Koszul algebras and Donaldson-Thomas invariants

Abstract

For a given symmetric quiver Q, we define a supercommutative quadratic algebra A Q whose Poincaré 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 A Q its Koszul dual Lie superalgebra g Q . We prove that the motivic Donaldson-Thomas invariants of the quiver Q may be computed using the Poincaré series of a certain Lie subalgebra of g Q that can be described, using an action of the first Weyl algebra on g Q , as the kernel of the operator ∂ t . This gives a new proof of positivity for motivic Donaldson-Thomas invariants. In addition, we prove that the algebra 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.

Generalized multiple q-zeta values and characters of vertex algebras

“…With this choice, the graph series is denoted by H Γ (q). Although unrelated to this work, it is worth noting that there is an interesting connection among the DT invariants of symmetric quivers, cohomological Hall algebras, and the dual of the principal subspace of W Γ [20,21] (see also [22]).…”