A brief survey will be given of the use of KP and 2D-Toda tau functions of special “hypergeometric type” as generating functions for weighted Hurwitz numbers (i.e. weighted enumerations of N-sheeted branched coverings of the Riemann sphere, or equivalently, weighted paths in the Cayley graph of the symmetric group S_N generated by transpositions). The weights depend on parametric families of auxiliary parameters, and consist of evaluations of basis elements of the algebra of symmetric functions of the latter. An alternative generating function is provided by certain correlation functions W_{n,g}(x_1,. …, x_n) depending on a pair of integers that play a role analogous to the multidifferentials in the Topological Recursion approach to intersection theory on moduli spaced of marked Riemann surfaces. As in that case, an associated invariant classical and quantum “spectral curve” is derived and a set of recursion relations that determine the general term quadratically in terms of finite sums over preceding ones.
Examples include: 1) the “simple” (double or single) Hurwitz numbers studied originally by Okounkov and Pandharipande, 2) The case of "Belyi curves”, having just three branch points, one of them weighted, and the related “dessins d’enfants”; 3) The “weakly monotonic” paths in the Cayley graph, for which the generating tau function is the Itzykson-Zuber-Harish-Chandra integral and (if time permits) 4) The case of simple "quantum Hurwitz numbers", in which the weighting is shown to coincide with that of a quantum Bose-Einstein gas. (Partly based on joint work with M. Guay-Paquet, A. Orlov, B. Eynard, A. Alexandrov and G. Chapuy)
