Geometry of the Hopf Bundle and spin-weighted Harmonics

TL;DR

This paper presents a geometric perspective on spin-weighted spherical harmonics by viewing them as vector-valued functions on the Hopf bundle, leveraging the bundle's invariant connection form for computational and conceptual advantages.

Contribution

It introduces a novel geometric framework for understanding spin-weighted harmonics using the Hopf bundle and its invariant connection form, enhancing both conceptual clarity and computational efficiency.

Findings

Spin-weighted harmonics can be modeled as functions on the Hopf bundle.

The invariant connection form simplifies the analysis of these harmonics.

The approach offers a unified geometric perspective for these functions.

Abstract

We demonstrate that it is conceptually and computationally favorable to regard spin-weighted spherical harmonics as vector valued functions on the total space $SO(3)$ of the Hopf bundle, satisfying a covariance condition with respect to the gauge group $U(1)$ of this bundle. A key role is played by the invariant connection form of the principle Hopf bundle, known to physicists from the geometry behind magnetic monopoles.