Coupling Constant Dependence in the Thermodynamics of N=4 Supersymmetric Yang-Mills Theory

TL;DR

This paper calculates the leading correction to the free energy of N=4 supersymmetric Yang-Mills theory at strong coupling, showing how it approaches the weak coupling limit and exploring similar effects in related conformal theories.

Contribution

It provides the first correction to the strong coupling free energy from string theory effects, revealing how the thermodynamics interpolates between weak and strong coupling regimes.

Findings

The correction to the free energy is positive and scales as (g_{YM}^2 N)^{-3/2}.

The free energy approaches the weak coupling value as the coupling decreases.

Similar corrections are discussed for other conformal brane theories.

Abstract

The free energy of the maximally supersymmetric SU(N) gauge theory at temperature T is expected to scale, in the large N limit, as N^2 T^4 times a function of the 't Hooft coupling, f(g_{YM}^2 N). 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 R^4 terms in the tree level effective action of type IIB string theory. The correction to 3/4 is positive and of order (g_{YM}^2 N)^{-3/2}. Thus, f(g_{YM}^2 N) 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 AdS_3\times S^3.