Euclidean Monopoles Research Articles

The Asymptotic Structure of the Centred Hyperbolic 2-Monopole Moduli Space

We construct an asymptotic metric on the moduli space of two centred hyperbolic monopoles by working in the point particle approximation, that is treating well-separated monopoles as point particles with an electric, magnetic and scalar charge and re-interpreting the dynamics of the 2-particle system as geodesic motion with respect to some metric. The corresponding analysis in the Euclidean case famously yields the negative mass Taub-NUT metric, which asymptotically approximates the L<sup>2</sup> metric on the moduli space of two Euclidean monopoles, the Atiyah-Hitchin metric. An important difference with the Euclidean case is that, due to the absence of Galilean symmetry, in the hyperbolic case it is not possible to factor out the centre of mass motion. Nevertheless we show that we can consistently restrict to a 3-dimensional configuration space by considering antipodal configurations. In complete parallel with the Euclidean case, the metric that we obtain is then the hyperbolic analogue of negative mass Taub-NUT. We also show how the metric obtained is related to the asymptotic form of a hyperbolic analogue of the Atiyah-Hitchin metric constructed by Hitchin.

Axion Mass from Magnetic Monopole Loops.

We show that axions interacting with Abelian gauge fields obtain a potential from loops of magnetic monopoles. This is a consequence of the Witten effect: the axion field causes the monopoles to acquire an electric charge and alters their energy spectrum. The axion potential can also be understood as a type of instanton effect due to a Euclidean monopole worldline winding around its dyon collective coordinate. We calculate this effect, which has features in common with both non-Abelian instantons and Euclidean brane instantons. To provide consistency checks, we argue that this axion potential vanishes in the presence of a massless charged fermion and that it is robust against the presence of higher-derivative corrections in the effective Lagrangian. Finally, as a first step toward connecting with particle phenomenology and cosmology, we discuss the regime in which this potential is important in determining the dark matter relic abundance in a hidden sector containing an Abelian gauge group, monopoles, and axions.

The charge $2$ monopole via the ADHMN construction

Recently we have shown how one may use use integrable systems techniques to implement the ADHMN construction and obtain general analytic formulae for the charge n su(2) Euclidean monopole. Here we do this for the case of charge 2: so answering an open problem of some 30 years standing. A comparison with known results and other approaches is made and new results presented.

Platonic Hyperbolic Monopoles

We construct a number of explicit examples of hyperbolic monopoles, with various charges and often with some platonic symmetry. The fields are obtained from instanton data in four-dimensional Euclidean space that are invariant under a circle action, and in most cases the monopole charge is equal to the instanton charge. A key ingredient is the identification of a new set of constraints on ADHM instanton data that are sufficient to ensure the circle invariance. Unlike for Euclidean monopoles, the formulae for the squared Higgs field magnitude in the examples we construct are rational functions of the coordinates. Using these formulae, we compute and illustrate the energy density of the monopoles. We also prove, for particular monopoles, that the number of zeros of the Higgs field is greater than the monopole charge, confirming numerical results established earlier for Euclidean monopoles. We also present some one-parameter families of monopoles analogous to known scattering events for Euclidean monopoles within the geodesic approximation.

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.

On the performance properties of the koch fractal and other bent wire monopoles

Koch fractal monopole antennas are known to exhibit lower resonant frequencies than Euclidean monopoles of the same height. It has been concluded that there exists a unique relationship between the antenna's fractal geometry and its electromagnetic behavior. Here, the performance properties of the Koch fractal monopole are examined and compared with the performance properties of other bent wire geometry monopoles having the same total wire length and overall height. It is demonstrated that monopoles with less complex shapes exhibit lower resonant frequencies because they are more effective at increasing the electrical volume of the antenna. When these antennas are made to be resonant at the same frequency, they exhibit virtually identical performance properties independent of differences in their geometric shape and total wire length. It is also demonstrated that the effective height of these monopoles converge to that of an electrically small Euclidean monopole near the small antenna limit and they exhibit virtually identical radiation resistance properties at low frequencies. Finally, it is shown that the fractal limit in lowering of resonant frequency is related to the limit in the increase in the antenna's effective volume.

On the resonant behavior of the small Koch fractal monopole antenna

AbstractThe resonant behavior of the Koch fractal monopole is considered as a function of its geometry, height and wire length. It is demonstrated that the resonant behavior of the Koch fractal is not uniquely determined by its geometry and that optimum resonant performance is exhibited by the simple Euclidean monopole. © 2002 Wiley Periodicals, Inc. Microwave Opt Technol Lett 35: 311–315, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10593

On the resonant properties of the Koch fractal and other wire monopole antennas

The Koch fractal monopole antenna has been shown to exhibit a lower resonant frequency than a simple Euclidean monopole having the same overall height. The performance properties of the Koch fractal monopole have primarily been attributed to its fractal geometry. Here, the performance properties of the Koch fractal monopole, normal mode helix, and two meander line configurations are considered. While the resonant frequency of these antennas is a function of both the antenna's geometry and total wire length, it is demonstrated that when these antennas are made to be resonant at the same frequency, they essentially exhibit the same performance characteristics independent of the differences in their geometry and total wire length. It is demonstrated that the electromagnetic behavior of the Koch fractal monopole is not uniquely defined by its geometry alone.

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.

Construction of Euclidian Monopoles

This paper describes a procedure for the construction of monopoles on three-dimensional Euclidean space, starting from their rational maps. A companion paper, ‘Euclidean monopoles and rational maps’, to appear in the same journal, describes the assignment to a monopole of a rational map, from CP1 to a suitable flag manifold. In describing the reverse direction, this paper completes the proof of the main theorem therein. A construction of monopoles from solutions to Nahm's equations (a system of ordinary differential equations) has been well-known for certain gauge groups for some time. These solutions are hard to construct however, and the equations themselves become increasingly unwieldy when the gauge group is not SU(2). Here, in contrast, a rational map is the only initial data. But whereas one can be reasonably explicit in moving from Nahm data to a monopole, here the monopole is only obtained from the rational map after solving a partial differential equation. A non-linear flow equation, essentially just the path of steepest descent down the Yang-Mills-Higgs functional, is set up. It is shown that, starting from an ‘approximate monopole’ - constructed explicitly from the rational map - a solution to the flow must exist, and converge to an exact monopole having the desired rational map. 1991 Mathematics Subject Classification: 53C07, 53C80, 58D27, 58E15, 58G11.

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.

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.