Geometry of Higgs and Toda Fields on Riemann Surfaces

TL;DR

This paper explores the geometric relationship between Higgs systems and Toda field theory on higher-genus Riemann surfaces, highlighting their equivalence through vector bundle connections and implications for complex geometry.

Contribution

It demonstrates the equivalence of Toda fields and Higgs systems via vector bundle connections and explores their role in Hodge structures and $W_n$-geometries.

Findings

Higgs systems can be represented as Toda fields on Riemann surfaces.

Variations of Hodge structures relate to embeddings into homogeneous spaces.

Insights into $W_n$-geometries and their geometric realizations.

Abstract

We discuss geometrical aspects of Higgs systems and Toda field theory in the framework of the theory of vector bundles on Riemann surfaces of genus greater than one. We point out how Toda fields can be considered as equivalent to Higgs systems -- a connection on a vector bundle $E$ together with an End($E$)--valued one form both in the standard and in the Conformal Affine case. We discuss how variations of Hodge structures can arise in such a framework and determine holomorphic embeddings of Riemann surfaces into locally homogeneous spaces, thus giving hints to possible realizations of $W_n$--geometries. }