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Twisted Dirac operators and generalized gradients

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Abstract

On Riemannian or spin manifolds, there are geometric first order differential operators called generalized gradients. In this article, we prove that the Dirac operator twisted with an associated bundle is a linear combination of some generalized gradients. This observation allows us to find all the homomorphism type Weitzenböck formulas. We also give some applications.

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Acknowledgments

This article was almost completed while the author stayed at Stuttgart University as a visiting researcher during his one year sabbatical. Thanks to all staffs of the institute of Geometry and Topology in Stuttgart University. Especially, the author is grateful to U. Semmelmann for his hospitality, fruitful discussions about spin geometry, and helpful comments to this article. The author also thanks D. Eelbode for a useful discussion about the twisted Dirac operator. This work was partially supported in part by JSPS KAKENHI Grant Number 15K04858.

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Authors and Affiliations

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

    Yasushi Homma

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  1. Yasushi Homma
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Correspondence to Yasushi Homma.

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Homma, Y. Twisted Dirac operators and generalized gradients. Ann Glob Anal Geom 50, 101–127 (2016). https://doi.org/10.1007/s10455-016-9503-7

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  • DOI: https://doi.org/10.1007/s10455-016-9503-7

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