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

The scale invariance of the O(3) sigma model can be broken by gauging a U (1) subgroup of the O(3) symmetry and including a Maxwell term for the gauge field in the Lagrangian. Adding also a suitable potential one obtains a field theory of Bogomol'nyi type with topological solitons. These solitons are stable against rescaling and carry magnetic flux which can take arbitrary values in some finite interval. The soliton mass is independent of the flux, but the soliton size depends on it. However, dynamically changing the flux requires infinite energy, so the flux, and hence the soliton size, remains constant during time evolution.

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.

We quantise a Poisson structure on H n+2g , where H is a semidirect product group of the form G ⋉ g * . This Poisson structure arises in the combinatorial description of the phase space of Chern-Simons theory with gauge group G ⋉ g * on R × S g,n , where S g,n is a surface of genus g with n punctures. The quantisation of this Poisson structure is a key step in the quantisation of Chern-Simons theory with gauge group G ⋉ g * . We construct the quantum algebra and its irreducible representations and show that the quantum double D(G) of the group G arises naturally as a symmetry of the quantum algebra.

We define and discuss classical and quantum gravity in 2+1 dimensions in the Galilean limit. Although there are no Newtonian forces between massive objects in (2+1)dimensional gravity, the Galilean limit is not trivial. Depending on the topology of spacetime there are typically finitely many topological degrees of freedom as well as topological interactions of Aharonov-Bohm type between massive objects. In order to capture these topological aspects we consider a two-fold central extension of the Galilei group whose Lie algebra possesses an invariant and non-degenerate inner product. Using this inner product we define Galilean gravity as a Chern-Simons theory of the doublyextended Galilei group. The particular extension of the Galilei group we consider is the classical double of a much studied group, the extended homogeneous Galilei group, which is also often called Nappi-Witten group. We exhibit the Poisson-Lie structure of the doubly extended Galilei group, and quantise the Chern-Simons theory using a Hamiltonian approach. Many aspects of the quantum theory are determined by the quantum double of the extended homogenous Galilei group, or Galilei double for short. We study the representation theory of the Galilei double, explain how associated braid group representations account for the topological interactions in the theory, and briefly comment on an associated non-commutative Galilean spacetime.