Chains of solitons

Abstract

We construct and analyse chains of solitons in various field theories. Particular emphasis is placed on the constituent structure, which appears to be be a generic feature of chains. In Yang-Mills theory, we construct axially symmetric chains of instantons (calorons) with instanton charge 2, making essential use of the Nahm transform. We show that there are two distinct families of caloron, which can be distinguished using representation theory. We also construct calorons on hyperbolic space with instanton charge 1 and monopole charge 0. This generalises earlier work of Garland and Murray, in the same way that non-integer-mass hyperbolic monopoles generalise the integer-mass hyperbolic monopoles of Atiyah. We study chains of skyrmions with charge 1 in both the Skyrme and planar Skyrme models, using various approximate analytic Ansätze. In the Skyrme model chains are argued to exist and to have an energy per baryon number lower than the charge 2 skyrmion. In the planar Skyrme model, we show that the stability of chains depends on the choice of potential function. We study chains and kinks in the CP(^n) sigma models analytically, in particular, we show that chains are kinks in a sigma model whose target is a homogeneous space for a loop group. This is the Sigma model analog of the statement that a caloron is a monopole whose gauge group is a loop group.