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 Kahler 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.