Supersymmetric Quotients of M-Theory and Supergravity Backgrounds
In this thesis we explore discrete quotients of maximally supersymmetric supergravity backgrounds. Our main focus will be on eleven-dimensional backgrounds preserving all 32 supercharges. We shall first consider quotients of the sphere part of the maximally supersymmetric Freund-Rubin background AdS4 ×S7. Our aim will be to determine the supersymmetry preserved in the resulting background. The quotients will be by freely acting discrete subgroups G, of the isometry group of S7. These subgroups have been classified as part of a wider classification of subgroups acting freely and properly discontinuously on the n-sphere. This classification was not easy: many partial results were obtained until Wolf's solution . For each possible quotient S7/G, called a spherical space form, we shall determine if it is a spin manifold and if so how much supersymmetry, v/32 , the corresponding background AdS4 ×(S7/G) preserves. This investigation leads us to the result that spin structure and orientation dictate supersymmetry, of the quotient S7/G, thus highlighting the importance of specifying these factors as part of the data defining a supergravity background. The second part of this thesis looks at discrete quotients of all the maximally supersymmetric supergravity backgrounds in ten and eleven dimensions. In this case, our aim is to see if some discrete subgroup G of the four-form-preserving isometries of the background preserves a fraction 31 32 of the supersymmetry. Such a background with 31 supercharges is called a preon. We shall boil down this problem to checking if some element , in the image of the exponential map from the Lie algebra to the symmetry group of the background, which preserves at least 30 supercharges will preserve 31. The motivation to consider such quotients 5 comes from , where it was shown that if such backgrounds exist then they are necessarily discrete quotients of maximally supersymmetric backgrounds. We shall show that ultimately no such quotients preserve 31 32 supercharges, thus ruling out the existence of preons. The bulk of our work is on the eleven-dimensional case, however we shall also derive results for the ten-dimensional case which follow from our investigation in eleven dimensions.