# Harmonic Forms and Spinors on the Taub-bolt Space

## TL;DR

This paper analyzes the space of harmonic forms and spinors on the Taub-bolt manifold, revealing a 2-dimensional space of harmonic 2-forms and explicit zero modes of the Dirac operator, with comparisons to related geometries.

## Contribution

It provides the first explicit construction of harmonic forms and spinor zero modes on Taub-bolt, extending known results from similar gravitational instantons.

## Key findings

- Harmonic 2-forms space is 2-dimensional with an explicit basis.
- Constructed a 2-parameter family of Dirac zero modes.
- Number of zero modes equals the index of the Dirac operator.

## Abstract

This paper studies the space of $L ^2 $ harmonic forms and $L ^2 $ harmonic spinors on Taub-bolt, a Ricci-flat Riemannian 4-manifold of ALF type. We prove that the space of harmonic square-integrable 2-forms on Taub-bolt is 2-dimensional and construct a basis. We explicitly find a 2-parameter family of $L ^2 $ zero modes of the Dirac operator twisted by an arbitrary $L ^2 $ harmonic connection. We also show that the number of zero modes found is equal to the index of the Dirac operator. We compare our results with those known in the case of Taub-NUT and Euclidean Schwarzschild as these manifolds present interesting similarities with Taub-bolt. In doing so, we slightly generalise known results on harmonic spinors on Euclidean Schwarzschild.