Domain Wall Equations, Hessian of Superpotential, and Bogomol'nyi Bounds

TL;DR

This paper investigates conditions under which all finite-energy solutions of domain wall equations in quantum field theories are BPS, showing that the Hessian of the superpotential plays a crucial role in this property.

Contribution

It proves that the definiteness of the Hessian of the superpotential guarantees all finite-energy solutions are BPS, and provides examples where this fails.

Findings

Definiteness of the Hessian ensures BPS solutions.

Indefinite Hessian allows non-BPS solutions.

Examples illustrate the conditions for BPS property.

Abstract

An important question concerning the classical solutions of the equations of motion arising in quantum field theories at the BPS critical coupling is whether all finite-energy solutions are necessarily BPS. In this paper we present a study of this basic question in the context of the domain wall equations whose potential is induced from a superpotential so that the ground states are the critical points of the superpotential. We prove that the definiteness of the Hessian of the superpotential suffices to ensure that all finite-energy domain-wall solutions are BPS. We give several examples to show that such a BPS property may fail such that non-BPS solutions exist when the Hessian of the superpotential is indefinite.