Journal of Noncommutative Geometry

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Volume 11, Issue 3, 2017, pp. 1115–1139
DOI: 10.4171/JNCG/11-3-10

Published online: 2017-09-26

Motivic Donaldson–Thomas invariants of some quantized threefolds

Alberto Cazzaniga[1], Andrew Morrison[2], Brent Pym[3] and Balázs Szendrői[4]

(1) Stellenbosch University, South Africa
(2) ETH Zürich, Switzerland
(3) University of Edinburgh, UK
(4) University of Oxford, UK

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 $A_n$-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.

Keywords: Donaldson–Thomas theory, motivic vanishing cycle, Calabi–Yau algebra, quiver representation, dimensional reduction

Cazzaniga Alberto, Morrison Andrew, Pym Brent, Szendrői Balázs: Motivic Donaldson–Thomas invariants of some quantized threefolds. J. Noncommut. Geom. 11 (2017), 1115-1139. doi: 10.4171/JNCG/11-3-10