Non-commutative Donaldson-Thomas theory and the conifold

TL;DR

This paper introduces invariants for non-commutative algebras related to Calabi-Yau threefolds, linking them to combinatorial models and revealing their factorization properties through localization techniques.

Contribution

It defines new invariants for quiver algebras with superpotentials, connecting non-commutative Donaldson-Thomas theory to combinatorial models and wall crossing phenomena.

Findings

Invariants count pyramid-shaped and dimer configurations.

Partition functions factorize into commutative DT partition functions.

Localization proves the counting interpretation for special cases.

Abstract

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.