Gravitational instantons, extra dimensions and form fields

A broad class of higher-dimensional instanton solutions are found for a the ory which contains gravity, a scalar field and antisymmetric tensor fields of arbitrary rank. The metric used, a warp product of an arbitrary number o f any compact Einstein manifolds, includes many of great interest in partic le physics and cosmology. For example 4D FRW universes with additional dime nsions compactified on a Calabi-Yau three fold, a torus, a compact hyperbol ic manifold or a sphere are all included. It is shown that the solution of this form which dominates the Hartle Hawking path integral is always a high er-dimensional generalisation of a Hawking Turok instanton when the potenti al of the scalar field is such that these instantons can exist. On continua tion to lorentzian signature such instantons give rise to a spacetime in wh ich all of the spatial dimensions are of equal size and where the spatial t opology is that of a sphere. The extra dimensions are thus not hidden. In t he case where the potential for the scalar field is generated solely by a d ilatonic coupling to the form fields we find no integrable instantons at al l. In particular we find no integrable solutions of the type under consider ation for the supergravity theories which are the low energy effective fiel d theories of superstrings.