Cohomogeneity one manifolds of Spin(7) and G(2) holonomy
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In this paper, we look for metrics of cohomogeneity one in D = 8 and 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 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 CP 3, with a nontrivial parameter that characterizes the homogeneous squashing of CP 3. We then consider the case where the principal orbits are the Aloff-Wallach spaces N(k,l) = SU(3)/U(1), 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. Cveti, G. W. Gibbons, H. L, and C. N. Pope, Nucl. Phys. B617, 151 (2001)]. These include certain explicit ALC metrics for all N(k,l), 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 nonsingular metrics with principal orbits that are S 3S 3. Additional topics include a detailed and explicit discussion of the Einstein metrics on N(k,l), and an explicit parametrization of SU(3). 2002 The American Physical Society.
