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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.
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