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

Abstract

We present fermionic sum representations of the characters ${\chi }_{r,s}^{(p,p^{\prime })}$ of the minimal $M(p,p^{\prime })$ models for all relatively prime integers $p^{\prime }>p$ 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 $\frac{1}{2}$ chain of anisotropy $-\Delta =-\cos (\pi \frac{p}{p^{\prime }})$. We use the Takahashi-Suzuki method to express the allowed values of $r$ (and $s$) in terms of the continued fraction decomposition of $\{\frac{p^{\prime }}{p}\}$ (and $\frac{p}{p^{\prime }}$) where $\{x\}$ stands for the fractional part of $x.$ These values are, in fact, the dimensions of the hermitian irreducible representations of $S{U}_{q_{-}}(2)$ (and $S{U}_{q_{+}}(2)$) with $q_{-}=\exp (i\pi \{\frac{p^{\prime }}{p}\})$ (and $q_{+}=\exp (i\pi \frac{p}{p^{\prime }})).$ We also establish the duality relation $M(p,p^{\prime })\leftrightarrow M(p^{\prime }-p,p^{\prime })$ 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.

Several references, one further explicit result and several discussion remarks added

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