Non-commutative Donaldson-Thomas theory and the conifold

By Balázs Szendrői. 2007

199 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/67ce409092c59cd241379d287f232d92e64d0cf4

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Citations in the corpus

Brane tilings and their applications (2008, background): “The connection with Donaldson-Thomas invariants [339, 340] is also interesting (see recent works of [341]).”
Derived categories of small toric Calabi-Yau 3-folds and counting invariants (2008, background): “..,q −1 N−1) = Z +(q). We set ζ(±) = (±1,...,±1). Note that M ζ(+)(v) is empty unless v = 0 and so Zζ(+) (q) = 1. The invariants Dζ(−)(v) are the non-commutative Donaldson-Thomas invariants deﬁned in [Sze]. We denote their generatingfunction Zζ(−)(q) by ZNCDT(q). We divide the set of positive real r”
Curve counting via stable pairs in the derived category (2007, background): “fully extended to the derived category and made to include the virtual class, our entire discussion here remains conjectural. Similar phenomena have been observed recently in a noncommutative example [49]. The paper [13] also studies counting invariants (and more generally BPS states) and wall cross”
Curve counting theories via stable objects Ⅱ : DT/ncDT flop formula (2009, methodology): “We also introduce an analogue of DT-invariant for the non-commutative scheme (Y,AY ), which generalizes Szendrői’s non-commutative DT (ncDT for short) invariant for a local (−1,−1)-curve example.”
A theory of generalized Donaldson–Thomas invariants (2008, methodology): “These quiver analogues of Mα,nstp (τ ′), P Iα,n(τ ′) are not new, similar things have been studied in quiver theory by Nakajima, Reineke, Szendrői and other authors for some years [24, 82–85, 88, 89, 99].”
On the noncommutative Donaldson-Thomas invariants arising from brane tilings (2008, methodology): “Ideals in �0 correspond to the pyramid partitions from [ 17 ].”
Refined, Motivic, and Quantum (2009, methodology): “For toric Calabi–Yau threefolds in the Szendr¨ oi region of moduli space, the generating functions of BPS states can be computed via statistical crystal melting models [ 8 ,10,11] that are seemingly distinct from the topological vertex formalism [12,13].”