Fermionic expressions for minimal model Virasoro characters

Abstract

Fermionic expressions for all minimal model Virasoro characters ${\chi }_{r,s}^{p,p^{\prime }}$ are stated and proved. Each such expression is a sum of terms of fundamental fermionic form type. In most cases, all these terms are written down using certain trees which are constructed for $s$ and $r$ from the Takahashi lengths and truncated Takahashi lengths associated with the continued fraction of $p^{\prime }/p$. In the remaining cases, in addition to such terms, the fermionic expression for ${\chi }_{r,s}^{p,p^{\prime }}$ contains a different character ${\chi }_{\hat{r},\hat{s}}^{\hat{p},{\hat{p}}^{\prime }}$, and is thus recursive in nature. Bosonic-fermionic $q$-series identities for all characters ${\chi }_{r,s}^{p,p^{\prime }}$ result from equating these fermionic expressions with known bosonic expressions. In the cases for which $p=2r$, $p=3r$, $p^{\prime }=2s$ or $p^{\prime }=3s$, Rogers-Ramanujan type identities result from equating these fermionic expressions with known product expressions for ${\chi }_{r,s}^{p,p^{\prime }}$. The fermionic expressions are proved by first obtaining fermionic expressions for the generating functions ${\chi }_{a,b,c}^{p,p^{\prime }}(L)$ of length $L$ Forrester-Baxter paths, using various combinatorial transforms. In the $L\to \infty$ limit, the fermionic expressions for ${\chi }_{r,s}^{p,p^{\prime }}$ emerge after mapping between the trees that are constructed for $b$ and $r$ from the Takahashi and truncated Takahashi lengths respectively.

153 pages, includes eps figures. v2: exceptional cases clarified, (1.45/6) corrected for d=1, Section 7.5 rewritten, reference added. v3: minor typos and clarifications. To appear in Memoirs of the American Mathematical Society

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