Multi-cover skeins, quivers, and 3d $\mathcal{N}=2$ dualities

TL;DR

This paper extends the knot-quiver correspondence to include multi-cover skein relations, revealing new dualities in 3d $ abla=2$ theories through quantum algebra and M-theory embedding.

Contribution

It introduces a generalized multi-cover skein relation for boundaries of holomorphic disks, leading to new dual quiver descriptions and 3d $ abla=2$ dualities.

Findings

Dual quiver descriptions of geometry via multi-cover skein relations

Formulation of relations using quantum torus algebras

Connection to wall-crossing identities of Kontsevich and Soibelman

Abstract

The relation between open topological strings and representation theory of symmetric quivers is explored beyond the original setting of the knot-quiver correspondence. Multiple cover generalizations of the skein relation for boundaries of holomorphic disks on a Lagrangian brane are observed to generate dual quiver descriptions of the geometry. Embedding into M-theory, a large class of dualities of 3d $\mathcal{N}=2$ theories associated to quivers is obtained. The multi-cover skein relation admits a compact formulation in terms of quantum torus algebras associated to the quiver and in this language the relations are similar to wall-crossing identities of Kontsevich and Soibelman.