Motivic Donaldson–Thomas invariants of some quantized threefolds

Journal article

This paper is motivated by the question of howmotivic 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 An-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.

Authors: Cazzaniga, A; Morrison, A; Pym, B; Szendroi, B

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: European Mathematical Society
• Journal: Journal of Noncommutative Geometry
• Volume: 11
• Issue: 3
• Pages: 1115–1139
• Publication date: 2017-09-26
• Acceptance date: 2016-04-01
• DOI: 10.4171/JNCG/11-3-10
• ISSN: 1661-6960
• Keywords: Calabi-Yau algebra; dimensional reduction; Donaldson-Thomas theory; motivic vanishing cycle; quiver representation