$\mathfrak{b}$-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.

v2: 41 pages, 2 figures, bibliography updated

References in corpus (25)

• Liouville Correlation Functions from Four-dimensional Gauge Theories
• Matrix Models for Beta Ensembles
• Hurwitz numbers and intersections on moduli spaces of curves
• Asymptotic expansion of beta matrix models in the one-cut regime
• Free energy topological expansion for the 2-matrix model
• Matrix eigenvalue model: Feynman graph technique for all genera
• Virasoro constraints and topological recursion for Grothendieck's dessin counting
• Orbifold Hurwitz numbers and Eynard-Orantin invariants
• Weighted Hurwitz numbers and topological recursion: an overview
• Generating functions for weighted Hurwitz numbers
• Topological recursion for Kadomtsev-Petviashvili tau functions of hypergeometric type
• A bijection for rooted maps on general surfaces
• Large $N$ expansions for the Laguerre and Jacobi $\beta$ ensembles from the loop equations
• Higher Airy structures and topological recursion for singular spectral curves
• Double Hurwitz numbers: polynomiality, topological recursion and intersection theory
• $b$-monotone Hurwitz numbers: Virasoro constraints, BKP hierarchy, and $O(N)$-BGW integral
• Cut-and-join equation for monotone Hurwitz numbers revisited
• Whittaker vectors for $\mathcal{W}$-algebras from topological recursion
• Quantum curves from refined topological recursion: the genus 0 case
• Symplectic duality for topological recursion
• Taking limits in topological recursion
• Refined topological recursion revisited -- properties and conjectures
• Deformation and quantisation condition of the $\mathcal{Q}$-top recursion
• Whittaker vectors at finite energy scale, topological recursion and Hurwitz numbers
• An irregular spectral curve for the generation of bipartite maps in topological recursion