Toeplitz Quantization of K\"ahler Manifolds and $gl(N)$ $N\to\infty$

TL;DR

This paper demonstrates that Toeplitz and geometric quantization methods for compact K"ahler manifolds produce a consistent classical limit, approximating Poisson algebras with finite-dimensional matrix algebras as N approaches infinity.

Contribution

It generalizes previous results to all compact K"ahler manifolds, establishing a well-defined classical limit via operator norm estimates.

Findings

Classical limit exists for general compact K"ahler manifolds.

Poisson algebra approximated by finite-dimensional matrix algebras.

Extension of earlier results from specific surfaces to all K"ahler manifolds.

Abstract

For general compact K\"ahler manifolds it is shown that both Toeplitz quantization and geometric quantization lead to a well-defined (by operator norm estimates) classical limit. This generalizes earlier results of the authors and Klimek and Lesniewski obtained for the torus and higher genus Riemann surfaces, respectively. We thereby arrive at an approximation of the Poisson algebra by a sequence of finite-dimensional matrix algebras $gl(N)$, $N\to\infty$.