A supermatrix model for N=6 super Chern-Simons-matter theory

TL;DR

This work constructs a 1/2-BPS Wilson loop in ${\cal N}=6$ super Chern-Simons-matter with gauge group ${U(N)\times U(M)}$, realized as the holonomy of a ${U(N|M)}$ superconnection and defined along both a straight line and a circle. Through a careful supersymmetry analysis, the authors show the loop preserves six Poincaré and six conformal supercharges (and, for the circle, a distinct 1/2-BPS configuration), with couplings to gauge fields, scalars, and bi-fundamental fermions arranged in a way that the supersymmetry variations cancel. They then apply localization to show the circular loop is computed by a supermatrix model, extending Kapustin–Lee–Gaiotto–Witten-type results to the ${U(N|M)}$ context, and connect this to pure Chern-Simons theory with a supergroup on lens space $S^3/\mathbb{Z}_2$. The resulting matrix model provides an exact interpolation in coupling and ranks, matching predictions from the AdS$_4$/CFT$_3$ dual and offering a framework to study other 1D defects in 3D supersymmetric theories. The work also outlines relations to vortex loops and possible generalizations to other 3D theories with lower supersymmetry.

Abstract

We construct the Wilson loop operator of N=6 super Chern-Simons-matter which is invariant under half of the supercharges of the theory and is dual to the simplest macroscopic open string in AdS_4 x CP^3. The Wilson loop couples, in addition to the gauge and scalar fields of the theory, also to the fermions in the bi-fundamental representation of the U(N) x U(M) gauge group. These ingredients are naturally combined into a superconnection whose holonomy gives the Wilson loop, which can be defined for any representation of the supergroup U(N|M). Explicit expressions for loops supported along an infinite straight line and along a circle are presented. Using the localization calculation of Kapustin et al. we show that the circular loop is computed by a supermatrix model and discuss the connection to pure Chern-Simons theory with supergroup U(N|M).