Cohomogeneity one manifolds of Spin(7) andG2holonomy

In this paper, we look for metrics of cohomogeneity one in $D=8\mathrm{}$ 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 ${\mathrm{CP}}^3,$ with a nontrivial parameter that characterizes the homogeneous squashing of ${\mathrm{CP}}^3.$ We then consider the case where the principal orbits are the Aloff-Wallach spaces $N(k,l)=\mathrm{SU}\mathrm{}\left(3\right)/\mathrm{U}\mathrm{}\left(1\right),$ 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 $N(k,l),$ and numerical and perturbative results for ALC families with AC limits. We then study $D=7\mathrm{}$ metrics of $G_2$ holonomy, and find new explicit examples, which, however, are singular, where the principal orbits are the flag manifold $\mathrm{SU}\mathrm{}\left(3\right)/\left[\mathrm{U}\mathrm{}\right(1)\mathrm{}\times \mathrm{U}\mathrm{}(1\left)\mathrm{}\right].$ We also obtain numerical results for new nonsingular metrics with principal orbits that are $S^3\times S^3.$ Additional topics include a detailed and explicit discussion of the Einstein metrics on $N(k,l),$ and an explicit parametrization of SU(3).