Killing and twistor spinors with torsion

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.