nLab
Hitchin fibration

Contents

Overview

Liouville integrable systems can always be phrased in the form of a Lax pair depending on a spectral parameter belonging to what is called a spectral curve. This way the complex analytic theory of Riemann surfaces comes into play.

In a landmark article in 1987, Nigel Hitchin introduced more refined viewpoint to a rather general class of integrable systems. The spectral curve is a hyperelliptic curve? (genus 2 or more) for Sp(m,C)-Higgs bundles. This system is defined on the cotangent bundle of moduli space of stable bundles studied by Ramanan, Narasimhan, Hitchin, Nitsure, Faltings, Simpson (introduction of Higgs bundles!) and others.

Some of the features of that system are more readily seen from consideration of a canonical map from the cotangent bundle of the moduli space to the so-called Hitchin base?, and the map is called the Hitchin fibration.

The elliptic Calogero-Moser system? is an example of a Hitchin system.

For other purposes than integrable systems, e.g. in representation theory, one can consider Hitchin fibration over other ground fields. Alexander Beilinson and Vladimir Drinfeld have shown the importance of Hitchin fibration in the geometric Langlands program. Ngô Bảo Châu (with Laumon) has studied a Hitchin fibration to prove an important step in Langlands program, so called Fundamental Lemma, which was conjectural from 1970s and is one of the main mathematical discoveries in last decade. For this achievement, Ngô was awarded the Fields medal at ICM 2010 in Hyderabad (link).

See also moduli space of bundles.

  • N.J. Hitchin, Stable bundles and integrable systems, Duke Math. J. , 54 (1987) pp. 91–114 MR89a:32021 doi euclid

  • N.J. Hitchin, Flat connections and geometric quantization, Comm. Math. Phys. 131, n 2 (1990), 347-380, euclid

  • Emma Previato, Hitchin system, Springer Online Enc. of Math.

  • A. Beilinson, V. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves, pdf

  • Ngô Bảo Châu, Le lemme fondamental pour les algebres de Lie, arxiv/0806.4566

  • Carlos T. Simpson, Higgs bundles and local systems, Publ. Mathématiques de l’IHÉS, 75 (1992), p. 5-95, numdam

  • R. Donagi, Spectral covers, MSRI series 28, pdf

  • Tamas Hausel, Global topology of the Hitchin system, arxiv/1102.1717

  • David Nadler, Springer theory via the Hitchin fibration, arxiv/0806.4566 (nnPOV emphasis to an example in representation theory)

  • James Arthur, The Work of Ngô Bảo Châu, Proc. ICM Hyderabad 2010, pdf

  • Ken-ichi Sugiyama, A quantization of the Hitchin hamiltonian system and the Beilinson-Drinfeld isomorphism, arxiv/0708.2957

Revised on March 20, 2013 22:09:41 by Blaavand? (130.225.20.53)