极小模型Virasoro特征标的费米表达式
Fermionic expressions for minimal model Virasoro characters
摘要：
给出并证明了所有极小模型Virasoro特征$χ^{p, p'}_{r, s}$的费米子表达式。每个这样的表达式都是基本费米子形式类型的项的和。在大多数情况下，所有这些项都是使用某些树来写下来的，这些树是根据与$p'/p$的连分数相关的高桥长度和截断的高桥长度为$s$和$r$构造的。在其余的情况下，除了这些项之外，$χ^{p, p'}_{r, s}$的费米子表达式包含不同的字符$χ^{\hat p, \hat p'}_{\hat r,\hat s}$，因此本质上是递归的。所有字符$χ^{p, p'}_{r, s}$的玻色-费米子$q$ -系列恒等式来自于将这些费米子表达式等同于已知的玻色表达式。在$p=2r$、$p=3r$、$p'=2s$或$p'=3s$、Rogers-Ramanujan型恒等式的情况下，这些费米子表达式与$χ^{p, p'}_{r, s}$的已知乘积表达式相等。首先利用各种组合变换得到长$L$ Forrester-Baxter路径的生成函数$χ^{p, p'}_{a, b, c}(L)$的费米子表达式，证明了费米子表达式.在$L\to\infty$极限下，分别从高桥长度和截断高桥长度为$b$和$r$构建的树之间映射后，出现了$χ^{p, p'}_{r, s}$的费米子表达式。
英文摘要：
Fermionic expressions for all minimal model Virasoro characters $χ^{p, p'}_{r, s}$ 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 $χ^{p, p'}_{r, s}$ contains a different character $χ^{\hat p, \hat p'}_{\hat r,\hat s}$, and is thus recursive in nature. Bosonic-fermionic $q$-series identities for all characters $χ^{p, p'}_{r, s}$ 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 $χ^{p, p'}_{r, s}$. The fermionic expressions are proved by first obtaining fermionic expressions for the generating functions $χ^{p, p'}_{a, b, c}(L)$ of length $L$ Forrester-Baxter paths, using various combinatorial transforms. In the $L\to\infty$ limit, the fermionic expressions for $χ^{p, p'}_{r, s}$ 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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