# Higgs Bundles
# Hitchin's Equations
Let be a smooth compact Riemann surface, of genus . Higgs bundles originally came to be as solutions to Hitchin's equations, or "self-duality" equations on . These are self-dual, dimensionally-reduced Yang-Mills equations written on a smooth Hermitian bundle of rank and degree on . Denote this bundle by and the metric by , then the equations take the form
Here, is a unitary connection on the bundle with respect to , is its curvature, and is a smooth bundle map from to , called a Higgs field.
Let be a smooth compact Riemann surface, with genus . Let be a holomorphic bundle on which has a holomorphic section . We refer to the pair as a Higgs bundle, and we call the holomorphic section a Higgs field.