Koszul algebras and Donaldson-Thomas invariants

TL;DR

This paper introduces a new algebraic framework linking Koszul duality and Donaldson-Thomas invariants for symmetric quivers, providing novel proofs of positivity and conjectures about algebraic properties.

Contribution

It defines a supercommutative quadratic algebra related to quivers, connects it to Lie superalgebras via Koszul duality, and uses this to compute motivic Donaldson-Thomas invariants with new positivity proofs.

Findings

Motivic DT invariants computed via Lie subalgebra Poincaré series.

Proved algebra $\\mathcal{A}_Q$ is numerically Koszul for all symmetric quivers.

Conjectured and partially proved that $\mathcal{A}_Q$ is Koszul.

Abstract

For a given symmetric quiver $Q$, we define a supercommutative quadratic algebra $\mathcal{A}_Q$ whose Poincar\'e series is related to the motivic generating function of $Q$ by a simple change of variables. The Koszul duality between supercommutative algebras and Lie superalgebras assigns to the algebra $\mathcal{A}_Q$ its Koszul dual Lie superalgebra $\mathfrak{g}_Q$. We prove that the motivic Donaldson-Thomas invariants of the quiver $Q$ may be computed using the Poincar\'e series of a certain Lie subalgebra of $\mathfrak{g}_Q$ that can be described, using an action of the first Weyl algebra on $\mathfrak{g}_Q$, as the kernel of the operator $\partial_t$. This gives a new proof of positivity for motivic Donaldson--Thomas invariants. In addition, we prove that the algebra $\mathcal{A}_Q$ is numerically Koszul for every symmetric quiver $Q$ and conjecture that it is in fact Koszul; we also prove this conjecture for quivers of a certain class.