Notes on supersymmetric Wilson loops on a two-sphere

TL;DR

The paper analyzes 1/8-BPS Wilson loops on a two-sphere in ${\cal N}=4$ SYM and their AdS duals, establishing that the regularized string action $S_{reg}$ is invariant under area-preserving boundary diffeomorphisms and depends only on the boundary areas via $S_{reg}=-(\sqrt{\lambda}/2\pi)\sqrt{A_1A_2}$. It reveals a deep bridge to an auxiliary ${\cal S}^3$ sigma-model, with a Pohlmeyer reduction to complex-sinh-Gordon structure, enabling construction of new solutions and understanding of stable/unstable pairs. The work also proves that connected correlators of non-coincident loops are captured by degenerate two-disk configurations exchanging supergravity modes, and demonstrates that correlators of coincident loops are described by Hermitian Gaussian two-matrix models in the zero-instanton sector, yielding exact large-$N$, strong-coupling predictions that match bulk saddle point structures. Together, these results provide a coherent, testable framework linking 2d Yang-Mills localization, matrix-model techniques, and string-theoretic constructions for a broad class of supersymmetric Wilson-loop observables. The findings offer practical tools to generate new 1/8-BPS string solutions and deepen the understanding of the strong-coupling dynamics of Wilson-loop correlators in AdS/CFT.

Abstract

We study a recently discovered family of 1/8-BPS supersymmetric Wilson loops in N=4 super Yang-Mills theory and their string theory duals. The operators are defined for arbitrary contours on a two-sphere in space-time, and they were conjectured to be captured perturbatively by 2d bosonic Yang-Mills theory. In the AdS dual, they are described by pseudo-holomorphic string surfaces living on a certain submanifold of AdS_5 x S^5. We show that the regularized area of these string surfaces is invariant under area preserving diffeomorphisms of the boundary loop, in agreement with the conjecture. Further, we find a connection between the pseudo-holomorphicity equations and an auxiliary sigma-model on S^3, which may help to construct new 1/8-BPS string solutions. We also show that the conjectured relation to 2d Yang-Mills implies that a connected correlator of two Wilson loops is computed by a Hermitian Gaussian two-matrix model. On the AdS dual side, we argue that the connected correlator is described by two disconnected disks interacting through the exchange of supergravity modes, and we show that this agrees with the strong coupling planar limit of the two-matrix model.