In mathematics, and in particular differential geometry and gauge theory, Hitchin's equations are a system of partial differential equations for a connection and Higgs field on a vector bundle or principal bundle over a Riemann surface, first written down by Nigel Hitchin in 1987. Hitchin's equations appear as a dimensional reduction of the self-dual Yang–Mills equations from four dimensions to two dimensions, and solutions to Hitchin's equations give examples of Higgs bundles. The existence of solutions to Hitchin's equations is equivalent to the stability of the corresponding Higgs bundle structure, and this is the simplest form of the Nonabelian Hodge correspondence for Higgs bundles. The moduli space of solutions to Hitchin's equations, the Higgs bundle moduli space, was constructed by Hitchin and was one of the first examples of a hyperkähler manifold constructed. Using the metric structure on this moduli space afforded by its description in terms of Hitchin's equations, Hitchin constructed the Hitchin system, a completely integrable system which was used by Ngô Bảo Châu in his proof of the fundamental lemma in the Langlands program, for which he was afforded the 2010 Fields medal. (Wikipedia).
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