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

Summary · qwen2.5:32b

The article presents a two-matrix model that generates higher-genus Fuss-Catalan (FC) numbers for any \( p \) as coefficients in its \( 1/N \)-expansion, extending the Gaussian matrix model's role in generating genus-\( g \) Catalan numbers for \( p=2 \). This work is significant as it provides exact sum rules and an explicit formula for higher-genus FC numbers, generalizing the Harer-Zagier formula to values of \( p > 2 \), and explores their relation to intersection numbers and the Euler characteristic of moduli spaces.

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Discovering a new matrix model for higher-genus Fuss--Catalan numbers in mathematics.

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arXiv:2605.24237v2 Announce Type: replace-cross 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.