$\mathfrak{b}$-Hurwitz numbers from refined topological recursion

Abstract

We prove that single $G$-weighted $\mathfrak{b}$-Hurwitz numbers with internal faces are computed by refined topological recursion on a rational spectral curve, for certain rational weights $G$. Consequently, the $\mathfrak{b}$-Hurwitz generating function analytically continues to a rational curve. In particular, our results cover the cases of $\mathfrak{b}$-monotone Hurwitz numbers, and the enumeration of maps and bipartite maps (with internal faces) on non-oriented surfaces. As an application, we prove that the correlators of the Gaussian, Jacobi and Laguerre $\beta$-ensembles are computed by refined topological recursion.