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Seminar by Leonid Chekhov

Title: Hypergeometric Hurwitz numbers: KP tau-functions and matrix models

2014.09.29 | Jane Jamshidi

Date Wed 15 Oct
Time 16:15 17:15
Location 1531-215 (Aud. D3)


We begin with constructing a matrix-model representation for the generating function of numbers of Belyi morphisms, clean Belyi morphisms, and two-profile Belyi morphisms. These generating functions can be reformulated in terms of fat graphs on Riemann surfaces and are correspondingly described by Hermitian one-matrix model with logarithmic addition to the potential, by the Kontsevich--Penner matrix model, and by the Generalized Kontsevich matrix model thus being tau-functions of the KP hierarchy. We extend these results to hypergeometric Hurwitz numbers with arbitrary (but fixed) number $n$ of branching points and two fixed profiles. Under some restrictions on the corresponding generating functions, we were able to present them in the form of a special chain of matrices, which admits solution in terms of the topological recursion procedure. As an example, I present the spectral curve calculation for the case $n=4$. (based on two recent joint works with J.Ambjorn, NBI, Copenhagen)