Seven-dimensional Einstein manifolds from Tod–Hitchin geometry

Abstract

We construct infinitely many seven-dimensional Einstein metrics of weak holonomy G 2 . These metrics are defined on principal SO(3) bundles over four-dimensional Bianchi IX orbifolds with the Tod–Hitchin metrics. The Tod–Hitchin metric has an orbifold singularity parametrized by an integer, and is shown to be similar near the singularity to the Taub-NUT de Sitter metric with a special charge. We show, however, that the seven-dimensional metrics on the total space are actually smooth. The geodesics on the weak G 2 manifolds are discussed. It is shown that the geodesic equation is equivalent to the Hamiltonian equation of an interacting rigid body system. We also discuss M-theory on the product space of AdS 4 and the seven-dimensional manifolds, and the dual gauge theories in three dimensions.