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We establish a Kobayashi-Hitchin correspondence between solutions of theextended Bogomolny equation with a Dirac type singularity and Heckemodifications of Higgs bundles. This correspondence was conjectured by Wittenand plays an important role in the physical description of the the geometricLanglands program in terms of S-duality for N=4 super Yang-Mills theory in fourdimensions.

Hecke modifications of Higgs bundles and the extended Bogomolny equation