Thermodynamics of large-N gauge theories on a sphere: weak versus strong coupling

Recently lattice simulation in pure Yang-Mills theory exposes significant quadratic corrections for both the thermodynamic quantities and the renormalized Polyakov loop in the deconfined phase. These terms are previously found to appear naturally for N=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=4 $$\end{document} Super Yang-Mills (SYM) on a sphere at strong coupling, through the gauge/gravity duality. Here we extend the investigation to the weak coupling regime, and for general large-N gauge theories. Employing the matrix model description, we find some novel behavior in the deconfined phase, which is not noticed in the literature. Due to the non-uniform eigenvalue distribution of the holonomy around the time circle, the deviation of the Polyakov loop from one starts from 1/T3 instead of 1/T2. Such a power is fixed by the space dimension and do not change with different theories. This statement is also true when perturbative corrections to the single-particle partition functions are included. The corrections to the Polyakov loop and higher moments of the distribution function combine to give a universal term, T/4, in the free energy. These differences between the weak and strong coupling regime could be easily explained if a strong/weak coupling phase transition occurs in the deconfined phase of large-N gauge theories on a compact manifold.