Metaplectic-c prequantizations and the Einstein--Brillouin--Keller quantizability condition

Abstract

We show that every symplectic manifold admits a symplectic form whose metaplectic-c prequantizations give quantizability conditions on all its Lagrangian submanifolds. Metaplectic-c prequantizations descend to prequantizations over the Lagrangian Grassmannian bundle that are fibrewise flat with holonomies primitive fourth roots of unity on the generators of the fundamental groups. This fibrewise holonomy is consistently interpreted as that of both Leray two-orientations and normalized Blattner--Kostant--Sternberg half-forms. The prequantizations pull back to flat circle bundles on each Lagrangian submanifold whose holonomy character defines a quantizability condition. The first and second Stiefel--Whitney classes are identified as splitting obstructions to the Lagrangianwise holonomies. These holonomies split into a product of holonomies on any orientable Lagrangian submanifold, and split consistently for all Lagrangian submanifolds of a manifold with metaplectic structure. When the first Chern class vanishes, the holonomy splits into the Bohr--Sommerfeld factor and a quarter phase Maslov correction, recovering the classical Einstein--Brillouin--Keller condition. We develop the underlying obstruction theory for circle extensions associated to general Lie group double coverings with first hypercohomology groupoids.