Lie groups | Differential geometry | Hamiltonian mechanics | Algebraic geometry | Dynamical systems
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 algebraic geometry, the 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 was discovered by R. Garnier somewhat earlier as a certain limit of the Schlesinger equations, and Garnier solved his system by defining spectral curves. (The Garnier system is the classical limit of the Gaudin model. In turn, the Schlesinger equations are the classical limit of the Knizhnik–Zamolodchikov equations). Almost all integrable systems of classical mechanics can be obtained as particular cases of the Garnier/Hitchin system or their common generalization defined by Bottacin and Markman in 1994. The Hitchin fibration is the map from the moduli space of Hitchin pairs to characteristic polynomials, a higher genus analogue of the map Garnier used to define the spectral curves. Ngô used Hitchin fibrations over finite fields in his proof of the fundamental lemma. (Wikipedia).
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