Geometric quantization of weak-Hamiltonian functions

The paper presents an extension of the geometric quantization procedure to integrable, big-isotropic structures. We obtain a generalization of the cohomology integrality condition, we discuss geometric structures on the total space of the corresponding principal circle bundle and we extend the notion of a polarization. 1 Big-isotropic structures Weak-Hamiltonian functions belong to the framework of big-isotropic structures and have been discussed in [11, 12]. For the convenience of the reader, we recall some basic facts here. All the manifolds and mappings are of C class and we denote by M an m-dimensional manifold, by χ(M) the space of k-vector fields, by Ω(M) the space of differential k-forms, by Γ the space of global cross sections of a vector bundle, by X,Y, .. either contravariant vectors or vector fields, by α, β, ... either covariant vectors or 1-forms, by d the exterior differential and by L the Lie derivative. The vector bundle T M = TM ⊕ T M is called the big tangent bundle. It has the natural, non degenerate metric of zero signature (neutral metric) g((X,α), (Y, β)) = 1 2 (α(Y ) + β(X)), (1.1) the non degenerate, skew-symmetric 2-form ω((X,α), (Y, β)) = 1 2 (α(Y )− β(X)) (1.2) and the Courant bracket of cross sections [(X,α), (Y, β)]C = ([X,Y ], LXβ − LY α+ 1 2 d(α(Y )− β(X))); (1.3) 2000 Mathematics Subject Classification: 53D17, 53D50.