## Supersymmetric Quotients of M-Theory and Supergravity Backgrounds

##### Abstract

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 [37]. 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 [24], 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.