CHIRAL DE RHAM COMPLEX

Fedor Malikov, Vadim Schechtman and Arkady Vaintrob

Introduction

0.1. The aim of this note is to define certain sheaves of vertex algebras on smooth manifolds. In this note, ”vertex algebra” will have the same meaning as in Kac’s book [K]. Recall that these algebras are by definition ℤ/(2)-graded. ”Smooth manifold” will mean a smooth scheme of finite type over ℂ.

For each smooth manifold X, we construct a sheaf Ω_{X}^{ch}, called the chiral de Rham complex of X. It is a sheaf of vertex algebras in the Zarisky topology, i.e. for each open U ⊂ X, Γ(U; Ω_{X}^{ch}) is a vertex algebra, and the restriction maps are morphisms of vertex algebras. It comes equipped with a ℤ-grading by fermionic charge, and the chiral de Rham differential d_{DR}^{ch}, which is an endomorphism of degree 1 such that (d_{DR}^{ch})^{2} = 0. One has a canonical embedding of the usual de Rham complex

(Ω_{X}, d_{DR}) ↪ (Ω_{X}^{ch}, d_{DR}^{ch})  (0.1)

The sheaf Ω_{X}^{ch} has also another ℤ-grading, by conformal weight, and the subsheaf Ω_{X} coincides with its conformal weight zero component. The wedge multiplication on Ω_{X} may be restored from the operator product in Ω_{X}^{ch}, see 1.3. The map (0.1) is a quasiisomorphism, cf. Theorem 4.4. Each component of Ω_{X}^{ch} of fixed conformal weight admits a canonical finite filtration whose graded factors are symmetric powers of the tangent bundle T_{X}, of the bundle of 1-forms Ω_{X}^{1}, and tensor products thereof.

The sheaf Ω_{X}^{ch} has the formal structure of the De Rham complex of a manifold with S^{1}-action. One may want to regard Ω_{X}^{ch} as a ”semiinfinite de Rham complex of the space of maps from the punctured formal disk to X”.

Similar sheaves exist in complex-analytic and C^{∞} settings.

If X is Calabi-Yau, then the sheaf Ω_{X}^{ch} has a structure of a topological vertex algebra (i.e. it admits N = 2 supersymmetry, cf. 2.1), cf. 4.5. (For an arbitrary X, the obstruction to the existence of this structure is expressible in terms of the first Chern class of T_{X}, cf. Theorem 4.2.)

One may hope that the vertex algebra RΓ(X; Ω_{X}^{ch}) defines the conformal field theory which is Witten’s ”A-model” associated with X.

0.2. One can also try to define a purely even sheaf O_{X}^{ch} of vertex algebras, which could be called chiral structure sheaf. Here the situation is more subtle than in the case of Ω_{X}^{ch}, where ”fermions cancel the anomaly”. On can define this sheaf for curves, cf. 5.5. If dim(X) > 1, then there exists a non-trivial obstruction of cohomological nature to the construction of O_{X}^{ch}. This obstruction can be expressed in terms of a certain homotopy Lie algebra, cf. §5 A.

However, one can define O_{X}^{ch} for the flag manifolds X = G/B (G being a simple algebraic group and B a Borel subgroup), cf. §5 B, C. This sheaf admits a structure of a 𝔤̂-module at the critical level (here 𝔤 = Lie(G) and 𝔤̂ is the corresponding affine Lie algebra). The space of global sections Γ(X; O_{X}^{ch}) is the irreducible vacuum 𝔤̂- module for 𝔤 = sl(2); conjecturally, it is also true for any 𝔤, cf. 5.13. The sheaf O_{X}^{ch} may be regarded as localization of Feigin-Frenkel bosonization. More generally, one may hope that ”chiral D-modules on X” provide localization of 𝔤̂-modules on the critical level.

0.3. A few words about the plan of the note. In §1 we recall the basic definitions, to fix terminology and notations. In §2, some ”free field representation” results are described. No doubt, they are all well known (although for some of them we do not know the precise reference). They are a particular case of the construction of Ω_{X}^{ch}, given in §3. In §4, we discuss the topological structure, and in §5 the chiral structure sheaf.

0.4. The idea of this note arose from reading the papers [LVW] and [LZ]. We have learned the first example of ”localization along the target space” from B. Feigin, to whom goes our deep gratitude.

The work was done while the authors were visiting the Max-Planck-Institut für Mathematik in Bonn. We are grateful to MPI for the excellent working conditions, and especially to Yu.I.Manin for highly stimulating environment.

§1. Recollections on vertex algebras

For more details on what follows, see [K]. For a coordinate-free exposition, see [BD1].

1.0. For a vector space V, V [[z, z^{−1}]] will denote the space of all formal sums ∑_{i∈ℤ} a_{i}z^{i}, a_{i} ∈ V. V ((z)) will denote the subspace of Laurent power series, i.e. the sums as above, with a_{i} = 0 for i << ∞. We denote by ∂_{z} the operator of differentiation by z acting on V [[z, z^{−1}]]. We set ∂_{z}^{(k)} := (∂_{z})^{k}/k!, for an integer k ≥ 0. For a(z) ∈ V [[z, z^{−1}]], we will also use the notation f(z)′ for ∂_{z}f(z), and ∫ a(z) for the coefficient at z^{−1} (i.e. the residue).

V^{∗} will denote the dual space. In the sequel, we will omit the prefix ”super” in the words like ”superalgebra”, etc. For example, ”a Lie algebra” will mean ”a Lie superalgebra”.

1.1. Let us define a vertex algebra following [K], 4.1. Thus, a vertex algebra is the data (V, Y, L_{−1}, 1). Here V is a ℤ/(2)-graded vector space V = V^{ev} ⊕ V^{odd}. The ℤ/(2)-grading is called parity; for an element a ∈ V, its parity will be denoted by ã. The space of endomorphisms End(V) inherits the ℤ/(2)-grading from V.

1 is an even element of V, called vacuum. Y is an even linear mapping

Y : V ⟶ End(V)[[z, z^{−1}]]  (1.1)

For a ∈ V, the power series Y (a) will be denoted a(z), and called the field corresponding to a. The coefficients of the series a(z) are called Fourier modes.

L_{−1} is an even endomorphism of V.

The following axioms must be satisfied.

Vacuum Axiom. L_{−1}(1) = 0; 1(z) = id (the constant power series). For all a ∈ V, a(z)(1)|_{z=0} = a (in particular, the components of a(z) at the negative powers of z act as zero on the vacuum).

Translation Invariance Axiom. For all a ∈ V, [L_{−1}, a(z)] = ∂_{z}a(z)

Locality Axiom. For all a, b ∈ V, (z − w)^{N}[a(z), b(w)] = 0 for N >> 0.

The meaning of this equality is explained in [K].

Given two fields a(z), b(z), their operator product expansion is defined as in [K], (2.3.7a)

a(z)b(w) = ∑_{j=1}^{N} \frac{c^{j}(w)}{(z − w)^{j}} + : a(z)b(w) :

where

: a(z)b(w) := a(z)_{+}b(w) + (−1)^{ãb̃}b(w)a(z)_{−}  (1.2)

Here

a(z)_{−} = ∑_{n≥0} a_{(n)}z^{−n−1}; a(z)_{+} = ∑_{n<0} a_{(n)}z^{−n−1}

for a(z) = ∑ a_{(n)}z^{−n−1}, cf. [K], (2.3.5), (2.3.3).

A morphism of vertex algebras (V, 1, L_{−1}, Y) ⟶ (V′, 1′, L′_{−1}, Y′) is an even linear map f : V ⟶ V′ taking 1 to 1′, such that f ∘ L_{−1} = L′_{−1} ∘ f, and for each a ∈ V, if a_{n} is a Fourier mode of the field a(z), we have f(a)_{n} ∘ f = f ∘ a_{n}.

1.2. A conformal vertex algebra (cf. [K], 4.10) is a vertex algebra (V, Y, L_{−1}, 1), together with an even element L ∈ V such that if we write the field L(z) as

L(z) = ∑_{n} L_{n}z^{−n−2},

the endomorphisms L_{n} satisfy the Virasoro commutation relations

[L_{n}, L_{m}] = (n − m)L_{n+m} + \frac{n^{3} − n}{12} · c · δ_{n,−m}  (1.3)

Here c is a complex number, to be called the Virasoro central charge of V (in [K] it is called the rank).

The component L_{−1} of the series (1.2) must coincide with the endomorphism L_{−1} from the definition of a vertex algebra.

The endomorphism L_{0} must be diagonalizable, with integer eigenvalues. Set V^{(n)} = {a ∈ V | L_{0}(a) = na}. We must have V = ⊕_{n∈ℤ} V^{(n)}. For a ∈ V^{(n)}, the number n is called the conformal weight of a and denoted |a|. This ℤ-grading induces a ℤ-grading on the space End(V), also to be called the conformal weight. By definition, an endomorphism has conformal weight n if it maps V^{(i)} to V^{(i+n)} for all i.

The conformal weight grading on V should be compatible with the parity grading, i.e. both gradings should come from a ℤ × ℤ/(2)-bigrading.

We require that if a has conformal weight n, then the field a(z) has the form

a(z) = ∑_{i} a_{i}z^{−i−n},  (1.4)

with a_{i} having the conformal weight −i.

We have the important Borcherds formula. If a, b are elements in a conformal vertex algebra, with |a| = n then

a(z)b(w) = ∑_{i} \frac{a_{i}(b)(w)}{(z − w)^{i+n}} + : a(z)b(w) :  (1.5)

cf. [K], Theorem 4.6.

The Virasoro commutation relations (1.3) are equivalent to the operator product

L(z)L(w) = \frac{c}{2(z − w)^{4}} + \frac{2L(w)}{(z − w)^{2}} + \frac{L(w)′}{z − w}  (1.6)

(we omit from now on the regular part).

1.3. Let V be a conformal vertex algebra, such that V^{(n)} = 0 for n < 0. For a ∈ V^{(n)}, b ∈ V^{(m)}, consider the operator product

a(z)b(w) = ∑_{i} \frac{a_{i}(b)(w)}{(z − w)^{i+n}}  (1.7)

We have |a_{i}(b)| = m−i. Therefore, a_{m}(b)z^{−n−m} is the most singular term in (1.5).

If n = m = 0 then a(z)b(w) is non-singular at z = w. Define the multiplication on the space V^{(0)} by the rule

a · b = (a(z)b(w)|_{z=w}1)|_{z=0}  (1.8)

This endows V^{(0)} with a structure of an associative and commutative algebra with unity 1.

1.4. Heisenberg algebra. In this Subsection, we will define a conformal vertex algebra, to be called Heisenberg vertex algebra.

Fix an integer N > 0. Let H_{N} be the Lie algebra which as a vector space has the base a_{n}^{i}, b_{n}^{i}, i = 1, …, N; n ∈ ℤ, and C, all these elements being even, with the brackets

[a_{m}^{j}, b_{n}^{i}] = δ_{ij}δ_{m,−n}C,  (1.9)

all other brackets being zero.

Our vertex algebra, to be denoted V_{N}, as a vector space is the vacuum representation of H_{N}. As an H_{N}-module, it is generated by one vector 1, subject to the relations

b_{n}^{i}1 = 0 if n > 0; a_{m}^{i}1 = 0 if m ≥ 0; C1 = 1  (1.10)

The mapping

P (b_{n}^{i}, a_{m}^{j}) ↦ P (b_{n}^{i}, a_{m}^{j}) · 1

identifies V_{N} with the ring of commuting polynomials on the variables b_{n}^{i}, a_{m}^{j}, n ≤ 0, m < 0, i, j = 1, …, N. We will use this identification below.

Let us define the structure of a vertex algebra on V_{N}. The ℤ/(2) grading is trivial: everything is even. The vacuum vector is 1.

The fields corresponding to the elements of V_{N} are defined by induction on the degree of a monomial. First we define the fields for the degree one monomials by setting

b_{0}^{i}(z) = ∑_{n∈ℤ} b_{n}^{i}z^{−n}; a_{−1}^{j}(z) = ∑_{n∈ℤ} a_{n}^{j}z^{−n−1}  (1.11)

Here b_{n}^{i}, a_{n}^{j} in the right hand side are regarded as operating on V_{N} by multiplication.

We set

b_{−n}^{i}(z) = ∂_{z}^{(n)}b_{0}^{i}(z), a_{−n−1}^{j}(z) = ∂_{z}^{(n)}a_{−1}^{i}(z)  (1.12)

for n > 0.

The fields corresponding to the monomials of degree > 1 are defined by induction, using the normal ordering. Let us call the operators b_{n}^{i}, n > 0, and a_{n}^{j}, n ≥ 0, acting on V_{N}, annihilation operators.

For x = b_{n}^{i} or a_{n}^{j}, n ∈ ℤ, and b ∈ End(V_{N}), the normal ordered product : xb : is given by

: xb := \{ \begin{matrix}bx & if x is an annihilation operator \\ xb & otherwise \end{matrix}  (1.13)

Define by induction

: x_{1} · . . . · x_{k} :=: x_{1} · (: x_{2} · . . . · x_{k} :) :,  (1.14)

for x_{p} = b_{n}^{i} or a_{n}^{j}, p = 1, …, k.

Given two series x(z) = ∑_{n∈ℤ} x_{n}z^{−n+p} and y(z) = ∑_{n∈ℤ} y_{n}z^{−n+q}, with x_{n} as above, we set

: x(z)y(w) := ∑_{n,m∈ℤ} : x_{n}y_{m} : z^{−n+p}w^{−m+q}  (1.15)

For any finite sequence x_{1}, …, x_{p}, where each x_{j} is equal to one of a_{n}^{i} or b_{n}^{i}, we define the series : x_{1}(z)x_{2}(z) · . . . · x_{p}(z) :∈ End(V_{N})[[z, z^{−1}]] by induction, as in (1.14). This expression does not depend on the order of x_{i}’s.

Given a monomial x_{1}·. . .·x_{p}1 ∈ V_{N}, with x_{i} as above, we define the corresponding field by

x_{1} · . . . · x_{p}(z) =: x_{1}(z) · . . . · x_{p}(z) :  (1.16)

Since every element of V_{N} is a finite linear combination of monomials as above, this completes the definition of the mapping (1.1).

We will use the shorthand notations

b^{i}(z) = b_{0}^{i}(z); a^{j}(z) = a_{−1}^{j}(z)  (1.17)

The operator products of these basic fields are

a^{j}(z)b^{i}(w) = \frac{δ_{ij}}{z − w} + (regular)  (1.18a)

b^{i}(z)b^{j}(w) = (regular); a^{i}(z)a^{j}(w) = (regular)  (1.18b)

where ”(regular)” means the part regular at z = w. These operator products are equivalent to the commutation relations (1.9).

Other operator products are computed by differentiation of (1.18), and using Wick theorem, cf. [K], 3.3.

One can say that the vertex algebra V_{N} is generated by the even fields b^{i}(z), a^{j}(z), of conformal weights 0 and 1 respectively, subject to the relations (1.18).

The Virasoro field is given by

L(z) = ∑_{i=1}^{N} : b^{i}(z)′ · a^{i}(z) :  (1.19)

The central charge is equal to 2N. Let us check this. Assume for simplicity that N = 1, and let us omit the index 1 at the fields a, b. Thus, we have

L(z) =: b(z)′a(z) :

Let us compute the operator product L(z)L(w) using the Wick theorem. We have

: b(z)′a(z) :: b(w)′a(w) := [b(z)′a(w)][a(z)b(w)′]+
+[b(z)′a(w)] : a(z)b(w)′ : +[a(z)b(w)′] : b(z)′a(w) :=

(we have b(z)′a(w) = a(z)b(w)′ = −1/(z − w)^{2})

= \frac{1}{(z − w)^{4}} − \frac{2 : b(w)′a(w) :}{(z − w)^{2}} − \frac{: b(w)′′a(w) : + : b(w)′a(w)′ :}{z − w}

Hence,

L(z)L(w) = \frac{1}{(z − w)^{4}} + \frac{2L(w)}{(z − w)^{2}} + \frac{L(w)′}{z − w}

which is (1.6) with c = 2.

1.5. Clifford algebra. Let Cl_{N} be the Lie algebra which as a vector space has the base ϕ_{n}^{i}, ψ_{n}^{i}, i = 1, …, N, n ∈ ℤ, and C, all these elements being odd, with the brackets

[ϕ_{m}^{j}, ψ_{n}^{i}] = δ_{ij}δ_{m,−n} · C  (1.20)

Clifford vertex algebra Λ_{N} is defined as in the previous subsection, starting with the odd fields

ψ^{i}(z) = ∑_{n∈ℤ} ψ_{n}^{i}z^{−n−1}; ϕ^{j}(z) = ∑_{n∈ℤ} ϕ_{n}^{j}z^{−n}  (1.21)

and repeating the definitions of loc. cit., with a (resp. b) replaced by ψ (resp. ϕ). One must put the obvious signs in the definition of the normal ordering. Thus, Λ_{N} is generated by the odd fields ϕ^{i}(z), ψ^{i}(z), subject to the relations

ϕ^{i}(z)ψ^{j}(w) = \frac{δ_{ij}}{z − w} + regular;  (1.22a)
ϕ^{i}(z)ϕ^{j}(w) = regular; ψ^{i}(z)ψ^{j}(w) = regular  (1.22b)

The Virasoro field is given by

L(z) = ∑_{i=1}^{N} : ϕ^{i}(z)′ · ψ^{i}(z) :  (1.23)

The central charge is equal to −2N.

1.6. If A and B are vertex algebras then their tensor product A ⊗ B admits a canonical structure of a vertex algebra, cf. [K], 4.3. The Virasoro element is given by

L_{A⊗B} = L_{A} ⊗ 1 + 1 ⊗ L_{B}  (1.24)

We will use in the sequel the tensor product of the Heisenberg and Clifford vertex algebras Ω_{N} = V_{N} ⊗ Λ_{N}. Its Virasoro central charge is equal to 0.

1.7. Let A be a vertex algebra. A linear map f : A ⟶ A is called derivation of A if for any a ∈ A,

f(a)(z) = [f, a(z)]  (1.25)

Note that an invertible map g : A ⟶ A is an automorphism of A iff

g(a)(z) = ga(z)g^{−1}  (1.26)

and (1.25) is the infinitesimal version of (1.26).

It follows from Borcherds formula that for every a ∈ A, the Fourier mode ∫ a(z) is a derivation, cf. Lemma 1.3 from [LZ]. Consequently, if the endomorphism exp(∫ a(z)) is well-defined, it is an automorphism of A.

§2. De Rham chiral algebra of an affine space

2.1. A topological vertex algebra of rank d is a conformal vertex algebra A of Virasoro central charge 0, equipped with an even element J of conformal weight 1, and two odd elements, Q, of conformal weight 1, and G, of conformal weight 2.

The following operator products must hold

L(z)L(w) = \frac{2L(w)}{(z − w)^{2}} + \frac{L(w)′}{z − w}  (2.1a)

J(z)J(w) = \frac{d}{(z − w)^{2}} ; L(z)J(w) = −\frac{1}{(z − w)^{3}} + \frac{J(w)}{(z − w)^{2}} + \frac{J(w)′}{z − w}  (2.1b)

G(z)G(w) = 0; L(z)G(w) = \frac{2G(w)}{(z − w)^{2}} + \frac{G(w)′}{z − w}; J(z)G(w) = −\frac{G(w)}{z − w} (2.1c)

Q(z)Q(w) = 0; L(z)Q(w) = \frac{Q(w)}{(z − w)^{2}} + \frac{Q(w)′}{z − w}; J(z)Q(w) = \frac{Q(w)}{z − w}  (2.1d)

Q(z)G(w) = −\frac{d}{(z − w)^{3}} + \frac{J(w)}{(z − w)^{2}} − \frac{L(w)}{z − w}  (2.1e)

Note the following consequence of (2.1e),

[Q_{0}, G(z)] = −L(z)  (2.2)

2.2. In this subsection, we will intorduce a structure of a topological vertex algebra of rank N on the vertex algebra Ω_{N} from 1.6. This topological vertex algebra will be called de Rham chiral algebra of the affine space 𝔸^{N}.

Recall that the Virasoro element is given by

L = ∑_{i=1}^{N} (b_{−1}^{i}a_{−1}^{i} + ϕ_{−1}^{i}ψ_{−1}^{i})  (2.3a)

Define the elements J, Q, G by

J = ∑_{i=1}^{N} ϕ_{0}^{i}ψ_{−1}^{i}; Q = ∑_{i=1}^{N} a_{−1}^{i}ϕ_{0}^{i}; G = ∑_{i=1}^{N} ψ_{−1}^{i}b_{−1}^{i}  (2.3b)

The corresponding fields are

L(z) = ∑ (: b^{i}(z)′a^{i}(z) : + : ϕ^{i}(z)′ψ^{i}(z) :)  (2.4a)

and

J(z) = ∑ : ϕ^{i}(z)ψ^{i}(z) :, Q(z) = ∑ : a^{i}(z)ϕ^{i}(z) :, G(z) = ∑ : ψ^{i}(z)b^{i}(z)′ :

(2.4b)

The relations (2.1) are readily checked using the Wick theorem.

2.3. Let us define the fermionic charge operator acting on Ω_{N}, by

F := J_{0} = ∑_{i} ∑_{n} : ϕ_{n}^{i}ψ_{−n}^{i} :  (2.5)

We have

F 1 = 0  (2.6)

and

[F, ϕ_{n}^{i}] = ϕ_{n}^{i}; [F, ψ_{n}^{i}] = −ψ_{n}^{i}; [F, a_{n}^{i}] = [F, b_{n}^{i}] = 0  (2.7)

We set

Ω_{N}^{p} = {ω ∈ Ω_{N}| F ω = pω}  (2.8)

Obviously,

Ω_{N} = ⊕_{p∈ℤ} Ω_{N}^{p}  (2.9)

We define an endomorphism d of the space Ω_{N} by

d := −Q_{0} = − ∑_{i,n} : a_{n}^{i}ϕ_{−n}^{i} :  (2.10)

(we could omit the normal ordering in the last formula since the letters a and ϕ commute anyway). We have d^{2} = 0. Indeed, by Wick theorem Q(z)Q(w) = regular, hence all Fourier modes of Q(z) (anti)commute. The map d is called chiral de Rham differential.

The map d increases the fermionic charge by 1, by (2.7). Thus, the space Ω_{N} equipped with the fermionic charge grading and the differential d, becomes a complex (infinite in both directions), called chiral de Rham complex of 𝔸^{N}.

Consider the usual algebraic de Rham complex Ω(𝔸^{N}) = ⊕_{p=0}^{N} Ω^{p}(𝔸^{N}) of the affine space 𝔸^{N}. We identify the coordinate functions with the letters b_{0}^{1}, …, b_{0}^{N}, and their differentials with the fermionic variables ϕ_{0}^{1}, …, ϕ_{0}^{N}.

Thus, we identify the commutative dg algebras

Ω(𝔸^{N}) = ℂ[b_{0}^{1}, …, b_{0}^{N}] ⊗ Λ(ϕ_{0}^{1}, …, ϕ_{0}^{N}),  (2.12)

the second factor being the exterior algebra. The grading is defined by assigning to the letters b_{0}^{i} (resp. ϕ_{0}^{j}) the degree 0 (resp. 1).

The usual de Rham differential is given by

d_{DR} = ∑_{i} a_{0}^{i}ϕ_{0}^{i},  (2.13)

as follows from the relations (1.9).

2.4. Theorem. The obvious embedding of complexes

i : (Ω(𝔸^{N}), d_{DR}) ⟶ (Ω_{N}, d)  (2.14)

is compatible with the differentials, and is a quasiisomorphism.

We identify the space Ω_{N} with the space of polynomials in the letters b_{n}^{i}, ϕ_{n}^{i} (n ≤ 0) and a_{n}^{i}, ψ_{n}^{i} (n < 0). One sees that on the subspace ℂ[b_{0}^{i}, ϕ_{0}^{i}], all the summands a_{n}^{i}ϕ_{−n}^{i} with n ≠ 0 act trivially. It follows that the map i is compatible with the differentials.

To prove that it is a quasiisomorphism, let us split d in two commuting summands d = d_{+} + d_{−} where

d_{+} = ∑_{i} ∑_{n≥0} a_{n}^{i}ϕ_{−n}^{i}, d_{−} = ∑_{i} ∑_{n<0} a_{n}^{i}ϕ_{−n}^{i}  (2.15)

We think of the space Ω_{N} as of the tensor product ℂ[a_{n}^{i}, ψ_{n}^{i}] ⊗ ℂ[b_{m}^{j}, ϕ_{m}^{j}]. The differential d_{−} acts trivially on the second factor, and on the first one it is the Koszul differential. So, the cohomology of d_{−} is ℂ[b_{m}^{j}, ϕ_{m}^{j}]_{m≤0}.

Now, we have to compute the cohomology of this space with respect to d_{+}. For this purpose, split d_{+} once again as d_{+} = d_{DR} + d_{+}^{>0}, and our space as ℂ[b_{0}^{j}, ϕ_{0}^{j}] ⊗ ℂ[b_{m}^{j}, ϕ_{m}^{j}]_{m<0}. The differential d_{+}^{>0} acts trivially on the first factor, and on the second one, it is the de Rham differential. Hence (by Poincaré lemma), taking the cohomology of d_{+}^{>0} kills all non-zero modes, and we are left precisely with the usual de Rham complex.

Alternatively, it follows from (2.2) that

[G_{0}, d] = L_{0}  (2.16)

The operator G_{0} commutes with L_{0}, and it follows from (2.16) that it gives a homotopy to 0 for the operator d on all the subcomplexes of non-zero conformal weight. Therefore, all cohomology lives in the conformal weight zero subspace. △

2.5. The vertex algebra Ω_{N} satisfies the assumptions of 1.3. The subspace Ω(𝔸^{N}) coincides with the conformal weight zero component of it. If we apply the definition of 1.3, we get the structure of a commutative algebra on Ω(𝔸^{N}) which is given by the usual wedge product of differential forms.

§3. Localization

3.1. Consider the Heisenberg vertex algebra V_{N} defined in 1.4. As in loc. cit., we will identify the space V_{N} with the space of polymonials ℂ[b_{−n}^{i}, a_{−m}^{j}]_{n≥0, m>0}. To simplify the notations below, let us denote the zero mode variables b_{0}^{i} by b^{i}.

Let A_{N} denote the algebra of polynomials ℂ[b^{1}, …, b^{N}]. The space A_{N} is identified with the subspace V_{N}^{(0)} ⊂ V_{N} of conformal weight zero. The space V_{N} has an obvious structure of an A_{N}-module. Let Â_{N} denote the algebra of formal power series ℂ[[b^{1}, …, b_{N}]]. Set

V̂_{N} = Â_{N} ⊗_{A_{N}} V_{N}  (3.1)

We are going to introduce a structure of a conformal vertex algebra on the space V̂_{N}. Let us define the map (1.1).

Let f(b^{1}, …, b^{N}) be a power series from Â_{N}. We claim that the expression f(b^{1}(z), …, b^{N}(z)) makes sense as an element of End(V̂_{N})[[z, z^{−1}]]. (We are grateful to Boris Feigin who has shown us a particular case of the following construction.) Let us express the power series b^{i}(z) as

b^{i}(z) = b^{i} + Δb^{i}(z)  (3.2)

Thus,

Δb^{i}(z) = ∑_{n>0} (b_{n}^{i}z^{−n} + b_{−n}^{i}z^{n})  (3.3)

Let us define f(b^{1}(z), …, b^{N}(z)) by the Taylor formula

f(b^{1}(z), …, b^{N}(z)) = ∑ Δb^{1}(z)^{i_{1}} · . . . · Δb^{N}(z)^{i_{N}} ∂^{(i_{1},... ,i_{N})}f(b^{1}, …, b^{N}) (3.4)

where

∂^{(i_{1},... ,i_{N})} = \frac{∂_{b^{1}}^{i_{1}}}{i_{1}!} · . . . · \frac{∂_{b^{N}}^{i_{N}}}{i_{N}!}  (3.5)

We will show that the series (3.4) gives a well-defined element of End(V̂_{N})[[z, z^{−1}]]. Let us write

Δb^{1}(z)^{i_{1}} · . . . · Δb^{N}(z)^{i_{N}} = ∑_{k} c_{k}^{i_{1},... ,i_{N}} z^{−k}

The coefficient c_{k}^{i_{1},... ,i_{N}} is an infinite sum of the monomials

b_{k_{1}}^{j_{1}} · . . . · b_{k_{I}}^{j_{I}} ,  (3.6)

with I = i_{1} + … + i_{N}, k_{1} + … + k_{I} = k. Pick an element v ∈ V̂^{N}. There exists M such that b_{l_{1}}^{j_{1}} · . . . · b_{l_{N}}^{j_{N}} v = 0 if ∑ l_{i} > M.

We have

k = ∑ k_{i} = ^{+} ∑ |k_{i}| − ^{−} ∑ |k_{i}|

where ^{+} ∑ (resp. ^{−} ∑) denotes the sum of all positive (resp. negative) summands. If b_{k_{1}}^{j_{1}} · . . . · b_{k_{I}}^{j_{I}} v ≠ 0, then

+ ∑ |k_{i}| ≤ M  (3.7a)

On the other hand,

− ∑ |k_{i}| = ^{+} ∑ |k_{i}| − k ≤ M − k  (3.7b)

There exists only a finite number of tuples (k_{1}, …, k_{I}) satisfying (3.7a) and (3.7b). Therefore, c_{k}^{i_{1},... ,i_{N}} are well-defined endomorphisms of V̂_{N}.

We have

f(b^{1}(z), …, b^{N}(z)) = ∑_{k} ( ∑_{i_{1},... ,i_{N}} c_{k}^{i_{1},... ,i_{N}} ∂^{(i_{1},... ,i_{N})}f(b^{1}, …, b^{N})) z^{−k} (3.8)

All numbers k_{i} are non-zero. Let I_{+} (resp., I_{−}) be the number of positive (resp. negative k_{i}’s). We have I_{−} ≤ ^{−} ∑ k_{i} ≤ M − k, hence M ≥ ^{+} ∑ k_{i} ≥ I_{+} = I − I_{−} ≥ I − M + k. Therefore,

I ≤ 2M − k  (3.9)

Therefore, when we apply the series (3.8) to the element v, only a finite number of terms in the sum over (i_{1}, …, i_{N}) survives. Therefore, the series (3.8) is a well-defined element of End(V̂_{N})[[z, z^{−1}]].

Every element of V̂_{N} is a finite sum of products g(a)f(b) where g(a) is a polynomial in the letters a and f(b) is a power series as above. We have already defined f(b)(z). The definition of g(a)(z) is the same as in the case of V_{N}. We define (g(a)f(b))(z) by

(g(a)f(b))(z) =: g(a)(z)f(b)(z) :  (3.10)

where the normal ordering is defined in (1.2). This completes the definition of the mapping

Y : V̂_{N} ⟶ End(V̂_{N})[[z, z^{−1}]]  (3.11)

The following version of the definition of the map (3.11) is helpful in practice. Every element c ∈ V̂_{N} is a limit of the elements of c_{i} ∈ V_{N} (in the obvious topology). We can regard the fields c_{i}(z) as the elements of End(V̂_{N})[[z, z^{−1}]]. The field c(z) is the limit of the fields c_{i}(z).

We define the vacuum and Virasoro element 1, L ∈ V̂_{N} as the image of the corresponding element of V_{N} under the natural map V_{N} ⟶ V̂_{N}.

3.2. Theorem. The construction of 3.1 defines a structure of a conformal vertex algebra on the space V̂_{N}.

This follows from [K], Theorem 4.5.

3.3. Let A_{N}^{an} ⊂ Â_{N} denote the subalgebra of power series, convergent in a neighbourhood of the origin. Set

V_{N}^{an} = A_{N}^{an} ⊗_{A_{N}} V_{N} ⊂ V̂_{N}  (3.12)

It is clear from the inspection of the Taylor formula (3.4) that for f(a, b) ∈ V_{N}^{an}, the Fourier modes of f(a, b)(z) which belong to End(V̂_{N}), respect the subspace V_{N}^{an}. Therefore, the conformal vertex algebra structure on V̂_{N} defined in 3.1 induces the structure of a conformal vertex algebra on V_{N}^{an}.

Let A_{N}^{sm} denote the algebra of germs of smooth (C^{∞}) functions. Set

V_{N}^{sm} = A_{N}^{sm} ⊗_{A_{N}} V_{N}  (3.13)

For every f ∈ A_{N}^{sm}, there exists a sequence {f_{i}} in A_{N}, converging to f together with all derivatives, up to any given order (this follows from the Stone-Weierstrass theorem). We can regard f_{i}(z) as elements of End(V_{N}^{sm})[[z, z^{−1}]]. Note that these fields are defined using the Taylor formula (3.4). It follows that the limit of {f_{i}(z)} exists in End(V_{N}^{sm})[[z, z^{−1}]]. We take this limit as the definition of the field f(z). This gives a conformal vertex algebra structure on V_{N}^{sm}.

3.4. Let f ∈ A_{N} be a nonzero polynomial. Let A_{N;f} denote the localization A_{N}[f^{−1}]. Set

V_{N;f} = A_{N;f} ⊗_{A_{N}} V_{N}  (3.14)

Consider the Taylor formula (3.4) applied to the function f^{−1}. We have evidently

[∂^{(i_{1},... ,i_{N})}f^{−1}](b^{1}, …, b^{N}) ∈ A_{N;f}

Therefore, the construction 3.1 provides a conformal vertex algebra structure on the space V_{N;f}.

Let X denote the affine space Spec(A_{N}). Let O_{X}^{ch} be the O_{X}-quasicoherent sheaf corresponding to the A_{N}-module V_{N}. We have just defined the structure of a conformal vertex algebra on the spaces V_{N;f} = Γ(U_{f}; O_{X}^{ch}) where U_{f} = Spec(A_{N;f}).

If U_{f} ⊂ U_{g} then the restriction map V_{N;g} ⟶ V_{N;f} is a morphism of conformal vertex algebras.

If U ⊂ X is an arbitrary open, we have U = ⋃ U_{f}, and

Γ(U; O_{X}^{ch}) = Ker(∏ V_{N;f} ⟶^{⟶} ∏ V_{N;fg})

Using this formula, we get a structure of a conformal vertex algebra on the space Γ(U; O_{X}^{ch}). Therefore, O_{X}^{ch} gets a structure of a sheaf of conformal vertex algebras.

3.5. We can add fermions to the picture. Consider the spaces Ω̂_{N} = Â_{N} ⊗_{A_{N}} Ω_{N}, Ω_{N}^{an} = A_{N}^{an} ⊗_{A_{N}} Ω_{N} ⊂ Ω̂_{N}, Ω_{N}^{sm} = A_{N}^{sm} ⊗_{A_{N}} Ω_{N}. The construction 3.1 provides a structure of a topological vertex algebras on these spaces.

Let X be as in 3.4; let Ω_{X}^{ch} denote the O_{X}-quasicoherent sheaf associated with the A_{N}-module Ω_{N}. The construction 3.4 provides a structure of a sheaf of topological vertex algebras of rank N on Ω_{X}^{ch}.

3.6. Now we want to study coordinate changes in our vertex algebras. Let X be the formal scheme Spf(ℂ[[b^{1}, …, b^{N}]]). Consider the formal N|N-dimensional superscheme X̃ = ΠT X (here T X is the total space of the tangent bundle, Π is the parity change functor). Thus, X̃ has the same underlying space as X, and the structure sheaf of X̃ coincides with the de Rham complex of X. On X̃, we have N even coordinates b^{1}, …, b^{N} and odd ones ϕ^{1} = db^{1}, …, ϕ^{N} = db^{N}.

To this superscheme X̃, with the above coordinates, we have assigned a (super)vertex algebra Ω̂_{N}, generated by the fields b^{i}(z), a^{i}(z) (even ones) and ϕ^{i}(z), ψ^{i}(z) (odd ones). The fields a^{i}(z) (resp. ψ^{i}(z)) correspond to the vector fields ∂_{b^{i}} (resp. ∂_{ϕ^{i}}) on X̃.

These fields satisfy the relations (cf. (1.18), (1.22))

a^{i}(w)b^{j}(w) = \frac{δ_{ij}}{z − w}  (3.15a)
b^{i}(z)b^{j}(w) = (regular); a^{i}(z)a^{j}(w) = (regular)  (3.15b)
ϕ^{i}(z)ψ^{j}(w) = \frac{δ_{ij}}{z − w}  (3.15c)
ϕ^{i}(z)ϕ^{j}(w) = (regular); ψ^{i}(z)ψ^{j}(w) = (regular)  (3.15d)
b^{i}(z)ϕ^{j}(w) = (regular); b^{i}(z)ψ^{j}(w) = (regular)  (3.15e)
a^{i}(z)ϕ^{j}(w) = (regular); a^{i}(z)ψ^{j}(w) = (regular)  (3.15f)

Consider an invertible coordinate transformation on X,

b̃^{i} = g^{i}(b^{1}, …, b^{N}); b^{i} = f^{i}(b̃^{1}, …, b̃^{N})  (3.16a)

where g^{i} ∈ ℂ[[b^{j}]]; f^{i} ∈ ℂ[[b̃^{j}]]. It induces the transformation of the odd coordinates ϕ^{i} = db^{i},

ϕ̃^{i} = \frac{∂g^{i}}{∂b^{j}} ϕ^{j}; ϕ^{i} = \frac{∂f^{i}}{∂b̃^{j}} ϕ̃^{j}  (3.16b)

(the summation over the repeating indices is tacitly assumed). The vector fields transform as follows,

∂b̃^{i} = \frac{∂f^{j}}{∂b̃^{i}} (g(b))∂_{b^{j}} + \frac{∂^{2}f^{k}}{∂b̃^{i}∂b̃^{l}} (g(b)) · \frac{∂g^{l}}{∂b^{r}} · ϕ^{r}∂_{ϕ^{k}}  (3.16c)

and

∂ϕ̃^{i} = \frac{∂f^{j}}{∂b̃^{i}} (g(b))∂_{ϕ^{j}}  (3.16d)

We want to lift the transformation (3.16a) to the algebra Ω_{N}. Define the tilded fields by

b̃^{i}(z) = g^{i}(b)(z)  (3.17a)
ϕ̃^{i}(z) = (\frac{∂g^{i}}{∂b^{j}} ϕ^{j})(z)  (3.17b)

ã^{i}(z) = (a^{j} \frac{∂f^{j}}{∂b̃^{i}} (g(b)))(z) + (\frac{∂^{2}f^{k}}{∂b̃^{i}∂b̃^{l}} (g(b))\frac{∂g^{l}}{∂b^{r}} ψ^{k}ϕ^{r})(z)  (3.17c)

ψ̃^{i}(z) = (\frac{∂f^{j}}{∂b̃^{i}} (g(b))ψ^{j})(z)  (3.17d)

3.7. Theorem. The fields b̃^{i}(z), ã^{i}(z), ϕ̃^{i}(z) and ψ̃^{i}(z) satisfy the relations (3.15).

We will use the relations

h(b)(z)a^{i}(w) = \frac{∂h/∂b^{i}(w)}{z − w}; a^{i}(z)h(b)(w) = \frac{∂h/∂b^{i}(w)}{z − w}  (3.18)

for each h ∈ Â_{N}, which follow from (3.15a) and the Wick theorem. Let us check (3.15a) for the tilded fields. We have

b̃^{i}(z)ã^{j}(w) = g^{i}(b)(z)a^{k} \frac{∂f^{k}}{∂b̃^{j}} (g(b))(w)+

+g^{i}(b)(z)\frac{∂^{2}f^{k}}{∂b̃^{j}∂b̃^{l}} (g(b))\frac{∂g^{l}}{∂b^{r}} ψ^{k}ϕ^{r}(w)

By the Wick theorem, the first summand is equal to

\frac{∂g^{i}}{∂b^{k}} \frac{∂f^{k}}{∂b̃^{j}} (g(b))(w) · \frac{1}{z − w} = \frac{δ_{ij}}{z − w} ,

by (3.18). The second summand is zero. Let us check (3.15b). The first identity is clear. We have

ã^{i}(z)ã^{j}(w) = a^{k} \frac{∂f^{k}}{∂b̃^{i}} (g(b))(z)a^{n} \frac{∂f^{n}}{∂b̃^{j}} (g(b))(w)+a^{k} \frac{∂f^{k}}{∂b̃^{i}} (g(b))(z)\frac{∂^{2}f^{n}}{∂b̃^{j}∂b̃^{l}} (g(b))\frac{∂g^{l}}{∂b^{r}} ψ^{n}ϕ^{r}(w)+

+\frac{∂^{2}f^{k}}{∂b̃^{i}∂b̃^{l}} (g(b))\frac{∂g^{l}}{∂b^{r}} ψ^{k}ϕ^{r}(z)a^{n} \frac{∂f^{n}}{∂b̃^{j}} (g(b))(w)+

+\frac{∂^{2}f^{k}}{∂b̃^{i}∂b̃^{l}} (g(b))\frac{∂g^{l}}{∂b^{r}} ψ^{k}ϕ^{r}(z)\frac{∂^{2}f^{n}}{∂b̃^{j}∂b̃^{l}} (g(b))\frac{∂g^{l}}{∂b^{r}} ψ^{n}ϕ^{r}(w)

When we compute each term using the Wick theorem, there appear single and double pairings. The part corresponding to the single pairings coincides with the expression of the bracket [∂b̃^{i}, ∂b̃^{j}] in old coordinates b^{p}, so it vanishes. The ”anomalous” part comes from the double pairings.

One double pairing appears in the first term and is equal to

−\frac{∂}{∂b^{n}} [\frac{∂f^{k}}{∂b̃^{i}} (g(b))](z)\frac{∂}{∂b^{k}} [\frac{∂f^{n}}{∂b̃^{j}} (g(b))](w) · \frac{1}{(z − w)^{2}}

another one appears in the fourth term and equals

δ_{km}δ_{rn} \frac{∂^{2}f^{k}}{∂b̃^{i}∂b̃^{l}} (g(b))\frac{∂g^{l}}{∂b^{r}} (z)\frac{∂^{2}f^{n}}{∂b̃^{j}∂b̃^{p}} (g(b))\frac{∂g^{p}}{∂b^{m}} (w) · \frac{1}{(z − w)^{2}} =

= \frac{∂^{2}f^{k}}{∂b̃^{i}∂b̃^{l}} (g(b))\frac{∂g^{l}}{∂b^{n}} (z)\frac{∂^{2}f^{n}}{∂b̃^{j}∂b̃^{p}} (g(b))\frac{∂g^{p}}{∂b^{k}} (w) · \frac{1}{(z − w)^{2}} .

We see that these terms cancel out.

The remaining relations, (3.15c - f), contain only single pairings, and are easily checked. △

Thus, for each automorphism g = (g^{1}, …, g^{N}) of ℂ[[b^{1}, …, b^{N}]], (3.16a), the formulas (3.17) determine a morphism of vertex algebras

g̃ : Ω̂_{N} ⟶ Ω̂_{N}  (3.19)

More precisely, each element c of V̂_{N} is an (infinite) sum of finite products of a_{−n}^{i}, b_{−m}^{j}, ψ_{−n}^{i}, ψ_{−m}^{j}. We have c = (c(z)1)(0). By definition,

g̃(c) = (g̃(c(z))1)(0)  (3.20)

Thus, we have to define the field g̃(c(z)) for each c ∈ V̂_{N}. If c is one of the generators a_{−1}^{i}, b_{0}^{i}, ϕ_{0}^{i}, ψ_{−1}^{i}, we define g̃(c(z)) by formulas (3.17). We set

g̃(a_{−1−n}^{i}(z)) = ∂_{z}^{(n)}g̃(a_{−1}^{i}(z)),  (3.21)

and the same with b, ϕ, ψ, cf. (1.12). Finally, if c = c^{1}c^{2} · . . . · c^{p} where each c^{i} is one of the letters a, b, ϕ or ψ, we set

g̃(c(z)) =: g̃(c^{1}(z))g̃(c^{2}(z)) · . . . · g̃(c^{p}(z)) :  (3.22)

where the normal ordered product of two factors is defined by (1.2), and if p > 2, we use the inductive formula (1.14).

Equivalently, if c^{j} = x_{k_{j}}^{j} where x^{j} = a^{i}, b^{i}, ϕ^{i} or ψ^{i}, we have

g̃(c^{1} · . . . · c^{p}1) = [g̃(x^{1}(z))]_{k_{1}} · . . . · [g̃(x^{p}(z))]_{k_{p}}1  (3.23)

(Here we use the following notation. If a(z) = ∑_{i} a_{i}z^{−i−n} is a field corresponding to an element of conformal weight n, we denoted the Fourier mode a_{i} by a(z)_{i}.)

Let G_{N} denote the group of automorphisms (3.16a).

3.8. Theorem. The assignment g ↦ g̃ defines the group homomorphism G_{N} ⟶ Aut(Ω_{N}).

Let us consider two coordinate transformations, ′b^{i} = g_{1}^{i}(b), and ′′b^{i} = g_{2}^{i}(′b). Let f_{j} denote the transformation inverse to g_{j}. We have to check that

g̃_{2}g_{1} = g̃_{2}g̃_{1}  (3.24)

By Theorem 4.5 from [K], it suffices to check this equality on the generators. Let us begin with a^{i}. The element ′′a_{−1}^{i}1 is expressed in the coordinates ′a, etc., as follows

′′a_{−1}^{i}1 = ( ′a_{−1}^{j} \frac{∂f_{2}^{j}}{∂′′b^{i}} (g_{2}(′b_{0})) + \frac{∂^{2}f_{2}^{k}}{∂′′b^{i}∂′′b^{l}} (g_{2}(′b_{0}))\frac{∂g_{2}^{l}}{∂′b^{r}} (′b_{0}) ′ψ_{−1}^{k} ′ϕ_{0}^{r})1

Expressing it in the coordinates a, etc., we get the element

g̃_{2}g̃_{1}(a^{i}) =

= [a^{p} \frac{∂f_{1}^{p}}{∂′b^{j}} (g_{1}(b))(z) + \frac{∂^{2}f_{1}^{p}}{∂′b^{j}∂′b^{q}} (g_{1}(b))\frac{∂g_{1}^{q}}{∂b^{s}} ψ^{p}ϕ^{s}(z)]_{−1} \frac{∂f_{2}^{j}}{∂′′b^{i}} (g_{2}g_{1}(b))(z)_{0}1+

+\frac{∂^{2}f_{2}^{k}}{∂′′b^{i}∂′′b^{l}} (g_{2}g_{1}(b))\frac{∂g_{2}^{l}}{∂′b^{r}} (g_{1}(b))(z)_{0}[\frac{∂_{1}^{p}}{∂′b^{k}} (g_{1}(b))ψ^{p}(z)]_{−1}[\frac{∂g_{1}^{r}}{∂b^{q}} ϕ^{q}(z)]_{0}1 (3.25)

(we have used (3.23)). Now, the action of our group G_{N} on the classical de Rham complex is associative. It follows that the expression (3.25) is equal to g̃_{2}g_{1}(a^{i}) plus two anomalous terms:

a_{0}^{p}[\frac{∂f_{1}^{p}}{∂′b^{j}} (g_{1}(b))(z)]_{−1} \frac{∂f_{2}^{j}}{∂′′b^{i}} (g_{2}g_{1}(b_{0}))1 = −[\frac{∂f_{1}^{p}}{∂′b^{j}} (g_{1}(b))(z)]_{−1} \frac{∂}{∂b_{0}^{k}} \frac{∂f_{2}^{j}}{∂′′b^{i}} (g_{2}g_{1}(b_{0}))1

coming from the first summand, and

\frac{∂^{2}f_{2}^{k}}{∂′′b^{i}∂′′b^{l}} (g_{2}g_{1}(b_{0}))\frac{∂g_{2}^{l}}{∂′b^{r}} (g_{1}(b_{0}))[\frac{∂f_{1}^{p}}{∂′b^{k}} (g_{1}(b))(z)]_{−1} \frac{∂g_{1}^{r}}{∂b^{q}} (b_{0})ψ_{0}^{p}ϕ_{0}^{q}1 =

= \frac{∂^{2}f_{2}^{k}}{∂′′b^{i}∂′′b^{l}} (g_{2}g_{1}(b_{0}))\frac{∂g_{2}^{l}}{∂′b^{r}} (g_{1}(b_{0}))[\frac{∂f_{1}^{p}}{∂′b^{k}} (g_{1}(b))(z)]_{−1} \frac{∂g_{1}^{r}}{∂b^{p}} (b_{0})1

coming from the second one. One sees that these two terms cancel out, which proves (3.24) for a^{i}. For the generators b, ψ, and ϕ, the anomaly does not appear at all. △

3.9. Theorems 3.7 and 3.8 allow one to define the sheaf of conformal vertex algebras Ω_{X}^{ch} for each smooth manifold X in an invariant way, by gluing the sheaves defined in 3.5. This can be done in each of the three settings: in algebraic, complex analytic or smooth one.

In the complex analytic situation, we have our sheaves of vertex algebras over the coordinate charts, and the formulas (3.17) allow to glue these sheaves in a sheaf over X.

In the algebraic situation, it is convenient to use the language of localization of (𝔤, K)-modules, cf. [BS], 4.2, 4.4; [BFM], §3. Let K be a (pro)algebraic group, and (𝔤, K) be a Harish-Chandra pair. Thus, 𝔤 is a Lie algebra, and 𝔨 = Lie(K) is embedded in 𝔤. Let X̂ be a (𝔤, K)-scheme, and X̂ ⟶ X be a K-torsor over a smooth scheme X. If M is a (𝔤, K)-module, i.e. a 𝔤-module, with the 𝔨-action integrated, then one assigns to it the K-equivariant DX̂-module ΔX̂(M) = DX̂ ⊗_{U𝔤} M, and the D_{X}-module Δ(M) = ΔX̂(M)^{K}.

If, in addition, M is a commutative algebra on which 𝔤 acts by derivations, then ΔX̂(M) is a DX̂-algebra, and Δ(M) is a D_{X}-algebra. In other words, Spec(Δ(M)) is a D_{X}-scheme, that is, a scheme over X equipped with an integrable connection. There arises a sheaf of sets on X such that to any open U ⊂ X there corresponds the set of horizontal sections of Spec(Δ(M)) over U. The latter set is in fact the set of ℂ-points of an ind-scheme: it equals Spec(H_{∇}(U; Δ(M))) where H_{∇}(U; Δ(M)) is a topological commutative algebra; for details, see [BD1], 2.3.15.

For example, let X̂ be the scheme of formal coordinate systems on a smooth N-dimensional manifold X. A point of X̂ is a pair (a point x ∈ X, an isomorphism Ô_{X;x} ⟶^{∼} ℂ[[b^{1}, …, b^{N}]]). Let K = Aut(ℂ[[b^{1}, …, b^{N}]]) be the group of formal coordinate changes (3.16a) preserving the origin, and 𝔤 = Der(ℂ[[b^{1}, …, b^{N}]]) the Lie algebra of infinitesimal coordinate changes not necessarily preserving the origin. The obvious projection X̂ ⟶ X is a K-torsor, and X̂ is a (𝔤, K)-scheme.

Let E be a finite dimensional GL(N)-module. Consider it as a K-module using the canonical projection K ⟶ GL(N). The space E ⊗ ℂ[[b^{1}, …, b^{N}]] is naturally a K-module, with the diagonal action of K; we equip it with a structure of a (𝔤, K)- module, by letting the constant vector fields act through the second factor. Let M_{E} be the algebra of functions on the last linear space; it is also a (𝔤, K)-module. Hence we get the commutative D_{X}-algebra Δ(M_{E}), cf. [BD1], Chapter 2, §3; [BD2], 1.2.5, and the sheaf of ind-schemes U ↦ Spec(H_{∇}(U; Δ(M_{E}))) on X. A little thought shows that this all is a fancy way to define the sheaf of sections of the bundle V_{E} = X̂×_{K}E ⟶ X of tensor fields associated with E: namely, the space of sections Γ(U; V_{E}) is equal to the set of ℂ-points of the scheme Spec(H_{∇}(U; Δ(M_{E}))). The D_{X}-module Δ(M_{E}) itself is equal to the sheaf of functions on the corresponding jet bundle.

Returning to our vertex algebras, we play the same game, starting with the algebra M_{N} of functions on Ω̂_{N} instead of M_{E}, equipped with the (𝔤, K)-module structure given by the formulas (3.17). We define Ω_{X}^{ch} to be the sheaf of ℂ-points of ind-schemes

U ↦ Γ(U; Ω_{X}^{ch}) = Spec(H_{∇}(U; Δ(M_{N})))(ℂ)  (3.26)

Note that the last schemes are in fact linear spaces by 3.10 below. Further, it is a sheaf of vertex algebras since the automorphisms (3.17) respect the vertex algebra structure. Let us show that this sheaf admits a canonical filtration whose graded factors are the sheaves of tensor fields discussed in the previous paragraph.

3.10. Placing the formulae (3.16) and (3.17), (3.23) on the desk next to each other, one realizes that the ”symbols of fields” transform in the same way as the corresponding geometric quantities: functions, 1-forms, and vector fields. To be more precise, introduce a filtration on Ω̂_{N} as follows.

The space Ω̂_{N} is a free Â_{N}-module with a base consisting of monomials in letters a_{n}^{i}, ψ_{n}^{i} (n < 0); b_{m}^{i}, ϕ_{m}^{i} (m ≤ 0). Define a partial ordering on this base by

(a) a > ϕ, a > ψ, a > b, ψ > ϕ, ψ > b, ϕ > b; x_{n}^{i} > x_{m}^{j} if n < m, x being a, b, ϕ or ψ;

(b) extending this order to the whole set of monomials lexicographically.

This partial order on the base naturally determines an increasing exhausting filtration on the spaces of fixed conformal weight,

F_{0}Ω̂_{N}^{(i)} ⊂ F_{1}Ω̂_{N}^{(i)} ⊂ …  (3.27)

For example, F_{0}Ω̂_{N}^{(0)} = Â_{N}, F_{0}Ω̂_{N}^{(−1)} = ⊕_{i=1}^{N} Â_{N}b_{−1}^{i}, etc. A glance at (3.17), (3.23) shows that the corresponding graded object Gr_{•}^{F} Ω_{N}^{(i)} is a direct sum of symmetric powers of the tangent bundle, symmetric powers of the bundle of 1-forms, and tensor products thereof. For example, the image of b_{0}^{i} in Gr_{•}^{F} Ω̂_{N}^{(0)} is a function, that of b_{−n}^{i} (n > 0) is a 1-form, that of a_{−n}^{i} is a vector field, etc.

It remains to notice that this filtration is stable under coordinate changes and therefore induces the desired filtration on Ω_{X}^{ch}.

§4. Conformal and topological structure

4.1. Let us return to the formal setting 3.6. Recall that we have in our vertex algebra Ω̂_{N} the fields L(z), J(z), Q(z) and G(z), defined by the formulas (2.4), which make it a topological vertex algebra. Let us study the effect of the coordinate changes (3.16a) on these fields.

Let us denote by L̃(z), etc., the field L(z), etc., written down using formulas (2.4), in terms of the tilded fields ã^{i}(z), etc., and then expressed in terms of the old fields a^{i}(z), etc.

4.2. Theorem. We have

L̃(z) = L(z)  (4.1a)
J̃(z) = J(z) + (Tr log(∂g^{i}/∂b^{j}(z)))′  (4.1b)
Q̃(z) = Q(z) + (\frac{∂}{∂b̃^{r}} [Tr log(∂f^{i}/∂b̃^{j})]ϕ̃^{r}(z))′  (4.1c)
G̃(z) = G(z)  (4.1d)

We have J = ϕ_{0}^{i}ψ_{−1}^{i}1. Therefore, (cf. (3.23)),

J̃ = [\frac{∂g^{i}}{∂b^{j}} (z)]_{0}[\frac{∂f^{k}}{∂b̃^{i}} ψ^{k}(z)]_{−1}1 = J + δ_{jk}[\frac{∂g^{i}}{∂b^{j}} (z)]_{−1}[\frac{∂f^{k}}{∂b̃^{i}} (z)]_{0}1 =

= J + (\frac{∂g^{i}}{∂b^{j}} (z))′_{0}(\frac{∂f^{j}}{∂b̃^{i}} (z))_{0}1,

which implies (4.1b). We have G = ψ_{−1}^{i}b_{−1}^{i}1. Therefore,

G̃ = [\frac{∂f^{j}}{∂b̃^{i}} ψ^{j}(z)]_{−1}[g^{i}(b)(z)]_{−1}1 = [\frac{∂f^{j}}{∂b̃^{i}} ψ^{j}(z)]_{−1}[\frac{∂g^{i}}{∂b^{k}} b^{k}(z)′]_{0}1 =

= δ_{jk}ψ_{−1}^{j}b_{−1}^{k}1 = G

This proves (4.1d). We have Q = a_{−1}^{i}ϕ_{0}^{i}1. Therefore,

Q̃ = [a^{j} \frac{∂f^{j}}{∂b̃^{i}} (z) + \frac{∂^{2}f^{k}}{∂b̃^{i}∂b̃^{l}} \frac{∂g^{l}}{∂b^{r}} ψ^{k}ϕ^{r}(z)]_{−1}[\frac{∂g^{i}}{∂b^{q}} ϕ^{q}(z)]_{0}1

The classical terms:

[a_{−1}^{j}(\frac{∂f^{j}}{∂b̃^{i}} )_{0} + (\frac{∂^{2}f^{k}}{∂b̃^{i}∂b̃^{l}} \frac{∂g^{l}}{∂b^{r}} )_{0}ψ_{−1}^{k}ϕ_{0}^{r}](\frac{∂g^{i}}{∂b^{q}} ϕ^{q})_{0}1 = Q,

since the second summand is zero, due to the anticommutation of ψ_{−1}^{k} and ϕ_{0}^{q}. Quantum corrections (anomalous terms):

[(\frac{∂f^{j}}{∂b̃^{i}} )_{−1}(\frac{∂^{2}g^{i}}{∂b^{j}∂b^{q}} )_{0}ϕ_{0}^{r} + (\frac{∂^{2}f^{k}}{∂b̃^{i}b̃^{l}} \frac{∂g^{l}}{∂b^{r}} ϕ^{r})_{−1}(\frac{∂g^{i}}{∂b^{k}} )_{0}]1 =

= ([(\frac{∂f^{j}}{∂b̃^{i}} )_{−1}(\frac{∂^{2}g^{i}}{∂b^{j}∂b^{r}} )_{0} + (\frac{∂^{2}f^{k}}{∂b̃^{i}∂b̃^{l}} \frac{∂g^{l}}{∂b^{r}} )_{−1}(\frac{∂g^{i}}{∂b^{k}} )_{0}]ϕ_{0}^{r}+

+(\frac{∂^{2}f^{k}}{∂b̃^{i}∂b̃^{l}} \frac{∂g^{l}}{∂b^{r}} \frac{∂g^{i}}{∂b^{k}} )_{0}ϕ_{−1}^{r})1 = (\frac{∂f^{k}}{∂b̃^{i}∂b̃^{l}} \frac{∂g^{l}}{∂b^{r}} \frac{∂g^{i}}{∂b^{k}} ϕ^{r})_{−1}1,  (4.2)

since

(\frac{∂^{2}f^{k}}{∂b̃^{i}∂b̃^{l}} \frac{∂g^{l}}{∂b^{r}} )_{0}(\frac{∂g^{i}}{∂b^{k}} )_{−1}1 = −(\frac{∂f^{k}}{∂b̃^{l}} \frac{∂^{2}g^{l}}{∂b^{t}∂b^{r}} \frac{∂f^{t}}{∂b̃^{i}} )_{0}(\frac{∂g^{i}}{∂b̃^{k}} )_{−1}1 =

= (\frac{∂f^{k}}{∂b̃^{l}} \frac{∂^{2}g^{l}}{∂b^{t}∂b^{r}} )_{0}(\frac{∂f^{t}}{∂b̃^{i}} )_{−1}(\frac{∂g^{i}}{∂b^{k}} )_{0}1 = (\frac{∂^{2}g^{i}}{∂b^{t}∂b^{r}} )_{0}(\frac{∂f^{t}}{∂b̃^{i}} )_{−1}1

Returning to (4.2), we have

\frac{∂f^{k}}{∂b̃^{i}∂b̃^{l}} \frac{∂g^{l}}{∂b^{r}} \frac{∂g^{i}}{∂b^{k}} ϕ^{r} = \frac{∂^{2}f^{k}}{∂b̃^{l}∂b̃^{i}} \frac{∂g^{i}}{∂g^{k}} ϕ̃^{l} = \frac{∂}{∂b̃^{l}} (Tr log(∂f^{i}/∂b̃^{j}))ϕ̃^{l},

which proves (4.1c). It follows from (4.1c) that the operator Q_{0} is invariant. Hence, (4.1a) follows from (4.1d) and (2.2). △

4.3. It follows from (4.1a) that for an arbitrary smooth manifold X, the field L(z) is a well-defined global section of the sheaf End(Ω_{X}^{ch})[[z, z^{−1}]], i.e. Ω_{X}^{ch} is canonically a sheaf of conformal vertex algebras.

It follows from (4.1b) and (4.1c) that the Fourier modes F = J_{0} (”fermionic charge”) and d_{DR}^{ch} = Q_{0} (”BRST charge”) are well-defined endomorphisms of the sheaf Ω_{X}^{ch}. Thus, (Ω_{X}^{ch}, d_{DR}^{ch}) becomes a complex of sheaves, graded by F. This is a localization of definition 2.3.

4.4. Theorem. For any smooth manifold X, the obvious embedding of complexes of sheaves

i : (Ω_{X}, d_{DR}) ⟶ (Ω_{X}^{ch}, d_{DR}^{ch})  (4.3)

is a quasiisomorphism. This is true in algebraic, analytic and C^{∞} settings.

Indeed, the problem is local along X, and we are done by Theorem 2.4.

4.5. If X is Calabi-Yau, i.e. c_{1}(T_{X}) = 0, then the fields J(z) and Q(z) are globally well defined, by (4.1b) and (4.1c). Here T_{X} denotes the tangent bundle. Therefore, in this case the sheaf Ω_{X}^{ch} is canonically a sheaf of topological vertex algebras.

§5. Chiral structure sheaf

A. OBSTRUCTION

5.1. Consider the formal setting 3.1, 3.2, 3.6. We have the vertex algebra V̂_{N} of ”chiral functions” over the formal disk D_{N} = Spf(Â_{N}) where Â_{N} = ℂ[[b^{1}, …, b^{N}]]. Let W_{N} denote the Lie algebra of vector fields f^{i}(b)∂_{b^{i}} on D_{N}. Let Ω^{1}(Â_{N}) denote the module of one-forms f^{i}(b)db^{i}. The spaces Â_{N} and Ω^{1}(Â_{N}) are naturally W_{N}- modules. Recall that the action of W_{N} on Ω^{1}(Â_{N}) is given by

f^{i}∂_{b^{i}} · g^{j}db^{j} = f^{i}∂_{b^{i}}g^{j}db^{j} + g^{j}df^{j}  (5.1)

The de Rham differential d : Â_{N} ⟶ Ω^{1}(Â_{N}) is compatible with the W_{N}-action.

Let us define a map

π : W_{N} ⟶ End(V̂_{N})  (5.2)

For a vector field τ = f^{i}(b)∂_{b^{i}}, let τ(z) denote the field f^{i}(b)a^{i}(z) (of conformal weight 1) of our vertex algebra V̂_{N}. Let π(τ) ∈ End(V̂_{N}) denote the Fourier mode

π(τ) := ∫ τ(z) = τ(z)_{0}  (5.3)

Note that by 1.7, the maps π(τ) are derivations of V̂_{N}.

The mapping π does not respect the Lie bracket. Let us compute the discrepancy. Let τ_{1} = f^{i}(b)∂_{b^{i}}, τ_{2} = g^{i}(b)∂_{b^{i}} be two vector fields. We have the operator product

τ_{1}(z)τ_{2}(w) = −\frac{∂_{b^{j}}f^{i}(b(z))∂_{b^{i}}g^{j}(b(w))}{(z − w)^{2}}+

+\frac{f^{i}(b(w))∂_{b^{i}}g^{j}(b(w))a^{j}(w) − g^{j}(b(w))∂_{b^{j}}f^{i}(b(w))a^{i}(w)}{z − w} =

= −\frac{∂_{b^{j}}f^{i}(b(z))∂_{b^{i}}g^{j}(b(w))}{(z − w)^{2}} + \frac{[τ_{1}, τ_{2}](w)}{z − w} − \frac{[∂_{b^{j}}f^{i}(b(w))]′∂_{b^{i}}g^{j}(b(w))}{z − w}  (5.4)

It follows that

[π(τ_{1}), τ_{2}(w)] = [τ_{1}, τ_{2}](w) − [∂_{b^{j}}f^{i}(b(w))]′∂_{b^{i}}g^{j}(b(w))  (5.5)

In particular,

[π(τ_{1}), π(τ_{2})] = π([τ_{1}, τ_{2}]) − ∫ (∂_{b^{j}}f^{i}(b(w)))′∂_{b^{i}}g^{j}(b(w))  (5.6)

5.2. For ω = f^{i}(b)db^{i} ∈ Ω^{1}(Â_{N}), let us denote by ω(z) the field f^{i}(b)b^{i}(z)′ of our vertex algebra. Denote by π(ω) the Fourier mode ω(z)_{0} = ∫ ω(z).

For f = f(b) ∈ Â^{N}, let f(z) denote the corresponding field of V̂_{N}. Its conformal weight is 0, and f(z)_{−1} = f. We have

df(z) = f(z)′  (5.7)

Given τ = f^{i}∂_{b^{i}} ∈ W_{N}, ω = g^{j}db^{j} ∈ Ω^{1}(Â_{N}), we have the operator product

τ(z)ω(w) = \frac{f^{i}(b(z))g^{i}(b(w))}{(z − w)^{2}} + \frac{f^{i}(b(w))∂_{b^{i}}g^{j}b^{j}(w)′}{z − w} =

= \frac{f^{i}(b(w))g^{i}(b(w))}{(z − w)^{2}} + \frac{f^{i}∂_{b^{i}}g^{j}(w)b^{j}(w)′ + f^{i}(b(w))′g^{i}(b(w))}{z − w}  (5.8)

It follows that

[π(τ), ω(z)] = (τω)(z)  (5.9)

and

[π(τ), π(ω)] = π(τω)  (5.10)

Let W̃_{N} denote the linear subspace of End(V̂_{N}) generated by the Fourier modes π(τ) (τ ∈ W_{N}) and π(ω) (ω ∈ Ω^{1}(Â_{N})). Let I_{N} ∈ W̃_{N} be the linear subspace generated by the Fourier modes π(ω).

It follows from (5.7) that if ω is exact then π(ω) = 0. Thus, π induces an epimorphic map

Ω^{1}(Â_{N})/dÂ_{N} ⟶ I_{N}  (5.11)

5.3. Lemma. The map (5.11) is an isomorphism.

This can be proved by writing down the Fourier mode as an infinite sum of monomials in b_{n}^{i} and comparing the coefficients of like terms. In fact, a more general statement, namely, that ∫ Q(z) = 0 if and only if Q = const · 1 or Q(z) = P (z)′ for some P, seems to be valid for a broad class of vertex algebras, cf. a similar statement in [FF3].

From our point of view, this phenomenon has topological nature. It is amusing to exhibit an example of a vertex algebra, for which the lemma above is false. Namely, take b^{−1}b_{−1} ∈ A_{1}[b^{−1}], see 3.4; then (b^{−1}b_{−1})(z) = b(z)^{−1}b(z)′ and ∫ b(z)^{−1}b(z)′ = 0, but b(z)^{−1}b(z)′ is not a total derivative. △ Content-Length: 20254 X-Lines: 512

5.4. Obviously, ω_{1}(z)ω_{2}(w) = 0 for all ω_{1}, ω_{2} ∈ Ω^{1}(Â_{N}). It follows from (5.6) and (5.10) that W̃_{N} is a Lie algebra, I_{N} is its abelian ideal, and we have the canonical extension

0 ⟶ I_{N} ⟶ W̃_{N} ⟶ W_{N} ⟶ 0  (5.12)

The action of W_{N} on I_{N} arising from this extension coincides with the canonical action of W_{N} on Ω^{1}(Â_{N})/dÂ_{N}, by (5.10). Note that we have defined this extension together with its splitting (5.2). It is given by the two-cocycle c ∈ Z^{2}(W_{N}; Ω^{1}(Â_{N})/dÂ_{N}) of W_{N} with values in Ω^{1}(Â_{N})/dÂ_{N}, read from (5.5),

c(f^{i}∂_{b^{i}}, g^{j}∂_{b^{j}}) = −∂_{b^{i}}g^{j}d(∂_{b^{j}}f^{i})(mod dÂ_{N})  (5.13)

5.5. Let us consider the truncated and shifted de Rham complex

Ω^{•} : 0 ⟶ Â_{N} ⟶ Ω^{1}(Â_{N}) ⟶ 0  (5.14)

where we place Ω^{1}(Â_{N}) in degree zero. It is a complex of W_{N}-modules. We have an obvious map of complexes of W_{N}-modules

Ω^{•} ⟶ Ω^{1}(Â_{N})/dÂ_{N}  (5.15)

where the target is regarded as a complex sitting in degree zero.

Let us write down a two-cocycle c̃ ∈ Z^{2}(W_{N}; Ω^{•}) which is mapped to c, (5.13), under the map (5.14). Such a cocycle is by definition a pair (c^{2}, c^{3}) where c^{2} ∈ C^{2}(W_{N}; Ω^{1}(Â_{N})) is a two-cochain, and c^{3} ∈ C^{3}(W_{N}; Â_{N}) is a three-cochain, such that

d_{Lie}(c^{2}) = d_{DR}(c^{3})  (5.16a)
d_{Lie}(c^{3}) = 0  (5.16b)

Let us define

c^{2}(f^{i}∂_{i}, g^{j}∂_{j}) = ∂_{i}g^{j}d(∂_{j}f^{i}) − ∂_{j}f^{i}d(∂_{i}g^{j})  (5.17)

and

c^{3}(f^{i}∂_{i}, g^{j}∂_{j}, h^{k}∂_{k}) = ∂_{j}f^{i}∂_{k}g^{j}∂_{i}h^{k} − ∂_{k}f^{i}∂_{i}g^{j}∂_{j}h^{k}  (5.18)

We write for brevity ∂_{i} instead of ∂_{b^{i}}. One checks the compatibilities (5.16) directly. Thus, we have defined c̃. One sees that c̃ is mapped to −2c under (5.15).

For N = 1 the space Ω^{1}(Â_{N})/dÂ_{N} is trivial. This allows one to define the sheaf O_{X}^{ch} for curves, acting as in 3.9, and starting from V̂_{N} instead of Ω̂_{N}.

Assume that N > 1. Using the computations of Gelfand-Fuchs, cf. [F], Theorems 2.2.7 and 2.2.4, one can show that the map in cohomology

H^{2}(W_{N}; Ω^{•}) ⟶ H^{2}(W_{N}; Ω^{1}(Â_{N})/dÂ_{N})  (5.19)

induced by (5.15) is an isomorphism. We have the canonical short exact sequence

0 ⟶ H^{2}(W_{N}; Ω^{1}(Â_{N})) ⟶ H^{2}(W_{N}; Ω^{•}) ⟶ H^{3}(W_{N}; Â_{N}) ⟶ 0  (5.20)

the left- and right-most terms being one-dimensional. Under the second map of this sequence, our cocycle c̃ is mapped to its second component c^{3} which is a canonical representative of a generator of the space H^{3}(W_{N}; Â_{N}), cf. [F], Theorem 2.2.7′ and Chapter 2, §1, no. 4. In particular, our cocycle c̃ is non-trivial. It follows that the cocycle c defining the extension (5.12) is also non-trivial.

What kind of an object does the cocycle c̃ define? Recall that a homotopy Lie algebra L^{•} is a complex of vector spaces equipped with a collection of brackets

[ , …, ]_{i} : Λ^{i}L^{•} ⟶ L^{•}[−i + 2], i ≥ 2,  (5.21)

satisfying certain compatibility conditions, cf. for example [HS], §4. In particular [ ,]_{2} is a skew symmetric map, satisfying the Jacobi identity up to the homotopy (given by the third bracket).

Let us define a complex L^{•} as follows. Set L^{0} = W_{N} ⊕ Ω^{1}(Â_{N}), L^{−1} = Â_{N}, the other components being trivial. The differential L^{−1} ⟶ L^{0} is the composition of the de Rham differential and the obvious embedding Ω^{1}(Â_{N}) ↪ L^{0}.

Let the second bracket [ , ]_{2} be given by the usual bracket of vector fields, and the action of vector fields on Â_{N} and Ω^{1}(Â_{N}). Define the third bracket with the only nontrivial component being the three-cocycle c^{3}, (5.18). We set the other brackets equal to zero. This way we get a structure of a homotopy Lie algebra on L^{•}.

We have a canonical extension of homotopy Lie algebras

0 ⟶ Ω^{•} ⟶ L^{•} ⟶ W_{N} ⟶ 0  (5.22)

Here Ω^{•} is an abelian ideal in L^{•} (all brackets are zero). This is a refinement of extension (5.12).

B. PROJECTIVE LINE

5.6. Let X be the projective line ℙ^{1}. Let us fix a coordinate b on ℙ^{1}, and consider the open covering X = U_{0} ∪ U_{1} where U_{0} = Spec(ℂ[b]), U_{1} = Spec(ℂ[b^{−1}]).

Consider the sheaves O_{U_{0}}^{ch} on U_{0} with coordinate b, and O_{U_{1}}^{ch} on U_{1} with coordinate b^{−1}, which were defined in 3.4. Let us glue them on the intersection U_{01} = U_{0} ∩ U_{1} using the transition function

b̃(z) = b(z)^{−1}  (5.23a)
ã(z) = b^{2}a(z) + 2b(z)′  (5.23b)

In this way, we get the sheaf on the X, to be denoted O_{X}^{ch}.

5.7. Theorem. The space of global sections Γ(X; O_{X}^{ch}) admits a natural structure of an irreducible vacuum ŝl_{2}-module on the critical level.

We have to compute O_{U_{0}}^{ch} ∩ O_{U_{1}}^{ch} where both O_{U_{i}}^{ch} are regarded as subspaces of O_{U_{01}}^{ch}. It is the essence of the Wakimoto construction, [W], that the fields a(z)b(z)^{2}+ 2b(z)′, a(z) (resp., ã(z)b̃(z)^{2} + 2b̃(z)′, ã(z)) generate an ŝl_{2}-action on O_{U_{0}}^{ch} (resp., on O_{U_{1}}^{ch}), and under this action, O_{U_{i}}^{ch}, i = 1, 2, become the restricted Wakimoto module with zero highest weight. (Restricted here means that the level is critical, and the Sugawara operators act by zero.) It follows from (5.23) that the ŝl_{2}-action comes from Γ(X; O_{X}^{ch}) and, therefore, O_{U_{0}}^{ch} ∩ O_{U_{1}}^{ch} is also an ŝl_{2}-module. It follows from [FF1] or [M] that each O_{U_{i}}^{ch} contains a unique proper submodule which is isomorphic to the irreducible vacuum representation. To complete the proof, it remains to show that O_{U_{0}}^{ch} ≠ O_{U_{0}}^{ch} ∩ O_{U_{1}}^{ch}, and this is obvious. △

5.8. In fact, the first cohomology space H^{1}(X; O_{X}^{ch}) is also isomorphic to the same irreducible ŝl_{2}-module.

To prove this, let us compute the Euler character

ch(X; O_{X}^{ch}) = ∑_{N=0}^{∞} χ(X; O_{X}^{ch(N)}) · q^{N}

in two different ways. First, by definition

ch(X; O_{X}^{ch}) = ch(Γ(X; O_{X}^{ch})) − ch(H^{1}(X; O_{X}^{ch}))

By Theorem 5.7 and [M],

ch(Γ(X; O_{X}^{ch})) = (1 − q)^{−1} ∏_{N=1}^{∞} (1 − q^{N})^{−2}

On the other hand, formulas (5.23) imply that O_{X}^{ch} carries a filtration F such that the image of a_{−n} (resp., b_{−n}) (n ≥ 1) in Gr^{F} is a vector field (resp., a 1-form). It follows that each monomial a_{−n_{1}} · . . . · a_{−n_{r}}b_{−m_{1}} · . . . · b_{−m_{s}} contributes 2s − 2r + 1 in χ(X; O_{X}^{ch(N)}), where N = ∑ n_{i} + ∑ m_{j}. Therefore,

ch(X; O_{X}^{ch}) = ∏_{N=1}^{∞} (1 − q^{N})^{−2},

hence

ch(H^{1}(X; O_{X}^{ch})) = q · ch(Γ(X; O_{X}^{ch}))

In other words, H^{1}(X; O_{X}^{ch}) has the same (up to the shift by q) character as Γ(X; O_{X}^{ch}). Again by [M], these two spaces are isomorphic as 𝔤̂-modules.

C. FLAG MANIFOLDS

5.9. Let G be a simple algebraic group, B ⊂ G a Borel subgroup, N ⊂ B the maximal nilpotent subgroup. The manifold N is isomorphic to the affine space, and is a (𝔤, B)-scheme, where B acts by conjugation. Consider the Heisenberg vertex algebra V associated with the affine space N. According to [FF2], V admits a structure of a 𝔤̂-module (Wakimoto module); in particular, V is a (𝔤, B)-module. Note that x ∈ 𝔤 acts on V as ∫ X(z) for some X ∈ V.

Consequently, considered as an affine space, V admits a structure of a (𝔤, B)- scheme. Let M be the algebra of functions on V. Proceeding as in 3.9, with K = B, X̂ = G, and X = G/B, we get the sheaf of ind-schemes

U ↦ Spec(H_{∇}(Δ(M)))

on X. The sheaf of its ℂ-points is called the chiral structure sheaf of X and denoted by O_{X}^{ch}.

5.10. If G = SL(n) then the sheaf O_{X}^{ch} admits a more explicit construction, using charts and gluing functions. In this case X = GL(n)/(B × ℂ^{∗}). The Weyl group is identified with the symmetric group S_{n} and can be realized as the subgroup of GL(n) consisting of permutation matrices. One checks that in terms of the Lie algebra gl(n), the simple permutation r_{i} (interchanging i and i + 1) can be written as follows

r_{i} = exp(π\sqrt{−1}E_{ii}) exp(E_{i+1,i}) exp(−E_{i,i+1}) exp(E_{i+1,i})  (5.24)

where E_{ij} (1 ≤ i, j ≤ n) form the standard base of gl(n).

The manifold X is covered by |S| = n! charts, the chart associated with an element w ∈ S_{n} being U_{w} = wNw_{0}B, where N ⊂ B is the unipotent subgroup consisting of all upper-triangular matrices and w_{0} ∈ S_{n} is the element of maximal length. Let us identify U_{w} with N using the bijection n ↦ wnw_{0}B. Under this identification, if x ∈ U_{w_{1}} ∩ U_{w_{2}}, then the change from the coordinates determined by U_{w_{1}} to the ones determined by U_{w_{2}}, is given by x ↦ w_{2}^{−1}w_{1}x.

Each U_{w} may be identified with the affine space ℂ^{n(n−1)/2}. To define O_{X}^{ch}, we declare that O_{X}^{ch}|_{U_{w}}= O_{U_{w}}^{ch} where the last sheaf is defined in 3.4. Now we have to glue these sheaves over the pairwise intersections in a consistent manner.

Let V denote the vertex algebra V_{n(n−1)/2} = Γ(U_{w}; O_{U_{w}}^{ch}). First, we extend the ŝl(n)-module structure on V to a ĝl(n)-module structure. For that, in addition to the formulae in [FF1], p. 279, define

E_{ii}(z) = − ∑_{j>i} b^{ij}a^{ij}(z) + ∑_{j<i} b^{ji}a^{ji}(z)  (5.25)

It is easily checked that in this way we indeed get an action of ĝl(n) on V.

For any A ∈ End(V), introduce the formal exponent exp(tA) : V ⟶ V ⊗ ℂ[[t]],

exp(tA)(v) = ∑_{i=0}^{∞} \frac{A^{i}(v)}{i!}t^{i}  (5.26)

Working over ℂ[[t]], we can easily compose such maps. Motivated by (5.24), set

r_{i}(t) = exp(tπ\sqrt{−1} ∫ E_{ii}(z))·

· exp(t ∫ E_{i+1,i}(z)) exp(−t ∫ E_{i,i+1}(z)) exp(t ∫ E_{i+1,i}(z))  (5.27)

Thus, r_{i}(t) is a map V ⟶ V [[z, z^{−1}]][[t]]. Note that V [[t]] is naturally a vertex algebra. It contains the vertex subalgebra ”generated by functions, rational in t”, that is,

V (t) = R(N × ℂ) ⊗_{A} V ⊂ V [[t]]  (5.28)

Here A = Γ(N; O_{N}), and R(N ×ℂ) denotes the ring of rational functions on N ×ℂ, regular at N × {0}.

5.11. Lemma. (a) r_{i}(t)V ⊂ V (t). Further, r_{i}(t)V is generated by functions well-defined for any value of t.

(b) r_{i}(1) is well defined (by (a)), and determines a map

r_{i}(1) : Γ(U_{w}; O_{U_{w}}^{ch}) ⟶ Γ(U_{w} ∩ U_{r_{i}w}, O_{U_{w}}^{ch})  (5.29)

As ∫ X(z), X ∈ V, is a derivation, the map r_{i}(t) is an embedding of vertex algebras. Therefore, it is enough to compute r_{i}(t) on generators.

There are two types of generators in V: (i) those coming from the subspace W ⊂ V, canonically isomorphic to 𝔤 = sl(n), such that the fields X(z) (X ∈ W) generate the action of 𝔤̂; (ii) those coming from ℂ[b_{n}^{ij}]1 ⊂ V. Obviously, the endomorphisms ∫ E_{ij}(z) (i ≠ j), act on W as E_{ij} on sl(n); in particluar, this action is nilpotent, and r_{i}(t) is polynomial in t. As for ℂ[b_{n}^{ij}]1, this subspace is identified with the symmetric algebra S^{•}(Ω^{1}(U_{w})). For any X ∈ gl(n) ⊂ Γ(X; T_{X}), the element ∫ X(z) acts on this space as the Lie derivative along X. Consequently, exp(∫ X(z)) maps S^{•}(Ω^{1}(U_{w})) into S^{•}(Ω^{1}(exp(X) · U_{w})). △

Repeated application of this lemma gives for any v = r_{i_{1}} · . . . r_{i_{k}} ∈ S_{n} the map

v(1) = r_{i_{1}} · . . . · r_{i_{k}} : Γ(U_{w}; O_{U_{w}}^{ch}) ⟶ Γ(U_{w} ∩ U_{v·w}; O_{U_{w}}^{ch})  (5.30)

Finally, to complete our construction of O_{X}^{ch}, glue the sheaves O_{U_{w_{1}}} and O_{U_{w_{2}}} using the maps

Γ(U_{w_{1}}; O_{U_{w_{1}}}^{ch}) ⟶ Γ(U_{w_{1}} ∩ U_{w_{2}}; O_{U_{w_{2}}}^{ch}) ⟵ Γ(U_{w_{2}}; O_{U_{w_{2}}}^{ch})  (5.31)

where the first (resp. second) arrow is w_{2}w_{1}^{−1}(1) (resp., w_{1}w_{2}^{−1}(1)). Since the gluing maps are induced by the action of S_{n}, they are transitive, and the sheaf O_{X}^{ch} is well defined.

5.12. Example. Let 𝔤 = sl(2). We have N = 𝔸^{1} ⊂ X = ℙ^{1}; b is the coordinate on N such that generators of gl(2) act as the following vector fields: E_{21} ↦ −∂_{b}, E_{12} ↦ b^{2}∂_{b}, E_{11} ↦ b∂_{b}.

One easily calculates that

exp(E_{21}) : b ↦ b − 1; exp(−E_{12}) : b ↦ b/(b + 1); exp(π\sqrt{−1}E_{11}) : b ↦ −b

The simple reflection is

r = exp(π\sqrt{−1}E_{11}) exp(E_{21}) exp(−E_{12}) exp(E_{21}) : b ↦ b^{−1}  (5.32)

To do the chiral analogue of this computation, recall (cf. [W]) that E_{21}(z) = −a(z), E_{12}(z) = b^{2}a(z)+2b(z)′, E_{11}(z) = ba(z). A direct computation using Wick theorem yields

r(1)a_{−1}1 = (b^{2}a_{−1} + 2b_{−1})1  (5.33)

which coincides with (5.23b). Computation of r(1)b1 does not differ from the classical one, see (5.32).

5.13. Theorem. The space Γ(X; O_{X}^{ch}) admits a natural structure of a 𝔤̂-module at the critical level, such that it is a submodule of the restricted Wakimoto module, and its conformal weight zero component is 1-dimensional.

Proof is not much different from that of Theorem 5.7. The space of sections W = Γ(N; O_{X}^{ch}) over the big cell is the restricted Wakimoto module with the zero highest weight. Restricted here again means that the center of U(𝔤̂)_{loc} acts trivially. As the action of B on V preserves 𝔤̂ ⊂ End(V), 𝔤̂ is spanned by the Fourier modes of fields associated with certain elements of Γ(X; O_{X}^{ch}). Therefore, Γ(X; O_{X}^{ch}) is a 𝔤̂-module at the critical level, and there arises a 𝔤̂-morphism

Γ(X; O_{X}^{ch}) ⟶ W  (5.34)

By construction, the sheaf O_{X} admits a canonical filtration whose associated quotients are free O_{X}-modules of finite rank. Therefore, the map (5.34) is an injection. By construction, the conformal weight zero component of Γ(X; O_{X}^{ch}) is equal to Γ(X; O_{X}) = ℂ. △

This theorem is less precise than 5.7, the reason being that the representationtheoretic result of [FF1], [M] mentioned in 5.7 is only available in the sl(2)-case. However, Theorem 5.1 of [FF1] makes it plausible that this representation-theoretic statement carries over to any 𝔤, and hence that Γ(X; O_{X}^{ch}) is in fact the irreducible vacuum 𝔤̂-module.

5.14. Note that the whole sheaf O_{X}^{ch} admits a structure of a 𝔤̂-module at the critical level.

5.15. Localization for non-zero highest weight. For any integral weight of 𝔤, there exists a twisted analogue of O_{X}^{ch}, to be denoted by L_{λ}^{ch}. Its construction repeats word for word 5.9, except that the action of (𝔤, B) on V is to be twisted by λ, see [W, FF1]. If λ is a regular dominant weight, then Theorems 5.7 and 5.13, along with their proofs, generalize in the obvious way. For example, the word ”vacuum” in the formulation of Theorem 5.7 is to be replaced with ”highest weight λ”. Likewise, the claim that ”conformal weight zero component is 1-dimensional” should be replaced with the following: ”conformal weight zero component is isomorphic to Γ(X; L_{λ})”, where L_{λ} is the standard line bundle on X.

One may want to regard L_{λ}^{ch} as a sheaf of modules over a sheaf of vertex algebras, just as O_{X}^{ch} is a sheaf of modules over itself. From this point of view, O_{X}^{ch} is a chiral analogue of the sheaf D_{X} of differential operators on X. Thus, one cannot expect L_{λ}^{ch} to be a module over O_{X}^{ch}. We believe (and have checked this for 𝔤 = sl(2)) that one can define a sheaf of vertex algebras O_{λ}^{ch} which acts on L_{λ}^{ch} and is locally isomorphic to O_{X}^{ch}. Therefore, this sheaf can be regarded as a chiral partner of the sheaf D_{λ} of twisted differential operators on X.

References

[BD1] A. Beilinson, V. Drinfeld, Chiral algebras I, Preprint.

[BD2] A. Beilinson, V. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves, Preprint, 1997.

[BFM] A. Beilinson, B. Feigin, B. Mazur, Introduction to algebraic field theory on curves, Preprint.

[BS] A. Beilinson, V. Schechtman, Determinant bundles and Virasoro algebras, Comm. Math.Phys. 118 (1988), 651-701.

[FF1] B. Feigin, E. Frenkel, Representations of affine Kac-Moody algebras and bosonization, in: V. Knizhnik Memorial Volume, L. Brink, D. Friedan, A.M. Polyakov (Eds.), 271-316, World Scientific, Singapore, 1990.

[FF2] B. Feigin, E. Frenkel, Affine Kac-Moody algebras and semi-infinite flag manifolds, Comm. Math. Phys. 128 (1990), 161-189.

[FF3] B. Feigin, E. Frenkel, Affine Kac-Moody algebras at the critical level and Gelfand-Dikii algebras, Infinite Analysis, Part A, B (Kyoto, 1991), 197-215, Adv. Ser. Math. Phys. 16, World Sci. Publishing, River Edge, NJ, 1992.

[F] D.B. Fuchs, Cohomology of infinite-dimensional Lie algebras, Contemp. Sov. Math., Consultants Bureau, New York, 1986.

[HS] V. Hinich, V. Schechtman, Homotopy Lie algebras, I.M. Gelfand Seminar, Adv. in Sov. Math., 16 (1993), Part 2, 1-28.

[K] V. Kac, Vertex algebras for beginners, University Lecture Series, 10, American Mathematical Society, Providence, RI, 1997.

[LVW] W. Lerche, C. Vafa, P. Warner, Chiral rings in N = 2 superconformal theories, Nucl. Phys. B324 (1989), 427-474.

[LZ] B. Lian, G. Zuckerman, New perspectives on the BRST-algebraic structure of string theory, Commun. Math. Phys. 154 (1993), 613-646; hep-th/9211072.

[M] F. Malikov, Verma modules over Kac-Moody algebras of rank 2, Leningrad Math. J. 2, No. 2 (1990), 269-286.

[W] M. Wakimoto, Fock representations of the affine Lie algebra A_{1}^{(1)}, Comm. Math. Phys. 104 (1986), 604-609.

F.M.: Max-Planck-Institut für Mathematik, Gottfried-Claren-Straße 26, 53225 Bonn, Germany; malikov@mpim-bonn.mpg.de

V.S.: Max-Planck-Institut für Mathematik, Gottfried-Claren-Straße 26, 53225 Bonn, Germany; vadik@mpim-bonn.mpg.de

A.V.: Department of Mathematics, New Mexico State University, Las Cruces, NM 88003-8001, USA; vaintrob@nmsu.edu