Fermionic expressions for minimal model Virasoro characters

Details Fermionic expressions for minimal model Virasoro characters Fermionic expressions for minimal model Virasoro characters

2002-12-11

arxiv.org/abs/math/0212154v3

Mem.Am.Math.Soc.175N827:1-160,2005

Mathematics

Combinatorics

153 pages, includes eps figures. v2: exceptional cases clarified, (1.45/6) corrected for d=1, Section 7.5 rewritten, reference

Scientific paper

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

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