Harmonics on the factored three-sphere and the Hopf map

TL;DR

This paper develops a new computational method for deriving Laplacian eigenmodes on certain three-sphere quotients, leveraging the Hopf map and binary invariants, with improved efficiency and confirmed accuracy.

Contribution

It introduces a novel approach using binary invariants for calculating harmonics on orbifolded spheres, simplifying the process and enhancing computational efficiency.

Findings

Method yields harmonics consistent with existing results.

Binary invariants facilitate easier computation of modes.

Approach improves over projection and Cartesian invariant methods.

Abstract

Laplacian eigenmodes on homogeneous Clifford--Klein factors of the three--sphere are obtained as pullbacks of harmonics on the orbifolded two--sphere using the Hopf map. A method of obtaining these polyhedral, or crystal, harmonics using binary invariants is presented which has computational advantages over those based on projection techniques, or those using invariants constructed in terms of Cartesian coordinates. In addition, modes transforming according to the irreps of the deck group are found in easy fashion using the covariants already conveniently calculated by Desmier and Sharp and by Bellon. Agreement is found with existing results.