Continued Fractions and Fermionic Representations for Characters of M(p,p') minimal models

TL;DR

This paper derives fermionic sum representations for minimal model characters using continued fractions, binomial identities, and quantum group symmetries, revealing new Rogers-Ramanujan type identities.

Contribution

It introduces a novel approach connecting continued fractions, quantum groups, and fermionic sums for minimal model characters, expanding the mathematical framework.

Findings

Fermionic sum representations for all minimal models $M(p,p')$.

New identities of Rogers-Ramanujan type.

Duality relation between models $M(p,p')$ and $M(p'-p,p')$.

Abstract

We present fermionic sum representations of the characters $\chi^{(p,p')}_{r,s}$ of the minimal $M(p,p')$ models for all relatively prime integers $p'>p$ for some allowed values of $r$ and $s$. Our starting point is binomial (q-binomial) identities derived from a truncation of the state counting equations of the XXZ spin ${1\over 2}$ chain of anisotropy $-\Delta=-\cos(\pi{p\over p'})$. We use the Takahashi-Suzuki method to express the allowed values of $r$ (and $s$) in terms of the continued fraction decomposition of $\{{p'\over p}\}$ (and ${p\over p'}$) where $\{x\}$ stands for the fractional part of $x.$ These values are, in fact, the dimensions of the hermitian irreducible representations of $SU_{q_{-}}(2)$ (and $SU_{q_{+}}(2)$) with $q_{-}=\exp (i \pi \{{p'\over p}\})$ (and $q_{+}=\exp ( i \pi {p\over p'})).$ We also establish the duality relation $M(p,p')\leftrightarrow M(p'-p,p')$ and discuss the action of the Andrews-Bailey transformation in the space of minimal models. Many new identities of the Rogers-Ramanujan type are presented.