Permutation combinatorics of worldsheet moduli space

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.

52 pages, 21 figures; minor corrections, "On the" dropped from title, matches published version

References in corpus (7)

• From Matrix Models and quantum fields to Hurwitz space and the absolute Galois group
• What is the Simplest Gauge-String Duality?
• Matrix string theory, contact terms, and superstring field theory
• Topological String Correlators from Matrix Models
• Thresholds of Large N Factorization in CFT4 : Exploring bulk spacetime in AdS5
• Metastring Theory and Modular Space-time
• Fatgraph Algorithms and the Homology of the Kontsevich Complex

Cited by in corpus (4)

• Metastring Theory and Modular Space-time
• Gauge Invariants and Correlators in Flavoured Quiver Gauge Theories
• Multi-trace Correlators from Permutations as Moduli Space
• The geometry of the light-cone cell decomposition of moduli space