Hyperbolic Monopoles Research Articles

Hyperbolic Calorons, Monopoles, and Instantons

We construct families of SO(3)-symmetric charge 1 instantons and calorons on the space \({\mathbb{H}^3 \times \mathbb{R}}\) . We show how the calorons include instantons and hyperbolic monopoles as limiting cases. We show how Euclidean calorons are the flat space limit of this family.

Hidden symmetry of hyperbolic monopole motion

Hyperbolic monopole motion is studied for well separated monopoles. It is shown that the motion of a hyperbolic monopole in the presence of one or more fixed monopoles is equivalent to geodesic motion on a particular submanifold of the full moduli space. The metric on this submanifold is found to be a generalisation of the multi-centre Taub-NUT metric introduced by LeBrun. The one centre case is analysed in detail as a special case of a class of systems admitting a conserved Runge–Lenz vector. The two centre problem is also considered. An integrable classical string motion is exhibited.

A new approach to monopole moduli spaces

We introduce a new way to study both Euclidean and hyperbolic monopole moduli spaces. The idea is to apply Kodaira's deformation theory to the spectral curve in an appropriate ambient space. Using this we are able to recover the usual hyperkähler structure on the Euclidean monopole moduli space and to identify a new type of geometry on the hyperbolic monopole moduli space.

Spectral Curves and the Mass of Hyperbolic Monopoles

The moduli spaces of hyperbolic monopoles are naturally fibred by the monopole mass, and this leads to a nontrivial mass dependence of the holomorphic data (spectral curves, rational maps, holomorphic spheres) associated to hyperbolic multi-monopoles. In this paper, we obtain an explicit description of this dependence for general hyperbolic monopoles of magnetic charge two. In addition, we show how to compute the monopole mass of higher charge spectral curves with tetrahedral and octahedral symmetries. Spectral curves of euclidean monopoles are recovered from our results via an infinite-mass limit.

Hyperbolic Monopoles and Holomorphic Spheres

We associate to an SU(2) hyperbolic monopole a holomorphicsphere embedded in projective space and use this to uncovervarious features of the monopole.

Bogomolny Yang–Mills–Higgs solutions in (2+1) anti-de Sitter space

This paper investigates an integrable system which is related to hyperbolic monopoles, i.e. the Bogomolny Yang–Mills–Higgs equations in (2+1) anti-de Sitter space which are integrable and whose solutions can be obtained using analytical methods. In particular, families of soliton solutions have been constructed explicitly and their dynamics has been investigated in some detail.

ASYMPTOTIC VALUES OF HYPERBOLIC MONOPOLES

It is shown that many hyperbolic monopoles can be distinguished from each other via their asymptotic values in contrast to the case of Euclidean monopoles.

Monopoles and harmonic maps

Recently Jarvis has proved a correspondence between SU(N) monopoles and rational maps of the Riemann sphere into flag manifolds. Furthermore, he has outlined a construction to obtain the monopole fields from the rational map. In this paper we examine this construction in some detail and provide explicit examples for spherically symmetric SU(N) monopoles with various symmetry breakings. In particular we show how to obtain these monopoles from harmonic maps into complex projective spaces. The approach extends in a natural way to monopoles in hyperbolic space and we use it to construct new spherically symmetric SU(N) hyperbolic monopoles.

On the Complete Integrability of the Discrete Nahm Equations

The discrete Nahm equations, a system of matrix valued difference equations, arose in the work of Braam and Austin on half-integral mass hyperbolic monopoles. We show that the discrete Nahm equations are completely integrable in a natural sense: to any solution we can associate a spectral curve and a holomorphic line-bundle over the spectral curve, such that the discrete-time DN evolution corresponds to walking in the Jacobian of the spectral curve in a straight line through the line-bundle with steps of a fixed size. Some of the implications for hyperbolic monopoles are also discussed.

Two integrable systems related to hyperbolic monopoles

Monopoles on hyperbolic 3-space were introduced by Atiyah in 1984. This article describes two integrable systems which are closely related to hyperbolic monopoles: a one-dimensional lattice equation (the Braam-Austin or discrete Nahm equation), and a soliton system in (2+1)-dimensional anti-deSitter space-time.

Zero and Infinite Curvature Limits of Hyperbolic Monopoles

We show that the zero curvature limit of the space of hyperbolic monopoles gives the Euclidean monopoles, settling a conjecture of Atiyah. We also study the infinite curvature limit of the space of hyperbolic monopoles, and show that the associated rational maps appear explicitly here. 1991 Mathematics Subject Classification 81T13, 53C07.

Compactification of hyperbolic monopoles

We prove that the space of SU(2) hyperbolic monopoles based at the centre of hyperbolic space is homeomorphic to the space of (unbased) rational maps of the two-sphere. The homeomorphism extends to a map of the natural compactifications of the two spaces. We also show that the scattering methods used in the study of monopoles apply to the configuration space for hyperbolic monopoles giving a homotopy equivalence of this space with the space of continuous self-maps of the two-sphere.

Spectral curves of non-integral hyperbolic monopoles

The twistor desription of non-integral -monopoles on hyperbolic 3-space is developed and some consequences discussed. It is shown that in a precise sense, any appropriately decaying hyperbolic monopole is determined by its asymptotic values, and also that such monopoles determine and are determined by their spectral data (which reduces to a complex algebraic curve when n = 2). This paper provides the hyperbolic analogue of the work of Hitchin and others on euclidean monopoles and generalizes that of Atiyah and Braam - Austin on integral hyperbolic monopoles.

Hilbert space for charged particles in perpendicular magnetic fields

We describe the quantum mechanics of two-dimensional charged particles in a perpendicular magnetic field in the planar Landau and spherical monopole configurations. These models, particularly in the work of Laughlin and Haldane, are crucial to the theoretical understanding of the quantum Hall effect. Here we present the full Hilbert space structure in each case, with special emphasis on the relationship between the two systems. The formulation in terms of stereographically projected complex coordinates makes the connection especially explicit and naturally generalizes to more complicated two-dimensional surfaces where the interaction of the particles with an external perpendicular magnetic field may be regarded as an interaction of the particles with the two-dimensional (Kähler) metric of the surface. This generalization is illustrated by the hyperbolic configuration of particles constrained to the upper sheet of a hyperboloid in the presence of a hyperbolic monopole.

Why instantons are monopoles

It is shown that instantons are hyperbolic monopoles for the loop group with non-maximal symmetry breaking at infinity.

Geometry of hyperbolic monopoles

The hyperbolic monopoles of Atiyah [M. F. Atiyah, Commun. Math. Phys. 93, 471 (1984); ‘‘Magnetic monopoles in hyperbolic space,’’ in Proceedings of the International Colloquium on Vector Bundles (Tata Institute, Bombay, 1984)] and Chakrabarti [A. Chakrabarti, J. Math. Phys. 27, 340 (1986)] are introduced and their geometric properties and relations to instantons and ordinary monopoles clarified. A key tool is the use of the ball model of hyperbolic space to construct and examine solutions.

Construction of hyperbolic monopoles

Methods are presented for explicit construction of hyperbolic monopoles as defined by Atiyah [M. F. Atiyah, Commun. Math. Phys. 93, 437 (1984); and ‘‘Magnetic monopoles in hyperbolic space, ’’ unpublished lecture notes, 1984]. The techniques permit the extraction of flat-space self-dual monopoles trivially through a simple rescaling. Relationships to previous works of the present author are indicated.