Energy Crisis or a New Soliton in the Noncommutative CP(1) Model?

TL;DR

This paper investigates the noncommutative CP(1) model, revealing a potential new soliton and an energy discrepancy due to inadequacies in the energy momentum tensor definitions, highlighting an 'energy crisis' in the theory.

Contribution

It introduces a new formalism for the NC CP(1) model, analyzes the energy bounds, and uncovers inconsistencies in the BPS equations versus equations of motion.

Findings

Identical static energy expressions in symmetric and canonical forms with spatial noncommutativity.

Existence of a lower energy bound indicating a potential new soliton.

Incompatibility between BPS equations and variational equations, leading to an energy crisis.

Abstract

The Non-Commutative (NC) CP(1) model is studied from field theory perspective. Our formalism and definition of the NC CP(1) model differs crucially from the existing one \cite{nccp}. Due to the U(1) gauge invariance, the Seiberg-Witten map is used to convert the NC action to an action in terms of ordinary spacetime degrees of freedom and the subsequent theory is studied. The NC effects appear as (NC parameter) $\theta $-dependent interaction terms. The expressions for static energy, obtained from both the symmetric and canonical forms of the energy momentum tensor, are {\it {identical}}, when only spatial noncommutativity is present. Bogomolny analysis reveals a lower bound in the energy in an unambiguous way, suggesting the presence of a new soliton. However, the BPS equations saturating the bound are not compatible to the full variational equation of motion. This indicates that the definitions of the energy momentum tensor for this particular NC theory, (the NC theory is otherwise consistent and well defined), are inadequate, thus leading to the "energy crisis". A collective coordinate analysis corroborates the above observations. It also shows that the above mentioned mismatch between the BPS equations and the variational equation of motion is small.