We study Ricci-flat metrics on non-compact manifolds with the exceptional holonomy $Spin(7), G_2$. We concentrate on the metrics which are defined on ${ \bf R} \times G/H$. If the homogeneous coset spaces $G/H$ have weak $G_2$, SU(3) holonomy, the manifold ${ \bf R} \times G/H$ may have $Spin(7), G_2$ holonomy metrics. Using the formulation with vector fields, we investigate the metrics with $Spin(7)$ holonomy on ${ \bf R}\times Sp(2)/Sp(1), { \bf R}\times SU(3)/U(1)$. We have found the explicit volume-preserving vector fields on these manifold using the elementary coordinate parameterization. This construction is essentially dual to solving the generalized self-duality condition for spin connections. We present most general differential equations for each coset. Then, we develop the similar formulation in order to calculate metrics with $G_2$ holonomy
Categories: hep-th