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

Abstract

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 the last two authors and has an interpretation in terms of generalized branched coverings of the sphere by non-oriented surfaces. We give an evolution (cut-and-join) equation for this model and we derive, by a method of independent interest, explicit Virasoro constraints from it, for arbitrary values of the deformation parameter $b$. We apply them to prove a conjecture of Féray on Jack characters. We also provide a combinatorial model of non-oriented monotone Hurwitz maps, which generalizes monotone transposition factorizations. In the case $b=1$ we show that the model obeys the BKP hierarchy of Kac and Van de Leur. As a consequence of our analysis we prove a recent conjecture of Oliveira and Novaes relating zonal polynomials with the dimensions of irreducible representations of $O(N)$. We also relate the model to an $O(N)$ version of the Brézin-Gross-Witten integral, which we solve explicitly in terms of Pfaffians in the case of even multiplicities.