On the Solutions of Generalized Bogomolny Equations

TL;DR

This paper studies solutions to generalized Bogomolny equations relevant in topological N=4 SYM theory, constructing model solutions for boundary operators and linking them to Toda systems.

Contribution

It constructs explicit model solutions for boundary 't Hooft and surface operators and relates these solutions to Toda chain and Toda system equations.

Findings

Reduced 't Hooft equations to open Toda chain for any simple gauge group.

Connected surface operator solutions to rational solutions of a periodic non-abelian Toda system.

Provided explicit boundary condition solutions in a half-space setting.

Abstract

Generalized Bogomolny equations are encountered in the localization of the topological N=4 SYM theory. The boundary conditions for 't Hooft and surface operators are formulated by giving a model solution with some special singularity. In this note we consider the generalized Bogomolny equations on a half space and construct model solutions for the boundary 't Hooft and surface operators. It is shown that for the 't Hooft operator the equations reduce to the open Toda chain for arbitrary simple gauge group. For the surface operators the solutions of interest are rational solutions of a periodic non-abelian Toda system.