Hecke modifications of Higgs bundles and the extended Bogomolny equation

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 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.

arXiv commentThe singularity analysis in Section 11 builds on and extends arXiv:1610.08283

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• Semi-abelian spectral data for singular fibres of the $\mathsf{SL}(2,\mathbb{C})$-Hitchin system
• The Kapustin--Witten equations on ALE and ALF gravitational instantons
• The Haydys monopole equation