Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter

TL;DR

The paper addresses the problem of computing exact Wilson loop expectation values in supersymmetric Chern-Simons theories with matter. It applies localization on S^3 to reduce the path integral to a matrix model, yielding a concise representation for Z and for the Wilson loop observable, with gauge and matter contributions encoded in determinants over the Cartan subalgebra. The key contributions include the explicit matrix-model reduction for general G and matter representations, exact results in pure Chern-Simons theory for U(N), and perturbative agreement with ABJM, plus a clear framework extendable to related superconformal theories. The work provides a powerful nonperturbative tool for testing holographic dualities and suggests avenues for exact or saddle-point analyses at large N and for general manifolds and loop configurations.

Abstract

We use localization techniques to compute the expectation values of supersymmetric Wilson loops in Chern-Simons theories with matter. We find the path-integral reduces to a non-Gaussian matrix model. The Wilson loops we consider preserve a single complex supersymmetry, and exist in any N=2 theory, though the localization requires superconformal symmetry. We present explicit results for the cases of pure Chern-Simons theory with gauge group U(N), showing agreement with the known results, and ABJM, showing agreement with perturbative calculations. Our method applies to other theories, such as Gaiotto-Witten theories, BLG, and their variants.