Integrability of Hurwitz partition functions

Abstract

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 = ∑Rd2 − kRχR(t(1))…χR(t(k))exp (∑nξnCR(n)) depend on two types of such time variables, t and ξ. KP/Toda integrability in t requires that k ⩽ 2 and also that CR(n) are selected in a rather special way, in particular the naive cut-and-join operators are not allowed for n > 2. Integrability in ξ further restricts the choice of CR(n), forbidding, for example, the free cumulants. It also requires that k ⩽ 1. The quasi-classical integrability (the WDVV equations) is naturally present in ξ variables, but also requires a careful definition of the generating function.