Quantization of compact symplectic manifolds

TL;DR

This paper develops a comprehensive theory of Berezin-Toeplitz operators on compact symplectic manifolds, simplifying existing frameworks and comparing with spin-c Dirac quantization to enhance understanding of geometric quantization methods.

Contribution

It introduces a simplified, self-contained development of Berezin-Toeplitz quantization on compact symplectic manifolds, including a comparative analysis with spin-c Dirac quantization.

Findings

Simplified the Boutet de Monvel-Guillemin theory for Berezin-Toeplitz operators

Established a clear framework for quantization on symplectic manifolds

Provided insights into the relationship between Berezin-Toeplitz and Dirac quantizations

Abstract

We develop the theory of Berezin-Toeplitz operator on any compact symplectic prequantizable manifold from scratch. Our main inspiration is the Boutet de Monvel-Guillemin theory, that we simplify in several ways to obtain a concise exposition. A comparison with the spin-c Dirac quantization is also included.