Thermal Yang-Mills Theory In the Einstein Universe

TL;DR

This paper analyzes the stability of non-Abelian chromomagnetic vacua in Yang-Mills theory on a Euclidean Einstein universe, identifying stable configurations and exploring thermodynamic properties at various scales.

Contribution

It provides a detailed heat kernel analysis for Yang-Mills fields on $S^3$, identifying conditions for stability and computing thermodynamic asymptotics.

Findings

Only fundamental SU(2) representations yield stable configurations.

All other configurations contain negative modes and are unstable.

Energy density exhibits a non-trivial minimum at finite sphere radius.

Abstract

We study the stability of a non-Abelian chromomagnetic vacuum in Yang-Mills theory in Euclidean Einstein universe $S^1\times S^3$. We assume that the gauge group is a simple compact group $G$ containing the group SU(2) as a subgroup and consider static covariantly constant gauge fields on $S^3$ taking values in the adjoint representation of the group $G$ and forming a representation of the group $SU(2)$. We compute the heat kernel for the Laplacian acting on fields on $S^3$ in an arbitrary representation of SU(2) and use this result to compute the heat kernels for the gluon and the ghost operators and the one-loop effective action. We show that the only configuration of the covariantly constant Yang-Mills background that is stable is the one that contains only spinor (fundamental) representations of the group SU(2); all other configurations contain negative modes and are unstable. For the stable configuration we compute the asymptotics of the effective action, the energy density, the entropy and the heat capacity in the limits of low/high temperature and small/large volume and show that the energy density has a non-trivial minimum at a finite value of the radius of the sphere $S^3$.