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

TL;DR

This paper investigates the algebraic structure of BPS Wilson loops in 3d N=2 supersymmetric Chern-Simons-matter theories, revealing quantum relations that often make the algebra finite-dimensional and exploring their duality mappings.

Contribution

It introduces new quantum relations in the Wilson loop algebra and proposes a duality map consistent with exact expectation values, extending connections to quantum K-theory.

Findings

Quantum relations make Wilson loop algebra finite-dimensional in many cases.

The proposed duality map matches exact Wilson loop expectation values.

Connections established between Wilson loop algebra and equivariant quantum K-theory.

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.