# The Haydys monopole equation

## TL;DR

This paper explores complexified Bogomolny monopoles, revealing their moduli space's rich geometric structure as a Kähler manifold containing the classical monopole moduli space as a Lagrangian submanifold, and constructs local neighborhoods using gluing techniques.

## Contribution

It introduces the concept of Haydys monopoles as solutions from dimensional reduction, analyzes their moduli space's geometry, and constructs local neighborhoods analogous to Higgs bundle theory.

## Key findings

- The moduli space of Haydys monopoles is a Kähler manifold.
- The classical Bogomolny moduli space is a minimal Lagrangian submanifold within it.
- A gluing construction models neighborhoods around the Bogomolny moduli space.

## Abstract

We study complexified Bogomolny monopoles using the complex linear extension of the Hodge star operator; these monopoles can be interpreted as solutions to the Bogomolny equation with a complex gauge group. Alternatively, these equations can be obtained from dimensional reduction of the Haydys instanton equations to three dimensions, thus we call them Haydys monopoles. We find that (under mild hypotheses) the smooth locus of the moduli space of finite energy Haydys monopoles on $\mathbb{R}^3$ is a K\"ahler manifold containing the ordinary Bogomolny moduli space as a minimal Lagrangian submanifold -- an $A$-brane. Moreover, using a gluing construction we construct an open neighborhood of this submanifold modeled on a neighborhood of the zero section in the tangent bundle to the Bogomolny moduli space. This is analogous to the case of Higgs bundles over a Riemann surface, where the (co)tangent bundle of holomorphic bundles canonically embeds into the Hitchin moduli space. These results contrast immensely with the case of finite energy Kapustin--Witten monopoles for which we have shown a vanishing theorem in [12].