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Harmonic spinors on a family of Einstein manifolds. (English) Zbl 1393.53043

Summary: 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 \(P^2(\mathbb{C})\) 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 \(\mathbb{C}^2\) and extend to \(P^2(\mathbb{C})\) 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.

MSC:

53C27 Spin and Spin\({}^c\) geometry
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C80 Applications of global differential geometry to the sciences
58J20 Index theory and related fixed-point theorems on manifolds
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