Instanton moduli and topological soliton dynamics

Abstract

It has been proposed by Atiyah and Manton that the dynamics of skyrmions in R 3+1 may be approximated by motion on a finite dimensional manifold obtained from the moduli space of SU(2) Yang-Mills instantons in R 4. Motivated by this work we describe how similar results exist for other soliton and instanton systems. We describe in detail two examples for the approximation of the infinite dimensional dynamics of sine-Gordon solitons in R 1+1 by finite dimensional dynamics on a manifold obtained from instanton moduli. In the first example we use the moduli space of CP 1 instantons in R 2 and in the second example we use the moduli space of SU(2) Yang-Mills instantons in R 4. The metric and potential functions on these manifolds are constructed and the resulting dynamics is compared with the explicit exact soliton solutions of the sine-Gordon theory.