On the energy crisis in noncommutative CP(1) model

TL;DR

This paper investigates the noncommutative CP(1) model in (2+1) dimensions, showing that instanton solutions are preserved under the Seiberg-Witten map and addressing the energy crisis hypothesis.

Contribution

It demonstrates that the noncommutative CP(1) system retains instanton solutions similar to the commutative case, challenging the energy crisis hypothesis.

Findings

Instanton solutions are identical in noncommutative and commutative models.

BPS equations are compatible with full equations of motion.

No energy crisis occurs in the noncommutative CP(1) model.

Abstract

We study the CP(1) system in (2+1)-dimensional noncommutative space with and without Chern-Simons term. Using the Seiberg-Witten map we convert the noncommutative CP(1) system to an action written in terms of the commutative fields. We find that this system presents the same infinite size instanton solution as the commutative Chern-Simons-CP(1) model without a potential term. Based on this result we argue that the BPS equations are compatible with the full variational equations of motion, rejecting the hypothesis of an "energy crisis". In addition we examine the noncommutative CP(1) system with a Chern-Simons interaction. In this case we find that when the theory is transformed by the Seiberg-Witten map it also presents the same instanton solution as the commutative Chern-Simons-CP(1) model.