# Harmonic Spinors on a Family of Einstein Manifolds

## TL;DR

This paper investigates harmonic spinors on a family of Einstein manifolds, providing explicit solutions and analyzing index theory implications, including boundary conditions for non-compact and edge-cone geometries.

## Contribution

It explicitly solves for harmonic spinors on a parameterized family of Einstein manifolds, connecting geometric analysis with index theory in non-compact and edge-cone settings.

## Key findings

- Explicit solutions for harmonic spinors on the manifolds.
- Analysis of boundary conditions for the Atiyah-Patodi-Singer index theorem.
- Confirmation of index calculations via explicit solutions.

## 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.