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.
Highlights
Duality is one of the most fascinating phenomena in quantum field theory
As an application and as a way to check the consistency of our formula, we study Seiberg duality on the ABJ Wilson loops
Rather than directly evaluating this ABJ matrix integral, we followed [17] and started instead with the Wilson loop in the lens space matrix integral, which is related to the ABJ one by the analytic continuation N2 → −N2
Summary
Duality is one of the most fascinating phenomena in quantum field theory. It provides an alternative, often non-perturbative, understanding of the theory that is not accessible in the original description. The localization method [2,3,4] is a powerful technique applicable to supersymmetric field theory and reduces the infinite dimensional path integral of quantum field theory to a finite integral which can be often regarded as a matrix model integral This method allows for exact computation of quantities such as partition function and Wilson loops at strong coupling, and is an ideal tool for studying non-perturbative physics such as Seiberg duality. These Wilson loops can be computed using the ABJ(M) matrix model [8], and the techniques developed for partition function to study non-perturbative effects can be generalized to Wilson loops, such as the large N analysis [9,46,47], the Fermi gas approach [48], and the cancellation between worldsheet and membrane instantons [49, 50]. Appendices contain further detail of computations in the main text, as well as a discussion of Wilson loops with general representation in Appendix C and an alternative derivation of the Seiberg duality rule using the algebraic approach of [51] in Appendix F
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