Deformation Quantization of Lagrangian Fiber Bundles

Abstract

Let (M, \om) be a symplectic manifold. A Lagrangian fiber bundle \pi : M -> B determines a completely integrable system on M. First integrals of this system are the pull-backs of functions on the base of the bundle. We show that for each Lagrangian fiber bundle \pi there exist star products on C^\infty(M)[[h]] which do not deform the pointwise multiplication on the subalgebra \pi^*(C^\infty (B)) [[h]]. The set of equivalence classes of such star products is in bijection with formal deformations of the symplectic structure \om for which \pi : M -> B remains Lagrangian taken modulo formal symplectomorphisms of M.