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

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

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Kapustin–Witten equations, Nahm pole, Kobayashi–Hitchin correspondence
Mathematical Subject Classification 2010
Primary: 53C07
Received: 3 March 2018
Revised: 20 August 2018
Accepted: 12 March 2019
Published: 13 October 2019
Proposed: Tomasz Mrowka
Seconded: Gang Tian, Ciprian Manolescu
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