A CalabiYau threefold X with torsion in H_2(X,Z) has a disconnected complexified
Kahler moduli space and multiple large volume limits. Bmodel techniques and mirror
symmetry need to be applied at all of these large volume limits in order to extract
the GromovWitten invariants of X. In this talk, I focus on the double cover of
degree 8 determinantal surfaces in P^3, their nonKahler 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 DonaldsonThomas invariants for these
noncommutative resolutions.
