Geometry & Topology

Non-commutative Donaldson–Thomas invariants and the conifold

Balázs Szendrői

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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
Received: 24 July 2007
Revised: 9 January 2008
Accepted: 7 February 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document
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)

Keywords
enumerative invariants Calabi-Yau algebra

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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