We develop a Kobayashi–Hitchin correspondence for the extended Bogomolny equations, that is, the dimensionally reduced Kapustin–Witten equations, on the product of a compact Riemann surface with , with generalized Nahm pole boundary conditions at . The correspondence is between solutions of these equations satisfying these singular boundary conditions and also limiting to flat connections as and certain holomorphic data consisting of triplets , where is a stable Higgs pair and is a holomorphic line bundle. This corroborates a prediction of Gaiotto and Witten and serves as an extension of our earlier article which treats only the case.
"The extended Bogomolny equations with generalized Nahm pole boundary conditions, II." Duke Math. J. 169 (12) 2281 - 2335, 1 September 2020. https://doi.org/10.1215/00127094-2020-0009