  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
and plays an important role in the physical description of the the geometric
Langlands program in terms of S-duality for N=4 super Yang-Mills theory in four
dimensions.
