Discrete gauge symmetries can arise in M-theory on Calabi-Yau threefolds with terminal singularities.
I will argue that the Gopakumar-Vafa invariants are encoded in partition functions associated to different values of a discrete flat B-field that stabilizes the singularities.
Locally, the effect of the B-field can be interpreted as replacing each node by a non-commutative conifold.
This picture allows us to derive closed expressions for the constant map contributions to the topological string partition functions on non-commutative crepant resolutions.
The main example in this talk will be singular torus fibered Calabi-Yau 3-folds that share the same Jacobian fibrations as smooth genus one fibered Calabi-Yaus.
Somewhat miraculously, all of the relevant partition functions can then be recovered at newly discovered non-commutative large volume limits in the stringy Kähler moduli spaces of a rich network of interconnected torus fibrations.
This makes it possible to extract the Gopakumar-Vafa invariants associated to discrete charged BPS states in M-theory on singular Calabi-Yau threefolds.
The talk is based on 2108.09311 by the speaker and on work in progress with Albrecht Klemm, Sheldon Katz and Eric Sharpe.