ABJ Wilson loops and Seiberg Duality

Abstract

We study supersymmetric Wilson loops in the $\mathcal{N}=6$ 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 $\min (N_1,N_2)$-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.

70 pages. v2: some text corrections, references added

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• Instanton Corrections of 1/6 BPS Wilson Loops in ABJM Theory
• Non-perturbative Tests of Duality Cascades in Three Dimensional Supersymmetric Gauge Theories
• Exact partition function in $U(2)\times U(2)$ ABJM theory deformed by mass and Fayet-Iliopoulos terms
• Geometric transition in Non-perturbative Topological string
• Exact results and Schur expansions in quiver Chern-Simons-matter theories
• Mass-deformed ABJ and ABJM theory, Meixner-Pollaczek polynomials, and $su(1,1)$ oscillators
• Dualities and loops on squashed $S^3$
• N=(4,4) Vector Multiplets on Curved Two-Manifolds