Spectra of Higher Spin Operators on the Sphere

TL;DR

This paper derives explicit spectral formulas for higher spin operators on spheres, including Dirac and Rarita-Schwinger operators, and extends to conformally invariant operators of all odd orders in various dimensions.

Contribution

It provides the first explicit eigenvalue formulas for a broad class of higher spin and conformally invariant operators on spheres in both even and odd dimensions.

Findings

Explicit spectra for higher spin operators on odd-dimensional spheres.

Eigenvalue formulas for conformally invariant operators of all odd orders.

Spectral results for Dirac and Rarita-Schwinger operators in even and odd dimensions.

Abstract

We present explicit formulas for the spectra of higher spin operators on the subbundle of the bundle of spinor-valued trace free symmetric tensors that are annihilated by the Clifford multiplication over the standard sphere in odd dimension. In even dimensional case, we give the spectra of the square of such operators. The Dirac and Rarita-Schwinger operators are zero-form and one-form cases, respectively. We also give eigenvalue formulas for the conformally invariant differential operators of all odd orders on the subbundle of the bundle of spinor-valued forms that are annihilated by the Clifford multiplication in both even and odd dimensions on the sphere.