Remarks on Two-Loop Free Energy in N=4 Supersymmetric Yang-Mills Theory at Finite Temperature

TL;DR

This paper computes the two-loop free energy of N=4 supersymmetric Yang-Mills theory at finite temperature in the weak coupling regime, revealing a smooth, monotonic behavior across coupling strengths.

Contribution

It provides a detailed two-loop calculation of the free energy at weak coupling, complementing previous strong coupling analyses and suggesting a continuous transition.

Findings

Two-loop free energy approximates to a specific formula at weak coupling.

Comparison indicates free energy varies smoothly with coupling.

Supports the idea of a monotonic free energy function across couplings.

Abstract

The strong coupling behavior of finite temperature free energy in N=4 supersymmetric SU(N) Yang-Mills theory has been recently discussed by Gubser, Klebanov and Tseytlin in the context of AdS-SYM correspondence. In this note, we focus on the weak coupling behavior. As a result of a two-loop computation we obtain, in the large N 't Hooft limit, $F(g^2N\to 0)\approx -\frac{\pi^2}{6}N^2V_3T^4(1-\frac{3}{2\pi^2}g^2N)$. Comparison with the strong coupling expansion provides further indication that free energy is a smooth monotonic function of the coupling constant.