The extended Bogomolny equations on $R^2\times R^{+}$ with real symmetry breaking

Abstract

In this paper, we construct solutions to the extended Bogomolny equations on $X=R^2\times R^{+}$ with certain boundary conditions and asymptotic conditions. Let $y$ be the coordinate of $R^{+}$. Roughly, both the boundary condition and the asymptotic condition say that a solution approaches to a certain model solution when $y\to 0$ and $y\to \infty$ resepctively. The boundary condition ($y\to 0$) is called generalized Nahm pole boundary condition and the asymptotic condition ($y\to \infty$) is called real symmetry breaking condition. The solutions should be thought as an analog of the instanton solutions that Taubes and Dimakis have created (using different methods), while their solutions satisfy a different asymptotic condition.