1/4 BPS circular loops, unstable world-sheet instantons and the matrix model

TL;DR

The paper analyzes a family of 1/4-BPS circular Wilson loops in N=4 SYM parameterized by $\theta_0$, uncovering a two-saddle structure both at weak and strong coupling. Perturbative gauge theory reduces to a Gaussian matrix model with $\lambda' = \lambda\cos^2\theta_0$, whose large-$\lambda'$ expansion exhibits two saddle points, mirroring the dual AdS analysis in which two supersymmetric string embeddings yield actions $\pm\cos\theta_0\sqrt{\lambda}$. The strong-coupling results from the string description reproduce the matrix-model two-saddle behavior, with the leading terms $\exp(\sqrt{\lambda'})$ and a subleading $\exp(-\sqrt{\lambda'})$ contribution, consistent with SUSY and fluctuation analyses. These findings suggest a BMN-like limit for nearly BPS Wilson loops and point toward a deeper localization-like mechanism governing these observables, uniting gauge theory perturbation theory, matrix-model results, and string theory predictions across regimes.

Abstract

The standard prescription for computing Wilson loops in the AdS/CFT correspondence in the large coupling regime and tree-level involves minimizing the string action. In many cases the action has more than one saddle point as in the simple example studied in this paper, where there are two 1/4 BPS string solutions, one a minimum and the other not. Like in the case of the regular circular loop the perturbative expansion seems to be captured by a free matrix model. This gives enough analytic control to extrapolate from weak to strong coupling and find both saddle points in the asymptotic expansion of the matrix model. The calculation also suggests a new BMN-like limit for nearly BPS Wilson loop operators.