Wilson Loop Correlator in the AdS/CFT Correspondence

TL;DR

The paper investigates the Gross-Ooguri phase transition for Wilson loop correlators in the AdS/CFT framework by solving the minimal-surface problem for two circular loops in AdS5 and analyzing the transition between connected and disconnected worldsheet topologies. It shows that at infinite coupling the transition is first order with a critical separation L* ≈ 1.04 R, and computes regularized areas to locate the transition point. It then uses a flat-space free-string model to demonstrate that worldsheet fluctuations (finite alpha') convert the semiclassical first-order transition into a crossover, which becomes sharper for larger loop radii. The work highlights potential universal aspects of stringy Wilson loop transitions and the role of alpha' corrections in smoothing semiclassical predictions, with implications for open vs closed string pictures in gauge theories.

Abstract

The AdS/CFT correspondence predicts a phase transition in Wilson loop correlators in the strong coupling N=4, D=4 SYM theory which arises due to instability of the classical string stretched between the loops. We study this transition in detail by solving equations of motion for the string in the particular case of two circular Wilson loops. The transition is argued to be smoothened at finite `t Hooft coupling by fluctuations of the string world sheet and to be promoted to a sharp crossover. Some general comments about Wilson loop correlators in gauge theories are made.

Figures (3)

• Figure 1: Minimal surface spanned by two concentric discs.
• Figure 2: Schematic form of the solution for the minimal surface.
• Figure 3: The superstring amplitude between two circular loops versus $L^2$ at $R^2=20$ (the units are such that $2\pi\alpha'=1$). The thick solid line is $-\ln{\cal A}$. Two other curves represent approximations for the amplitude valid in different phases. The thin solid line is the area of the catenoid with $O(\alpha'^0)$ correction added. The line terminates at the point of instability, $L=L_*^2$. The dashed line is $2\pi R^2-\ln(\hbox{supergraviton exchange})$. The estimate for the crossover point based on matching of the classical actions in the two phases is $L_c\simeq 22$ for the chosen value of $R$. Account of the supergraviton exchange and of the first order in the semiclassical expansion shift the crossover point by about $20\%$. The reason for numerically large deviation from the classical estimate is that the first order corrections are logarithmic in $L^2$.