Donaldson-Thomas invariants versus intersection cohomology of quiver moduli

TL;DR

This paper establishes a deep connection between Donaldson-Thomas invariants and intersection cohomology for quiver moduli, proving a conjecture relating these invariants to intersection complexes using motivic methods.

Contribution

It proves that the Hodge theoretic Donaldson-Thomas invariant for certain quivers equals the intersection cohomology of the stable locus closure, and confirms the relative integrality conjecture in the motivic setting.

Findings

Donaldson-Thomas invariant equals intersection cohomology for zero potential quivers.

Proves the intersection complex corresponds to the Donaldson-Thomas function.

Establishes the relative integrality conjecture in motivic Donaldson-Thomas theory.

Abstract

The main result of this paper is the statement that the Hodge theoretic Donaldson-Thomas invariant for a quiver with zero potential and a generic stability condition agrees with the compactly supported intersection cohomology of the closure of the stable locus inside the associated coarse moduli space of semistable quiver representations. In fact, we prove an even stronger result relating the Donaldson-Thomas "function" to the intersection complex. The proof of our main result relies on a relative version of the integrality conjecture in Donaldson-Thomas theory. This will be the topic of the second part of the paper, where the relative integrality conjecture will be proven in the motivic context.