Effective Action and Phase Transitions in Thermal Yang-Mills Theory on Spheres

TL;DR

This paper investigates the phase structure of finite-temperature Yang-Mills theory on curved spacetime, revealing a second-order phase transition characterized by specific heat behavior and monopole-antimonopole configurations.

Contribution

It provides an exact computation of heat kernels, analyzes the gluon operator's negative modes, and characterizes the phase transition in Yang-Mills theory on $S^1\times S^1\times S^2$ with monopole-antimonopole backgrounds.

Findings

Heat capacity exhibits a second-order phase transition near $T_c$.

Gluon operator has negative modes on the considered background.

Monopole-antimonopole pairs are essential in covariantly constant fields.

Abstract

We study the covariantly constant Savvidy-type chromomagnetic vacuum in finite-temperature Yang-Mills theory on the four-dimensional curved spacetime. Motivated by the fact that a positive spatial curvature acts as an effective gluon mass we consider the compact Euclidean spacetime $S^1\times S^1\times S^2$, with the radius of the first circle determined by the temperature $a_1=(2\pi T)^{-1}$. We show that covariantly constant Yang-Mills fields on $S^2$ cannot be arbitrary but are rather a collection of monopole-antimonopole pairs. We compute the heat kernels of all relevant operators exactly and show that the gluon operator on such a background has negative modes for any compact semi-simple gauge group. We compute the infrared regularized effective action and apply the result for the computation of the entropy and the heat capacity of the quark-gluon gas. We compute the heat capacity for the gauge group SU(2N) for a field configuration of $N$ monopole-antimonopole pairs. We show that in the high-temperature limit the heat capacity is well defined in the infrared limit and exhibits a typical behavior of second-order phase transition $\sim (T-T_c)^{-3/2}$ with the critical temperature $T_c=(2\pi a)^{-1}$, where $a$ is the radius of the 2-sphere $S^2$.