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

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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, G2 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.

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Acknowledgements

We would like to thank Anand Dessai for several helpful comments and for his interest in our work. We also thank the anonymous referee for helpful suggestions. The first author was partially supported by JSPS KAKENHI Grant Number JP15K04858. Both authors were partially supported by the DAAD-Waseda University Partnership Programme.

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Authors and Affiliations

  1. Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Tokyo, Shinjuku-ku, 169-8555, Japan

    Yasushi Homma

  2. Institut für Geometrie und Topologie, Fachbereich Mathematik, Universität Stuttgart, Pfaffenwaldring 57, 70569, Stuttgart, Germany

    Uwe Semmelmann

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  1. Yasushi Homma
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  2. Uwe Semmelmann
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Correspondence to Uwe Semmelmann.

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Communicated by P. Chrusciel

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Homma, Y., Semmelmann, U. The Kernel of the Rarita–Schwinger Operator on Riemannian Spin Manifolds. Commun. Math. Phys. 370, 853–871 (2019). https://doi.org/10.1007/s00220-019-03324-8

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