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.
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In this paper all the manifolds, tensor fields and other geometric objects under consideration, are assumed to be smooth.
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Acknowledgments
The author acknowledges support by GAČR (Czech Science Foundation), post-doctoral grant GP14-24642P. He warmly thanks Ilka Agricola (Marburg) and Thomas Friedrich (Berlin) for valuable discussions that improved this work, as well as, Ivan Minchev (Brno) and Arman Taghavi-Chabert (Brno) for very useful comments. He also acknowledges FB12 at Philipps-Universität Marburg for its hospitality during a research stay in April 2015.
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Appendix A. The endomorphism \(\sigma _{T}\) on the spinor bundle \(\Sigma \)
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,
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,
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
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
Similarly, for a 7-dimensional (proper) nearly parallel \({{\mathrm{G}}}_2\)-manifold \((M^{7}, g, \omega )\) we easily conclude that
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])
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.
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Chrysikos, I. Killing and twistor spinors with torsion. Ann Glob Anal Geom 49, 105–141 (2016). https://doi.org/10.1007/s10455-015-9483-z
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DOI: https://doi.org/10.1007/s10455-015-9483-z
Keywords
- Characteristic connection
- Parallel spinor
- Twistor spinor
- Killing spinor with torsion
- \(\nabla \)-Einstein structure
- Cubic Dirac operator