^{+} = Λ^{•}(U) ⊗ S^{•}(V)  (2.12)

as a graded coalgebra, where we assign the degree −1 to U, and the degree 0 to V.

Consider the cocommutative counital coalgebra B = S^{•}(V). The action of the Lie algebra U on V induces the U-action on all symmetric powers

U ⊗ S^{n}(V) ⟶ S^{n}(V)

On the other hand, the differential d : U ⟶ V induces the maps

U ⊗ S^{n}(V) ⟶^{d⊗id} V ⊗ S^{n}(V) ⟶ S^{n+1}(V),  (2.13)

the last arrow being the multiplication. Adding up, we get the maps

U ⊗ S^{n}(V) ⟶ S^{n}(V) ⊕ S^{n+1}(V)

Adding over n, we get a map

U ⊗ B ⟶ B  (2.14)

which defines the action of the Lie algebra U by coderivations on the coalgebra B.

Geometrically, this action can be interpreted as follows. Assume that the space V is finite dimensional. Consider the commutative algebra A = S^{•}(V^{∗}), and regard V as an affine scheme Spec(A). The Lie algebra U acts on V by affine vector fields; namely, to an element u ∈ U, there corresponds the vector field τ_{u}, whose value at a point v ∈ V is equal to du + u · v. Here we have identified the tangent space T_{V;v} with V. Therefore, we get the action of U on the functions

U ⊗ A ⟶ A  (2.15)

Now, the action (2.10) is nothing but the dual to (2.11).

Consider the homological Chevalley-Eilenberg complex of the Lie algebra U with coefficients in the B, where the action is (2.10),

C_{homol}(U; B) : … ⟶ Λ^{2}(U) ⊗ B ⟶ U ⊗ B ⟶ B ⟶ 0  (2.16)

Let us regard it as living in degrees ≤ 0.

2.7. Lemma. The complex (2.16) coincides with C_{Cocom}(𝔤^{•})^{+}, if we use the identification (2.12).

2.8. Corollary. Assume that V is finite dimensional. Then we have a canonical isomorphism of pro-artinian algebras,

Ô_{O_{V} ;0}^{U} = (H^{0}(C_{Cocom}(𝔤^{•})^{+}))^{∗}  (2.17)

Here the superscript (·)^{U} means the subspace of U-invariants.

2.9. Let us introduce a shorthand notation

C(𝔤^{•}) = C_{Cocom}(𝔤^{•})^{+}  (2.18)

Assume that 𝔤^{i} = 0 for i < 0. The dg Lie algebra 𝔤^{≥1} is a dg module over the Lie algebra 𝔤^{0}, therefore its symmetric powers are also dg 𝔤^{0}-modules, i.e. we have the action maps

𝔤^{0} ⊗ S^{n}(𝔤^{•}) ⟶ S^{n}(𝔤^{•})  (2.19)

On the other hand, we have the maps

𝔤^{0} ⊗ S^{n}(𝔤^{1}) ⟶ S^{n+1}(𝔤^{1})  (2.20)

defined as in (2.13). Adding the maps (2.19) and (2.20) and summing up over n ≥ 0 and p ≥ 1, we get a map

𝔤^{0} ⊗ S^{•}(𝔤^{≥1}[1]) ⟶ S^{•}(𝔤^{≥1}[1])  (2.21)

which defines a structure of a 𝔤^{0}-module on S^{•}(𝔤^{≥1}[1]) = C(𝔤^{≥1}).

2.10. Theorem. The cocommutative dg coalgebra C(𝔤^{•}) is canonically isomorphic to the Chevalley-Eilenberg complex C_{homol}(𝔤^{0}; C(𝔤^{≥1})), where the action of 𝔤^{0} on C(𝔤^{≥1}) is defined in (2.21).

The coalgebra structure on the Chevalley-Eilenberg complex comes from the cocommutative coalgebra structure on C(𝔤^{≥1}).

2.11. For a more general discussion, I refer the reader to the very nice paper [H].

§3. Deformations

3.1. Let C^{•} be a graded coalgebra. Recall that a coderivation of degree d of C^{•} is a linear mapping D : C^{•} ⟶ C^{•}[d] satisfying the co-Leibnitz rule

Δ(D(x)) = ∑ (D(a) ⊗ b + (−1)^{d·|a|}a ⊗ D(b))  (3.1)

if Δ(x) = ∑ a ⊗ b. Here Δ is the comultiplication in B^{•}. The coderivations form a fraded Lie algebra, to be denoted Coder(C^{•}). The bracket is given by

[D_{1}, D_{2}] = D_{1}D_{2} − (−1)^{|D_{1}||D_{2}|}D_{2}D_{1}  (3.2)

Let V be a vector space. Consider the cofree graded coalgebra F_{GCoass}(V [−1]). Set

𝔤_{V}^{•} = Coder(F_{GCoass}(V [−1]))  (3.3)

As a graded vector space,

𝔤_{V}^{•} = ⊕_{i≥0} Hom(V^{⊗i}, V),  (3.4)

where 𝔤^{i} = Hom(V^{⊗(i+1)}, V).

3.2. Lemma. Suppose we are given a map

f : V ⊗ V ⟶ V

It is associative if and only if, when considered as an element of 𝔤_{V}^{1} , it satisfies the equation

\frac{1}{2}[f, f] = 0

Given such f, if we extend it to the coderivation of degree 1 of

F_{Coass} (V [−1]) = ⊕_{i≥1} V^{⊗i}

its component

f_{3} : V^{⊗3} ⟶ V^{⊗2}

is given by

f_{3}(a ⊗ b ⊗ c) = f(a ⊗ b) ⊗ c − a ⊗ f(b ⊗ c),

by the co-Leibnitz rule. Therefore the component of \frac{1}{2}[f, f] = f ·f acting from V^{⊗3} to V, sends a ⊗ b ⊗ c to f(f(a ⊗ b) ⊗ c) − f(a ⊗ f(b ⊗ c)). The Lemma follows. △.

3.3. Let (A, f : A ⊗ A ⟶ A) be an associative algebra. Let us define the dg Lie algebra 𝔤_{f}^{•}, which is equal to

Coder(F_{GCoass}(A[1])) = ⊕_{i≥0} Hom(A^{⊗i}, A)  (3.5)

as a graded Lie algebra, and with the differential d_{f} = ad(f).

Here we consider f as an element of 𝔤_{f}^{1}, and by the previous Lemma, [f, f] = 0. Now, let f′ = f + h : A ⊗ A ⟶ A be another candidate for the multiplication.

We have

\frac{1}{2}[f + h, f + h] = [f, h] + \frac{1}{2}[h, h] = d_{f}h + \frac{1}{2}[h, h]

Let M_{A} denote the set of associative multiplications on A.

More generally, define M_{A} as a functor Com^{+} ⟶ Sets which assigns to a commutative unital k-algebra B the set of associative multiplications on A ⊗ B.

We have proven

3.4. Lemma. Assume that A is finite dimensional. Then M_{A} is represented by an affine scheme. We have a canonical isomorphism of schemes

M_{A} = MC(𝔤_{f}^{•})

which takes f ∈ M_{A}(k) to 0 ∈ MC(𝔤_{f}^{•})(k).

△

Applying Theorem 2.5, we get

3.5. Theorem. We have a canonical isomorphism of formal commutative algebras

Ô_{M_{A};f} = (H^{0}(C_{Cocom}(𝔤_{f}^{≥1})^{+}))^{∗}  (3.6)

PART II. MILNOR RING AND BATALIN-VILKOVISKY ALGEBRAS

§1. Koszul complex and Milnor ring

1.1. Koszul complex. Let A be a commutative ring, and f_{1}, …, f_{r} a sequence of its elements.

We denote by K^{•}(A; f_{1}, …, f_{r}) the complex

0 ⟶ Λ^{r}(A^{r}) ⟶ Λ^{r−1}(A^{r}) ⟶ … ⟶ A^{r} ⟶ A ⟶ 0  (1.1)

The exterior powers are over A. The differential is given by

d(a_{i_{1}} ∧ … ∧ a_{i_{p}}) = ∑_{j=1}^{p} (−1)^{j−1}f_{i_{j}}a_{i_{1}} ∧ … ∧ â_{i_{j}} ∧ … ∧ a_{i_{p}}  (1.2)

Here for a ∈ A and 1 ≤ i ≤ r, we denote by a_{i} ∈ A^{r} the element (0, …, a, …, 0), with a on i-th place. We consider this complex as living in degrees from −r to 0.

We have

K^{•}(A; f_{1}, …, f_{r}) = K^{•}(A; f_{1}) ⊗_{A} … ⊗_{A} K^{•}(A; f_{r})  (1.3)

Equipped with the obvious wedge product, K^{•}(A; f_{1}, …, f_{r}) becomes a commutative dg A-algebra.

In fact, the differential (1.2) is the unique differential on the exterior algebra Λ^{•}(A^{r}), whose component A^{r} ⟶ A is given by

d(a_{1}, …, a_{r}) = ∑_{i=1}^{r} f_{i}a_{i},  (1.4)

and making it a commutative dg algebra.

1.2. Assume that (f_{1}, …, f_{r}) is a regular sequence, i.e. the operator of multiplication by f_{i} on A/(f_{1}, …, f_{i−1}) is injective, for all i. Then the Koszul complex K^{•}(A; f_{1}, …, f_{r}) is acyclic in negative degrees.

This is proved easily by induction on r, using (1.3), cf. [EGA III], Prop. 1.1.4.

Obviously,

H^{0}(K^{•}(A; f_{1}, …, f_{r})) = A/(f_{1}, …, f_{r})  (1.5)

1.3. From now on, until mentioned otherwise, the ring A will be equal to a polynomial algebra k[x_{1}, …, x_{n}], k being a fixed ground commutative ring. We denote by

T (A) = Der_{k}(A)

the Lie algebra of k-linear derivations of A. We identify T (A) with A^{n} using the basis ∂_{1}, …, ∂_{n}, where ∂_{i} := ∂/∂x_{i}.

We fix a function f ∈ A such that f(0) = 0. We denote by M^{•}(A; df) the Koszul complex

M^{•}(A; df) = K^{•}(A; ∂_{1}f, …, ∂_{n}f)  (1.6)

Thus,

M^{•}(A; df) = Λ^{•}T (A)  (1.7)

as a graded commutative algebra, with T (A) having the degree −1. The (−1)−component of the differential T (A) ⟶ A sends τ ∈ T (A) to τ(f), cf. the remark at the end of 1.1.

1.4. Lemma. The following conditions are equivalent:

(i) dim(A/(∂_{1}f, …, ∂_{n}f)) < ∞;

(ii) M^{•}(A; df) is acyclic in negative degrees;

(iii) (∂_{1}f, …, ∂_{n}f) is a regular sequence.

This is a standard fact of Commutative Algebra, and follows from [S], Ch. IV, B.2, Théorème 2, Ch. III, B.3.

If these equivalent conditions are fulfilled, we say that f has an isolated singularity at 0. The only non-zero cohomology space,

A/(∂_{1}f, …, ∂_{n}f) = H^{0}(M^{•}(A; df))  (1.8)

is called the Milnor ring of f at 0, cf. [M], §7.

Thus, the commutative dg algebra M^{•}(A; df) may be considered as a resolution of the Milnor ring.

1.5. Definition. A Gerstenhaber algebra is a commutative ℤ-graded algebra G^{•}, equipped with a bracket

[ , ] : G^{•} ⊗ G^{•} ⟶ G^{•}[1]

of degree 1, which makes G^{•}[−1] a graded Lie algebra.

The multiplication and the bracket must satisfy the following compatibility,

[a, b · c] = [a, b] · c + (−1)^{(|a|−1)|b|}b · [a, c]  (1.9)

for all homogeneous a, b, c.

Note the following equivalent form of (1.9),

[a · b, c] = a · [b, c] + (−1)^{(|c|−1)|b|}[a, c] · b  (1.10)

The equivalence of (1.9) and (1.10) is proved using the symmetry relations

a · b = (−1)^{|a||b|}b · a  (1.11)

and

[a, b] = −(−1)^{(|a|−1)(|b|−1)}[b, a]  (1.12)

1.6. Theorem. There exists a unique bracket of degree 1 on the algebra of polyvector fields Λ^{•}T (A), where to T (A) the degree −1 and to A the degree 0 is assigned, such that

(a) for a ∈ A, τ ∈ T (A), [τ, a] = τ(a);

(b) the bracket on T (A) coincides with the usual bracket of vector fields;

(c) together with the wedge product, this algebra becomes a Gerstenhaber algebra.

It is called the Schouten bracket, [Sch]. The Gerstenhaber algebra defined in this Theorem, will be called the Schouten algebra, and denoted T^{•}(A).

Now, let us identify Λ^{•}T (A) with M^{•}(A; df).

1.7. Theorem. Equipped with the Schouten bracket, the complex M^{•}(A; df)[−1] is a dg Lie algebra.

We have to prove that

d([a, b]) = [da, b] + (−1)^{|a|−1}[a, db]  (1.13)

for a ∈ Λ^{|a|}. Using (1.12), it is easy to see that (1.13) is true for (a, b) if and only if it is for (b, a).

1.8. Lemma. If (1.13) holds true for (a, b) and (a, c) then it holds true for (a, bc).

This Lemma is proved by a straightforward computation, which we omit. One uses (1.10) and

d(a · b) = da · b + (−1)^{|a|}a · db  (1.14)

△

Returning to the Theorem, every element in Λ^{•}T (A) is a sum of elements, which are either functions, or products of vector fields. Therefore, due to the Lemma, we have to prove (1.13) only in the case when a, b are either functions or vector fields. The only non-trivial identity comes out when both are vector fields. We have

d([τ_{1}, τ_{2}]) = [τ_{1}, τ_{2}](f),

τ_{i} ∈ T (A). On the other hand,

[dτ_{1}, τ_{2}] + (−1)^{−1−1}[τ_{1}, dτ_{2}] = [τ_{1}(f), τ_{2}] + [τ_{1}, τ_{2}(f)] =
= −τ_{2}τ_{1}(f) + τ_{1}τ_{2}(f) = [τ_{1}, τ_{2}](f)

The Theorem is proved. △

1.9. Let us call a degenerate Batalin-Vilkovisky algebra a commutative dg algebra G^{•}, together with a bracket G^{•} ⊗ G^{•} ⟶ G^{•}[1] of degree 1 making G^{•}[−1] a dg Lie algebra, such that with the differential forgotten, G^{•} is a Gerstenhaber algebra.

The previous considerations show that the complex M^{•}(A; df), together with the wedge product of polyvector fields and the Schouten bracket, is a degenerate BV algebra.

§2. Batalin-Vilkovisky algebras on Calabi-Yau manifolds

2.1. Definition. A Batalin-Vilkovisky algebra is a Gerstenhaber algebra B^{•}, together with a differential d of degree 1 on it, such that

(−1)^{|a|}[a, b] = d(a · b) − da · b − (−1)^{|a|}a · db  (2.1)

for all homogeneous a, b.

It follows from (2.1) that

d([a, b]) = [da, b] + (−1)^{|a|−1}[a, db],  (2.2)

i.e. B^{•}[−1] is a dg Lie algebra.

2.2. Let us return to the situation of the last Section. Consider the de Rham complex

Ω^{•}(A) : 0 ⟶ A ⟶^{d_{DR}} Ω^{1}(A) ⟶^{d_{DR}} … ⟶^{d_{DR}} Ω^{n}(A) ⟶ 0  (2.3)

Let us fix the top differential form

ω = dx_{1} ∧ … ∧ dx_{n}  (2.4)

Using ω, one gets the isomorphisms

ω_{∗} : Λ^{p}T (A) ⟶^{∼} Ω^{n−p}(A)  (2.5)

defined by

ω_{∗}(a) = aω  (2.6)

for a ∈ A = Λ^{0}T (A), and

ω_{∗}(τ_{1} ∧ … ∧ τ_{p}) = i_{τ_{1}} · . . . · i_{τ_{p}}(ω)  (2.7)

for p ≥ 1. Explicitely,

ω_{∗}(a∂_{i_{1}} ∧ … ∧ ∂_{i_{p}}) = (−1)^{(∑_{j=1}^{p} i_{j})−p}adx_{1} ∧ … ∧ d̂x_{i_{1}} ∧ … ∧ d̂x_{i_{p}} ∧ … ∧ dx_{n} (2.8)

for i_{1} < … < i_{p}.

Let us identify the shifted de Rham complex Ω^{•}(A)[n] with Λ^{•}T (A), using the isomorphisms (2.5). We can transfer the de Rham differential d_{DR} to Λ^{•}T (A). Here T (A) will have the degree −1.

2.3. Theorem. Equipped with the differential coming from −d_{DR}, the wedge product and the Schouten bracket, the algebra of polyvector fields Λ^{•}T (A) becomes a BV algebra.

We have to check the relation (2.1). First note that if (2.1) holds true for a pair (a, b) then it does for (b, a).

2.4. Lemma. Suppose that (2.1) is true for pairs (a, b), (b, c) and (ab, c). Then it holds true the pair (a, bc).

Indeed, we have

(−1)^{|a|}[a, bc] = (−1)^{|a|}[a, b] · c + (−1)^{|a||b|+|a|+|b|}b · [a, c],  (2.9)

by (1.9). On the other hand,

d(a(bc)) − da · bc − (−1)^{|a|}a · d(bc) =
= (−1)^{|a|+|b|}[ab, c] + d(ab) · c + (−1)^{|a|+|b|}ab · d(c)−
−da · bc − (−1)^{|a|}a · ((−1)^{|b|}[b, c] + db · c + (−1)^{|b|}b · dc)

(we have applied (2.1) for (ab, c) and (b, c))

= (−1)^{|a|+|b|}[ab, c] + ((−1)^{|a|}[a, b] + da · b + (−1)^{|a|}a · db) · c−
−da · bc − (−1)^{|a|+|b|}a · [b, c] − (−1)^{|a|}a · db · c

(we have applied (2.1) for (a, b))

= (−1)^{|a|+|b|}[ab, c] + (−1)^{|a|}[a, b] · c − (−1)^{|a|+|b|}a · [b, c]  (2.10)

The right hand sides of (2.9) and (2.10) are equal, by the relation (1.10). This proves the Lemma. △

In view of this Lemma, to prove the Theorem, it is enough to check (2.1) when a is either a function, or a vector field. Let us do that.

If both a and b are functions, (2.1) is trivial.

Let us check that

[a, b_{1}∂_{i_{1}} ∧ … ∧ b_{p}∂_{i_{p}}] =
= d(ab_{1}∂_{i_{1}} ∧ b_{2}∂_{i_{2}} ∧ … ∧ ∂_{i_{p}}) − a · d(b_{1}∂_{i_{1}} ∧ … ∧ b_{p}∂_{i_{p}}),  (2.11)

where a ∈ A, i_{1} < … < i_{j}. Using the definition, one checks that

d(b_{1}∂_{i_{1}} ∧ … ∧ b_{p}∂_{i_{p}}) = ∑_{j=1}^{p} (−1)^{j}∂_{i_{j}}(b_{1} · . . . · b_{p})∂_{i_{1}} ∧ … ∧ ˆ∂_{i_{j}} ∧ … ∧ ∂_{i_{p}} (2.12)

On the other hand, the repeated application of (1.9) gives

[a, b_{1}∂_{i_{1}} ∧ … ∧ b_{p}∂_{i_{p}}] = ∑_{j=1}^{p} (−1)^{j}b_{j}∂_{i_{j}}(a)b_{1}∂_{i_{1}} ∧ … ∧ b̂_{j}∂_{i_{j}} ∧ … ∧ b_{p}∂_{i_{p}} (2.13)

These two formulas easily imply (2.11). This completes the check of (2.1) for a ∈ A.

The check of (2.1) for a ∈ T (A) is similar; we leave it to the reader. The Theorem is proven. △

2.5. Now let us turn the function f on. First of all, note that when we identify Ω^{•}(A) with Λ^{•}T (A) using ω_{∗} as in (2.2), the multiplication by df becomes the Koszul differential in M^{•}(A; df), (1.6).

2.6. Theorem. The Schouten algebra T^{•}(A) (cf. 1.6) equipped with the differential coming from −d_{DR} + df, is a Batalin-Vilkovisky algebra.

2.7. Lemma. Let S^{•} be a Gerstenhaber algebra. Let d_{0}, d_{1} be two anticommuting differentials on S^{•}, such that (S^{•}, d_{0}) is a degenerate BV-algebra (cf. 1.9), and (S^{•}, d_{1}) is a BV-algebra. Then (S^{•}, d_{0} + d_{1}) is a BV-algebra.

The proof is obvious. The anticommutation of d_{0} and d_{1} is needed for the identity (d_{0} + d_{1})^{2} = 0. △

Applying the Lemma to the Schouten algebra T^{•}(A), to d_{0} induced from df, and d_{1} induced from −d_{DR}, we get the Theorem, in view of 2.5 and Theorems 1.7 and 2.3. △

The BV-algebra defined in Theorem 2.6 will be denoted M(A, ω; df).

2.8. Let us make a base change k ⟶ k[λ] where λ is an independent variable. Using 2.6, we get a BV-algebra over k[λ], M(A[λ], ω; λ · df). The degenerate BValgebra M^{•}(A; df) may be regarded as a ”limit” of M(A, ω; λ · df) at λ ⟶ ∞.

2.9. We can apply the same construction by taking as an algebra A a localization of the polynomial algebra k[x_{1}, …, x_{n}], and as a form ω an arbitrary generator

ω = c(x)dx_{1} ∧ … ∧ dx_{n},  (2.14)

of the A-module Ω^{n}(A). Here c(x) is a unit in A.

Theorem 2.3 remains true, with the same proof. We should apply the following modifications of the formulas (2.11) and (2.13).

d(a∂_{i}) = −∂_{i}(a) − a · ∂_{i} log c,  (2.15)

and

d(a∂_{i} ∧ b∂_{j}) = −(∂_{i}(ab) + ab · ∂_{i} log c)∂_{j} + (∂_{j}(ab) + ab · ∂_{j} log c)∂_{i}  (2.16)

Here

∂_{i} log c = ∂_{i}c · c^{−1}  (2.17)

Theorem 2.6 remains true, with the same proof.

2.10. Let A be an arbitrary commutative k-algebra, smooth over k. Assume that A is étale over a polynomial algebra B = k[x_{1}, …, x_{n}]. By definition, cf. [SGA 1], Exposé II, Déf. 1.1, this holds true for any smooth k-algebra, locally over Spec(A).

The derivations ∂/∂_{x_{i}} ∈ T (B) extend uniquely to derivations ∂_{i} ∈ T (A), which form a basis of T (A) as an A-module, and anticommute. Their duals dx_{i} form a basis of Ω^{n}(A).

Choose a generator

ω = cdx_{1} ∧ … ∧ dx_{n}  (2.17)

of Ω^{n}(A). It induces the isomorphisms

ω_{∗} : Λ^{•}T (A) ⟶^{∼} Ω^{n−•}(A)  (2.18)

Theorems 2.3 and 2.6 remain true, with the same proof. This follows from 2.9.

2.11. Now let us pass to the global situation. Suppose we are given a scheme X, smooth of relative dimension n over k. Assume that the canonical bundle Ω_{X}^{n} is trivial.

Let us choose a trivialization

ω : O_{X} ⟶^{∼} Ω_{X}^{n}  (2.19)

It induces the isomorphisms of graded O_{X}-modules

ω_{∗} : Λ^{•}T_{X} ⟶^{∼} Ω_{X}^{n−•}  (2.20)

2.12. Theorem. Given a closed one-form ϕ ∈ Γ(X; Ω_{X}^{1}), consider the Schouten algebra Λ^{•}T_{X}, equipped with the differential obtained, using the identification (2.18), from the differential

−d_{DR} + ϕ  (2.21)

on Ω_{X}^{•}. This way we get a sheaf of Batalin-Vilkovisky algebras on X.

Indeed, the statement is local, and we can assume we are in the situation 2.10. Now, consider the situation of Theorem 1.7, with df replaced by an arbitrary oneform ϕ ∈ Ω^{1}(A). In the corresponding complex M^{•}(A; ϕ), the differential is equal to the multiplication by ϕ, if we use the identification (2.18).

In particular the differential T (A) ⟶ A is given by

τ ↦ ⟨τ, ϕ⟩  (2.22)

The conclusion of Theorem 1.7 remains true in these more general assumptions. Indeed, we can use the same Lemma 1.8, and replace the last part of the proof of 1.7 by the Lemma below.

2.13. Lemma. For all τ_{1}, τ_{2} ∈ T (A), and ϕ ∈ Ω^{1}(A), we have

⟨[τ_{1}, τ_{2}], ϕ⟩ = τ_{1}(⟨τ_{2}, ϕ⟩) − τ_{2}(⟨τ_{1}, ϕ⟩)  (2.23)

In fact, we can assume that τ_{1} = a∂_{i}, τ_{2} = b∂_{j} and ϕ = dx_{k}. Then the left hand side of (2.23) is equal to

⟨a∂_{i}(b)∂_{j} − b∂_{j}(a)∂_{i}, dx_{k}⟩ = a∂_{i}(b)δ_{jk} − b∂_{j}(a)δ_{ik},

which is equal to the right hand side of (2.23). △

Using this generalization of Theorem 1.7, we get a generalization of Theorem 2.6, with df replaced by an arbitrary closed one form ϕ, and A as in 2.10. (The closedness of ϕ is necessary and sufficient condition for the equality (−d_{DR} + ϕ)^{2} = 0.) This generalization of 2.6 implies, in turn, Theorem 2.12. △

The sheaf of BV algebras defined in this Theorem, will be denoted M^{•}(X, ω; ϕ).

§3. Recollections on D-modules

For more details about the generalities below, cf. [BD1], Chapter 2, §1, and [B].

3.1. From now on, we assume that our ground ring k is a field of characteristic 0. Let X be a scheme, smooth over k, of dimension n. We will consider the categories Mod(D_{X})^{ℓ} and Mod(D_{X})^{r} of left (resp., right) D_{X}-modules (:= sheaves of left (resp., right) D_{X}-modules, quasicoherent over O_{X}).

The sheaf of rings D_{X} is generated by two O_{X}-modules, O_{X} and T_{X}, and relations

τ · a − a · τ = τ(a)  (3.1)

and

τ · τ′ − τ′ · τ = [τ, τ′],  (3.2)

for τ, τ′ ∈ T_{X}, a ∈ O_{X}.

Therefore, a left D_{X} module is a quasicoherent O_{X}-module M, equipped with the left multiplication by T_{X}, such that

a(τm) = (aτ) · m  (3.3a)
τ(am) = a(τm) + τ(a) · m  (3.3b)
τ(τ′m) − τ′(τm) = [τ, τ′] · m,  (3.3c)

for m ∈ M.

Similarly, a right D_{X}-module is a quasicoherent O_{X}-module N, equipped with the right multiplication by T_{X}, such that

(na)τ = n · (aτ)  (3.4a)
(nτ)a = n(aτ) + n · τ(a)  (3.4b)
(nτ)τ′ − (nτ′)τ = n · [τ, τ′],  (3.4c)

for n ∈ N. Here we agree that

n · a = a · n  (3.5)

3.2. For M, M′ ∈ Mod(D_{X})^{ℓ}, the tensor product M ⊗_{O_{X}} N admits a canonical structure of a left D_{X}-module, defined by the Leibnitz rule

τ · (m ⊗ m′) = τm ⊗ m′ + m ⊗ τm′,  (3.6)

for τ ∈ T_{X}. This way we get a tensor structure on Mod(D_{X})^{ℓ} (it is denoted by ⊗^{!} in [BB], cf. op. cit., 2.2.1).

The sheaf Hom_{O_{X}} (M, M′) is also canonically a left D_{X}-module, with the T_{X}- action given by

(τf)(m) = τ(f(m)) − f(τm),  (3.7)

for f ∈ Hom_{O_{X}} (M, M′). The rule (3.7) is invented in such a way that the canonical morphism,

Hom_{O_{X}} (M, M′) ⊗ M ⟶ M′  (3.8)

would be a morphism of D_{X}-modules.

3.3. For M ∈ Mod(D_{X})^{ℓ}, N ∈ Mod(D_{X})^{r}, the tensor product M ⊗ N is canonically a right D_{X}-module, with the multiplication by T_{X} defined by

(m ⊗ n) · τ = m ⊗ nτ − τm ⊗ n  (3.9)

For N′ ∈ Mod(D_{X})^{r}, the O_{X}-module Hom_{O_{X}} (N, N′) is canonically a left D_{X}- module, with the multiplication by T_{X} given by

(τf)(n) = f(nτ) − f(n) · τ  (3.10)

Again, this rule is written in such a way that the canonical adjunction morphism is a morphism of right D_{X}-modules.

3.4. The structure sheaf O_{X} has an obvious structure of a left D_{X}-module. The canonical bundle Ω_{X}^{n} is canonically a right D_{X}-module. The multiplication

by T_{X} is given by

ω · τ = −Lie_{τ} (ω),  (3.11)

for ω ∈ Ω_{X}^{n}.

We have the Cartan formula

i_{τ} ∘ d + d ∘ i_{τ} = Lie_{τ}  (3.12)

Consequently,

ω · τ = −d(i_{τ} ω)  (3.13)

Let us check the relations (3.4a-c). We have

ω · (aτ) = −di_{aτ} (ω) = −di_{τ} (aω) = (ωa) · τ,

which proves (3.4a). We have

ω · (aτ) = −Lie_{τ} (aω) = −τ(a)ω − aLie_{τ(ω)},

which proves (3.4b). The relation (3.4c) is obvious.

Using 3.3, we get the functors

Mod(D_{X})^{ℓ} ⟶ Mod(D_{X})^{r}, M ↦ M ⊗ Ω_{X}^{n},  (3.14)

and

Mod(D_{X})^{r} ⟶ Mod(D_{X})^{ℓ}, N ↦ Hom(Ω_{X}^{n}, N)  (3.15)

which are mutually inverse equivalences.

3.5. De Rham complex. We have the De Rham complex of our scheme X,

Ω_{X}^{•} : 0 ⟶ O_{X} ⟶ Ω_{X}^{1} ⟶ … ⟶ Ω_{X}^{n} ⟶ 0  (3.16)

We can regard Ω_{X}^{•} as a twisted version of the standard cochain complex of the Lie algebra of vector fields T_{X} with coefficients in the module of functions O_{X}. Namely, let us identify

Ω_{X}^{i} = Hom_{O_{X}} (Λ^{i}T_{X}, O_{X})  (3.17)

Thus, we get the identification

Ω_{X}^{•} = Hom_{O_{X}} (Λ^{•}T_{X}, O_{X}),  (3.18)

with the differential defined by

dω(τ_{1} ∧ … ∧ τ_{p}) = ∑_{i=1}^{p} (−1)^{i+1}τ_{i}(ω(τ_{1} ∧ … ∧ τ̂_{i} ∧ … ∧ τ_{p}))+

+ ∑_{1≤i<j≤p} (−1)^{i+j+1} ω([τ_{i}, τ_{j}] ∧ τ_{1} ∧ … ∧ τ̂_{i} ∧ … ∧ τ̂_{j} ∧ … ∧ τ_{p}),  (3.19)

for ω ∈ Hom(Λ^{p}T_{X}, O_{X}). cf. [KN], Chapter I, Prop. 3.11.

Twisted, since the bracket in T_{X} is not O_{X}-linear. The differentials are not O_{X}-linear, but are differential operators of the first order.

In the same sense, D_{X} is the twisted enveloping algebra of T_{X},

D_{X} = U_{O_{X}} T_{X}  (3.20)

More generally, for a left D_{X}-module M ∈ Mod(D_{X})^{ℓ}, we have the De Rham complex

DR^{ℓ}(M) : 0 ⟶ M ⟶ Ω_{X}^{1} ⊗ M ⟶ … ⟶ Ω_{X}^{n} ⊗ M ⟶ 0  (3.21)

Using the identifications (3.17), we can identify this complex with the twisted cochain complex of the Lie algebra T_{X} with coefficients in M,

DR^{ℓ}(M) = Hom_{O_{X}} (Λ^{•}T_{X}, M),  (3.22)

the differential being defined by the same formula (3.19).

For a right D_{X}-module N ∈ Mod(D_{X})^{r}, we have the De Rham complex

DR(N)^{r} : 0 ⟶ N ⊗ Λ^{n}T_{X} ⟶ … ⟶ N ⊗ T_{X} ⟶ N ⟶ 0  (3.23)

We consider it as living in degrees from −n to 0. It may be regarded as the twisted version of the standard chain complex of the Lie algebra T_{X} with coefficients in N,

DR(N)^{r} = N ⊗_{O_{X}} Λ^{•}T_{X}  (3.24)

The differential acts as follows

d(n ⊗ τ_{1} ∧ … ∧ τ_{p}) = ∑_{i=1}^{p} (−1)^{i−1}nτ_{i} ⊗ τ_{1} ∧ … ∧ τ̂_{i} ∧ … ∧ τ_{p}+

+ ∑_{1≤i<j≤p} (−1)^{i+j}n ⊗ [τ_{i}, τ_{j}] ∧ … ∧ τ̂_{i} ∧ … ∧ τ̂_{j} ∧ … ∧ τ_{p}  (3.25)

Let us check that (3.25) is a well defined map. For example, we have to check that

d(n ⊗ aτ_{1} ∧ τ_{2}) = d(n ⊗ τ_{1} ∧ aτ_{2}),  (3.26)

for a ∈ O_{X}. We have

d(n ⊗ aτ_{1} ∧ τ_{2}) = naτ_{1} ⊗ τ_{2} − nτ_{2} ⊗ aτ_{1} − n ⊗ [aτ_{1}, τ_{2}] =
= nτ_{1}a ⊗ τ_{2} − nτ_{1}(a) ⊗ τ_{2} − nτ_{2} ⊗ aτ_{1} − n ⊗ a[τ_{1}, τ_{2}] + n ⊗ τ_{2}(a)τ_{1}

We have used the formula

[aτ_{1}, τ_{2}] = a[τ_{1}, τ_{2}] − τ_{2}(a)τ_{1}  (3.27)

which is a particular case of (1.10). On the other hand,

d(n ⊗ τ_{1} ∧ aτ_{2}) = nτ_{1} ⊗ aτ_{2} − naτ_{2} ⊗ τ_{1} − n ⊗ [τ_{1}, aτ_{2}] =
= nτ_{1} ⊗ aτ_{2} − nτ_{2}a ⊗ τ_{1} + nτ_{2}(a) ⊗ τ_{1} − n ⊗ a[τ_{1}, τ_{2}] − n ⊗ τ_{1}(a)τ_{2}

which is equal to the left hand side of (3.26). In the same manner, the analog of (3.26) with an arbitrary number of τ’s is checked. This proves that the map (3.25) is well defined.

The complexes DR^{ℓ} and DR^{r} are mapped to each other under the canonical equivalence between left and right D_{X}-modules, (3.14), (3.15).

§4. Connections and Batalin-Vilkovisky structures

4.1. We keep the assumptions of the last Section. Let us call a Calabi-Yau data on X an integrable connection

∇ : Ω_{X}^{n} ⟶ Ω_{X}^{1} ⊗ Ω_{X}^{n}  (4.1)

on the canonical bundle Ω_{X}^{n}.

In other words, a CY data is a structure of a left D_{X}-module on Ω_{X}^{n}. By 3.4, this is the same as a structure of a right D_{X}-module on the structure sheaf O_{X} = Hom_{O_{X}} (Ω_{X}^{n}, Ω_{X}^{n}), i.e. a map

∇^{r} : T_{X} = O_{X} ⊗_{O_{X}} T_{X} ⟶ O_{X}  (4.2)

such that

∇^{r}(aτ) = a∇^{r}(τ) − τ(a),  (4.3)

for a ∈ O_{X}, τ ∈ T_{X}.

4.2. Let us call a Batalin-Vilkovisky data on X a differential on the Schouten algebra Λ^{•}T_{X}, making it a BV algebra.

4.3. Theorem. Given a CY data on X, the De Rham complex DR^{r}(O_{X}) of the corresponding right D_{X}-module O_{X} is a BV data on X.

Conversely, given a BV data on X, the component T_{X} ⟶ O_{X} of the corresponding differential on Λ^{•}T_{X} gives a map (4.2) which defines a CY data on X.

These two procedures establish the mutually inverse bijections between the sets of CY data and BV data on X.

Suppose we are given a CY data. We have to show the DR^{r}(O_{X}) is a BV algebra. This is a generalization of Theorem 2.12. We have to check the identity (2.1). We will use the formula

d(τ_{1} ∧ … ∧ τ_{p}) = ∑_{j=1}^{p} (−1)^{j−1}∇^{r}(τ_{j})τ_{1} ∧ … ∧ τ̂_{j} ∧ … ∧ τ_{p}+

+ ∑_{i<j} (−1)^{i+j}[τ_{i}, τ_{j}] ∧ τ_{1} ∧ … ∧ τ̂_{i} ∧ … ∧ τ̂_{j} ∧ … ∧ τ_{p}  (4.4)

which is a version of (3.25).

Application of Lemma 2.4 shows that it is enough to check (2.1) when a is either a function or a vector field. If both a, b are functions, (2.1) is trivial.

Let a ∈ O_{X}. We have

[a, τ_{1} ∧ … ∧ τ_{p}] = ∑_{j=1}^{p} (−1)^{p}τ_{j}(a)τ_{1} ∧ … ∧ τ̂_{j} ∧ … ∧ τ_{p}

On the other hand, it follows from (4.4) that

d(aτ_{1} ∧…∧τ_{p}) = ∇^{r}(aτ_{1})∧τ_{2} ∧…∧τ_{p} +∑_{j=2}^{p} (−1)^{j−1}∇^{r}(τ_{j})aτ_{1} ∧…∧τ̂_{j} ∧…∧τ_{p}+

+ ∑_{1<j} (−1)^{1+j}[aτ_{1}, τ_{j}]∧τ_{2}∧…∧τ̂_{j}∧…∧τ_{p}+ ∑_{1<i<j} (−1)^{i+j}a[τ_{i}, τ_{j}]∧τ_{1}∧…∧τ̂_{i}∧…∧τ̂_{j}∧…∧τ_{p} =

= (a∇^{r}(τ_{1}) − τ_{1}(a)) ∧ τ_{2} ∧ … ∧ τ_{p} + ∑_{j=2}^{p} (−1)^{j−1}a∇^{r}(τ_{j})τ_{1} ∧ … ∧ τ̂_{j} ∧ … ∧ τ_{p}+

+ ∑_{1<j} (−1)^{1+j}(a[τ_{1}, τ_{j}] − τ_{j}(a)) ∧ τ_{2} ∧ … ∧ τ̂_{j} ∧ … ∧ τ_{p}+

+ ∑_{1<i<j} (−1)^{i+j}a[τ_{i}, τ_{j}] ∧ τ_{1} ∧ … ∧ τ̂_{i} ∧ … ∧ τ̂_{j} ∧ … ∧ τ_{p}

Subtracting from this a · d(τ_{1} ∧ … ∧ τ_{p}), we get (2.1) in this case.

It remains to check the following particular case of (2.1),

−[τ_{0}, τ_{1} ∧ … ∧ τ_{p}] = d(τ_{0} ∧ … ∧ τ_{p}) − dτ_{0} ∧ τ_{1} ∧ … ∧ τ_{p} + τ_{0} ∧ d(τ_{1} ∧ … ∧ τ_{p}) (4.5)

The left hand side of (4.5) is equal to

∑_{j=1}^{p} (−1)^{j}[τ_{0}, τ_{j}] ∧ τ_{1} ∧ … ∧ τ̂_{j} ∧ … ∧ τ_{p}

The right hand side is equal to

∑_{j=0}^{p} (−1)^{j}∇^{r}(τ_{j}) ∧ τ_{0} ∧ … ∧ τ̂_{j} ∧ … ∧ τ_{p} + ∑_{1≤j} (−1)^{j}[τ_{0}, τ_{j}] ∧ τ_{0} ∧ … ∧ τ̂_{j} ∧ … ∧ τ_{p}+

+ ∑_{1≤i<j} (−1)^{i+j}[τ_{i}, τ_{j}] ∧ τ_{0} ∧ … ∧ τ̂_{i} ∧ … ∧ τ̂_{j} ∧ … ∧ τ_{p} − ∇^{r}(τ_{0}) ∧ τ_{1} ∧ … ∧ τ_{p}+

+ ∑_{j=1}^{p} (−1)^{j−1}∇^{r}(τ_{j})τ_{0}∧…∧τ̂_{j}∧…∧τ_{p}+ ∑_{1≤i<j} (−1)^{i+j}τ_{0}∧[τ_{i}, τ_{j}]∧τ_{1}∧…∧τ̂_{i}∧…∧τ̂_{j}∧…∧τ_{p}

which is equal to the left hand side. This proves (4.5) and completes the construction of the arrow α : (CY data) ⟶ (BV data).

Conversely, given a BV data, a particular case of (2.1) is

[a, τ] = ∇^{r}(aτ) − a∇^{r}(τ)

which is nothing but (4.3). This gives the arrow β : (BV data) ⟶ (CY data). It is clear that the composition βα is the identity. On the other hand, given a BV data, the differential in Λ^{•}T_{X} is uniquely determined by its last component, and (2.1). This implies that αβ is the identity.

The Theorem is proven. △

4.4. Let A be a commutative D_{X}-algebra, i.e. a commutative (implied: associative, unital) algebra in the category of Mod(D_{X})^{ℓ} , with the ⊗^{!} tensor structure, cf. 3.2. On the sheaf of A-modules

T_{A} := A ⊗_{O} T_{X},  (4.6)

one defines a Lie algebra structure by

[aτ, bτ′] = aτ(b)τ′ − bτ′(a)τ + ab[τ, τ′],  (4.7)

Here by aτ we denoted a ⊗ τ, for a ∈ A, τ ∈ T_{X}, and by τ(a) we denoted τ · a ∈ A, defined by the left D_{X}-module structure on A.

The given O_{X}-linear map T_{X} ⟶ Der(A) is extended by linearity to the A-linear map

T_{A} ⟶ Der(A)

which defines on A a structure of a module over the Lie algebra T_{A}. The action of T_{A} on A will also be denoted by τ(a), a ∈ A, τ ∈ A. We have

[τ, aτ′] = a[τ, τ′] + τ(a)τ′  (4.8)

for all τ, τ′ ∈ T_{A}, a ∈ A. Thus, T_{A} is a Lie algebroid over A, in the terminology of [BD2], 3.5.10.

Let Λ^{•}(T_{A}) denote the exterior algebra of T_{A} over A. We regard it as a graded commutative algebra, with A (resp. T_{A}) having degree 0 (resp. −1).

We have the following generalization of Theorem 1.6.

4.5. Theorem. There exists a unique Lie bracket of degree 1 on Λ^{•}(T_{A}) such that

(a) for a ∈ A, τ ∈ T_{A}, [τ, a] = τ(a);

(b) the bracket on T_{A} is given by (4.7);

(c) equipped with this bracket, Λ^{•}(T_{A}) becomes a Gerstenhaber algebra.

4.6. Let D_{A} denote the sheaf of associative algebras generated by the algebra A and the sheaf T_{X}, subject to relations

(a) τ · a − a · τ = τ(a);

(b) τ · τ′ − τ′ · τ = [τ, τ′].

We have a canonical filtration {D_{A}^{≤i}} on D_{A}, with the graded algebra

gr^{•}D_{A} = S^{•}(T_{A}),  (4.9)

the symmetric algebra over A.

One has a canonical isomorphism of left A-modules

D_{A} = A ⊗_{O} D_{X}  (4.10)

We have canonical embeddings of O_{X}-algebras

D_{X} ↪ D_{A}; A ↪ D_{A}

(We have just described the enveloping algebra construction from [BB], 1.2.5.)

4.7. The algebra A has an obvious structure of a left D_{A}-module. Assume we are given a structure of a right D_{A}-module on A which induces an obvious A-module structure. After restriction, A becomes a right D_{X}-module, such that

(a · b) · τ = a · (b · τ) − b · τ(a),  (4.11)

for all a, b ∈ A, τ ∈ T_{X}. Define a map ∇^{r} : T_{A} ⟶ A by

∇^{r}(aτ) = a · τ  (4.12)

We have

∇^{r}(aν) = a∇^{r}(ν) − ν(a)  (4.13)

for all a ∈ A, ν ∈ T_{A}.

4.8. Lemma. The previous definitions establish bijections between the data (i)

— (iv) below.

(i) The structures of a right D_{A}-module on A, inducing the obvious A-module structure.

(ii) The structures of a right D_{X}-module on A inducing the given O_{X}-module structure, such that (4.11) is satisfied.

(iii) The maps

∇^{r} : T_{X} ⟶ A

such that (4.13) is satisfied, for all a ∈ O_{X}, ν ∈ T_{X}.

(iv) The maps

∇^{r} : T_{A} ⟶ A

such that (4.13) is satisfied.

A structure described in this Lemma will be called a CY_{A}-structure.

4.9. Theorem. Given a CY_{A}-structure, there exists a unique differential of degree 1 on the Gerstenhaber algebra Λ^{•}T_{A}, whose first component coincides with (4.13) and making Λ^{•}T_{A} a Batalin-Vilkovisky algebra.

Let us call a differential on Λ^{•}T_{A} making it a BV algebra a BV_{A}-structure. Given a BV_{A}-structure, the first component of the differential is a map T_{A} ⟶ A satisfying (4.13).

4.10. Theorem. The above constructions establish the two inverse bijections between the set of CY_{A}-data and the set of BV_{A}-data.

This is a generalization of Theorem 4.3. The proof is the same. On just notes that in the proof of 4.3, we have used only the identity (4.13).

4.11. What is a Gerstenhaber algebra (over X) living in degrees −1, 0? It is the same as a sheaf of commutative algebras A^{0}, a sheaf of Lie algebras A^{−1}, which is also an A^{0}-module, and acts on A^{0} by derivations. The corresponding map A^{−1} ⟶ Der(A^{0}) should be A^{0}-linear, and the identity (4.8) should hold for all a ∈ A^{0}, τ ∈ A^{−1}. In other words, A^{−1} is a Lie algebroid over A^{0}.

Let Gerst denote the category of Gerstenhaber algebras over X, and let

Gerst^{[−1,0]} ⊂ Gerst  (4.14)

be the full subcategory consisting of Gerstenhaber algebras living in degrees −1, 0. We claim that the obvious trucation functor

t : Gerst ⟶ Gerst^{[−1,0]}  (4.15)

admits a left adjoint

S : Gerst^{[−1,0]} ⟶ Gerst  (4.16)

In fact, given A ∈ Gerst^{[−1,0]}, set

S(A) = S_{A^{0}}^{•}(A^{−1}[1]),  (4.17)

the symmetric algebra over A^{0}. In other words, as a graded algebra

S(A) = Λ_{A^{0}}^{•}(A^{−1}),  (4.18)

with A^{−1} having degree −1.

4.12. Theorem. There exists a unique Lie bracket of degree 1 on S(A), making it a Gerstenhaber algebra, such that the obvious embedding A ↪ S(A) is a morphism of graded Lie algebras.

This is a version of Theorems 4.5 and 1.6. This bracket is called the Schouten bracket. This way we get a structure of a Gerstenhaber algebra on S(A), called the Schouten algebra of A.

The functor S is the left adjoint to (4.15). Obviously, the composition t ∘ S is the identity.

4.13. Given A ∈ Gerst^{[−1,0]}, assume we have a BV structure on S(A), i.e. a differential of degree 1 on this algebra, making it a BV algebra. Its first component is a mapping

∇^{r} : A^{−1} ⟶ A^{0}  (4.19)

such that

∇^{r}(aτ) = a∇^{r}(τ) − τ(a)  (4.20)

for all a ∈ A^{0}, τ ∈ A^{−1}.

Conversely, given a mapping (4.19) satisfying (4.20), there exists a unique extension of this mapping to a differential d on S(A), making it a BV algebra. In fact, the uniqueness of d is obvious. To prove the existence, one defines d by the formula (4.4); the computations after (4.4) show that we get indeed a BV structure.

Thus, we have proven

4.14. Theorem. The above constructions establish two inverse bijections between the set of mappings (4.19) satisfying (4.20) and the set of BV structures on S(A). △

This Theorem generalizes Theorems 4.10 and 4.3.

References

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[KN] S. Kobayashi, K. Nomizu, Foundations of differential geometry, Volume I, Interscience Publishers, New York-London, 1963.

[M] J. Milnor, Singular points of complex hypersurfaces, Princeton University Press, Princeton, New Jersey, 1968.

[R] Z. Ran, Thickening Calabi-Yau spaces, in: Mirror Symmetry II, AMS/IP Stud. Adv. Math., 1, Amer. Math. Soc., Providence, Rhode Island, 1997, 393-400.

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Max-Planck-Institut für Mathematik, Gottfried-Claren-Straße 26, 53225 Bonn e-mail: vadik@mpim-bonn.mpg.de