On the eigenvalues of the twisted Dirac operator

TL;DR

This paper investigates how the eigenvalues of the twisted Dirac operator behave under certain connections on compact Riemannian spin manifolds with positive scalar curvature, revealing conditions for arbitrarily small eigenvalues and establishing lower bounds.

Contribution

It introduces a family of connections that make the first eigenvalue arbitrarily small and shows that restricting to Hermitian-Einstein connections yields nonzero lower bounds.

Findings

First eigenvalue can be made arbitrarily small with specific connections.

Nonzero lower bounds exist when restricting to Hermitian-Einstein connections.

The results apply to Riemann surfaces with positive scalar curvature.

Abstract

Given a compact Riemannian spin manifold with positive scalar curvature, we find a family of connections $\nabla^{A_t}$ for $t\in[0,1]$ on a trivial vector bundle of sufficiently high rank, such that the first eigenvalue of the twisted Dirac operator $D_{A_t}$ is nonzero and becomes arbitrarily small as $t\to1$. However, if one restricts the class of twisting connections considered, then nonzero lower bounds do exist. We illustrate this fact by establishing a nonzero lower bound for the Dirac operator twisted by Hermitian-Einstein connections over Riemann surfaces.