Complex Matrix Models and Statistics of Branched Coverings of 2D Surfaces

TL;DR

This paper introduces a complex matrix gauge model on 2D lattices that counts branched covers of Riemann surfaces and connects to 2D Yang-Mills theory and string descriptions.

Contribution

It formulates a matrix model that solves the combinatorial problem of counting branched covers and relates it to known 2D gauge theories and string theories.

Findings

Provides a matrix model for counting branched covers

Reproduces 2D chiral U(N) Yang-Mills theory in a continuum limit

Connects branched cover enumeration to string theory descriptions

Abstract

We present a complex matrix gauge model defined on an arbitrary two-dimensional orientable lattice. We rewrite the model's partition function in terms of a sum over representations of the group U(N). The model solves the general combinatorial problem of counting branched covers of orientable Riemann surfaces with any given, fixed branch point structure. We then define an appropriate continuum limit allowing the branch points to freely float over the surface. The simplest such limit reproduces two-dimensional chiral U(N) Yang-Mills theory and its string description due to Gross and Taylor.