Advertisement

Archiv der Mathematik

, Volume 73, Issue 1, pp 50–55 | Cite as

A note on the complete integrability of the Hitchin system

  • Renata Scognamillo
Article
  • 74 Downloads

Abstract.

We consider the Hitchin map \( {\cal H}\!: T^{*}{\cal M}_{G}\longrightarrow {\cal K},\) defined on the cotangent bundle of the moduli space of stable principal G-bundles over a compact Riemann surface, G being any reductive complex Lie group. In [5] Hitchin showed that the \(m=\dim{\cal M}_{G}\) components of \(\cal H\) are analytic functions that commute with respect to the natural Poisson structure on \(T^{*}{\cal M}_{G}\). In order to do this, Hitchin used a description of \(T^{*}{\cal M}_{G}\) as a Marsden-Weinstein quotient \(\mu ^{-1}(0)/{\cal G}, \mu \) being the moment map for the action of the Gauge group \(\cal G\) on the infinite dimensional space of stable holomorphic structures on a fixed principal G-bundle. In this note we obtain the same result as an application of a few elementary properties satisfied by the homogeneous AdG-invariant polynomials on the Lie algebra of G.

Keywords

Modulus Space Gauge Group Riemann Surface Poisson Structure Cotangent Bundle 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Birkhäuser Verlag, Basel 1999

Authors and Affiliations

  • Renata Scognamillo
    • 1
  1. 1.Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, ItalyItaly

Personalised recommendations