On the quantization of a transversely symplectic foliation (Q2459022)

The paper aims to extend the methods of quantizing symplectic manifolds to quantizing manifolds with a transversely symplectic foliation. The notions of symplectic structure and holomorphic structure are generalized to foliations not along the leaves, but in the transversal direction. The notion of the quantization of a transversally symplectic quantization is defined as a deformation quantization. The main question raised in the paper is: does every transversely symplectic structure admit a quantization? The answer is still unknown, in contrast to the case of symplectic manifolds, which all admit deformation quantizations. The author only solves the case of a transversely projective structure, in which situation the Weyl-Moyal quantizations of the fibers of the normal bundle to the foliation can be patched together to give a formula for the quantization.