On the energy crisis in noncommutative CP(1) model

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.


Introduction
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. Afterwards Seiberg and Witten showed that the low energy dynamics of string theory can be described in terms of the noncommutative Yang-Mills theory [2]. Since then many papers appeared covering diverse applications of noncommutative theory in physical problems. One important aspect in these investigations has been the study of noncommutative instantons and solitons. These play a fundamental role in understanding of the nonperturbative effects in noncommutative gauge theory. In four-dimensional Yang-Mills theory it was shown the existence of instantons even in the U (1) case [3], and nontrivial solitons were found in noncommutative scalar theory [4].
The two-dimensional CP (N ) sigma model presents many similarities to four-dimensional Yang-Mills theory. An important one is the existence of the instanton solutions [5]. The extension of this similarity to the case of noncommutative space was first analyzed by B. Lee, K. Lee and S. Yang in Ref. [6]. After that many interesting properties of the CP (1) model in noncommutative space were found [7]. The noncommutative CP (1) model was investigated by S. Ghosh using the Seiberg-Witten map in Ref. [8]. The author utilizes there the Seiberg-Witten map to convert the noncommutative CP (1) action to an action written in terms of the commutative fields. He claims that in the subsequent theory the BPS equations are not compatible with the variational equation of motion, attributing this particular feature to an inadequate definition of energymomentum tensor in noncommutative space. In this paper we show, using the results recently obtained in Ref. [9], that the BPS and Euler-Lagrange equations obtained in Ref. [8] are the same as the well-know BPS and motion equations of the commutative CP (1) theory. Using the method explored in Ref. [8] we also study the case of noncommutative Chern-Simons-CP (1) theory without a potential term. We find that both models share the same instanton solution, which is the infinite size solution found in Ref. [9] for commutative Chern-Simons-CP (1) in the absence of potential term.
The paper is organized as follows: In Section 2 we review the result obtained in Ref. [9]. Section 3 is advocated to show that the BPS and variational motion equation present in the model study by Ghosh are compatible. Finally we analyze in Section 4 the Chern-Simons-CP (1) case.

The commutative Chern-Simons-CP (1) instanton
In this Section we shall describe briefly the characteristics of the Chern-Simons-CP (1) model in commutative space, which was studies in Ref. [9]. We begin by considering a (2 + 1)-dimensional Chern-Simons model coupled to a complex two component field n(x) described by the action Here D µ = ∂ µ − iA µ (µ = 0, 1, 2) is the covariant derivative and S cs is the Chern-Simons action given by where The metric signature is (1, −1, −1) and the two component field n(x) is subject to the constraint n † n = 1. The constraint can be introduced in the variational process with a Lagrange multiplier. Then we extremise the following action The obtained field equations are and From the first equation we can deduce to yield The equation of notion for the gauge field can be rewritten as κǫ µνρ F νρ = J µ where is Gauss's law of the Chern-Simons dynamics [10]. Using the Gauss's law, the energy functional for a static field configuration can be expressed as where B = F 12 . Using the identity |D i n| 2 = |(D 1 ± iD 2 )n| 2 ∓ B ± ǫ ij 2 ∂ i J j , the energy (10) becomes where Q CP (1) is the topological bound of the pure CP(1) system [5], [9]. The saturation of the bound take place when the first-order Bogomolnyi self-duality equations are satisfied, It was shown in Ref. [9] that the N-solution for these equations are instantons of infinite size. 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.

The noncommutative CP (1) model and the Seiberg-Witten map
We shall consider the Chern-Simons-CP (1) model before introduced, but now on (2+1)-dimensional noncommutative space. The coordinates of the noncommutative space obey the relations with θ > 0. It will be convenient introduce complex variables z andz which can be related to an annihilation and creation operatorsâ andâ † acting on a Fock space,â so that (13) becomes â,â † = 1 (16) In this way, through the action ofâ † on the vacuum state |0 , eigenstates of the number operatorN are generated. With our conventions, derivatives in the Fock space are given by and integration on the noncommutative plane should be interpreted as a trace The action of the noncommutative CP (1) model [8] iŝ where the covariant derivative is defined aŝ and withn andÂ µ we indicate that the fields are noncommutative. The new gauge fields transforms as followŝ The noncommutative fieldsn,Â µ and the noncommutative gauge parameterα, can be expressed as a function of the ordinary fields n, A µ and the commutative gauge parameter α by the so called Seiberg-Witten [2] map, whose expression at the lowest order in θ isÂ this map transforms a noncommutative gauge orbit in a commutative one. Following the reference [8] we substitute the above form ofn andÂ µ in the action (20), which readsŜ As in reference [8] we assumen †n = n † n = 1 (25) The equation of motion, derived from the action (24), for the gauge field is This equation can be rewritten in a reduced form as where a µ (θ) represent the O(θ) correction obtained from (26). Nevertheless, as it is mentioned in the paper of Ghosh [8], the term a µ (θ) does not play any role when we introduce (27) in (24) because, here, we are only interested on the first order correction in θ. Thus we drop the term a µ (θ) in (27) and identify Here we are interested in time-independent instanton solutions to the field equations that ensure the finiteness of the action (24). These are stationary points of the energy, which for static field configuration reads as By varying with respect to A 0 we obtain the relation which, in virtue of our considerations, may be rewritten as A 0 = 0. Replacing this relation in the action (29) we arrive to Since we are not interested in O(θ 2 ) contribution, the last expression may be written as which show that the energy is zero or a positive number. The finiteness of the energy requires that the covariant derivative must vanish asymptotically. This fixes the asymptotic behavior of the fields where n 0 is a fixed complex vector with (n 0 ) † n 0 = 1 and α is common phase angle. This α depend on φ, the angle in coordinate space that parameterizes the boundary of the space. With these conditions the magnetic flux reads where N is a topological invariant which takes only integer values. As in the commutative CP (1) model we can use to explore the minimum of the energy the following identity Using this identity the energy functional becomes Here we choose the upper signs for positive flux and for negative flux we choose the lower signs. Choosing the upper sign we obtain the following lower bound on the energy which is saturated by fields satisfying the first-order Bogomolnyi self-duality equation [11] This is the result arrived in Ref. [8]. However the term Φ + d 2 x 1 2 θ 12 B 2 is not really a lower bound on the energy. We will show that the integral in the equation (37) goes to zero when the region of integration becomes infinity. We begin the proof by the considering the following ansatz for the N -instanton solution where r = x 2 1 + x 2 2 and φ = tan −1 ( x 2 x 1 ) . Introducing this ansatz in the equation (36) and taken the upper sign, we obtain whereas the substitution in the equation (38) gives In (40) we have integrated over a two-dimensional disc D R of radius R. Also we have considered only the corrections to first order in θ. From here it follows that the appropriate boundary conditions for finite energy solution are [9] lim r→0 f (r) = π , lim r→R f (r) = 0 Following the reference [9] we shall denote the solution of Eq.(41) subject to the boundary conditions (42) on a disc D R 1 as f R 1 (r). Consider now the following configuration defined on a disc D λR 1f where λ > 1. According to (42) this configuration is subject to the following boundary conditions We can evaluate the energy (40) for the configuration (43) on the disc D λR 1 E(λR 1 ) = 2π Sincef λR 1 (r) satisfies the same boundary conditions as the solution of Eq.(41) on the disc D λR 1 , which we denote here as f λR 1 (r), we arrive to where E(λR 1 ) denote the functional energy evaluated on the disc D λR 1 for the solution f λR 1 (r). Under the transformation r = xλ the functional (45) becomes Since f R 1 is a solution of (41) on the disc D R 1 , the functionalẼ(λR 1 ) reduce tõ E(λR 1 ) = 2π The limit for arbitrary large λ gives lim λ→∞Ẽ (λR 1 ) = Φ (49) which implies, in virtue of equation (46), that Thus the topological bound is saturated for arbitrary large R and for the fields satisfying the condition (38). This also shows that both the magnetic field and the integral of its square should be zero for arbitrary large R. Nevertheless the magnetic flux remains constant for all two-dimensional disc. This is because the magnetic flux depends only on the boundary conditions which are fixed One important aspect here is that the solution of our model, as R → ∞, is the same as the infinite size solution found in Ref. [9] for the commutative Chern-Simons-CP (1) system. Both systems are subject to identical boundary conditions and satisfied the same BPS equations. Thus the effect of mapping to a first order in θ the noncommutative CP (1) theory by Seiberg-Witten map is the same as to add a Chern-Simons term in the commutative CP (1) theory.
By varying the action (24) with respect to CP (1) field n yields Following the reference [8] we may eliminate the lagrange multiplier λ Inserting λ in the equation of motion we get As we mentioned the magnetic field F 12 becomes zero as R → ∞. So the equation (55) reduce to This is the Euler-Lagrange equation of the CP (1) theory [5]. Hence the BPS and the Euler-Lagrange equations are identical to those of the commutative CP (1) system and there are no incompatibility between them. It is interesting to note that our result is valid only when we are working in two-dimensional infinite disc. For finite space the BPS and the field equations are no compatible and the result obtained by Ghosh is valid.

Noncommutative Chern-Simons-CP (1) case
In this section we analyze the model obtained by the application of the Seiberg-Witten map to the noncommutative Chern-Simons-CP (1) action. We show that this model admit the same instanton solution as the system studied in the previous section. The action that describe this model iŝ Under the Seiberg-Witten map (23), this action transforms as followŝ This action differs from the action (24) only on the commutative Chern-Simons term [12]. Using the notation of equation (27) we can write the equation of motion for the gauge field as whereas the the dynamical equation for the CP (1) field is the same as the equation obtained in the previous section In the static field configuration the action (58) leads to the following form of the functional energy The substitution of the time component of (59) in the expression of the energy yields After integration by parts, the last equation can be rewritten as Here J is the topological charge density of CP (1) model. For κ = 0 the last expression reduce to the functional energy (31) of the mapped pure CP (1) model. The energy (64) is further rewritten by using the identity is connected with magnetic flux by following formula [9] 2π d 2 xJ The line integral vanishes for any finite solution on the plane. We will choose the upper sings for positive flux and the lower signs for negative flux. We shall show that the magnetic flux is lower bound on the energy as the region of integration becomes infinity. This bound will be saturated by static configuration obeying the self-duality equation Let us consider the following ansatz for the N -instanton solution [9] n(φ, r) = where r and φ are defined as in the Section 3. Using this ansatz it is not very difficult to show that the N -instanton energy functional (66) reads as where we integrate over a two-dimensional disc D R . The equation (68) takes the form It is reasonable to impose for finiteness of the energy the following boundary conditions [9]  Notice that the boundary condition for f (r) are the same as the conditions (42), whereas the A φ (r) defined in the ansatz (39) satisfies the boundary conditions (72) and (73) imposed to a(r). As in Ref. [9] we consider a R 1 (r) and f R 1 (r) the solutions of (71) on a disc D R 1 , subject to the boundary conditions (72) and (73). With these solutions we construct the following configuration defined on a disc D λR 1 [9] again λ > 1. The boundary conditions for this configuration are The energy functional (70) evaluated in the configuration (74) gives ∂ r (rã λR 1 (r)) r 2 sin(f λR 1 (r))∂ rfλR 1 (r) + |Φ|(77) We will denote the solution of (71) corresponding to the disc D λR 1 and subject to the boundary conditions (72)-(73) as a λR 1 (r) and f λR 1 (r) and its energy as E(λR 1 ). Then it is clear that the configuration (74) do not minimize the energy on the disc D λR 1 . From this and the equation (66) we can write Again we introduce the transformation r = xλ E(λR 1 ) = 2π The limit for large λ gives lim λ→∞Ẽ (λR 1 ) = 2π Since f R 1 is a solution of (71) on the disc D R 1 the last expression reduce to lim λ→∞Ẽ (λR 1 ) = |Φ| In order to complete the proof we show that d 2 xJ CP (1) 0 F 12 becomes zero as λ → ∞. In terms of the solution on the disc D λR 1 this integral reads Introducing the change of variable r = λx, we obtain In a shorten form this integral may be rewritten as where K(R 1 ) is finite since Therefore for arbitrary large λ the equation (83) In the same form as the model studied in Section 3, here the saturation of topological bound requires that the field satisfy the self-duality equation (68) as the region of integration in (70) becomes infinite. In addition, notice that since the boundary conditions are the same in both models, the solution of the model analyzed in Section 3 is also the N -instanton solution for this model.

Conclusions
In this article we have discussed a model obtained by the application of the Seiberg-Witten map to noncommutative CP (1) system. This model was studied previously by Ghosh in reference [8]. The author showed there that the BPS equations for this model are the same as those found in the commutative CP (1) model. Nevertheless he claims that the variational equations in both models are different. The difference consists in the θ-term 1 2 θ 12 ∂ i F 12 D i n. In the present paper we showed that this term vanishes if boundary conditions are in an infinite plane. More specifically we showed that This allows us to correct the result present in [8] and also show that for the fields satisfying first order self-duality equations |(D 1 ± iD 2 )n| 2 = 0 (89) the magnetic flux is the lower bound on the energy It is also remarkable that this model admits, as R → ∞, the same infinite size N -instanton solutions found in [9]. Finally, using the method of Section 3, we have studied the noncommutative Chern-Simons-CP (1) model, finding that this model also presents the infinite size instanton solutions of the commutative Chern-Simons-CP (1) system.