Killing and twistor spinors with torsion

TL;DR

This paper investigates twistor and Killing spinors with torsion on Riemannian spin manifolds, establishing conditions for their existence, properties, and relations to Einstein metrics, with applications to eigenvalue estimates of Dirac operators.

Contribution

It introduces a new perspective on twistor spinors with torsion as parallelism conditions, and links Killing spinors with torsion to Einstein geometry and Dirac operator eigenvalues.

Findings

Twistor spinors with torsion have isolated zero points.

Existence of certain torsion eigenspinors implies Einstein and $ abla^{c}$-Einstein conditions.

Constructs families of Killing spinors with torsion on special manifolds.

Abstract

We study twistor spinors (with torsion) on Riemannian spin manifolds $(M^{n}, g, T)$ carrying metric connections with totally skew-symmetric torsion. We consider the characteristic connection $\nabla^{c}=\nabla^{g}+\frac{1}{2}T$ and under the condition $\nabla^{c}T=0$, we show that the twistor equation with torsion w.r.t. the family $\nabla^{s}=\nabla^{g}+2sT$ can be viewed as a parallelism condition under a suitable connection on the bundle $\Sigma\oplus\Sigma$, where $\Sigma$ is the associated spinor bundle. Consequently, we prove that a twistor spinor with torsion has isolated zero points. Next we study a special class of twistor spinors with torsion, namely these which are $T$-eigenspinors and parallel under the characteristic connection; we show that the existence of such a spinor for some $s\neq 1/4$ implies that $(M^{n}, g, T)$ is both Einstein and $\nabla^{c}$-Einstein, in particular the equation ${\rm Ric}^{s}=\frac{{\rm Scal}^{s}}{n}g$ holds for any $s\in\mathbb{R}$. In fact, for $\nabla^{c}$-parallel spinors we provide a correspondence between the Killing spinor equation with torsion and the Riemannian Killing spinor equation. This allows us to describe 1-parameter families of non-trivial Killing spinors with torsion on nearly K\"ahler manifolds and nearly parallel ${\rm G}_2$-manifolds, in dimensions 6 and 7, respectively, but also on the 3-dimensional sphere ${\rm S}^{3}$. We finally present applications related to the universal and twistorial eigenvalue estimate of the square of the cubic Dirac operator.