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.

Appendix A. The endomorphism \(\sigma _{T}\) on the spinor bundle \(\Sigma \)

Relations (4.1) and (4.3) have another important consequence, related with the eigenspinors of the endomorphism defined by the 4-form $\sigma _{T}$ (or equivalently dT) in the Clifford algebra. In particular, the results that we describe below are known (see [31, Lem. 10.7], [11, Thm. 1.1]), but our proofs are different.

Proposition 6.6

([31, Lem. 10.7], [11, Thm. 1.1]) On a nearly Kähler manifold $(M^{6}, g, J)$ the spinor fields $\varphi ^{\pm }\in \Sigma _{\pm 2\Vert T\Vert }$ are eigenspinors of the endomorphism defined by the 4-form $\sigma _{T}$. In particular,

$$\begin{aligned} \sigma _{T}\cdot \varphi ^{\pm } =-\frac{{{\mathrm{Scal}}}^{c}}{4}\cdot \varphi ^{\pm }=-\frac{3}{2}\Vert T\Vert ^{2}\cdot \varphi ^{\pm }=-3\tau _{0}\cdot \varphi ^{\pm }. \end{aligned}$$

Similarly, on a proper nearly parallel ${{\mathrm{G}}}_2$-manifold the spinor field $\varphi _{0}\in \Sigma _{-\sqrt{7}\Vert T\Vert }$ is eigenspinor of the endomorphism defined by the 4-form $\sigma _{T}$. In particular,

$$\begin{aligned} \sigma _{T}\cdot \varphi _{0} =-\frac{{{\mathrm{Scal}}}^{c}}{4}\cdot \varphi _{0}=-3\Vert T\Vert ^{2}\cdot \varphi _{0}=-\frac{7}{12}\tau _{0}^{2}\cdot \varphi _{0}. \end{aligned}$$

Proof

We present a direct proof based on (4.1) and (4.3), respectively. Consider a local orthonormal frame $\{e_{i}\}$. In the nearly Kähler case $(M^{6}, g, J)$ and due to (4.1), it holds that

$$\begin{aligned} (e_{i}\lrcorner T)\cdot \varphi ^{\pm }=\mp \Vert T\Vert e_{i}\cdot \varphi ^{\pm }, \quad \forall i\in \{1, \ldots , 6\}. \end{aligned}$$

Then, because $(2\sigma _{T}-3\Vert T\Vert ^{2})\cdot \varphi ^{\pm }=\sum _{i}(e_{i}\lrcorner T)\cdot (e_{i}\lrcorner T)\cdot \varphi ^{\pm }$ (see Appendix C in [3]), we immediately get

$$\begin{aligned} (2\sigma _{T}-3\Vert T\Vert ^{2})\cdot \varphi ^{\pm }= & {} \mp \Vert T\Vert \sum _{i}(e_{i}\lrcorner T)\cdot e_{i}\cdot \varphi ^{\pm } =\mp 3\Vert T\Vert T\cdot \varphi ^{\pm }\\= & {} \mp 3\Vert T\Vert \Big (\pm 2\Vert T\Vert \Big )\cdot \varphi ^{\pm }=-6\Vert T\Vert ^{2}\cdot \varphi ^{\pm }. \end{aligned}$$

Similarly, for a 7-dimensional (proper) nearly parallel ${{\mathrm{G}}}_2$-manifold $(M^{7}, g, \omega )$ we easily conclude that

$$\begin{aligned} (2\sigma _{T}-3\Vert T\Vert ^{2})\cdot \varphi _{0}= & {} \sum _{i}(e_{i}\lrcorner T)\cdot (e_{i}\lrcorner T)\cdot \varphi _{0}\overset{(4.3)}{=}\frac{3\Vert T\Vert }{\sqrt{7}}\sum _{i}(e_{i}\lrcorner T)\cdot e_{i}\cdot \varphi _{0}\\= & {} \frac{9\Vert T\Vert }{\sqrt{7}} T\cdot \varphi _{0}\overset{(4.2)}{=} -9 \Vert T\Vert ^{2}\cdot \varphi _{0}, \end{aligned}$$

and the claim follows. $\square$

Combining the expression for $\sigma _{T}$ and the equality $T^{2}=-2\sigma _{T}+\Vert T\Vert ^{2}$, it is easy to compute also the action of $T^{2}$ on $\nabla ^{c}$-parallel spinors lying in $\Sigma _{\gamma }$. In particular, for a $\nabla ^{c}$-parallel spinor $\varphi$ the action $T^{2}\cdot \varphi$ in encrypted in the kernel of the Casimir operator $\Omega :=\Delta _{T}+\frac{1}{16}\Big [2{{\mathrm{Scal}}}^{g}+\Vert T\Vert ^{2}\Big ]-\frac{1}{4}T^{2}$ [7]. Any $\nabla ^{c}$-parallel belongs in the kernel of $\Omega$, hence it satisfies the equation (see for example [8])

$$\begin{aligned} T^{2}\cdot \varphi =\frac{1}{4}\Big [2{{\mathrm{Scal}}}^{g}+\Vert T\Vert ^{2}\Big ]\cdot \varphi . \end{aligned}$$

This formula in combination with $T^{2}=-2\sigma _{T}+\Vert T\Vert ^{2}$, gives rise to another way for the computation of $\sigma _{T}\cdot \varphi$. Finally, for the action $T^{2}\cdot \varphi$ the relation $T\cdot \varphi =\gamma \varphi$ can be applied twice which yields the same result, i.e. $T^{2}\cdot \varphi =\gamma ^{2}\varphi$ with $\gamma ^{2}=\frac{1}{4}\Big [2{{\mathrm{Scal}}}^{g}+\Vert T\Vert ^{2}\Big ]$ by Theorem 3.1.