Fermionic solution of the Andrews-Baxter-Forrester model II: proof of Melzer's polynomial identities

TL;DR

This paper proves polynomial identities for finitizations of Virasoro characters in the ABF model using fermionic techniques, confirming conjectures by Melzer and connecting to Rogers--Ramanujan identities.

Contribution

It provides a rigorous proof of Melzer's polynomial identities for the ABF model's configuration sums, linking fermionic methods with Virasoro character identities.

Findings

Proof of Melzer's polynomial identities.

Reproduction of Rogers--Ramanujan type identities in the thermodynamic limit.

List of additional Virasoro character identities derived from the proof.

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 $\chi_{b,a}^{(r-1,r)}(q)$ 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.