nLab
Hitchin fibration

Contents

Overview

Liouville integrable systems may 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.

The landmark Hitchin 87 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).

References

Discussion in the context of the topological recursion includes

reviewed in

See also