Toda Field Theory as a Clue to the Geometry of $W_n$--Gravity

Abstract

We discuss geometrical aspects of Toda Fields generalizing the links between Liouville gravity and uniformization of Riemann surfaces of genus greater than one. The framework is the interplay between the hermitian and the holomorphic geometry of vector bundles on such Riemann surfaces. Pointing out how Toda fields can be considered as equivalent to Higgs systems, we show how the theory of Variations of Hodge Structures enters the game inducing local holomorphic embeddings of Riemann surfaces into homogeneous spaces. The relations of such constructions with previous realizations of $W_n$--geometries are briefly discussed.