The extended Bogomolny equations and generalized Nahm pole boundary condition

He, Siqi¹ ; Mazzeo, Rafe²

¹ Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, NY, United States
² Department of Mathematics, Stanford University, Stanford, CA, United States

Geometry & topology, Tome 23 (2019) no. 5, pp. 2475-2517.

Résumé

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

Détail

DOI : 10.2140/gt.2019.23.2475

Classification : 53C07
Keywords: Kapustin–Witten equations, Nahm pole, Kobayashi–Hitchin correspondence

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