Interpolating factorizations for acyclic Donaldson--Thomas invariants

Abstract

We prove a family of factorization formulas for the combinatorial Donaldson--Thomas invariant for an acyclic quiver. A quantum dilogarithm identity due to Reineke, later interpreted by Rimanyi by counting codimensions of quiver loci, gives two extremal cases of our formulation in the Dynkin case. We establish our interpolating factorizations explicitly with a dimension counting argument by defining certain stratifications of the space of representations for the quiver and calculating Betti numbers in the corresponding equivariant cohomology algebras.