Higgs bundles arise as solutions to noncompact analog of the Yang-Mills equation. Hitchin showed that irreducible solutions of the GL(2,C) Yang-Mills equation on a compact Riemann surface X are precisely the polystable Higgs vector bundles on X of rank two and degree zero. Subsequently Simpson proved that irreducible solutions of the GL(r,C) Yang-Mills equation on a compact Kaehler manifold X are precisely the polystable Higgs vector bundles on X of rank r and vanishing Chern classes.Hitchin showed that the moduli spaces of stable Higgs bundles give examples of hyper-Kaehler manifolds and provide examples of completely integrable systems. Simpson proved basic theorems on fundamental group of Kaehler manifolds using the identification of Higgs bundles with the solutions of the Yang-Mills equation.The moduli space of Higgs bundles is then full of rich geometric structures, but the most interesting part of the story is not just this but the different points of view that one can use for s...
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