Trivial Isotropy Group Sample Clauses

Trivial Isotropy Group. Now, if the isotropy group of the parallel spinors is the trivial group {1}, the inte- grability condition in (2.24) means that Rˆ = 0 , F = 0 . (2.27) The spacetime is parallelisable with respect to a connection with skew-symmetric torsion and admits eight parallel spinors. Moreover, the torsion is anti-self-dual. All such spacetimes are group manifolds with anti-self-dual structure constants. We will discuss these backgrounds in more detail in section 2.9.

Trivial Isotropy Group. Backgrounds with parallel spinors which have a trivial isotropy group admit eight parallel spinors. These are maximally supersymmetric backgrounds. The spacetime is a Lorentzian Lie group with anti-self-dual structure constants. They have been classified in a similar context in [58]. In particular, the spacetime is locally isometric to R5,1 , AdS3 × S3 , CW6 , (2.169) where the radii of AdS3 and S3 are equal, and the structure constants of CW6 are given by a constant self-dual 2-form on R4. Moreover, F (C) = 0 , (2.170) which we infer from the integrability condition (2.24). This concludes the conditions which arise from the gravitino KSE. The gaugino KSE implies the gauge field strength vanishes and µr' = 0. The tensorini KSE implies that the 3-form field strengths vanish and the tensor multiplet scalars are constants. Similarly, the hyperini KSE implies that the scalars of the hypermultiplet are constant. In turn using (2.7), the latter gives C = 0.