  In this paper we propose a definition of torsion refined Gopakumar-Vafa (GV)
invariants for Calabi-Yau threefolds with terminal nodal singularities that do
not admit K\"ahler crepant resolutions. Physically, the refinement takes into
account the charge of five-dimensional BPS states under a discrete gauge
symmetry in M-theory. We propose a mathematical definition of the invariants in
terms of the geometry of all non-K\"ahler crepant resolutions taken together.
The invariants are encoded in the A-model topological string partition
functions associated to non-commutative (nc) resolutions of the Calabi-Yau. Our
main example will be a singular degeneration of the generic Calabi-Yau double
cover of $\mathbb{P}^3$ and leads to an enumerative interpretation of the
topological string partition function of a hybrid Landau-Ginzburg model. Our
results generalize a recent physical proposal made in the context of torus
fibered Calabi-Yau manifolds by one of the authors and clarify the associated
enumerative geometry.
