Abstract
This thesis is devoted to the study of geometric structures in supergravity. In the first part of the thesis, we classify extreme near-horizon geometries in D =11 supergravity preserving four supersymmetries, utilising spinorial geometry techniques. In particular, we show that the Killing spinors fall into three possible orbits, corresponding to pairs of spinors defned on the spatial cross-sections of the horizon which have isotropy groups SU(3), G_2, or SU(4). In each case, we determine the conditions on the geometry and the 4-form flux imposed by supersymmetry. We also analyse the integrability conditions arising from the Killing spinor equations of D=11 supergravity. Subsequently, we study supersymmetric near-horizon geometries in the context of heterotic supergravity. Firstly, we obtain a necessary and sufficient condition for a solution to preserve more than the minimal N = 2 supersymmetry. Then, we construct a supersymmetric near-horizon solution which is a U(1) fibration of AdS_3 over a particular Aloff-Wallach space. We prove that this solution preserves the conditions required for N=2 supersymmetry, but does not satisfy the necessary condition required for further supersymmetry enhancement. Hence, we conclude that there exist supersymmetric near-horizon heterotic solutions preserving exactly two supersymmetries.
In the second part of the thesis, we initiate a systematic classification of warped product de-Sitter solutions in D=11 and heterotic supergravity, the latter up to two loops in sigma model perturbation theory. Firstly, we classify all warped product dSn solutions for n >= 5 in D= 11 supergravity. In particular, we find that all D=11 warped product dSn backgrounds for 7 <= n <= 10 are flat, with vanishing 4-form. Moreover, we show that D=11 warped product dS_6 backgrounds are either the maximally supersymmetric AdS_7 x S^4 solution, or R^{1;6} x N_4 where N_4 is a 4-dimensional hyperKahler manifold, with vanishing 4-form. Furthermore, we prove that D=11 warped product dS5 backgrounds are generalized M5-brane geometries, for which the transverse space is R x N_4, where N_4 is a 4-dimensional hyperKahler manifold. Subsequently, we show that warped product dSn geometries in heterotic supergravity up to two loops are very restricted. In fact, we prove that for n>= 3 all such solutions are R^{1;n} xM_{9-n} , where M is a (9-n)-dimensional Riemannian manifold and for n= 2 all such solutions are AdS_3xM_7, where M_7 is a 7-dimensional Riemannian manifold.