Motivic Donaldson--Thomas invariants of some quantized threefolds

TL;DR

This paper computes motivic Donaldson--Thomas invariants for certain noncommutative Calabi--Yau threefolds, revealing their behavior in families and providing explicit formulas, with implications for more complex cases.

Contribution

It provides the first explicit calculations of motivic DT invariants for families of noncommutative Calabi--Yau threefolds defined by quivers with potentials.

Findings

Invariants are generically constant but jump at special parameters.

Closed-form generating series as plethystic exponentials of rational functions.

Conjectures for invariants of more complex structures like elliptic Sklyanin algebras.

Abstract

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.