Spectra of the Rarita-Schwinger operator on some symmetric spaces

The kernel of the Rarita-Schwinger operator on Riemannian spin manifolds

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 classific…

Manifolds with many Rarita-Schwinger fields

The Rarita-Schwinger operator is the twisted Dirac operator restricted to 3/2-spinors. Rarita-Schwinger fields are solutions of this operator which are in addition divergence-free. This is an overdetermined problem and solutions are rare; it is even more unexpected for there to be large dimensional spaces of solutions. In this paper we prove the existence of a sequence of compact manifolds in any given dimension greater than or equal to 4 for which the dimension of the space of Rarita-Schwinger fields tends to infinity. These manifolds are either simply connected Kähler-Einstein spin with nega…

Rarita-Schwinger fields on nearly Kähler manifolds

We study Rarita-Schwinger fields on 6-dimensional compact strict nearly Kähler manifolds. In order to investigate them, we clarify the relationship between some differential operators for the Hermitian connection and the Levi-Civita connection. As a result, we show that the space of the Rarita-Schwinger fields coincides with the space of the harmonic 3-forms. Applying the same technique to a deformation theory, we also find that the space of the infinitesimal deformations of Killing spinors coincides with the direct sum of a certain eigenspace of the Laplace operator and the space of the Killi…

三维李群上的左不变Rarita-Schwinger场

Left-invariant Rarita-Schwinger fields on 3-dimensional Lie groups

本文分类了带左不变黎曼度量的二维、三维李群上的左不变Rarita-Schwinger场，将对应方程简化为代数系统，得出二维非平凡例仅在平坦阿贝尔情形、三维额外例在带特殊左不变度量的SU(2)上且场空间复维数为2的结论。

Rarita-Schwinger-Seiberg-Witten 方程的 Pin(2)-等变性

Pin(2)-equivariance of the Rarita-Schwinger-Seiberg-Witten Equations

本文在闭单连通自旋光滑 4-流形上引入含 Rarita-Schwinger 算子的 Seiberg-Witten 方程变体，利用 Pin(2) 对称性作有限维逼近，证明特定拓扑条件下解模空间非紧且非空。

Conformally Invariant Powers of Spin Operators on the Sphere

We give explicit formulas for all odd order differential intertwinors on the subbundle of the bundle of spinor-$k$-forms that are annihilated by the Clifford multiplication over the odd dimensional standard sphere. The Dirac and Rarita-Schwinger operators appear in the case of $k=0$ and $k=1$, respectively.