The colored Jones polynomials as vortex partition functions

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.

35 pages, 9 figures; v2: minor corrections, references added; v3: minor changes, references added, published version

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• Multi-cover skeins, quivers, and 3d $\mathcal{N}=2$ dualities
• On the quantum sl_2 invariants of knots and integral homology spheres
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