2007/05/31 by Balazs Szendroi, Balázs Szendrői · 136 citations
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #Boundary (topology) #Commutative property #Conifold #Infinite product #Partition (number theory) #Partition function (quantum field theory) #Quiver #Singularity #Torus #hep-th #math.AG
paper · pdf · doi:10.2140/gt.2008.12.1171
published in Geometry & Topology 12(2), 1171-1202 (Mathematical Sciences Publishers) · Infinite product form, conjectured in v1, now a theorem of Ben Young. Additional discussion of small-volume expansion related to Eisenstein-like series
openalex publication_date 2008/05/24 · arxiv created 2008/09/19 · arxiv updated 2009/12/01 · openalex created_date 2019/06/27 · openalex updated_date 2026/08/05
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.