Wilson Loops and Minimal Surfaces

TL;DR

This paper explores the relationship between Wilson loops in N=4 supersymmetric gauge theory and minimal surfaces in AdS_5 x S^5, analyzing their properties, equations, and symmetries within the AdS/CFT framework.

Contribution

It formulates the loop equation for BPS Wilson loops and demonstrates how minimal surfaces in AdS_5 x S^5 satisfy these equations, elucidating the geometric dual of gauge theory loops.

Findings

Minimal surfaces in AdS_5 x S^5 solve the loop equation for BPS loops.

Zig-zag symmetry is preserved when scalar coupling is absent.

BPS loops are free from ultra-violet divergences.

Abstract

The AdS/CFT correspondence suggests that the Wilson loop of the large N gauge theory with N=4 supersymmetry in 4 dimensions is described by a minimal surface in AdS_5 x S^5. We examine various aspects of this proposal, comparing gauge theory expectations with computations of minimal surfaces. There is a distinguished class of loops, which we call BPS loops, whose expectation values are free from ultra-violet divergence. We formulate the loop equation for such loops. To the extent that we have checked, the minimal surface in AdS_5 x S^5 gives a solution of the equation. We also discuss the zig-zag symmetry of the loop operator. In the N=4 gauge theory, we expect the zig-zag symmetry to hold when the loop does not couple the scalar fields in the supermultiplet. We will show how this is realized for the minimal surface.