The colored Jones polynomials as vortex partition functions

TL;DR

This paper establishes a novel connection between 3D $ abla=2$ abelian gauge theories and $SU(2)$ Chern-Simons theories, showing that vortex partition functions encode colored Jones polynomials for knots.

Contribution

It constructs explicit 3D abelian gauge theories whose vortex partition functions reproduce colored Jones polynomials, linking gauge theory and knot invariants.

Findings

Vortex partition functions match colored Jones polynomials.

Constructed gauge theories correspond to knot diagrams.

Verified for 2-bridge knots with specific twists.

Abstract

We construct 3D $\mathcal{N}=2$ abelian gauge theories on $\mathbb{S}^2 \times \mathbb{S}^1$ labeled by knot diagrams whose K-theoretic vortex partition functions, each of which is a building block of twisted indices, give the colored Jones polynomials of knots in $\mathbb{S}^3$. The colored Jones polynomials are obtained as the Wilson loop expectation values along knots in $SU(2)$ Chern-Simons gauge theories on $\mathbb{S}^3$, and then our construction provides an explicit correspondence between 3D $\mathcal{N}=2$ abelian gauge theories and 3D $SU(2)$ Chern-Simons gauge theories. We verify, in particular, the applicability of our constructions to a class of tangle diagrams of 2-bridge knots with certain specific twists.