Super Yang-Mills Theory at Finite Temperature

The AdS/CFT correspondence provides valuable constraints on the possible exact form of various physical quantities in the N = 4 super Yang-Mills theory in the large N limit. We examine the free energy as the expansions in a small as well as in a large ’t Hooft parameter λ. We argue that it is impossible to smoothly extrapolate from the weak coupling regime to the strong coupling regime, thus there must exist a large N phase transition in λ at a finite temperature. We also argue that there is no world-sheet instanton in the background of the Euclidean anti-de Sitter black hole.