The Kernel of the Rarita–Schwinger Operator on Riemannian Spin Manifolds

Abstract

We study the Rarita-Schwinger operator on compact Riemannian spin manifolds. In particular, we find examples of compact Einstein manifolds with positive scalar curvature where the Rarita-Schwinger operator has a non-trivial kernel. For positive quaternion Kähler manifolds and symmetric spaces with spin structure we give a complete classification of manifolds admitting Rarita-Schwinger fields. In the case of Calabi-Yau, hyperkähler, G 2 and Spin(7) manifolds we find an identification of the kernel of the Rarita-Schwinger operator with certain spaces of harmonic forms. We also give a classification of compact irreducible spin manifolds admitting parallel Rarita-Schwinger fields.2000 Mathematics Subject Classification: Primary 32Q20, 57R20, 53C26, 53C27 53C35, 53C15.

Manifolds with Many Rarita–Schwinger Fields

“…This lemma and Theorem 2.9 combine to give to Proposition 4.6 in [5]. We do not know whether the bound is also sharp in higher dimensions.…”

“…Wang [11] studied the role of Rarita-Schwinger fields in the deformation theory of Einstein metrics with parallel spinors, see also his earlier paper [10]. The problem of counting Rarita-Schwinger fields was considered only quite recently by Homma and Semmelmann [5], with related work more recently still in Homma-Tomihisa [6]. The goal of [5] was to find manifolds admitting any nontrivial Rarita-Schwinger fields; as part of this they obtain a complete classification of positive quaternion-Kähler manifolds and spin symmetric spaces admitting such fields, but their mention of the negative Einstein case is rather brief.…”

“…The problem of counting Rarita-Schwinger fields was considered only quite recently by Homma and Semmelmann [5], with related work more recently still in Homma-Tomihisa [6]. The goal of [5] was to find manifolds admitting any nontrivial Rarita-Schwinger fields; as part of this they obtain a complete classification of positive quaternion-Kähler manifolds and spin symmetric spaces admitting such fields, but their mention of the negative Einstein case is rather brief. We refer to all of these papers for a more extended discussion and some description of the use of Rarita-Schwinger fields in physics.…”

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

“…After a decade, the first author of this paper showed an explicit method to construct the Weitzenböck formulas for the generalized gradients in [16], which produce a lot of applications, vanishing theorem, eigenvalue estimates and so on. There are also many articles to study the generalized gradients and their applications to mathematics and physics ( [15,17,18,22], etc.). Moving on to analysis, we know that one of the main topics in Clifford analysis is to generalize spherical harmonic analysis on Euclidean space to such spinor and tensor fields.…”

“…To clarify the meaning of the factorization, we show how the spinor fields with spin j + 1∕2 are influenced by the spinor fields with lower spin in Theorem 2.11. Remark that, in the case of j = 1 , we need only the assumption of Einstein manifold and can develop fruitful geometry and analysis in [1,18] and [19]. Next, we study harmonic analysis on spinor fields with spin j + 1∕2 on the standard sphere as a model case for spinor analysis on a curved space in Sect.…”