On the Combinatorics of Forrester-Baxter Models

Abstract

We provide further boson-fermion q-polynomial identities for the 'finitised' Virasoro characters χ p,p ′ r,s of the Forrester-Baxter minimal models M (p, p ′ ), for certain values of r and s. The construction is based on a detailed analysis of the combinatorics of the set P p,p ′ a,b,c (L) of q-weighted, length-L Forrester-Baxter paths, whose generating function χ p,p ′ a,b,c (L) provides a finitisation of χ p,p ′ r,s . In this paper, we restrict our attention to the case where the startpoint a and endpoint b of each path both belong to the set of 'Takahashi lengths'. In the limit L → ∞, these polynomial identities reduce to q-series identities for the corresponding characters.We obtain two closely related fermionic polynomial forms for each (finitised) character. The first of these forms uses the classical definition of the Gaussian polynomials, and includes a term that is a (finitised) character of a certain M (p,p ′ ) wherep ′ < p ′ . We provide a combinatorial interpretation for this form using the concept of 'particles'. The second form, which was first obtained using different methods by the Stony-Brook group, requires a modified definition of the Gaussian polynomials, and its combinatorial interpretation requires not only the concept of particles, but also the additional concept of 'particle annihilation'.

Paths, Crystals and Fermionic Formulae

“…Solvable RSOS models at a generic primitive root of unity give rise to yet further 1dsums associated with complicated energy functions. A typical example is [FB] corresponding to A (1) 1 , and as noted in Section 1.1, intricate arithmetic structures have been worked out on the fermionic formula and relation to fractional level representations [BMS,BMSW,FW2]. The crystal theory has not yet been adapted to such situations.…”

“…It was followed by many papers, most typically [Ber, Wa, Sc, BMS], where the fermionic formulae were established for a polynomial analogue of A (1) 1 coset Virasoro characters including the fractional level cases. See also [FW1,FW2]. An essential question in finitization is the meaning of the consequent q-polynomials.…”

Non-unitary minimal models, Bailey's Lemma and N=1,2 Superconformal algebras

“…As shown by Foda and Quano [15], the Bose-Fermi character identities [5,6,16,26] for the minimal models M (p, p ′ ) are of the form…”

“…In this section we summarize Bailey's original lemma [2,3] and the Bose-Fermi identities for the M (p, p ′ ) minimal models [5,6,16,26].…”

Fermi-gas interpretation of the RSOS path representation of the superconformal unitary minimal models

“…A frontal attack of the difficult combinatorics of the paths for the generic Forrester-Baxter models is considered in [14,15,16,36] (albeit using variables that are characteristics of the XXZ spectrum analysis and whose path interpretation is not immediate). The key preliminary step is the discovery of a new (manifestly positive definite) characterization of the weight of a Forrester-Baxter RSOS path (cf.…”