Fermionic Expressions for Minimal Model Virasoro Characters

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.