Motivic Donaldson-Thomas invariants and Kac conjecture

TL;DR

This paper explores the positivity of Donaldson-Thomas invariants for symmetric quivers, deriving combinatorial consequences and proving the Kac conjecture for quivers with loops at every vertex.

Contribution

It establishes the Kac conjecture for a broad class of quivers by leveraging the positivity of Donaldson-Thomas invariants recently proven by Efimov.

Findings

Positivity of Donaldson-Thomas invariants implies new combinatorial results.

Proof of the Kac conjecture for quivers with loops at each vertex.

Connections between invariants and quiver properties are clarified.

Abstract

We derive some combinatorial consequences from the positivity of Donaldson-Thomas invariants for symmetric quivers conjectured by Kontsevich and Soibelman and proved recently by Efimov. These results are used to prove the Kac conjecture for quivers having at least one loop at every vertex.