On the existence of minima in the Skyrme model

Well-separated Skyrme solitons of arbitrary degree attract after a suitable relative rotation in space and iso-space, provided the orders of the solitons' leading multipoles do not differ by more than two. I summarise the derivation of this result, obtained jointly with Manton and Singer, and discuss to what extent its combination with earlier results of Esteban allows one to deduce the existence of minima of the Skyrme energy functional.Comment: 11 pages amslatex, talk given at the workshop on Integrable theories, Solitons and Duality, Sao Paulo, July 200

Unstable manifolds and Schroedinger dynamics of Ginzburg-Landau vortices

The time evolution of several interacting Ginzburg-Landau vortices according to an equation of Schroedinger type is approximated by motion on a finite-dimensional manifold. That manifold is defined as an unstable manifold of an auxiliary dynamical system, namely the gradient flow of the Ginzburg-Landau energy functional. For two vortices the relevant unstable manifold is constructed numerically and the induced dynamics is computed. The resulting model provides a complete picture of the vortex motion for arbitrary vortex separation, including well-separated and nearly coincident vortices.Comment: 23 pages amslatex, 5 eps figures, minor typos correcte

S-duality in SU(3) Yang-Mills Theory with Non-abelian Unbroken Gauge Group

It is observed that the magnetic charges of classical monopole solutions in Yang-Mills-Higgs theory with non-abelian unbroken gauge group $H$ are in one-to-one correspondence with coherent states of a dual or magnetic group $\tilde H$. In the spirit of the Goddard-Nuyts-Olive conjecture this observation is interpreted as evidence for a hidden magnetic symmetry of Yang-Mills theory. SU(3) Yang-Mills-Higgs theory with unbroken gauge group U(2) is studied in detail. The action of the magnetic group on semi-classical states is given explicitly. Investigations of dyonic excitations show that electric and magnetic symmetry are never manifest at the same time: Non-abelian magnetic charge obstructs the realisation of electric symmetry and vice-versa. On the basis of this fact the charge sectors in the theory are classified and their fusion rules are discussed. Non-abelian electric-magnetic duality is formulated as a map between charge sectors. Coherent states obey particularly simple fusion rules, and in the set of coherent states S-duality can be formulated as an SL(2,Z)-mapping between sectors which leaves the fusion rules invariant.Comment: 27 pages, harvmac, amssym, one eps figure; minor misprints corrected and title amende

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

Poisson structure and symmetry in the Chern-Simons formulation of (2+1)-dimensional gravity

In the formulation of (2+1)-dimensional gravity as a Chern-Simons gauge theory, the phase space is the moduli space of flat Poincar\'e group connections. Using the combinatorial approach developed by Fock and Rosly, we give an explicit description of the phase space and its Poisson structure for the general case of a genus g oriented surface with punctures representing particles and a boundary playing the role of spatial infinity. We give a physical interpretation and explain how the degrees of freedom associated with each handle and each particle can be decoupled. The symmetry group of the theory combines an action of the mapping class group with asymptotic Poincar\'e transformations in a non-trivial fashion. We derive the conserved quantities associated to the latter and show that the mapping class group of the surface acts on the phase space via Poisson isomorphisms