Harmonic spinors on the Davis hyperbolic 4-manifold

TL;DR

This paper demonstrates that the Davis hyperbolic 4-manifold admits harmonic spinors using the G-spin theorem, marking the first such example among closed hyperbolic 4-manifolds, and provides explicit descriptions of spinor bundles and sign calculations.

Contribution

It provides the first example of a closed hyperbolic 4-manifold with harmonic spinors and details the computation of sign terms in the G-spin theorem for hyperbolic manifolds.

Findings

Davis hyperbolic 4-manifold admits harmonic spinors

Explicit description of spinor bundles for hyperbolic manifolds

Calculation method for sign terms in G-spin theorem

Abstract

In this paper we use the G-spin theorem to show that the Davis hyperbolic 4-manifold admits harmonic spinors. This is the first example of a closed hyperbolic 4-manifold that admits harmonic spinors. We also explicitly describe the Spinor bundle of a spin hyperbolic 2- or 4-manifold and show how to calculated the subtle sign terms in the G-spin theorem for an isometry, with isolated fixed points, of a closed spin hyperbolic 2- or 4-manifold.