$L^2$ geometry of hyperbolic monopoles

TL;DR

This paper introduces a new gauge-fixing approach inspired by supersymmetry that resolves divergence issues in the $L^2$ metric on hyperbolic monopole moduli spaces, revealing a hyperbolic hyperk"ahler structure.

Contribution

It proposes an alternative gauge-fixing condition that cures divergence in the $L^2$ metric, unveiling a hyperbolic analogue of Euclidean monopole moduli space geometry.

Findings

The divergence in the $L^2$ metric is resolved with a supersymmetry-inspired gauge choice.

The resulting moduli space exhibits hyperbolic hyperk"ahler geometry.

The approach parallels the Euclidean case, extending geometric understanding of monopoles.

Abstract

It is well-known that the $L^2$ metric on the moduli space of hyperbolic monopoles, defined using the Coulomb gauge-fixing condition, diverges. This article shows that an alternative gauge-fixing condition inspired by supersymmetry cures this divergence. The resulting geometry is a hyperbolic analogue of the hyperk\"ahler geometry of Euclidean monopole moduli spaces.