A category of modules for the full toroidal Lie algebra.

Yuly Billig

To Robert Moody

Introduction.

Toroidal Lie algebras are very natural multi-variable generalizations of affine Kac-Moody algebras. The theory of affine Lie algebras is rich and beautiful, having connections with diverse areas of mathematics and physics. Toroidal Lie algebras are also proving themselves to be useful for the applications. Frenkel, Jing and Wang [FJW] used representations of toroidal Lie algebras to construct a new form of the McKay correspondence. Inami et al., studied toroidal symmetry in the context of a 4-dimensional conformal field theory [IKUX], [IKU]. There are also applications of toroidal Lie algebras to soliton theory. Using representations of the toroidal algebras one can construct hierarchies of non-linear PDEs [B2], [ISW]. In particular, the toroidal extension of the Korteweg-de Vries hierarchy contains the Bogoyavlensky’s equation, which is not in the classical KdV hierarchy [IT]. One can use the vertex operator realizations to construct n-soliton solutions for the PDEs in these hierarchies. We hope that further development of the representation theory of toroidal Lie algebras will help to find new applications of this interesting class of algebras.

The construction of a toroidal Lie algebra is totally parallel to the well-known construction of an (untwisted) affine Kac-Moody algebra [K1]. One starts with a finite-dimensional simple Lie algebra 𝔤̇ and considers Fourier polynomial maps from an N + 1-dimensional torus into 𝔤̇. Setting t_{k} = e^{ix_{k}}, we may identify the algebra of Fourier polynomials on a torus with the Laurent polynomial algebra R = ℂ[t_{0}^{±}, t_{1}^{±}, …, t_{N}^{±}], and the Lie algebra of the 𝔤̇-valued maps from a torus with the multi-loop algebra ℂ[t_{0}^{±}, t_{1}^{±}, …, t_{N}^{±}] ⊗ 𝔤̇. When N = 0, this yields the usual loop algebra.

Just as for the affine algebras, the next step is to build the universal central extension (R ⊗ 𝔤̇) ⊕ K of R ⊗ 𝔤̇. However unlike the affine case, the center K is infinite-dimensional when N ≥ 1. The infinite-dimensional center makes this Lie algebra highly degenerate. One can show, for example, that in an irreducible bounded weight module, most of the center should act trivially. To eliminate this degeneracy, we add the Lie algebra of vector fields on a torus, D = Der (R) to (R ⊗ 𝔤̇) ⊕ K. The resulting algebra,

𝔤 = (R ⊗ 𝔤̇) ⊕ K ⊕ D

is called the full toroidal Lie algebra (see Section 1 for details). The action of D on K is nontrivial, making the center of the toroidal Lie algebra 𝔤 finite-dimensional. This enlarged algebra will have a much better representation theory.

2000 Mathematics Subject Classification. Primary 17B65, 17B69; Secondary 17B66.

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

The most important class of modules for the affine Lie algebras are the highest weight modules, and one would certainly want to construct their toroidal analogs. The first problem that arises here is that one needs a triangular decomposition for the Lie algebra in order to introduce the notion of the highest weight module. Toroidal Lie algebras are graded by ℤ^{N+1}, and for N > 0, there is no canonical way of dividing this lattice into positive and negative parts. This difficulty is not present for the affine Lie algebras, which are graded by ℤ, and for ℤ such a splitting is natural.

One way to split ℤ^{N+1} is to cut it with a hyperplane that intersects with the lattice only at zero. The corresponding class of the highest weight modules was studied by Berman and Cox [BC], where it was found that the Verma modules constructed in this way will have infinitedimensional weight spaces and do not produce any representations with interesting realizations. Modules corresponding to other decompositions of the lattice were studied in [DFP].

An extremal way of dividing the lattice is to cut it with a hyperplane that intersects ℤ^{N+1} at a sublattice of rank N. This approach was taken by Moody, Rao and Yokonuma, who constructed a homogeneous vertex operator realization of the basic module for the universal central extension of the multi-loop algebra [MRY]. In [EM], Rao and Moody showed how to get

a representation of a bigger algebra on the same space, adding a subalgebra D^{∗} = ⊕_{p=1}^{N} R\frac{∂}{∂t_{p}} of

the Lie algebra of vector fields. One unanticipated development in [EM] was the appearance of a K-valued 2-cocycle τ_{1} on the Lie algebra of vector fields, which is an abelian generalization of the Virasoro cocycle. A principal realization for the basic module was given in [B1].

Developing further these ideas, Larsson succeeded in constructing a wider class of representations for the toroidal Lie algebras [L]. He showed that the basic module for the affine Lie algebra ˆ 𝔤̇ may be replaced with an arbitrary highest weight module. Larsson also discovered that affine ĝl_{N}-modules can be used as an ingredient in these constructions. In Larsson’s paper a combination of 2-cocycles τ_{1} and τ_{2} had appeared.

Berman and Billig developed a categorical approach to the representation theory of toroidal Lie algebras, introducing the generalized Verma modules for toroidal Lie algebras [BB]. They developed a theory of Lie algebras with polynomial multiplication, which allowed them to prove that simple quotients of the generalized Verma modules always have finite-dimensional weight spaces. Realizations of these irreducible quotients were obtained using a modified version of Larsson’s construction.

A vertex algebra interpretation of these results was given by Berman, Billig and Szmigielski in [BBS].

Although the generalized Verma modules could be defined for the full toroidal algebras and the result of [BB] concerning finite-dimensional weight spaces also holds in full generality, there was one substantial difficulty unresolved in all of these publications. It was not known how to construct realizations for the modules over the full toroidal algebra – the piece R\frac{∂}{∂t_{0}} was always missing. This piece corresponds to the energy-momentum tensor in quantum field theories, so it is important to have it represented.

A class of modules for the full toroidal Lie algebras was constructed in [B4] (unpublished), and in the present paper we completely solve this problem.

Extended affine Lie algebras (EALAs) are another important family of algebras closely related to the toroidal Lie algebras. The main feature of the extended affine Lie algebras is the existence of a non-degenerate symmetric invariant bilinear form. Such a form does not exist on the full toroidal algebra, but it can be defined on its subalgebra (R ⊗ 𝔤̇) ⊕ K ⊕ D_{div}, where D_{div} is the Lie algebra of divergence zero vector fields on a torus. The results of the present paper make it possible to develop the representation theory for the toroidal EALA using restriction from the full toroidal algebras [B4], [B6].

It is shown in [ABFP] that most of the extended affine Lie algebras may be realized as twisted toroidal EALAs. The representation theory of the twisted toroidal EALAs is studied by Billig and Lau in [BL].

In the present paper we define a rather natural category B_{χ} of bounded 𝔤-modules with finite-dimensional weight spaces with the central character χ. Our goal is to study irreducible modules in this category. We show that every irreducible module is characterized by its top T – the highest eigenspace for the operator d_{0} = t_{0} \frac{∂}{∂t_{0}} . The space T is a submodule with respect to the subalgebra 𝔤_{0} consisting of elements of 𝔤 of degree zero with respect to t_{0}. Following [BB], we define a generalized Verma module M(T) and its irreducible quotient L(T). We show (Theorem 2.5) that every irreducible module in category B_{χ} is isomorphic to L(T) for some irreducible 𝔤_{0}-module T with finite-dimensional weight spaces. Using the results of [JM], [E] and [B5], we see that such 𝔤_{0}-modules are precisely those considered in [BB] – they are multi-loop modules with respect to R_{N} ⊗ 𝔤̇ and tensor modules with respect to Der (R_{N}).

Once we get a description of the tops T, we wish to completely determine the structure of the 𝔤-modules L(T). This is done by constructing the vertex operator realizations of these modules. The crucial observation here is that the full toroidal Lie algebras are vertex Lie algebras. This allows us to construct the universal enveloping VOA V_{𝔤}. We show (Proposition 4.2) that for a particular top T_{0}, the irreducible module L(T_{0}) is a factor-VOA of V_{𝔤}. Using the methods developed in [BB], we study the kernel of the projection V_{𝔤} → L(T_{0}). This kernel gives us valuable information about L(T_{0}). Once we determine that a vector v ∈ V_{𝔤} belongs to the kernel, we apply the state-field correspondence Y and conclude that Y_{L(T_{0})}(v, z) = 0. In this way we derive important relations that hold in L(T_{0}). We use these relations to define a toroidal VOA V (T_{0}) as a tensor product of a sub-VOA V_{Hyp}^{+} of a lattice VOA and a VOA V_{𝔣} corresponding to the twisted Virasoro-affine Lie algebra with 𝔣̇ = 𝔤̇ ⊕ gl_{N}.

In the previous papers on this subject, the vertex operator realization for the toroidal modules had to be essentially guessed. The significant difference in the present approach is that we are able to derive all the properties of the vertex operator realizations from inside, using only the relations in the toroidal Lie algebra 𝔤 and its universal enveloping vertex algebra V_{𝔤}.

The VOA V (T_{0}) controls the representation theory of 𝔤. We show (Theorem 5.3) that every irreducible module L(T) in category B_{χ} is a simple VOA module for V (T_{0}) and can be constructed as a tensor product of a simple module M_{Hyp}^{+}(α) for the VOA V_{Hyp}^{+} with an irreducible highest weight module L_{𝔣} for the twisted Virasoro-affine algebra 𝔣. For a generic level c, we can further factor L_{𝔣} in a tensor product and get the following decomposition for L(T):

L(T) ≅ M_{Hyp}^{+}(α) ⊗ L_{𝔤^{}}˙ ⊗ L_{_{sl}_{N}} ⊗ L_{Hei} ⊗ L_{Vir},

where the last four factors are certain irreducible highest weight modules for the affine algebras ˆ 𝔤̇, ŝl_{N}, the infinite-dimensional Heisenberg algebra and the Virasoro algebra. In this way we reduce the representation theory of toroidal Lie algebras to the representation theory of affine, Heisenberg and Virasoro algebras. Whenever explicit realizations are available for the components in the tensor product decomposition above, we get a realization for the irreducible module for the full toroidal Lie algebra.

This leads to the following open problem: while the explicit expressions for the characters of irreducible modules may be known, there is no Weyl-type character formula for the toroidal Lie algebras. Obtaining such a formula may yield interesting number-theoretic identities.

The structure of the paper is the following. In Section 1 we review the construction of the toroidal Lie algebras. In Section 2 we introduce a category B_{χ} of 𝔤-modules and show that every irreducible module in B_{χ} is characterized by its top T. We also describe the structure of the top T. In Section 3 we recall the definition of the vertex operator algebra and the construction of the universal enveloping vertex algebra of a vertex Lie algebra. We introduce twisted Virasoroaffine Lie algebras, show that these algebras are in fact vertex Lie algebras and describe the structure of the corresponding enveloping vertex algebras and their simple modules. At the end of Section 3 we describe the hyperbolic lattice VOA V_{Hyp} and its sub-VOA V_{Hyp}^{+}. In Section 4 we show that the full toroidal Lie algebra is vertex Lie algebra and define its enveloping VOA V_{𝔤}. Next we establish a series of relations that hold in the simple quotient L(T_{0}) and use them to decompose L(T_{0}) into a tensor product of two VOAs, V_{Hyp}^{+} and L_{𝔣}(γ_{0}). In the final Section 5 we prove that a slightly bigger VOA V (T_{0}) = V_{Hyp}^{+} ⊗ V_{𝔣}(γ_{0}) also admits the structure of a module over the full toroidal algebra 𝔤. We show that every irreducible 𝔤-module in category B_{χ} is a simple VOA module for V (T_{0}), which then allows us to obtain a complete description of these irreducible 𝔤-modules.

Acknowledgements: I thank Stephen Berman for the stimulating discussions and encouragement. I have greatly benefited from Chongying Dong’s lectures on vertex operator algebras given at the Fields Institute.

1. Toroidal Lie algebras.

Toroidal Lie algebras are the natural multi-variable generalizations of affine Lie algebras. In this review of the toroidal Lie algebras we follow the work [BB]. Let 𝔤̇ be a simple finitedimensional Lie algebra over ℂ with a non-degenerate invariant bilinear form (·|·) and let N ≥ 1 be an integer. We consider the Lie algebra R ⊗ 𝔤̇ of maps from an N + 1 dimensional torus into 𝔤̇, where R = ℂ[t_{0}^{±}, t_{1}^{±}, …, t_{N}^{±}] is the algebra of Fourier polynomials on a torus. The universal central extension of this Lie algebra may be described by means of the following construction

which is due to Kassel [Kas]. Let Ω_{R}^{1} be the space of 1-forms on a torus: Ω_{R}^{1} = ⊕_{p=0}^{N} Rdt_{p}. We

will choose the forms {k_{p} = t_{p}^{−1}dt_{p}|p = 0, …, N} as a basis of this free R module. There is a

natural map d from the space of functions R into Ω_{R}^{1}: d(f) = ∑_{p=0}^{N} \frac{∂f}{∂t_{p}} dt_{p} = ∑_{p=0}^{N} t_{p} \frac{∂f}{∂t_{p}} k_{p}. The

center K for the universal central extension (R ⊗ 𝔤̇) ⊕ K of R ⊗ 𝔤̇ is realized as

K = Ω_{R}^{1}/d(R),

and the Lie bracket is given by the formula

[f_{1}(t)g_{1}, f_{2}(t)g_{2}] = f_{1}(t)f_{2}(t)[g_{1}, g_{2}] + (g_{1}|g_{2})f_{2}d(f_{1}).

Here and in the rest of the paper we will denote elements of K by the same symbols as elements of Ω_{R}^{1}, keeping in mind the canonical projection Ω_{R}^{1} → Ω_{R}^{1}/d(R).

Next we add to (R ⊗ 𝔤̇) ⊕ K the algebra D of vector fields on the torus

D = ⊕_{p=0}^{N} Rd_{p},

where d_{p} = t_{p} \frac{∂}{∂t_{p}}. We will use the multi-index notation writing t^{r} = t_{0}^{r_{0}}t_{1}^{r_{1}} … t_{N}^{r_{N}} for r = (r_{0}, r_{1}, …, r_{N}), etc.

The natural action of D on R ⊗ 𝔤̇

[t^{r}d_{a}, t^{m}g] = m_{a}t^{r+m}g  (1.1)

uniquely extends to the action on the universal central extension (R ⊗ 𝔤̇) ⊕ K by

[t^{r}d_{a}, t^{m}k_{b}] = m_{a}t^{r+m}k_{b} + δ_{ab} ∑_{p=0}^{N} r_{p}t^{r+m}k_{p}.  (1.2)

This corresponds to the Lie derivative action of the vector fields on 1-forms.

It turns out that there is still an extra degree of freedom in defining the Lie algebra structure on (R ⊗ 𝔤̇) ⊕ K ⊕ D. The Lie bracket on D may be twisted with a K-valued 2-cocycle:

[t^{r}d_{a}, t^{m}d_{b}] = m_{a}t^{r+m}d_{b} − r_{b}t^{r+m}d_{a} + τ(t^{r}d_{a}, t^{m}d_{b}).  (1.3)

In order to compute the second cohomology space H^{2}(D, K), one could use the Gelfand-Fuks cohomology theory [F], [T]. Unfortunately this theory only allows one to do the computations in the C^{∞} setup, i.e., when R is replaced with the algebra of infinitely differentiable functions on a torus, and not for the algebra of Fourier polynomials that we consider here. For the C^{∞} situation the calculation of H_{C^{∞}}^{2}(D, K) has been carried out in [BN]. For the (N +1)-dimensional torus with N + 1 ≥ 2, the dimension of the second cohomology space is

dim H_{C^{∞}}^{2}(D, K) = 2 + ( \begin{matrix}N + 1 \\ 3 \end{matrix} ) ,

and the following cocycles form the basis of this space:

τ_{1}(t^{r}d_{a}, t^{m}d_{b}) = m_{a}r_{b} ∑_{p=0}^{N} m_{p}t^{r+m}k_{p},

τ_{2}(t^{r}d_{a}, t^{m}d_{b}) = r_{a}m_{b} ∑_{p=0}^{N} m_{p}t^{r+m}k_{p},

together with a family {η_{abc}|0 ≤ a < b < c ≤ N}, where the cocycle η_{abc} is defined by the conditions that

η_{abc}(t^{r}d_{σ(a)}, t^{m}d_{σ(b)}) = (−1)^{σ}t^{r+m}k_{σ(c)},

for any permutation σ : {a, b, c} → {a, b, c} and η_{abc}(t^{r}d_{i}, t^{m}d_{j}) = 0 if i = j or {i, j} ⊄ {a, b, c}. It is clear that H^{2}(D, K) in the algebraic setup contains the space H_{C^{∞}}^{2}(D, K). After twisting with a cocycle η_{abc}, the vector fields d_{a} and d_{b} no longer commute. For this reason we will be considering only the cocycles τ_{1} and τ_{2} in this paper.

We will write τ = μτ_{1} + ντ_{2}. The resulting algebra (or rather a family of algebras) is called the full toroidal Lie algebra

𝔤 = 𝔤(μ, ν) = (R ⊗ 𝔤̇) ⊕ K ⊕ D.

Note that after adding the algebra of derivations D, the center Z of the toroidal Lie 𝔤 becomes finite-dimensional with the basis {k_{0}, k_{1}, …, k_{N}}. This can be seen from the action (1.2) of D on K, which is non-trivial.

The study representation theory of toroidal Lie algebras has begun in [MRY] and [EM], with further developments in [B1], [L], [BB], [BBS]. In all of these papers there was one common difficulty that has not been resolved – the representations constructed there were not for the full toroidal algebra 𝔤, but only for a subalgebra

𝔤^{∗} = (R ⊗ 𝔤̇) ⊕ K ⊕ ( ⊕_{p=1}^{N} Rd_{p}) ,

where the piece Rd_{0} that corresponds to the toroidal energy-momentum tensor was missing. This left the theory in a somewhat incomplete form, and the goal of the present paper is to study representations for the full toroidal Lie algebra.

2. A category of bounded modules for toroidal Lie algebras.

In this section we will introduce a category of bounded modules for toroidal Lie algebras that could be regarded as analogs of the highest weight modules for affine Kac-Moody algebras. The difference from the affine case is that the highest weight space is not 1-dimensional, but rather forms a multi-loop module for a smaller toroidal subalgebra.

These bounded modules are quite promising from the point of view of applications. In [B2] a module of this type was used to construct a toroidal extension of the Korteweg-de Vries hierarchy. The vertex operator realizations of the toroidal modules allow one to construct soliton solutions to these non-linear PDEs.

The variable t_{0} will play a special role in our construction. From the physics perspective, it may be interpreted as time, whereas t_{1}, … t_{N} are the space variables.

The algebra 𝔤 has a ℤ^{N+1}-grading by the eigenvalues of the adjoint action of d_{0}, d_{1}, … d_{N}. We will denote by {ϵ_{0}, …, ϵ_{N}} the standard basis of ℤ^{N+1}. We will be also considering its ℤ-grading just with respect to the action of d_{0}:

𝔤 = ⊕_{n∈ℤ} 𝔤_{n},

and define subalgebras 𝔤_{±} = ⊕_{n≷0} 𝔤_{n}, which yields the decomposition 𝔤 = 𝔤_{−} ⊕ 𝔤_{0} ⊕ 𝔤_{+}.

We recall that the center Z of 𝔤 is N + 1-dimensional: Z = Span ⟨k_{0}, k_{1}, …, k_{N}⟩. Clearly, in any irreducible weight module with finite-dimensional weight spaces, these central elements will act as multiplications by scalars. Let us fix a non-zero central character χ : Z → ℂ and define a category of bounded modules with central character χ for the toroidal Lie algebra.

Definition. A category B_{χ} of bounded modules for the toroidal Lie algebra is a category whose objects are 𝔤-modules B satisfying the following axioms:

(B1) B has a weight decomposition with respect to the subalgebra ⟨d_{0}, d_{1}, …, d_{N}⟩ :

B = ⊕_{m∈ℂ^{N+1}} B_{m},

where B_{m} = {v ∈ B | d_{j}(v) = m_{j}v, j = 0, …, N}; (B2) All weight spaces B_{m} are finite-dimensional; (B3) The action of the center Z on B is given by the central character χ: k_{j}v = χ(k_{j})v

for all v ∈ B, j = 0, …, N; (B4) Real parts of eigenvalues of d_{0} on B are bounded from above.

The physical meaning of the last axiom is that the spectrum of the energy operator E = −d_{0} has a lower bound, i.e., there exist states of the lowest energy.

The goal of this paper is to describe irreducible modules in category B_{χ} and find their characters.

First of all we are going to show that in order for B_{χ} to be non-trivial, the central character must satisfy the condition χ(k_{1}) = 0, …, χ(k_{N}) = 0. Clearly, χ must vanish on an N-dimensional subspace in Z, and it turns out that this N-dimensional nullspace must be “aligned” with the choice of the operator d_{0} in axiom (B4).

Lemma 2.1. Suppose that B_{χ} is a non-trivial category. Then χ(k_{j}) = 0 for all j = 1, …, N.

Proof. Let B be a non-zero module in B_{χ}. Let us reason by contradiction and assume that χ(k_{j}) = c_{j} ≠ 0 for some j, 1 ≤ j ≤ N.

Since the spectrum of d_{0} is bounded, we can find a weight space B_{m} such that m_{0} + 1 is not an eigenvalue of d_{0} on B. Let v be a non-zero vector in B_{m}, and consider the following family of vectors:

(t_{0}^{−1}t_{j}^{−n}k_{j})(t_{0}^{−1}t_{j}^{n}k_{j})v, n = 1, 2, …

Clearly, all these vectors belong to the same weight space B_{m−2ϵ_{0}}, and we claim that they are all linearly independent. Indeed, suppose

∑_{n>0} a_{n}(t_{0}^{−1}t_{j}^{−n}k_{j})(t_{0}^{−1}t_{j}^{n}k_{j})v = 0.

Since

d_{0}(t_{0}t_{j}^{r}d_{0})v = (m_{0} + 1)(t_{0}t_{j}^{r}d_{0})v,

and m_{0} + 1 is not an eigenvalue of d_{0} on B, we conclude that (t_{0}t_{j}^{r}d_{0})v = 0 for all r ∈ ℤ. We also note that

[t_{0}t_{j}^{r}d_{0}, t_{0}^{−1}t_{j}^{s}k_{j}] = −t_{j}^{r+s}k_{j} = −δ_{r,−s}k_{j}.

Taking these two facts into account, we get that for r > 0,

0 = (t_{0}t_{j}^{r}d_{0})(t_{0}t_{j}^{−r}d_{0}) ∑_{n>0} a_{n}(t_{0}^{−1}t_{j}^{−n}k_{j})(t_{0}^{−1}t_{j}^{n}k_{j})v = a_{r}c_{j}^{2}v.

Since c_{j} ≠ 0, we conclude that a_{r} = 0 for all r > 0. Thus the vectors {(t_{0}^{−1}t_{j}^{−n}k_{j})(t_{0}^{−1}t_{j}^{n}k_{j})v} with n > 0 are linearly independent, which contradicts (B2). This proves that χ(k_{1}) = 0, …, χ(k_{N}) = 0.

For the rest of this paper we fix a non-zero constant c ∈ ℂ and let χ = (c, 0, …, 0). From now on, the multivariable t will not include t_{0}, that is t^{r} will stand for t_{1}^{r_{1}} … t_{N}^{r_{N}}, etc.

Consider an irreducible module L in category B_{χ}. It is clear that the eigenvalues of d_{0} on L belong to a single ℤ-coset in ℂ. Let d be the eigenvalue of d_{0} with the highest real part, and let T be the corresponding eigenspace.

Obviously, T is a 𝔤_{0}-module and 𝔤_{+}T = 0. It is easy to see that irreducibility of L implies the irreducibility of T as a 𝔤_{0}-module. We will call the subspace T the top of L. Next we are going to describe the structure of T. We will be using a result of [JM] for this.

Theorem 2.2 ([JM]). Suppose χ(k_{0}) = c ≠ 0, χ(k_{1}) = 0, …, χ(k_{N}) = 0. Let L be an irreducible module in category B_{χ} with the top T. Then

T ≅ ℂ[q_{1}^{±}, … q_{N}^{±}] ⊗ U,

where U is a finite-dimensional space, and the action of 𝔤_{0} on T satisfies

(t^{r}k_{0})(q^{m} ⊗ u) = cq^{m+r} ⊗ u,  (t^{r}k_{j})(q^{m} ⊗ u) = 0,  (2.1)
d_{0}(q^{m} ⊗ u) = dq^{m} ⊗ u, d_{j}(q^{m} ⊗ u) = (m_{j} + α_{j})q^{m} ⊗ u, u ∈ U, j = 1, …, N,  (2.2)

for some fixed α = (α_{1}, …, α_{N}) ∈ ℂ^{N}, d ∈ ℂ.

If we take the quotient of 𝔤_{0} by the ideal J = Span {t^{r}k_{j}|r ∈ ℤ^{N}, j = 1, …, N}, which annihilates T, then we will get a semi-direct product of the Lie algebra of vector fields D_{N} = Der ℂ[t_{1}^{±}, …, t_{N}^{±}] on N-dimensional torus with a multi-loop algebra:

𝔤_{0}/J ≅ D_{N} ⋉ ℂ[t_{1}^{±}, …, t_{N}^{±}] ⊗ (𝔤̇ ⊕ ℂd_{0} ⊕ ℂk_{0}) .

Since \frac{1}{c}(t^{r}k_{0}) acts on T as multiplication by q^{r}, we can derive from (1.2) the following compatibility relations between the action of 𝔤_{0} and the operators of multiplication by q^{r}:

(t^{s}d_{j})q^{r} − q^{r}(t^{s}d_{j}) = r_{j}q^{s+r},  (2.3)
(t^{s}d_{0})q^{r} = q^{r}(t^{s}d_{0}),  (t^{s}g)q^{r} = q^{r}(t^{s}g), g ∈ 𝔤̇.  (2.4)

Eswara Rao [E] classified irreducible D_{N}-modules with a compatible action of the algebra of Laurent polynomials, proving that any such module is a tensor module. We will use a version of this result for the semidirect product of D_{N} with a multi-loop algebra given in [B5], Theorem 4(c):

Theorem 2.3 ([E], [B5]). Let α ∈ ℂ^{N}, c, d ∈ ℂ, c ≠ 0. Let T be an irreducible 𝔤_{0}- module satisfying the conclusion of Theorem 2.2, as well as (2.3), (2.4). Then there exist a finite-dimensional irreducible 𝔤̇-module V and a finite-dimensional irreducible gl_{N}-module W, such that

T ≅ ℂ[q_{1}^{±}, … q_{N}^{±}] ⊗ V ⊗ W,  (2.5)

and the action of 𝔤_{0} on T is given by (2.1) and

(t^{r}d_{j})(q^{m} ⊗ v ⊗ w) = (m_{j} + α_{j})q^{m+r} ⊗ v ⊗ w + ∑_{p=1}^{N} r_{p}q^{m+r} ⊗ v ⊗ E_{pj}w, j = 1, …, N, (2.6)

(t^{r}d_{0})(q^{m} ⊗ v ⊗ w) = dq^{m+r} ⊗ v ⊗ w,  (2.7)
(t^{r}g)(q^{m} ⊗ v ⊗ w) = q^{m+r} ⊗ gv ⊗ w, g ∈ 𝔤̇.  (2.8)

Here in (2.6) E_{pj} denotes a matrix with 1 in position (p, j), and zeros elsewhere.

Combining these two theorems, we conclude that an irreducible module in category B_{χ} yields the following data – a finite-dimensional irreducible 𝔤̇-module V, a finite-dimensional irreducible gl_{N}-module W, a constant d ∈ ℂ and α ∈ ℂ^{N}. It is easy to see that the choice of α is not canonical and α can be changed to any value in the coset α + ℤ^{N} by choosing a different weight space for the generators of T as a free ℂ[q_{1}^{±}, …, q_{N}^{±}]-module. An irreducible gl_{N}-module W is determined by the action of sl_{N} and a scalar h, by which the identity matrix acts on W.

It was shown in [BB] that for any 𝔤_{0}-module T, corresponding to the data (V, W, h, d, α) as above, there exists an irreducible module in B_{χ} with T as a top. Let us review this construction.

First we let 𝔤_{+} act on T trivially, and define the generalized Verma module as the induced module

M(T) = Ind𝔤_{𝔤_{0}⊕𝔤_{+}}(T).

Note that the module M(T) does not belong to category B_{χ} since its weight spaces lying below T are infinite-dimensional. Nonetheless, the following result holds:

Theorem 2.4 ([BB]). (a) The 𝔤-module M(T) has a unique maximal submodule M^{rad}.

(b) The factor-module L(T) = M(T)/M^{rad} is an irreducible 𝔤-module.

(c) All weight spaces of L(T) are finite-dimensional, and L(T) belongs to the category B_{χ}.

Summarizing, we get the following

Theorem 2.5. (a) Let χ be a non-zero central character χ : Z → ℂ. A category B_{χ} is non-trivial if and only if χ(k_{0}) = c, χ(k_{1}) = 0, …, χ(k_{N}) = 0 for some non-zero c ∈ ℂ.

Let now χ = (c, 0, …, 0) with c ≠ 0.

(b) Irreducible 𝔤-modules in category B_{χ} are in 1-1 correspondence with the data (V, W, h, d, α), where V is a finite-dimensional irreducible 𝔤̇-module, W is a finite-dimensional irreducible sl_{N}-module, α ∈ ℂ^{N}/ℤ^{N}, h, d ∈ ℂ.

(c) Every irreducible module in category B_{χ} is isomorphic to L(T) where

T = ℂ[q_{1}^{±}, … q_{N}^{±}] ⊗ V ⊗ W

with the action of 𝔤_{0} given by (2.1) and (2.6)-(2.8).

Proof. Part (a) has been already proved in Lemma 2.1. Let us prove part (c). As we have seen above, an irreducible module L in category B_{χ} has a top T, the structure of which is described by Theorem 2.3. Thus L is a factor-module of M(T). However M(T) has a unique irreducible factor, which is isomorphic to L(T). This proves L ≅ L(T). Part (b) follows from (c) and Theorem 2.4.

Our main goal will be to completely determine the structure of the irreducible modules L(T), and in particular, to find their characters. This will be done using the theory of vertex operator algebras (VOAs). We will show that for a particular choice of the data (V, W, h, d, α), namely, with V and W being trivial 1-dimensional modules for 𝔤̇ and sl_{N}, α = 0, h = Nνc and d = \frac{1}{2}(μ + ν)c, yielding the top

T_{0} = ℂ[q_{1}^{±}, …, q_{N}^{±}],

the module L(T_{0}) is a vertex operator algebra, while all irreducible modules L(T) are VOAmodules for a slightly bigger VOA V (T_{0}). Once we determine the structure of V (T_{0}) as a VOA, we will immediately get the structure of all the modules L(T) using the principle of preservation of identities in the VOA theory.

3. Vertex operator algebras and vertex Lie algebras.

3.1. Definitions and properties of a VOA.

Let us recall the basic notions of the theory of the vertex operator algebras. Here we are following [K2] and [Li].

Definition. A vertex algebra is a vector space V with a distinguished vector 1 (vacuum vector) in V, an operator D (infinitesimal translation) on the space V, and a linear map Y (state-field correspondence)

Y (·, z) : V → (EndV)[[z, z^{−1}]],
a ↦ Y (a, z) = ∑_{n∈ℤ} a_{(n)}z^{−n−1} (where a_{(n)} ∈ EndV),

such that the following axioms hold: (V1) For any a, b ∈ V, a_{(n)}b = 0 for n sufficiently large; (V2) [D, Y (a, z)] = Y (D(a), z) = \frac{d}{dz} Y (a, z) for any a ∈ V; (V3) Y (1, z) = Id_{V}; (V4) Y (a, z)1 ∈ V [[z]] and Y (a, z)1|_{z=0} = a for any a ∈ V (self-replication); (V5) For any a, b ∈ V, the fields Y (a, z) and Y (b, z) are mutually local, that is,

(z − w)^{n} [Y (a, z), Y (b, w)] = 0, for n sufficiently large.

A vertex algebra V is called a vertex operator algebra (VOA) if, in addition, V contains a vector ω (Virasoro element) such that (V6) The components L(n) = ω_{(n+1)} of the field

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

satisfy the Virasoro algebra relations:

[L(n), L(m)] = (n − m)L(n + m) + δ_{n,−m} \frac{n^{3} − n}{12}(rank V)Id, where rank V ∈ ℂ; (3.1)

(V7) D = L(−1); (V8) V is graded by the eigenvalues of L(0): V = ⊕_{n∈ℤ} V_{n} with L(0)|_{V_{n}} = nId.

This completes the definition of a VOA.

As a consequence of the axioms of the vertex algebra we have the following important commutator formula:

[Y (a, z_{1}), Y (b, z_{2})] = ∑_{n≥0} \frac{1}{n!}Y (a_{(n)}b, z_{2}) [z_{1}^{−1} (\frac{∂}{∂z_{2}} )^{n} δ (\frac{z_{2}}{z_{1}} )] .  (3.2)

As usual, the delta function is

δ(z) = ∑_{n∈ℤ} z^{n}.

By (V1), the sum in the right hand side of the commutator formula is actually finite.

All the vertex operator algebras that appear in this paper have the gradings by non-negative integers (degree): V = ⊕_{n=0}^{∞} V_{n}. In this case the sum in the right hand side of the commutator

formula (3.2) runs from n = 0 to n = deg (a) + deg (b) − 1, because

deg (a_{(n)}b) = deg (a) + deg (b) − n − 1,  (3.3)

and the elements of negative degree vanish.

It follows from (V7) and (V8) that

ω_{(0)}a = D(a), ω_{(1)}a = deg (a)a  for a homogeneous.  (3.4)

Another consequence of the axioms of a vertex algebra is the Borcherds’ identity:

∑_{j≥0} ( \begin{matrix}m \\ j \end{matrix} ) (a_{(k+j)}b)_{(m+n−j)}c

= ∑_{j≥0}(−1)^{k+j+1} ( \begin{matrix}k \\ j \end{matrix} ) b_{(n+k−j)}a_{(m+j)}c + ∑_{j≥0}(−1)^{j} ( \begin{matrix}k \\ j \end{matrix} ) a_{(m+k−j)}b_{(n+j)}c, k, m, n ∈ ℤ.

(3.5)

We will particularly need its special case when m = 0:

(a_{(k)}b)_{(n)}c = ∑_{j≥0}(−1)^{k+j+1} ( \begin{matrix}k \\ j \end{matrix} ) b_{(n+k−j)}a_{(j)}c + ∑_{j≥0}(−1)^{j} ( \begin{matrix}k \\ j \end{matrix} ) a_{(k−j)}b_{(n+j)}c, k, n ∈ ℤ.

(3.6)

The last formula that we quote here is the skew-symmetry identity:

a_{(n)}b = ∑_{j≥0}(−1)^{n+j+1} \frac{1}{j!}D^{j}(b_{(n+j)}a).  (3.7)

3.2. Tensor products of VOAs. Let us review here the definition of the tensor product of two VOAs (V′, Y′, ω′, 1) and

(V′′, Y′′, ω′′, 1) (the case of an arbitrary number of factors is a trivial generalization). The tensor product space V = V′ ⊗ V′′ has the VOA structure under

Y (a ⊗ b, z) = Y′(a, z) ⊗ Y′′(b, z),  (3.8)
ω = ω′ ⊗ 1 + 1 ⊗ ω′′,  (3.9)

and 1 = 1 ⊗ 1 being the identity element.

It follows from (3.9) that the rank of V (see V6) is the sum of the ranks of the tensor factors.

3.3. Vertex Lie algebras. An important source of the vertex algebras is provided by the vertex Lie algebras. In

presenting this construction we will be following [DLM] (see also [P], [R], [K2], [FKRW]).

Let L be a Lie algebra with the basis {u(n), c(−1)|u ∈ U, c ∈ C, n ∈ ℤ} (U, C are some index sets). Define the corresponding fields in L[[z, z^{−1}]]:

u(z) = ∑_{n∈ℤ} u(n)z^{−n−1}, c(z) = c(−1)z^{0}, u ∈ U, c ∈ C.

Let F be a subspace in L[[z, z^{−1}]] spanned by all the fields u(z), c(z) and their derivatives of all orders.

Definition. A Lie algebra L with the basis as above is called a vertex Lie algebra if the following two conditions hold:

(VL1) for all u_{1}, u_{2} ∈ U,

[u_{1}(z_{1}), u_{2}(z_{2})] = ∑_{j=0}^{n} f_{j}(z_{2}) [z_{1}^{−1} (\frac{∂}{∂z_{2}} )^{j} δ (\frac{z_{2}}{z_{1}} )] ,  (3.10)

where f_{j}(z) ∈ F, n ≥ 0 and depend on u_{1}, u_{2}, (VL2) for all c ∈ C, the elements c(−1) are central in L.

Let L^{(+)} be a subspace in L with the basis {u(n)|u ∈ U, n ≥ 0} and let L^{(−)} be a subspace with the basis {u(n), c(−1)|u ∈ U, c ∈ C, n < 0}. Then L = L^{(+)} ⊕ L^{(−)} and L^{(+)}, L^{(−)} are in fact subalgebras in L.

The universal enveloping vertex algebra V_{L} of a vertex Lie algebra L is defined as an induced module

V_{L} = Ind_{L^{(+)}}^{L}(ℂ1) = U(L^{(−)}) ⊗ 1,

where ℂ1 is a trivial 1-dimensional L^{(+)} module.

Theorem 3.1. ([DLM], Theorem 4.8) Let L be a vertex Lie algebra. Then

(a) V_{L} has a structure of a vertex algebra with the vacuum vector 1, infinitesimal translation D being a natural extension of the derivation of L given by D(u(n)) = −nu(n−1), D(c(−1)) = 0, u ∈ U, c ∈ C, and the state-field correspondence map Y defined by the formula:

Y (a_{1}(−1 − n_{1}) … a_{k−1}(−1 − n_{k−1})a_{k}(−1 − n_{k})1, z)

=: (\frac{1}{n_{1}!} (\frac{∂}{∂z} )^{n_{1}} a_{1}(z)) …: (\frac{1}{n_{k−1}!} (\frac{∂}{∂z} )^{n_{k−1}} a_{k−1}(z)) (\frac{1}{n_{k}!} (\frac{∂}{∂z} )^{n_{k}} a_{k}(z)) : …: ,

(3.11)

where a_{j} ∈ U, n_{j} ≥ 0 or a_{j} ∈ C, n_{j} = 0.

(b) Any bounded L-module is a vertex algebra module for V_{L}.

(c) For an arbitrary character γ : C → ℂ, the factor module

V_{L}(γ) = U(L^{(−)})1/U(L^{(−)})⟨(c(−1) − γ(c))1⟩_{c∈C}

is a quotient vertex algebra.

(d) Any bounded L-module in which c(−1) act as γ(c)Id, for all c ∈ C, is a vertex algebra module for V_{L}(γ).

In the formula (3.11) above, the normal ordering of two fields : a(z)b(z) : is defined as

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

Note the following relation that we will implicitly use throughout the paper: (a(−1)1)_{(n)} = a(n) for a ∈ U, n ∈ ℤ.

Theorem 3.2. Any D-invariant L-submodule U in V_{L} is a vertex algebra ideal in V_{L}. Conversely, every vertex algebra ideal in V_{L} is a D-invariant L-submodule.

Proof. Let us prove the first part of the Theorem. We need to show that for any u ∈ U, a ∈ V_{L}, n ∈ ℤ, we have a_{(n)}u ∈ U and u_{(n)}a ∈ U. By (3.7), it is enough to prove that a_{(n)}u ∈ U. It is sufficient to consider a of the form a = a_{1}(−1 − n_{1}) … a_{k}(−1 − n_{k})1, where a_{j} ∈ U ∪ C, n_{j} ≥ 0. We use induction on k. For k = 0 we get that a = 1 and 1_{(n)}u = δ_{n,−1}u. The inductive step follows from the Borcherds’ formula (3.6) and L-invariance of U. The second part of the Theorem follows immediately from the definition of V_{L}.

Corollary 3.3. If U is a maximal D-invariant L-submodule in V_{L} then the quotient L_{L} = V_{L}/U is a simple vertex algebra.

Remark 3.4. In case when the set U contains an element ω generating the Virasoro field ω(z) in L, satisfying [ω(0), a(n)] = −na(n − 1) for all a ∈ U, the vertex algebra V_{L} becomes a VOA, and in the statement of Theorem 3.2 the condition of D-invariance of U will automatically follow from its L-invariance.

3.4. VOA associated with the twisted Virasoro-affine algebra.

The toroidal VOA that will be constructed in Section 4, decomposes into a tensor product of two VOAs. One of these factors is a VOA associated with a twisted Virasoro-affine Lie algebra, which we introduce here.

Let 𝔣̇ be a finite-dimensional reductive Lie algebra. Consider a semi-direct product of the Lie algebra of vector fields on a circle with a loop algebra:

𝔣̃ = Der ℂ[t_{0}, t_{0}^{−1}] ⋉ ℂ[t_{0}, t_{0}^{−1}] ⊗ 𝔣̇.

A twisted Virasoro-affine algebra 𝔣 is the universal central extension of the Lie algebra 𝔣̃. Using the results on the central extensions of the loop algebras and the Lie algebra of vector fields on a circle, one can show that the second cohomology of 𝔣̃ has the following description:

H^{2}(𝔣̃) = S^{2}(𝔣̇)^{inv} ⊕ 𝔣̇^{inv} ⊕ ℂ,

where the last 1-dimensional component corresponds to the Virasoro cocycle on Der ℂ[t_{0}, t_{0}^{−1}], and its generator will be denoted by C_{Vir}. Since 𝔣̇ is reductive, we have the following canonical projections of 𝔣̇-modules:

φ : 𝔣̇ ⊗ 𝔣̇ → S^{2}(𝔣̇)^{inv}

and

ψ : 𝔣̇ → Z(𝔣̇) = 𝔣̇^{inv}.

We will use this maps to write down the Lie bracket in the twisted Virasoro-affine algebra 𝔣 = 𝔣̃ ⊕ S^{2}(𝔣̇)^{inv} ⊕ 𝔣̇^{inv} ⊕ ℂ:

[L(n), L(m)] = (n − m)L(n + m) + \frac{n^{3} − n}{12}δ_{n,−m}C_{Vir},  (3.12)
[L(n), f(m)] = −mf(n + m) − (n^{2} + n)δ_{n,−m}ψ(f),  (3.13)
[f(n), g(m)] = [f, g](n + m) + nδ_{n,−m}φ(f ⊗ g), f, g ∈ 𝔣̇.  (3.14)

Here and below we are using the notations L(n) = −t_{0}^{n+1} \frac{d}{dt_{0}} and f(n) = t_{0}^{n} ⊗ f for f ∈ 𝔣̇. Consider the following fields in 𝔣:

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

and

f(z) = ∑_{n∈ℤ} f(n)z^{−n−1}, for f ∈ 𝔣̇.

Proposition 3.5. Twisted Virasoro-affine Lie algebra 𝔣 is a vertex Lie algebra.

Proof. Take for a set U the element ω together with a basis of 𝔣̇, and for a set C a basis of S^{2}(𝔣̇)^{inv} ⊕ 𝔣̇^{inv} ⊕ ℂ. Then the defining relations (3.12)-(3.14) may be rewritten as follows:

\begin{aligned}[ω(z_{1}), ω(z_{2})] = (\frac{∂}{∂z_{2}} ω(z_{2})) [z_{1}^{−1}δ (\frac{z_{2}}{z_{1}} )] & + 2ω(z_{2}) [z_{1}^{−1} \frac{∂}{∂z_{2}} δ (\frac{z_{2}}{z_{1}} )] \\ & + \frac{C_{Vir}}{12} [z_{1}^{−1} (\frac{∂}{∂z_{2}} )^{3} δ (\frac{z_{2}}{z_{1}} )] , \end{aligned}  (3.15)

\begin{aligned}[ω(z_{1}), f(z_{2})] = (\frac{∂}{∂z_{2}} f(z_{2})) [z_{1}^{−1}δ (\frac{z_{2}}{z_{1}} )] & + f(z_{2}) [z_{1}^{−1} \frac{∂}{∂z_{2}} δ (\frac{z_{2}}{z_{1}} )] \\ & − ψ(f) [z_{1}^{−1} (\frac{∂}{∂z_{2}} )^{2} δ (\frac{z_{2}}{z_{1}} )] , \end{aligned}  (3.16)

[f(z_{1}), g(z_{2})] = [f, g](z_{2}) [z_{1}^{−1}δ (\frac{z_{2}}{z_{1}} )] + φ(f ⊗ g) [z_{1}^{−1} \frac{∂}{∂z_{2}} δ (\frac{z_{2}}{z_{1}} )] .  (3.17)

This shows that 𝔣 is indeed a vertex Lie algebra, and the claim of the Proposition is established. From now on we fix 𝔣̇ to be 𝔣̇ = 𝔤̇ ⊕ gl_{N}. A linear map S^{2}(𝔣̇)^{inv} → ℂ defines a symmetric invariant bilinear form on 𝔣̇. Thus S^{2}(𝔣̇)^{inv}

is the dual space to the space of symmetric invariant forms on 𝔣̇ and has dimension 3. Let us fix an invariant form on 𝔤̇ normalized by the condition that (α|α) = 2 for the long roots of 𝔤̇, an invariant form on sl_{N} with the same normalization, and a form on the space of scalar matrices normalized by its value on the identity matrix: (I|I) = 1. Denote by {C𝔤^{}˙ , C_{sl_{N}} , C_{Hei}} the dual basis in S^{2}(𝔣̇)^{inv}.

The space 𝔣̇^{inv} ≅ Z(𝔣̇) is one-dimensional, and we will denote its generator ψ(I) by C_{VH}. Hence the space H^{2}(𝔣̃) is 5-dimensional with the basis {C𝔤^{}˙ , C_{sl_{N}} , C_{Hei}, C_{VH}, C_{Vir}}.

The Lie algebra 𝔣 contains four subalgebras – a Virasoro algebra, two affine algebras, ˆ𝔤̇ = ℂ[t_{0}, t_{0}^{−1}] ⊗ 𝔤̇ ⊕ ℂC𝔤^{}˙ and ŝl_{N} = ℂ[t_{0}, t_{0}^{−1}] ⊗ sl_{N} ⊕ ℂC_{sl_{N}}, and an infinite-dimensional Heisenberg algebra Hei = ℂ[t_{0}, t_{0}^{−1}] ⊗ I ⊕ ℂC_{Hei}.

Fix a central character γ : H^{2}(𝔣̃) → ℂ:

γ(C𝔤^{}˙ ) = c𝔤^{}˙ , γ(C_{sl_{N}}) = c_{sl_{N}} , γ(C_{Hei}) = c_{Hei}, γ(C_{VH}) = c_{VH}, γ(C_{Vir}) = c_{Vir},

and consider the corresponding quotient V_{𝔣}(γ) of the universal enveloping vertex algebra. Using the commutator formula (3.2), we derive from (3.16), (3.17) the following relations for the n-th products.

Lemma 3.6. The following relations hold in V_{𝔣}(γ):

\begin{aligned}(a) E_{ab}(0)E_{cd}(−1)1 & = δ_{bc}E_{ad}(−1)1 − δ_{ad}E_{cb}(−1)1, \\ E_{ab}(1)E_{cd}(−1)1 & = δ_{ad}δ_{bc}c_{sl_{N}} 1 + δ_{ab}δ_{cd} (\frac{c_{Hei}}{N^{2}} − \frac{c_{sl_{N}}}{N} ) 1, \\ E_{ab}(n)E_{cd}(−1)1 & = 0 for n ≥ 2. \end{aligned}

\begin{aligned}(b) ω_{(0)}E_{ab}(−1)1 & = D(E_{ab}(−1)1), ω_{(1)}E_{ab}(−1)1 = E_{ab}(−1)1, \\ ω_{(2)}E_{ab}(−1)1 & = −δ_{ab} \frac{2c_{VH}}{N} 1, ω_{(n)}E_{ab}(−1)1 = 0 for n ≥ 3. \end{aligned}

Let us now discuss bounded weight modules for 𝔣. This Lie algebra is ℤ-graded by degree in t_{0}. We associate with this grading a decomposition 𝔣 = 𝔣_{−} ⊕ 𝔣_{0} ⊕ 𝔣_{+}, where 𝔣_{0} = ℂd_{0} ⊕ 𝔤̇ ⊕ gl_{N} ⊕ H^{2}(𝔣̃). Let V be a finite-dimensional irreducible 𝔤̇-module, and W be a finitedimensional irreducible module for sl_{N}. Fix a central character γ : H^{2}(𝔣̃) → ℂ and two constants h_{Vir}, h_{Hei} ∈ ℂ. We define on V ⊗ W the structure of an irreducible 𝔣_{0}-module on which L(0) = −d_{0} acts as multiplication by h_{Vir}, I acts as multiplication by h_{Hei}, and the action of H^{2}(𝔣̃) is determined by γ. Let 𝔣_{+} act on V ⊗ W trivially and consider the induced module

M_{𝔣}(V, W, h_{Hei}, h_{Vir}, γ) = Ind𝔣_{𝔣_{0}⊕𝔣_{+}} (V ⊗ W) .

This module has a unique maximal submodule, and the factor-module by the maximal submodule is an irreducible 𝔣-module, which we denote as L_{𝔣}(V, W, h_{Hei}, h_{Vir}, γ).

Remark 3.7. Similar constructions may be applied to Virasoro, affine, and Heisenberg algebras, yielding the corresponding vertex algebras V_{Vir}(c_{Vir}), V_{𝔤^{}}˙ (c𝔤^{}˙ ), V_{_{sl}_{N}} (c_{sl_{N}}), V_{Hei}(c_{Hei}) and irreducible highest weight modules L_{Vir}(h_{Vir}, c_{Vir}) for the Virasoro algebra, L_{𝔤^{}}˙ (V, c𝔤^{}˙ ) for

the affine algebra ˆ 𝔤̇, L_{_{sl}_{N}} (W, c_{sl_{N}}) for the affine algebra ŝl_{N} and L_{Hei}(h_{Hei}, c_{Hei}) for the infinitedimensional Heisenberg algebra.

Note that for the trivial 1-dimensional modules V = ℂ, W = ℂ and h_{Vir} = h_{Hei} = 0, the irreducible module L_{𝔣}(ℂ, ℂ, 0, 0, γ) is precisely the simple VOA L_{𝔣}(γ).

For a generic γ (γ not at a critical level), we may apply the Sugawara construction to decompose the irreducible module L_{𝔣}(V, W, h_{Hei}, h_{Vir}, γ) into a tensor product of irreducible Virasoro, affine and Heisenberg modules.

Proposition 3.8. Let c𝔤^{}˙ ≠ −h^{∨}, c_{sl_{N}} ≠ −N, c_{Hei} ≠ 0, where h^{∨} is the dual Coxeter number for ˆ 𝔤̇. Then the VOA V_{𝔣}(γ) decomposes into a tensor product of four VOAs:

V_{𝔣}(γ) ≅ V_{𝔤^{}}˙ (c𝔤^{}˙ ) ⊗ V_{_{sl}_{N}} (c_{sl_{N}}) ⊗ V_{Hei}(c_{Hei}) ⊗ V_{Vir}(c′_{Vir}),

where

c′_{Vir} = c_{Vir} − \frac{c𝔤^{}˙ dim(𝔤̇)}{c𝔤^{}˙ + h^{∨}} − \frac{c_{sl_{N}} (N^{2} − 1)}{c_{sl_{N}} + N} − 1 + 12\frac{c_{VH}^{2}}{c_{Hei}} ,  (3.18)

and the Heisenberg VOA V_{Hei}(c_{Hei}) is taken with a non-standard Virasoro element

ω_{Hei} = \frac{1}{2c_{Hei}} I(−1)I(−1)1 + \frac{c_{VH}}{c_{Hei}} I(−2)1,  (3.19)

so that its rank is 1 − 12\frac{c_{VH}^{2}}{c_{Hei}}.

Proof. This result is obtained by applying the Sugawara construction three times – to the affine ˆ 𝔤̇-subalgebra, affine ŝl_{N}-subalgebra, and the twisted Virasoro-Heisenberg subalgebra (see e.g. [FLM] and [ACKP] for details). Since this construction is well-known, we only sketch the proof.

Let {u_{i}}, {u^{i}} be dual bases of 𝔤̇, and {v_{j}}, {v^{j}} be dual bases of sl_{N} with respect to the chosen invariant bilinear forms. Consider a new Virasoro field

\begin{matrix}ω′(z) = & ω(z) − \frac{1}{2(c𝔤^{}˙ + h^{∨})} ∑_{i} : u_{i}(z)u^{i}(z) : −\frac{1}{2(c_{sl_{N}} + N)} ∑_{j} : v_{j}(z)v^{j}(z) : \\ − \frac{1}{2c_{Hei}} : I(z)I(z) : −\frac{c_{VH}}{c_{Hei}} \frac{∂}{∂z} I(z). \end{matrix}  (3.20)

It is possible to verify that the moments of ω′(z) satisfy the Virasoro algebra relations, with the action of the central element given by (3.18). Moreover, this new Virasoro field ω′(z) commutes with the fields of the affine ˆ 𝔤̇ and ŝl_{N} subalgebras, as well the Heisenberg subalgebra field.

The formula (3.20) defines a homomorphism of vertex algebras

V_{𝔤^{}}˙ (c𝔤^{}˙ ) ⊗ V_{_{sl}_{N}} (c_{sl_{N}}) ⊗ V_{Hei}(c_{Hei}) ⊗ V_{Vir}(c′_{Vir}) → V_{𝔣}(γ),

which is in fact an isomorphism. Moreover, if we choose the Virasoro element in V_{Hei}(c_{Hei}) to be given by (3.19), the above map becomes the isomorphism of the VOAs.

Corollary 3.9. Under the same restriction on the central charges as in Proposition 3.8, the irreducible highest weight 𝔣-module L_{𝔣}(V, W, h_{Hei}, h_{Vir}, γ) decomposes into a tensor product of irreducible highest weight modules for the affine ˆ 𝔤̇, ŝl_{N}, infinite-dimensional Heisenberg and the Virasoro modules:

L_{𝔣}(V, W, h_{Hei}, h_{Vir}, γ) ≅ L_{𝔤^{}}˙ (V, c𝔤^{}˙ ) ⊗ L_{_{sl}_{N}} (W, c_{sl_{N}}) ⊗ L_{Hei}(h_{Hei}, c_{Hei}) ⊗ L_{Vir}(h′_{Vir}, c′_{Vir}),

where c′_{Vir} is given by (3.18) and

h′_{Vir} = h_{Vir} − \frac{Ω_{V}}{2(c𝔤^{}˙ + h^{∨})} − \frac{Ω_{W}}{2(c_{sl_{N}} + N)} − \frac{h_{Hei}^{2} − 2c_{VH}h_{Hei}}{2c_{Hei}} .  (3.21)

Here Ω_{V} and Ω_{W} are the eigenvalues of the Casimir operators of 𝔤̇ and sl_{N} on V and W respectively. These are given by Ω_{V} = (λ_{V} |λ_{V} + 2ρ), Ω_{W} = (λ_{W} |λ_{W} + 2ρ), where λ_{V} is the highest weight of the irreducible 𝔤̇-module V, and λ_{W} is the highest weight of the sl_{N}-module W [K1].

Remark 3.10. If one of the inequalities in the statement of the above proposition fails, we still can apply a partial Sugawara construction to the remaining components. For example, if c𝔤^{}˙ ≠ −h^{∨}, c_{sl_{N}} ≠ −N, but c_{Hei} = 0, the irreducible highest weight 𝔣-module is isomorphic to the tensor product of the affine ˆ 𝔤̇ and ŝl_{N}-modules and an irreducible highest weight module for the twisted Heisenberg-Virasoro algebra at level zero. The characters of such modules for the twisted Heisenberg-Virasoro algebra were computed in [B3].

As we mentioned earlier, the toroidal VOA decomposes in a tensor product, where one factor is a twisted Virasoro-affine VOA. The other factor in this decomposition is a sub-VOA of a lattice VOA, which we will describe next.

3.5. Hyperbolic lattice VOA.

Here we present the construction of a hyperbolic lattice VOA. The general construction of a VOA corresponding to an arbitrary even lattice may be found in [FLM] or [K2].

Consider a hyperbolic lattice Hyp, which is a free abelian group on 2N generators {u_{i}, v_{i}|i = 1, …, N} with the symmetric bilinear form

(·|·) : Hyp × Hyp → ℤ,

defined by

(u_{i}|v_{j}) = δ_{ij}, (u_{i}|u_{j}) = (v_{i}|v_{j}) = 0.

Note that the form (·|·) is non-degenerate and Hyp is an even lattice, i.e., (x|x) ∈ 2ℤ. The construction of the VOA associated to Hyp proceeds as follows. First we complexify Hyp:

H = Hyp ⊗ℤ ℂ,

and extend (·|·) by linearity on H. Next, we “affinize” H by defining a Lie algebra Ĥ = ℂ[t, t^{−1}] ⊗ H ⊕ ℂK with the bracket

[x(n), y(m)] = n(x|y)δ_{n,−m}K, x, y ∈ H, [Ĥ, K] = 0.  (3.22)

Here and in what follows, we are using the notation x(n) = t^{n} ⊗ x. The algebra Ĥ has a triangular decomposition Ĥ = Ĥ_{−}⊕Ĥ_{0}⊕Ĥ_{+}, where Ĥ_{0} = ⟨1 ⊗ H, K⟩ and Ĥ_{±} = t^{±1}ℂ[t^{±1}]⊗H.

We also need a twisted group algebra of Hyp, denoted by ℂ[Hyp], which we now describe. The basis of ℂ[Hyp] is {e^{x}|x ∈ Hyp}, and the multiplication is twisted with the 2-cocycle ϵ:

e^{x}e^{y} = ϵ(x, y)e^{x+y}, x, y ∈ Hyp,  (3.23)

where ϵ is a multiplicatively bilinear map

ϵ : Hyp × Hyp → {±1},

defined on the generators by ϵ(v_{i}, u_{j}) = (−1)^{δ_{ij}}, ϵ(u_{i}, v_{j}) = ϵ(u_{i}, u_{j}) = ϵ(v_{i}, v_{j}) = 1, i, j = 1, …, N.

We define the structure of Ĥ_{0} ⊕ Ĥ_{+}-module on ℂ[Hyp], letting Ĥ_{+} act on ℂ[Hyp] trivially and Ĥ_{0} act by

x(0)e^{y} = (x|y)e^{y}, Ke^{y} = e^{y}.  (3.24)

Finally let V_{Hyp} be the induced Ĥ module:

V_{Hyp} = Ind_{H} ; _{H_{0}}⊕_{H_{+}} (ℂ[Hyp]) .

This is the VOA attached to the lattice Hyp. As a space V_{Hyp} is isomorphic to the tensor product of the symmetric algebra S(Ĥ_{−}) with the twisted group algebra ℂ[Hyp]:

V_{Hyp} = S(Ĥ_{−}) ⊗ ℂ[Hyp].

The Y -map is defined on the basis elements of ℂ[Hyp] by

Y (e^{x}, z) = exp (∑_{j≥1} \frac{x(−j)}{j}z^{j}) exp (− ∑_{j≥1} \frac{x(j)}{j}z^{−j}) e^{x}z^{x},  (3.25)

where e^{x} acts by twisted multiplication (3.23) and z^{x}e^{y} = z^{(x|y)}e^{y}. For a general basis element a = x_{1}(−1 − n_{1}) … x_{k}(−1 − n_{k}) ⊗ e^{y}, with x_{i}, y ∈ Hyp, n_{i} ≥ 0, one defines (cf. (3.11))

Y (a, z) =: (\frac{1}{n_{1}!} (\frac{∂}{∂z} )^{n_{1}} x_{1}(z)) … (\frac{1}{n_{k}!} (\frac{∂}{∂z} )^{n_{k}} x_{k}(z)) Y (e^{y}, z) :,  (3.26)

where x(z) = ∑_{j∈ℤ} x(j)z^{−j−1}.

The Virasoro element in V_{Hyp} is ω_{Hyp} = ∑_{p=1}^{N} u_{p}(−1)v_{p}(−1)1, where 1 = e^{0} is the identity

element of V_{Hyp}. The rank of V_{Hyp} is 2N.

In the construction of the toroidal VOAs we would need not V_{Hyp} itself, but its sub-VOA V_{Hyp}^{+}:

V_{Hyp}^{+} = S(Ĥ_{−}) ⊗ ℂ[Hyp^{+}],

where Hyp^{+} (resp. Hyp^{−}) is the isotropic sublattice of Hyp generated by {u_{i}|i = 1, …, N} (resp. {v_{i}|i = 1, …, N}). One can verify immediately by inspecting (3.25) and (3.26) that V_{Hyp}^{+} is indeed a sub-VOA of V_{Hyp}. Also note that the cocycle ϵ trivializes on ℂ[Hyp^{+}], making ℂ[Hyp^{+}] the usual (untwisted) group algebra. The Virasoro element of V_{Hyp}^{+} is the same as in V_{Hyp}, and so the rank of V_{Hyp}^{+} is also 2N.

Let us describe a class of modules for V_{Hyp}^{+}. Consider the group algebra ℂ[H^{+}] of the vector space H^{+} = Hyp^{+} ⊗ℤ ℂ. The space S(Ĥ_{−}) ⊗ ℂ[H^{+}] ⊗ ℂ[Hyp^{−}] has a structure of a VOA module for V_{Hyp}^{+}, where the action of V_{Hyp}^{+} is still given by (3.24),(3.25) and (3.26). Fix α ∈ ℂ^{N}, β ∈ ℤ^{N}. Then the subspace

M_{Hyp}^{+}(α, β) = S(Ĥ_{−}) ⊗ e^{αu+βv}ℂ[Hyp^{+}]

in S(Ĥ_{−}) ⊗ ℂ[H^{+}] ⊗ ℂ[Hyp^{−}] is an irreducible VOA module for V_{Hyp}^{+}. Here we are using the notations αu = α_{1}u_{1} + … + α_{N}u_{N}, etc. For β = 0 we will denote the module M_{Hyp}^{+}(α, 0) simply by M_{Hyp}^{+}(α).

4. Toroidal vertex operator algebras.

In this section we will construct several VOAs associated with the toroidal Lie algebras. We will construct a universal enveloping VOA V_{𝔤}, its “level c” quotient V_{𝔤}(c) and the simple quotient L(T_{0}). As 𝔤-modules, V_{𝔤}(c) does not belong to category B_{χ}, but L(T_{0}) does. We will establish several important relations that hold in L(T_{0}), which will allow us to show that L(T_{0}) factors into the tensor product of two VOAs discussed in the previous section, V_{Hyp}^{+} and the twisted Virasoro-affine VOA L_{𝔣}(γ_{0}).

The key fact which makes it possible to construct these VOAs, is the observation that toroidal Lie algebras 𝔤(μ, ν) are in fact vertex Lie algebras for all values of μ, ν.

This observation is not quite trivial, since it requires a rather delicate choice of a basis in 𝔤(μ, ν), in order to exhibit the vertex Lie algebra structure.

Theorem 4.1. Toroidal Lie algebras 𝔤(μ, ν) are vertex Lie algebras.

Proof. Consider the following generating series in 𝔤[[z, z^{−1}]]:

k_{0}(r, z) = ∑_{j=−∞}^{∞} t_{0}^{j}t^{r}k_{0}z^{−j},  k_{p}(r, z) = ∑_{j=−∞}^{∞} t_{0}^{j}t^{r}k_{p}z^{−j−1},  (4.1)

g(r, z) = ∑_{j=−∞}^{∞} t_{0}^{j}t^{r}gz^{−j−1}, g ∈ 𝔤̇,  (4.2)

d̃_{p}(r, z) = ∑_{j=−∞}^{∞} t_{0}^{j}t^{r}d̃_{p}z^{−j−1},  d̃_{0}(r, z) = ∑_{j=−∞}^{∞} t_{0}^{j}t^{r}d̃_{0}z^{−j−2},  (4.3)

where for p = 1, …, n,

t_{0}^{j}t^{r}d̃_{p} = t_{0}^{j}t^{r}d_{p} − νr_{p}t_{0}^{j}t^{r}k_{0},  (4.4)

and

t_{0}^{j}t^{r}d̃_{0} = −t_{0}^{j}t^{r}d_{0} + (μ + ν)(j + \frac{1}{2})t_{0}^{j}t^{r}k_{0}.  (4.5)

Although the moments of the above series are not linearly independent, all linear dependencies may be encoded as relations between the fields:

\frac{∂}{∂z} k_{0}(r, z) = ∑_{p=1}^{N} r_{p}k_{p}(r, z).

Using these relations we can eliminate from the above list for each non-zero r the field k_{p}(r, z) with the smallest p such that r_{p} ≠ 0. Non-zero moments of the remaining fields will form a basis of 𝔤(μ, ν).

Verification of the axioms of a vertex Lie algebra is now quite straightforward. The set C consists of a single element that corresponds to the central field k_{0}(0, z) = k_{0}z^{0}, so (VL2) holds.

Before we check the property (VL1), let us record the commutator relations between the newly introduced elements t_{0}^{j}t^{r}d̃_{p}, t_{0}^{j}t^{r}d̃_{0}. Note that their brackets with the elements of R ⊗ 𝔤̇ and K are essentially given by (1.1) and (1.2) (with an obvious change of sign for t_{0}^{j}t^{r}d̃_{0}), while the rest of the commutator relations are given by the following formulas with a, b = 1, …, N:

\begin{matrix}[t_{0}^{i}t^{r}d̃_{a}, t_{0}^{j}t^{s}d̃_{b}] = & s_{a}t_{0}^{i+j}t^{r+s}d̃_{b} − r_{b}t_{0}^{i+j}t^{r+s}d̃_{a} \\ + & (μs_{a}r_{b} + νr_{a}s_{b})jt_{0}^{i+j}t^{r+s}k_{0} + (μs_{a}r_{b} + νr_{a}s_{b}) ∑_{p=1}^{N} s_{p}t_{0}^{i+j}t^{r+s}k_{p}. \end{matrix}  (4.6)

\begin{matrix}[t_{0}^{i}t^{r}d̃_{0}, t_{0}^{j}t^{s}d̃_{b}] = & − jt_{0}^{i+j}t^{r+s}d̃_{b} − r_{b}t_{0}^{i+j}t^{r+s}d̃_{0} − (μr_{b}(j − 1) + νs_{b}(i + 1))jt_{0}^{i+j}t^{r+s}k_{0} \\ − & (μr_{b}j + νs_{b}(i + 1)) ∑_{p=1}^{N} s_{p}t_{0}^{i+j}t^{r+s}k_{p}. \end{matrix} (4.7)

\begin{matrix}[t_{0}^{i}t^{r}d̃_{0}, t_{0}^{j}t^{s}d̃_{0}] = & (i − j)t_{0}^{i+j}t^{r+s}d̃_{0} + (μ + ν)j(j + 1)(i + 1)t_{0}^{i+j}t^{r+s}k_{0} \\ + & (μ + ν)(j + 1)(i + 1) ∑_{p=1}^{N} s_{p}t_{0}^{i+j}t^{r+s}k_{p}. \end{matrix}  (4.8)

Using these formulas together with (1.1) and (1.2), we can derive the commutator relations for the fields in 𝔤:

[k_{a}(r, z_{1}), k_{b}(m, z_{2})] = 0,  (4.9)
[g(r, z_{1}), k_{a}(m, z_{2})] = 0,  (4.10)

[g_{1}(r, z_{1}), g_{2}(m, z_{2})] = [g_{1}, g_{2}](z_{2}) [z_{1}^{−1}δ (\frac{z_{2}}{z_{1}} )]

+(g_{1}|g_{2})k_{0}(r + m, z_{2}) [z_{1}^{−1} \frac{∂}{∂z_{2}} δ (\frac{z_{2}}{z_{1}} )] + (g_{1}|g_{2}) ∑_{p=1}^{N} r_{p}k_{p}(r + m, z_{2}) [z_{1}^{−1}δ (\frac{z_{2}}{z_{1}} )] , (4.11)

[d̃_{j}(r, z_{1}), g(m, z_{2})] = m_{j}g(r + m, z_{2}) [z_{1}^{−1}δ (\frac{z_{2}}{z_{1}} )] ,  (4.12)

[d̃_{0}(r, z_{1}), g(m, z_{2})] = \frac{∂}{∂z_{2}} \{g(r + m, z_{2}) [z_{1}^{−1}δ (\frac{z_{2}}{z_{1}} )]\} ,  (4.13)

\begin{matrix}[d̃_{i}(r, z_{1}), d̃_{j}(m, z_{2})] = & (m_{i}d̃_{j}(r + m, z_{2}) − r_{j}d̃_{i}(r + m, z_{2})) [z_{1}^{−1}δ (\frac{z_{2}}{z_{1}} )] \\ − & (μm_{i}r_{j} + νr_{i}m_{j}) ∑_{p=1}^{N} r_{p}k_{p}(r + m, z_{2}) [z_{1}^{−1}δ (\frac{z_{2}}{z_{1}} )] \\ − & (μm_{i}r_{j} + νr_{i}m_{j})k_{0}(r + m, z_{2}) [z_{1}^{−1} \frac{∂}{∂z_{2}} δ (\frac{z_{2}}{z_{1}} )] , \end{matrix}  (4.14)

[d̃_{0}(r, z_{1}), d̃_{j}(m, z_{2})] = \frac{∂}{∂z_{2}} \{d̃_{j}(r + m, z_{2}) [z_{1}^{−1}δ (\frac{z_{2}}{z_{1}} )]\} − r_{j}d̃_{0}(r + m, z_{2}) [z_{1}^{−1}δ (\frac{z_{2}}{z_{1}} )]

+νm_{j} ∑_{p=1}^{N} r_{p}k_{p}(r + m, z_{2}) [z_{1}^{−1} \frac{∂}{∂z_{2}} δ (\frac{z_{2}}{z_{1}} )] + νm_{j}k_{0}(r + m, z_{2}) [z_{1}^{−1} (\frac{∂}{∂z_{2}} )^{2} δ (\frac{z_{2}}{z_{1}} )]

−μr_{j} \frac{∂}{∂z_{2}} \{∑_{p=1}^{N} r_{p}k_{p}(r + m, z_{2}) [z_{1}^{−1}δ (\frac{z_{2}}{z_{1}} )] + k_{0}(r + m, z_{2}) [z_{1}^{−1} \frac{∂}{∂z_{2}} δ (\frac{z_{2}}{z_{1}} )]\} , (4.15)

[d̃_{0}(r, z_{1}), d̃_{0}(m, z_{2})] = \{\frac{∂}{∂z_{2}} d̃_{0}(r + m, z_{2})\} [z_{1}^{−1}δ (\frac{z_{2}}{z_{1}} )] + 2d̃_{0}(r + m, z_{2}) [z_{1}^{−1} \frac{∂}{∂z_{2}} δ (\frac{z_{2}}{z_{1}} )]

+(μ + ν)\frac{∂}{∂z_{2}} \{∑_{p=1}^{N} r_{p}k_{p}(r + m, z_{2}) [z_{1}^{−1} \frac{∂}{∂z_{2}} δ (\frac{z_{2}}{z_{1}} )] + k_{0}(r + m, z_{2}) [z_{1}^{−1} (\frac{∂}{∂z_{2}} )^{2} δ (\frac{z_{2}}{z_{1}} )]\} ,

(4.16)

where g, g_{1}, g_{2} ∈ 𝔤̇, a, b = 0, 1, …, N, i, j = 1, …, N.

Now, the right-hand sides of the above commutators are precisely in the format required by (VL1). Thus both (VL1) and (VL2) hold, we conclude that 𝔤(μ, ν) is a vertex Lie algebra, which allows us to consider its universal enveloping vertex algebra V_{𝔤}. Moreover, d̃_{0}(0, z) is the Virasoro field with the central element C_{Vir} = 12(μ + ν)k_{0} and D = t_{0}^{−1}d̃_{0}.

The subalgebra 𝔤^{(−)} of the vertex Lie algebra 𝔤 (not to be confused with its subalgebra 𝔤_{−} with respect to ℤ-grading) is generated by the following elements: t_{0}^{j}t^{r}k_{0} with j ≤ 0, t_{0}^{j}t^{r}k_{p}, t_{0}^{j}t^{r}g, t_{0}^{j}t^{r}d̃_{p} with j ≤ −1, p = 1, …, N, and t_{0}^{j}t^{r}d̃_{0} with j ≤ −2. The subalgebra 𝔤^{(+)} is spanned by the complementary moments of the fields (4.1)-(4.3). We recall that 𝔤^{(+)}1 = 0 in V_{𝔤}.

It follows from Theorem 3.1 that Y (t^{r}k_{0}, z) = k_{0}(r, z), Y (t_{0}^{−1}t^{r}k_{j}, z) = k_{j}(r, z), Y (t_{0}^{−1}t^{r}g, z) = g(r, z), Y (t_{0}^{−1}t^{r}d̃_{j}, z) = d̃_{j}(r, z), Y (t_{0}^{−2}t^{r}d̃_{0}, z) = d̃_{0}(r, z). For the sake of simplicity of notations, we are writing Y (t^{r}k_{0}, z) for Y ((t^{r}k_{0})1, z), etc. Also, when r = 0, we will simply write g(z) for g(0, z), etc.

We are going to show that for a particular irreducible 𝔤_{0}-module T_{0}, the irreducible 𝔤module L(T_{0}) is a factor-VOA of V_{𝔤}. This will be done in two steps. First we will construct a factor-VOA of V_{𝔤} that has an irreducible 𝔤_{0}-module as its top. After that we will show that the irreducible quotient of this 𝔤-module is a VOA, and determine the structure of this vertex algebra.

The operator d̃_{0} = −d_{0} + \frac{1}{2}(μ + ν)k_{0} induces the ℤ-grading of the universal enveloping vertex algebra V_{L}. Since V_{𝔤} = U(𝔤^{(−)}) ⊗ 1, we see that its zero component is spanned by the elements (t^{r_{1}}k_{0}) … (t^{r_{s}}k_{0})1.

Proposition 4.2. Fix a non-zero c ∈ ℂ and consider a 𝔤-submodule R(S) in V_{𝔤} generated by the set S = {k_{0}1 − c1, (t^{r}k_{0})(t^{m}k_{0})1 − c(t^{r+m}k_{0})1 | r, m ∈ ℤ^{N}}.

(a) The quotient V_{𝔤}(c) = V_{𝔤}/R(S) is a factor-module of the generalized Verma module M(T_{0}) with the top

T_{0} = ℂ[q_{1}^{±}, …, q_{N}^{±}] ⊗ V ⊗ W = ℂ[q_{1}^{±}, …, q_{N}^{±}]

defined as in (2.5) with α = 0, d = \frac{1}{2}(μ + ν)c, V being the trivial 1-dimensional 𝔤̇-module, W being a 1-dimensional gl_{N}-module on which sl_{N} acts trivially and I acts a multiplication by h = Nνc.

(b) V_{𝔤}(c) inherits a vertex algebra structure from V_{𝔤}.

(c) V_{𝔤}(c) is a VOA of rank 12(μ + ν)c with the Virasoro field ω(z) = d̃_{0}(z).

(d) The projection from M(T_{0}) to L(T_{0}) factors through V_{𝔤}(c):

M(T_{0}) → V_{𝔤}(c) → L(T_{0}).  (4.17)

This defines a VOA structure on L(T_{0}) as a factor-VOA of V_{𝔤}(c).

Proof. Let us prove part (a). Consider a ℤ-grading on V_{𝔤}. We claim that the zero component R(S)_{0} coincides with

Span ⟨(t^{r_{1}}k_{0}) … (t^{r_{s}}k_{0})1 − c^{s−1}(t^{r_{1}+...+r_{s}}k_{0})1, k_{0}1 − c1 | r_{1}, …, r_{s} ∈ ℤ^{N}⟩ .  (4.18)

First of all it is easy to see by repeated multiplication of the elements in S by t^{r_{j}}k_{0}, that the elements (4.18) are indeed in R(S). Next, we write R(S) = U(𝔤)S. We have a triangular decomposition U(𝔤) = U(𝔤_{−}) ⊗ U(𝔤_{0}) ⊗ U(𝔤_{+}), and 𝔤_{+} acts on S trivially because the elements of S are of degree zero. This implies that R(S)_{0} = U(𝔤_{0})S. Let us show that (4.18) is invariant under the action of 𝔤_{0}. The subalgebra 𝔤_{0} is spanned by the elements t^{m}k_{0}, t^{m}k_{p}, t^{m}g, t^{m}d̃_{p}, t^{m}d̃_{0}, m ∈ ℤ^{N}, g ∈ 𝔤̇, p = 1, …, N. We have already verified the invariance of (4.18) under the action of t^{m}k_{0}. We note that the remaining generators of 𝔤_{0} belong to 𝔤^{(+)} and thus annihilate 1. In addition to this, the elements t^{m}k_{p} and t^{m}g commute with t^{r}k_{0}, which implies that they also annihilate (4.18). It follows from (1.2) that t^{m}d̃_{0} annihilate (4.18) as well.

Using the commutator relation [t^{m}d̃_{p}, t^{r}k_{0}] = r_{p}t^{r+m}k_{0}, we get that

(t^{m}d̃_{p}) ((t^{r_{1}}k_{0}) … (t^{r_{s}}k_{0})1 − c^{s−1}(t^{r_{1}+...+r_{s}}k_{0})1)

= ∑_{j=1}^{s} r_{j} ((t^{r_{1}}k_{0}) … (t^{r_{j}+m}k_{0}) … (t^{r_{s}}k_{0})1 − c^{s−1}(t^{r_{1}+...+r_{s}+m}k_{0})1) ,

and the right hand side is in (4.18). This proves our claim that R(S)_{0} is given by (4.18).

It follows from this that the top of the module V_{𝔤}(c) = V_{𝔤}/R(S) may be identified with the space of Laurent polynomials T_{0} = ℂ[q_{1}^{±}, …, q_{N}^{±}], under the isomorphism

(t^{r}k_{0})1 ↦ cq^{r}.

Let us describe the action of the subalgebra 𝔤_{0} on the top T_{0}. It follows from the relations (4.18) that

(t^{m}k_{0})q^{r} = cq^{r+m}.  (4.19)

Next, we have seen that t^{m}k_{p}, t^{m}g and t^{m}d̃_{0} annihilate T_{0}:

(t^{m}k_{p})q^{r} = 0, (t^{m}g)q^{r} = 0, (t^{m}d̃_{0})q^{r} = 0.  (4.20)

Since t^{m}d̃_{0} = −t^{m}d_{0} + \frac{1}{2}(μ + ν)t^{m}k_{0}, we get that

(t^{m}d_{0})q^{r} = \frac{1}{2}(μ + ν)cq^{r+m}.  (4.21)

Finally,

(t^{m}d̃_{p})q^{r} = \frac{1}{c}(t^{m}d̃_{p})(t^{r}k_{0})1 = \frac{1}{c} [t^{m}d̃_{p}, t^{r}k_{0}] 1 = \frac{1}{c}r_{p}(t^{r+m}k_{0})1 = r_{p}q^{r+m}.  (4.22)

Taking into account that t^{m}d̃_{p} = t^{m}d_{p} − m_{p}νt^{m}k_{0}, we obtain

(t^{m}d_{p})q^{r} = (r_{p} + νcm_{p})q^{r+m},  (4.23)

which corresponds to the tensor module action (2.6) with the trivial action of sl_{N} and I acting as multiplication by Nνc. This completes the proof of part (a). Part (b) follows from Theorem 3.2. The claim of part (c) has been already established and finally, for part (d) we note that L(T_{0}) is a unique irreducible factor of M(T_{0}), thus the projection M(T_{0}) → L(T_{0}) factors through V_{𝔤}(c) as in (4.17). Let us point out that the kernel of the homomorphism M(T_{0}) → V_{𝔤}(c) is the submodule generated by {(t_{0}^{−1}t^{m}d̃_{0})1|m ∈ ℤ^{N}}. Applying Theorem 3.2 again, we conclude that L(T_{0}) inherits the vertex operator algebra structure from V_{𝔤}(c) given by formula (3.11). This completes the proof of the Proposition.

Next we are going to study the structure of the VOA L(T_{0}).

Theorem 4.3.

(a) The VOA L(T_{0}) is generated by the following elements: q^{m} = \frac{1}{c}(t^{m}k_{0})1, (t_{0}^{−1}g)1, (t_{0}^{−1}k_{a})1 , (t_{0}^{−1}d̃_{a})1, E_{ab}, (t_{0}^{−2}d̃_{0})1, with m ∈ ℤ^{N}, g ∈ 𝔤̇, a, b = 1, …, N, where

E_{ab} = \frac{1}{c}(t_{0}^{−1}t^{ϵ_{a}} d̃_{b})(t^{−ϵ_{a}}k_{0})1 − (t_{0}^{−1}d̃_{b})1 + \frac{1}{c}δ_{ab}(t_{0}^{−1}k_{a})1,  (4.24)

with ϵ_{a} being a standard basis vector in ℤ^{N} with 1 in a-th position.

(b) The module L(T_{0}) has a structure of a ℂ[q_{1}^{±}, …, q_{N}^{±}]-module, which is compatible with the action of the algebra of Laurent polynomials on T_{0}. The action of the field k_{0}(m, z) is given by the following vertex operator:

\frac{1}{c}k_{0}(m, z) = Y (q^{m}, z) = q^{m} exp (∑_{p=1}^{N} m_{p} ∑_{j∈ℤ\\{0\}} \frac{1}{cj} (t_{0}^{−j}k_{p})z^{j}) .  (4.25)

(c) The action on L(T_{0}) of the remaining fields in (4.1)-(4.3) is expressed in the following way

g(m, z) = g(z)Y (q^{m}, z),  (4.26)
k_{a}(m, z) = k_{a}(z)Y (q^{m}, z),  (4.27)
d̃_{a}(m, z) =: d̃_{a}(z)Y (q^{m}, z) : + ∑_{p=1}^{N} m_{p}Y (E_{pa}, z)Y (q^{m}, z),  (4.28)

d̃_{0}(m, z) =: d̃_{0}(z)Y (q^{m}, z) : +\frac{1}{c} ∑_{a,b=1}^{N} m_{a}k_{b}(z)Y (E_{ab}, z)Y (q^{m}, z)

+(μ − \frac{1}{c}) ∑_{p=1}^{N} m_{p} (\frac{∂}{∂z} k_{p}(z)) Y (q^{m}, z).  (4.29)

(d) The vertex operator algebra L(T_{0}) is isomorphic to the tensor product of two VOAs:

L(T_{0}) = V_{Hyp}^{+} ⊗ L_{𝔣}(γ_{0}),

where V_{Hyp}^{+} is a sub-VOA of the lattice VOA described in section 3.5, and L_{𝔣}(γ_{0}) is the simple VOA corresponding to the twisted Virasoro-affine Lie algebra constructed from the reductive Lie algebra 𝔣̇ = 𝔤̇ ⊕ gl_{N}, and the central character γ_{0} given by the following values:

c𝔤^{}˙ = c, c_{sl_{N}} = 1 − μc,

c_{Hei} = N(1 − μc) − N^{2}νc, c_{VH} = N(\frac{1}{2} − νc),

c_{Vir} = 12c(μ + ν) − 2N.  (4.30)

The fields k_{0}(m, z), k_{p}(z), d̃_{p}(z), p = 1, …, N, act on V_{Hyp}^{+} by

k_{0}(m, z) = cY (e^{mu}, z), k_{p}(z) = cu_{p}(z), d̃_{p}(z) = v_{p}(z),

while the fields g(z) and Y (E_{ab}, z) act on L_{𝔣}(γ_{0}). The field d̃_{0}(z) is the Virasoro field of the tensor product V_{Hyp}^{+} ⊗ L_{𝔣}(γ_{0}).

Proof. We are going to determine the structure of the VOA L(T_{0}) using the following strategy. The technique developed in [BB] provides a method to calculate any given homogeneous component of the kernel of the epimorphism π : V_{𝔤}(c) → L(T_{0}), though of course such a computation is feasible only for the components of low degrees. Any v ∈ Ker π yields a relation between the fields in L(T_{0}):

Y_{L(T_{0})}(v, z) = 0.

It turns out that it is sufficient to know the elements of Ker (π) of degrees 1 and 2 in order to completely determine the structure of L(T_{0}) as a VOA.

Let us illustrate the technique of [BB] with the following example. Fix m ∈ ℤ^{N} and g ∈ 𝔤̇. Consider the subspace P_{1} = Span ⟨(t_{0}^{−1}t^{r}g)q^{m−r} | r ∈ ℤ^{N}⟩ ⊂ V_{𝔤}(c). This subspace belongs to the homogeneous component of weight (−1, m) in V_{𝔤}(c). We are going to find the intersection of P_{1} with Ker π. We note that a vector v in component (−1, m) of V_{𝔤}(c) belongs to Ker π if and only if U_{1}(𝔤_{+})v = 0.

Lemma 4.4. Let g, g′ ∈ 𝔤̇, b = 1, …, N. Then

(t_{0}t^{s}k_{0})(t_{0}^{−1}t^{r}g)q^{m−r} = 0, (t_{0}t^{s}k_{b})(t_{0}^{−1}t^{r}g)q^{m−r} = 0,  (4.31)
(t_{0}t^{s}g′)(t_{0}^{−1}t^{r}g)q^{m−r} = (g′|g)cq^{m+s},  (4.32)
(t_{0}t^{s}d̃_{0})(t_{0}^{−1}t^{r}g)q^{m−r} = 0, (t_{0}t^{s}d̃_{b})(t_{0}^{−1}t^{r}g)q^{m−r} = 0.  (4.33)

Proof. Let us prove (4.32):

(t_{0}t^{s}g′)(t_{0}^{−1}t^{r}g)q^{m−r} = [t_{0}t^{s}g′, t_{0}^{−1}t^{r}g]q^{m−r} + (t_{0}^{−1}t^{r}g)(t_{0}t^{s}g′)q^{m−r}.

The second term vanishes because 𝔤_{+} acts trivially on the top of V_{𝔤}(c). For the first term we use the relations in 𝔤, (4.19) and (4.20):

\begin{aligned}[t_{0}t^{s}g′, t_{0}^{−1}t^{r}g]q^{m−r} & = (t^{r+s}[g′, g])q^{m−r} + (g′|g)(t^{r+s}k_{0})q^{m−r} + (g′|g) ∑_{p=1}^{N} s_{p}(t^{r+s}k_{p})q^{m−r} \\ & = (g′|g)cq^{m+s} \end{aligned} .

All other equalities in the statement of this Lemma are obtained in a similar way.

Let us analyze the results of this Lemma. We see that the right hand sides in (4.31)- (4.33) are independent of r. Thus U_{1}(𝔤_{+}) ((t_{0}^{−1}t^{r}g)q^{m−r} − (t_{0}^{−1}g)q^{m}) = 0, which implies that (t_{0}^{−1}t^{r}g)q^{m−r} − (t_{0}^{−1}g)q^{m} ∈ Ker π. Applying the state-field correspondence, we get that the following relation holds in L(T_{0}):

: Y (t_{0}^{−1}t^{r}g, z)Y (q^{m−r}, z) : = : Y (t_{0}^{−1}g, z)Y (q^{m}, z) : .  (4.34)

Since these vertex operators commute, we may drop the normal ordering symbol. In particular, for m = r, we get that

Y (t_{0}^{−1}t^{m}g, z) = Y (t_{0}^{−1}g, z)Y (q^{m}, z).  (4.35)

This factorization property means that the fields Y (t_{0}^{−1}t^{m}g, z) = g(m, z) reduce to more elementary fields Y (t_{0}^{−1}g, z) = g(z) and Y (q^{m}, z). The fields g(z) correspond to the affine subalgebra ℂ[t_{0}, t_{0}^{−1}] ⊗ 𝔤̇ ⊕ ℂk_{0} ⊂ 𝔤. Below we establish analogous factorization formulas for other fields. We shall also see that the affine fields Y (t_{0}^{−1}g, z) commute with other elementary fields, except for Y (t_{0}^{−2}d̃_{0}, z). This will imply that the affine VOA generated by the affine fields splits off as a tensor factor in L(T_{0}) when c is not the critical level for this affine subalgebra. In this way we will obtain a tensor product decomposition of L(T_{0}).

Now let us get the formula for the vertex operators Y (t_{0}^{−1}t^{r}k_{a}, z) and Y (t^{r}k_{0}, z). For a fixed m ∈ ℤ^{N} and 1 ≤ a ≤ N, consider the subspace P_{2} = Span ⟨(t_{0}^{−1}t^{r}k_{a})q^{m−r}|r ∈ ℤ^{N}⟩ ⊂ V_{𝔤}(c). Again we are going to find the intersection of P_{2} with Ker π.

Lemma 4.5. Let 1 ≤ a, b ≤ N. Then

(t_{0}t^{s}k_{0})(t_{0}^{−1}t^{r}k_{a})q^{m−r} = 0, (t_{0}t^{s}k_{b})(t_{0}^{−1}t^{r}k_{a})q^{m−r} = 0,  (4.36)
(t_{0}t^{s}g)(t_{0}^{−1}t^{r}k_{a})q^{m−r} = 0, g ∈ 𝔤̇, (t_{0}t^{s}d̃_{0})(t_{0}^{−1}t^{r}k_{a})q^{m−r} = 0,  (4.37)
(t_{0}t^{s}d̃_{b})(t_{0}^{−1}t^{r}k_{a})q^{m−r} = cδ_{ab}q^{m+s}.  (4.38)

The proof of this Lemma is the same as for Lemma 4.4, and we omit these calculations.

We see again that the right hand sides in (4.36)-(4.38) are independent of r. Thus (t_{0}^{−1}t^{r}k_{a})q^{m−r} − (t_{0}^{−1}k_{a})q^{m} ∈ Ker π, or equivalently,

(t_{0}^{−1}t^{r}k_{a})q^{m−r} = (t_{0}^{−1}k_{a})q^{m} in L(T_{0}).  (4.39)

Taking r = m and applying the state-field correspondence, we get that the following relation holds in L(T_{0}):

Y (t_{0}^{−1}t^{m}k_{a}, z) = Y (t_{0}^{−1}k_{a}, z)Y (q^{m}, z).  (4.40)

Since these vertex operators commute, we dropped the normal ordering symbol in the right

hand side. Also, taking into account that t_{0}^{−1}t^{m}k_{0} = ∑_{p=1}^{N} m_{p}t_{0}^{−1}t^{m}k_{p}, we obtain

c\frac{∂}{∂z} Y (q^{m}, z) = Y (t_{0}^{−1}t^{m}k_{0}, z) = ∑_{p=1}^{N} m_{p}Y (t_{0}^{−1}k_{p}, z)Y (q^{m}, z).  (4.41)

Next we will use the results of Section 3 of [BB]. It is proved there that L(T_{0}) is a module over a commutative associative algebra ℂ[q_{1}^{±}, …, q_{N}^{±}] and the vertex operator Y (q^{m}, z) is given by the expression (4.25). We can see that this formula is compatible with (4.41), and in fact it is not too difficult to derive (4.25) from (4.41).

We will later need another relation in L(T_{0}) which can be derived either from (4.39) or (4.25):

(t_{0}^{−1}t^{r}k_{0})q^{s} = ∑_{p=1}^{N} r_{p}(t_{0}^{−1}k_{p})q^{r+s}.  (4.42)

Our next goal is to derive a formula for the vertex operator Y (t_{0}^{−1}t^{m}d̃_{a}, z). We will use the same strategy as above.

Lemma 4.6. Let 1 ≤ a, b ≤ N. Then

(t_{0}t^{s}k_{0})(t_{0}^{−1}t^{r}d̃_{a})q^{m−r} = −s_{a}cq^{m+s},  (4.43)
(t_{0}t^{s}k_{b})(t_{0}^{−1}t^{r}d̃_{a})q^{m−r} = δ_{ab}cq^{m+s},  (4.44)
(t_{0}t^{s}g)(t_{0}^{−1}t^{r}d̃_{a})q^{m−r} = 0, g ∈ 𝔤̇,  (4.45)
(t_{0}t^{s}d̃_{0})(t_{0}^{−1}t^{r}d̃_{a})q^{m−r} = ((m_{a} − r_{a}) − 2(μs_{a} − νr_{a})c) q^{m+s},  (4.46)
(t_{0}t^{s}d̃_{b})(t_{0}^{−1}t^{r}d̃_{a})q^{m−r} = (r_{b}(m_{a} − r_{a}) − s_{a}(m_{b} − r_{b}) − (μr_{b}s_{a} + νr_{a}s_{b})c) q^{m+s}.  (4.47)

Proof. Let us show the calculations for (4.46) and leave rest as an exercise to the reader. We are going to use (4.7), (4.19) and (4.22):

(t_{0}t^{s}d̃_{0})(t_{0}^{−1}t^{r}d̃_{a})q^{m−r} = [t_{0}t^{s}d̃_{0}, t_{0}^{−1}t^{r}d̃_{a}]q^{m−r}
= (t^{r+s}d̃_{a})q^{m−r} − s_{a}(t^{r+s}d̃_{0})q^{m−r}

+(−2μs_{a} + 2νr_{a})(t^{r+s}k_{0})q^{m−r} − (−μs_{a} + 2νr_{a}) ∑_{p=1}^{N} r_{p}(t^{r+s}k_{p})q^{m−r}

= (m_{a} − r_{a} − 2μcs_{a} + 2νcr_{a})q^{m+s}.

Unlike the previous cases, the right hand sides in Lemma 4.6 do depend on r. Note, however, that this dependence is polynomial, and we may separate constant, linear and quadratic components. Clearly, the constant term is obtained by setting r = 0, and is produced by the element (t_{0}^{−1}d̃_{a})q^{m}.

Next, comparing (4.43)-(4.47) with (4.36)-(4.38) we notice that the quadratic term is given

by the vector −\frac{r_{a}}{c} ∑_{p=1}^{N} r_{p}(t_{0}^{−1}k_{p})q^{m}:

(t_{0}t^{s}d̃_{b}) (−\frac{r_{a}}{c} ∑_{p=1}^{N} r_{p}(t_{0}^{−1}k_{p})q^{m}) = −r_{a}r_{b}q^{m+s},

while the other raising operators annihilate this vector.

Finally, the linear in r component is given by the vector ∑_{p=1}^{N} r_{p}E_{pa}^{m}, where

E_{pa}^{m} = (t_{0}^{−1}t^{ϵ_{p}} d̃_{a})q^{m−ϵ_{p}} − (t_{0}^{−1}d̃_{a})q^{m} + \frac{1}{c}δ_{ap}(t_{0}^{−1}k_{p})q^{m}.  (4.48)

Using Lemmas 4.6 and 4.5, we can easily see that

(t_{0}t^{s}k_{0}) ∑_{p=1}^{N} r_{p}E_{pa}^{m} = 0, (t_{0}t^{s}k_{b}) ∑_{p=1}^{N} r_{p}E_{pa}^{m} = 0, (t_{0}t^{s}g) ∑_{p=1}^{N} r_{p}E_{pa}^{m} = 0,  (4.49)

(t_{0}t^{s}d̃_{0}) ∑_{p=1}^{N} r_{p}E_{pa}^{m} = r_{a}(−1 + 2νc)q^{m+s},  (4.50)

(t_{0}t^{s}d̃_{b}) ∑_{p=1}^{N} r_{p}E_{pa}^{m} = (r_{b}m_{a} + (1 − μc)s_{a}r_{b} − νcr_{a}s_{b})q^{m+s}.  (4.51)

It follows from the above computations that the following vector is annihilated by U_{1}(𝔤_{+}) and thus vanishes in L(T_{0}):

(t_{0}^{−1}t^{r}d̃_{a})q^{m−r} − (t_{0}^{−1}d̃_{a})q^{m} − ∑_{p=1}^{N} r_{p}E_{pa}^{m} + \frac{r_{a}}{c} ∑_{p=1}^{N} r_{p}(t_{0}^{−1}k_{p})q^{m} = 0.  (4.52)

Setting r = m, and applying the map Y we obtain a relation for the vertex operators in L(T_{0}):

d̃_{a}(m, z) =: d̃_{a}(z)Y (q^{m}, z) : + ∑_{p=1}^{N} m_{p}Y (E_{pa}^{m}, z) − \frac{m_{a}}{c} ∑_{p=1}^{N} m_{p}k_{p}(z)Y (q^{m}, z).  (4.53)

Let us now establish a relation between E_{ab}^{m} and

E_{ab} = E_{ab}^{0} = (t_{0}^{−1}t^{ϵ_{a}} d̃_{b})q^{−ϵ_{a}} − (t_{0}^{−1}d̃_{b})1 + \frac{1}{c}δ_{ab}(t_{0}^{−1}k_{a})1.  (4.54)

Lemma 4.7. (a) The fields Y (q^{m}, z), g(z), k_{p}(z), d̃_{p}(z) commute with Y (E_{ab}, z), p, a, b = 1, …, N.

(b) The following relation holds:

E_{ab}^{m} = (E_{ab})_{(−1)}q^{m} + \frac{m_{b}}{c} (t_{0}^{−1}k_{a})q^{m}.  (4.55)

Proof. Let us show that

(q^{m})_{(n)}E_{ab} = 0 for all n ≥ 0.  (4.56)

Since deg ((q^{m})_{(n)}E_{ab}) = −n, we only need to consider the case of n = 0. But (q^{m})_{(0)} = \frac{1}{c}(t_{0}t^{m}k_{0}), and we get the desired claim from (4.49). Now applying the commutator formula (3.2) we get that the fields Y (q^{m}, z) and Y (E_{ab}, z) commute. Using a similar argument we can derive from (4.49)-(4.51) that g(z), k_{p}(z), d̃_{p}(z) also commute with Y (E_{ab}, z).

Taking into account the skew symmetry identity (3.7) we obtain as a consequence of (4.56) the equality

(E_{ab})_{(−1)}q^{m} = (q^{m})_{(−1)}E_{ab}.

If we substitute (4.54) in the right hand side of this equality, we will get

(E_{ab})_{(−1)}q^{m} = \frac{1}{c}(t^{m}k_{0}) ((t_{0}^{−1}t^{ϵ_{a}} d̃_{b})q^{−ϵ_{a}} − (t_{0}^{−1}d̃_{b})1 + \frac{1}{c}δ_{ab}(t_{0}^{−1}k_{a})1)

= \frac{1}{c}(t_{0}^{−1}t^{ϵ_{a}} d̃_{b})(t^{m}k_{0})q^{−ϵ_{a}} − \frac{1}{c}(t_{0}^{−1}d̃_{b})(t^{m}k_{0})1 + \frac{1}{c^{2}} δ_{ab}(t_{0}^{−1}k_{a})(t^{m}k_{0})1

−\frac{1}{c}[t_{0}^{−1}t^{ϵ_{a}} d̃_{b}, t^{m}k_{0}]q^{−ϵ_{a}} + \frac{1}{c}[t_{0}^{−1}d̃_{b}, t^{m}k_{0}]1

= (t_{0}^{−1}t^{ϵ_{a}} d̃_{b})q^{m−ϵ_{a}} − (t_{0}^{−1}d̃_{b})q^{m} + \frac{1}{c}δ_{ab}(t_{0}^{−1}k_{a})q^{m} − \frac{m_{b}}{c} (t_{0}^{−1}t^{m+ϵ_{a}}k_{0})q^{−ϵ_{a}} + \frac{m_{b}}{c} (t_{0}^{−1}t^{m}k_{0})1

= E_{ab}^{m} − \frac{m_{b}}{c} (t_{0}^{−1}k_{a})q^{m}.

To get the last equality we used (4.48) and (4.42). This completes the proof of the Lemma. Combining (4.53) with (4.55), we obtain (4.28). We also get from (4.52) and (4.55) that

(t_{0}^{−1}t^{r}d̃_{a})q^{m} = (t_{0}^{−1}d̃_{a})q^{m+r} + ∑_{p=1}^{N} r_{p}(E_{pa})_{(−1)}q^{m+r} + \frac{m_{a}}{c} ∑_{p=1}^{N} r_{p}(t_{0}^{−1}k_{p})q^{m+r}.  (4.57)

Next we are going to determine the commutator relations between Y (E_{ab}, z), 1 ≤ a, b ≤ N.

Lemma 4.8. (a) (E_{ab})_{(0)}E_{sp} = δ_{bs}E_{ap} − δ_{ap}E_{sb},
(b) (E_{ab})_{(1)}E_{sp} = (1 − μc)δ_{bs}δ_{ap}1 − νcδ_{ab}δ_{sp}1,
(c) (E_{ab})_{(n)}E_{sp} = 0 for n ≥ 2.

Proof. Since deg (E_{ab}) = 1, we have that deg ((E_{ab})_{(n)}E_{sp}) = 1 − n, from which part (c) immediately follows. Let us carry out the computations for part (b) of the Lemma. We have

(E_{ab})_{(1)}E_{sp} = ((t_{0}^{−1}t^{ϵ_{a}} d̃_{b})_{(−1)}q^{−ϵ_{a}})_{(1)} E_{sp} − (t_{0}d̃_{b})E_{sp} + \frac{1}{c}δ_{ab}(t_{0}k_{a})E_{sp}.  (4.58)

The last two terms vanish by (4.49) and (4.51). To evaluate the first term in the right hand side of (4.58), we use the Borcherds’ identity (3.6). Noting that by (4.56), (q^{−ϵ_{a}})_{(n)}E_{sp} = 0 for all n ≥ 0, we get

((t_{0}^{−1}t^{ϵ_{a}} d̃_{b})_{(−1)}q^{−ϵ_{a}})_{(1)} E_{sp} = \frac{1}{c}(t_{0}t^{−ϵ_{a}}k_{0})(t^{ϵ_{a}} d̃_{b})E_{sp} + \frac{1}{c}(t^{−ϵ_{a}}k_{0})(t_{0}t^{ϵ_{a}} d̃_{b})E_{sp}.

The first term is equal to zero since

\frac{1}{c}(t_{0}t^{−ϵ_{a}}k_{0})(t^{ϵ_{a}} d̃_{b})E_{sp} = \frac{1}{c}(t^{ϵ_{a}} d̃_{b})(t_{0}t^{−ϵ_{a}}k_{0})E_{sp} − \frac{1}{c}[t^{ϵ_{a}} d̃_{b}, t_{0}t^{−ϵ_{a}}k_{0}]E_{sp}

and we may apply (4.49) to the right hand side. Finally, using (4.51) and (4.19), we get

\frac{1}{c}(t^{−ϵ_{a}}k_{0})(t_{0}t^{ϵ_{a}} d̃_{b})E_{sp} = ((1 − μc)δ_{bs}δ_{ap} − νcδ_{ab}δ_{sp}) 1.

This completes the proof of part (b) of the Lemma, and we leave part (a) as an exercise for the reader.

Comparing Lemmas 4.8 and 3.6 (a), we conclude that the operators (E_{ab})_{(n)} produce a representation of affine ĝl_{N}.

Lemma 4.9. The following relations hold in L(T_{0}):

(a)  (t_{0}^{−1}t^{r}d̃_{0})q^{m} = \frac{1}{c} ∑_{p=1}^{N} m_{p}(t_{0}^{−1}k_{p})q^{m+r},

(b)  (t_{0}t^{r}d̃_{0})(E_{ab})_{(−1)}q^{m} = δ_{ab}(−1 + 2νc).

Proof.

(t_{0}^{−1}t^{r}d̃_{0})q^{m} = \frac{1}{c}(t_{0}^{−1}t^{r}d̃_{0})(t^{m}k_{0})1 = \frac{1}{c}[t_{0}^{−1}t^{r}d̃_{0}, t^{m}k_{0}]1

= \frac{1}{c} ∑_{p=1}^{N} m_{p}(t_{0}^{−1}t^{r+m}k_{p})1 = \frac{1}{c} ∑_{p=1}^{N} m_{p}(t_{0}^{−1}k_{p})q^{r+m}.

This proves the claim (a). Part (b) follows from Lemma 4.7(b), (4.50) and (4.37).

Finally let us study the properties of the field d̃_{0}(r, z). This will require carrying out calculations with certain elements of degree 2. Let v ∈ L(T_{0}), deg (v) = 2. Since L(T_{0}) is an irreducible 𝔤-module, it is generated by any non-zero vector. If v ≠ 0 then U_{2}(𝔤_{+})v = T_{0}. However it is easy to see that 𝔤_{+} is generated by 𝔤_{1}. Thus 𝔤_{1}v = 0 implies that v = 0 . This observation will help us find a relation in L(T_{0}) involving (t_{0}^{−2}t^{m}d̃_{0})1, which in turn will yield an expression for the field d̃_{0}(m, z).

Lemma 4.10. The following relation holds in L(T_{0}):

(t_{0}^{−2}t^{m}d̃_{0})1 = (t_{0}^{−2}d̃_{0})q^{m} + \frac{1}{c} ∑_{p,j=1}^{N} m_{p}(t_{0}^{−1}k_{j})(E_{pj})_{(−1)}q^{m} − \frac{1}{c}(1 − μc) ∑_{p=1}^{N} m_{p}(t_{0}^{−2}k_{p})q^{m}.

Proof. We shall prove this Lemma by showing that the vector

v = (t_{0}^{−2}t^{m}d̃_{0})1 − (t_{0}^{−2}d̃_{0})q^{m} − \frac{1}{c} ∑_{p,j=1}^{N} m_{p}(t_{0}^{−1}k_{j})(E_{pj})_{(−1)}q^{m} + \frac{1}{c}(1 − μc) ∑_{p=1}^{N} m_{p}(t_{0}^{−2}k_{p})q^{m}

is annihilated in L(T_{0}) by 𝔤_{1}. Let us show that (t_{0}t^{s}d̃_{0})v = 0 in L(T_{0}):

(t_{0}t^{s}d̃_{0})v = [t_{0}t^{s}d̃_{0}, t_{0}^{−2}t^{m}d̃_{0}]1 − [t_{0}t^{s}d̃_{0}, t_{0}^{−2}d̃_{0}]q^{m}

−\frac{1}{c} ∑_{p,j=1}^{N} m_{p}[t_{0}t^{s}d̃_{0}, t_{0}^{−1}k_{j}](E_{pj})_{(−1)}q^{m} − \frac{1}{c} ∑_{p,j=1}^{N} m_{p}(t_{0}^{−1}k_{j})(t_{0}t^{s}d̃_{0})(E_{pj})_{(−1)}q^{m}

+\frac{1}{c}(1 − μc) ∑_{p=1}^{N} m_{p}[t_{0}t^{s}d̃_{0}, t_{0}^{−2}k_{p}]q^{m}

= 3(t_{0}^{−1}t^{m+s}d̃_{0})1 + 4(μ + ν)(t_{0}^{−1}t^{m+s}k_{0})1 − 2(μ + ν) ∑_{p=1}^{N} m_{p}(t_{0}^{−1}t^{m+s}k_{p})1

−3(t_{0}^{−1}t^{s}d̃_{0})q^{m} − 4(μ + ν)(t_{0}^{−1}t^{s}k_{0})q^{m}

−\frac{1}{c} ∑_{p,j=1}^{N} m_{p}(t^{s}k_{j})(E_{pj})_{(−1)}q^{m} − \frac{1}{c}(−1 + 2νc) ∑_{p=1}^{N} m_{p}(t_{0}^{−1}k_{p})q^{m+s}

+\frac{2}{c}(1 − μc) ∑_{p=1}^{N} m_{p}(t_{0}^{−1}t^{s}k_{p})q^{m}

= 4(μ + ν) ∑_{p=1}^{N}(m_{p} + s_{p})(t_{0}^{−1}k_{p})q^{m+s} − 2(μ + ν) ∑_{p=1}^{N} m_{p}(t_{0}^{−1}k_{p})q^{m+s} − \frac{3}{c} ∑_{p=1}^{N} m_{p}(t_{0}^{−1}k_{p})q^{m+s}

−4(μ + ν) ∑_{p=1}^{N} s_{p}(t_{0}^{−1}k_{p})q^{m+s} − \frac{1}{c} ∑_{p,j=1}^{N} m_{p}(E_{pj})_{(−1)}(t^{s}k_{j})q^{m}

−\frac{1}{c}(−1 + 2νc) ∑_{p,j=1}^{N} m_{p}(t_{0}^{−1}k_{p})q^{m+s} + \frac{2}{c}(1 − μc) ∑_{p=1}^{N} m_{p}(t_{0}^{−1}k_{p})q^{m+s} = 0.

The cases of other elements spanning 𝔤_{1} are treated similarly, and are left as an exercise.

Applying the state-field correspondence Y to both sides of the equality in Lemma 4.10, we will get the formula (4.29).

Let us complete the proof of Theorem 4.3. We have now established all the relations between the fields stated in part (c) of the Theorem. The universal enveloping vertex algebra V_{𝔤} is generated by the fields (4.1)-(4.3). The same is true for L(T_{0}), since L(T_{0}) is a factor vertex algebra of V_{𝔤}. Taking into account the relations of part (c), the claim of part (a) of the Theorem follows. As we mentioned above, the claim of part (b) follows from the results of [BB].

Let us establish the claim of part (d) of the Theorem. First we construct a homomorphism of vertex algebras

φ : V_{Hyp}^{+} → L(T_{0}),

defined by φ(e^{mu}) = q^{m}, φ(u_{p}(−1)1) = \frac{1}{c}(t_{0}^{−1}k_{p})1, φ(v_{p}(−1)1) = (t_{0}^{−1}d̃_{p})1. Using (1.2), (4.20), (4.23) and (4.25), we can see that the images of the generators of V_{Hyp}^{+} satisfy the required relations (3.22)-(3.25), thus φ is indeed a homomorphism of vertex algebras. Since V_{Hyp}^{+} is a simple vertex algebra, the map φ is injective. The image of the Virasoro field of V_{Hyp}^{+} is

φ(ω_{Hyp}(z)) = \frac{1}{c} ∑_{p=1}^{N} : d_{p}(z)k_{p}(z) :,

and the central charge of this Virasoro field is equal to rank (V_{Hyp}^{+}) = 2N.

We know that the fields g(z) generate an affine subalgebra ˆ 𝔤̇ in the toroidal algebra 𝔤, which commutes with the fields generated by the image of φ. This affine subalgebra has the central charge c𝔤^{}˙ = c on L(T_{0}). Next, comparing Lemmas 4.8 and 3.6(a), we conclude that the fields Y (E_{ab}, z) yield a representation of the affine ĝl_{N} on L(T_{0}) with central charges c_{sl_{N}} = 1 − μc and c_{Hei} = N(1 − μc) − N^{2}νc. It follows from Lemma 4.7 (a) that the fields Y (E_{ab}, z) also commute with the image of φ.

The relation (4.8) implies that the field d̃_{0}(z) generates a Virasoro algebra with the central charge 12(μ + ν)c. The fact that the element t_{0}^{−1}d̃_{0} is an infinitesimal translation operator D follows from the relations (1.1), (1.2), (4.7) and (4.8). Thus L(T_{0}) is a VOA of rank 12(μ + ν)c with the Virasoro field d̃_{0}(z). The field d̃_{0}(z) − φ(ω_{Hyp}(z)) yields another Virasoro algebra with central charge 12(μ + ν)c − 2N. Comparing Lemmas 3.6 (b) and 4.9 (b), and noting that φ(ω_{Hyp}(z)) commutes with g(z) and Y (E_{ab}, z), we get that the fields d̃_{0}(z) − φ(ω_{Hyp}(z)), g(z) and Y (E_{ab}, z) yield a representation of the twisted Virasoro-affine algebra 𝔣 with 𝔣̇ = 𝔤̇ ⊕ gl_{N} and central character given by (4.30).

This allows us to define a homomorphism of vertex algebras

ψ : V_{𝔣}(γ_{0}) → L(T_{0})

by ψ(g(−1)1) = (t_{0}^{−1}g)1, ψ(E_{ab}(−1)1) = E_{ab}, ψ(ω_{𝔣}) = (t_{0}^{−2}d̃_{0})1 − φ(ω_{Hyp}). The sub-VOAs φ(V_{Hyp}^{+}) and ψ(V_{𝔣}) commute in L(T_{0}). Thus we have a homomorphism

θ : V_{Hyp}^{+} ⊗ V_{𝔣}(γ_{0}) → L(T_{0}).

Since θ(ω_{Hyp} + ω_{𝔣}) = (t_{0}^{−2}d̃_{0})1, this is in fact a homomorphism of VOAs. Moreover, by part (a), θ is an epimorphism.

The 𝔣-module V_{𝔣}(γ_{0}) has a unique maximal submodule and a unique irreducible quotient L_{𝔣}(γ_{0}). By Theorem 3.2, L_{𝔣}(γ_{0}) is a unique quotient vertex algebra of V_{𝔣}(γ_{0}). Since V_{Hyp}^{+} is a simple vertex algebra, we conclude that V_{Hyp}^{+} ⊗ L_{𝔣}(γ_{0}) is a unique simple quotient vertex algebra of V_{Hyp}^{+} ⊗ V_{𝔣}(γ_{0}). However L(T_{0}) is also a simple quotient of V_{Hyp}^{+} ⊗ V_{𝔣}(γ_{0}). Thus

L(T_{0}) ≅ V_{Hyp}^{+} ⊗ L_{𝔣}(γ_{0}).

This completes the proof of Theorem 4.3.

5. Realizations for the irreducible modules in category B_{χ}.

In this section we are going to use the theory of VOA modules to give realizations for all irreducible 𝔤(μ, ν)-modules in category B_{χ}. By the principle of preservation of identities [Li], every VOA module for V_{Hyp}^{+} ⊗ L_{𝔣}(γ_{0}) is also a module for the Lie algebra 𝔤(μ, ν). However in order to get all irreducible modules in B_{χ}, we need to use a larger VOA

V (T_{0}) = V_{Hyp}^{+} ⊗ V_{𝔣}(γ_{0}),

which has more irreducible modules than V_{Hyp}^{+} ⊗ L_{𝔣}(γ_{0}). In order to carry out this plan, we should first prove that V_{Hyp}^{+} ⊗ V_{𝔣}(γ_{0}) also admits a structure of a 𝔤(μ, ν)-module.

First, we need to establish the following technical lemma:

Lemma 5.1. For a Zariski dense set of triples (c, μ, ν), the modules V_{𝔣}(γ_{0}) and L_{𝔣}(γ_{0}) coincide.

Proof. It follows from Proposition 3.8 and (4.30) that whenever c ≠ 0, c ≠ −h^{∨}, c_{sl_{N}} = 1 − μc ≠ −N, c_{Hei} = N(1 − μc) − N^{2}νc ≠ 0, the VOA V_{𝔣}(γ_{0}) factors into a tensor product of four VOAs: V_{𝔤^{}}˙ (c𝔤^{}˙ ), V_{_{sl}_{N}} (c_{sl_{N}}), V_{Hei}(c_{Hei}) and V_{Vir}(c′_{Vir}). It is clear that V_{𝔣}(γ_{0}) is a simple VOA if and only if each of these four VOAs is simple. First of all, under the assumption that c_{Hei} ≠ 0, the Heisenberg VOA is simple. We are going to show that the remaining affine and Virasoro VOAs are simple for (c, μ, ν) in a dense subset of ℂ^{3}.

We note that affine and Virasoro VOAs are the generalized Verma modules for the respective Lie algebras, and a generalized Verma module admits a Shapovalov form [J]. Using the Shapovalov determinant argument, it is not difficult to see that the VOA V_{𝔤^{}}˙ (c𝔤^{}˙ ) (resp. V_{_{sl}_{N}} (c_{sl_{N}}), V_{Vir}(c′_{Vir})) is simple outside a countable set of values of the central charge c𝔤^{}˙ (resp. c_{sl_{N}}, c′_{Vir}). In fact, an explicit formula for the Shapovalov determinant for the generalized Verma modules for the affine algebras may be found in [KK], while for the Virasoro VOA, it follows from the description of the irreducible modules for the Virasoro algebra [FF] that V_{Vir}(c′_{Vir}) is simple if and only if c′_{Vir} ≠ 1−6\frac{(r−s)^{2}}{rs}, where r and s are relatively prime integers, r, s > 1.

Each inequality on the values of the central charges defines a dense Zariski open subset of values of (c, μ, ν). Since a countable intersection of dense Zariski open subsets in ℂ^{3} is Zariski dense, we establish the claim of the Lemma.

Proposition 5.2. Let c ≠ 0, and let γ_{0} be given by (4.30). Then V (T_{0}) = V_{Hyp}^{+} ⊗ V_{𝔣}(γ_{0}) has a structure of a 𝔤(μ, ν)-module given by the formulas of Theorem 4.3 (b)-(d).

Proof. We proved in Theorem 4.3 (d) that V_{Hyp}^{+} ⊗L_{𝔣}(γ_{0}) is a 𝔤(μ, ν)-module. Now we want to prove the same for V_{Hyp}^{+} ⊗ V_{𝔣}(γ_{0}). This amounts to verifying relations (4.9)-(4.16) in this vertex algebra. It is possible to do this directly, and this was the approach taken, for example, in [B4], but we are going to present here an alternative argument that allows us to circumvent these rather tedious computations.

By Lemma 5.1, for a Zariski dense set of triples (c, μ, ν), the modules V_{𝔣}(γ_{0}) and L_{𝔣}(γ_{0}) coincide. Thus for the generic values of (c, μ, ν) the VOAs V_{Hyp}^{+} ⊗ V_{𝔣}(γ_{0}) are indeed 𝔤(μ, ν)- module, and the relations (4.9)-(4.16) hold. However the commutator formula (3.2) applied to the left hand sides of the relations (4.9)-(4.16) in V_{Hyp}^{+} ⊗ V_{𝔣}(γ_{0}) will yield expressions with coefficients that are polynomials in c^{±1}, μ, ν. Since these agree with the right hand sides of (4.9)-(4.16) on a Zariski dense set of parameters, the equalities must hold for all values of (c, μ, ν) with c ≠ 0. This concludes the proof of the Proposition.

Now we are ready to give realizations for all irreducible 𝔤-modules in category B_{χ} using highest weight 𝔣-modules.

Theorem 5.3. Let c ≠ 0, and let L(T) be an irreducible module in category B_{χ} determined by the data (V, W, h, d, α) as in Theorem 2.5, where V is a finite-dimensional irreducible 𝔤̇- module, W is a finite-dimensional irreducible sl_{N}-module, α ∈ ℂ^{N} and h, d ∈ ℂ. Then

L(T) ≅ M_{Hyp}^{+}(α) ⊗ L_{𝔣}(V, W, h_{Hei}, h_{Vir}, γ_{0}),

where γ_{0} is the same as in Theorem 4.3,

h_{Hei} = h − Nνc, h_{Vir} = −d + \frac{1}{2}(μ + ν)c.  (5.1)

Proof. First of all, we are going to show that M_{Hyp}^{+}(α) ⊗ L_{𝔣}(V, W, h_{Hei}, h_{Vir}, γ_{0}) is a 𝔤(μ, ν)-module. Indeed, it is a module for the vertex algebra V_{Hyp}^{+} ⊗ V_{𝔣}(γ_{0}), and by principle of preservation of identities [Li], the 𝔤(μ, ν)-module structure on V_{Hyp}^{+} ⊗ V_{𝔣}(γ_{0}) gets transferred to its VOA module M_{Hyp}^{+}(α) ⊗ L_{𝔣}(V, W, h_{Hei}, h_{Vir}, γ_{0}).

Next, let us show that M_{Hyp}^{+}(α) ⊗ L_{𝔣}(V, W, h_{Hei}, h_{Vir}, γ_{0}) is irreducible as a 𝔤-module. This is not difficult to see. The fields that define the 𝔤-module structure on V_{Hyp}^{+} ⊗ V_{𝔣}(γ_{0}) generate this VOA. Thus any 𝔤-submodule in M_{Hyp}^{+}(α) ⊗ L_{𝔣}(V, W, h_{Hei}, h_{Vir}, γ_{0}) is also a VOA submodule. However as a VOA module, M_{Hyp}^{+}(α)⊗L_{𝔣}(V, W, h_{Hei}, h_{Vir}, γ_{0}) is irreducible. Thus it is also irreducible as a 𝔤-module.

It is also easy to see that the 𝔤(μ, ν)-module M_{Hyp}^{+}(α) ⊗ L_{𝔣}(V, W, h_{Hei}, h_{Vir}, γ_{0}) belongs to the category B_{χ}, and its top is

ℂ[q_{1}^{±}, …, q_{N}^{±}] ⊗ V ⊗ W.

We can derive from (4.25)-(4.29) that 𝔤_{0} acts on this top according to (2.1), (2.6)-(2.8). Taking into account the relations (4.4) and (4.5), we conclude that the top of M_{Hyp}^{+}(α)⊗L_{𝔣}(V, W, h_{Hei}, h_{Vir}, γ_{0}) is isomorphic to T as a 𝔤_{0}-module. Since two simple modules with the same top are isomorphic, we obtain the claim of the Theorem.

Finally, applying Corollary 3.9, we obtain the following result: Theorem 5.4. Let L(T) be the irreducible 𝔤(μ, ν)-module in category B_{χ} determined by

the data (V, W, h, d, α) as in the statement of Theorem 2.5. Suppose

c ≠ 0, c ≠ −h^{∨}, c_{sl_{N}} = 1 − μc ≠ −N,
c_{Hei} = N(1 − μc) − N^{2}νc ≠ 0,

c′_{Vir} = 12c(μ + ν) − 2N − \frac{c dim(𝔤̇)}{c + h^{∨}} − \frac{c_{sl_{N}} (N^{2} − 1)}{c_{sl_{N}} + N} − 1 + 12\frac{N^{2}(\frac{1}{2} − νc)^{2}}{c_{Hei}} .

Let

h_{Hei} = h − Nνc,

h′_{Vir} = −d + \frac{1}{2}(μ + ν)c − \frac{Ω_{V}}{2(c + h^{∨})} − \frac{Ω_{W}}{2(c_{sl_{N}} + N)} − \frac{h_{Hei}(h_{Hei} − N(1 − 2νc))}{2c_{Hei}}.

Then

(a) L(T) ≅ M_{Hyp}(α) ⊗ L_{𝔤^{}}˙ (V, c) ⊗ L_{_{sl}_{N}} (W, c_{sl_{N}}) ⊗ L_{Hei}(h_{Hei}, c_{Hei}) ⊗ L_{Vir}(h′_{Vir}, c′_{Vir}),
(b) char L(T) =char q^{α}ℂ[q_{1}^{±}, …, q_{N}^{±}] × ∏_{j≥1}(1 − t^{j})^{−(2N+1)}

× char L_{𝔤^{}}˙ (V, c) × char L_{_{sl}_{N}} (W, c_{sl_{N}}) × char L_{Vir}(h′_{Vir}, c′_{Vir}).

Remark 5.5. In case of 2-toroidal Lie algebras (N = 1), the sl_{N} piece will not be present, and should be omitted from all the statements in this paper.

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School of Mathematics & Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, K1S 5B6, Canada

E-mail address: billig@math.carleton.ca