Hurwitz配分函数的可积性。I.总结
Integrability of Hurwitz Partition Functions. I. Summary
摘要：
如果把配分函数看作依赖于被称为时间变量的无穷多个参数集，那么它们常常成为可积族的τ-函数。Hurwitz配分函数Z = \sum_R d_R^{2-k} x_R（t^{（1）}）.χ_R（t^{（k）}）\exp（\sum_n n_nC_R（n））取决于两种类型的这样的时间变量t和n。t中的KP/Toda可积性要求k \leq 2和C_R（n）以一种相当特殊的方式选择，特别是对于n>2，不允许使用朴素割并算子。可积性进一步限制了C_R（n）的选择，例如禁止自由累积量。它还要求k \leq 1.准经典可积性（WDVV方程）自然存在于双变量中，但也需要仔细定义生成函数。
英文摘要：
Partition functions often become τ-functions of integrable hierarchies, if they are considered dependent on infinite sets of parameters called time variables. The Hurwitz partition functions Z = \sum_R d_R^{2-k}χ_R(t^{(1)})...χ_R(t^{(k)})\exp(\sum_n ξ_nC_R(n)) depend on two types of such time variables, t and ξ. KP/Toda integrability in t requires that k\leq 2 and also that C_R(n) are selected in a rather special way, in particular the naive cut-and-join operators are not allowed for n>2. Integrability in ξfurther restricts the choice of C_R(n), forbidding, for example, the free cumulants. It also requires that k\leq 1. The quasiclassical integrability (the WDVV equations) is naturally present in ξvariables, but also requires a careful definition of the generating function.
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