Abstract
In this paper, we look for metrics of cohomogeneity one in and 7 dimensions with Spin(7) and holonomy, respectively. In we first consider the case of principal orbits that are viewed as an bundle over with triaxial squashing of the fibers. 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 nonsingular asymptotically locally conical (ALC) Spin(7) metrics on line bundles over with a nontrivial parameter that characterizes the homogeneous squashing of We then consider the case where the principal orbits are the Aloff-Wallach spaces where the integers k and l characterize the embedding of U(1). We find new ALC and asymptotically conical (AC) metrics of Spin(7) holonomy, as solutions of the first-order equations that we obtained previously [M. Cvetič, G. W. Gibbons, H. Lü, and C. N. Pope, Nucl. Phys. B617, 151 (2001)]. These include certain explicit ALC metrics for all and numerical and perturbative results for ALC families with AC limits. We then study metrics of holonomy, and find new explicit examples, which, however, are singular, where the principal orbits are the flag manifold We also obtain numerical results for new nonsingular metrics with principal orbits that are Additional topics include a detailed and explicit discussion of the Einstein metrics on and an explicit parametrization of SU(3).
- Received 20 November 2001
DOI:https://doi.org/10.1103/PhysRevD.65.106004
©2002 American Physical Society