The spinor and tensor fields with higher spin on spaces of constant curvature

TL;DR

This paper derives formulas and spectral properties of higher-spin spinor and tensor fields on constant curvature spaces, connecting geometric operators, representation theory, and differential forms.

Contribution

It provides explicit formulas, spectral calculations, and decompositions for higher-spin fields and tensor fields on constant curvature manifolds, advancing understanding of their geometric and algebraic structures.

Findings

Explicit Weitzenb"ock formulas for higher-spin fields

Spectral formulas on the standard sphere

Hodge-de Rham decomposition for coupled spinor and form fields

Abstract

In this article, we give all the Weitzenb\"ock-type formulas among the geometric first order differential operators on the spinor fields with spin $j+1/2$ over Riemannian spin manifolds of constant curvature. Then we find an explicit factorization formula of the Laplace operator raised to the power $j+1$ and understand how the spinor fields with spin $j+1/2$ are related to the spinors with lower spin. As an application, we calculate the spectra of the operators on the standard sphere and clarify the relation among the spinors from the viewpoint of representation theory. Next we study the case of trace-free symmetric tensor fields with an application to Killing tensor fields. Lastly we discuss the spinor fields coupled with differential forms and give a kind of Hodge-de Rham decomposition on spaces of constant curvature.