Twistorial eigenvalue estimates for generalized Dirac operators with torsion

TL;DR

This paper derives optimal eigenvalue bounds for Dirac operators with torsion on compact spin manifolds using twistor theory, extending classical estimates and exploring special solutions like twistor and Killing spinors with torsion.

Contribution

It introduces new eigenvalue estimates for Dirac operators with torsion and develops associated twistor and Killing equations incorporating torsion effects.

Findings

Established an optimal lower bound for the first eigenvalue with torsion

Derived new twistor and Killing equations with torsion

Characterized cases where the bound is attained

Abstract

We study the Dirac spectrum on compact Riemannian spin manifolds $M$ equipped with a metric connection $\nabla$ with skew torsion $T\in\Lambda^3M$ by means of twistor theory. An optimal lower bound for the first eigenvalue of the Dirac operator with torsion is found that generalizes Friedrich's classical Riemannian estimate. We also determine a novel twistor and Killing equation with torsion and use it to discuss the case in which the minimum is attained in the bound.