Fermionic counting of RSOS-states and Virasoro character formulas for the unitary minimal series M(\nu,\nu+1). Exact results

TL;DR

This paper introduces a fermionic basis for RSOS-model states, proves binomial identities related to it, and derives character formulas for the unitary minimal series M(ν,ν+1) by connecting bosonic and fermionic sums.

Contribution

It presents a new fermionic basis for RSOS-models, proves q-binomial identities, and derives exact character formulas for minimal models, linking bosonic and fermionic representations.

Findings

Fermionic basis has the same dimension as RSOS basis.

Q-binomial identities are proved using elementary recurrences.

Derived explicit character formulas for M(ν,ν+1) minimal series.

Abstract

The Hilbert space of an RSOS-model, introduced by Andrews, Baxter, and Forrester, can be viewed as a space of sequences (paths) {a_0,a_1,...,a_L}, with a_j-integers restricted by 1<=a_j<=\nu, |a_j-a_{j+1}|=1, a_0=s, a_L=r. In this paper we introduce different basis which, as shown here, has the same dimension as that of an RSOS-model. Following McCoy et al, we call this basis -- fermionic (FB). Our first theorem Dim(FB)=Dim(RSOS-basis) can be succinctly expressed in terms of some identities for binomial coefficients. Remarkably, these binomial identities can be q-deformed. Here, we give a simple proof of these q-binomial identities in the spirit of Schur's proof of the Rogers-Ramanujan identities. Notably, the proof involves only the elementary recurrences for the q-binomial coefficients and a few creative observations. Finally, taking the limit L --> \infty in these q-identities, we derive an expression for the character formulas of the unitary minimal series M(\nu,\nu+1) "Bosonic Sum = Fermionic Sum". Here, Bosonic Sum denotes Rocha-Caridi representation (\chi_{r,s=1}^{\nu,\nu+1}(q)) and Fermionic Sum stands for the companion representation recently conjectured by the Stony Brook group.