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An irregular spectral curve for the generation of bipartite maps in topological recursion

Johannes Branahl, Alexander Hock

TL;DR

The paper addresses the lack of a spectral-curve formulation for bipartite map enumeration across genus and boundary lengths. It derives an irregular spectral curve within the topological recursion framework by connecting to the Chapuy-Fang TR-like recursion and performing coordinate transformations, yielding explicit curves $x_{bip}$ and $y_{bip}$ and the key identifications $x_{bip}(z^2)=x_{ord}(z)^2$ and $y_{bip}(z^2)=\frac{y_{ord}(z)}{x_{ord}(z)}$, which enable TR generating functions for bipartite maps with $n$ boundaries. The work situates bipartite map enumeration in the irregular-curve regime of Do and Norbury and links it to Grothendieck’s dessins d’enfant, while demonstrating connections to complex vs Hermitian matrix models and providing concrete quadrangulation examples. It culminates in a robust framework for obtaining generating functions via TR for bipartite maps and suggests avenues for exploring irregular curves beyond the current setting. The results extend the reach of TR to a broader class of combinatorial geometries and offer a unifying perspective on map enumeration across different graph-coloring constraints.

Abstract

We derive an efficient way to obtain generating functions of bipartite maps of arbitrary genus and boundary length using a spectral curve as initial data for the framework of topological recursion. Based on an earlier result of Chapuy and Fang counting these maps and having a structural proximity to topological recursion, we deduce the corresponding spectral curve which has a strong relation to the spectral curve giving rise to generating functions of ordinary maps. In contrast to ordinary maps, the spectral curve is an irregular one in the sense of Do and Norbury. It generalises the irregular curve for the enumeration of Grothendieck's dessins d'enfant.

An irregular spectral curve for the generation of bipartite maps in topological recursion

TL;DR

The paper addresses the lack of a spectral-curve formulation for bipartite map enumeration across genus and boundary lengths. It derives an irregular spectral curve within the topological recursion framework by connecting to the Chapuy-Fang TR-like recursion and performing coordinate transformations, yielding explicit curves and and the key identifications and , which enable TR generating functions for bipartite maps with boundaries. The work situates bipartite map enumeration in the irregular-curve regime of Do and Norbury and links it to Grothendieck’s dessins d’enfant, while demonstrating connections to complex vs Hermitian matrix models and providing concrete quadrangulation examples. It culminates in a robust framework for obtaining generating functions via TR for bipartite maps and suggests avenues for exploring irregular curves beyond the current setting. The results extend the reach of TR to a broader class of combinatorial geometries and offer a unifying perspective on map enumeration across different graph-coloring constraints.

Abstract

We derive an efficient way to obtain generating functions of bipartite maps of arbitrary genus and boundary length using a spectral curve as initial data for the framework of topological recursion. Based on an earlier result of Chapuy and Fang counting these maps and having a structural proximity to topological recursion, we deduce the corresponding spectral curve which has a strong relation to the spectral curve giving rise to generating functions of ordinary maps. In contrast to ordinary maps, the spectral curve is an irregular one in the sense of Do and Norbury. It generalises the irregular curve for the enumeration of Grothendieck's dessins d'enfant.
Paper Structure (7 sections, 4 theorems, 17 equations, 1 table)

This paper contains 7 sections, 4 theorems, 17 equations, 1 table.

Table of Contents

  1. Introduction and Main Result
  2. Proof
  3. Reminder of Previous Results
  4. Proof of the Spectral Curve
  5. Discussion of the Result
  6. On the Irregularity
  7. Example of Quadrangulations

Key Result

Theorem 1.1

The spectral curve $(\overline{\mathbb{C}},x_{ord},y_{ord}, \frac{dz_1\, dz_2}{(z_1-z_2)^2})$ with where computes via TR (see formula BTR-intro) generating functions for the enumeration of ordinary maps with $n$ marked faces of even boundary lengths. The faces have even degrees up to $2d$, where a face of degree $2k$ is weighted by $t_{2k}$.

Theorems & Definitions (4)

  • Theorem 1.1: Eynard:2016yaa
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 2.1: Chapuy2016