Wilson loops in supersymmetric Chern-Simons-matter theories and duality

TL;DR

This work constructs and analyzes the algebra of BPS Wilson loops in 3d ${\mathcal N}=2$ Chern–Simons–matter theories, showing that quantum relations truncate the naive representation-ring picture and often yield finite-dimensional algebras. Using $S^3$ localization, it derives explicit relations and shows that the Wilson-loop algebra for a theory with matter is a quotient of the representation ring, with dimension $\binom{k+N_f}{N_c}$ in unitary theories. The authors then prove that GK duality induces an isomorphism between the Wilson-loop algebras of dual pairs, and provide an explicit map of Wilson-loop operators (extending the level-rank transpose rule) that preserves exact circle Wilson-loop vevs. They further connect these algebras to geometric objects via connections to quantum cohomology and equivariant quantum K-theory of Grassmannian-related bundles, offering a unified framework for dualities, loop operators, and topological invariants. The results hold across ${\mathcal N}\ge 3$ theories and extend to ${\mathcal N}=2$ cases (with numerical checks), and they survive deformations to a squashed sphere, underscoring the robustness of the loop-algebra structure and its duality maps.

Abstract

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.