Coupling constant dependence in the thermodynamics of script N = 4 supersymmetric Yang-Mills theory

The free energy of the maximally supersymmetric SU(N) gauge theory at temperature T is expected to scale, in the large-N limit, as N²T⁴ times a function of the 't Hooft coupling, f(g²_YMN). In the strong coupling limit the free energy has been deduced from the near-extremal 3-brane geometry, and its normalization has turned out to be 3/4 times that found in the weak coupling limit. In this paper we calculate the leading correction to this result in inverse powers of the coupling, which originates from the R⁴ terms in the tree-level effective action of type IIB string theory. The correction to 3/4 is positive and of order (g²_YMN)^-3/2. Thus, f(g²_YMN) increases as the 't Hooft coupling is decreased, in accordance with the expectation that it should be approaching 1 in the weak coupling limit. We also discuss similar corrections for other conformal theories describing coincident branes. In particular, we suggest that the coupling independence of the near extremal entropy for D1-branes bound to D5-branes is related to the vanishing of the Weyl tensor of AdS₃ × S³.