A supermatrix model for $$ \mathcal{N} $$ = 6 super Chern-Simons-matter theory

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 × 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 ) × 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 ).

BPS Wilson loops and Bremsstrahlung function in ABJ(M): a two loop analysis

“…When M is constant, M = diag(1, 1, −1, −1) and the path is chosen to be a maximal circle on S 2 , we obtain the well studied 1/6−BPS Wilson loop W 1/6 [23][24][25]. Adding local couplings to the fermions allows to generalize the Wilson operator to the holonomy of a superconnection of the U(N |M ) supergroup, leading to an enhanced 1/2−BPS operator W 1/2 [26] (see also [27] for an alternative derivation and [28] for previous attempts).…”

“…The fermionic 1/2-BPS Wilson loop has been proved to be cohomologically equivalent to a linear combination of 1/6-BPS Wilson loops, since their difference is expressible as an exact Q-variation, where Q is the SUSY charge used in localizing the functional integral of the 1/6−BPS operator [26]. Therefore, its vev localizes to the same matrix model and a prediction for its exact value can be easily obtained from the 1/6−BPS vev.…”

“…They generalize the corresponding four-dimensional operators constructed in [43] and are in general 1/6−BPS. 2 For α = π 4 we are back to the 1/2−BPS operator of [26], whereas for α = 0 a new class of three-dimensional Zarembo-like Wilson loops [45] are obtained.…”

Wilson loops in N $$ \mathcal{N} $$ = 6 superspace for ABJM theory

“…However, this is not the only option; one could instead consider the supertrace (see the discussions in refs. [52][53][54][55][56][57][58]). We will restrict to the planar limit, where N → ∞, M → ∞, k → ∞ such that the ratios N k and M k are constant.…”

“…Another way, inspired by the construction (see ref. [55]) of the 1 2 -BPS Wilson loop in ABJM theory, is to use a super-connection 9 in the sense that the gauge part is a (N |M ) × (N |M ) supermatrix.…”