A foliation of a symplectic manifold is Lagrangian when every leaf has half the ambient dimension and the restricted symplectic form vanishes. The tangent distribution is a real polarization in geometric quantization. Restricted prequantum connections are flat on individual leaves, but nontrivial holonomy can obstruct global parallel sections.
A leaf of a Lagrangian foliation is an immersed Lagrangian submanifold tangent to its integrable distribution. A prequantum line bundle restricts to a flat connection along such a leaf, because its curvature is proportional to the symplectic form. Nonzero global parallel sections require trivial holonomy along every leaf loop.
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Lagrangian foliation is a concept that arises in the field of symplectic geometry, which is a branch of differential geometry and mathematics concerned with structures that allow for a generalization of classical mechanics. In this context, a foliation is a decomposition of a manifold into a collection of submanifolds, called leaves, which locally look like smaller, simpler pieces of the original manifold.