#### Volume 12, issue 2 (2008)

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Non-commutative Donaldson–Thomas invariants and the conifold

### Balázs Szendrői

Geometry & Topology 12 (2008) 1171–1202
##### 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.

##### Keywords
enumerative invariants, Calabi-Yau algebra
Primary: 14J32
Secondary: 14N10
##### Publication
Revised: 9 January 2008
Accepted: 7 February 2008
Published: 24 May 2008
Proposed: Jim Bryan
Seconded: Lothar Goettsche, Simon Donaldson
##### Authors
 Balázs Szendrői Mathematical Institute University of Oxford 24-29 St Giles’ Oxford OX1 3LB UK http://www.maths.ox.ac.uk/~szendroi