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Gauge theory in physics and mathematics refers to a field theory whose fields include principal bundles with connection.
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What is the precise statement of the correspondence between stable Higgs bundles on a Rieman...
See Ó. Garcia-Prada's appendix to the third edition of R.O. Wells' book Differential Analysis on Complex Manifolds. It addresses most of what you are asking quite explicitly, in the context of Riemann …