Perturbations of principal submodules in the Drury-Arveson space

TL;DR

This paper investigates how small changes in principal submodules of the Drury-Arveson space create smooth vector bundles with Hermitian connections, analyzing their parallel transport and monodromy properties.

Contribution

It introduces a geometric framework for understanding perturbations of principal submodules in the Drury-Arveson space, including the construction of vector bundles and computation of connections.

Findings

Perturbations induce smooth vector bundles with Hermitian connections.

Parallel transport operators are explicitly computed.

Monodromy properties of these bundles are explored.

Abstract

We study the geometry in the perturbations of principal submodules in the Drury-Arveson space. We show that the perturbations give rise to smooth vector bundles of Hilbert spaces which are equipped with natural Hermitian connections. We compute the associated parallel transport operators and explore properties of the monodromy.