Eight-manifolds with G-structure in eleven dimensional supergravity

Abstract

We extend the refined G-structure classification of supersymmetric solutions of eleven dimensional supergravity. We derive necessary and sufficient conditions for the existence of an arbitrary number of Killing spinors whose common isotropy group contains a compact factor acting irreducibly in eight spatial dimensions and which embeds in $(Spin(7)\ltimes\mathbb{R}^8)\times\mathbb{R}$. We use these conditions to explicitly derive the general local bosonic solution of the Killing spinor equation admitting an N=4 SU(4) structure embedding in a $(Spin(7)\ltimes\mathbb{R}^8)\times\mathbb{R}$ structure, up to an eight-manifold of SU(4) holonomy. Subject to very mild assumptions on the form of the metric, we explicitly derive the general local bosonic solutions of the Killing spinor equation for N=6 Sp(2) structures and N=8 $SU(2)\times SU(2)$ structures embedding in a $(Spin(7)\ltimes\mathbb{R}^8)\times\mathbb{R}$ structure, again up to eight-manifolds of special holonomy. We construct several other classes of explicit solutions, including some for which the preferred local structure group defined by the Killing spinors does not correspond to any holonomy group in eleven dimensions. We also give a detailed geometrical characterisation of all supersymmetric spacetimes in eleven dimensions admitting G-structures with structure groups of the form $(G\ltimes\mathbb{R}^8)\times\mathbb{R}$.