The extended Bogomolny equations with generalized Nahm pole boundary conditions, II

Abstract

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 $\Sigma$ with ${\mathbb{R}}_y^{+}$, with generalized Nahm pole boundary conditions at $y=0$. The correspondence is between solutions of these equations satisfying these singular boundary conditions and also limiting to flat connections as $y\to \infty$ and certain holomorphic data consisting of triplets $(\mathcal{E},\phi ,L)$, where $(\mathcal{E},\phi )$ is a stable $SL(n+1,\mathbb{C})$ Higgs pair and $L\subset \mathcal{E}$ 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 $SL(2,\mathbb{R})$ case.