Dynamical Quantization of Contact Structures

TL;DR

This paper develops a novel dynamical quantization method for contact manifolds using tractor bundles, linking Reeb dynamics, parabolic geometries, and Einstein metrics, with explicit example on the 3-sphere.

Contribution

It introduces a contact quantization framework independent of contact form, utilizing tractor connections and Reeb dynamics, and provides a detailed quantization of the 3-sphere's contact structure.

Findings

Quantization is independent of contact form.

Reeb dynamics are quantized via tractor connections.

Explicit quantization of the 3-sphere's contact structure.

Abstract

We construct a dynamical quantization for contact manifolds in terms of a flat connection acting on a Hilbert tractor bundle. We show that this contact quantization, which is independent of the choice of contact form, can be obtained by quantizing the Reeb dynamics of an ambient strict contact manifold equivariantly with respect to an R+-action. The contact quantization further determines a certain contact tractor connection whose parallel sections determine a distinguished choice of Reeb dynamics and their quantization. This relationship relies on tractor constructions from parabolic geometries and mirrors the tight relationship between Einstein metrics and conformal geometries. Finally, we construct in detail the dynamical quantization of the unique tight contact structure on the 3-sphere, where the Holstein-Primakoff transformation makes a surprising appearance.