Toric G 2 and Spin(7) holonomy spaces from gravitational instantons and other examples

Abstract

Non-compact G 2 holonomy metrics that arise from a T 2 bundle over a hyper-Kähler space are constructed. These are one parameter deformations of certain metrics studied by Gibbons, Lü, Pope and Stelle in [1]. 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 using the Apostolov-Salamon theorem [2], we construct a new example that, still being a T 2 bundle over hyper-Kähler, represents a non-trivial two parameter deformation of the metrics studied in [1]. We then review the Spin(7) metrics arising from a T 3 bundle over a hyper-Kähler and we find a two parameter deformation of such spaces as well. We show that if the hyper-Kähler base satisfies certain properties, a non-trivial three parameter deformation is also possible. The relation between these spaces with half-flat and almost G 2 holonomy structures is briefly discussed. © 2007 Springer-Verlag.