nLab
prequantum geometry

Contents

Context

Symplectic geometry

Geometric quantization

Contents

Idea

Traditional prequantum geometry is the differential geometry of smooth manifolds which are equipped with a twist in the form of a circle group-principal bundle and a circle-principal connection. In the context of geometric quantization of symplectic manifolds these arise as prequantum bundles. Equivalently, prequantum geometry is the contact geometry of the total spaces of these bundles, equipped with their Ehresmann connection differential 1-form and thought of as regular contact manifolds. Prequantum geometry notably studies the automorphisms of prequantum bundles covering diffeomorphisms of the base – the prequantum operators or contactomorphisms – and the action of these on the space of sections of the associated line bundle – the prequantum states. This is an intermediate step in the genuine geometric quantization of the curvature differential 2-form of these bundles, which is obtained by “dividing the above data in half” (polarization), important for instance in the the orbit method.

But prequantum geometry is of interest in its own right. For instance the above automorphism group naturally provides the Lie integration of the Poisson bracket Lie algebra of the underlying symplectic manifold, together with the canonical injection into the group of bisections of the Lie integration of the Atiyah Lie algebroid which is associated with the given circle bundle, all of which are fundamental objects of interest in the study of line bundles over manifolds.

Created on February 21, 2013 at 02:10:50. See the history of this page for a list of all contributions to it.