Spectral curves of hyperbolic monopoles from ADHM

TL;DR

This paper derives explicit formulas for the spectral curves and rational maps of hyperbolic monopoles using constrained ADHM data, extending previous results beyond the JNR class and providing new examples.

Contribution

It introduces new formulae linking ADHM instanton data to spectral curves of hyperbolic monopoles, broadening the class of explicit solutions available.

Findings

Derived explicit spectral curve formulas from constrained ADHM data.

Extended previous results to include non-JNR instantons.

Provided new explicit examples of hyperbolic monopoles.

Abstract

Magnetic monopoles in hyperbolic space are in correspondence with certain algebraic curves in mini-twistor space, known as spectral curves, which are in turn in correspondence with rational maps between Riemann spheres. Hyperbolic monopoles correspond to circle-invariant Yang-Mills instantons, with an identification of the monopole and instanton numbers, providing the curvature of hyperbolic space is tuned to a value specified by the asymptotic magnitude of the Higgs field. In previous work, constraints on ADHM instanton data have been identified that provide a non-canonical realization of the circle symmetry that preserves the standard action of rotations in the ball model of hyperbolic space. Here formulae are presented for the spectral curve and the rational map of a hyperbolic monopole in terms of its constrained ADHM matrix. This extends earlier results that apply only to the subclass of instantons of JNR type. The formulae are applied to obtain new explicit examples of spectral curves that are beyond the JNR class.