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

We study a -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 Féray on Jack characters. A combinatorial model of non-oriented monotone Hurwitz maps that generalize monotone transposition factorizations is provided. In the case, 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. We also relate the model to an version of the Brézin-Gross-Witten integral, which we solve explicitly in terms of Pfaffians in the case of even multiplicities.