Notes on supersymmetric Wilson loops on a two-sphere

TL;DR

This paper investigates a family of supersymmetric Wilson loops on a two-sphere in N=4 super Yang-Mills theory, exploring their string duals, invariance properties, and connections to 2D Yang-Mills and matrix models, revealing new insights into their structure and correlations.

Contribution

It demonstrates the invariance of string surface areas under area-preserving diffeomorphisms and links the Wilson loop correlators to a two-matrix model, advancing understanding of their dual descriptions.

Findings

String surface areas are invariant under area-preserving diffeomorphisms.

Connected Wilson loop correlators are computed by a Hermitian two-matrix model.

Strong coupling limit matches the two-matrix model predictions.

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.