The second Dirac eigenvalue of a nearly parallel $G_2$-manifold

TL;DR

This paper studies the second eigenvalue of the Dirac operator on certain 7-dimensional manifolds with special geometric structures, providing explicit formulas and insights into its spectral properties.

Contribution

It derives explicit formulas for the second Dirac eigenvalue on nearly parallel G2-manifolds, expanding understanding of Dirac spectra in special geometric contexts.

Findings

Explicit formulas for the second Dirac eigenvalue in dimension 7

Dependence of the spectrum on eigenvalues of functions and 1-forms

Insights into spectral geometry of G2-manifolds

Abstract

We investigate the second Dirac eigenvalue on Riemannian manifolds admitting a Killing spinor. In small dimensions the whole Dirac spectrum depends on special eigenvalues on functions and 1-forms. We compute and discuss the formulas in dimension $n = 7$.