b-Monotone Hurwitz Numbers: Virasoro Constraints, BKP Hierarchy, and O(N)-BGW Integral

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 Feray on Jack characters. A combinatorial model of non-oriented monotone Hurwitz maps that generalize 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 Brezin-Gross-Witten integral, which we solve explicitly in terms of Pfaffians in the case of even multiplicities.