ABJ Wilson loops and Seiberg Duality

TL;DR

This paper analyzes supersymmetric Wilson loops in the ABJ theory, providing exact calculations, a nonperturbative completion, and proofs of Seiberg duality mappings between dual theories.

Contribution

It introduces a nonperturbative framework for Wilson loops in ABJ theory and explicitly demonstrates the duality mappings, extending previous work on the partition function.

Findings

Exact Wilson loop calculations in the lens space matrix model.

A nonperturbative completion via integrals that makes series well-defined.

Explicit duality maps exchanging perturbative and nonperturbative contributions.

Abstract

We study supersymmetric Wilson loops in the ${\cal N} = 6$ supersymmetric $U(N_1)_k\times U(N_2)_{-k}$ Chern-Simons-matter (CSM) theory, the ABJ theory, at finite $N_1$, $N_2$ and $k$. This generalizes our previous study on the ABJ partition function. First computing the Wilson loops in the $U(N_1) \times U(N_2)$ lens space matrix model exactly, we perform an analytic continuation, $N_2$ to $-N_2$, to obtain the Wilson loops in the ABJ theory that is given in terms of a formal series and only valid in perturbation theory. Via a Sommerfeld-Watson type transform, we provide a nonperturbative completion that renders the formal series well-defined at all couplings. This is given by ${\rm min}(N_1,N_2)$-dimensional integrals that generalize the "mirror description" of the partition function of the ABJM theory. Using our results, we find the maps between the Wilson loops in the original and Seiberg dual theories and prove the duality. In our approach we can explicitly see how the perturbative and nonperturbative contributions to the Wilson loops are exchanged under the duality. The duality maps are further supported by a heuristic yet very useful argument based on the brane configuration as well as an alternative derivation based on that of Kapustin and Willett.