In 2007 and prompted by a question of George Papadopoulos, I turned my attention to quotients again. George and collaborators had just shown that if an eleven-dimensional supergravity background preserves a fraction 31/32 of the supersymmetry, then it is locally maximally supersymmetric, so that the restricted holonomy of the superconnection is trivial. This means that its simply-connected universal cover is maximally supersymmetric, whence the background in question is a quotient of a simply-connected maximally supersymmetric background by a discrete subgroup of symmetries. In order to rule out this supersymmetry fraction, one would have to analyse the possible quotients of every maximally supersymmetric background and check that no such quotients preserves exactly a fraction 31/32.
Of course, in our work we had only looked at quotients by cyclic subgroups in the image of the exponential map, so it was not immediately clear that we could apply those results "as is". Luckily this turns out to be the case, following from the following two observations. The first is that, certainly, if there is a discrete group Γ which preserves a fraction 31/32 of the supersymmetry, then some element γ ∈ Γ must preserve such a fraction, but we do not know that γ lies in the image of the exponential map. Because the action of Γ on the Killing spinors of the backgrounds is via real matrices with unit determinant, it follows that it is enough to show that γN preserves all Killing spinors to deduce that γ does the same. For the maximally supersymmetric supergravity backgrounds, it follows moreover that for every element γ of the symmetry group, there is some some integer N for which γN is in the image of the exponential map. Therefore if we manage to show that no γ in the image of the exponential map preserves exactly a fraction 31/32, we can conclude that no such backgrounds exist.
This reduced the problem to something tractable using the classifications of possible quotients that Joan Simón and I had obtained for the cases of the Minkowski and Freund–Rubin backgrounds, as well as a new classification that was necessary for the Kowalski-Glikman wave. The resulting analysis is recorded in a paper with my student Sunil Gadhia. A consequence of this result is that there are no supergravity backgrounds (in any dimension) which preserve exactly 31/32 of the supersymmetry.
Such backgrounds are known as (BPS) preons in analogy with the preons of the Pati–Salam model of particle physics, since just as the Pati–Salam preons were meant to be the elementary constituents of quarks and leptons, so are the BPS-preons meant to be the building blocks, in a yet unspecified way, of the other BPS backgrounds.
George and collaborators have recently ruled out the existence of ten-dimensional IIB supergravity backgrounds preserving a fraction 1 < ν < 7/8 of the supersymmetry via a mixture of their local methods to deduce that any such BPS background is locally maximally supersymmetric and then an analysis of possible quotients as in my paper with Sunil.