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

References in corpus (14)

• N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals
• Seiberg Duality in Chern-Simons Theory
• Summing Up All Genus Free Energy of ABJM Matrix Model
• Wilson Loops in Superconformal Chern-Simons Theory and Fundamental Strings in Anti-de Sitter Supergravity Dual
• Brane Tilings and M2 Branes
• Supersymmetric Wilson Loops in N=6 Super Chern-Simons-matter theory
• Three-dimensional N=6 SCFT's and their membrane dynamics
• Exact results on ABJ theory and the refined topological string
• Instanton Effects in Orbifold ABJM Theory
• ABJM Mirrors and a Duality of Dualities
• Wilson loops in supersymmetric Chern-Simons-matter theories and duality
• Chern-Simons and RG Flows: Contact with Dualities
• Vortices and 3 dimensional dualities
• Spin(7) duality for N=1 CS-matter theories

Cited by in corpus (3)

• Probing non-perturbative effects in M-theory
• Instanton Corrections of 1/6 BPS Wilson Loops in ABJM Theory
• Geometric transition in Non-perturbative Topological string