Cohomogeneity One Manifolds of Spin(7) and G(2) Holonomy

TL;DR

This paper constructs and analyzes new cohomogeneity one metrics with Spin(7) and G(2) holonomy in 7 and 8 dimensions, using numerical and explicit methods to find non-singular and singular solutions with various principal orbits.

Contribution

It introduces new explicit and numerical cohomogeneity one metrics with Spin(7) and G(2) holonomy, expanding known solutions and exploring their geometric properties.

Findings

Established existence of new non-singular asymptotically locally conical Spin(7) metrics.

Found explicit and numerical examples of Spin(7) metrics on line bundles over P^3 and Aloff-Wallach spaces.

Discovered new explicit G(2) holonomy metrics, including singular solutions and non-singular examples with S^3 tion S^3 or flag manifold orbits.

Abstract

In this paper, we look for metrics of cohomogeneity one in D=8 and D=7 dimensions with Spin(7) and G_2 holonomy respectively. In D=8, we first consider the case of principal orbits that are S^7, viewed as an S^3 bundle over S^4 with triaxial squashing of the S^3 fibres. This gives a more general system of first-order equations for Spin(7) holonomy than has been solved previously. Using numerical methods, we establish the existence of new non-singular asymptotically locally conical (ALC) Spin(7) metrics on line bundles over \CP^3, with a non-trivial parameter that characterises the homogeneous squashing of CP^3. We then consider the case where the principal orbits are the Aloff-Wallach spaces N(k,\ell)=SU(3)/U(1), where the integers k and \ell characterise the embedding of U(1). We find new ALC and AC metrics of Spin(7) holonomy, as solutions of the first-order equations that we obtained previously in hep-th/0102185. These include certain explicit ALC metrics for all N(k,\ell), and numerical and perturbative results for ALC families with AC limits. We then study D=7 metrics of $G_2$ holonomy, and find new explicit examples, which, however, are singular, where the principal orbits are the flag manifold SU(3)/(U(1)\times U(1)). We also obtain numerical results for new non-singular metrics with principal orbits that are S^3\times S^3. Additional topics include a detailed and explicit discussion of the Einstein metrics on N(k,\ell), and an explicit parameterisation of SU(3).