#### Volume 23, issue 5 (2019)

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The extended Bogomolny equations and generalized Nahm pole boundary condition

### Siqi He and Rafe Mazzeo

Geometry & Topology 23 (2019) 2475–2517
##### Abstract

We develop a Kobayashi–Hitchin-type correspondence between solutions of the extended Bogomolny equations on $\Sigma ×{ℝ}^{+}$ with Nahm pole singularity at $\Sigma ×\left\{0\right\}$ 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 $ℂ×{ℝ}^{+}$.

##### Keywords
Kapustin–Witten equations, Nahm pole, Kobayashi–Hitchin correspondence
Primary: 53C07
##### Publication
Revised: 20 August 2018
Accepted: 12 March 2019
Published: 13 October 2019
Proposed: Tomasz Mrowka
Seconded: Gang Tian, Ciprian Manolescu
##### Authors
 Siqi He Simons Center for Geometry and Physics Stony Brook University Stony Brook, NY United States Rafe Mazzeo Department of Mathematics Stanford University Stanford, CA United States