On the eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces

TL;DR

This work computes the Dirac operator's eigenfunctions on S^N and H^N by separating variables in geodesic coordinates, then derives the spinor heat kernel for the iterated Dirac operator on these spaces. It unifies analytic separation methods with a group-theoretic approach via Harish-Chandra's radial Casimir framework, showing that spinor fields on these symmetric spaces are cross sections of homogeneous vector bundles and that the heat kernel can be expressed through spinor-spherical functions and parallel transport. The results cover both compact and noncompact cases, providing explicit eigenvalues, degeneracies, and spectral functions, and they demonstrate consistency between direct analytic methods and harmonic analysis on symmetric spaces. The paper also highlights a duality between S^N and H^N and shows how analytic continuation connects the two settings, with implications for field theory on curved backgrounds and for understanding heat-kernel structures on homogeneous bundles.

Abstract

The eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces of arbitrary dimension are computed by separating variables in geodesic polar coordinates. These eigenfunctions are used to derive the heat kernel of the iterated Dirac operator on these spaces. They are then studied as cross sections of homogeneous vector bundles, and a group-theoretic derivation of the spinor spherical functions and heat kernel is given based on Harish-Chandra's formula for the radial part of the Casimir operator.