Wilson loops on three-manifolds and their M2-brane duals

TL;DR

This paper calculates the large N limit of Wilson loop expectation values in N=2 supersymmetric gauge theories on three-manifolds, revealing a simple geometric dependence and matching supergravity duals involving M2-branes.

Contribution

It provides a closed-form formula for Wilson loops on general three-manifolds and establishes a precise match with M2-brane supergravity duals, highlighting geometric dependence.

Findings

Wilson loop expectation values depend only on a supersymmetric Killing vector.

The supergravity dual involves M2-branes wrapping the M-theory circle and a complex curve.

The regularized M2-brane action matches the field theory computation.

Abstract

We compute the large N limit of Wilson loop expectation values for a broad class of N=2 supersymmetric gauge theories defined on a general class of background three-manifolds M_3, diffeomorphic to S^3. We find a simple closed formula which depends on the background geometry only through a certain supersymmetric Killing vector field. The supergravity dual of such a Wilson loop is an M2-brane wrapping the M-theory circle, together with a complex curve in a self-dual Einstein manifold M_4, whose conformal boundary is M_3. We show that the regularized action of this M2-brane also depends only on the supersymmetric Killing vector, precisely reproducing the large N field theory computation.