The extended Bogomolny equations and generalized Nahm pole boundary condition

Abstract

We develop a Kobayashi–Hitchin-type correspondence between solutions of the extended Bogomolny equations on $\mathrm{\Sigma }\times {ℝ}^{+}$ with Nahm pole singularity at $\mathrm{\Sigma }\times \left\{0\right\}$ and the Hitchin component of the stable $SL\left(2,ℝ\right)$ 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 $SL\left(2,ℝ\right)$ Higgs bundles. We also prove existence and uniqueness of solutions with knot singularities on $ℂ\times {ℝ}^{+}$.