Berezin-type quantization on even-dimensional compact manifolds

TL;DR

This paper develops a Berezin-type quantization method for even-dimensional compact manifolds by removing lower-dimensional skeletons to obtain a structure similar to complex Euclidean space, enabling a star product and quantization.

Contribution

It introduces a novel quantization approach on compact manifolds using cell decomposition and embedding into complex projective space, extending Berezin quantization techniques.

Findings

Constructed a Berezin-type quantization on compact manifolds.

Defined a star product satisfying the correspondence principle.

Illustrated the method with the example of the torus.

Abstract

In this article we show that a Berezin-type quantization can be achieved on a compact even dimensional manifold $M^{2d}$ by removing a skeleton $M_0$ of lower dimension such that what remains is diffeomorphic to $R^{2d}$ (cell decomposition) which we identify with $C^d$ and embed in $ CP^d$. A local Poisson structure and Berezin-type quantization are induced from $ CP^d$. Thus we have a Hilbert space with a reproducing kernel. The symbols of bounded linear operators on the Hilbert space have a star product which satisfies the correspondence principle outside a set of measure zero. This construction depends on the diffeomorphism. One needs to keep track of the global holonomy and hence the cell decomposition of the manifold. As an example, we illustrate this type of quanitzation of the torus. We exhibit Berezin-Toeplitz quantization of a complex manifold in the same spirit as above.