The extended Bogomolny equations and generalized Nahm pole boundary condition

He, Siqi; Mazzeo, Rafe

doi:10.2140/gt.2019.23.2475

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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

DOI: 10.2140/gt.2019.23.2475

Abstract

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

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Keywords

Kapustin–Witten equations, Nahm pole, Kobayashi–Hitchin correspondence

Mathematical Subject Classification 2010

Primary: 53C07

References

Publication

Received: 3 March 2018

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

Volume 23, issue 5 (2019)