Harmonic Spinors on a Family of Einstein Manifolds

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.

Updated to match the published version

References in corpus (3)

• Taub-NUT Dynamics with a Magnetic Field
• Harmonic forms on ALF gravitational instantons
• Spectral Properties of Schwarzschild Instantons

Cited by in corpus (4)

• Axial anomaly in nonlinear conformal electrodynamics
• Gravitating anisotropic merons and squashed spheres in the three-dimensional Einstein-Yang-Mills-Chern-Simons theory
• Harmonic Forms and Spinors on the Taub-bolt Space
• Gravitational instantons and the quality problem of the QCD axion: Facts, speculations, and statements in between