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

TL;DR

The paper addresses how the free energy of large-$N$ ${ m N}=4$ SYM at finite temperature depends on the 't Hooft coupling. It computes the leading strong-coupling correction from the Type IIB $ ext{R}^4$ term by evaluating on the near-extremal D3-brane background, finding $f(g_{YM}^2N)=rac{3}{4}+rac{45}{32} ext{ζ}(3)(2g_{YM}^2N)^{-3/2}+...$ and showing the correction is positive, driving $f$ toward the weak-coupling value $1$ as coupling decreases. A perturbative check confirms that including backreaction does not change this first-order result. The authors also explore corrections for other coincident-brane CFTs: the D1–D5 system shows vanishing leading corrections due to Weyl-flat AdS$_3 imes S^3$, while M-brane theories exhibit explicit $1/N$ corrections, e.g., $F=-V_5T^6(a_0N^3+a_1N)$ for M5 and $F=-V_2T^3(b_0N^{3/2}+b_1N^{1/2})$ for M2 with provided coefficients. These results extend the holographic understanding of coupling and finite-$N$ effects in strongly coupled CFTs.

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.