COINVARIANTS OF NILPOTENT SUBALGEBRAS OF THE VIRASORO ALGEBRA AND PARTITION IDENTITIES

BORIS FEIGIN AND EDWARD FRENKEL Dedicated to I.M. Gelfand on his 80th birthday

1. Introduction

Let V_{p,q}^{m,n} be the irreducible representation of the Virasoro algebra L with central charge c_{p,q} = 1 − 6(p − q)^{2}/pq and highest weight h_{m,n} = [(np − mq)^{2} − (p − q)^{2}]/4pq, where p, q > 1 are relatively prime integers, and m, n are integers, such that 0 < m < p, 0 < n < q. For fixed p and q the representations V_{p,q}^{m,n} form the (p, q) minimal model of the Virasoro algebra [1].

For N > 0 let L_{N} be the Lie subalgebra of the Virasoro algebra, generated by L_{i}, i < −N. There is a map from the Virasoro algebra to the Lie algebra of polynomial vector fields on ℂ^{∗}, which takes L_{i} to z^{−i+1} \frac{∂}{∂z}, where z is a coordinate. This map identifies L_{N} with the Lie algebra of vector fields on ℂ, which vanish at the origin along with the first N + 1 derivatives. The Lie algebra L_{2N} has a family of deformations L(p_{1}, ..., p_{N+1}), which consist of vector fields, vanishing at the points p_{i} ∈ ℂ along with the first derivative.

For a Lie algebra g and a g−module M we denote by H(g, M) the space of coinvariants (or 0th homology) of g in M, which is the quotient M/g · M, where g · M is the subspace of M, linearly spanned by vectors a · x, a ∈ g, x ∈ M. We will prove the following result.

Theorem 1. For any irreducible representation V of the (2, 2r + 1) minimal model

dim H(L_{2N}, V) = dim H(L(p_{1}, ..., p_{N+1}), V).

This statement was proved in [2] for N = 2 and irreducible representations of general minimal models.

As an application, we will give a new proof of the Gordon identities, which relies on two different computations of the characters of the irreducible representations of the (2, 2r + 1) minimal models.

Let us note that a priori dim H(L_{2N}, V) ≥ dim H(L(p_{1}, …, p_{N+1}), V). Indeed, if we have a family of Lie algebras, then the dimension of homology is the same for generic points of the space of parameters of the family, but it may increase at special points.

Research of the second author was supported by a Junior Fellowship from the Harvard Society of Fellows and by NSF grant DMS-9205303

Each minimal model of the Virasoro algebra associates to a punctured complex curve, with a representation inserted at each puncture, a linear space, which is called the space of conformal blocks. This space can be defined as the space of coinvariants of the Lie algebra of meromorphic vector fields on this curve, which are allowed to have poles only at the punctures, in the tensor product of these representations (cf. [1], mathematical aspects of this correspondence are treated in detail in [2], [3]).

In this language, the space H(L(p_{1}, …, p_{N+1}), V_{p,q}^{m,n}) is isomorphic to the direct sum of the spaces of conformal blocks, associated to the projective line with punctures p_{1}, …, p_{N+1} with all possible insertions from the (p, q) minimal model, and ∞ with the insertion of V_{p,q}^{m,n}. The dimension of this space can be calculated by a Verlinde type argument by deforming our curve – the projective line with N +2 punctures – to a joint union of N projective lines, each with 3 punctures. Under this deformation the dimension of the homology does not change [2], [3], and can therefore be reduced to the well-known result on the dimensions of the spaces of conformal blocks, associated to the projective line with 3 insertions – the fusion coefficients.

This gives an explicit formula for dim H(L(p_{1}, …, p_{N+1}), V_{p,q}^{m,n}), which represents a lower bound for dim H(L_{2N}, V_{p,q}^{m,n}). We can also obtain an upper bound by a different method.

This method is based on the calculation of the annihilating ideal of a minimal model [4]. Let us recall the definition of the annihilating ideal. For c ∈ ℂ let M_{c} be the vacuum Verma module with central charge c. It is generated by a vector v_{c}, such that L_{i}v_{c} = 0, i ≥ −1. This representation has the structure of a vertex operator algebra (VOA) [5], which acts on any irreducible module M of the Virasoro algebra with the same central charge. It means that each vector of M_{c} defines a local current (or field), which is a formal power series in z and z^{−1}, whose Fourier coefficients are linear operators, acting on M. These local currents can be constructed as follows. Consider the projective line ℂℙ^{1} with three punctures: 0 and ∞ with the insertions of M, and z with the insertion of M_{c}. The space of conformal blocks is one-dimensional in this case. In other words, the space of coinvariants of the Lie algebra A(0, z) of meromorphic vector fields on ℂℙ^{1}, which are allowed to have poles only at 0, ∞ and z, in M ⊗ M ⊗ M_{c}, is one-dimensional. It has a canonical generator – the projection of the tensor product of the highest weight vectors. The dual to this generator defines for each vector of M_{c} a linear operator from M to M, depending on z, which is our local current.

These local currents can be constructed explicitly. The monomials

{L_{−m_{1}} … L_{−m_{l}}v_{c}| m_{1} ≥ m_{2} ≥ ⋯ ≥ m_{l} > 1}

linearly span M_{c}. The corresponding local currents are equal to

\frac{1}{(m_{1} − 2)!} … \frac{1}{(m_{l} − 2)!} : ∂_{z}^{m_{1}−2}T (z) … ∂_{z}^{m_{l}−2}T (z) :, (1)

where T (z) = ∑_{i∈ℤ} L_{i}z^{−i−2}. The Fourier components of thus defined local currents form a Lie algebra U_{c}(L)_{loc}, which is called the local completion of the universal enveloping algebra of the Virasoro algebra with central charge c [6],[4]. They act on any representation of L, which has the same central charge.

The module M_{c} is irreducible, if and only if c is not equal to c_{p,q}. However, if c = c_{p,q}, then M_{c} contains a (unique) singular vector. The quotient of M_{c_{p,q}} by the submodule, generated by this singular vector, is isomorphic to V_{p,q}^{1,1}. This representation, which is called the vacuum representation of the (p, q) minimal model, defines another VOA, which is the quotient of the VOA of M_{c_{p,q}} in the appropriate sense [7]. An irreducible representation M of the Virasoro algebra with central charge c_{p,q} is a module over this VOA [7], if and only if the space of coinvariants of the Lie algebra A(0, z) in M ⊗ M ⊗ V_{p,q}^{1,1} is one-dimensional. An explicit computation of the dimension of this space, which was made in [2], tells us that it is so, if and only if M is an irreducible representation of the (p, q) minimal model.

The Fourier components of the local current, corresponding to the singular vector of M_{c_{p,q}}, generate an ideal in U_{c_{p,q}}(L)_{loc}, which is called the annihilating ideal of the (p, q) minimal model. Any element of this ideal acts trivially on any representation of the (p, q) minimal model.

Imposing this condition with respect to the generators of the annihilating ideal immediately leads to certain linear relations among the monomial elements of V_{p,q}^{m,n}. This allows to estimate the dimension of the coinvariants H_{0}(L_{2N}, V_{p,q}^{m,n}) from above.

The peculiarity of the (2, 2r + 1) models is that this upper bound coincides with the lower bound, and is therefore exact. This proves Theorem 1 and leads to a nice combinatorial description of the minimal representations of these models. Namely, as a ℤ−graded linear space, such a representation is isomorphic to the quotient of the space of polynomials in infinitely many variables by a certain monomial ideal. Thus we obtain an expression for the character of this module, which coincides with the right hand side of one of the Gordon identities. But it is known that this character is equal to the left hand side of the identity. Hence, we obtain a new proof of the Gordon identities.

Implementing this program for other minimal models would lead to a result, analogous to Theorem 1, as well as to nice character formulas for irreducible representations. However, for general minimal models, for which the representations are “smaller” and the structure of the annihilating ideal is more complicated, one should impose extra conditions, corresponding to other elements of the annihilating ideal, to obtain the exact upper bound.

2. Lower bound: genus 0 conformal blocks

In this section we will calculate the dimension of H(L(p_{1}, …, p_{N+1}), V) for an irreducible representation V of a (p, q) minimal model.

Denote by A(p_{1}, …, p_{N+1}) the Lie algebra of meromorphic vector fields on ℂℙ^{1}, which are allowed to have poles only at the distinct points p_{1}, …, p_{N+1} ∈ ℂ, and ∞. There is an embedding of this Lie algebra into the direct sum of N + 2 Virasoro algebras, and thus it acts on tensor products V_{1} ⊗…⊗V_{N+2} of N +2 reprersentations of the Virasoro algebra with the same central charge.

Let us fix central charge c_{p,q} and an irreducible representation V = V_{N+2} of the (p, q) minimal model. The following statement follows from the general results on computation of conformal blocks, outlined in Section 5 of [2] (cf. [3] for details).

Proposition 2. (1) dim H(L(p_{1}, …, p_{N+1}), V) = ∑ dim H(A(p_{1}, …, p_{N+1}), V_{1} ⊗ … ⊗ V_{N+1} ⊗ V), where the sum is taken over all (N + 1)−tuples V_{1}, …, V_{N+1} of irreducible representations of the (p, q) minimal model.

(2) dim H(A(p_{1}, …, p_{N+1}), V_{1} ⊗ … ⊗ V_{N+1} ⊗ V) = ∑ dim H(A(p_{1}, …, p_{N}), V_{1} ⊗ … ⊗ V_{N} ⊗ W) · dim H(A(0, 1), W ⊗ V_{N+1} ⊗ V), where the sum is taken over all irreducible representations W of the (p, q) minimal model.

Using this result, we can calculate dim H(L(p_{1}, …, p_{N+1}), V) by induction.

Let V_{1}, …, V_{s} be the set of all irreducible representations of the (p, q) minimal model (s = (p−1)(q−1)/2), and let u_{N} be the s−vector, whose components are u_{N}^{i} = dim H(L(p_{1}, …, p_{N+1}), V_{i}), i = 1, …, s. Put c_{ijk} = dim H(A(0, 1), V_{i} ⊗ V_{j} ⊗ V_{k}). The numbers c_{ijk} are usually called the fusion coefficients. They define the fusion algebra of this minimal model, which has generators g_{i}, 1 ≤ i ≤ s and relations g_{i} · g_{j} = ∑_{k} c_{ijk}g_{k}.

Let us introduce the s × s matrix M = M_{p,q}, whose (i, j)th entry is equal to ∑_{k} c_{ijk}. This matrix can be interpreted as the matrix of action of the sum ∑_{k} g_{k} on the fusion algebra.

According to Proposition 2, u_{N}^{i} = ∑_{j,k} c_{ijk}u_{N−1}^{j}. This gives us the recursion relation for u_{N}:

u_{N} = Mu_{N−1}.

For any i, the representation V_{i} is generated from the highest weight vector v_{i} by the action of the Lie algebra L_{0}, hence u_{0}^{i} = 1. Therefore, we obtain the formula

u_{N} = M^{N}u_{0},

where u_{0}^{t} = [1, 1, …, 1].

This formula enables us to calculate the dimensions u_{N}^{i} explicitly by diagonalizing the matrix M.

In the rest of the paper we will focus on the (2, 2r + 1) models. In such a model we have r irreducible representations V_{i} = V_{2,2r+1}^{1,i}, i = 1, …, r. The (i, j)th entry of the corresponding r × r matrix M is equal to min{i, j}. This matrix is equal to the square of the matrix

M_{r}(x) = (\begin{matrix}0 & & 0 & & … & & 0 & & 1 \\ 0 & & 0 & & … & & 1 & & 1 \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 0 & & 1 & & … & & 1 & & 1 \\ 1 & & 1 & & … & & 1 & & 1 \end{matrix}) . (2)

This fact has the following interpretation: in the fusion algebra of this model ∑_{k} g_{k} is equal to the square of g_{r}, and the action of g_{r} is given by matrix M_{r}.

Let us introduce vectors x_{i}, whose ith component is equal to 1, and all other component are equal to 0. Since u_{0} = M_{r}^{2}x_{1}, we have u_{N}^{i} = x_{i}^{t}M_{r}^{2N+2}x_{1}. This gives us a lower bound for v_{N}^{i} = dim H(L_{2N}, V_{i}).

Proposition 3. v_{N}^{i} ≥ x_{i}^{t}M_{r}^{2N+2}x_{1}.

In the next Section we will show that this bound is exact.

3. Upper bound: the annihilating ideal

The annihilating ideal of the (2, 2r+1) minimal model is the ideal of the Lie algebra U_{c_{2,2r+1}}(L)_{loc}, generated by the Fourier components of the local current, corresponding to the singular vector of the vacuum Verma module M_{c_{2,2r+1}}. All elements of this ideal act by 0 on any irreducible representation of the (2, 2r + 1) minimal model. In particular, the generators of the ideal act by 0. This leads to certain relations between vectors in the irreducible minimal representations, as explained in [4].

Introduce a filtration on the irreducible representation V_{i} of the (2, 2r + 1) models as follows: 0 ⊂ V_{i}^{0} ⊂ V_{i}^{1} ⊂ … ⊂ V_{i}^{∞} = V_{i}, where V_{i}^{k} is linearly spanned by the monomials L_{−m_{1}} … L_{−m_{l}}v_{i}, such that m_{1} ≥ m_{2} ≥ ⋯ ≥ m_{l} ≥ 1, and l ≤ k. Let Ω_{i} be the adjoint graded space:

Ω_{i} = ⊕_{k≥0}Ω_{i}^{k}, Ω_{i}^{k} = V_{i}^{k}/V_{i}^{k−1}.

The space Ω_{i} is a polynomial algebra in variables a_{j}, j > 0, where a_{j} is the image of L_{−j}v_{i}.

The symbol of the singular vector of M_{c_{2,2r+1}} is equal to L_{−2}^{r}v_{0}. According to formula (1), the symbol of the corresponding local current is equal to : T (z)^{r} :. Hence the symbols of the Fourier components of this current are given by the formula

∑_{j_{1}+...+j_{r}=n} : L_{−j_{1}} … L_{−j_{r}} : .

Their action on Ω_{i} is given by the multiplication by

S_{n} = ∑_{j_{k}>0;j_{1}+...+j_{r}=n} a_{j_{1}} … a_{j_{r}}, (3)

for n ≥ r, and 0 for n < r.

It is known that V_{i} is the quotient of the Verma module with highest weight h_{1,i} by the maximal submodule, generated by two singular vectors. The symbol of one of them is equal to L_{−1}^{i}v_{i}. This gives us a surjective map from the quotient Ω′_{i} of ℂ[a_{j}]_{j>0} by the ideal I_{i} generated by S_{n}, n > r, and a_{1}^{i}, to Ω_{i}.

Let Ω_{i}^{mon} be the quotient of ℂ[a_{j}]_{j>0} by the ideal I_{i}^{mon}, generated by the monomials a_{j}^{u}a_{j+1}^{v}, 0 ≤ u < r, u + v = r, j > 0, and a_{1}^{i}.

We can introduce a ℤ−grading on the module V_{i} by putting deg v_{i} = 0, deg L_{−j} = j. The space Ω_{i} inherits this grading. We can also introduce a compatible ℤ−grading on the space ℂ[a_{j}]_{j>0} by putting deg a_{−j} = j. The spaces Ω′_{i} and Ω_{i}^{mon} inherit this grading.

For any ℤ−graded linear space V = ⊕_{n≥0}V (n), such that dim V (n) ≤ ∞, let ch V = ∑_{n≥0} dim V (n)q^{n} be its character. We will write ch V ≤ ch V′, if dim V (n) ≤ dim V′(n) for any n ≥ 0.

We have

ch V_{i} = ch Ω_{i} ≤ ch Ω′_{i} ≤ ch Ω_{i}^{mon}.

The last inequality follows from the fact that each of the generators S_{n} of the ideal I_{i} has as a summand one and only one generator of the ideal I_{i}^{mon}, namely, a_{j}^{u}a_{j+1}^{v} is a summand of S_{rj+v}.

Now, by the construction of our filtration, the character of coinvariants H(L_{2N}, V_{i}) is equal to the character of the quotient Ω_{i,N} of the space Ω_{i} by the ideal, generated by a_{j} with j > 2N. Again, we have:

(4) ch H(L_{2N}, V_{i}) = ch Ω_{i,N} ≤ ch Ω_{i,N}^{mon},

where Ω_{i,N}^{mon} is the quotient of Ω_{i}^{mon} by the ideal, generated by a_{j}, j > 2N. Therefore,

v_{N}^{i} = dim H(L_{2N}, V_{i}) ≤ ω_{N}^{i} = dim Ω_{i,2N}^{mon}.

We can easily calculate ω_{N}^{i} by induction [8].

Let C_{i}^{r} = {(m_{1}, …, m_{l})| m_{1} ≥ ⋯ ≥ m_{l} ≥ 1, m_{i} ≥ m_{i+r−1} + 2, m_{l−i+1} > 1}. The monomials

{a_{m_{1}} … a_{m_{l}}| (m_{1}, …, m_{l}) ∈ C_{i}^{r}}

constitute a linear basis in Ω_{i}^{mon}.

For every integer N > 0, introduce the subspaces W_{k,N}^{i}, 1 ≤ k ≤ r, of Ω_{i}^{mon}, which are linearly spanned by the monomials

{a_{m_{1}} … a_{m_{l}}| (m_{1}, …, m_{l}) ∈ C_{i}^{r}, m_{1} ≤ N, m_{k} ≤ N − 1}.

Clearly, W_{1,N}^{i} is isomorphic to Ω_{i,N−1}^{mon}.

The dimension of the space W_{k,N}^{i}, which is spanned by our monomials, in which a_{N} is allowed in the power less than k, is equal to the sum of dimensions of the spaces W_{l,N−1}^{i} with l = 1, …, r − k + 1, because we have the relations a_{N−1}^{u}a_{N}^{v} = 0 for u + v = r. Let us introduce the r−vector w_{N}^{i}, whose components are w_{k,N}^{i} = dim W_{k,N}^{i}, k = 1, …, r. We obtain the formula [8]:

w_{N}^{i} = M_{r}w_{N−1}^{i},

where M_{r} is given by (2), which shows that w_{2N+1}^{i} = M_{r}^{2N}w_{1}^{i}.

Remark 1. As shown in [8], there is a q−deformation of this formula, which gives an expression for the characters of W_{k,N}^{i}.

By definition, w_{k,1}^{i} = min{i, k}. One can check that w_{1}^{i} = M_{r}^{2}x_{i}, and so w_{2N+1}^{i} = M_{r}^{2N+2}x_{i}. Therefore, ω_{N}^{i} = w_{1,2N+1}^{i} = x_{1}^{t}M_{r}^{2N+2}x_{i}, and this gives us an upper bound for v_{N}^{i}.

Proposition 4.

v_{N}^{i} ≤ x_{1}^{t}M_{r}^{2N+2}x_{i}.

But since M_{r}^{t} = M_{r}, this upper bound coincides with the lower bound from Proposition 3. Therefore, we have the equality

u_{N}^{i} = v_{N}^{i} = ω_{N}^{i} = x_{1}^{t}M_{r}^{2N+2}x_{i}.

This completes the proof of Theorem 1.

Note that we have also proved that Ω_{i} is isomorphic to Ω′_{i} = ℂ[a_{j}]_{j>0}/I_{i}.

Remark 2. For general minimal models the graded space Ω of an irreducible representation is also isomorphic to the quotient of ℂ[a_{j}]_{j>0} by a certain ideal I. This ideal contains the operators S_{n}, given by formula (3) with r = (p − 1)(q − 1)/2. They correspond to the action of the symbols of generators of the annihilating ideal of the minimal model. In general, however, these elements do not generate the ideal I. There are other generators, corresponding to the action of the symbols of other elements of the annihilating ideal on Ω. It is an interesting problem to find explicit formulas for them. This will hopefully lead to a nice combinatorial description of general irreducible representations.

4. Application: Gordon identities

By formula (4), dim H(L_{2N}, V_{i})(n) ≤ dim Ω_{i,2N}^{mon}(n) for any n and N. In the previous Section we proved that dim H(L_{2N}, V_{i}) = dim Ω_{i,2N}^{mon}, therefore dim H(L_{2N}, V_{i})(n) = dim Ω_{i,2N}^{mon}(n) for any n and N. But, clearly, dim H(L_{2N}, V_{i})(n) = dim V_{i}(n) and dim Ω_{i,2N}^{mon}(n) = dim Ω_{i}^{mon}(n) for N large enough. Hence dim V_{i}(n) = dim Ω_{i}^{mon}(n) for any n, and ch V_{i} = ch Ω_{i}^{mon}. This can be interpreted as follows.

Proposition 5. The monomials

{L_{−m_{1}} … L_{−m_{l}}v_{i}| (m_{1}, …, m_{l}) ∈ C_{i}^{r}}

constitute a linear basis in the irreducible representation V_{i} of the (2, 2r + 1) minimal model.

Thus, we obtain the following formula for the character of the module V_{i}:

ch V_{i} = ∑_{n≥0} |C_{i}^{r}(n)|q^{n}, (5)

where C_{i}^{r}(n) is the subset of C_{i}^{r}, which consists of the elements (m_{1}, …, m_{l}) ∈ C_{i}^{r}, for which ∑ m_{j} = n. The right hand side of the formula (5) is known to be equal to

∑_{n_{1},... ,n_{r−1}≥0} \frac{q^{N_{1}^{2}+...+N_{r−1}^{2}+N_{i}+...+N_{r−1}}}{(q)_{n_{1}} … (q)_{n_{r−1}}} ,

where N_{j} = n_{j} + … + n_{r−1}, and (q)_{n} = ∏_{l=1}^{l=n}(1 − q^{l}) (cf. [9]).

On the other hand, it is known [10, 11] that

ch V_{i} = ∏_{n>0,n≠0,±imod(2r+1)}(1 − q^{n})^{−1}.

This formula follows from the Weyl-Kac type character formula for irreducible minimal representations of the Virasoro algebra.

Thus, we have obtained a new proof of the Gordon identities:

∏_{n>0,n≠0,±imod(2r+1)}(1 − q^{n})^{−1} = ∑_{n_{1},... ,n_{r−1}≥0} \frac{q^{N_{1}^{2}+...+N_{r−1}^{2}+N_{i}+...+N_{r−1}}}{(q)_{n_{1}} … (q)_{n_{r−1}}} .

For r = 2 these are the Rogers-Ramanujan identities:

∏_{n>0,n≠0,±1mod5}(1 − q^{n})^{−1} = ∑_{n≥0} \frac{q^{n(n+1)}}{(q)_{n}} and ∏_{n>0,n≠0,±2mod5}(1 − q^{n})^{−1} = ∑_{n≥0} \frac{q^{n^{2}}}{(q)_{n}} .

They correspond to two irreducible representations of the (2, 5) minimal model.

Acknowledgements The main part of this work was done while B.F. was visiting the Isaac Newton Institute of the University of Cambridge. He would like to the thank the Institute for hospitality. E.F. thanks A.Szenes for valuable discussions.

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Landau Institute for Theoretical Physics, Moscow 117334, Russia

Department of Mathematics, Harvard University, Cambridge, MA 02138, USA