Two loop partition function for large N pure Yang-Mills theory on a small three-sphere

TL;DR

This paper computes the two-loop partition function of large N pure Yang-Mills theory on a small three-sphere, revealing the shift in deconfinement temperature and anomalous dimensions, thus validating previous theoretical methods.

Contribution

It provides a direct path-integral two-loop calculation of the partition function, confirming previous results and offering new insights into deconfinement transition and operator dimensions.

Findings

Deconfinement temperature increases with coupling.

Reproduces anomalous dimension sums independently.

Validates methods used for phase transition analysis.

Abstract

We give a direct path-integral calculation of the partition function for pure 3+1 dimensional U(N) Yang-Mills theory at large N on a small three-sphere, up to two-loop order in perturbation theory. From this, we calculate the one-loop shift in the Hagedorn/deconfinement temperature for the theory at small volume, finding that it increases (in units of the inverse sphere radius) as we go to larger coupling (larger volume). Our results also allow us to read off the sum of one-loop anomalous dimensions for all operators with a given engineering dimension in planar Yang-Mills theory on R^4. As checks on our calculation, we reproduce both the Hagedorn shift and some of the anomalous dimension sums by independent methods using the results of hep-th/0412029 and hep-th/0408178. The success of our calculation provides a significant check of methods used in hep-th/0502149 to establish a first order deconfinement transition for pure Yang-Mills theory on a small three-sphere.