Coset construction of Spin (7), G 2 gravitational instantons

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Abstract
We study Ricci-flat metrics on non-compact manifolds with the exceptional holonomy Spin(7), G2. We concentrate on the metrics which are defined on R × G/H. If the homogeneous coset spaces G/H have weak G2, SU(3) holonomy, the manifold R × G/H may have Spin(7), G2 holonomy metrics. Using the formulation with vector fields, we investigate the metrics with Spin(7) holonomy on R × Sp(2)/Sp(1), R × SU(3)/U(1). We have found the explicit volume-preserving vector fields on these manifolds using the elementary coordinate parametrization. This construction is essentially dual for solving the generalized self-duality condition for spin connections. We present the most general differential equations for each coset. Then, we develop a similar formulation in order to calculate metrics with G2 holonomy.

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