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

@misc{kirillov2026matrix,
  title = {{A Matrix Model for Higher-Genus Fuss--Catalan Numbers}},
  author = {Anatol Kirillov and Ivan Kostov},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.24237},
  eprint = {2605.24237},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.24237},
}

TY  - UNPB
TI  - A Matrix Model for Higher-Genus Fuss--Catalan Numbers
AU  - Anatol Kirillov
AU  - Ivan Kostov
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2605.24237
UR  - https://arxiv.org/abs/2605.24237
AB  - 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.
ER  -