Weighted Hurwitz numbers, tau functions and topological recursion

04/12/2016 - 15:30
04/12/2016 - 16:30
Speaker: 
John Harnad, Concordia University et CRM
Location: 
CRM, UdeM, Pavillon André-Aisenstadt, 2920, ch. de la Tour, salle 4336
Abstract: 

 Hurwitz numbers are classical combinatorial/geometric invariants that enumerate branched coverings of the Riemann sphere (or other Riemann surfaces). Combinatorially, they are understood equivalently as enumerating factorizations of elements of the symmetric group with given cycle structure. Extending results of Okounkov and Pandharipande on the use of KP tau-functions as generating functions for (single or double) Hurwitz numbers with only simple branching at all but one or two specified branch points to the general weighted case, with weights determined by a generating function with infinite parametric dependence, a KP tau-function of hypergeometric type is constructed which is shown to be the generating function for the latter.

This may be further generalized to include “quantum weighted” Hurwitz numbers, in terms of Macdonald polynomials, in which the weighted distribution is closely related to that for a quantum Bose gas with linear energy spectrum. An alternative type of generating function for weighted Hurwitz numbers may be constructed from the associated current correlation functions. These are shown to satisfy the equations of the Eynard-Orantin topological recursion scheme. The associated quantum spectral curve is derived, and the KP flows shown to be isomonodromic deformations of the corresponding differential operator in the spectral parameter.Hurwitz numbers are classical combinatorial/geometric invariants that enumerate branched coverings of the Riemann sphere (or other Riemann surfaces). Combinatorially, they are understood equivalently as enumerating factorizations of elements of the symmetric group with given cycle structure. Extending results of Okounkov and Pandharipande on the use of KP tau-functions as generating functions for (single or double) Hurwitz numbers with only simple branching at all but one or two specified branch points to the general weighted case, with weights determined by a generating function with infinite parametric dependence, a KP tau-function of hypergeometric type is constructed which is shown to be the generating function for the latter.

This may be further generalized to include “quantum weighted” Hurwitz numbers, in terms of Macdonald polynomials, in which the weighted distribution is closely related to that for a quantum Bose gas with linear energy spectrum. An alternative type of generating function for weighted Hurwitz numbers may be constructed from the associated current correlation functions. These are shown to satisfy the equations of the Eynard-Orantin topological recursion scheme. The associated quantum spectral curve is derived, and the KP flows shown to be isomonodromic deformations of the corresponding differential operator in the spectral parameter.

Last edited by on Thu, 04/07/2016 - 13:57