Foliated Cohomology and Geometric Quantization

Abstract

It often turns out that the bundle L^ω ⊗ N ^1/2_D may not have any smooth global sections which are covariantly constant along vector fields in polarization. Then the construction of the Hilbert representation space via the standard technique of geometric quantization becomes impossible. In these pathological cases Kostant [1975] has suggested the using of higher cohomology groups of M for the quantization process. Then the Dolbeault-Kostant complex gives a convenient representation of these cohomological groups in terms of L^ω ⊗ N ^1/2_D -valued forms defined on M.

In this chapter we shall develop some aspects of the theory of foliated cohomology of a smooth manifold and we shall point out some of its applications in geometric quantization.