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. }

26 pages, LaTeX+AMSFonts (One section and a few comments added) Aar. Mat. Inst. Prep. 1993/32 -- Montpellier PM/93--43

References in corpus (5)

• Classical A_n--W-Geometry
• Covariant $W$ Gravity \& its Moduli Space from Gauge Theory
• Light Cone $W_n$ Geometry and its Symmetries and Projective Field Theory
• Folding the W Algebras
• Free field representation of Toda field theories

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• Higher dimensional uniformisation and W-geometry
• Toda lattice field theories, discrete W algebras, Toda lattice hierarchies and quantum groups
• A proposal for the geometry of W_n gravity
• Minimal surfaces in AdS space and Integrable systems
• Solving loop equations by Hitchin systems via holography in large-N QCD_4
• Generalized Kazdan-Warner equations associated with a linear action of a torus on a complex vector space
• Emergent Spacetime and Cosmic Inflation
• Higher Complex Structures and Flat Connections
• The geometric Toda equations for noncompact symmetric spaces