Two-dimensional gauge theories of the symmetric group S(n) and branched n-coverings of Riemann surfaces in the large-n limit

TL;DR

This paper explores the large-n limit of 2D lattice gauge theories based on the symmetric group S(n), revealing a complex phase diagram with multiple transitions and connections to random walk and graph theories.

Contribution

It introduces a novel analysis of branched n-coverings via S(n) gauge theories, uncovering rich phase structures and links to other mathematical fields.

Findings

Identified multiple phase transitions in the large-n limit.

Characterized phases by connectivity properties of coverings.

Connected gauge theory behavior to random walks and graph theory.

Abstract

Branched n-coverings of Riemann surfaces are described by a 2d lattice gauge theory of the symmetric group S(n) defined on a cell discretization of the surface. We study the theory in the large-n limit, and we find a rich phase diagram with first and second order transition lines. The various phases are characterized by different connectivity properties of the covering surface. We point out some interesting connections with the theory of random walks on group manifolds and with random graph theory.