Noncommutative vector bundles over fuzzy CP^N and their covariant derivatives

TL;DR

This paper develops a comprehensive framework for noncommutative vector bundles over fuzzy complex projective spaces, introducing covariant derivatives and generalizing classical constructions to the noncommutative setting.

Contribution

It provides a new construction of fuzzy CP^N with access to all equivariant vector bundles and introduces noncommutative covariant derivatives and generalized Schwinger-Jordan methods.

Findings

Constructed noncommutative covariant derivatives for fuzzy CP^N

Generalized polarization tensors from S^2 to complex projective space

Identified a Heisenberg algebra structure for composite oscillators

Abstract

We generalise the construction of fuzzy CP^N in a manner that allows us to access all noncommutative equivariant complex vector bundles over this space. We give a simplified construction of polarization tensors on S^2 that generalizes to complex projective space, identify Laplacians and natural noncommutative covariant derivative operators that map between the modules that describe noncommuative sections. In the process we find a natural generalization of the Schwinger-Jordan construction to su(n) and identify composite oscillators that obey a Heisenberg algebra on an appropriate Fock space.