On the maximal superalgebras of supersymmetric backgrounds

TL;DR

This paper defines and constructs maximal superalgebras for certain supersymmetric backgrounds, showing their structure, classification, and explicit examples, including AdS_4 x S^7, and explores their properties and limitations.

Contribution

It provides a precise geometric construction and classification of maximal superalgebras for supersymmetric backgrounds, extending known constructions and analyzing specific cases.

Findings

Maximal superalgebras are classified into finitely many isomorphism classes.

Maximally supersymmetric waves lack maximal superalgebras.

The maximal superalgebra of AdS_4 x S^7 is isomorphic to osp(1|32).

Abstract

In this note 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 AdS_4 x S^7 and find that is isomorphic to osp(1|32). We propose an algebraic construction of the maximal superalgebra of any background asymptotic to AdS_4 x S^7 and we test this proposal by computing the maximal superalgebra of the M2-brane in its two maximally supersymmetric limits, finding agreement.