Abstract
In the first part of the Thesis, we investigate some background material which is utilized in the second part, where the original research is presented. We present the proof of how the supercovariant derivative acting on a spinor is Spin-gauge covariant. We introduce the spinorial geometry techniques used to analyze the Killing spinor equations (KSEs), and give an example in the case of the gaugino KSE of certain warped product solutions of D = 10 heterotic supergravity. We describe the isometries of de Sitter space and we give the ansatz for warped product dS4 solutions. Then, we briefly prove some classical no-go theorems for warped product de Sitter solutions.
In the second part of the Thesis, the necessary and sufficient conditions for warped product dS4 solutions in D =11 supergravity to preserve the minimal N = 8 supersymmetry are determined. We find, on integrating the KSE along the dS4 directions, that the necessary and sufficient conditions for supersymmetry are encoded in a single gravitino-type equation, which is satisfied by a spinor ψ whose components depend only on the co-ordinates of the internal space. The spinor ψ is associated with two possible stabilizer groups, SU(3) or G2. We derive explicitly the Spin(7) gauge transformations which are used to write ψ in simple canonical forms with stabilizer subgroups SU(3) or G2. We then solve the linear system obtained from the KSEs. In particular, we show that the linear system implies there are no solutions for which the stabilizer of ψ is G2. For the case of SU(3) stabilizer subgroup, the KSEs determine all components of the 4-form flux in terms of the geometry of the internal manifold, and we present the geometric conditions and the components of the flux, written in a SU(3) covariant fashion.