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.
Highlights
The study of field theories in noncommutative space has received much attention in the last few years [1]. The connection between these theories and string theory was first considered by Connes, Douglas and Schwartz, who observed that noncommutative geometry arise as a possible scenario for certain low energy of string theory and M-theory
In this article we have discussed a model obtained by the application of the Seiberg-Witten map to noncommutative CP (1) system
The author showed there that the BPS equations for this model are the same as those found in the commutative CP (1) model
Summary
The study of field theories in noncommutative space has received much attention in the last few years [1]. The infinite size of the solution is due to the condition B = 0 derived from the introduction of the Chern-Simons term in the CP (1) model. We shall consider the Chern-Simons-CP (1) model before introduced, but on (2+1)-dimensional noncommutative space. We are interested in time-independent instanton solutions to the field equations that ensure the finiteness of the action (24).
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