The Skyrme model can be generalised to a situation where static fields are maps from one Riemannian manifold to another. Here we study a Skyrme model where physical space is twodimensional euclidean space and the target space is the two-sphere with its standard metric. The model has topological soliton solutions which are exponentially localised. We describe a superposition procedure for solitons in our model and derive an expression for the interaction potential of two solitons which only involves the solitons' asymptotic fields. If the solitons have topological degree 1 or 2 there are simple formulae for their interaction potentials which we use to prove the existence of solitons of higher degree. We explicitly compute the fields and energy distributions for solitons of degrees between one and six and discuss their geometrical shapes and binding energies. A Skyrme Model in Two DimensionsThe Skyrme model is a non-linear theory for SU(2) valued fields in 3 (spatial) dimensions which has soliton solutions. Each soliton has an associated integer topological charge or degree which Skyrme identified with the baryon number [1]. A soliton with topological charge one is called a Skyrmion; suitably quantised it is a model for a physical nucleon.Solitons of higher topological charge, called multisolitons, are classical models for higher nuclei.In this paper we study multisolitons in a two-dimensional version of the Skyrme model. The model was first considered in [2], but the motivation there is somewhat different from the approach taken here. For the present purpose it is important to be clear in what sense our model resembles Skyrme's model. In this section we will therefore briefly review a general framework for the Skyrme model due to Manton [3] and explain how our model fits into that framework. In [3] the usual Skyrme energy functional is interpreted in terms of elasticity theory and a static Skyrme field is a map π : S → Σ (1.1) from physical space S to the target space Σ. Both S and Σ are assumed to be Riemannian manifolds with metrics t and τ respectively. The energy of a configuration π is expressed in terms of its strain tensor D. To calculate the strain tensor one introduces coordinates p i on S and π α on Σ and orthonormal frame fields s m on S and σ µ on Σ (1 ≤ i, m ≤ dimS, 1 ≤ α, µ ≤ dimΣ). The Jacobian of the map π is, in orthonormal coordinates, J mµ = s i m ∂π α ∂p i σ µα (1.2) and the strain tensor D is defined via
Baby Skyrmions are topological solitons in a (2+1)-dimensional field theory which resembles the Skyrme model in important respects. We apply some of the techniques and approximations commonly used in discussions of the Skyrme model to the dynamics of baby Skyrmions and directly test them against numerical simulations. Specifically we study the effect of spin on the shape of a single baby Skyrmion, the dependence of the forces between two baby Skyrmions on the baby Skyrmions' relative orientation and the forces between two baby Skyrmions when one of them is spinning.
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A gauged (2+1)-dimensional version of the Skyrme model is investigated. The gauge group is U (1) and the dynamics of the associated gauge potential is governed by a Maxwell term. In this model there are topologically stable soliton solutions carrying magnetic flux which is not topologically quantized. The properties of rotationally symmetric solitons of degree one and two are discussed in detail. It is shown that the electric field for such solutions is necessarily zero. The solitons' shape, mass and magnetic flux depend on the U (1) coupling constant, and this dependence is studied numerically from very weak to very strong coupling. PACS number(s): 11.10.Kk, 11.10.Lm, 11.27.+d, 12.39.Dc I IntroductionThe Skyrme model is a generalized non-linear sigma model in (3+1) dimensions [1]. It has soliton solutions which, after suitable quantization, are models for physical nucleons [2]. The theory is invariant under the group SO(3) iso of iso-rotations, and electromagnetism is introduced into the model by gauging a U(1) subgroup of SO(3) iso , see [3] for details. The resulting fully coupled Skyrme-Maxwell system is mathematically hard to analyze, but of considerable physical interest: it is here that one should compute the Skyrme model's 1
We use a prescription to gauge the SU (2) Skyrme model with a U (1) field, characterised by a conserved Baryonic current. This model reverts to the usual Skyrme model in the limit of the gauge coupling constant vanishing. We show that there exist axially symmetric static solutions with zero magnetic charge, which can be electrically either charged or uncharged. The energies of the (uncharged) gauged Skyrmions are less than the energy of the (usual) ungauged Skyrmion. For physical values of the parameters the impact of the U (1) field is very small, so that it can be treated as a perturbation to the (ungauged) spherically symmetric Hedgehog. This allows the perturbative calculation of the magnetic moment.
Alternative SU(N) Skyrme models and their solutionsHarmonic maps from S 2 to CP NϪ1 are introduced to construct low-energy configurations of the SU(N) Skyrme model. We show that one of such maps gives an exact, topologically trivial, solution of the SU(3) model. We study various properties of these maps and show that, in general, their energies are only a little higher than the energies of the corresponding SU(2) embeddings. Moreover, we show that the baryon and energy densities of the SU(3) configurations with baryon number Bϭ3Ϫ6 are more symmetrical than their SU(2) analogs, thus suggesting that there exist solutions of the model with these symmetries. We also show that any SU(2) solution embedded into the SU(4) Skyrme model becomes a topologically trivial solution of this model.
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