Twisted chiral de Rham algebras on ℙ^{1}

Vassily Gorbounov, Fyodor Malikov^{1}, Vadim Schechtman

To Dennis Sullivan on his 60-th birthday

§1. Introduction

1.1. Let G be a semisimple complex Lie group, B ⊂ G a Borel subgroup, X = G/B the flag space. Let 𝔤 = Lie(G) and 𝔤̂ the corresponding Kac-Moody Lie algebra. B.Feigin and E.Frenkel [FF1]-[FF3] have constructed (generalizing the previous construction by M.Wakimoto [W]) a family of representations of 𝔤̂, called Wakimoto modules, which are ”loop space” analogs of contragradient Verma modules over 𝔤, realized in the spaces of differential operators over an open Schubert cell in X.

Let us recall this construction. From now on we restrict ourselves to the case G = SL(2), so 𝔤 = sl(2), X = ℙ^{1}.

Let M be a vertex superalgebra generated by even fields a(z), b(z) of conformal weight 1 and 0 respectively and odd fields ϕ(z), ψ(z) of conformal weight 0 and 1 respectively, with nontrivial OPE’s

a(z)b(w) ∼ \frac{1}{z − w}; ψ(z)ϕ(w) ∼ \frac{1}{z − w}  (1.1.1)

(this vertex algebra is what is classically called a ”ϕ, ψ, a, b-system”). As a vector space, M is a polynomial algebra in infinite set of variables,

M = ℂ[a_{−i−1}1, b_{−i}1, ϕ_{−i}1, ψ_{−i−1}1]_{i≥0}  (1.1.2)

(where 1 denotes the vacuum vector). It is ℤ-graded by ”fermion number”: M = ⊕_{i∈ℤ} M^{i} where M^{i} ⊂ M is the subspace generated by all monomials having the number of letters ϕ - the number of letters ψ equal to i.

Let e, f, h be the standard generators of 𝔤. Let V_{0,k} denote the vacuum module over 𝔤̂ of level k. This is a vertex algebra generated by the fields e(z), h(z), f(z) subject to standard relations, cf. [K], (2.5.6) and Example 4.9(b).

Let α be a complex number. A correspondence

e(z) ↦ −a(z), h(z) ↦ −2 : a(z)b(z) : −2α : ϕ(z)ψ(z) :,
f(z) ↦: b(z)^{2}a(z) : +2α : b(z)ϕ(z)ψ(z) : −2(1 − α^{2})b(z)′  (1.1.3)

defines a morphism of vertex algebras

V_{0,2α^{2}−2} ⟶ M  (1.1.4)

1partially supported by an NSF grant

and makes M (in fact, all M^{i}) a 𝔤̂-module of central charge 2α^{2} − 2. In particular, the fermion number zero component M^{0} is a Wakimoto module to be denoted W_{0,2α^{2}−2}.

The above construction coincides with the one from [FF1], except for one modification: we have replaced the ”Cartan” boson from op. cit. by the product of two fermions ϕψ using the boson-fermion correspondence (cf. for example [K], 5.2).

1.2. As was mentioned above, the classical analogs of Wakimoto modules are contragradient Verma modules over 𝔤. According to Beilinson-Bernstein, they may be realized as spaces of sections of certain ”standard” D-modules over X corresponding to Schubert cells - in our case they correspond to open cells.

The first aim of the present note is to ”localize” the definition from 1.1, more precisely to construct sheaves of vertex algebras over X = ℙ^{1} whose spaces of sections over an open Schubert cell would contain the Wakimoto modules W_{0,2α^{2}−2}.

In the case α = 1 this problem is solved by the chiral de Rham complex Ω_{X}^{ch} introduced in [MSV]. Let us explain this. Choose a coordinate b on X = ℙ^{1} and consider the covering X = U_{0} ∪ U_{1} by two affine lines, where U_{0} = {x ∈ X| b(x) ≠

∞}, U_{1} = {x ∈ X| b(x) ≠ 0}, so b : U_{0} ⟶^{∼} 𝔸^{1} and b^{−1} : U_{1} ⟶^{∼} 𝔸^{1}.

We identify b = b_{0}1 and consider the space M from 1.1 as a ℂ[b] = O(U_{0})- module in the obvious way. Let M_{0} = M^{∼} be the corresponding quasicoherent sheaf over U_{0}. According to the Zariski localization procedure described in [MSV] (cf. also [GMS1], 1.10), M_{0} admits a structure of a sheaf of vertex (super)algebras over U_{0}. Similarly using the coordinate b^{−1} we get a sheaf of vertex algebras M_{1} over U_{1}. To get a sheaf over X, we want to glue M_{0} and M_{1} over the intersection U_{01} = U_{0} ∩ U_{1} ⟶^{∼} 𝔾_{m}, i.e. to define an isomorphism M_{0}(U_{01}) ⟶^{∼} M_{1}(U_{01}). This is done by the formulas

b(z) ↦ b(z)^{−1}, a(z) ↦ − : b(z)^{2}a(z) : −2 : b(z)ϕ(z)ψ(z) :
ϕ(z) ↦: b(z)^{−2}ϕ(z) :, ψ(z) ↦: b(z)^{2}ψ(z) :  (1.2.1)

The meaning of these formulas is very simple. We consider the superalgebra ℂ[a, b, ϕ, ψ] as a superalgebra of derivations of the de Rham algebra Ω_{ℂ[b]}^{·} where a = ∂_{b}, ϕ = db and ψ = ∂_{ϕ}. Then we make a change of variables b ↦ b^{−1} and see how the variables a, ϕ, ψ transform. The formulas (1.2.1) are nothing but the same transformation law translated into the universe of fields. The fact that they define an automorphism of the vertex algebra M[b^{−1}] = M_{0}(U_{01}) (”absence of quantum anomaly”) is not banal - it is one of the main observations in [MSV].

By definition, the chiral de Rham algebra Ω_{X}^{ch} is a sheaf of vertex superalgebras over X obtained by the above gluing. By a general fact (cf. [MS1], Part III, Corollary 1.3) the Kac-Moody algebra 𝔤̂ acts on Ω_{X}^{ch} by derivations, with central charge zero. The space of sections of the fermionic zero component Ω_{X}^{ch,0} is the Wakimoto module W_{0,0}. A description of the spaces Γ(X, Ω_{X}^{ch,i}) as 𝔤̂-modules is given in op. cit., Part. III, 2.2 (cf. Proposition 3.3 below).

In order to obtain the other Wakimoto modules W_{2α^{2}−2} we want to ”twist” the definition of Ω_{X}^{ch}. More precisely, in Ω_{X}^{ch} the fields ϕ (resp. ψ) are sections of the sheaf of one-forms Ω_{X}^{1} = O_{ℙ^{1}}(−2) (resp. of the tangent sheaf T_{X} = O_{ℙ^{1}}(2)).

Assume that 2α ∈ ℤ. In a ”twisted” de Rham algebra Ω_{X,(α)}^{ch} they will be sections of O_{ℙ^{1}}(−2α) and O_{ℙ^{1}}(2α) respectively. In fact, Ω_{X,(α)}^{ch} is an example of a twisted sheaf of chiral differential operators appeared in [GMS2], cf. 2.5 below.

The Lie algebra 𝔤̂ acts on Ω_{X,(α)}^{ch} with central charge k = 2α^{2} − 2. Similarly to the case α = 1 these sheaves are ℤ-graded by fermionic number. We describe the spaces Γ(X, Ω_{X,(α)}^{ch,i}) as extensions of Weyl modules.

We also introduce, following the pattern of [MS2], a sort of chiral Serre duality on the category of Ω_{X,(α)}^{ch}-modules and use it to get a similar description of

H^{1}(X, Ω_{X,(α)}^{ch,i}). This allows to compute their characters.

1.3. The next topic of this note will be a ”free field resolution” of the vertex superalgebra H^{∗}(X, Ω_{X,(α)}^{ch}), i.e. a description of the last algebra (which is a complicated object) as a cohomology of a simple ”free field” algebra with respect to certain square zero odd derivation (which is a Fourier component of a field, and called ”a screening charge”). At this point we use the fact that ℙ^{1} is a toric variety. In [B] L.Borisov constructs for an arbitrary smooth toric variety Y a remarkable free field resolution of H^{∗}(Y, Ω_{Y}^{ch}). We show how to twist the Borisov construction to obtain a free field resolution of H^{∗}(X, Ω_{X,(α)}^{ch}).

It turns out that this ”twisted” Borisov resolution essentially coincides with a resolution discussed by B. Feigin in an unpublished manuscript [F]. Introduced in [F] are several free field resolutions of the format from [FF4]. According to the philosophy of op. cit., these should be considered as ”quantizations” of standard Bernstein-Gelfand-Gelfand complexes of a maximal nilpotent Lie subalgebra 𝔫′ in a semisimple Lie algebra 𝔤′ (with coefficients in a Heisenberg module). ”Quantization” means that instead of 𝔤′ one deals with corresponding quantum group. B. Feigin has discovered that in the case at hand the quantum group that governs the resolution corresponds to the Lie superalgebra sl(2, 1), 𝔫′ being generated by 2 odd elements. In our text this manifests in that the operators D, D̄ appearing in (5.9.3) generate an algebra isomorphic to 𝔫′, and the complex (5.9.3) is exactly the Bernstein-Gelfand-Gelfand complex associated with V_{L}^{0}.

To be more precise, Feigin discusses the analog of this resolution in the case of a generic α (and here the quantum group arises). In our picture, the full sheaf Ω_{X,(α)}^{ch} makes no sense for generic α; however, its component of fermionic charge 0 does (and has a vacuum ŝl_{2}-module as the space of global sections). So we can argue that the present construction gives a natural extension of Feigin’s resolution in the case of non-generic α.

The ”chiral Serre duality” makes it clear why Feigin’s resolution is indeed a resolution and exactly what higher cohomology appear for non-generic α. This is probably the only real advantage of our approach over Feigin’s.

The ”linear algebra data” underlying our resolution is a 3-dimensional lattice L equipped with a one-parameter family of ”hyperbolic reflections” ϕ_{α}, cf. infra (5.7.1)^{2}.

2These data are in some sense associated with the root system of sl(2). Can one define similar data for other root systems?

Remark. One can define similar twisted de Rham algebras which localize the Wakimoto modules for an arbitrary semisimple Lie group G. However, we do not know the analogs of Feigin-Borisov resolutions for an arbitrary G — this is a very interesting open question.

1.4. During the work on this note the authors enjoyed the hospitality of Max-Planck-Institut für Mathematik in Bonn and Korteweg-de Vries Institute in Amsterdam. We are grateful to Don Zagier who taught us the computations 3.5-3.6.

§2. The sheaf Ω_{X,(α)}^{ch}

.

We use all the notations of the Introduction.

2.1. Consider the vertex algebra M[b^{−1}] = M_{0}(U_{01}). It is generated by the fields b(z), b(z)^{−1}, a(z), ϕ(z), ψ(z). By composing the Wakimoto map (1.1.4) with the canonical localisation map M ⟶ M[b^{−1}] we get a morphism of vertex algebras

V_{0,2α^{2}−2} ⟶ M[b^{−1}]  (2.1.1)

Each vacuum module V_{0,k} affords an involution defined by

e(z) ↦ f(z), f(z) ↦ e(z)  (2.1.2)

(note that V_{0,k} is generated by two fields e(z), f(z)).

On the classical level, if we realize 𝔤 by means of vector fields acting on X and identify e with −∂_{b} and f with b^{2}∂_{b} then the interchage of e with f corresponds to the coordinate change b ↦ b^{−1}. The next lemma chiralizes this automorphism.

2.2. Lemma. Assume that α ∈ \frac{1}{2}ℤ. The assignment

b(z) ↦ b(z)^{−1}, a(z) ↦ − : b(z)^{2}a(z) : −2α : b(z)ϕ(z)ψ(z) : −2(1 − α^{2})b(z)′,

ϕ(z) ↦: b(z)^{−2α}ϕ(z) :, ψ(z) ↦: b(z)^{2α}ψ(z) :  (2.2.1)

defines an automorphism of M[b^{−1}] which is compatible with the involution (2.1.2) via the map (2.1.1).

If α is not necessarily a half-integer, the involution (2.2.1) is still well defined on the fermionic number zero component M[b^{−1}]^{0}. △

2.3. Remark. Talking about fields is equivalent to talking about the states. In the language of states (2.2.1) is equivalent to

b_{0}1 ↦ b_{0}^{−1}1, a_{−1}1 ↦ −b_{0}^{2}a_{−1}1 − 2αb_{0}ϕ_{0}ψ_{−1}1 − 2(1 − α^{2})b_{−1}1

ϕ_{0}1 ↦ b_{0}^{−2α}ϕ_{0}1, ψ_{−1}1 ↦ b_{0}^{2α}ψ_{−1}1  (2.3.1)

2.4. On the neighbourhood of infinity U_{1} with coordinate b̄ = b^{−1} we have the sheaf M_{1} generated by the fields ā(z), b̄(z), ϕ̄(z), ψ̄(z). By sending these barred fields to the corresponding right hand side expressions from (2.2.1) we get an isomorphism

g : M_{1}|_{U_{01}} ⟶^{∼} M_{0}|_{U_{01}}  (2.4.1)

By the way, to get g^{−1} one simply puts the bars over the fields in the rhs of (2.2.1).

Gluing M_{0} with M_{1} over U_{01} by means of g we get a sheaf of vertex superalgebras on X, to be denoted by Ω_{X,(α)}^{ch}. This sheaf is defined if α ∈ \frac{1}{2}ℤ. For α = 1 we get the old chiral de Rham algebra Ω_{X,(1)}^{ch} = Ω_{X}^{ch}.

For an arbirtary α the whole sheaf is not defined but the fermion number zero component Ω_{X,(α)}^{ch,0} is.

2.5. One can easily explain the meaning of the above sheaves in the framework of [GMS2]. Namely, consider a supervariety X_{O(−2α)} with the same underlying space as X and with the structure sheaf Ω_{X,(α)} = O_{X} ⊕ O_{X}(−2α) being the symmetric (in the super sense) algebra of the line bundle O_{X}(−2α) where this line bundle is equipped the odd parity. So this superalgebra is a twisted analog of the de Rham algebra.

The sheaf Ω_{X,(α)}^{ch} is nothing but the sheaf of chiral differential operators on X_{O(−2α)}. Note that since X is a curve, such a sheaf is unique up to a unique isomorphism, by the general cohomological formalism of op. cit.

2.6. By definition, Ω_{X,(α)}^{ch} is a sheaf of 𝔤̂-modules of level k = 2α^{2} −2. Therefore it carries a conformal structure where the Virasoro field is provided by the Sugawara construction. Locally on U_{0} it is given by

L(z) = \frac{1}{k + 2}\{: e(z)f(z) : + : f(z)e(z) : +\frac{1}{2} : h(z)^{2} :\} =

=: b(z)′a(z) : +\frac{1 − α}{2α} : ϕ(z)ψ(z)′ : +\frac{1 + α}{2α} : ϕ(z)′ψ(z) :  (2.6.1)

The grading given by the operator L_{0} of this field differs from the obvious one: e.g.,

L_{0}(ϕ_{0}1) = \frac{α − 1}{2α}ϕ_{0}1, L_{0}(ψ_{−1}1) = −\frac{α + 1}{2α}ψ_{−1}1  (2.6.2)

However, the old grading (but not the old conformal structure) which assigns to x_{i}1 the degree −i, still makes sense for the whole sheaf Ω_{X,(α)}^{ch}. Below when we speak about ”conformal weight” we mean always this old grading. Thus, the conformal weight zero component Ω_{X,(α)0}^{ch} = O_{X} ⊕ O_{X}(−2α), ϕ_{0}1 being a local section of O_{X}(−2α). Similarly, Ω_{X,(α)1}^{ch} admits O_{X}(2α) as a canonical quotient, ψ_{−1}1 being its local generator, etc.

The fermionic charge grading on Ω_{X,(α)}^{ch} is defined as in the untwisted case. Our algebra admits a canonical filtration whose graded quotient is given by (cf. [BL], [GMS2], 7.6)

[GrΩ_{X,(α)}^{ch}] = ∏_{n≥0} [S_{q^{n}}Ω_{X}^{1}][S_{q^{n+1}}T_{X}][Λ_{yq^{n}}O_{X}(−2α)][Λ_{y^{−1}q^{n+1}}O_{X}(2α)] (2.6.3)

Here the square brackets denote the class in the K-theory; the power at q (resp.

y) counts the conformal weight (resp. fermionic charge). In the right hand side we use the notation

[S_{x}(E)] = ∑_{i=0}^{∞} [S_{O_{X}}^{i} (E)]x^{i} ∈ K(X)[[x]], [Λ_{x}(E)] = ∑_{i=0}^{∞} [Λ^{i}(E)]x^{i} ∈ K(X)[x]

(2.6.4)

for a vector bundle E over X.

§3. Global sections

3.1. In this section we give a description of Γ(X, Ω_{X,(α)}^{ch}) as a 𝔤̂-module and compute its character.

Let m be a nonnegative integer and k an arbitrary complex number. Let L_{m} denote the (m + 1)-dimensional irreducible 𝔤-module. The Kac-Moody algebra 𝔤̂ = 𝔤[t, t^{−1}] ⊕ ℂc contains the Lie subalgebra 𝔤̂_{+} = 𝔤[t] ⊕ ℂc equipped with an obvious ”evaluation at 0” projection p : 𝔤̂_{+} ⟶ 𝔤 ⊕ ℂc. Let L_{m,k} denote a 𝔤 ⊕ ℂcmodule equal to L_{m} as a 𝔤-module, where c acts as multiplication by k; using p we can regard L_{m,k} as a 𝔤̂_{+}-module. By definition, the Weyl module V_{m,k} is the induced 𝔤̂-module

V_{m,k} = Ind𝔤̂ ; 𝔤_{+}L_{m,k}  (3.1.1)

3.2. Let Ω_{X,(α)}^{ch,i} ⊂ Ω_{X,(α)}^{ch} denote the component of fermionic charge i ∈ ℤ. By construction,

Γ(U_{0}, Ω_{X,(α)}^{ch,0}) = W_{0,2α^{2}−2}  (3.2.1)

Similarly,

Γ(U_{0}, Ω_{X,(α)}^{ch,i}) = W_{−2αi,2α^{2}−2}  (3.2.2)

where in the right hand side we have the Wakimoto module having the highest weight vector on which h acts as 2αi.

It is clear that just as in the untwisted (α = 1) case discussed in [MS1], Part III, 2, we have

Γ(X, Ω_{X,(α)}^{ch}) = Γ(U_{0}, Ω_{X,(α)}^{ch})^{𝔤−int}  (3.2.3)

where the superscript in the right hand side denotes the maximal 𝔤-integrable submodule.

The diagram on p. 301 of [FF3] says that

3.3. Proposition. (i) If α ∈ ℤ_{≥0}, i ≤ 0, then Γ(X, Ω_{X,(α)}^{ch,i}) admits a filtration with quotients equal to Weyl modules V_{−2iα+4jα^{2},2α^{2}−2}, j = 0, 1, 2, …;

(ii) if α ∈ ℤ_{≥0}, i > 0, then Γ(X, Ω_{X,(α)}^{ch,i}) admits a filtration with quotients equal to dual Weyl modules V_{2iα−2+4(j+1)α^{2},2α^{2}−2}^{∗}, j = 0, 1, 2, …; (iii) if α is irrational then Ω_{X,(α)}^{ch} does not exist but Ω_{X,(α)}^{ch,0} does, and we have

Γ(X, Ω_{X,(α)}^{ch,0}) = W_{0,2α^{2}−2}^{𝔤−int} = V_{0,2α^{2}−2}  (3.3.1)

△

3.4. Let us write down the character of our module. First consider the case α = 1. We define

ch Γ(X, Ω_{X}^{ch}) := ∑_{i≥0} dim Γ(X, Ω_{X}^{ch})_{i} · q^{i}  (3.4.1)

where the subscript _{i} denotes the component of conformal weight i.

Due to 3.3 (i), (ii) we have

ch Γ(X, Ω_{X}^{ch}) = { ∑_{m=0}^{∞} (4m+1)(2m+1)q^{(2m+1)m}+ ∑_{m=0}^{∞} (4m+3)(2m+1)q^{(2m+1)(m+1)}}·

· ∏_{n=1}^{∞} (1 − q^{n})^{−3} = ∑_{m∈ℤ} (4m + 1)(2m + 1)q^{(2m+1)m} · ∏_{n=1}^{∞} (1 − q^{n})^{−3}  (3.4.2)

We want to transform this to a more familiar expression.

3.5. Introduce the standard notation

η(q) = q^{1/24} ∏_{n=1}^{∞} (1 − q^{n})  (3.5.1)
G_{2}(q) = −\frac{1}{24} + ∑_{n=1}^{∞} σ_{1}(n)q^{n}, σ_{1}(n) := ∑_{d|n} d  (3.5.2)

and (with D. Zagier, [Z]) a less standard one

θ(q) = ∑_{m=0}^{∞} q^{(2m+1)^{2}/8}  (3.5.3)

for the generating function counting the triangular numbers.

We have the Gauß formula (cf. [K2], Exercise 12.4, p. 176)

ϕ(q^{2})^{2}/ϕ(q) = ∑_{m∈ℤ} q^{m(2m+1)}  (3.5.3)

where

ϕ(q) = ∏_{n=1}^{∞} (1 − q^{n})  (3.5.4)

from which one deduces easily, using the identity

m(2m + 1) = \frac{(4m + 1)^{2} − 1}{8}  (3.5.5)

that

θ(q) = η(q^{2})^{2}/η(q)  (3.5.6)

Similarly, using the Jacobi formula (cf. loc. cit)

ϕ(q)^{3} = ∑_{m∈ℤ} (4m + 1)q^{m(2m+1)}  (3.5.7)

one deduces that

η(q)^{3} = ∑_{m∈ℤ,m=1(mod4)} mq^{m^{2}/8}  (3.5.8)

where we have used that

∑_{m odd,m>0} m^{i}q^{m^{2}/8} = ∑_{m=1(mod4), m∈ℤ} m^{i}q^{m^{2}/8}  (3.5.9)

We also have

q∂_{q} log η(q) = −G_{2}(q)  (3.5.10)

(see [HBJ], Appendix I, Lemma 4.9).

3.6. It follows from (3.5.6) and (3.5.10) that

q∂_{q} log θ(q) = G_{2}(q) − 2G_{2}(q^{2})  (3.6.1)

On the other hand, using again (3.5.9),

q∂_{q} θ(q) = \frac{1}{8} ∑_{m∈ℤ,m=1(mod4)} m^{2}q^{m^{2}/8}  (3.6.2)

Returning to our character (3.4.2) we deduce (using (3.5.5))

ch Γ(X, Ω_{X}^{ch}) = q^{−1/8} ∑_{n=1(mod4), n∈ℤ} \frac{n(n + 1)}{2}q^{n^{2}/8} =

= \frac{1}{2}\{8q∂_{q} θ(q) + η^{3}(q)\}η(q)^{−3} = \frac{1}{2} + \frac{4θ(q)}{η(q)^{3}}\{G_{2}(q) − 4G_{2}(q^{2})\}  (3.6.3)

Note that

(θ(q)/η(q)^{3})^{4} = η(q^{2})^{8}/η(q)^{16} = (16ϵ(q))^{−1}  (3.6.4)

where ϵ = (e_{1} − e_{2})(e_{1} − e_{3}) is a standard modular form of weight 4 with respect to Γ_{0}(2), cf. [HBJ], Appendix I, Cor. 4.11.

3.7. Now let us consider the case of an arbitrary nonnegative half-integer α. From 3.3 we have

ch H^{0}(X, Ω_{X,(α)}^{ch}) = ∑_{(m,j)∈Q_{+}∪−Q_{−}} (4jα^{2}+2mα+1)q^{(mα+2jα^{2})(1+mα+2jα^{2})/2α^{2}}·ϕ(q)^{−3} =

where Q_{+} = {(m, j) ∈ ℤ^{2}|m ≥ 0, j ≥ 0}; Q_{−} = {(m, j) ∈ ℤ^{2}|m < 0, j < 0}; now make change of variables m + 2αj = d + 2αk, 0 ≤ d < 2α,

= q^{−1/8α^{2}} ∑_{d=0}^{2α−1} ∑_{k∈ℤ} (k + 1)(2α(2αk + d) + 1)q^{2α^{2}(k+ \frac{2αd+1}{4α^{2}} )^{2}} · ϕ(q)^{−3} =

= q^{−1/8α^{2}} ∑_{d=0}^{2α−1} ∑_{l∈ℤ+ \frac{2αd+1}{4α^{2}}} \{4α^{2}l + 4α^{2} − 2αd − 1\}lq^{2α^{2}l^{2}} · ϕ(q)^{−3}  (3.7.1)

Introduce the Jacobi theta functions

θ_{n,m}(τ, z) = ∑_{l∈ℤ+n/2m} e^{2πim(l^{2}τ−lz)}, m ∈ \frac{1}{2}ℤ, m ≥ 0, n ∈ ℤ/2mℤ  (3.7.2)

cf. [K2], Example 13.3; as usually q = e^{2πiτ}.

We see that the numerator in (3.7.1) is a finite linear combination of the values ∂_{z}θ_{2αd+1,2α^{2}}(τ, 0) and ∂_{τ} θ_{2αd+1,2α^{2}}(τ, 0). This implies that it is a quasimodular form with respect to a finite index subgroup Γ ⊂ SL(2; ℤ), cf. [KZ].

§4. The first cohomology and a chiral Serre duality

In this section we give a similar description H^{1}(X, Ω_{X,(α)}^{ch}) as a 𝔤̂-module, the tool of trade being the observation that the chiral Serre duality [MS2] carries over to the twisted case word for word. Let us explain the last point very briefly.

4.1. Recall that we have defined in 2.5 a sheaf Ω_{X,(α)} = O_{X} ⊕ O_{X}(−2α) of supercommutative O_{X}-algebras. Let D_{Ω_{X,(α)}} be the algebra of superdifferential operators acting on Ω_{X,(α)}.

Consider the category of left D_{Ω_{X,(α)}}-modules, to be denoted D_{Ω_{X,(α)}} − Mod^{l}, and of restricted left Ω_{X,(α)}^{ch}-modules, to be denoted Ω_{X,(α)}^{ch}−Mod^{l}; restricted means that the subsheaves of negative conformal weights are equal to 0.

To compare D_{Ω_{X,(α)}} − Mod^{l} with Ω_{X,(α)}^{ch} − Mod^{l} introduce, by analogy with [MSV], 6.11, or [MS1], I.4.5., the Weyl functor

𝕎^{l} : D_{Ω_{X,(α)}} − Mod^{l} → Ω_{X,(α)}^{ch} − Mod^{l}  (4.1.1)

to be the left adjoint to the functor of taking the conformal weight zero component.

4.2. Example. 𝕎^{l}(Ω_{X,(α)}) = Ω_{X,(α)}^{ch}.

4.3. Proposition. 𝕎^{l} is an equivalence of categories. △

4.4. Now we want to play a similar game with right modules. Consider the category of right D_{Ω_{X,(α)}}-modules, to be denoted D_{Ω_{X,(α)}} − Mod^{r}, and of restricted right Ω_{X,(α)}^{ch}-modules, to be denoted Ω_{X,(α)}^{ch} − Mod^{r}; here restricted means that the subsheaves of positive conformal weights are equal to 0. (We leave the definition of a right module over a vertex algebra to the reader.)

Define the right Weyl functor

𝕎^{r} : D_{Ω_{X,(α)}} − Mod^{r} → Ω_{X,(α)}^{ch} − Mod^{r}  (4.4.1)

to be the left adjoint to the functor of taking the conformal weight zero component. 4.5. Proposition. 𝕎^{r} is an equivalence of categories. △ 4.6. Let us call a D_{Ω_{X,(α)}}-module smooth if it is a locally free O_{X}-module of

finite rank. Given a smooth D_{Ω_{X,(α)}}-module N, set

N^{∨} = Hom_{O_{X}} (N, Ω_{X}^{1})  (4.6.1)

In virtue of the Serre duality

H^{1}(X, N) ⟶^{∼} H^{0}(X, N^{∨})^{∗}  (4.6.2)

The O_{X}-module N^{∨} admits a canonical structure of a right D_{Ω_{X,(α)}}-module. Property (4.6.2) chiralizes as follows (c.f. [MS2] formula (11.6)). 4.7. Proposition. If N is a smooth D_{Ω_{X,(α)}}-module, then

H^{1}(X, 𝕎(N)) ⟶^{∼} H^{0}(X, 𝕎^{r}(N^{∨}))^{∗}  (4.7.1)

△

4.8. Denote Ω_{X,(α)}^{ch^{∨}} = 𝕎^{r}(Ω_{X,(α)}^{∨}). Example 4.2 and Proposition 4.7 combined give

H^{1}(X, Ω_{X,(α)}^{ch}) = H^{0}(X, Ω_{X,(α)}^{ch∨})^{∗}  (4.8.1)

It is clear that the dual sheaf Ω_{X,(α)}^{ch∨} is very similar to Ω_{X,(α)}^{ch} because both are defined as left adjoint functors. Ω_{X,(α)}^{ch∨} inherits the fermion number grading from Ω_{X,(α)}^{ch} if we place Ω_{X}^{1} in, say, the fermion number 1 component:

Ω_{X,(α)}^{ch∨} = ⊕_{i∈ℤ} Ω_{X,(α)}^{ch∨,i}  (4.8.2)

Further, Ω_{X,(α)}^{ch∨} carries a right action of ŝl_{2}, and this action precomposed with a canonical antiinvolution of ŝl_{2} makes Ω_{X,(α)}^{ch∨} a sheaf of ŝl_{2}-modules of level 2α^{2} − 2.

The analogs of (3.2.1-3.2.3) are as follows:

Γ(U_{0}, Ω_{X,(α)}^{ch∨,1}) = W_{−2,2α^{2}−2},  (4.8.3)
Γ(U_{0}, Ω_{X,(α)}^{ch∨,i}) = W_{−2(i−1)αi−2,2α^{2}−2},  (4.8.4)
Γ(X, Ω_{X,(α)}^{ch∨}) = Γ(U_{0}, Ω_{X,(α)}^{ch∨})^{𝔤−int}  (4.8.5)

Comparing duality (4.8.1) with (4.8.3-5) we arrive at the following analog of Proposition 3.3.

4.9. Proposition. (i) If α ∈ ℤ_{>0}, i ≥ 1, then H^{1}(X, Ω_{X,(α)}^{ch,i}) admits a filtration with quotients equal to the dual Weyl modules V_{2iα−2+4jα^{2},2α^{2}−2}^{∗}, j = 0, 1, 2, …;

(ii) if α ∈ ℤ_{>0}, i < 0, then H^{1}(X, Ω_{X,(α)}^{ch,i}) admits a filtration with quotients equal to the Weyl modules V_{−2iα+4(j+1)α^{2},2α^{2}−2}, j = 0, 1, 2, …;

(iii) if α is irrational then Ω_{X,(α)}^{ch} does not exist but Ω_{X,(α)}^{ch,0} does, and we have

H^{1}(X, Ω_{X,(α)}^{ch,0}) = W_{−2,2α^{2}−2}^{𝔤−int} = 0.  (4.9.1)

△

§5. Free field realization

In this section we want to write down a free field resolution of the vertex superalgebra H^{∗}(X, Ω_{X,(α)}^{ch}). From now on unless mentioned otherwise assume that α ∈ ℤ.

5.1. Consider a 3-dimensional integer lattice L with generators μ, β, ρ and an inner product (μ, β) = (β, μ) = (ρ, ρ) = 1, the other products of basic elements being zero. One associates with it a lattice vertex superalgebra V_{L}. Recall (cf. [K], 5.5) that as a vector space

V_{L} = V ac_{𝔥} ⊗ ℂ[L]  (5.1.1)

where V ac_{𝔥} is the vacuum module over the Heisenberg Lie algebra 𝔥̂ = 𝔥[t, t^{−1}]⊕ℂc, 𝔥 := ℂ ⊗_{ℤ} L, ℂ[L] being the group algebra of L whose basic elements are denoted by e^{γ}, (γ ∈ L). The parity of e^{γ} is equal to

p(e^{γ}) = (γ, γ)mod 2  (5.1.2)

the factor V ac_{𝔥} is purely even.

According to loc. cit., the definition of V_{L} depends also on a choice of certain 2cocycle ϵ : L × L ⟶ ℂ^{∗}. However, the algebras corresponding to different choices of ϵ are canonically isomorphic. We fix one ϵ and banish it from the notation.

As a vertex algebra V_{L} is generated by fields γ(z) := (γ ⊗ 1)(z) and e^{γ}(z) := (1 ⊗ e^{γ})(z) (γ ∈ L).

Thus, V_{L} is ℤ^{3}-graded

V_{L} = ⊕_{(i,j,k)∈ℤ^{3}} V_{L}^{ijk}, V_{L}^{ijk} := V ac_{𝔥} ⊗ e^{iβ+jμ+kρ}  (5.1.3)

We will use the notation V_{L}^{γ} for V_{L}^{ijk} where γ = iβ + jμ + kρ, so

V_{L} = ⊕_{γ∈L} V_{L}^{γ}  (5.1.4)

We also use the ℤ-grading

V_{L} = ⊕_{j∈ℤ} V_{L}^{j}, V_{L}^{j} := ⊕_{(i,k)∈ℤ^{2}} V_{L}^{ijk}  (5.1.5)

The subspace V ac_{𝔥} ⊗ 1 = V_{L}^{000} is purely even vertex subalgebra of V_{L}, to be denoted simply by V ac_{𝔥}. Each V_{L}^{γ} is a vertex module over V ac_{𝔥}; it is an irreducible Heisenberg 𝔤̂-module on whose highest weight 𝔥 acts by means of the character (γ,?).

For γ′ = i′β + j′μ + k′ρ ∈ L, the field e^{γ}(z) has degree (i′, j′, k′), that is, all its Fourrier components act from V_{L}^{γ} = V_{L}^{ijk} to V_{L}^{γ+γ′} = V_{L}^{i+i′,j+j′,k+k′}.

5.2. Remark. The Heisenberg modules V_{L}^{γ} make perfect sense for arbitrary γ ∈ 𝔥 = L_{ℂ}. Also one can write down the usual formula (cf. [K] (5.5.9)) for vertex operators e^{γ′}(z) for an arbitrary γ′ ∈ 𝔥.

These ”fields” act also from V_{L}^{γ} to V_{L}^{γ+γ′}. However, they are multivalued: they have ”monodromy” (whose logarithm is equal to 2πi(γ, γ′)). (Note that for γ = 0, or more generally for such γ that (γ, γ′) ∈ ℤ this monodromy disappears). Their composition acquires monodromy as well. These monodromies is a source of appearance of quantum groups in the picture.

5.3. According to [B], the assignment

b(z) ↦ e^{β}(z), b(z)^{−1} ↦ e^{−β}(z),
a(z) ↦: (μ(z) − ρ(z))e^{−β}(z) :, ϕ(z) ↦ e^{β+ρ}(z), ψ(z) ↦ e^{−β−ρ}(z)  (5.3.1)

determines an embedding of vertex superalgebras

M[b^{−1}] ↪ V_{L}  (5.3.2)

which induces an isomorphism

M[b^{−1}] ⟶^{∼} V_{L}^{0}  (5.3.3)

5.4. Consider an odd (since (μ − ρ, μ − ρ) = 1) field e^{μ−ρ}(z) and denote its zeroth component

d_{M} = ∫ e^{μ−ρ}(z)  (5.4.1)

Here and below we will use the notation ∫ c(z) for the component c_{(0)} of a field c(z) = ∑ c_{(i)}z^{−i−1}.

The operator d_{M} has degree 1 with respect to the grading (5.1.5), and d_{M}^{2} = 0.

5.5. Lemma, [B]. The sequence

0 ⟶ M ⟶ V_{L}^{0} ⟶^{d_{M}} V_{L}^{1} ⟶^{d_{M}} …  (5.5.1)

is exact. △

Here the first arrow is the composition of the obvious embedding M ↪ M[b^{−1}] with the isomorphism (5.2.3).

5.6. We can compute the cohomology superalgebra H^{∗}(X, Ω_{X,(α)}^{ch}) by means of the Cech complex of the covering X = U_{0} ∪ U_{1}:

0 ⟶ M ⊕ M̄ ⟶ M[b^{−1}] ⟶ 0  (5.6.1)

Here M̄ = Γ(U_{1}, Ω_{X,(α)}^{ch}) is simply another copy of M (whose fields we will denote by barred letters). The differential is the obvious embedding on the first component and (taken with the minus sign) composition of the obvious embedding M̄ ↪ M[b^{−1}] and the automorphism of M[b^{−1}] described in Lemma 2.2.

Now we want to replace M, M̄ by their Borisov resolutions (5.5.1) (denote them V_{L} and VL̄) and M[b^{−1}] by the isomorphic algebra V_{L}^{0}, cf. (5.2.3).

The only problem is to lift the differential (5.5.1) to a map of resolutions

V_{L} ⊕ VL̄ ⟶ V_{L}^{0}  (5.6.2)

where we consider V_{L}^{0} as a complex sitting in degree zero.

5.7. Consider an automorphism ϕ_{α} : L ⟶^{∼} L given by

ϕ_{α}(β) = −β, ϕ_{α}(μ) = −μ − 2(α − 1)ρ + 2(α − 1)^{2}β, ϕ_{α}(ρ) = ρ − 2(α − 1)β (5.7.1)

One checks that ϕ_{α} respects the inner product defined in 5.1 and that ϕ_{α}^{2} = Id.

Hence it defines an involution of V_{L} which we will denote by the same letter

ϕ_{α} : V_{L} ⟶^{∼} V_{L}  (5.7.2)

5.8. Lemma. If we identify M[b^{−1}] with V_{L}^{0} by means of the isomorphism (4.2.3), then the involution ϕ_{α} (5.7.2) induces the automorphism (2.2.1) of M[b^{−1}]. △

5.9. This lemma provides the necessary tool for defining (5.5.2). We have to define a morphism

d̂: V_{L}^{0} ⊕ V 0 ; L̄ ⟶ V_{L}^{0}  (5.9.1)

It will be the identity on the first component and taken with the minus sign composition of the tautological identification VL̄ = V_{L} and the involution (5.7.2).

Since bicomplex (5.6.2) is a resolution of the Cech complex (5.6.1), the cohomology algebra H^{∗}(X, Ω_{X,(α)}^{ch}) is canonically identified with the cohomology of the complex

0 ⟶ Ker d̂ ⟶^{d_{M} ⊕dM̄} V_{L}^{1} ⊕ V 1 ; L̄ ⟶^{d_{M} ⊕dM̄} V_{L}^{2} ⊕ V 2 ; L̄ ⟶ …  (5.9.2)

Now let us identify VL̄ with V_{L} using the composition of the tautological identification and of (5.6.2). Then (5.8.2) becomes

0 ⟶ V_{L}^{0} ⟶^{(D,D̄)} V_{L}^{1} ⊕ V_{L}^{−1} ⟶^{D⊕D̄} V_{L}^{2} ⊕ V_{L}^{−2} ⟶ …  (5.9.3)

where

D = d_{M} = ∫ e^{μ−ρ}(z) : V_{L}^{i} ⟶ V_{L}^{i+1}  (5.9.4)

cf. (5.3.1), and

D̄ = ∫ e^{μ̄−ρ̄}(z) = ∫ e^{−μ+(1−2α)ρ+2α(α−1)β}(z) : V_{L}^{i} ⟶ V_{L}^{i−1}  (5.9.5)

5.10. It follows from (5.9.3) that

H^{0}(ℙ^{1}, Ω_{X,(α)}^{ch}) = Ker(D : V_{L}^{0} ⟶ V_{L}^{1}) ∩ Ker(D̄ : V_{L}^{0} ⟶ V_{L}^{−1})  (5.10.1)

Thus our space of global sections is the intersection of kernels of two screening operators connected with lattice vectors μ − ρ and μ̄ − ρ̄ respectively. Let us compute the angle between these two vectors:

(μ − ρ, μ̄ − ρ̄) = (μ − ρ, −μ + (1 − 2α)ρ + 2α(α − 1)β) = 2α^{2} − 1 = k + 1 (5.10.2)

(Recall that k = 2α^{2} − 2 is the central charge of 𝔤̂ acting on our modules.)

5.11. Set V_{L}^{′0} = ⊕_{i∈ℤ} V_{L}^{i00} ⊂ V_{L}^{0}. The operators D, D̄ make sense on V_{L}^{′0} for all α ∈ ℂ not necessarily integer, cf. 4.2. Intersection of their kernels is equal to H^{0}(X, Ω_{X,(α)}^{ch,0}), which due to Proposition 3.3 (iii) is V_{0,2α^{2}−2}; as far as we can judge, this is Feigin’s result [F]. The restriction of (5.9.2) to V_{L}^{′0} is the resolution appearing, perhaps implicitly, in [F]. It is indeed a resolution due to Proposition 4.9 (iii). If α is not generic, then at most the 1st cohomology arise and its structure is provided by Proposition 4.9 (i,ii). The size of the 0th cohomology group also jumps at special values of α as Proposition 3.3 (i-ii) shows.

References

[B] L. Borisov, Vertex algebras and mirror symmetry, Comm. Math. Phys., 215 (2001), 517-557.

[BL] L. Borisov, A. Libgober, Elliptic genera of toric varieties and applications to mirror symmetry, Inv. Math., 140 (2000), 453-485.

[F] B. Feigin, Super quantum groups and the algebra of screening for ŝl_{2}-algebra, Preprint.

[FF1] B. Feigin, E. Frenkel, A family of representations of affine Lie algebras, Uspechi Math. Nauk, 43, no. 5 (263) (1988), 227-228 (russian).

[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, Representations of affine Kac-Moody algebras and bosonisation, V.G.Knizhnik Memorial volume, L. Brink, D. Friedan, A.M. Polyakov (eds.), World Scientific, Singapore (1990), 271-316.

[FF4] B. Feigin, E. Frenkel, Integrals of motion and quantum groups, Integrable systems and quantum groups, R. Donagi, B. Dubrovin, E. Frenkel, E. Previato, Lect. Notes in Math., 1620 (1995), 349-418.

[GMS1] V. Gorbounov, F. Malikov, V. Schechtman, Gerbes of chiral differential operators. II, math.AG/0003170.

[GMS2] V. Gorbounov, F. Malikov, V. Schechtman, Gerbes of chiral differential operators. III, math.AG/0005201, to appear in A.A. Kirillov Festschrift (Birkhäuser).

[HBJ] F. Hirzebruch, Th. Berger, R. Jung, Manifolds and modular forms, 2nd ed., Aspects of Mathematics, Vieweg, 1994.

[K1] V. Kac, Vertex algebras for beginners, University Lecture Series, 10, American Mathematical Society, Providence, Rhode Island, 1997.

[K2] V. Kac, Infinite dimensional Lie algebras, 2nd ed., Cambridge University Press, Cambridge et al., 1985.

[KZ] M. Kaneko, D. Zagier, A generalized Jacobi theta function and quasimodular forms, The moduli space of curves (Texel Island, 1994), Progr. Math., 129 (1995), Birkhäuser Boston, MA, 165-172.

[MS1] F. Malikov, V. Schechtman, Chiral de Rham complex. II, Amer. Math. Soc. Transl., 194 (1999), 149-188.

[MS2] F. Malikov, V. Schechtman, Chiral Poincaré duality, Math. Res.Lett 6 (1999), 533-546

[MSV] F. Malikov, V. Schechtman, A. Vaintrob, Chiral de Rham complex, Comm. Math. Phys., 204 (1999), 439-473.

[W] M. Wakimoto, Fock representations of the affine Lie algebra A_{1}^{1}, Comm. Math. Phys., 104 (1986), 605-609.

[Z] D. Zagier, A proof of the Kac-Wakimoto affine denominator formula for the strange series, Math. Res. Let., 7 (2000), 597-604.

V.G.: Department of Mathematics, University of Kentucky, Lexington, KY 40506, USA; vgorb@ms.uky.edu

F.M.: Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA; fmalikov@mathj.usc.edu

V.S.: Laboratoire Emile Picard, UFR MIG, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France; schechtman@picard.ups-tlse.fr