Two-dimensional gauge theories of the symmetric group S_n in the large-n limit

TL;DR

This paper investigates the large-n limit of 2D gauge theories with the symmetric group S_n, revealing phase transitions related to surface connectivity, random walks, and connections to Yang-Mills theory.

Contribution

It introduces a novel analysis of phase structures in S_n gauge theories on Riemann surfaces and links these to random walk cutoff phenomena and phase transitions in Yang-Mills theory.

Findings

Identification of phase transitions related to surface connectivity.

Connection between gauge theory phases and random walk cutoff phenomena.

Discovery of a cutoff transition in large N 2D Yang-Mills with scaled coupling.

Abstract

We study the two-dimensional gauge theory of the symmetric group S_n describing the statistics of branched n-coverings of Riemann surfaces. We consider the theory defined on the disk and on the sphere in the large-n limit. A non trivial phase structure emerges, with various phases corresponding to different connectivity properties of the covering surface. We show that any gauge theory on a two-dimensional surface of genus zero is equivalent to a random walk on the gauge group manifold: in the case of S_n, one of the phase transitions we find can be interpreted as a cutoff phenomenon in the corresponding random walk. A connection with the theory of phase transitions in random graphs is also pointed out. Finally we discuss how our results may be related to the known phase transitions in Yang-Mills theory. We discover that a cutoff transition occurs also in two dimensional Yang-Mills theory on a sphere, in a large N limit where the coupling constant is scaled with N with an extra log N compared to the standard 't Hooft scaling.