Abstract
We establish a Kobayashi–Hitchin correspondence between solutions of the extended Bogomolny equation with a Dirac type singularity and Hecke modifications of Higgs bundles. This correspondence was conjectured by Witten (2015), p. 668 and plays an important role in the physical description of the geometric Langlands program in terms of S–duality for N=4 super Yang–Mills theory in four dimensions.
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