Geometry of Higgs and Toda fields on Riemann surfaces

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.

Higher dimensional uniformisation and W-geometry

“…Since the moduli of these spaces correspond to variation of Hodge structures [9,15], there must exist a description of these structures directly in the higher dimension. This can be done via Higgs bundles in higher dimensions which has been studied by Simpson [28].…”

“…But their work does not seem to be related to the Fuchsian uniformisation presented in this paper. 3 Alvarez-Gaumé discusses 3 The difference can be seen by considering the well understood case of Fuchsian uniformithe problem of uniformisation in his ICTP lectures [14] and more recently, Aldrovandi and Falqui have also attempted to understand uniformisation using period maps [15].…”

Toda fields on riemann surfaces: Remarks on the Miura transformation

“…Let C be a compact Riemann surface of genus g > 1. It has been shown that (1.1) defines a holomorphic connection on the vector bundle E = J n−1 (K C − n−1 2 ) of (n − 1)-jets of holomorphic sections of K C − n−1 2 , where K C is the canonical line bundle [3,16]. On the other hand, (1.2) defines a meromorphic connection on the vector bundle E = n−1 r=0 K C − n−1 2 +r [1,17].…”

“…We consider the pair (E, ∇) where E → C is the jet-bundle mentioned before. It is determined by the following condition [3]. Let us temporarily append the subscript "n", so that E (n) ≡ E; then E n is characterized as the extension…”

“…(We follow [3,6,17], a different naming convention was adopted in [1].) This is true in particular for the zero curvature representation of the Toda equations in the holomorphic gauge [3,17]. On the other hand, there are other relevant gauges [6], and in particular the "diagonal gauge"…”