Quiver diagonalization and open BPS states

TL;DR

The paper demonstrates that motivic Donaldson-Thomas invariants of symmetric quivers can be represented by a special infinite quiver with only loops, enabling new insights into BPS state counting and invariants.

Contribution

It introduces a method to encode invariants of any symmetric quiver using an infinite loop-only quiver, generalizing previous combinatorial interpretations.

Findings

Invariants of symmetric quivers can be expressed via infinite loop quivers.

The approach generalizes proofs of integrality for BPS state invariants.

Motivic invariants of arbitrary symmetric quivers relate to those of m-loop quivers.

Abstract

We show that motivic Donaldson-Thomas invariants of a~symmetric quiver $Q$, captured by the generating function $P_Q$, can be encoded in another quiver $Q^{(\infty)}$ of (almost always) infinite size, whose only arrows are loops, and whose generating function $P_{Q^{(\infty)}}$ is equal to $P_Q$ upon appropriate identification of generating parameters. Consequences of this statement include a generalization of the proof of integrality of Donaldson-Thomas and Labastida-Mari\~{n}o-Ooguri-Vafa invariants that count open BPS states, as well as expressing motivic Donaldson-Thomas invariants of an arbitrary symmetric quiver in terms of invariants of $m$-loop quivers. In particular, this means that the already known combinatorial interpretation of invariants of $m$-loop quivers extends to arbitrary symmetric quivers.