Eleven-dimensional symmetric supergravity backgrounds, their geometric superalgebras, and a common reduction

We present two different families of eleven-dimensional manifolds that admit non-restricted extensions of their isometry algebras to geometric superalgebras. Both families admit points for which the superalgebra extends to a super Lie algebra; on the one hand, a family of N = 1, ν = 3/4 supergravity backgrounds and, on the other hand, an N = 1, ν = 1 supergravity background. Furthermore, both families admit a point that can be identified with an N = 4, ν = 1/2 six-dimensional supergravity background