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 AQ, and study the Lie superalgebra gQ that corresponds to AQ under Koszul duality. We introduce an action of the first Weyl algebra on gQ and prove that the motivic Donaldson–Thomas invariants of Q may be computed via the Poincaré series of the kernel of the operator ∂t. This gives a new proof of positivity for motivic Donaldson–Thomas invariants. Along the way, we prove that the algebra AQ is numerically Koszul 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.
All Science Journal Classification (ASJC) codes
- !!Statistical and Nonlinear Physics
- !!Mathematical Physics