Bianchi IX Self-dual Einstein Metrics and Singular G_2 Manifolds

TL;DR

This paper constructs explicit G_2 holonomy metrics foliated by twistor spaces over Bianchi IX Einstein metrics, analyzing their singularities and implications for M-theory.

Contribution

It provides new explicit cohomogeneity two G_2 metrics with detailed analysis of their singularities and global properties, especially in the biaxial case.

Findings

Metrics have orbifold or cosmic-string type singularities.

Complete analysis of biaxial Bianchi IX metrics.

Relation to Tod and Hitchin's equations in the triaxial case.

Abstract

We construct explicit cohomogeneity two metrics of G_2 holonomy, which are foliated by twistor spaces. The twistor spaces are S^2 bundles over four-dimensional Bianchi IX Einstein metrics with self-dual (or anti-self-dual) Weyl tensor. Generically the 4-metric is of triaxial Bianchi IX type, with SU(2) isometry. We derive the first-order differential equations for the metric coefficients, and obtain the corresponding superpotential governing the equations of motion, in the general triaxial Bianchi IX case. In general our metrics have singularities, which are of orbifold or cosmic-string type. For the special case of biaxial Bianchi IX metrics, we give a complete analysis their local and global properties, and the singularities. In the triaxial case we find that a system of equations written down by Tod and Hitchin satisfies our first-order equations. The converse is not always true. A discussion is given of the possible implications of the singularity structure of these spaces for M-theory dynamics.