Tangles, Generalized Reidemeister Moves, and Three-Dimensional Mirror Symmetry

TL;DR

This paper develops a geometric framework using tangles and generalized Reidemeister moves to understand mirror symmetries in three-dimensional superconformal field theories derived from M5-branes on three-manifolds.

Contribution

It introduces a novel approach linking tangles, singularities, and geometric transitions to dualities and mirror symmetries in 3D N=2 theories.

Findings

Generalized Reidemeister moves capture mirror symmetries.

Singularities in tangles correspond to massless matter.

Resolutions of singularities relate to massive deformations.

Abstract

Three-dimensional N=2 superconformal field theories are constructed by compactifying M5-branes on three-manifolds. In the infrared the branes recombine, and the physics is captured by a single M5-brane on a branched cover of the original ultraviolet geometry. The branch locus is a tangle, a one-dimensional knotted submanifold of the ultraviolet geometry. A choice of branch sheet for this cover yields a Lagrangian for the theory, and varying the branch sheet provides dual descriptions. Massless matter arises from vanishing size M2-branes and appears as singularities of the tangle where branch lines collide. Massive deformations of the field theory correspond to resolutions of singularities resulting in distinct smooth manifolds connected by geometric transitions. A generalization of Reidemeister moves for singular tangles captures mirror symmetries of the underlying theory yielding a geometric framework where dualities are manifest.