Quantum Hurwitz Numbers
Hurwitz numbers are defined geometrically as enumerations of inequivalent branched covers of the Riemann sphere with specified ramification structure at the branch points. Weighted Hurwitz numbers include and generalize all previously studied special cases such as, e.g., only simple branch points (Okounkov and Pandharipande), or Belyi curves (3 branch points) or monotonic paths in the Cayley graph of the symmetric group generated by transpositions. The general classical weight consists of monomial sum symmetric functions of an arbitrary number of parameters $(c_1, c_2, \dots)$. The generating functions for such weighted Hurwitz numbers are KP or $2d$-Toda $\tau$-functions of hypergeometric type. The simplest quantum Hurwitz numbers are obtained by specialization to $c_i = q^i$. The weights for this case may be normalized to a probability measure that coincides with that for a Bose gas with linear energy spectrum. A further generalization consists of replacing the monomial sum symmetric functions with their Macdonald polynomial analogs. The semiclassical asymptotics $(q \to 1)$ of the simple quantum Hurwitz numbers are shown to reproduce, as leading term, the case studied by Pandharipande and Okounkov, while the zero temperature asymptotics $(q \to 0)$ have as leading term the Hurwitz numbers for Belyi curves.