Permutation combinatorics of worldsheet moduli space

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.