Integrability of Hurwitz Partition Functions. I. Summary

Abstract: Partition functions often become \tau-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}\chi_R(t^{(1)})...\chi_R(t^{(k)})\exp(\sum_n \xi_nC_R(n)) depend on two types of such time variables, t and \xi. 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 \xi further 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 \xi variables, but also requires a careful definition of the generating function.