RKHS, Berezin and Odzijewicz-type quantizations on arbitrary compact smooth manifold

TL;DR

This paper develops Berezin-type and Odzijewicz-type quantizations for arbitrary compact smooth manifolds by embedding them into complex projective space and inducing quantizations from there, generalizing previous work.

Contribution

It introduces a method to define Berezin and Odzijewicz quantizations on any compact smooth manifold via embedding into complex projective space and pullback of ambient quantizations.

Findings

Defined Berezin-type quantization on compact manifolds.

Extended Odzijewicz-type quantization to arbitrary compact manifolds.

Explicitly constructed quantizations on complex projective space and induced on submanifolds.

Abstract

In this article we define Berezin-type and Odzijewicz-type quantizations on compact smooth manifolds. The method is we embed the smooth manifold of real dimension $n$ into ${\mathbb C}P^n$ and induce the quantizations from there. The standard way by which reproducing kernel Hilbert spaces are defined on submanifolds gives a way to define the pullback coherent states. In Berezin-type quantization the Hilbert space of quantization is the pullback (by the embedding) of the Hilbert space of geometric quantization of ${\mathbb C}P^n$. In the Odzijewicz-type quantization one has to consider a tensor product of the geometric quantization line bundle with holomorphic $n$-forms. In the Berezin case, the operators that are quantized are those induced from the ambient space ${\mathbb C}P^n$. The Berezin-type quantization exhibited here is a generalization of an earlier work of the author and Ghosh. In both Berezin and Odzijewicz-type quantizations we first exhibit this quantization explicitly on ${\mathbb C}P^n$ and we induce the quantization on the smooth compact embedded manifold from ${\mathbb C}P^n$.