qq-identities and affinized projective varieties, I. Quadratic monomial ideals

We define the concept of an affinized projective variety and show how one can, in principle, obtain q-identities by different ways of computing the Hilbert series of such a variety. We carry out this program for projective varieties associated to quadratic monomial ideals. The resulting identities have applications in describing systems of quasi-particles containing null-states and can be interpreted as alternating sums of quasi-particle Fock space characters.Comment: AMSTeX, 22 page

Yangian Symmetries in the SU(N)1SU(N)_1 WZW Model and the Calogero-Sutherland Model

We study the $SU(N)$, level $1$ Wess-Zumino-Witten model, with affine primary fields as spinon fields of fundamental representation. By evaluating the action of the Yangian generators $Q_{0}^{a}, Q_{1}^{a}$ and the Hamiltonian $H_2$ on two spinon states we get a new connection between this conformal field theory and the Calogero-Sutherland model with $SU(N)$ spin. This connection clearly confirms the need for the $W_3$ generator in $H_2$ and an additional term in the $Q^{a}_{1}$. We also evaluate some energy spectra of $H_2$, by acting it on multi-spinon states.Comment: 12 pages, Latex; Errors in the previous version are corrected and new results on spinons are include

Graded parafermions: standard and quasi-particle bases

Two bases of states are presented for modules of the graded parafermionic conformal field theory associated to the coset \osp(1,2)_k/\uh(1). The first one is formulated in terms of the two fundamental (i.e., lowest dimensional) parafermionic modes. In that basis, one can identify the completely reducible representations, i.e., those whose modules contain an infinite number of singular vectors; the explicit form of these vectors is also given. The second basis is a quasi-particle basis, determined in terms of a modified version of the \ZZ_{2k} exclusion principle. A novel feature of this model is that none of its bases are fully ordered and this reflects a hidden structural $\Z_3$ exclusion principle.Comment: Harvmac 24 p; minor corrections in eqs 5.2 and 5.

On the Cohomology of the Noncritical WW-string

We investigate the cohomology structure of a general noncritical $W_N$-string. We do this by introducing a new basis in the Hilbert space in which the BRST operator splits into a ``nested'' sum of nilpotent BRST operators. We give explicit details for the case $N=3$. In that case the BRST operator $Q$ can be written as the sum of two, mutually anticommuting, nilpotent BRST operators: $Q=Q_0+Q_1$. We argue that if one chooses for the Liouville sector a $(p,q)$ $W_3$ minimal model then the cohomology of the $Q_1$ operator is closely related to a $(p,q)$ Virasoro minimal model. In particular, the special case of a (4,3) unitary $W_3$ minimal model with central charge $c=0$ leads to a $c=1/2$ Ising model in the $Q_1$ cohomology. Despite all this, noncritical $W_3$ strings are not identical to noncritical Virasoro strings.Comment: 38 pages, UG-7/93, ITP-SB-93-7

q-Identities and affinized projective varieties, II. Flag varieties

In a previous paper we defined the concept of an affinized projective variety and its associated Hilbert series. We computed the Hilbert series for varieties associated to quadratic monomial ideals. In this paper we show how to apply these results to affinized flag varieties. We discuss various examples and conjecture a correspondence between the Hilbert series of an affinized flag variety and a modified Hall-Littlewood polynomial. We briefly discuss the application of these results to quasi-particle character formulas for affine Lie algebra modules.Comment: AMSTeX, 25 pages, 3 figure

The SU(n)_1 WZW Models: Spinon Decomposition and Yangian Structure

We present a `spinon formulation' of the $SU(n)_1$ Wess-Zumino-Witten models. Central to this approach are a set of massless quasi-particles, called `spinons', which transform in the representation ${\bf \bar{n}}$ of $su(n)$ and carry fractional statistics of angle $\theta = \pi/n$. Multi-spinon states are grouped into irreducible representations of the yangian $Y(sl_n)$. We give explicit results for the $su(n)$ content of these yangian representations and present $N$-spinon cuts of the WZW character formulas. As a by-product, we obtain closed expressions for characters of the $su(n)$ Haldane-Shastry spin chains.Comment: 38 pages, LaTeX, no figure

The deformed Virasoro algebra at roots of unity

We discuss some aspects of the representation theory of the deformed Virasoro algebra \virpq. In particular, we give a proof of the formula for the Kac determinant and then determine the center of \virpq for $q$ a primitive N-th root of unity. We derive explicit expressions for the generators of the center in the limit $t=qp^{-1}\to \infty$ and elucidate the connection to the Hall-Littlewood symmetric functions. Furthermore, we argue that for q=\sqrtN{1} the algebra describes `Gentile statistics' of order $N-1$, i.e., a situation in which at most $N-1$ particles can occupy the same state.Comment: 51 pages, TeX (with amssym.def

On deformed W-algebras and quantum affine algebras

We discuss some aspects of the deformed W-algebras W_{q,t}[g]. In particular, we derive an explicit formula for the Kac determinant, and discuss the center when t^2 is a primitive k-th root of unity. The relation of the structure of W_{q,t}[g] to the representation ring of the quantum affine algebra U_q(\hat g), as discovered recently by Frenkel and Reshetikhin, is further elucidated in some examples.Comment: 40 pages, plain TeX with amssym.def, improved referencin