The Extended Bogomolny Equations and Generalized Nahm Pole Boundary Condition

TL;DR

This paper establishes a correspondence between solutions of extended Bogomolny equations with singularities and the Hitchin component, confirming a conjecture and exploring solutions with knot singularities in the context of Higgs bundles.

Contribution

It develops a Kobayashi-Hitchin type correspondence for extended Bogomolny equations with Nahm pole and knot singularities, verifying a conjecture and expanding understanding of Higgs bundle moduli spaces.

Findings

Verified a conjecture of Gaiotto and Witten.

Established existence and uniqueness of solutions with knot singularities.

Developed partial correspondences for non-Hitchin components.

Abstract

In this paper we develop a Kobayashi-Hitchin type correspondence between solutions of the extended Bogomolny equations on $\Sigma\times \RP$ with Nahm pole singularity at $\Sigma \times \{0\}$ 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 $\mathbb{C}\ti\RP$.