Quivers for 3-manifolds: the correspondence, BPS states, and 3d $\mathcal{N}$=2 theories

TL;DR

This paper establishes a novel correspondence between quivers and 3-manifolds, particularly knot complements, linking topological invariants, BPS spectra, and 3d $ ext{N}=2$ theories, with a step towards categorification via $t$-deformation.

Contribution

It introduces a new quiver-3-manifold correspondence, extending knots-quivers ideas to Gukov-Manolescu invariants and exploring physical and categorification aspects.

Findings

Quivers assigned to torus knot complements.

Physical interpretation in terms of BPS states and 3d theories.

Proposal of a $t$-deformation for categorification.

Abstract

We introduce and explore the relation between quivers and 3-manifolds with the topology of the knot complement. This idea can be viewed as an adaptation of the knots-quivers correspondence to Gukov-Manolescu invariants of knot complements (also known as $F_K$ or $\hat{Z}$). Apart from assigning quivers to complements of $T^{(2,2p+1)}$ torus knots, we study the physical interpretation in terms of the BPS spectrum and general structure of 3d $\mathcal{N}=2$ theories associated to both sides of the correspondence. We also make a step towards categorification by proposing a $t$-deformation of all objects mentioned above.