The extended Bogomolny equations on $\mathbb {R}^2 \times \mathbb {R}^+$ I: Nilpotent Higgs field

Abstract

We study the extended Bogomolny equations with gauge group $SU(2)$ on $\mathbb {R}^2 \times \mathbb {R}^+$ with generalized Nahm pole boundary conditions and nilpotent Higgs field. We completely classify solutions by relating them to certain holomorphic data through a Kobayashi-Hitchin correspondence.