A first order deconfinement transition in large N Yang-Mills theory on a small 3-sphere

TL;DR

This paper analytically demonstrates that large N SU(N) Yang-Mills theory on a small 3-sphere undergoes a first order deconfinement transition at finite temperature, using explicit 3-loop calculations and a novel regularization method.

Contribution

It provides the first analytic proof of a first order deconfinement transition in this setting, extending previous numerical and heuristic results.

Findings

Confirmed the first order nature of the transition through 3-loop calculations.

Introduced a new regularization method for nonabelian gauge theories on 3-spheres.

Conjectured the phase structure's independence from the sphere size.

Abstract

We give an analytic demonstration that the 3+1 dimensional large N SU(N) pure Yang-Mills theory, compactified on a small 3-sphere so that the coupling constant at the compactification scale is very small, has a first order deconfinement transition as a function of temperature. We do this by explicitly computing the relevant terms in the canonical partition function up to 3-loop order; this is necessary because the leading (1-loop) result for the phase transition is precisely on the borderline between a first order and a second order transition. Since numerical work strongly suggests that the infinite volume large N theory also has a first order deconfinement transition, we conjecture that the phase structure is independent of the size of the 3-sphere. To deal with divergences in our calculations, we are led to introduce a novel method of regularization useful for nonabelian gauge theory on a 3-sphere.