Fermionic solution of the Andrews-Baxter-Forrester model. II. Proof of Melzer's polynomial identities

Abstract

We compute the one-dimensional configuration sums of the ABF model using the fermionic technique introduced in part I of this paper. Combined with the results of Andrews, Baxter and Forrester, we find proof of polynomial identities for finitizations of the Virasoro characters χ (r−1,r) b,a (q) as conjectured by Melzer. In the thermodynamic limit these identities reproduce Rogers-Ramanujan type identities for the unitary minimal Virasoro characters, conjectured by the Stony Brook group. We also present a list of additional Virasoro character identities which follow from our proof of Melzer's identities and application of Bailey's lemma.

Paths for $${\mathcal{Z}}_k$$ Parafermionic Models

“…This duality allows us to translate results derived in the context of the unitary minimal models [19,20,11] to the parafermionic case.…”

A new path description for the {\cal M} (k+1,2k+3) models and the dual {\cal Z}_k graded parafermions

“…The structural similarity between the path description of the M(k + 1, 2k + 3) and the M(k + 1, k + 2) models allows us to generalize rather immediately the combinatorial analysis of the latter models by Warnaar [40,41] and derive the generating function for the former paths in the form of a positive multiple sum. This happens to generate novel fermionic expressions.…”

The perturbations φ2,1 and φ1,5 of the minimal models M(p, p′) and the trinomial analogue of Bailey's lemma

“…We also note that an identity somewhat similar to (3.8) appeared in [42] as eqn. (5.1) and was proven in [43].…”

“…The right hand side of that equation is indeed the same as the right hand side of (3.8). However, the left hand side of the identity of [42] is not the same as (3.9). In particular we note: (1) for finite L the fermionic form of [42] is not in the canonical form (3.4) which has a quasi-particle interpretation; and (2) in the limit L → ∞ the fermionic form of [42] is identical with (1.5) and not with the left hand side of (3.3).…”