Seven-dimensional Einstein Manifolds from Tod-Hitchin Geometry

TL;DR

This paper constructs an infinite family of smooth seven-dimensional Einstein manifolds with weak G_2 holonomy, derived from Tod-Hitchin geometries on principal SO(3) bundles over Bianchi IX orbifolds, with implications for M-theory and dual gauge theories.

Contribution

It introduces new smooth seven-dimensional Einstein G_2 manifolds based on Tod-Hitchin metrics, expanding the known examples and exploring their geometric and physical properties.

Findings

Infinite family of smooth G_2 Einstein metrics constructed

Geodesic equations linked to rigid body Hamiltonian systems

Potential applications in M-theory and dual gauge theories

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 parameterized 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.