Abstract: In general, a projective Calabi-Yau threefold with nodal singularities does not admit any Kaehler small resolution.This happens in particular if the exceptional curves are torsion in the homology of some small resolution. I will first introduce a class of examples, constructed as double covers of P3 with symmetric determinantal ramification locus. The proof that the exceptional curves are torsion is based on a so-called conifold transition to another smooth CY 3-fold. From a physical perspective, this can be interpreted as a Higgs transition that leads to a discrete gauge symmetry in M-theory on the singular CY. The M-theory interpretation then motivates a proposal for torsion refined Gopakumar-Vafa invariants that are associated to the singular CY 3-fold and, in a certain sense, capture the enumerative geometry of non-Kaehler small resolutions. Finally, I will discuss the relation to topological strings on "Calabi-Yau gerbes" and how one can calculate these invariants using mirror symmetry.