Invariance of Polarization Induced by Symplectomorphisms

A variant of the Kirillov-Kostant-Souriau approach to quantizing a symplectic manifold \((M,\omega )\) requires associating a prequantum line bundle \((L,\nabla )\to M\) and a Lagrangian foliation to M. One then uses these data to define a vector space called the quantization. In this paper, I introduce an action of the symplectomorphisms of \((M,\omega )\) on the Lagrangian foliations of M. I then show that a symplectomorphism \(\Phi :M\to M\) will preserve the quantization if it admits a connection-preserving lift to the prequantum line bundle. Finally, I give a topological condition on M which guarantees the existence of such a lift of a symplectomorphism.