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Weighted Hurwitz numbers, topological recursion and isomonodromic deformations Harnad, John


The family of 2D-Toda tau functions of hypergeometric type that serve as generating functions for weighted Hurwitz numbers with polynomial weight generating functions have an associated family of spectral curves that are rational. The corresponding quantum spectral curves are given by a family of ODE's with rational coefficients whose monodromy is invariant under the deformations generated by the underlying KP flows. An alternative generating function for the weighted Hurwitz numbers is provided by the multicurrent correlators, which are expressible both as fermionic vacuum expectation values, and directly in terms of the tau function. The WKB series for the Baker function leads to a series of recursion relations between the weighted Hurwitz numbers, fitting within the general framework of the Topological Recursion program. (Based on joint work with A. Alexandrov, G. Chapuy and B. Eynard)

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