Abstract
In this thesis we explore discrete quotients of maximally sup ersymmetric sup er- gravity backgrounds. Our main fo cus will b e on eleven-dimensional backgrounds preserving all 32 sup ercharges. We shall rst consider quotients of the sphere part of the maximally sup er- symmetric Freund-Rubin background AdS 4 × S 7 . Our aim will b e to determine the sup ersymmetry preserved in the resulting background. The quotients will b e by freely acting discrete subgroups G , of the isometry group of S 7 . These subgroups have b een classied as part of a wider classication of subgroups act- ing freely and prop erly discontinuously on the n -sphere. This classication was not easy: many partial results were obtained until Wolf 's solution [37]. For each p ossible quotient S 7 /G , called a spherical space form , we shall determine if it is a spin manifold and if so how much sup ersymmetry, ν 32 , the corresp onding background AdS 4 × ( S 7 /G ) preserves. This investigation leads us to the result that spin structure and orientation dictate sup ersymmetry, of the quotient S 7 /G , thus highlighting the imp ortance of sp ecifying these factors as part of the data dening a sup ergravity background. The second part of this thesis lo oks at discrete quotients of all the maximally sup ersymmetric sup ergravity 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 sup ersymmetry. Such a background with 31 sup ercharges is called a preon. We shall b oil down this problem to checking if some element γ , in the image of the exp onential map from the Lie algebra to the symmetry group of the background, which preserves at least 30 sup ercharges will preserve 31 . The motivation to consider such quotients comes from [24], where it was shown that if such backgrounds exist then they are necessarily discrete quotients of maximally sup ersymmetric backgrounds. We shall show that ultimately no such quotients preserve 31 32 sup ercharges, 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.