The Extended Bogomolny Equations and Generalized Nahm Pole Boundary Condition

In this paper we develop a Kobayashi-Hitchin type correspondence between solutions of the extended Bogomolny equations on $\mathrm{\Sigma }\times \text{RP}$ with Nahm pole singularity at $\mathrm{\Sigma }\times \{0\}$ and the Hitchin component of the stable $SL(2,\mathbb{R})$ Higgs bundle; this verifies a conjecture of Gaiotto and Witten. We also develop a partial Kobayashi-Hitchin correspondence for solutions with a knot singularity in this program, corresponding to the non-Hitchin components in the moduli space of stable $SL(2,\mathbb{R})$ Higgs bundles. We also prove existence and uniqueness of solutions with knot singularities on $\mathbb{C}\text{ti}\text{RP}$.