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arXiv · 1103.4100

Integrability of Hurwitz Partition Functions. I. Summary

A. Alexandrov·A. Mironov·A. Morozov·S. Natanzon
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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.

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A. Alexandrov, A. Mironov, A. Morozov, S. Natanzon. 2011-03-21. Integrability of Hurwitz Partition Functions. I. Summary. https://doi.org/10.1088/1751-8113/45/4/045209

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