Wilson Loops in ${\cal N}=4$ Supersymmetric Yang-Mills Theory from Random Matrix Theory

TL;DR

The paper investigates the universality of circular Wilson loops in ${\cal N}=4$ SYM by recasting the problem in a Random Matrix Theory framework with potential $V(M)=\sum_{k} \frac{g_k}{k}M^k$ and using the loop insertion operator to generate all genus $1/N^2$ corrections. It shows that the leading $\lambda$-growth, $W_0(x) \sim c\, \lambda^{-3/4} e^{x\sqrt{\lambda}}$, is universal across a broad class of potentials, while higher-genus corrections depend only on a finite set of moments $M_k$ and $J_k$, allowing exact computation of all $k$-point Wilson loop correlators to all orders in $1/N^2$. The authors provide explicit universal expressions for Wilson-loop kernels in terms of modified Bessel functions, e.g. $W_0(x,y)=\frac{a x y}{2(x+y)}\left[I_0(a y) I_1(a x) + I_0(a x) I_1(a y)\right]$ and $W_1(x)=\frac{x^2}{12 M_1} I_2(ax) - \frac{M_2}{8(M_1)^2} x I_1(ax)$, with the Gaussian case recovering known results. They further discuss multicritical universality, where $W_0(x) \propto I_{k+1}(ax)/(ax)^{k+1}$, preserving the exponential growth and indicating potential connections to different conformal points, though no definitive interpretation is provided.

Abstract

Based on the AdS/CFT correspondence, string theory has given exact predictions for circular Wilson loops in U(N) ${\cal N}=4$ supersymmetric Yang-Mills theory to all orders in a 1/N expansion. These Wilson loops can also be derived from Random Matrix Theory. In this paper we show that the result is generically insensitive to details of the Random Matrix Theory potential. We also compute all higher $k$-point correlation functions, which are needed for the evaluation of Wilson loops in arbitrary irreducible representations of U(N).