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

TL;DR

This paper introduces a $b$-deformation of monotone Hurwitz numbers using Jack functions, deriving Virasoro constraints, and connecting to integrable hierarchies and matrix models, thus proving several conjectures and providing explicit solutions.

Contribution

It develops a new $b$-deformed model of monotone Hurwitz numbers, establishes Virasoro constraints, and links to BKP hierarchy and $O(N)$ matrix integrals, confirming multiple conjectures.

Findings

Derived evolution equations and Virasoro constraints for the $b$-deformed model.

Proved a conjecture of Féray on Jack characters.

Connected the model to the BKP hierarchy and $O(N)$ matrix integrals, providing explicit Pfaffian solutions.

Abstract

We study a $b$-deformation of monotone Hurwitz numbers, obtained by deforming Schur functions into Jack symmetric functions. We give an evolution equation for this model and derive from it Virasoro constraints, thereby proving a conjecture of F\'eray on Jack characters. A combinatorial model of non-oriented monotone Hurwitz maps which generalizes monotone transposition factorizations is provided. In the case $b=1$ we obtain an explicit Schur expansion of the model and show that it obeys the BKP integrable hierarchy. This Schur expansion also proves a conjecture of Oliveira--Novaes relating zonal polynomials with irreducible representations of $O(N)$. We also relate the model to an $O(N)$ version of the Br\'ezin--Gross--Witten integral, which we solve explicitly in terms of Pfaffians in the case of even multiplicities.