The dynamics of vortices on S^2 near the Bradlow limit

The explicit solutions of the Bogomolny equations for N vortices on a sphere of radius R^2 > N are not known. In particular, this has prevented the use of the geodesic approximation to describe the low energy vortex dynamics. In this paper we introduce an approximate general solution of the equations, valid for R^2 close to N, which has many properties of the true solutions, including the same moduli space CP^N. Within the framework of the geodesic approximation, the metric on the moduli space is then computed to be proportional to the Fubini- Study metric, which leads to a complete description of the particle dynamics.Comment: 17 pages, 9 figure

Monopole Planets and Galaxies

Spherical clusters of SU(2) BPS monopoles are investigated here. A large class of monopole solutions is found using an abelian approximation, where the clusters are spherically symmetric, although exact solutions cannot have this symmetry precisely. Monopole clusters generalise the Bolognesi magnetic bag solution of the same charge, but they are always larger. Selected density profiles give structures analogous to planets of uniform density, and galaxies with a density decaying as the inverse square of the distance from the centre. The Bolognesi bag itself has features analogous to a black hole, and this analogy between monopole clusters and astrophysical objects with or without black holes in their central region is developed further. It is also shown that certain exact, platonic monopoles of small charge have sizes and other features consistent with what is expected for magnetic bags.Comment: 23 pages. Revised version to appear in Physical Review D. New introduction and conclusions; analogy between monopoles and astrophysical objects developed furthe

Calogero-Moser Models V: Supersymmetry and Quantum Lax Pair

It is shown that the Calogero-Moser models based on all root systems of the finite reflection groups (both the crystallographic and non-crystallographic cases) with the rational (with/without a harmonic confining potential), trigonometric and hyperbolic potentials can be simply supersymmetrised in terms of superpotentials. There is a universal formula for the supersymmetric ground state wavefunction. Since the bosonic part of each supersymmetric model is the usual quantum Calogero-Moser model, this gives a universal formula for its ground state wavefunction and energy, which is determined purely algebraically. Quantum Lax pair operators and conserved quantities for all the above Calogero-Moser models are established.Comment: LaTeX2e, 31 pages, no figure

Symetric Monopoles

We discuss $SU(2)$ Bogomolny monopoles of arbitrary charge $k$ invariant under various symmetry groups. The analysis is largely in terms of the spectral curves, the rational maps, and the Nahm equations associated with monopoles. We consider monopoles invariant under inversion in a plane, monopoles with cyclic symmetry, and monopoles having the symmetry of a regular solid. We introduce the notion of a strongly centred monopole and show that the space of such monopoles is a geodesic submanifold of the monopole moduli space. By solving Nahm's equations we prove the existence of a tetrahedrally symmetric monopole of charge $3$ and an octahedrally symmetric monopole of charge $4$, and determine their spectral curves. Using the geodesic approximation to analyse the scattering of monopoles with cyclic symmetry, we discover a novel type of non-planar $k$-monopole scattering process

The interaction energy of well-separated Skyrme solitons

We prove that the asymptotic field of a Skyrme soliton of any degree has a non-trivial multipole expansion. It follows that every Skyrme soliton has a well-defined leading multipole moment. We derive an expression for the linear interaction energy of well-separated Skyrme solitons in terms of their leading multipole moments. This expression can always be made negative by suitable rotations of one of the Skyrme solitons in space and iso-space.We show that the linear interaction energy dominates for large separation if the orders of the Skyrme solitons' multipole moments differ by at most two. In that case there are therefore always attractive forces between the Skyrme solitons.Comment: 27 pages amslate

On the constraints defining BPS monopoles

We discuss the explicit formulation of the transcendental constraints defining spectral curves of SU(2) BPS monopoles in the twistor approach of Hitchin, following Ercolani and Sinha. We obtain an improved version of the Ercolani-Sinha constraints, and show that the Corrigan-Goddard conditions for constructing monopoles of arbitrary charge can be regarded as a special case of these. As an application, we study the spectral curve of the tetrahedrally symmetric 3-monopole, an example where the Corrigan-Goddard conditions need to be modified. A particular 1-cycle on the spectral curve plays an important role in our analysis.Comment: 29 pages, 7 eps figure

Sigma Model BPS Lumps on Torus

We study doubly periodic Bogomol'nyi-Prasad-Sommerfield (BPS) lumps in supersymmetric CP^{N-1} non-linear sigma models on a torus T^2. Following the philosophy of the Harrington-Shepard construction of calorons in Yang-Mills theory, we obtain the n-lump solutions on compact spaces by suitably arranging the n-lumps on R^2 at equal intervals. We examine the modular invariance of the solutions and find that there are no modular invariant solutions for n=1,2 in this construction.Comment: 15 pages, 3 figures, published versio

Electrically Charged Sphalerons

We investigate the possibility that the Higgs sector of the Weinberg-Salam model admits the existence of electrically charged, sphaleron states. Evidence is provided through an asymptotic and numerical perturbative analysis about the uncharged sphaleron. By introducing a toy model in two dimensions we demonstrate that such electrically charged, unstable states can exist. Crucially, they can have a comparable mass to their uncharged counterparts and so may also play a role in electroweak baryogenesis, by opening up new channels for baryon number violating processes.Comment: 12 pages, 4 Postscript figure

New Integrable Sectors in Skyrme and 4-dimensional CP^n Model

The application of a weak integrability concept to the Skyrme and $CP^n$ models in 4 dimensions is investigated. A new integrable subsystem of the Skyrme model, allowing also for non-holomorphic solutions, is derived. This procedure can be applied to the massive Skyrme model, as well. Moreover, an example of a family of chiral Lagrangians providing exact, finite energy Skyrme-like solitons with arbitrary value of the topological charge, is given. In the case of $CP^n$ models a tower of integrable subsystems is obtained. In particular, in (2+1) dimensions a one-to-one correspondence between the standard integrable submodel and the BPS sector is proved. Additionally, it is shown that weak integrable submodels allow also for non-BPS solutions. Geometric as well as algebraic interpretations of the integrability conditions are also given.Comment: 23 page