Koszul algebras and Donaldson-Thomas invariants

We propose a new method for computing motivic Donaldson-Thomas invariants of a symmetric quiver which relies on Koszul duality between supercommutative algebras and Lie superalgebras and completely bypasses cohomological Hall algebras. Specifically, we define, for a given symmetric quiver Q, a supercommutative quadratic algebra A(Q), and study the Lie superalgebra g(Q) that corresponds to A(Q) under Koszul duality. We introduce an action of the first Weyl algebra on g(Q) and prove that the motivic Donaldson-Thomas invariants of Q may be computed via the Poincare series of the kernel of the operator delta(t). This gives a new proof of positivity for motivic Donaldson-Thomas invariants. Along theway, we prove that the algebra A(Q) is numericallyKoszul for every symmetric quiver Q and conjecture that it is in fact Koszul; we show that this conjecture holds for a certain class of quivers.

Automated summary

We propose a new method for computing motivic Donaldson-Thomas invariants of a symmetric quiver using Koszul duality between supercommutative algebras and Lie superalgebras, bypassing cohomological Hall algebras. The method involves defining a supercommutative quadratic algebra and studying the corresponding Lie superalgebra. The positivity of the invariants is proven using the Poincare series of a certain operator's kernel.