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 $s\ne 1/4$ implies that $(M^n,g,T)$ is both Einstein and ${\nabla }^c$-Einstein, in particular the equation ${\mathrm{Ric}}^s=\frac{{\mathrm{Scal}}^s}{n}g$ 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 Kähler manifolds and nearly parallel ${\mathrm{G}}_2$-manifolds, in dimensions 6 and 7, respectively, but also on the 3-dimensional sphere ${\mathrm{S}}^3$. We finally present applications related to the universal and twistorial eigenvalue estimate of the square of the cubic Dirac operator.

to appear in Annals of Global Analysis and Geometry, the number of pages has been reduced to 28, minor changes

References in corpus (2)

• Spinorial description of $\mathrm{SU}(3)$- and $G_2$-manifolds
• Invariant connections with skew-torsion and $\nabla$-Einstein manifolds

Cited by in corpus (5)

• Invariant connections and $\nabla$-Einstein structures on isotropy irreducible spaces
• Deformations of ${\mathrm{G}}_2$-instantons on nearly ${\mathrm{G}}_2$ manifolds
• A new $\frac{1}{2}$-Ricci type formula on the spinor bundle and applications
• Revisiting 3-Sasakian and $G_2$-structures
• A note on the volume of $\nabla$-Einstein manifolds with skew-torsion