Description
In general, a Kaehler Calabi-Yau threefold with nodal singularities does not admit a Kaehler small resolution.
This happens in particular if the exceptional curves are torsion in homology.
However, the presence of torsion also leads to the possibility of turning on a flat, topologically non-trivial B-field that stabilizes the singularities.
Using conifold transitions, we will describe a large family of examples for this phenomenon and explain how the resulting backgrounds can be studied using hybrid phases of gauged linear sigma models.
Using the sphere partition function, we can then extract periods of the mirror Calabi-Yaus, which allows us to study the topological string partition functions.
We argue that the latter encode Gopakumar-Vafa invariants associated to BPS states with discrete charges and that the invariants capture the enumerative geometry of the non-Kaehler small resolutions.
On the other hand, it turns out that the periods correspond to well known Q-variations of Hodge structure but with a new integral structure.
