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

TL;DR

This paper explores the geometric interpretation of Toda fields in relation to $W_n$-gravity, linking Higgs systems, vector bundles, and Hodge structures to understand the geometry of these theories.

Contribution

It introduces a geometric framework connecting Toda fields with Higgs systems and Hodge structures, extending the understanding of $W_n$-gravity and its relation to Riemann surface uniformization.

Findings

Toda fields are equivalent to Higgs systems.

Variations of Hodge Structures induce embeddings of Riemann surfaces.

Connections between $W_n$-geometries and vector bundle theory.

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.