NDLI: Killing and twistor spinors with torsion

Content Provider	Springer Nature Link
Author	Chrysikos, Ioannis
Copyright Year	2015
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.
Ending Page	141
Page Count	37
Starting Page	105
File Format	PDF
ISSN	0232704X
e-ISSN	15729060
Journal	Annals of Global Analysis and Geometry
Issue Number	2
Volume Number	49
Language	English
Publisher	Springer Netherlands
Publisher Date	2015-11-19
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Killing spinor with torsion Relations with special manifold structures (Riemannian, Finsler, etc.) Characteristic connection Homogeneous manifolds Theoretical, Mathematical and Computational Physics Parallel spinor $$\nabla $$ -Einstein structure Geometry Statistics for Business/Economics/Mathematical Finance/Insurance Cubic Dirac operator Analysis Twistor spinor Group Theory and Generalizations
Content Type	Text
Resource Type	Article
Subject	Analysis Geometry and Topology

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