Kink Moduli Spaces -- Collective Coordinates Reconsidered

TL;DR

This paper reexamines the structure of moduli spaces for kinks and antikinks in certain field theories, proposing improved coordinate choices and demonstrating how reduced models capture key dynamics.

Contribution

It introduces refined moduli space constructions for kink configurations, resolving coordinate issues and linking geometry with exact solutions in sine-Gordon theory.

Findings

Constructed examples of moduli spaces for kink-antikink systems.

Resolved null-vector problem via better coordinate choices.

Reduced dynamics accurately match exact solutions across energies.

Abstract

Moduli spaces - finite-dimensional, collective coordinate manifolds - for kinks and antikinks in $\phi^4$ theory and sine-Gordon theory are reconsidered. The field theory Lagrangian restricted to moduli space defines a reduced Lagrangian, combining a potential with a kinetic term that can be interpreted as a Riemannian metric on moduli space. Moduli spaces should be metrically complete, or have an infinite potential on their boundary. Examples are constructed for both kink-antikink and kink-antikink-kink configurations. The naive position coordinates of the kinks and antikinks sometimes need to be extended from real to imaginary values, although the field remains real. The previously discussed null-vector problem for the shape modes of $\phi^4$ kinks is resolved by a better coordinate choice. In sine-Gordon theory, moduli spaces can be constructed using exact solutions at the critical energy separating scattering and breather (or wobble) solutions; here, energy conservation relates the metric and potential. The reduced dynamics on these moduli spaces accurately reproduces properties of the exact solutions over a range of energies.