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arXiv · 1705.02666

Harmonic Spinors on a Family of Einstein Manifolds

Guido Franchetti
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Abstract

The purpose of this paper is to study harmonic spinors defined on a 1-parameter family of Einstein manifolds which includes Taub-NUT, Eguchi-Hanson and $P^2(C)$ with the Fubini-Study metric as particular cases. We discuss the existence of and explicitly solve for spinors harmonic with respect to the Dirac operator twisted by a geometrically preferred connection. The metrics examined are defined, for generic values of the parameter, on a non-compact manifold with the topology of $C^2$ and extend to $P^2(C)$ as edge-cone metrics. As a consequence, the subtle boundary conditions of the Atiyah-Patodi-Singer index theorem need to be carefully considered in order to show agreement between the index of the twisted Dirac operator and the result obtained by counting the explicit solutions.

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Guido Franchetti. 2017-05-07. Harmonic Spinors on a Family of Einstein Manifolds. https://doi.org/10.1088/1361-6544%2Faab0bd

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