Hitchin system

In mathematics, the Hitchin integrable system is an integrable system depending on the choice of a complex reductive group and a compact Riemann surface, introduced by Nigel Hitchin in 1987. It lies on the crossroads of the algebraic geometry, theory of Lie algebras and integrable system theory. It also plays an important role in geometric Langlands correspondence over the field of complex numbers; related to conformal field theory. A genus zero analogue of the Hitchin system arises as a certain limit of the Knizhnik–Zamolodchikov equations. Almost all integrable systems of classical mechanics can be obtained as particular cases of the Hitchin system (or its meromorphic generalization or in a singular limit).

The Hitchin fibration is the map from the moduli space of Hitchin pairs to characteristic polynomials. Ngô (2006, 2010) used Hitchin fibrations over finite fields in his proof of the fundamental lemma.

Using the language of algebraic geometry, the phase space of the system is a partial compactification of the cotangent bundle to the moduli space of stable G-bundles for some reductive group G, on some compact algebraic curve. This space is endowed with a canonical symplectic form. Suppose for simplicity that G=GL(n), the general linear group; then the hamiltonians can be described as follows: the tangent space to G-bundles at the bundle F is

which by Serre duality is dual to

so a pair

called a Hitchin pair or Higgs bundle, defines a point in the cotangent bundle. Taking

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