Boundary values of hyperbolic monopoles

Abstract

We show that a magnetic monopole on hyperbolic space is determined, up to gauge equivalence, by its asymptotic boundary value. Our basic tool is the ADHM construction which specializes to a discrete Nahm equation. We also construct a complete metric on the moduli spaces.

One-parameter family of selfdual solutions in classical Yang-Mills theory

“…where the matricesx, ∂x ∂θ , and 1 sin θ ∂x ∂φ span a representation of su (2). From these components we can explicitly confirm the anti-selfduality F µν = −F µν , and the hermiticity F † µν = F µν .…”

“…This transformation has the form g = e χ·1 with χ(ρ ± ; q) being real function, which means (47) does not belong to U (2). Of course our curvature transforms covariantly under an ordinary, i.e., q-independent, SU(2) transformation.…”

Supersymmetry of hyperbolic monopoles

“…A BPS monopole in hyperbolic space is labelled by a mass m ∈ R + and a charge k ∈ Z + given by m = lim r→∞ |φ(r)| 5) and it is known [8] that hyperbolic monopoles exist for all values of m and k. In contrast to the euclidean monopoles, m cannot be rescaled to unity in the hyperbolic case, as the value of m affects the monopole solutions [9]. Alternatively, one can normalise the mass to unity, but only at the price of rescaling the hyperbolic metric to one of curvature −1/m 2 .…”

“…Most of the progress in the study of hyperbolic monopoles was focused on finding methods of constructing multimonopole solutions, either by building a hyperbolic version of the Nahm transform [9,[13][14][15] or by studying the spectral curves associated with hyperbolic monopoles [16][17][18]. Progress on the geometry of the moduli space was hindered by the early realisation [9] that the natural L 2 metric, which in the euclidean case induces upon reduction a hyperkähler metric on the moduli space, does not converge in the case of hyperbolic monopoles, suggesting that the geometry of the moduli space is not in fact riemannian. Nevertheless, Hitchin [19] constructed a family g m of self-dual Einstein metrics on the moduli space of centered hyperbolic monopoles with mass m ∈ Z, which in the flat limit m → ∞ recovers the Atiyah-Hitchin metric.…”

Aq-analogue of the ADHMN construction and the axisymmetric multi-instantons

“…In particular, it turns out to be a differential equation in Nahm's case. Further application was made by Braam and Austin [8], who gave a discretized version of the Nahm formalism and pointed out the correspondence between the discrete Nahm equation and the hyperbolic monopoles [9], i.e., monopoles on hyperbolic three-space H 3 . In ref.…”