Coset Construction of $Spin(7), G_2$ Gravitational Instantons

We study Ricci-flat metrics on non-compact manifolds with the exceptional holonomy $Spin(7), G_2$. We concentrate on the metrics which are asymptotically conical and defined on ${\bf R} \times G/H$. Using the formulation with vector fields based on the higher-dimensional genaralization of Ashtekar gravity, we investigate the metrics with $Spin(7)$ holonomy on ${\bf R}\times Sp(2)/Sp(1), {\bf R}\times SU(3)/U(1)$ recently found by Cveti\v{c}, Gibbons, L\"u and Pope [hep-th/0103155]. We have found the explicit volume-preserving vector fields on these manifold using the elementary coordinate parametrization. Furthermore, we develop the similar formulation in order to find metrics with $G_2$ holonomy. We also note that there should be no more explicit metrics with $Spin(7)$ holonomy defined by coset spaces $G/H$.