Integrability of Hurwitz Partition Functions. I. Summary

Understand

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.