Vector bundles on fuzzy K\"{a}hler manifolds

TL;DR

This paper introduces a matrix regularization method for vector bundles on K"ahler manifolds, generalizing Berezin-Toeplitz quantization, and explores its asymptotic behavior with explicit examples.

Contribution

It presents a novel matrix regularization framework for vector bundles on K"ahler manifolds, extending quantization techniques and establishing algebraic correspondences in the large-N limit.

Findings

Established a correspondence between section algebra and matrix algebra in the large-N limit.

Provided explicit examples for monopole bundles over complex projective space and tori.

Analyzed the asymptotic behavior of the matrix regularization method.

Abstract

We propose a matrix regularization of vector bundles over a general closed K\"ahler manifold. This matrix regularization is given as a natural generalization of the Berezin-Toeplitz quantization and gives a map from sections of a vector bundle to matrices. We examine the asymptotic behaviors of the map in the large-$N$ limit. For vector bundles with algebraic structure, we derive a beautiful correspondence of the algebra of sections and the algebra of corresponding matrices in the large-$N$ limit. We give two explicit examples for monopole bundles over a complex projective space $CP^n$ and a torus $T^{2n}$.