Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 2 (2006), 077, 28 pages      math.DG/0611288      http://dx.doi.org/10.3842/SIGMA.2006.077

The Torsion of Spinor Connections and Related Structures

Frank Klinker
University of Dortmund, 44221 Dortmund, Germany

Received August 25, 2006, in final form November 03, 2006; Published online November 09, 2006

Abstract
In this text we introduce the torsion of spinor connections. In terms of the torsion we give conditions on a spinor connection to produce Killing vector fields. We relate the Bianchi type identities for the torsion of spinor connections with Jacobi identities for vector fields on supermanifolds. Furthermore, we discuss applications of this notion of torsion.

Key words: spinor connection; torsion; Killing vector; supermanifold.

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References

  1. Alekseevsky D.V., Cortés V., Devchand C., van Proeyen A., Polyvector super-Poincaré algebras, Comm. Math. Phys., 2005, V.253, 385-422, hep-th/0311107.
  2. Alekseevsky D.V., Cortés V., Devchand C., Semmelmann U., Killing spinors are Killing vector fields in Riemannian supergeometry, J. Geom. Phys., 1998, V.26, 37-50, dg-ga/9704002.
  3. Alekseevsky D.V., Cortés V., Classification of N-extended Poincaré algebras and bilinear invariants of the spinor representation of Spin(p,q), Comm. Math. Phys., 1997, V.183, 477-510, math.RT/9511215.
  4. Alishahiha M., Mohammad A. Ganjali M.A., Ghodsi A., Parvizi S., On type IIA string theory on the PP-wave background, Nuclear Phys. B, 2003, V.661, 174-190, 2003, hep-th/0207037.
  5. Alishahiha M., Kumar A., D-brane solutions from new isometries of pp-waves, Phys. Lett. B, 2002, V.542, 130-136, hep-th/0205134.
  6. Atiyah M.F., Hitchin N.J., Singer I.M., Self-duality in four-dimensional Riemannian geometry. Proc. Roy. Soc. London Ser. A, 1978, V.362, N 1711, 425-461.
  7. Bär C., Real Killing spinors and holonomy, Comm. Math. Phys., 1993, V.154, 509-521.
  8. Baum H., Complete Riemannian manifolds with imaginary Killing spinors, Ann. Global Anal. Geom., 1989, V.7, 205-226.
  9. Baum H., Friedrich T., Grunewald R., Kath I., Twistors and Killing spinors on Riemannian manifolds, Teubner-Texte zur Mathematik, Vol. 124, Stuttgart, B.G. Teubner Verlagsgesellschaft, 1991.
  10. Bergshoeff E., de Roo M., Janssen B., Ortín T., The super D9-brane and its truncations, Nuclear Phys. B, 1999, V.550, 289-302, 1999, hep-th/9901055.
  11. Bernhardt N., Nagy P.-A., On algebraic torsion forms and their spin holonomy algebra, math.DG/0608509.
  12. Blau M., Supergravity solitons, Lecture available at http://www.unine.ch/phys/string/mblau/mblau.html.
  13. Chevalley C., The algebraic theory of spinors and Clifford algebras, Collected works, Vol. 2, Berlin, Springer-Verlag, 1997 (Edited and with a foreword by P. Cartier and C. Chevalley, postface by J.-P. Bourguignon).
  14. Cremmer E., Julia B., The SO(8) supergravity, Nuclear Phys. B, 1979, V.159, 141-212.
  15. Cremmer E., Julia B., Scherk J., Supergravity theory in 11 dimensions, Phys. Lett. B, 1979, V.76, 409-412.
  16. Friedrich T., Stefan Ivanov S., Parallel spinors and connections with skew-symmetric torsion in string theory, Asian J. Math., 2002, V.6, 303-335, math.DG/0102142.
  17. Green M.B., Schwarz J.H., Witten E., Superstring theory, Vols. 1-2, 2nd ed., Cambridge Monographs on Mathematical Physics, Cambridge, Cambridge University Press, 1988.
  18. Ivanov P., Ivanov S., SU(3)-instantons and G2, Spin(7)-heterotic string solitons, Comm. Math. Phys., 2005, V.259, 79-102, math.DG/0312094.
  19. Ivanov S., Connections with torsion, parallel spinors and geometry of Spin(7) manifolds, Math. Res. Lett., 2004, V.11, 171-186, math.DG/0111216.
  20. Ivanov S., Papadopoulos G., Vanishing theorems and string backgrounds, Class. Quant. Grav., 2001, V.18, 1089-1110, math.DG/0010038.
  21. Kath I., Killing spinors on pseudo-Riemannian manifolds, Habilitation Thesis, Berlin, Humboldt University, 2000.
  22. Kath I., Parallel pure spinors on pseudo-Riemannian manifolds, in Geometry and Topology of Submanifolds X, Proceedings of the Conference on Differential Geometry in Honor of Prof. S.S. Chern (1999, Beijing - Berlin), Editors W.H. Chen et al., Singapore, World Scientific, 2000, 87-103.
  23. Kennedy A.D., Clifford algebras in 2w dimensions, J. Math. Phys, 1981, V.22, 1330-1337.
  24. Klinker F., Supersymmetric Killing structures, PhD Thesis, University Leipzig, 2003.
  25. Klinker F., Supersymmetric Killing structures, Comm. Math. Phys., 2005, V.255, 419-467.
  26. Kostant B., Graded manifolds, graded Lie theory, and prequantization. in Differential Geometrical Methods in Mathematical Physics (1975, University of Bonn), Lecture Notes in Math., Vol. 570, Berlin, Springer, Berlin, 1977, 177-306.
  27. Lawson H.B.Jr., Michelsohn M.-L., Spin geometry, Princeton Mathematical Series, Vol. 38, Princeton, NJ, Princeton University Press, 1989.
  28. Mangiarotti L., Sardanashvily G., Connections in classical and quantum field theory, Singapore, World Scientific, 2000.
  29. Moroianu A., On the infinitesimal isometries of manifolds with Killing spinors, J. Geom. Phys., 2000, V.35, 63-74.
  30. Papadopoulos G., Spin cohomology, J. Geom. Phys., 2006, V.56, 1893-1919, math.DG/0410494.
  31. Papadopoulos G., Tsimpis D., The holonomy of the supercovariant connection and Killing spinors, JHEP, 2003, V.7, 018, 28 pages, hep-th/0306117.
  32. van Proeyen A., Tools for supersymmetry, hep-th/9910030.

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