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

The strong coupling behavior of finite temperature free energy in N = 4 supersym-metric 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 2 N → 0) ≈ − π 2 6 N 2 V 3 T 4 (1 − 3 2π 2 g 2 N). Comparison with the strong coupling expansion provides further indication that free energy is a smooth monotonic function of the coupling constant.