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

Nitin Kumar Chidambaram, Maciej Dołęga, Kento Osuga

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 $β$-ensembles are computed by refined topological recursion.

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

Abstract

We prove that single -weighted -Hurwitz numbers with internal faces are computed by refined topological recursion on a rational spectral curve, for certain rational weights . Consequently, the -Hurwitz generating function analytically continues to a rational curve. In particular, our results cover the cases of -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 -ensembles are computed by refined topological recursion.

Paper Structure

This paper contains 26 sections, 31 theorems, 122 equations, 2 figures.

Table of Contents

  1. Introduction
  2. $\mathfrak{b}$-Hurwitz numbers
  3. Jack polynomials
  4. $\mathfrak{b}$-Hurwitz numbers: combinatorial interpretation
  5. $\mathfrak{b}$-Hurwitz numbers with internal faces
  6. Constraints for $\tau^{(\mathfrak{b})}_{G}$
  7. Refined topological recursion
  8. Refined spectral curve
  9. Refined topological recursion
  10. Variational formula
  11. $\mathfrak{b}$-Hurwitz numbers from refined topological recursion
  12. Formal loop equations
  13. The spectral curve
  14. Refined topological recursion
  15. Extension: inserting internal faces
  16. ...and 11 more sections

Key Result

Theorem 1.1

For each of the weights $G(z)$ given in eq:allowedweights, the refined topological recursion correlators $\omega_{g,n}$ on the corresponding refined spectral curve $\mathcal{S}_{\bm \mu}$ are generating functions for $G$-weighted $\mathfrak{b}$-Hurwitz numbers. More precisely, for any $(g,n) \in \fr where the $F_{g,n}$ are the expansion coefficients of the tau function $\tau^{(\mathfrak{b})}_G$ in

Figures (2)

  • Figure 1: On the left hand side we show a graphical representation of the conditions HM4:(i)+(ii) and HM4:(iii) for a monotone Hurwitz map represented as a ribbon graph. In the middle we represent a monotone Hurwitz map embedded into the Klein bottle. On the right hand side we show the same map as a ribbon graph.
  • Figure 2: A bipartite map on the torus with three boundaries of degrees $4,9,9$, and with internal faces of degree at most $3$, so that it contributes to the coefficient of $[X_1^{-5}X_2^{-10}X_3^{-10}]$ in $\phi_{1,3}^{{\rm bip},3}$. The marked corners are indicated by blue arrows, and the root is indicated by the yellow oriented arrow.

Theorems & Definitions (64)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Remark 2.1
  • Definition 2.2: Colored monotone Hurwitz map
  • Example 2.3
  • Definition 2.4
  • Example 2.5
  • Theorem 2.6
  • Definition 2.7
  • ...and 54 more