Supersymmetric AdS backgrounds and weak generalised holonomy

Abstract

We study supersymmetric AdS$_D$ backgrounds of eleven-dimensional or type II supergravity preserving $\mathcal{N}$ supersymmetries using generalised geometry. We show that a large class correspond precisely to spaces admitting a generalised $G_{D,\mathcal{N}}$ structure with a weak integrability condition, which we call "weak generalised holonomy". This class contains all such backgrounds with odd $D$, but for even $D$ we must impose a technical assumption concerning the inner products of Killing spinors. It is known that all compact smooth solutions are included, though our analysis is purely local. We show that the Killing superalgebras of such backgrounds are isomorphic to the standard $\mathcal{N}$-extended AdS$_D$ superalgebras, and give a geometric proof of the existence of isometries of the internal spaces which realise the AdS R-symmetry, as often assumed in explicit studies of such solutions.