Non-commutative Donaldson–Thomas invariants and the conifold

Szendrői, Balázs

doi:10.2140/gt.2008.12.1171

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Non-commutative Donaldson–Thomas invariants and the conifold

Balázs Szendrői

Geometry & Topology 12 (2008) 1171–1202

DOI: 10.2140/gt.2008.12.1171

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

Mathematical Subject Classification 2000

Primary: 14J32

Secondary: 14N10

References

Publication

Received: 24 July 2007

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

Volume 12, issue 2 (2008)