# 𝔟-Hurwitz numbers from refined topological recursion

## TL;DR

This paper shows how a mathematical technique called refined topological recursion can compute certain types of Hurwitz numbers and other related quantities.

## Contribution

The paper proves that refined topological recursion computes b-Hurwitz numbers and their generating functions for specific weights.

## Key findings

- b-Hurwitz numbers with internal faces are computed using refined topological recursion on a rational spectral curve.
- The generating function for b-Hurwitz numbers analytically continues to a rational curve.
- Correlators of Gaussian, Jacobi, and Laguerre β-ensembles are computed using refined topological recursion.

## Abstract

We prove that single G-weighted \documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak {b}$$\end{document}b-Hurwitz numbers with internal faces are computed by refined topological recursion on a rational spectral curve, for certain rational weights G. Consequently, the \documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak {b}$$\end{document}b-Hurwitz generating function analytically continues to a rational curve. In particular, our results cover the cases of \documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak {b}$$\end{document}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 \documentclass[12pt]{minimal}
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\begin{document}$$\beta $$\end{document}β-ensembles are computed by refined topological recursion.