BPS states, knots and quivers

TL;DR

This paper establishes a novel connection between knots and quivers, showing how knot invariants can be expressed through quiver representation moduli spaces, leading to a new categorification and proof of integrality of LMOV invariants.

Contribution

It introduces a new framework linking knots to quiver representations, enabling the expression of knot invariants via motivic Donaldson-Thomas invariants and proving their integrality.

Findings

Knot invariants can be expressed in terms of quiver moduli space characteristics.

LMOV invariants are shown to be motivic Donaldson-Thomas invariants of associated quivers.

The integrality of LMOV invariants is proven based on this quiver correspondence.

Abstract

We argue how to identify supersymmetric quiver quantum mechanics description of BPS states, which arise in string theory in brane systems representing knots. This leads to a surprising relation between knots and quivers: to a given knot we associate a quiver, so that various types of knot invariants are expressed in terms of characteristics of a moduli space of representations of the corresponding quiver. This statement can be regarded as a novel type of categorification of knot invariants, and among its various consequences we find that Labastida-Mari\~no-Ooguri-Vafa (LMOV) invariants of a knot can be expressed in terms of motivic Donaldson-Thomas invariants of the corresponding quiver; this proves integrality of LMOV invariants, conjectured originally based on string theory and M-theory arguments.