Vertex operator algebras and the representation theory of toroidal algebras

Stephen Berman^{∗}

Department of Mathematics and Statistics University of Saskatchewan, Saskatoon, Canada

Yuly Billig^{∗}

School of Mathematics and Statistics Carleton University, Ottawa, Canada

Jacek Szmigielski^{∗}

Department of Mathematics and Statistics

University of Saskatchewan Saskatoon, Canada

February 22, 2001

Abstract

An explicit vertex operator algebra construction is given of a class of irreducible modules for toroidal Lie algebras. AMS (MOS) Subject Classifications:17B69, 17B68, 17B66, 17B10. Key words: vertex operators, toroidal Lie algebras, Heisenberg algebras, Fock space.

∗Research supported by the Natural Sciences and Engineering Research Council of Canada

1 Introduction

This paper is about representations of toroidal Lie algebras and vertex operator algebras which are naturally associated to them. In fact, we find the picture is much like the case of the affine Kac-Moody Lie algebras, where, for quite some time now, VOAs have played an important role in the study of their representations. Indeed, this link with VOAs and affine Kac-Moody Lie algebras has been one of the driving forces for both the representation theory of these algebras as well as for the development of the theory of VOAs. Now toroidal Lie algebras are multi-variable generalizations of the affine algebras and so it is extremely natural to consider VOAs associated to their representation theory. In fact, roughly speaking, one obtains a toroidal Lie algebra associated to any finite dimensional simple Lie algebra over ℂ, call it 𝔤̇, as follows. One tensors 𝔤̇ with the Laurent polynomials in several, say N + 1 where N ≥ 1, variables to get a “multi-loop” algebra and then takes the universal central extension of this algebra. Next, just as in the affine case, one adds some derivations. The result is a toroidal algebra. Here however, the reader should understand that already this process is quite a bit more complicated than the affine case due to the fact the universal central extension will always be infinite dimensional as N + 1 ≥ 2 and hence the derivation algebra we need to add will also be infinite dimensional. Letting R denote this ring of Laurent polynomials one knows,[11], that the space of 1-forms modulo exact forms, we will denote this space by K, needs to be added to 𝔤̇ ⊗ R to get the universal central extension and that the algebra D^{∗} which we need to add is a subalgebra of Der(R). Thus, our toroidal Lie algebra looks like

𝔤̇ ⊗ R ⊕ K ⊕ D^{∗}.  (1.1)

In the first section of this paper we will define all of this rigorously. Here, we want to mention that, in fact, there is a possible cocycle

τ : D^{∗} × D^{∗} → K,  (1.2)

which enters into the definition of the toroidal algebra. Thus, we denote the resulting toroidal algebra by

𝔤_{τ} = 𝔤̇ ⊗ R ⊕ K ⊕ D^{∗}.  (1.3)

In our set-up these possible cocycles come from a 2-dimensional space and hence parametrize the class of toroidal algebras which we will work with here. One of these cocycle was first discovered in [5] where the authors gave a representation, via vertex operators, for one of the algebras, 𝔤_{τ}. This cocycle was produced by the representation itself and the authors of this work did the appropriate computations necessary to see exactly what this cocycle was when a particular algebra of derivations, D^{∗}, was used. This has influenced the entire theory as right from its start both vertex operators and cocycles were present.

In the work [3] the author gave what could be called the principal realization by vertex operators for this same toroidal algebra with the cocycle discovered in [5]. This time the associated representation was irreducible and the space of the representation, the so-called Fock space, was just a ring of Laurent polynomials in N variables tensored with a symmetric algebra in infinitely many variables. The work [2], building on [3] as well as on [4, 12], produced infinite families of irreducible representations for all of the algebras 𝔤_{τ}. Moreover, these representations have finite dimensional weight spaces relative to a natural Cartan subalgebra of the toroidal algebra 𝔤, and besides this, they are as faithful as is possible. What this means is the following. The toroidal algebra has an N + 1 dimensional centre (most of the elements from K do not remain central when D^{∗} is added) and so in any representation which is irreducible and has finite dimensional weight spaces one must have a kernel which is at least N dimensional by Schur’s Lemma. The representations constructed in [2] have kernels which are exactly N dimensional. Thus, they are as faithful as they possibly can be. We proceed to describe them in a little more detail here.

Let V = K ⊕ D^{∗} and let W be any finite dimensional gl_{N}-module. Next, let T (W) = R_{N} ⊗ W where R_{N} is the ring of Laurent polynomials in N variables. Then, as seen from [4, 12], T (W) becomes a module for Der(R_{N}). This so called tensor module corresponds geometrically to the action of vector fields on tensor fields by the Lie derivative. Now V has a natural ℤ gradation which corresponds to the first variable, t_{0} in the natural ℤ^{N+1}-gradation of V and hence this allows us to write

V = V_{−} ⊕ V_{0} ⊕ V_{+}  (1.4)

where V_{0} is all elements of degree 0 and V_{+} (respectively V_{−}) is all elements of positive (respectively negative) degree. Now it is not hard to see that V_{0} acts on T (W) and this introduces a scalar c, called the central charge, into this construction. Moreover, it is true that T (W) is an irreducible V_{0} module provided only that W is an irreducible gl_{N} module and c ≠ 0, see [2] Proposition 2.25. Just as in the case of affine Kac-Moody Lie algebras one forms the V module

\overline{U}(W, c) = Ind_{V_{+}⊕V_{0}}^{V}T (W)  (1.5)

(these modules were called M(W, c) in [2]).

It is not hard to see that this V module has a unique submodule intersecting the top T (W) trivially and if W is chosen to be irreducible then the corresponding quotient module, denoted L(W, c), is irreducible and has finite dimensional weight spaces (see [2],Theorem 4.9). The finite dimensionality is quite non-trivial and is seen because of the notion of polynomial multiplication which is introduced in [2].

In the final stage of this construction one needs to take the special derivation algebra

D^{∗} = {∑_{p=1}^{N} f_{p}d_{p}|f_{p} ∈ R},  (1.6)

where d_{p} denotes the degree derivation of R corresponding to the pth variable t_{p} but where any of the cocycles τ are allowed. Then we tensor one of the modules obtained above for V with a highest weight module for the nontwisted affine Lie algebra associated to 𝔤̇. Here both modules must have the same central charges. We then obtain modules for the toroidal algebra 𝔤_{τ} which again, if our initial data satisfies that W is irreducible, c ≠ 0 and the module for the affine algebra is irreducible, then yields an irreducible module for our toroidal algebra with finite dimensional weight spaces and which is as faithful as possible in the sense already described.

In many ways the above picture resembles the representation theory of highest weight modules for the affine Kac-Moody Lie algebras and hence, it became clear that one should look for VOAs here as well. Looking at the representation given in [3] one certainly expects an affine VOA, as well as a sub-VOA of a a hyperbolic lattice VOA to play a role here. There is also the possibility of duplicating the theory of affine VOAs as presented in [9, 14] in this toroidal setting and in fact , in our first attempt we did exactly this. However, this approach only identifies some of the above modules as VOAs and so does not at all address the possibility of giving realizations for these modules using VOA theory. This is in fact what we accomplish in this work. We are able to give VOAs V which appear as a tensor product of three fairly well-known VOAs and then we show that this VOA is isomorphic to one of the above irreducible toroidal modules. Moreover, just as in the affine case, we are able to show that the toroidal modules introduced in [2], will in fact be modules for this VOA. Since the characters of the three pieces in our realizations are already known, it becomes clear that the characters of the modules from [2] are just products of these known characters.

We next describe the three types of VOAs which will make up V. Here we only describe these VOAs roughly but later in the body of the paper we shall develop precise notation for all of them. Let ˆ 𝔤̇ be the untwisted affine algebra, with central element denoted k_{0}, built on 𝔤̇ and let Vˆ𝔤̇ be the Verma VOA associated to ˆ 𝔤̇ (Vˆ𝔤̇ is a generalized Verma module for 𝔤̇) with central charge c. Here we assume c ≠ −h^{∨} where h^{∨} is the dual Coxeter number of 𝔤̇. Then one knows that the quotient module Lˆ𝔤̇ is an irreducible highest weight module of ˆ 𝔤̇. Next, let Lat be the hyperbolic lattice with basis {a_{p}, b_{p}|1 ≤ p ≤ N} where the a_{p}’s and b_{p}’s span isotropic subspaces and satisfy (a_{p}, b_{q}) = δ_{p,q}. We let V_{Lat} be the lattice VOA associated to this even lattice. Then we have

V_{Lat} = S_{H} ⊗ ℂ[Lat],  (1.7)

where S_{H} is a symmetric algebra and ℂ[Lat] is a twisted group algebra of the group algebra of Lat. We then let

V_{Lat}^{+} = S_{H} ⊗ ℂ[Lat^{+}]  (1.8)

be the sub-VOA associated to the isotropic subspace, Lat^{+}, of Lat spanned by the elements a_{p}, 1 ≤ p ≤ N. Our third VOA is associated to the Lie algebras ĝl_{N} which we take to be the direct sum of ŝl_{N} with the central element C_{1} and a Heisenberg Lie algebra, Hei generated by the identity matrix and the central element C_{2} so that

Hei = I ⊗ ℂ[t_{0}, t_{0}^{−1}] ⊕ ℂC_{2}.  (1.9)

Then the VOA associated to this algebra ĝl_{N} is taken to be the tensor product of the Verma VOA associated to the affine Kac-Moody Lie algebra ŝl_{N} with the simple Heisenberg VOA associated to the Heisenberg algebra Hei above. We denote the first of these by V_{ŝl_{N}} and the second one by L_{Hei}. Thus, we are letting

V_{ĝl_{N}} = V_{ŝl_{N}} ⊗ L_{Hei},  (1.10)

be the VOA associated to the algebra ĝl_{N}. Then our VOA V is given as

V = Vˆ𝔤̇ ⊗ V_{Lat}^{+} ⊗ V_{ĝl_{N}} .  (1.11)

Actually, because of the various choices for central charges, namely c, c_{1}, c_{2} from above, we obtain a family of VOAs. Moreover, we will show that c_{1}, c_{2} correspond to different choices of the cocycle τ, while c determines a cocycle associated with the action of the Virasoro algebra on V. All of this is explained later in the paper when the proper notation and results have been established.

The proof of our result depends on carrying out certain computations in the VOA V. What we need to do is to define the proper vertex operators, Y (v, z) for certain v ∈ V which will be the images of the specified generating fields of the toroidal algebra, and then to show they satisfy the same commutation relations. This involves some fairly delicate computations, but these computations are straightforward enough for those well versed in VOAs. After this is done, the way is clear to show that modules for the VOA V correspond to modules for the toroidal algebra 𝔤_{τ} and to then explain how the fairly well-known irreducible modules for the VOAs involved provide a model for the modules from [2]. It is really only in this last part that a good knowledge of [2] is necessary. We have recalled here, in this Introduction, much from that work only to show what our motivation for the present work is.

In the second section of this work we give the definition of the toroidal algebras as well as the cocycles τ. We then present the generating fields which we use and display their commutation relations. One of these fields K_{0}(m, z) is special and it has conformal weight different from that of the other central fields. We close this section with a simple Lemma which allows us to conclude certain commutation relations hold once we know that some others do. This lets us cut down on the amount of computation we must do.

In the next section we begin by recalling the definition of a VOA. Here, we are dropping the requirement that the grading consists of finite dimensional homogeneous spaces but we do require all other axioms. But note that when we identify the modules from [2] we will know that then the homogeneous pieces involved do have a ℤ^{N} gradation which has finite dimensional homogeneous spaces. We then go on to discuss the particular types of VOAs we need. As above, these include affine VOAs, lattice VOAs, and Heisenberg VOAs. Most of this material is by now quite well known, so we are brief, and only develop the material to the point where we have enough notation and information to accomplish the computations which we do in the following section.

The fourth section is the heart of the paper and here is where we present the details of our computation. We must check the various commutation relations case by case. We do this in such a way so as to give the reader a feeling for working with the various VOAs involved and to show how the various pieces from the tensor product of our three VOAs fit together to produce the toroidal algebra. Almost everything important that is taking place here is involved with negative powers of z so we often just work with the “anti-holomorphic” part of our series. We indicate this when we do this. After accomplishing this we go on, in the final section , to tie up our results to the modules from [2]. We point out how irreducible modules for our VOA V coincide with irreducible modules for 𝔤_{τ}.

We close this introduction with several remarks. First, we want to mention that there is another very nice and quite different work by K. Saito and D. Yoshii on toroidal algebras and VOA’s, [16]. They work with a lattice VOA attached to the full hyperbolic lattice and use the realization given in [5, 6]. They then look at a certain Lie algebra attached to this VOA, namely the VOA factored by L(−1) of it, and find the toroidal algebra is isomorphic to a subalgebra of this Lie algebra. They do not use any derivations of the toroidal algebra and so just work with the algebra 𝔤̇ ⊗ R ⊕ K. One feature of their work is to give different descriptions of the same toroidal Lie algebra. The second remark we want to make here is that the part of our VOA V associated to ĝl_{N} may seem a bit mysterious to some. It was in the insightful work of T. Larsson [12, 13] that this part seems to have been first understood. This insight of Larsson was built into [2] since, at the very beginning of the construction there, one begins with a finite dimensional gl_{N} module. This then naturally leads to the VOA V_{ĝl_{N}}.

Regarding the references to a fast growing literature on VOAs we mention only those which we used directly in this work, to wit, the references [7, 8, 10, 14, 15] were all quite helpful to us at various stages of this work and served as our basic sources for information on VOAs. In addition, the course given at the Fields Institute by Chongying Dong during the fall term of 2000 was extremely valuable to all three of us.

While working on this project the authors greatly benefited from many very helpful discussions with both Chongying Dong and Yi-Zhi Huang. This project certainly benefited from their help, encouragement, and insights, and the authors want to take this opportunity to offer thanks to them for this.

2 Toroidal algebras

For information and notation on toroidal algebras we will follow the work [2]. However we recall what we need here.

Let N ≥ 1 be an integer and let R = ℂ[t_{0}^{±1}, t_{1}^{±1}, ..., t_{N}^{±1}] be the ring of Laurent polynomials in N + 1 variables over the complex field and let 𝔤̇ be a finite dimensional Lie algebra over ℂ. We let K = Ω_{R}^{1}/dR be the space of 1forms modulo the exact forms. For notation we let fdg denote the element of K corresponding to the pair of elements f, g from R. We let k_{i} = t_{i}^{−1}dt_{i} .Then we have that K is spanned by elements of the form t^{m}k_{i} where m ∈ ℤ^{N+1}, 0 ≤ i ≤ N. Here we are using for notation that t^{m} = t_{0}^{m_{0}}t_{1}^{m_{1}} … t_{N}^{m_{N}} where m = (m_{0}, m_{1}, ..., m_{N}) ∈ ℤ^{N+1}. The relations staisfied by these elements are that

∑_{p=0}^{N} m_{p}t^{m}k_{p} = 0, m ∈ ℤ^{N+1}.  (2.1)

The space K is important for giving the universal central extension of the Lie algebra 𝔤̇ ⊗ R, where the bracket in 𝔤̇ ⊗ R ⊕ K is given by

[g_{1} ⊗ f_{1}(t), g_{2} ⊗ f_{2}(t)] = [g_{1}, g_{2}] ⊗ (f_{1}f_{2}) + (g_{1}, g_{2})f_{2}d(f_{1}),  (2.2)

where (·, ·) is a symmetric non-degenerate invariant form on 𝔤̇ normalized by (θ, θ) = 2, where θ is the longest root of 𝔤̇.

In agreement with the affine case, we add certain outer derivations to this algebra. More precisely, we consider the following algebra of derivations:

D^{∗} = ∑_{p=1}^{N} Rd_{p},  (2.3)

where d_{j} = t_{j} \frac{∂}{∂t_{j}}. The derivations of R naturally extend to derivations of the Lie algebra 𝔤̇ ⊗ R. Since the above algebra, 𝔤̇ ⊗ R ⊕ K, is the universal central extension of 𝔤̇ ⊗ R we can lift these derivations to this universal central extension by using a result of Benkart-Moody [1], which says that every derivation of a perfect Lie algebra uniquely extends to a derivation of its universal central extension. The action of D^{∗} on 𝔤̇ ⊗ R ⊕ K is given by

[t^{m}d_{j}, g ⊗ t^{r}] = r_{j}g ⊗ t^{m+r},  (2.4)
[t^{m}d_{j}, t^{r}k_{i}] = r_{j}t^{m+r}k_{i} + δ_{ji} ∑_{p=0}^{N} m_{p}t^{m+r}k_{p}.  (2.5)

The first formula corresponds to the action of vector fields on functions, while the second is the Lie derivative action of vector fields on 1-forms.

The formulas (2.4), (2.5) identify the bracket in D^{∗} only up to a K-valued 2-cocycle τ ∈ H^{2}(D^{∗}, K):

[t^{m}d_{i}, t^{r}d_{j}] = r_{i}t^{m+r}d_{j} − m_{j}t^{m+r}d_{i} + τ(t^{m}d_{i}, t^{r}d_{j}).  (2.6)

We will use a two-dimensional space of cocycles as in [2] (see the remarks between (2.12) and (2.13) there). It is spanned by the two cocycles

τ_{1}(t^{m}d_{i}, t^{r}d_{j}) = m_{j}r_{i} ∑_{p=0}^{N} r_{p}t^{m+r}k_{p} = −m_{j}r_{i} ∑_{p=0}^{N} m_{p}t^{m+r}k_{p},  (2.7)

and

τ_{2}(t^{m}d_{i}, t^{r}d_{j}) = m_{i}r_{j} ∑_{p=0}^{N} r_{p}t^{m+r}k_{p} = −m_{i}r_{j} ∑_{p=0}^{N} m_{p}t^{m+r}k_{p}.  (2.8)

Thus we obtain a two-parametric family of algebras 𝔤 = 𝔤_{τ} = 𝔤̇⊗R⊕K⊕D^{∗}, where the cocylce in (2.6) is a linear combination of (2.7) and (2.8): τ = μτ_{1} + ντ_{2}. A consequence of the fact that the action (2.5) of D^{∗} on K is nontrivial, is that the center of the algebra 𝔤̇ ⊗ R ⊕ K ⊕ D^{∗} is finite-dimensional and is spanned by k_{0}, k_{1}, …, k_{N}.

As in the affine case, we can also consider a semidirect product of 𝔤 with the Virasoro algebra

V ir = ℂ[t_{0}, t_{0}^{−1}]d_{0} ⊕ ℂC_{V ir}.  (2.9)

The Lie bracket in the Virasoro algebra is given by the well-known formula

[t_{0}^{n}d_{0}, t_{0}^{m}d_{0}] = (m − n)t_{0}^{n+m}d_{0} + \frac{n^{3} − n}{12}δ_{n,−m}C_{V ir}  (2.10)

and the element C_{V ir} is central.

The action of V ir on 𝔤̇ ⊗ R ⊕ K is given by (2.4), (2.5), where we let j = 0 and t^{m} = t_{0}^{m}, while the action on D^{∗} is similar to (2.6), only we use a different cocycle:

[t_{0}^{n}d_{0}, t^{m}d_{i}] = m_{0}t_{0}^{n}t^{m}d_{i} − ρm_{i}n(n + 1)t_{0}^{n}t^{m}k_{0}, ρ ∈ ℂ.  (2.11)

This new cocycle emerged from the representations of 𝔤 ⊕ V ir that we study below. Later we will see that we need to specify an exact value of ρ and this will depend on the particular value of the central charge we are using.

We next introduce the following formal fields in (𝔤 ⊕ V ir)[[z, z^{−1}]] whose moments together with C_{V ir} span 𝔤 ⊕ V ir. Notice that in the single field K_{0}(m, z) below the coefficient of z^{−j−1} involves t^{j+1} while in all other fields it only involves t^{j}. It turns out that this is important when we begin to line up our fields with elements in a VOA as it is tied up with the conformal weights, (see our comment below).

g(m, z) = ∑_{j} g ⊗ t_{0}^{j}t^{m}z^{−j−1}, g ∈ 𝔤  (2.12)
K_{0}(m, z) = ∑_{j} t_{0}^{j}t^{m}k_{0}z^{−j}  (2.13)
K_{s}(m, z) = ∑_{j} t_{0}^{j}t^{m}k_{s}z^{−j−1}, 1 ≤ s ≤ N  (2.14)
D_{s}(m, z) = ∑_{j} t_{0}^{j}t^{m}d_{s}z^{−j−1}, 1 ≤ s ≤ N  (2.15)
L(z) = − ∑_{j} t_{0}^{j}d_{0}z^{−j−2}  (2.16)

Here and in what follows, m ∈ ℤ^{N} and t^{m} = t_{1}^{m_{1}} … t_{N}^{m_{N}}.

The Lie algebra structure of 𝔤 ⊕ V ir is encoded in the following relations for the above fields:

\frac{d}{dz} K_{0}(m, z) = ∑_{p=1}^{N} m_{p}K_{p}(m, z)  (2.17)

[g_{1}(m, z_{1}), g_{2}(r, z_{2})] =\{[g_{1}, g_{2}](m + r, z_{2}) + (g_{1}, g_{2}) ∑_{p=1}^{N} m_{p}K_{p}(m + r, z_{2})\}z_{1}^{−1}δ(\frac{z_{2}}{z_{1}} ) +
(g_{1}, g_{2})K_{0}(m + r, z_{2})z_{1}^{−2}δ^{(1)}(\frac{z_{2}}{z_{1}} )  (2.18)
[g(m, z_{1}), K_{i}(r, z_{2})] = 0  (2.19)
[K_{i}(m, z_{1}), K_{j}(r, z_{2})] = 0  (2.20)
[D_{s}(m, z_{1}), g(r, z_{2})] = r_{s}g(m + r, z_{2})z_{1}^{−1}δ(\frac{z_{2}}{z_{1}} )  (2.21)
[L(z_{1}), g(m, z_{2})] = (\frac{∂}{∂z_{2}} (g(m, z_{2}))) z_{1}^{−1}δ(\frac{z_{2}}{z_{1}} ) + g(m, z_{2})z_{1}^{−2}δ^{(1)}(\frac{z_{2}}{z_{1}} ) (2.22)

[D_{s}(m, z_{1}), K_{0}(r, z_{2})] = r_{s}K_{0}(m + r, z_{2})z_{1}^{−1}δ(\frac{z_{2}}{z_{1}} )  (2.23)
[D_{s}(m, z_{1}), K_{q}(r, z_{2})] =\{r_{s}K_{q}(m + r, z_{2}) + δ_{s,q} ∑_{p=1}^{N} m_{p}K_{p}(m + r, z_{2})\}z_{1}^{−1}δ(\frac{z_{2}}{z_{1}} )
+ δ_{s,q}K_{0}(m + r, z_{2})z_{1}^{−2}δ^{(1)}(\frac{z_{2}}{z_{1}} )  (2.24)
[L(z_{1}), K_{0}(m, z_{2})] = (\frac{∂}{∂z_{2}} (K_{0}(m, z_{2}))) z_{1}^{−1}δ(\frac{z_{2}}{z_{1}} )  (2.25)
[L(z_{1}), K_{s}(m, z_{2})] = (\frac{∂}{∂z_{2}} (K_{s}(m, z_{2}))) z_{1}^{−1}δ(\frac{z_{2}}{z_{1}} ) + K_{s}(m, z_{2})z_{1}^{−2}δ^{(1)}(\frac{z_{2}}{z_{1}} ) (2.26)

[D_{s}(m, z_{1}), D_{q}(r, z_{2})] =\{r_{s}D_{q}(m + r, z_{2}) − m_{q}D_{s}(m + r, z_{2})\}z_{1}^{−1}δ(\frac{z_{2}}{z_{1}} )

− (μr_{s}m_{q} + νr_{q}m_{s}) ∑_{p=1}^{N} m_{p}K_{p}(m + r, z_{2})z_{1}^{−1}δ(\frac{z_{2}}{z_{1}} )

− (μr_{s}m_{q} + νr_{q}m_{s})K_{0}(m + r, z_{2})z_{1}^{−2}δ^{(1)}(\frac{z_{2}}{z_{1}} ) (2.27)

[L(z_{1}), D_{s}(m, z_{2})] =\frac{∂}{∂z_{2}} (D_{s}(m, z_{2}))z_{1}^{−1}δ(\frac{z_{2}}{z_{1}} ) + D_{s}(m, z_{2})z_{1}^{−2}δ^{(1)}(\frac{z_{2}}{z_{1}} )
+ ρm_{s}K_{0}(m, z_{2})z_{1}^{−3}δ^{(2)}(\frac{z_{2}}{z_{1}} )  (2.28)
[L(z_{1}), L(z_{2})] = (\frac{∂}{∂z_{2}} (L(z_{2}))) z_{1}^{−1}δ(\frac{z_{2}}{z_{1}} ) + 2L(z_{2})z_{1}^{−2}δ^{(1)}(\frac{z_{2}}{z_{1}} ) +
\frac{C_{V ir}}{12} z_{1}^{−4}δ^{(3)}(\frac{z_{2}}{z_{1}} )  (2.29)

where 1 ≤ s, q ≤ N, 0 ≤ i, j ≤ N, m, r ∈ ℤ^{N} and δ^{(n)}(z) is the n-th derivative of the delta series δ(z) = ∑_{j∈ℤ} z^{j}.

Next we list the conformal weights of all the fields involved. We recall that the conformal weights are defined relative to the operator L(0) which appears in the Virasoro field:

L(z) = ∑_{n∈ℤ}L(n)z^{−n−2}

by declaring that the local field b(z) has conformal weight Δ if

[L(0), b(z)] = (z \frac{d}{dz} + Δ)b(z).  (2.30)

Since the field L(z) satisfies the Virasoro commutation relations we will use the coefficient at the −2nd power of z in the expansion of L(z). Thus to read off the conformal weights so defined from (2.25), (2.26), we extract the coefficient of z_{1}^{−2}. We obtain that K_{0}(m, z) has conformal weight 0, D_{s}(m, z), K_{s}(m, z), g(m, z) have conformal weight 1 while L(z) has conformal weight 2.

In order to prove that a map ϕ : 𝔤 → End(V) is a representation of 𝔤, we would need to verify that the relations (2.18)-(2.29) hold in End(V). We note that (2.23)-(2.26) follow from (2.18)-(2.22) since a derivation of a Lie algebra is uniquely determined by its action on the generators, and the space 𝔤̇⊗R generates the algebra 𝔤̇⊗R⊕K. To make this more rigorous, we prove the following simple Lemma:

Lemma 2.1 Let ϕ : G → End(V) be a representation of a Lie algebra G and d be a derivation of G. Let the subspace X ⊂ G generate G as a Lie algebra. If d′ ∈ EndV satisfies ϕ ∘ d|_{X} = ad(d′) ∘ ϕ|_{X} then

(1) ϕ ∘ d = ad(d′) ∘ ϕ on G and

(2) The representation ϕ can be extended to a representation of a semidirect product ϕ : G ⊕ ℂd → End(V) by ϕ(d) = d′.

Proof. Since X generates G as a Lie algebra, every element g ∈ G may be written as g = ∑_{s}[u_{s}, v_{s}] + w, where u_{s}, v_{s}, w ∈ X. Then

ϕ(d(g)) = ϕ (∑_{s}[d(u_{s}), v_{s}] + [u_{s}, d(v_{s})] + d(w)) =

∑_{s}[ϕ(d(u_{s})), ϕ(v_{s})] + [ϕ(u_{s}), ϕ(d(v_{s}))] + ϕ(d(w)) =

∑_{s}[ad(d′)ϕ(u_{s}), ϕ(v_{s})] + [ϕ(u_{s}), ad(d′)ϕ(v_{s})] + ad(d′)ϕ(w) = ad(d′)ϕ(g).

This proves (1), while (2) immediately follows from (1).

3 Vertex operator algebras

We begin by reviewing some basic facts about vertex operator algebras (VOAs) and begin by recalling the definition. For notation, we follow [7], [8].

Definition 3.1 A vertex operator algebra is a 4-tuple (V,Y,1,ω) where V = ⊕_{n∈ℤ}V_{n} is a ℤ-graded vector space equipped with a linear map

V → (End V)[[z, z^{−1}]] :

v → Y (v, z) = ∑_{n∈ℤ}v_{n}z^{−n−1}, v_{n} ∈ End V

and two distinguished vectors 1 and ω ∈ V satisfying the following conditions:

u_{n}v = 0, for n >> 0 and u, v ∈ V arbitrary,  (3.1)
Y (1, z) = Id  (3.2)
Y (v, z)1 ∈ V [[z]] and v_{−1}1 = v  (3.3)

z_{0}^{−1}δ(\frac{z_{1} − z_{2}}{z_{0}} )Y (v, z_{1})Y (w, z_{2}) − z_{0}^{−1}δ(\frac{z_{2} − z_{1}}{−z_{0}} )Y (w, z_{2})Y (v, z_{1})

= z_{2}^{−1}δ(\frac{z_{1} − z_{0}}{z_{2}} )Y (Y (v, z_{0})w, z_{2}) (Jacobi identity),  (3.4)
V_{n} = {0} for n << 0  (3.5)

where as usual δ(z) = ∑_{n∈ℤ} z^{n} and where all binomial expressions are to be expanded in nonnegative powers of the second variable. The following Virasoro algebra relations must also hold:

[L(m), L(n)] = (m − n)L(m + n) + \frac{1}{12}(m^{3} − m)δ_{m+n,0}(rankV)Id, (3.6)

where m, n ∈ ℤ, rankV ∈ ℂ, L(n) = ω_{n+1} so that

Y (ω, z) = ∑_{n∈ℤ}ω_{n}z^{−n−1} = ∑_{n∈ℤ}L(n)z^{−n−2}

and

L(0)v = nv, for v ∈ V_{n}  (3.7)
\frac{d}{dz} Y (v, z) = Y (L(−1)v, z).  (3.8)

This completes the definition. By the above mentioned usual conventions we have that the three-variable delta function is given by:

δ(\frac{z_{1} − z_{2}}{z_{0}} ) = ∑_{n∈ℤ}z_{0}^{−n}(z_{1} − z_{2})^{n} = ∑_{n∈ℤ} ∑_{k∈ℕ}(−1)^{k}\binom{n}{k}z_{0}^{−n}z_{1}^{n−k}z_{2}^{k}  (3.9)

Note that the above is not quite the definition of VOA as found in [7] or [8] as we have not required that the spaces V_{n} be finite dimensional. This causes no problems in what follows and it is well known that most of the basic results in the theory of VOAs remain true without this extra hypothesis.

The following important commutator formula is a consequence of the Jacobi identity:

[Y (u, z_{1}), Y (v, z_{2})] = ∑_{n≥0} \frac{1}{n!} (z_{1}^{−1} (\frac{∂}{∂z_{2}} )^{n} δ (\frac{z_{2}}{z_{1}} )) Y (u_{n}v, z_{2}), (3.10)

Now we review the construction of the tensor product of VOAs that will be useful to us later.

Definition 3.2 Given finitely many vertex operator algebras

(V_{1}, Y_{1}, 1_{1}, ω_{1}), ..., (V_{n}, Y_{n}, 1_{n}, ω_{n})

we call

V = V_{1} ⊗ ⋯ ⊗ V_{n}  (3.11)

the tensor product of vertex algebras V_{1}, ..., V_{n} provided

Y (v_{1} ⊗ ⋯ ⊗ v_{n}, z) = Y_{1}(v_{1}, z) ⊗ ⋯ ⊗ Y_{n}(v_{n}, z), v_{i} ∈ V_{i}  (3.12)
1 = 1_{1} ⊗ ⋯ 1_{n}  (3.13)
ω = ω_{1} ⊗ 1_{2}⊗ ⋯ 1_{n} + ⋯ + 1_{1} ⊗ ⋯ ⊗ 1_{n−1} ⊗ ω_{n}  (3.14)

Then the following theorem holds [7]

Theorem 3.3 The tensor product V of the vertex operator algebras

(V_{1}, Y_{1}, 1_{1}, ω_{1},), ..., (V_{n}, Y_{n}, 1_{n}, ω_{n},)

is again a vertex operator algebra. The rank of V is equal to the sum of ranks of the V_{i}’s. Moreover V is a tensor product of Virasoro algebra modules.

We next go on to review some particular VOA’s. We begin with a lattice VOA and its subVOA which we need.

Lattice vertex algebras

Let Lat be a free abelian group on 2N generators {a_{i}, b_{i} : 1 ≤ i, j ≤ N} We give Lat the structure of a lattice by defining a symmetric bilinear form:

(., .) : Lat × Lat →ℤ by
(a_{i}, b_{j}) = (a_{i}, b_{j}) = 0 (a_{i}, b_{j}) = δ_{i,j}  (3.15)

Note that the form (., .) is non-degenerate and Lat is even, that is (x, x) ∈ 2ℤ. The construction of the VOA associated to Lat proceeds as follows.

First we consider the complexification of Lat:

H = Lat ⊗_{ℤ} ℂ,

and extend (., .) by linearity obtaining a non-degenerate bilinear form on H.

We next “affinize” H as follows. Thus, we define a Lie algebra:

Ĥ = H ⊗ ℂ[t, t^{−1}] ⊕ ℂK

with bracket:

[x ⊗ t^{i}, y ⊗ t^{j}] = i(x, y)δ_{i+j,0}K, [x ⊗ t^{i}, K] = 0.

We denote by S_{H} the symmetric algebra of the space H ⊗t^{−1}ℂ[t^{−1}]. Thus writing α(n) for the element α ⊗ t^{n} for α ∈ H, n ∈ ℤ we have that

S_{H} =< α_{1}(−n_{1})...α_{k}(−n_{k})|α_{i} ∈ H, n_{i} ∈ ℕ, 1 ≤ i ≤ k > .

Here, and throughout, we use pointed brackets, < > to denote the space spanned by the displayed vectors inside the brackets.

We also need a twisted group algebra of Lat, denoted ℂ[Lat], which we now describe. For this we use a two cocycle

ϵ : Lat × Lat → {±1}  (3.16)

which satisfies for all α, β, γ ∈ Lat

ϵ(α, β)ϵ(α + β, γ) = ϵ(β, γ)ϵ(α, β + γ),  (3.17)

ϵ(α, 0) = ϵ(0, α) = 1,  (3.18)

and

ϵ(α, β)ϵ(β, α) = (−1)^{(α,β)}.  (3.19)

The twisted group algebra of Lat which we need will be denoted ℂ[Lat] and it is spanned by elements e^{x} with multiplication given by

e^{α}e^{β} = ϵ(α, β)e^{α+β}.  (3.20)

We have ℂ[Lat] =< e^{x}, x ∈ Lat >. Then the space V_{Lat} = S_{H} ⊗ ℂ[Lat] becomes the VOA attached to the lattice Lat. The details can be found in [8]. For example, we have

[a_{p}(n), b_{q}(m)] = nδ_{p,q}δ_{n+m,0}K,  (3.21)

where K acts as 1. We will need an explicit form of ω as well as L(−1). With this in mind we briefly show below how these two elements are constructed. Let {α_{1}, ..., α_{2N}} be an orthonormal basis of H. We set

1 =1 ⊗ 1,  (3.22)
α(z) = ∑_{n∈ℤ}α(n)z^{−n−1}  (3.23)

ω =\frac{1}{2} ∑_{k=1}^{2N} α_{k}(−1)^{2}  (3.24)

Y (ω, z) =\frac{1}{2} ∑_{k=1}^{2N} : α_{k}(z)α_{k}(z) :  (3.25)

Y (1 ⊗ e^{α}, z) = exp(∑_{n≥1} \frac{α(−n)z^{n}}{n}) exp(− ∑_{n≥1} \frac{α(n)z^{−n}}{n})e^{α}z^{α}  (3.26)

where : : denotes the normally ordered product and e^{α} acts by our twisted multiplication so that e^{α}e^{β} = ϵ(α, β)e^{α+β}. Also z^{α}e^{β} = z^{(α,β)}e^{β}. In particular from (3.22) one obtains, by expanding in powers of z, that

L(m) = \frac{1}{2} ∑_{k=1}^{2N} ∑_{n∈ℤ} : α_{k}(m − n)α_{k}(n) : m ∈ ℤ  (3.27)

The rank of V_{Lat} is equal to 2N.

One can give an explicit form of the map Y. Set

∂_{n} =\frac{1}{n!}(\frac{d}{dz})^{n}  (3.28)
v =β_{1}(−n_{1})...β_{k}(−n_{k}) ⊗ e^{α}  (3.29)

Then

Y (v, z) =: (∂_{n_{1}−1}β_{1}(z)...∂_{n_{k}−1}β_{k}(z))Y (1 ⊗ e^{α}) :  (3.30)

Consider the following sub-lattice Lat^{+} of Lat where Lat^{+} is generated by < a_{i}, i = 1, ..., N >. We now define a subspace of V_{Lat} by saying

V_{Lat}^{+} = S_{H} ⊗ ℂ[Lat^{+}].  (3.31)

We are going to need the fact that V_{Lat}^{+} is a sub-VOA of V_{Lat} and so we now indicate why this is true.

The fact that V_{Lat}^{+} is a sub-VOA of V_{Lat} will follow from us showing that if v ∈ V_{Lat}^{+} then the moments of the series Y (v, z) map V_{Lat}^{+} to itself. Here the moments are just the v_{n}’s where Y (v, z) = ∑_{n∈ℤ} v_{n}z^{−n−1}. However there are only three types of operators which act nontrivially on 1 ⊗ ℂ[Lat]: {α(0), α ∈ Lat}, {z^{α}, α ∈ Lat} and the multiplication operators {e^{α}, α ∈ Lat}. From among them α(0) acts diagonally on 1 ⊗ ℂ[Lat], while z^{α} acts trivially on 1 ⊗ ℂ[Lat^{+}] if α ∈ Lat^{+}. Finally, since Lat^{+} is a sub-lattice the twisted multiplication operator e^{α} maps 1 ⊗ ℂ[Lat^{+}] into itself if α ∈ Lat^{+}. Finally, since (3.30) contains products of these three types of operators and operators which act trivially on 1 ⊗ ℂ[Lat] the fact that V_{Lat}^{+} is a sub-VOA of V_{Lat} follows.

One can also give an explicit form of the map Y of this sub-VOA V_{Lat}^{+}. Note that here, as usual, we are using the same symbol, Y, for both VOA’s V_{Lat} and V_{Lat}^{+} but this should cause no confusion. In fact, for v ∈ V_{Lat}^{+} we see that

Y (v, z) = ∑_{n∈ℤ}v_{n}|_{V_{Lat}^{+}}z^{−n−1}.  (3.32)

Since it is cumbersome to keep the notation for restriction we drop it and simply write

Y (v, z) = ∑_{n∈ℤ}v_{n}z^{−n−1}.  (3.33)

Then using the same notation as in (3.30) and assuming α ∈ Lat^{+} we get:

Y (1 ⊗ e^{α}, z) = exp(∑_{n≥1} \frac{α(−n)z^{n}}{n}) exp(− ∑_{n≥1} \frac{α(n)z^{−n}}{n})e^{α}  (3.34)
Y (v, z) = : (∂_{n_{1}−1}β_{1}(z)...∂_{n_{k}−1}β_{k}(z))Y (1 ⊗ e^{α}, z) :  (3.35)

where now e^{α} acts by a standard multiplication. Note also that since z^{α} acts trivially on V_{Lat}^{+}, it disappeared from the formula for Y (1⊗e^{α}, z) (cf. (3.26)).

We also need the following notation as we will have need to deal with the vertex operators Y (1 ⊗ e^{α}, z) with α = ∑_{p=1}^{N} m_{p}a_{p}. Thus for m = (m_{1}, ..., m_{N}) ∈ ℤ^{N} we let m · a = ∑_{p=1}^{N} m_{p}a_{p}. That is, we think of a as the N-tuple (a_{1}, ..., a_{N}). Then we have that

Y (1 ⊗ e^{∑_{p=1}^{N} m_{p}a_{p}}, z) = Y (1 ⊗ e^{m·a}, z).  (3.36)

Also, in keeping with this notation we also let

m · a(−1) = ∑_{p=1}^{N} m_{p}a_{p}(−1).  (3.37)

Affine vertex algebras

For an affine VOA one lets 𝔤̇ be any finite dimensional simple Lie algebra and we let (·, ·) be an invariant symmetric non-degenerate bilinear form on 𝔤̇ which is normalized so that (θ, θ) = 2 where θ is the longest root of 𝔤̇. We let

ˆ𝔤̇ = 𝔤̇ ⊗ ℂ[t_{0}, t_{0}^{−1}] ⊕ ℂk_{0}  (3.38)

be the non-twisted affine Lie algebra associated to this data with multiplication given by saying k_{0} is central and

[x ⊗ t_{0}^{m}, y ⊗ t_{0}^{n}] = [x, y] ⊗ t_{0}^{m+n} + mδ_{m+n,0}(x, y)k_{0}.  (3.39)

We write x(n) for the element x ⊗ t_{0}^{n}. Let c be an arbitrary constant in ℂ and let ℂ1 be the one dimensional module for 𝔤̇ ⊗ ℂ[t_{0}] ⊕ ℂk_{0} where 𝔤̇ ⊗ ℂ[t_{0}] acts as zero on ℂ1 and where k_{0} acts as the constant c on 1. We form a generalized Verma module by letting

V_{c} = V_{ˆ𝔤̇,c} = Indˆ𝔤̇ ; 𝔤̇⊗ℂ[t_{0}]⊕ℂk_{0}ℂ1  (3.40)

This module has a unique irreducible quotient module which we denote by L_{c} = L_{ˆ𝔤̇,c}. Then it is well known [9], [14] that as long as c ≠ −h^{∨} then both V_{c} and L_{c} are VOA’s. Here −h^{∨} is the negative of the dual Coxeter number of the Lie algebra 𝔤̇.

We quickly review this so as to establish some notation. We work with V_{c} first and define Y (x(−1), z) = ∑_{n∈ℤ} x(n)z^{−n−1} ∈ End(V_{c})[[z, z^{−1}]] and let D ∈ Der(ˆ𝔤̇) be given by [D, x(m)] = −mx(m − 1), [D, k_{0}] = 0. Then the set

{Y (x(−1), z)|x ∈ 𝔤̇}  (3.41)

consists of mutually local fields satisfying \frac{d}{dz}Y (x(−1), z) = [D, Y (x(−1), z)] and so it follows that V_{c} has the structure of a vertex algebra. Letting x_{1}, ..., x_{d} be an orthonormal basis of 𝔤̇ we let

ω = \frac{1}{2(c + h^{∨})} ∑_{i=1}^{d} x_{i}(−1)^{2}.  (3.42)

As usual we let Y (ω, z) = ∑_{n∈ℤ} L(n)z^{−n−2} and let

(V_{c})_{n} = {v ∈ V_{c}|L(0)v = nv}.  (3.43)

Then we have V_{c} = ⊕_{n∈ℤ}(V_{c})_{n} and also

(V_{c})_{n} =< y_{1}(−n_{1}) ⋯ y_{k}(−n_{k})| ∑_{i=1}^{k} n_{i} = n, n_{i} ∈ ℕ, y_{i} ∈ 𝔤̇ >  (3.44)

Then as long as c ≠ −h^{∨} we obtain the VOA (V_{c}, Y, 1, ω). We denote its rank by r_{c} and have that

r_{c} = \frac{c dim(𝔤̇)}{c + h^{∨}}.  (3.45)

Finally, one knows that the usual radical of the module V_{c} is an ideal of the VOA V_{c} so we obtain the VOA L_{c} as a simple quotient of V_{c}.

ĝl_{N} vertex algebras

The Lie algebra gl_{N} is reductive and decomposes into a direct sum gl_{N} = sl_{N} ⊕ ℂI. Let ψ_{1} denote the projection on the traceless matrices and ψ_{2} be the projection on the scalar matrices in this decomposition. Accordingly, we define the affine algebra ĝl_{N} to be a direct sum of the affine algebra ŝl_{N} = sl_{N}(ℂ[t_{0}, t_{0}^{−1}]) ⊕ ℂC_{1} and a (degenerate) Heisenberg algebra Hei = I ⊗ ℂ[t_{0}, t_{0}^{−1}] ⊕ ℂC_{2}.

The Lie bracket in ĝl_{N} is given by

[g_{1} ⊗ t_{0}^{i}, g_{2} ⊗ t_{0}^{j}] =[g_{1}, g_{2}] ⊗ t_{0}^{i+j} + iδ_{i,−j}tr(ψ_{1}(g_{1})ψ_{1}(g_{2}))C_{1} +
iδ_{i,−j}(ψ_{2}(g_{1}), ψ_{2}(g_{2}))C_{2},  (3.46)

where for the last term we will use the normalization (I, I) = 1.

However, if we fix the basis of gl_{N} consisting of elementary matrices, then the formula above will become

[E_{pq} ⊗ t_{0}^{i}, E_{rs} ⊗ t_{0}^{j}] =δ_{qr}E_{ps} ⊗ t_{0}^{i+j} − δ_{ps}E_{rq} ⊗ t_{0}^{i+j} +
iδ_{i,−j} (δ_{qr}δ_{ps}C_{1} + +δ_{pq}δ_{rs} (\frac{C_{2}}{N^{2}} − \frac{C_{1}}{N} ))  (3.47)

To construct a VOA corresponding to ĝl_{N}, we will take a tensor product of a VOA for affine Lie algebra ŝl_{N} and a Heisenberg VOA corresponding to Hei. We have just seen the construction of an affine VOA and hence we obtain a Verma VOA V_{ŝl_{N} ,c_{1}} and a simple VOA L_{ŝl_{N} ,c_{1}} for an arbitrary complex number c_{1} ≠ −N (the dual Coxeter number for sl(N) is N). Let us review the construction of the Heisenberg VOA.

Let c_{2} be an arbitrary constant in ℂ and consider a one-dimensional module ℂ1 for the subalgebra I ⊗ ℂ[t_{0}]⊕ < C_{2} > where I ⊗ ℂ[t_{0}] acts on 1 trivially and C_{2} acts as multiplication by c_{2}.

If c_{2} ≠ 0 then the induced module

Ind_{I⊗ℂ[t_{0}]⊕<C_{2}>}^{Hei}ℂ1  (3.48)

has a structure of a VOA [10, 15], which we denote by L_{Hei,c_{2}}. As a vector space, L_{Hei,c_{2}} is isomorphic to the symmetric algebra S[I ⊗ t_{0}^{−1}ℂ[t_{0}^{−1}]]. To construct the state-field correspondence Y, we take

Y (I(−1), z) = ∑_{n∈ℤ}I(n)z^{−n−1},  (3.49)

Y (I(−j − 1), z) =\frac{1}{j!} (\frac{∂}{∂z} )^{j} Y (I(−1), z),  (3.50)
Y (I(−n_{1}) … I(−n_{k}), z) = : Y (I(−n_{1}), z) … Y (I(−n_{k}), z) :,  (3.51)

where we again use the notation I(n) = I ⊗ t_{0}^{n}.

The Virasoro element is given by ω = \frac{1}{2c_{2}} I(−1)^{2}.

If c_{2} = 0 then the induced module has a one-dimensional factor L_{Hei,0} = ℂ1. The VOA L_{Hei,c_{2}} is simple and its rank is equal to 1 if c_{2} ≠ 0 and to 0 if c_{2} = 0.

Finally, for a pair of complex numbers c_{1}, c_{2}, c_{1} ≠ −N, we define two VOAs for ĝl_{N} as tensor products:

V_{ĝl_{N}} =V_{ĝl_{N} ,c_{1},c_{2}} := V_{ŝl_{N} ,c_{1}} ⊗ L_{Hei,c_{2}}  (3.52)
L_{ĝl_{N}} =L_{ĝl_{N} ,c_{1},c_{2}} := L_{ŝl_{N} ,c_{1}} ⊗ L_{Hei,c_{2}}  (3.53)

Of course, both VOAs V_{ĝl_{N}} and L_{ĝl_{N}} are modules for the Lie algebra ĝl_{N}.

4 Toroidal VOAs

Let c, c_{1}, c_{2} be complex numbers such that c ≠ 0, c ≠ −h^{∨}, c_{1} ≠ −N. We consider two tensor products of VOAs,V = V_{ˆ𝔤̇,c} ⊗ V_{Lat}^{+} ⊗ V_{ĝl_{N} ,c_{1},c_{2}} and L = L_{ˆ𝔤̇,c}⊗V_{Lat}^{+}⊗L_{ĝl_{N} ,c_{1},c_{2}} with the following assignment of vertex operators, which is the same for V and L. We will take m ∈ ℤ^{N}, 1 ≤ s ≤ N. Then we let:

K_{0}(m, z) →cY (1 ⊗ e^{m·a} ⊗ 1, z)  (4.1)
D_{s}(m, z) →Y (1 ⊗ b_{s}(−1)e^{m·a} ⊗ 1 + 1 ⊗ e^{m·a} ⊗ (∑_{p=1}^{N} m_{p}E_{ps}(−1)), z) (4.2)

K_{s}(m, z) →cY (1 ⊗ a_{s}(−1)e^{m·a} ⊗ 1, z)  (4.3)
g(m, z) →Y (g(−1) ⊗ e^{m·a} ⊗ 1, z)  (4.4)
L(z) →Y (ω_{1} ⊗ 1 ⊗ 1 + 1 ⊗ ω_{2} ⊗ 1 + 1 ⊗ 1 ⊗ ω_{3}, z)  (4.5)
C_{V ir} → rank(L)id,  (4.6)
where rank(L) =\frac{cdim(𝔤̇)}{c + h^{∨}} + 2N + \frac{c_{1}(N^{2} − 1)}{c_{1} + N} + rank(L_{Hei,c_{2}})  (4.7)

We will denote by Y_{i}, i = 1, 2, 3 the restriction of the state-field map Y to the first, second, and third factor respectively in the tensor product.

Theorem 4.1 Under the correspondence (4.1)-(4.7), both V and L are modules for the Lie algebra 𝔤_{τ} ⊕V ir = 𝔤̇⊗R⊕K⊕D^{∗} ⊕V ir, where τ = μτ_{1} +ντ_{2} with μ = \frac{1−c_{1}}{c}, ν = ^{\frac{\frac{c_{1}}{N} − \frac{c_{2}}{N^{2}}}{c}} and ρ = \frac{1}{2c}. Moreover, C_{V ir} equals rank(L) as given by (4.7).

Proof. The proof is identical for V and L, so we will carry it out for V. Our first claim is that

\frac{d}{dz} Y (1 ⊗ e^{m·a} ⊗ 1, z) = ∑_{p=1}^{N} m_{p}Y (1 ⊗ a_{p}(−1)e^{m·a} ⊗ 1, z)  (4.8)

It suffices to prove this claim for Y_{2}. From the VOA axioms we have

\frac{d}{dz} Y_{2}(e^{m·a}, z) = Y_{2}(L(−1)e^{m·a}, z).  (4.9)

To compute L(−1)e^{m·a} we observe that

L(−1)e^{m·a} = lim_{w→0}[L(−1), Y_{2}(e^{m·a}, w)]1 =
lim_{w→0} \frac{d}{dw}Y_{2}(e^{m·a}, w)1  (4.10)

¿From (3.26) we obtain ∑_{p=1}^{N} m_{p}a_{p}(−1)e^{m·a} for the limit and the claim follows.

In order to prove that V is a module for the toroidal Lie algebra, we need to show that the relations 2.18 - 2.29 hold in End(V)[[z, z^{−1}]]. We begin with the commutator corresponding to (2.18). The first step here is to verify all commutators involving the affine part. We observe that we can ignore the the third factor in the tensor product. So we will suppress the third factor and will restore it only in the final answers. First, in our affine VOA we have the following relation:

[Y_{1}(g_{1}(−1), z_{1}), Y_{1}(g_{2}(−1), z_{2})] =Y_{1}([g_{1}, g_{2}](−1), z_{2})z_{1}^{−1}δ(\frac{z_{2}}{z_{1}} ) +
z_{1}^{−1}(g_{1}, g_{2})c\frac{∂}{∂z_{2}} δ(\frac{z_{2}}{z_{1}} )  (4.11)

Using this we get

[Y_{1}(g_{1}(−1), z_{1}) ⊗ Y_{2}(e^{m·a}, z_{1}), Y_{1}(g_{2}(−1), z_{2}) ⊗ Y_{2}(e^{r·a}, z_{2})] =

\{Y_{1}([g_{1}, g_{2}](−1), z_{2})z_{1}^{−1}δ(\frac{z_{2}}{z_{1}} ) + z_{1}^{−1}(g_{1}, g_{2})c\frac{∂}{∂z_{2}} δ(\frac{z_{2}}{z_{1}} )\} ⊗ Y_{2}(e^{m·a}, z_{1})Y_{2}(e^{r·a}, z_{2}) =

Y_{1}([g_{1}, g_{2}](−1), z_{2}) ⊗ Y_{2}(e^{(m+r)·a}, z_{2})z_{1}^{−1}δ(\frac{z_{2}}{z_{1}} ) +

(g_{1}, g_{2})z_{1}^{−1}c\frac{∂}{∂z_{2}} δ(\frac{z_{2}}{z_{1}} ) ⊗ Y_{2}(e^{m·a}, z_{1})Y_{2}(e^{r·a}, z_{2})  (4.12)

The second term above can be dealt with using properties of the δ-function, yielding

[Y_{1}(g_{1}(−1), z_{1}) ⊗ Y_{2}(e^{m·a}, z_{1}), Y_{1}(g_{2}(−1), z_{2}) ⊗ Y_{2}(e^{r·a}, z_{2})] =

\{Y_{1}([g_{1}, g_{2}](−1), z_{2}) ⊗ Y_{2}(e^{(m+r)·a}, z_{2}) + (g_{1}, g_{2})c1 ⊗ Y_{2}(m · a(−1)e^{(m+r)·a}, z_{2})\}z_{1}^{−1}δ(\frac{z_{2}}{z_{1}} ) +

(g_{1}, g_{2})z_{1}^{−1}c1 ⊗ Y_{2}(e^{(m+r)·a}, z_{2})\frac{∂}{∂z_{2}} δ(\frac{z_{2}}{z_{1}} )  (4.13)

which, in turn, implies

[Y (g_{1}(−1) ⊗ e^{m·a} ⊗ 1, z_{1}), Y_{2}(g_{2}(−1) ⊗ e^{r·a} ⊗ 1, z_{2})] =

\{Y ([g_{1}, g_{2}](−1) ⊗ e^{(m+r)·a} ⊗ 1, z_{2}) + (g_{1}, g_{2})cY (1 ⊗ m · a(−1)e^{(m+r)·a} ⊗ 1, z_{2})\}z_{1}^{−1}δ(\frac{z_{2}}{z_{1}} ) +

(g_{1}, g_{2})z_{1}^{−1}cY (1 ⊗ e^{(m+r)·a} ⊗ 1, z_{2})\frac{∂}{∂z_{2}} δ(\frac{z_{2}}{z_{1}} )  (4.14)

thus reproducing (2.18).

To establish 2.20 we note that the operators Y_{2}(e^{m·a}, z) and Y_{2}(a_{s}(−1)e^{m·a}, z) correspond to elements from the totally isotropic subspace in Lat and thus all commute. This also implies that

[Y_{1}(g(−1), z_{1}) ⊗ Y_{2}(e^{m·a}, z_{1}) ⊗ 1, 1 ⊗ Y_{2}(e^{r·a}, z_{2}) ⊗ 1] = 0  (4.15)

and

[Y_{1}(g(−1), z_{1}) ⊗ Y_{2}(e^{m·a}, z_{1}) ⊗ 1, 1 ⊗ Y_{2}(a_{s}(−1)e^{r·a}, z_{2}) ⊗ 1] = 0, (4.16)

which establishes (2.19).

We turn to verifying (2.21). We start with

[1 ⊗ Y_{2}(b_{s}(−1)e^{m·a}, z_{1}) ⊗ 1 + 1 ⊗ Y_{2}(e^{m·a}, z_{1}) ⊗ Y_{3}(∑_{p=1}^{N} m_{p}E_{ps}(−1), z_{1}),

Y_{1}(g(−1), z_{2}) ⊗ Y_{2}(e^{r·a}, z_{2}) ⊗ 1] =

Y_{1}(g(−1), z_{2}) ⊗ [Y_{2}(b_{s}(−1)e^{m·a}, z_{1}), Y_{2}(e^{r·a}, z_{2})] ⊗ 1

We compute the commutator [Y_{2}(e^{r·a}, z_{2}), Y_{2}(b_{s}(−1)e^{m·a}, z_{1})] using the commutator formula (3.10):

[Y_{2}(e^{r·a}, z_{2}), Y_{2}(b_{s}(−1)e^{m·a}, z_{1})] =
∑_{n≥0} \frac{1}{n!} (z_{2}^{−1} (\frac{∂}{∂z_{1}} )^{n} δ (\frac{z_{1}}{z_{2}} )) Y (e^{r·a}_{(n)}b_{s}(−1)e^{m·a}, z_{1}) .  (4.17)

The terms e_{(n)}^{r·a}b_{s}(−1)e^{m·a}, n ≥ 0, are given by the antiholomorphic part of Y_{2}(e^{r·a}, z)b_{s}(−1)e^{m·a}. Here is what we mean by this. For an arbitrary expression x(z) = ∑_{k∈ℤ} x_{k}z^{k} ∈ End(V)[[z, z^{−1}]], we denote its antiholomorphic part ∑_{k<0} x_{k}z^{k} by x(z)^{−}. Since

Y_{2}(e^{r·a}, z)b_{s}(−1)e^{m·a} = [Y_{2}(e^{r·a}, z), b_{s}(−1)]e^{m·a} + b_{s}(−1)Y_{2}(e^{r·a}, z)e^{m·a} (4.18) and the antiholomorphic part of the last term is 0, while the commutator in the first term can be easily computed using the explicit expression (3.26) for Y_{2}(e^{r·a}, z):

[Y_{2}(e^{r·a}, z), b_{s}(−1)] = −\frac{r_{s}}{z} Y_{2}(e^{r·a}, z),  (4.19)

we get that

(Y_{2}(e^{r·a}, z)b_{s}(−1)e^{m·a})^{−} = (−\frac{r_{s}}{z} Y_{2}(e^{r·a}, z)e^{m·a})^{−} = −\frac{r_{s}}{z} e^{(m+r)·a}. (4.20)

This means that e_{(0)}^{r·a}b_{s}(−1)e^{m·a} = −r_{s}e^{(m+r)·a} and e_{(n)}^{r·a}b_{s}(−1)e^{m·a} = 0 for n >

0. Substituting this into (4.17) we obtain the desired commutator relation (2.21):

[Y (1 ⊗ b_{s}(−1)e^{m·a} ⊗ 1 + 1 ⊗ e^{m·a} ⊗ (∑_{p=1}^{N} m_{p}E_{ps}(−1)), z_{1}), Y (g(−1) ⊗ e^{r·a} ⊗ 1, z_{2})] =

r_{s}z_{2}^{−1}δ (\frac{z_{1}}{z_{2}} ) Y_{1}(g(−1), z_{2}) ⊗ Y_{2}(e^{(m+r)·a}, z_{1}) ⊗ 1 =

r_{s}z_{1}^{−1}δ (\frac{z_{2}}{z_{1}} ) Y (g(−1) ⊗ e^{(m+r)·a} ⊗ 1, z_{2}).  (4.21)

We turn now to (2.22):

[Y (ω, z_{1}), Y (g(−1) ⊗ e^{m·a} ⊗ 1, z_{2})] =

∑_{n≥0} \frac{1}{n!} (z_{1}^{−1} (\frac{∂}{∂z_{2}} )^{n} δ (\frac{z_{2}}{z_{1}} )) Y (ω_{n}(g(−1) ⊗ e^{m·a} ⊗ 1), z_{2}) .  (4.22)

Using the general formula for the weight of the n-th product in a VOA

wt(u_{n}v) = wt(u) + wt(v) − n − 1  (4.23)

and using the fact that wt(ω) = 2 and wt(g(−1) ⊗ e^{m·a} ⊗ 1) = 1 we see that ω_{n}(g(−1)⊗e^{m·a}⊗1) = 0 for n ≥ 3. We thus need to compute ω_{n}(g(−1)e^{m·a}⊗

1) = L(n − 1)(g(−1) ⊗ e^{m·a} ⊗ 1), for n = 0, 1, 2.

Observing that L(0)v = nv for v ∈ V_{n}, and taking into account that wt(g(−1) ⊗ e^{m·a} ⊗ 1) = 1, we get L(0)(g(−1) ⊗ e^{m·a} ⊗ 1) = g(−1) ⊗ e^{m·a} ⊗ 1.

For L(−1) we use the relation Y (L(−1)v, z) = \frac{∂}{∂z}Y (v, z).

To compute the term with L(1) we consider

L(1)(g(−1) ⊗ e^{m·a} ⊗ 1) =(L(1)g(−1))1 ⊗ e^{m·a} ⊗ 1 + g(−1) ⊗ L(1)e^{m·a} ⊗ 1
g(−1) ⊗ e^{m·a} ⊗ L(1)1.  (4.24)

However all three terms here are zero. The last two terms vanish because of (4.23), while for the first term we use the following relation in the affine VOA:

[L(n), g(k)] = −kg(n + k), for g ∈ 𝔤̇,  (4.25)

which gives [L(1), g(−1)]1 = g(0)1 = 0.

Summarizing, we get the relation (2.22):

[L(z_{1}), Y (g(−1) ⊗ e^{m·a} ⊗ 1, z_{2})] =z_{1}^{−1}δ(\frac{z_{2}}{z_{1}} ) (\frac{∂}{∂z_{2}} Y (g(−1) ⊗ e^{m·a} ⊗ 1, z_{2})) +

z_{1}^{−1} (\frac{∂}{∂z_{2}} ) δ(\frac{z_{2}}{z_{1}} )Y (g(−1) ⊗ e^{m·a} ⊗ 1, z_{2}). (4.26)

By Lemma (2.1), the relations (2.23) - (2.26) follow automatically from the ones we just established.

To verify (2.27) we write out the whole commutator:

[1 ⊗ Y_{2}(b_{s}(−1)e^{m·a}, z_{1}) ⊗ 1 + 1 ⊗ Y_{2}(e^{m·a}, z_{1}) ⊗ Y_{3}(∑_{p=1}^{N} m_{j}E_{ps}(−1), z_{1}),

1 ⊗ Y_{2}(b_{q}(−1)e^{r·a}, z_{2}) ⊗ 1 + 1 ⊗ Y_{2}(e^{r·a}, z_{2}) ⊗ Y_{3}(∑_{k=1}^{N} r_{k}E_{kq}(−1), z_{2})] =

1 ⊗ [Y_{2}(b_{s}(−1)e^{m·a}, z_{1}), Y_{2}(b_{q}(−1)e^{r·a}, z_{2})] ⊗ 1 +

1 ⊗ [Y_{2}(b_{s}(−1)e^{m·a}, z_{1}), Y_{2}(e^{r·a}, z_{2})] ⊗ Y_{3}(∑_{k=1}^{N} r_{k}E_{kq}(−1), z_{2}) +

1 ⊗ [Y_{2}(e^{m·a}, z_{1}), Y_{2}(b_{q}(−1)e^{r·a}, z_{2})] ⊗ Y_{3}(∑_{p=1}^{N} m_{p}E_{ps}(−1), z_{1})

1 ⊗ Y_{2}(e^{m·a}, z_{1})Y_{2}(e^{r·a}, z_{2}) ⊗ [Y_{3}(∑_{p=1}^{N} m_{p}E_{ps}(−1), z_{1}), Y_{3}(∑_{k=1}^{N} r_{k}E_{kq}(−1), z_{2})],

where in the very last line we used that [Y_{2}(e^{m·a}, z_{1}), Y_{2}(e^{r·a}, z_{2})] = 0. To compute the first commutator above we will need (Y_{2}(b_{s}(−1)e^{m·a}, z)b_{q}(−1)e^{r·a})^{−} We claim that

(Y_{2}(b_{s}(−1)e^{m·a}, z)b_{q}(−1)e^{r·a})^{−} =

\frac{1}{z}(r_{s}b_{q}(−1) − m_{q}b_{s}(−1))e^{(m+r)·a} − \frac{1}{z}r_{s}m_{q} ∑_{p=1}^{N} m_{p}a_{p}(−1)e^{(m+r)·a} − \frac{1}{z^{2}} r_{s}m_{q}e^{(m+r)·a}

(4.27)

To prove this statement we write:

(Y_{2}(b_{s}(−1)e^{m·a}, z)b_{q}(−1)e^{r·a})^{−} = (: b_{s}(z)Y_{2}(e^{m·a}, z) : b_{q}(−1)e^{r·a})^{−} =

(b_{s}(z)^{+}Y_{2}(e^{m·a}, z)b_{q}(−1)e^{r·a}1)^{−} + (Y_{2}(e^{m·a}, z)b_{s}(z)^{−}b_{s}(−1)e^{r·a}1)^{−} (4.28)

Now the first term in (4.28)

(b_{s}(z)^{+}Y_{2}(e^{m·a}, z)b_{q}(−1)e^{r·a}1)^{−} =

(b_{s}(z)^{+}[Y_{2}(e^{m·a}, z), b_{q}(−1)]e^{r·a}1)^{−} + (b_{s}(z)^{+}b_{q}(−1)Y_{2}(e^{m·a}, z)e^{r·a}1)^{−} (4.29)

Since the second term is 0 we get using (4.19):

(b_{s}(z)^{+}Y_{2}(e^{m·a}, z)b_{q}(−1)e^{r·a}1)^{−} = (b_{s}(z)^{+}[Y_{2}(e^{m·a}, z), b_{q}(−1)]e^{r·a}1)^{−}
= − (\frac{m_{q}}{z} b_{s}(z)^{+}Y_{2}(e^{m·a}, z)e^{r·a}1)^{−} , (4.30)

which in turn, after noticing that only lim_{z→0} b_{s}(z)^{+} = b_{s}(−1) contributes to the antiholomorphic part, gives

(b_{s}(z)^{+}Y_{2}(e^{m·a}, z)b_{q}(−1)e^{r·a}1)^{−} = −\frac{m_{q}b_{s}(−1)e^{(m+r)·a}}{z}  (4.31)

Likewise, to compute the second term in (4.28), we make use of the commutator

[b_{s}(z)^{−}, e^{r·a}] = \frac{r_{s}e^{r·a}}{z}  (4.32)

as well as of (4.19):

(Y_{2}(e^{m·a}, z)b_{s}(z)^{−}b_{q}(−1)e^{r·a}1)^{−} = (Y_{2}(e^{m·a}, z)b_{q}(−1)\frac{r_{s}}{z} e^{r·a}1)^{−} =

(\frac{r_{s}}{z} [Y_{2}(e^{m·a}, z)b_{q}(−1)]e^{r·a}1)^{−} + (\frac{r_{s}}{z} b_{q}(−1)Y_{2}(e^{m·a}, z)e^{r·a}1)^{−} =

− (\frac{r_{s}m_{q}}{z^{2}} Y_{2}(e^{m·a}, z)e^{r·a}1)^{−} + (\frac{r_{s}}{z} b_{q}(−1)Y_{2}(e^{m·a}, z)e^{r·a}1)^{−} =

− \frac{r_{s}m_{q}}{z^{2}} e^{(m+r)·a} − \frac{r_{s}m_{q}}{z} ∑_{p=1}^{N} m_{p}a_{p}(−1)e^{(m+r)·a} + \frac{r_{s}}{z} b_{q}(−1)e^{(m+r)·a}, (4.33)

from which the (4.27) follows. Using (4.32) we get

(Y_{2}(b_{s}(−1)e^{m·a}, z)e^{r·a})^{−} = \frac{r_{s}e^{(m+r)·a}}{z}  (4.34)

and, using (4.19) we get

(Y_{2}(e^{m·a}, z)b_{q}(−1)e^{r·a})^{−} = −\frac{m_{q}e^{(m+r)·a}}{z}  (4.35)

There is one more commutator, namely,

1 ⊗ Y_{2}(e^{m·a}, z_{1})Y_{2}(e^{r·a}, z_{2}) ⊗ [Y_{3}(∑_{p=1}^{N} m_{p}E_{ps}(−1), z_{1}), Y_{3}(∑_{k=1}^{N} r_{k}E_{kq}(−1), z_{2})]

Using that V_{ĝl(N)} is a ĝl(N) module we obtain

[Y_{3}(∑_{p=1}^{N} m_{p}E_{ps}(−1), z_{1}), Y_{3}(∑_{k=1}^{N} r_{k}E_{kq}(−1), z_{2})] =

z_{1}^{−1}δ(\frac{z_{2}}{z_{1}} )\{r_{s}Y_{3}(∑_{p=1}^{N} m_{p}E_{pq}(−1), z_{2}) − m_{q}Y_{3}(∑_{k=1}^{N} r_{k}E_{ks}(−1), z_{2})\} +

(r_{s}m_{q}c_{1} + r_{q}m_{s} (\frac{c_{2}}{N^{2}} − \frac{c_{1}}{N} ))z_{1}^{−1} \frac{∂}{∂z_{2}} δ(\frac{z_{2}}{z_{1}} )

We compute the contribution of the very last term, leaving out the factor r_{s}m_{q}c_{1} + r_{q}m_{s} (\frac{c_{2}}{N^{2}} − \frac{c_{1}}{N} ), with the help of the identity z_{1}^{−1} \frac{∂}{∂z_{2}} δ(\frac{z_{2}}{z_{1}} ) = −z_{2}^{−1} \frac{∂}{∂z_{1}} δ(\frac{z_{2}}{z_{1}} )

1 ⊗ Y_{2}(e^{m·a}, z_{1})Y_{2}(e^{r·a}, z_{2})z_{1}^{−1} \frac{∂}{∂z_{2}} δ(\frac{z_{2}}{z_{1}} ) ⊗ 1 =

− 1 ⊗ Y_{2}(e^{m·a}, z_{1})Y_{2}(e^{r·a}, z_{2})z_{2}^{−1} \frac{∂}{∂z_{1}} δ(\frac{z_{2}}{z_{1}} )⊗ =

−1 ⊗ (\frac{∂}{∂z_{1}} (Y_{2}(e^{m·a}, z_{1})Y_{2}(e^{r·a}, z_{2})z_{2}^{−1}δ(\frac{z_{2}}{z_{1}} ))) ⊗ 1 +

z_{2}^{−1}1 ⊗ Y_{2}(e^{r·a}, z_{2})(\frac{∂}{∂z_{1}} Y_{2}(e^{m·a}, z_{1}))δ(\frac{z_{2}}{z_{1}} )) ⊗ 1 =

− 1 ⊗ Y_{2}(e^{(m+r)·a}, z_{2})z_{2}^{−1} \frac{∂}{∂z_{1}} δ(\frac{z_{2}}{z_{1}} ) ⊗ 1 +

z_{2}^{−1}1 ⊗ Y_{2}(e^{r·a}, z_{2}) ∑_{p=1}^{N} m_{p}Y_{2}(a_{p}(−1)e^{m·a}, z_{1})δ(\frac{z_{2}}{z_{1}} )) ⊗ 1 =

1 ⊗ Y_{2}(e^{(m+r)·a}, z_{2})z_{1}^{−1} \frac{∂}{∂z_{2}} δ(\frac{z_{2}}{z_{1}} ) ⊗ 1 +

z_{2}^{−1}1 ⊗ ∑_{p=1}^{N} m_{p}Y_{2}(a_{p}(−1)e^{(m+r)·a}, z_{2})δ(\frac{z_{2}}{z_{1}} )) ⊗ 1  (4.36)

Here we used the fact that Y_{2}(e^{r·a}, z_{1})Y_{2}(e^{m·a}, z_{2})z_{1}^{−1}δ(\frac{z_{2}}{z_{1}} ) = Y_{2}(e^{(m+r)·a}, z_{2})z_{1}^{−1}δ(\frac{z_{2}}{z_{1}} ). For simplicity, we set

d_{s}(m, z) = Y (1 ⊗ b_{s}(−1)e^{m·a} ⊗ 1 + 1 ⊗ e^{m·a} ⊗ (∑_{p=1}^{N} m_{p}E_{ps}(−1)), z) (4.37)

Combining together (4.27), (4.34), (4.35), (4.36) we get that

[d_{s}(m, z_{1}), d_{q}(r, z_{2})] = {r_{s}d_{q}(m + r, z_{2}) − m_{q}d_{s}(m + r, z_{2}) +

(r_{s}m_{q}(c_{1} − 1) + r_{q}m_{s} (\frac{c_{1}}{N} − \frac{c_{2}}{N^{2}} )) ∑_{p=1}^{N} m_{p}Y (1 ⊗ a_{p}(−1)e^{(m+r)·a} ⊗ 1, z_{2})\}z_{1}^{−1}δ(\frac{z_{2}}{z_{1}} ) +

(r_{s}m_{q}(c_{1} − 1) + r_{q}m_{s} (\frac{c_{1}}{N} − \frac{c_{2}}{N^{2}} )) Y (1 ⊗ e^{(m+r)·a} ⊗ 1, z_{2})z_{1}^{−1} \frac{∂}{∂z_{2}} δ(\frac{z_{2}}{z_{1}} )(4.38)

which coincides with (2.27) with μ = \frac{1−c_{1}}{c} and ν = ^{\frac{\frac{c_{1}}{N} − \frac{c_{2}}{N^{2}}}{c}} .

To verify (2.28) we need to inspect the following commutators

[Y_{2}(ω_{2}, z_{1}), Y_{2}(b_{s}(−1)e^{m·a}, z_{2})] [Y_{2}(ω_{2}, z_{1}), Y_{2}(e^{m·a}, z_{2})]

[Y_{3}(ω_{3}, z_{1}), Y_{3}(∑_{p=1}^{N} m_{p}E_{ps}(−1), z_{2})]

In all three cases the verification is similar to (4.22). We use the fact that the weight of b_{s}(−1)e^{m·a} and E_{ps}(−1) is one, and is zero for e^{m·a}. The results for the anti-holomorphic parts have the same form in all three cases

(Y (ω, z)v)^{−} = \frac{L(−1)v}{z} + \frac{L(0)v}{z^{2}} + \frac{L(1)v}{z^{3}}

and in a way analogous to the computation for (4.22), we obtain

(Y_{2}(ω, z)b_{s}(−1)e^{m·a})^{−} = \frac{1}{z}L(−1)b_{s}(−1)e^{m·a} + \frac{1}{z^{2}} b_{s}(−1)e^{m·a} + \frac{m_{s}}{z^{3}} e^{m·a},

(Y_{2}(ω, z)e^{m·a})^{−} = \frac{1}{z}L(−1)e^{m·a},

(Y_{3}(ω, z) ∑_{p=1}^{N} m_{p}E_{ps}(−1))^{−} = \frac{1}{z}L(−1) ∑_{p=1}^{N} m_{p}E_{ps}(−1) + \frac{1}{z^{2}} ∑_{p=1}^{N} m_{p}E_{ps}(−1)

which after assembling all the terms yields (2.28), provided ρ = \frac{1}{2c}.

Finally, the field Y (ω, z) satisfies the Virasoro commutation relations by definition, with the proviso C_{V ir} = rank(L).

5 Modules for toroidal VOAs

In [2] a family of irreducible 𝔤-modules was constructed. In this section we will identify these modules as irreducible modules for the toroidal VOAs, thus giving their explicit realizations.

We begin by recalling the construction of the 𝔤_{τ} modules from [2]. Fix as before a non-zero complex number c. The input data for building the module consists of a finite-dimensional irreducible 𝔤̇-module V̇, a finite-dimensional irreducible sl_{N}-module W and a complex number c_{3}. The pair (W, c_{3}) actually defines a gl_{N}-module structure on the space W by letting the identity matrix act as multiplication by c_{3}.

The toroidal Lie algebra 𝔤 = 𝔤̇ ⊗ R ⊕ K ⊕ D^{∗} has a ℤ-grading by degree in the variable t_{0}. We also consider a decomposition 𝔤 = 𝔤_{−} ⊕ 𝔤_{0} ⊕ 𝔤_{+} associated with this grading. The subalgebra 𝔤_{0} is spanned by the elements {g ⊗ t^{m}, t^{m}k_{0}, t^{m}k_{s}, t^{m}d_{s}|m ∈ ℤ^{n}, g ∈ 𝔤̇, s = 1, …, N}. We begin by constructing a 𝔤_{0}-module T from the input data:

T = V̇ ⊗ ℂ[Lat^{+}] ⊗ W.  (5.1)

The action of 𝔤_{0} on T is given by the formulas (see [2]):

g ⊗ t^{m} (v ⊗ e^{r·a} ⊗ w) =(gv) ⊗ e^{(m+r)·a} ⊗ w,  (5.2)
t^{m}k_{0} (v ⊗ e^{r·a} ⊗ w) =cv ⊗ e^{(m+r)·a} ⊗ w,  (5.3)
t^{m}k_{s} (v ⊗ e^{r·a} ⊗ w) =0,  (5.4)
t^{m}d_{s} (v ⊗ e^{r·a} ⊗ w) =r_{s}v ⊗ e^{(m+r)·a} ⊗ w + ∑_{p=1}^{N} m_{p}v ⊗ e^{(m+r)·a} ⊗ E_{ps}w, (5.5)

Next we let 𝔤_{+} act trivially on T and we build the induced module

\overline{U} = Ind_{𝔤_{0}⊕𝔤_{+}}^{𝔤}(T).  (5.6)

The module \overline{U} inherits a natural ℤ^{N+1}-grading from 𝔤. We point out that the ℤ^{N+1}-homogeneous components of \overline{U} lying below the top T are infinitedimensional. The module \overline{U} has a unique maximal homogeneous submodule \overline{U}_{rad} intersecting T trivially. The factor-module

U = U(c, V̇ , W, c_{3}) = \overline{U}/\overline{U}_{rad}  (5.7)

is a graded simple module (every ℤ^{N+1}-graded submodule is trivial). Applying Theorem 1.12 of [2], we get that the homogeneous components of U are finite-dimensional.

Now we will use the same input data to construct a module for the toroidal VOA V_{c,c_{1},c_{2}}, where c is as above and c_{1}, c_{2} are related to the cocycle τ

by μ = \frac{1−c_{1}}{c}, ν = ^{\frac{\frac{c_{1}}{N} − \frac{c_{2}}{N^{2}}}{c}}. We also impose additional restrictions, namely c ≠ −h^{∨}, c_{1} ≠ −N. The toroidal VOA V_{c,c_{1},c_{2}} is a tensor product of three VOAs: V_{c,c_{1},c_{2}} = V_{ˆ𝔤̇,c} ⊗ V_{Lat}^{+} ⊗ V_{ĝl_{N} ,c_{1},c_{2}}. We will construct a VOA module by taking a tensor product of modules for each factor:

M_{ˆ𝔤̇,c,V̇} ⊗ V_{Lat}^{+} ⊗ M_{ĝl_{N} ,c_{1},c_{2},c_{3},W}  (5.8)

We view V_{Lat}^{+} as a module over itself. Let us briefly describe M_{ˆ𝔤̇,c,V̇} and M_{ĝl_{N} ,c_{1},c_{2},c_{3},W}.

Extend the action of 𝔤̇ on V̇ to an action of 𝔤̇ ⊗ ℂ[t_{0}] ⊕ ℂk_{0} by letting 𝔤̇⊗t_{0}ℂ[t_{0}] act on V̇ trivially and letting k_{0} act as multiplication by c. Consider the induced ˆ 𝔤̇-module:

\overline{M}_{ˆ𝔤̇,c,V̇} = Indˆ𝔤̇ ; 𝔤̇⊗ℂ[t_{0}]⊕ℂk_{0}(V̇)  (5.9)

and its irreducible factor M_{ˆ𝔤̇,c,V̇}. The ˆ 𝔤̇-module M_{ˆ𝔤̇,c,V̇} is also an irreducible VOA module for affine VOA V_{ˆ𝔤̇,c}.

The construction for M_{ĝl_{N} ,c_{1},c_{2},c_{3},W} is analogous, but we should also take into account the Heisenberg part. Recall that V_{ĝl_{N} ,c_{1},c_{2}} is itself a tensor product of two VOAs: V_{ĝl_{N} ,c_{1},c_{2}} = V_{ŝl_{N} ,c_{1}} ⊗V_{Hei,c_{2}}. The module M_{ĝl_{N} ,c_{1},c_{2},c_{3},W} that we are going to introduce will be a tensor product of a module M_{ŝl_{N} ,c_{1},W} for V_{ŝl_{N} ,c_{1}} and a module M_{Hei,c_{2},c_{3}} for V_{Hei,c_{2}}. The construction for M_{ŝl_{N} ,c_{1},W} is completely parallel to the one we used above for M_{ˆ𝔤̇,c,V̇}. Let us describe the Heisenberg VOA module M_{Hei,c_{2},c_{3}}.

We take a one-dimensional space ℂ1 and make it into a module for I ⊗ ℂ[t_{0}] ⊕ ℂC_{2} by I1 = c_{3}1, C_{2}1 = c_{2}1, I ⊗ t_{0}ℂ[t_{0}]1 = 0. The induced module

Ind_{I⊗ℂ[t_{0}]⊕ℂC_{2}}^{Hei}  (5.10)

is an irreducible module over the Heisenberg algebra and will be denoted by M_{Hei,c_{2},c_{3}}, unless c_{2} = 0, in which case we set M_{Hei,0,c_{3}} to be the onedimensional factor ℂ1 of the induced module. The module M_{Hei,c_{2},c_{3}} is an irreducible module for the Heisenberg VOA V_{Hei,c_{2}} and

M_{ĝl_{N} ,c_{1},c_{2},c_{3},W} = M_{ŝl_{N} ,c_{1},W} ⊗ M_{Hei,c_{2},c_{3}}  (5.11)

is an irreducible VOA module for V_{ĝl_{N} ,c_{1},c_{2}}. Finally,

M = M_{c,c_{1},c_{2},c_{3},V̇ ,W} = M_{ˆ𝔤̇,c,V̇} ⊗ V_{Lat}^{+} ⊗ M_{ĝl_{N} ,c_{1},c_{2},c_{3},W}  (5.12)

is an irreducible module for the toroidal VOA V_{c,c_{1},c_{2}}.

Now we are ready to state the main result of this section:

Theorem 5.1 Let c, c_{1}, c_{2}, c_{3} be complex numbers such that c ≠ 0, c ≠ −h^{∨}, c_{1} ≠ −N. Let V̇ be an irreducible 𝔤̇-module and W be an irreducible sl_{N}-module.

(1) the module M_{c,c_{1},c_{2},c_{3},V̇ ,W} is a module for the toroidal Lie algebra 𝔤_{τ},

τ = μτ_{1} + ντ_{2}, where μ = \frac{1−c_{1}}{c} , ν = ^{\frac{\frac{c_{1}}{N} − \frac{c_{2}}{N^{2}}}{c}}.

(2) M_{c,c_{1},c_{2},c_{3},V̇ ,W} is irreducible as a 𝔤_{τ}-module.

(3) M_{c,c_{1},c_{2},c_{3},V̇ ,W} ≅ U(c, V̇ , W, c_{3}) as 𝔤_{τ}-modules.

Proof. The module M = M_{c,c_{1},c_{2},c_{3},V̇ ,W} is a module for the toroidal VOA V = V_{c,c_{1},c_{2}}. Under the correspondence (4.1)-(4.7) the commutator relations (2.18)-(2.29) of 𝔤 hold in V. Thus, by preservation of identities in VOA modules(see [14] Lemma 2.3.5), the same relations hold in M, which proves that M is a 𝔤_{τ}-module.

The module M is an irreducible module for the toroidal VOA V. In order to show that it is also irreducible as a 𝔤-module, it is enough to show that V is generated as a VOA by the set

S = {g(−1)e^{m·a}, e^{m·a}, a_{j}(−1)e^{m·a},b_{j}(−1)e^{m·a} + ∑_{p=1}^{N} m_{p}E_{pj}(−1)e^{m·a}

|g ∈ 𝔤̇, m ∈ ℤ^{N}, j = 1, …, N} (5.13)

which corresponds to (4.1)-(4.4). Once this is established, every 𝔤-invariant subspace in a VOA module is V -invariant, since, by the associativity property of VOAs every concatenation of n-th products may be expressed through the composition of operators from 𝔤. Thus irreducibility with respect to V will imply irreducibility with respect to 𝔤.

It is sufficient to show that the set S generates each of the factors V_{ˆ𝔤̇,c} ⊗ 1 ⊗ 1, 1 ⊗ V_{Lat}^{+} ⊗ 1 and 1 ⊗ 1 ⊗ V_{ĝl_{N} ,c_{1},c_{2}}. To get the first factor, we note that g(−1) ∈ S for all g ∈ 𝔤̇. The VOA V_{ˆ𝔤̇,c} is spanned by the elements

g_{1}(−n_{1}) … g_{k}(−n_{k})1, g_{i} ∈ 𝔤̇, h_{i} ∈ ℕ.  (5.14)

However, since

g_{1}(−n_{1}) … g_{k}(−n_{k})1 = g_{1}(−1)_{(−n_{1})} (… (g_{k}(−1)_{(−n_{k})}1) …) ,  (5.15)

we get that S generates V_{ˆ𝔤̇,c} ⊗ 1 ⊗ 1. Since a_{j}(−1), b_{j}(−1), e^{m·a} ∈ S, we also get 1⊗V_{Lat}^{+} ⊗1 by a similar argument. In particular we see that b_{j}(−1)e^{m·a} is in the space generated by S for any m ∈ ℤ^{N}. Choosing m with m_{p} = δ_{kp}, p = 1, …, N, we get that both b_{j}(−1)e^{a_{k}} and b_{j}(−1)e^{a_{k}} + E_{kj}(−1)e^{a_{k}} are in the space generated by S, and so is E_{kj}(−1)e^{a_{k}}. Finally (E_{kj}(−1)e^{a_{k}})_{(−1)}e^{−a_{k}} = E_{kj}(−1), and hence we can generate 1 ⊗ 1 ⊗ V_{ĝl_{N} ,c_{1},c_{2}} as well. This completes the proof of part (2) of the theorem.

To prove (3) we note that M_{c,c_{1},c_{2},c_{3},V̇ ,W} has a natural ℤ-grading corresponding to the degree in the t_{0}-variable and moreover its zero component M_{0} = V̇ ⊗ℂ[Lat^{+}]⊗W is isomorphic to T. One can easily see that the action of 𝔤_{0} on M_{0} coincides with (5.2)-(5.5). Thus M is a homomorphic image of U. However, since the ℤ^{N+1}-gradings on both modules are compatible, and U is graded simple, we get an isomorphism M ≅ U. This also proves that for c ≠ −h^{∨}, μ ≠ N + 1, U is a simple module and not just graded simple.

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