Spin(7)Spin(7)-manifolds with three-torus symmetry

Metrics of exceptional holonomy are vacuum solutions to the Einstein equation. In this paper we describe manifolds with holonomy contained in Spin(7)Spin(7) preserved by a three-torus symmetry in terms of tri-symplectic geometry of four-manifolds. These complement examples that have appeared in the context of domain wall problems in supergravity.

► Metrics of exceptional holonomy are vacuum solutions to the Einstein equation.
► We describe manifolds with holonomy in Spin(7)Spin(7) preserved by 3-torus symmetry.
► This is done in terms of tri-symplectic geometry of 4-manifolds.
► We complement examples that appeared in context of domain wall problems in sugra.