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arXiv · 2605.24237

A Matrix Model for Higher-Genus Fuss--Catalan Numbers

Anatol Kirillov·Ivan Kostov
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Abstract

The genus-g Fuss--Catalan (FC) number counts the number of ways to obtain a genus-g surface by identifying the edges of a pn-gon via p-valent hyperedges. For p=2 these are the genus--g Catalan numbers which are generated as the trace correlations in the Gaussian matrix model (GUE). Here we construct a simple two-matrix model which generates the higher-genus Fuss-Catalan numbers for any p as the coefficients of its 1/N-expansion. We obtain exact sum rules and an explicit formula for the higher-genus Fuss--Catalan numbers which generalises the Harer--Zagier formula to p>2. We discuss the relation of the higher-genus FC numbers to the intersection numbers and the Euler characteristic of the moduli space of spin-p curves.

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Anatol Kirillov, Ivan Kostov. 2026-05-22. A Matrix Model for Higher-Genus Fuss--Catalan Numbers. https://arxiv.org/abs/2605.24237

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