On the maximal superalgebras of supersymmetric backgrounds

In this paper we give a precise definition of the notion of a maximal superalgebra of certain types of supersymmetric supergravity backgrounds, including the Freund-Rubin backgrounds, and propose a geometric construction extending the well-known construction of its Killing superalgebra. We determine the structure of maximal Lie superalgebras and show that there is a finite number of isomorphism classes, all related via contractions from an orthosymplectic Lie superalgebra. We use the structure theory to show that maximally supersymmetric waves do not possess such a maximal superalgebra, but that the maximally supersymmetric Freund-Rubin backgrounds do. We perform the explicit geometric construction of the maximal superalgebra of AdS4 X S7 and find that it is isomorphic to osp(1|32). We propose an algebraic construction of the maximal superalgebra of any background asymptotic to AdS4 X S7 and we test this proposal by computing the maximal superalgebra of the M2-brane in its two maximally supersymmetric limits, finding agreement.