Toric G_2 and Spin(7) holonomy spaces from gravitational instantons and other examples

TL;DR

This paper constructs and analyzes new non-compact G_2 and Spin(7) holonomy spaces as deformations of known gravitational instanton-based metrics, expanding the landscape of special holonomy manifolds with potential applications in string theory.

Contribution

It introduces new explicit examples of G_2 and Spin(7) holonomy spaces as multi-parameter deformations of existing metrics derived from hyper-Kahler bases, utilizing T^2 and T^3 bundles.

Findings

Found G_2 holonomy spaces fibered over Taub-Nut and Eguchi-Hanson instantons.

Constructed a two-parameter deformation of previously known G_2 metrics.

Identified two-parameter deformations of Spin(7) metrics and conditions for three-parameter deformations.

Abstract

Non-compact G_2 holonomy metrics that arise from a T^2 bundle over a hyper-Kahler space are discussed. These are one parameter deformations of the metrics studied by Gibbons, Lu, Pope and Stelle in hep-th/0108191. Seven-dimensional spaces with G_2 holonomy fibered over the Taub-Nut and the Eguchi-Hanson gravitational instantons are found, together with other examples. By considering the Apostolov-Salamon theorem math.DG/0303197, we construct a new example that, still being a T^2 bundle over hyper-Kahler, represents a non trivial two parameter deformation of the metrics studied in hep-th/0108191. We then review the Spin(7) metrics arising from a T^3 bundle over a hyper-Kahler and we find two parameter deformation of such spaces as well. We show that if the hyper-Kahler base satisfies certain properties, a non trivial three parameter deformations is also possible. The relation between these spaces with the half-flat structures and almost G_2 holonomy spaces is briefly discussed.