Comments on knotted 1/2 BPS Wilson loops

TL;DR

This paper explores the relationship between 3D N=2 supersymmetric Chern-Simons theory on squashed spheres and torus knots, analyzing Wilson loops' topology and confirming known BPS conditions with additional insights on level shifts and framing anomalies.

Contribution

It establishes a connection between Wilson loops in supersymmetric Chern-Simons theory and torus knot topology, extending understanding of their geometric and topological properties.

Findings

Wilson loops correspond to U(1) fibers in Seifert fibrations.

The topology of Wilson loops depends on squashing parameters.

Confirmed BPS conditions match known results, with insights on level shift and framing anomaly.

Abstract

In this paper, we show that the localization of three-dimensional N = 2 supersymmetric Chern-Simons theory on the ellipsoid-like squashed sphere is related to a nontrivial knot structure called torus knot. More precisely, we can capture the three sphere as the nontrivial so-called Seifert fibrations by regarding 1/2 BPS Wilson loops as U(1) fibers. The topology of knotted 1/2 BPS Wilson loops is controlled by squashing parameters. We calculate the 1/2 BPS condition of the Wilson loop and find perfect agreement with known results. We also remark on the level shift and framing anomaly.