We propose a new approach to define theories of random metrics in two and higher dimensions, based on recent methods in Kahler geometry. The main idea is to use symmetric spaces of Bergman metrics, parameterized by large N hermitian matrices and converging to the full space of Kahler metrics in large N limit. This approach suggests the relevance of a new gravitational effective action, the Mabuchi functional, which plays a key role in the Kahler geometry. It appears naturally when a matter field theory is coupled to gravity, and generalizes the standard Liouville model in two dimensions.

Universite Libre de Bruxelles