Non-commutative Donaldson–Thomas invariants and the conifold

Szendrői, Balázs¹

¹ Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB, UK

Geometry & topology, Tome 12 (2008) no. 2, pp. 1171-1202.

Résumé

Given a quiver algebra $A$ with relations defined by a superpotential, this paper defines a set of invariants of $A$ counting framed cyclic $A$–modules, analogous to rank–$1$ Donaldson–Thomas invariants of Calabi–Yau threefolds. For the special case when $A$ is the non-commutative crepant resolution of the threefold ordinary double point, it is proved using torus localization that the invariants count certain pyramid-shaped partition-like configurations, or equivalently infinite dimer configurations in the square dimer model with a fixed boundary condition. The resulting partition function admits an infinite product expansion, which factorizes into the rank–$1$ Donaldson–Thomas partition functions of the commutative crepant resolution of the singularity and its flop. The different partition functions are speculatively interpreted as counting stable objects in the derived category of $A$–modules under different stability conditions; their relationship should then be an instance of wall crossing in the space of stability conditions on this triangulated category.

Détail

DOI : 10.2140/gt.2008.12.1171

Keywords: enumerative invariants, Calabi-Yau algebra

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