The extended Bogomolny equations with generalized Nahm pole boundary conditions, II

Abstract

We develop a Kobayashi-Hitchin correspondence for the extended Bogomolny equations, i.e., the dimensionally reduced Kapustin-Witten equations, on the product of a compact Riemann surface $\Sigma$ with ${\mathbb{R}}_y^{+}$, with generalized Nahm pole boundary conditions at $y=0$. The correspondence is between solutions of these equations satisfying these singular boundary conditions and also limiting to flat connections as $y\to \infty$, and certain holomorphic data consisting of effective triplets $(\mathcal{E},\phi ,L)$ where $(\mathcal{E},\phi )$ is a stable $\mathrm{SL}(n+1,\mathbb{C})$ Higgs pair and $L\subset \mathcal{E}$ is a holomorphic line bundle. This corroborates a prediction of Gaiotto and Witten, and is an extension of our earlier paper \cite{HeMazzeo2017} which treats only the $\mathrm{SL}(2,\mathbb{R})$ case.

38 pages

References in corpus (3)

• Inverses of Cartan Matrices of Lie Algebras and Lie Superalgebra
• A Gluing Theorem for the Kapustin-Witten Equations with a Nahm Pole
• The Extended Bogomolny Equations and Generalized Nahm Pole Boundary Condition

Cited by in corpus (1)

• Supersymmetry of the D3/D5 Defect Field Theory