b-monotone Hurwitz numbers: Virasoro constraints, BKP hierarchy, and O(N)-BGW integral

Abstract: My talk will be an introduction to (part of) my recent paper arXiv:2109.01499 written jointly with Valentin Bonzom and Maciej Dołęga. We study a b-deformation of monotone Hurwitz numbers, obtained by deforming Schur functions into Jack symmetric functions. It is a special case of the b-deformed weighted Hurwitz numbers recently introduced by Dołęga and myself and is related branched coverings of the sphere by non-oriented surfaces.

We give an evolution (cut-and-join) equation for the model and we derive, by a method of independent interest, explicit Virasoro constraints from it, for arbitrary values of the deformation parameter b. For b=1 the model is related to the (large) BKP hierarchy and an O(N) version of the BGW integral. The talk will not assume previous knowledge, I will try in particular to explain where the interest of combinatorialists for these deformations come from, and in particular the Goulden-Jackson b-conjecture.

HEP - theorymathematical physicsalgebraic geometrycombinatoricsexactly solvable and integrable systems

Audience: researchers in the discipline

IBS-CGP Mathematical Physics Seminar

Series comments: Registration is required at cgp.ibs.re.kr/activities/talkregistration