Supersymmetric backgrounds, the Killing superalgebra, and generalised special holonomy

A bstract We prove that, for M theory or type II, generic Minkowski flux backgrounds preserving N supersymmetries in dimensions D ≥ 4 correspond precisely to integrable generalised G N structures, where G N is the generalised structure group defined by the Killing spinors. In other words, they are the analogues of special holonomy manifolds in E d d × ℝ + generalised geometry. In establishing this result, we introduce the Kosmann-Dorfman bracket, a generalisation of Kosmann’s Lie derivative of spinors. This allows us to write down the internal sector of the Killing superalgebra, which takes a rather simple form and whose closure is the key step in proving the main result. In addition, we find that the eleven-dimensional Killing superalgebra of these backgrounds is necessarily the supertranslational part of the N -extended super-Poincaré algebra.