I'm looking for a reference for a special type of Lagrangian embedding in a prequantizable symplectic manifold.

The setting is a symplectic manifold $(M, \omega)$, whose symplectic form is the curvature of a connection $\vartheta$ in a principal circle-bundle $L \to M$. Let $\iota: L \to M$ be a Lagrangian embedding. By assumption, the curvature $ \iota^* \omega$ of the pull-back connection $ \iota^* (L, \vartheta)$ vanishes for the Lagrangian embedding. In particular, the Chern class of the pull-back bundle is trivial in real cohomology.
However, the Chern class may still have a torsion part.

Does this torsion class has a name? Is there a paper (or book) discussing Lagrangian embeddings for which also the torsion of the pull-back has to vanish? What are those embeddings called? Literally every source discussing some aspect of this would be helpful.