Eguchi-Hanson harmonic spinors revisited

TL;DR

This paper revisits the zero modes of the Dirac operator on Eguchi-Hanson space, introducing a simplified formalism that reproduces known results and extends to other Ricci-flat Kähler manifolds.

Contribution

It develops a spin-$c$ spinor formalism simplifying calculations of Dirac zero modes and generalizes results to Calabi's Ricci-flat Kähler manifolds.

Findings

Reproduces known normalisable zero modes on Eguchi-Hanson space

Simplifies calculations by avoiding spin connection computations

Extends analysis to Ricci-flat Kähler manifolds from Calabi's construction

Abstract

We revisit the problem of determining the zero modes of the Dirac operator on the Eguchi-Hanson space. It is well known that there are no normalisable zero modes, but such zero modes do appear when the Dirac operator is twisted by a $U(1)$ connection with $L^2$ normalisable curvature. The novelty of our treatment is that we use the formalism of spin-$c$ spinors (or spinors as differential forms), which makes the required calculations simpler. In particular, to compute the Dirac operator we never need to compute the spin connection. As a result, we are able to reproduce the known normalisable zero modes of the twisted Eguchi-Hanson Dirac operator by relatively simple computations. We also collect various different descriptions of the Eguchi-Hanson space, including its construction as a hyperk\"ahler quotient of $\mathbb{C}^4$ with the flat metric. The latter illustrates the geometric origin of the connection with $L^2$ curvature used to twist the Dirac operator. To illustrate the power of the formalism developed, we generalise the results to the case of Dirac zero modes on the Ricci-flat K\"ahler manifolds obtained by applying Calabi's construction to the canonical bundle of $\mathbb{C} P^n $.