Permutation combinatorics of worldsheet moduli space

TL;DR

This paper explores the combinatorics of Nakamura graphs related to light-cone string diagrams, connecting them to permutation tuples and Belyi maps, enabling high-genus cell counting via matrix models and group theory.

Contribution

It introduces a new permutation-based combinatorial framework for Nakamura graphs, linking them to branched covers and Belyi maps, and develops efficient enumeration methods.

Findings

Established a relation between Nakamura graphs and Belyi maps.

Derived analytic formulas for cell counting at high genus.

Implemented enumeration algorithms using GAP software.

Abstract

Light-cone string diagrams have been used to reproduce the orbifold Euler characteristic of moduli spaces of punctured Riemann surfaces at low genus and with few punctures. Nakamura studied the meromorphic differential introduced by Giddings and Wolpert to characterise light-cone diagrams and introduced a class of graphs related to this differential. These Nakamura graphs were used to parametrise the cells in a light-cone cell decomposition of moduli space. We develop links between Nakamura graphs and realisations of the worldsheet as branched covers. This leads to a development of the combinatorics of Nakamura graphs in terms of permutation tuples. For certain classes of cells, including those of top dimension, there is a simple relation to Belyi maps, which allows us to use results from Hermitian and complex matrix models to give analytic formulae for the counting of cells at arbitrarily high genus. For the most general cells, we develop a new equivalence relation on Hurwitz classes which organises the cells and allows efficient enumeration of Nakamura graphs using the group theory software GAP.