Given a quiver algebra
with relations defined by a superpotential, this paper defines a set of invariants of
counting framed
cyclic –modules,
analogous to rank–
Donaldson–Thomas invariants of Calabi–Yau threefolds. For the special case when
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–
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
–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.
