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Register a new userRoadmap on Wilson loops in 3d Chern-Simons-matter theories
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This is a compact review of recent results on supersymmetric Wilson loops in ABJ(M) and related theories. It aims to be a quick introduction to the state of the art in the field and a discussion of open problems. It is divided into short chapters devoted to different questions and techniques. Some new results, perspectives and speculations are also presented. We hope this might serve as a baseline for further studies of this topic.
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We study the algebra of BPS Wilson loops in 3d gauge theories with N=2 supersymmetry and Chern-Simons terms. We argue that new relations appear on the quantum level, and that in many cases this makes the algebra finite-dimensional. We use our results to propose the mapping of Wilson loops under Seiberg-like dualities and verify that the proposed map agrees with the exact results for expectation values of circular Wilson loops. In some cases we also relate the algebra of Wilson loops to the equivariant quantum K-ring of certain quasi projective varieties. This generalizes the connection between the Verlinde algebra and the quantum cohomology of the Grassmannian found by Witten.
In $mathcal N geq 2$ superconformal Chern-Simons-matter theories we construct the infinite family of Bogomolnyi-Prasad-Sommerfield (BPS) Wilson loops featured by constant parametric couplings to scalar and fermion matter, including both line Wilson loops in Minkowski spacetime and circle Wilson loops in Euclidean space. We find that the connection of the most general BPS Wilson loop cannot be decomposed in terms of double-node connections. Moreover, if the quiver contains triangles, it cannot be interpreted as a supermatrix inside a superalgebra. However, for particular choices of the parameters it reduces to the well-known connections of 1/6 BPS Wilson loops in Aharony-Bergman-Jafferis-Maldacena (ABJM) theory and 1/4 BPS Wilson loops in $mathcal N = 4$ orbifold ABJM theory. In the particular case of $mathcal N = 2$ orbifold ABJM theory we identify the gravity duals of a subset of operators. We investigate the cohomological equivalence of fermionic and bosonic BPS Wilson loops at quantum level by studying their expectation values, and find strong evidence that the cohomological equivalence holds quantum mechanically, at framing one. Finally, we discuss a stronger formulation of the cohomological equivalence, which implies non-trivial identities for correlation functions of composite operators in the defect CFT defined on the Wilson contour and allows to make novel predictions on the corresponding unknown integrals that call for a confirmation.
We construct new families of 1/4 BPS Wilson loops in circular quiver $mathcal N=4$ superconformal Chern-Simons-matter (SCSM) theories in three dimensions. They are defined as the holonomy of superconnections that contain non-trivial couplings to scalar and fermions, and cannot be reduced to block-diagonal matrices. Consequently, the new operators cannot be written in terms of double-node Wilson loops, as the ones considered so far in the literature. For particular values of the couplings the superconnection becomes block-diagonal and we recover the known fermionic 1/4 and 1/2 BPS Wilson loops. The new operators are cohomologically equivalent to bosonic 1/4 BPS Wilson loops and are then amenable of exact evaluation via localization techniques. Moreover, in the case of orbifold ABJM theory we identify the corresponding gravity duals for some of the 1/4 and 1/2 BPS Wilson loops.
We investigate phases of 3d ${cal N}=2$ Chern-Simons-matter theories, extending to three dimensions the celebrated correspondence between 2d gauged Wess-Zumino-Witten (GWZW) models and non-linear sigma models (NLSMs) with geometric targets. We find that although the correspondence in 3d and 2d are closely related by circle compactification, an important subtlety arises in this process, changing the phase structure of the 3d theory. Namely, the effective theory obtained from the circle compactification of a phase of a 3d ${cal N}=2$ gauge theory is, in general, different from the phase of the 3d ${cal N}=2$ theory on ${mathbb R}^2times S^{1}$, which means taking phases of a 3d gauge theory does not necessarily commute with compactification. We compute the Witten index of each effective theory to check this observation. Furthermore, when the matter fields have the same non-minimal charges, the 3d ${cal N}=2$ Chern-Simons-matter theory with a proper Chern-Simons level will decompose into several identical 2d gauged linear sigma models (GLSMs) for the same target upon reduction to 2d. To illustrate this phenomenon, we investigate how vacua of the 3d gauge theory for a weighted projective space $Wmathbb{P}_{[l,cdots,l]}$ move on the field space when we change the radius of $S^{1}$.
We present new circular Wilson loops in three-dimensional N=4 quiver Chern-Simons-matter theory on S^3. At any given node of the quiver, a two-parameter family of operators can be obtained by opportunely deforming the 1/4 BPS Gaiotto-Yin loop. Including then adjacent nodes, the coupling to the bifundamental matter fields allows to enlarge this family and to construct loop operators based on superconnections. We discuss their classification, which depends on both discrete data and continuous parameters subject to an identification. The resulting moduli spaces are conical manifolds, similar to the conifold of the 1/6 BPS loops of the ABJ(M) theory.
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