A Calabi-Yau threefold X with torsion in H_2(X,Z) has a disconnected complexified
Kahler moduli space and multiple large volume limits. B-model techniques and mirror
symmetry need to be applied at all of these large volume limits in order to extract
the Gromov-Witten invariants of X. In this talk, I focus on the double cover of
degree 8 determinantal surfaces in P^3, their non-Kahler small resolutions possessing
Z_2 torsion, and their noncommutative resolutions. There is a derived equivalence
between sheaves on the noncommutative resolutions and twisted sheaves on the small
resolutions, suggesting a theory of Donaldson-Thomas invariants for these
noncommutative resolutions.
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