Curvature Corrections to Dynamics of Domain Walls

TL;DR

This paper critically reexamines the derivation of curvature corrections to domain wall dynamics, rectifies a fundamental step, and derives a new exact effective action including quadratic curvature terms.

Contribution

It provides a corrected, exact analytic expression for the effective action of domain walls with curvature corrections, clarifying previous assumptions and extending the theoretical framework.

Findings

Derived a new exact effective action including quadratic curvature terms

Identified and corrected a key step in the derivation process

Showed that previous zero rigidity assumptions remain physically valid

Abstract

The most usual procedure for deriving curvature corrections to effective actions for topological defects is subjected to a critical reappraisal. A logically unjustified step (leading to overdetermination) is identified and rectified, taking the standard domain wall case as an illustrative example. Using the appropriately corrected procedure, we obtain a new exact (analytic) expression for the corresponding effective action contribution of quadratic order in the wall width, in terms of the intrinsic Ricci scalar $R$ and the extrinsic curvature scalar $K$. The result is proportional to $cK^2-R$ with the coefficient given by $c\simeq 2$. The resulting form of the ensuing dynamical equations is obtained in terms of the second fundamental form and the Dalembertian of its trace, K. It is argued that this does not invalidate the physical conclusions obtained from the "zero rigidity" ansatz $c=0$ used in previous work.