## Geometry & Topology

### Non-commutative Donaldson–Thomas invariants and the conifold

Balázs Szendrői

#### 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.

#### Article information

Source
Geom. Topol., Volume 12, Number 2 (2008), 1171-1202.

Dates
Revised: 9 January 2008
Accepted: 7 February 2008
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513800070

Digital Object Identifier
doi:10.2140/gt.2008.12.1171

Mathematical Reviews number (MathSciNet)
MR2403807

Zentralblatt MATH identifier
1143.14034

Subjects
Primary: 14J32: Calabi-Yau manifolds
Secondary: 14N10: Enumerative problems (combinatorial problems)

#### Citation

Szendrői, Balázs. Non-commutative Donaldson–Thomas invariants and the conifold. Geom. Topol. 12 (2008), no. 2, 1171--1202. doi:10.2140/gt.2008.12.1171. https://projecteuclid.org/euclid.gt/1513800070

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