Motivic Donaldson-Thomas invariants of some quantized threefolds

This paper is motivated by the question of how motivic Donaldson--Thomas
invariants behave in families. We compute the invariants for some simple
families of noncommutative Calabi--Yau threefolds, defined by quivers with
homogeneous potentials. These families give deformation quantizations of affine
three-space, the resolved conifold, and the resolution of the transversal
$A_n$-singularity. It turns out that their invariants are generically constant,
but jump at special values of the deformation parameter, such as roots of
unity. The corresponding generating series are written in closed form, as
plethystic exponentials of simple rational functions. While our results are
limited by the standard dimensional reduction techniques that we employ, they
nevertheless allow us to conjecture formulae for more interesting cases, such
as the elliptic Sklyanin algebras.