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, ) = SU(3)/U (1), where the integers k and 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, ), 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) × U (1)). We also obtain numerical results for new non-singular metrics with principal orbits that are S 3 × S 3 . Additional topics include a detailed and explicit discussion of the Einstein metrics on N (k, ), and an explicit parameterisation of SU(3). C 2002 Elsevier Science (USA) CONTENTS 1. Introduction. 2. New Spin(7) metrics with triaxial S 3 bundle over S 4 . 2.1. Ansatz and first-order equations. 2.2. Some properties of the equations. 2.3. Numerical analysis. 3. Spin( ) metrics with SU(3)/U (1) principal orbits. 3.1. The principal orbits: Aloff-Wallach spaces. 3.2. An explicit Spin(7) Solution for All N (k, ). 3.3. Small distance behaviour and numerical analysis. 4. Analytic results for the Spin(7) equations for N (k, ) orbits. 4.1. The general case N (k, ). 4.2. The Case N (1, -1). 4.3.
Perturbative construction of AC metrics.