The colored Jones polynomials as vortex partition functions

Abstract

Abstract We construct 3D N $$ \mathcal{N} $$ = 2 abelian gauge theories on S $$ \mathbbm{S} $$ 2 × S $$ \mathbbm{S} $$ 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 S $$ \mathbbm{S} $$ 3. The colored Jones polynomials are obtained as the Wilson loop expectation values along knots in SU(2) Chern-Simons gauge theories on S $$ \mathbbm{S} $$ 3, and then our construction provides an explicit correspondence between 3D N $$ \mathcal{N} $$ = 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.

Keywords

• Chern-Simons Theories
• Duality in Gauge Field Theories
• Supersymmetric Gauge Theory
• Field Theories in Lower Dimensions