Some remarks on the geometry of superspace supergravity

JOURNAL OF GEOMETRY AND PHYSICS

卷:62

Article

Egeileh, Michel¹  El Chami, Fida² 

[1] Amer Univ Beirut, Dept Math, Beirut 11072020, Lebanon

[2] Lebanese Univ, Dept Math, Fac Sci 2, Fanar Maten 90656, Lebanon

关键词: Supergravity;    Superspace;    Killing spinors;    Cartan connections;    G-structures;   

DOI  :  10.1016/j.geomphys.2011.09.008

来源: Elsevier

【 摘 要 】

In this note, we first give a quick presentation of the supergeometry underlying supergravity theories, using an intrinsic differential geometric language. For this, we adopt the point of view of Cartan geometries, and rely as well on the work of John Lott, who has found a unified geometrical interpretation of the torsion constraints for many supergravity theories, based on the use of H-structures. In this framework, the constraints amount to requiring first-order integrability of H-structures, for a specific supergroup H. The supergroup H used by Lott is not the usual diagonal representation of the Lorentz group on superspace, but an extension of the latter. This extension appears to be natural and it can be related to the super-Poincare group. We also observe that the constraints arising from the requirement of first-order integrability have basically the same form, in any spacetime dimension. Looking at supergravity from an affine viewpoint (i.e. as a gauge theory for the super-Poincare group), we show that requiring first-order integrability amounts to requiring the equivalence, up to gauge transformations, between infinitesimal gauge supertranslations acting on the supervielbein and infinitesimal superdiffeomorphisms acting on the supervielbein. The latter action is performed through a covariant Lie derivative, whose expression involves naturally the supertorsion tensor. We use this expression to show that the term added to the spin connection, in the supercovariant derivative of d = 11 supergravity, has a natural superspace origin. In particular, the 4-form field strength is related to a specific component of the supertorsion tensor. We conclude by some general remarks concerning Killing spinors in geometry and supergravity, discussing their possible interpretations, as Killing vector fields on a specific supermanifold on one hand, and as parallel spinors for an appropriate connection on the other hand. We show that this last interpretation is very natural from the point of view of Klein and Cartan geometries. (C) 2011 Elsevier B.V. All rights reserved.

【 授权许可】

Free