Integrability of Hurwitz Partition Functions. I. Summary

TL;DR

This paper investigates the conditions under which Hurwitz partition functions exhibit integrability, focusing on dependencies on different parameter sets and restrictions imposed by KP/Toda hierarchies and WDVV equations.

Contribution

It provides a detailed analysis of the integrability conditions for Hurwitz partition functions, highlighting restrictions on parameters and operators for different types of integrability.

Findings

Integrability in t variables requires k ≤ 2 and specific C_R(n) choices.

Integrability in ξ variables restricts C_R(n), excluding free cumulants.

Quasiclassical integrability (WDVV equations) is naturally present in ξ variables.

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.