Riemannian Metrics and Harmonic Sections of Spinor Bundles

TL;DR

This paper investigates how the lowest eigenvalue of the Dirac operator varies with the metric on spin manifolds and proves the existence of metrics with non-trivial harmonic spinors on certain manifolds.

Contribution

It introduces new insights into the spectral behavior of the Dirac operator under metric variations and establishes the existence of harmonic spinors on higher-dimensional spin manifolds.

Findings

Lowest eigenvalue clustering analyzed under metric changes

Existence of non-trivial harmonic spinors on closed spin manifolds of dimension ≥ 4

Metrics can be constructed to admit harmonic spinors

Abstract

We study the clustering of the lowest non negative eigenvalue of the Dirac operator on a general Dirac bundle when the metric structure is varied. In the classical case we show that any closed spin manifold of dimension greater than or equal to four has a Riemannian metric admitting non trivial harmonic spinors.