The geometry of the light-cone cell decomposition of moduli space

TL;DR

This paper explores a geometric cell decomposition of the moduli space of Riemann surfaces with punctures, using Nakamura graphs and permutation classes to parametrize cells as convex polytopes.

Contribution

It introduces a detailed geometric framework for moduli space decomposition based on light-cone string parameters and graph automorphisms, with explicit low-genus examples.

Findings

Cells are convex polytopes defined by linear systems.

Parametrization uses permutation classes and graph automorphisms.

Explicit low-genus examples demonstrate the decomposition.

Abstract

The moduli space of Riemann surfaces with at least two punctures can be decomposed into a cell complex by using a particular family of ribbon graphs called Nakamura graphs. We distinguish the moduli space with all punctures labelled from that with a single labelled puncture. In both cases, we describe a cell decomposition where the cells are parametrised by graphs or equivalence classes of finite sequences (tuples) of permutations. Each cell is a convex polytope defined by a system of linear equations and inequalities relating light-cone string parameters, quotiented by the automorphism group of the graph. We give explicit examples of the cell decomposition at low genus with few punctures.