/home/www/ftp/data/hep-th/dir_0306045/0306045.dvi ENERGY CRISIS OR A NEW SOLITON IN THE NONCOMMUTATIVE CP(1) MODEL? Subir Ghosh Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Calcutta 700108, India. Abstract: The Non-Commutative (NC) CP(1) model is studied from field theory perspective. Our for- malism and definition of the NC CP(1) model differs crucially from the existing one [7]. 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) θ-dependent interaction terms. The expressions for static energy, obtained from both the symmetric and canonical forms of the energy momentum tensor, are 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. Keywords: CP(1) model, Noncommutative field theory, Seiberg-Witten map. 1 Introduction Non-Commutative (NC) field theories have turned into a hotbed of research activity af- ter its connection to low energy string physics was elucidated by Seiberg and Witten [1, 2]. Specifically, the open string boundaries, attached to D-branes [3], in the presence of a two-form background field, turns into NC spacetime coordinates [1]. (This phenomenon has been recov- ered from various alternative viewpoints [4].) The noncommutativity induces an NC D-brane world volume and hence field theories on the brane become NC field theories. Studies in NC field theories have revealed unexpected features, such as UV-IR mixing [5], soliton solutions in higher dimensional scalar theories [6], to name a few. The inherent non- locality, (or equivalently the introduction of a length scale by θ - the noncommutativity param- eter), of the NC field theory is manifested through these peculiar properties, which are absent in the corresponding ordinary spacetime theories. Also, solitons in NC CP(1) model have been found [7], very much in analogy to their counterpart in ordinary spacetime. The present work also deals with the search for the solitons in the NC CP(1) model. The difference between our field theoretic analysis and the existing framework [7] is explained below. In fact, we closely follow the conventional field theoretic approach in ordinary spacetime [8]. The Seiberg-Witten map [1] plays a pivotal role in our scheme. The Bogolmolny analysis of the static energy reveals a lower bound, protected by topological considerations. However, we encounter a small discrep- ancy between the BPS equations and the variational equation of motion. Although ”small” in an absolute sense, the mismatch is conceptually significant for the reasons elaborated below. The above conclusions, drawn from field theoretic analysis, will be corroborated and quantified explicitly in a collective coordinate framework. It appears natural to attribute the above mentioned problem to the definition of the energy functional of the NC CP(1) model in particular, and of the NC field theories in general, (since the BPS equations are derived directly from the energy of the system). As it is well known, there are complications in the definition of the Energy-Momentum (EM) tensor in NC field theory [9, 10]. In general, it is not possible to obtain a symmetric, gauge invariant and conserved EM tensor. There are two forms of EM tensor in vogue: a manifestly symmetric form [10], obtained from the variation of the action with respect to the metric, and the canonical form [9], following the Noether prescription. The former is covariantly conserved whereas the latter is conserved. Interestingly, we find that in the particular case that we are considering, that is NC CP(1) model in 2+1-dimensions, with only spatial noncommutativity, expressions for the static energy, obtained from both the derivations, are identical. Moreover the expression for energy is gauge invariant. We show that there is a Bogomolny like lower bound in the energy. However, the subsequently derived BPS equations (that saturate the lower bound), does not fully satisfy the equation of motion. Let us put our work in its proper perspective. As such, our result in no way questions the consistency of the existing literature [7] on CP(1) solitons since our model differs crucially from the one considered in [7]. In particular, we have adopted a different NC generalization of the CP(1) constraint. (We have provided conceptual and technical reasons for allowing such a difference.) Thus it is not expected that the results of [7] will be reproduced. In fact, the energy profile of the localized structure that we uncover, is more sharply peaked and has an O(θ) correction, with respect to the ordinary spacetime CP(1) soliton. (This will be made explicit in the collective coordinate analysis at the end.) Surprisingly, the BPS equations remain 2 unchanged, although the variational equation of motion is altered. All the same, we emphasize that, we have provided a well defined NC gauge theory, which conforms to the expected features of such a system. Hence, the clash between the BPS equations and the equation of motion that is revealed here, can have a deeper bearing on the structure of a general NC gauge theory, indicating that the traditional lore of field theory in ordinary spacetme should be applied with greater care in the context of NC field theory. Our methodology and its difference from the existing one [7] is explained below. There are two basic approaches in studying an NC field theory: (i) The appropriate NC field theory is constructed in terms of NC analogue fields (ψ̂) of the fields (ψ) with the replacement of ordinary products of fields (ψϕ), by the Moyal-Weyl ∗-product (ψ̂ ∗ ϕ̂), ψ̂(x)∗ϕ̂(x) = e i2 θµν ∂σµ ∂ξν ψ̂(x+σ)ϕ̂(x+ξ) |σ=ξ=0= ψ̂(x)ϕ̂(x) + i 2 θρσ∂ρψ̂(x)∂σϕ̂(x) + O(θ 2). (1) The hatted variables are NC degrees of freedom. We take θρσ to be a real constant antisymmet- ric tensor, as is customary [1], (but this need not always be the case [11]). The NC spacetime follows from the above definition, [xρ,xσ]∗ = iθ ρσ. (2) Note that the effects of spacetime noncommutativity has been accounted for by the introduc- tion of thr ∗-product. For gauge theories the Seiberg-Witten Map [1] plays a crucial role in connecting φ̂(x) to φ(x). This formalism allows us to study the effects of noncommutativity as θρσ dependent interaction terms in an ordinary spacetime field theory format. This is the prescription we will follow. (ii) An alternative framework is to treat the NC theories as systems of operator valued fields and to directly work with operators on the quantum phase space, characterized by the non- commutativity condition (2). On the NC plane, the coordinates satisfy a Heisenberg algebra [x1,x2]∗ = x1 ∗x2 −x2 ∗x1 = iθ12 = i�12θ = iθ which in the complex coordinates reduces to the creation annihilation operator algebra for the simple Harmonic Oscillator. Thus to a function in the NC spacetime, through Weyl transform, one associates an operator acting on the Hilbert space, in a basis of a simple Harmonic Oscillator eigenstates. The investigations on the NC CP 1 solitons carried out so far [7, 12] exploit the latter method. As it turns out, a major advantage is that structurally, the NC system with its dynamical equations, energy functionals etc., are similar to their ordinary spacetime counterpart. This happens because the ∗-products of (i) are replaced by operator products in (ii) and the spacetime integrals are replaced by trace over the basis states in the hilbert space. In the present work, our aim is to study the NC CP 1 solitons in the former field theoretic approach. From past experiences [13] we know this to be a perfectly viable formalism. Indeed, since the NC spacetime physics is not that much familiar or well understood, it is imperative that one explores different avenues to reach the same goal, to gain further insights. Also we would like to point out that since solitons are already present in the CP 1 model at θ = 0 (i.e. ordinary spacetime), unlike the noncommutative solitons of the scalar theory [6] it is natural to analyze the fate of the solitons under a small perturbation, (which is a small value of θ in the present case). The small θ-results of [7, 12] are perfectly well defined. The paper is organized as follows: Section II contains a short recapitulation of the CP(1) solitons. This will help us fix the notations and in fact, identical procedure will be pursued 3 in the NC theory as well. The detailed construction of our version of the NC CP(1) model is provided in section III. Section IV discusses the energy momentum tensor of the model. Section V consists of the Bogomolny analysis in the NC theory. Section VI is devoted to the collective coordinate analysis. The paper ends with a conclusion in Section VII. Section II - CP(1) Soliton: a brief digression Let us digress briefly on the BPS solitons of CP(1) model in ordinary spacetime. Later we will proceed with the NC theory in an identical fashion. The gauge invariant action, S = ∫ d3x [(Dµφ)†Dµφ + λ(φ †φ − 1)], (3) where Dµφ = (∂µ − iAµ)φ defines the covariant derivative and the multiplier λ enforces the constraint, the equation of motion for Aµ leads to the identification, Aµ = −iφ†∂µφ. (4) Since the ”gauge field” Aµ does not have any independent dynamics one is allowed to make the above replacement directly in the action. Obviously the infinitesimal gauge transformation of the variables are, δφ† = −iλφ† ; δφ = iλφ ; δAµ = ∂µλ. (5) From the EM tensor Tµν = (Dµφ) †Dνφ + (Dνφ) †Dµφ − gµν (Dσφ)†Dσφ, (6) the total energy can be expressed in the form, E = ∫ d2x (| D0φ |2 + | (D1 ± iD2)φ |2) ± 2πN, (7) where the last term denotes the topological charge N ≡ ∫ d2x n(x) = 1 2πi ∫ d2x �ij (Diφ) †Djφ = ∫ d2x 1 4π �ijFij = ∫ d2x 1 2π F12, (8) corresponding to the conserved topological current. The Bogomolny bound follows from (7), E ≥ 2π | N |, (9) with the following saturation conditions (BPS equations) obeyed by the soliton, | D0φ |2=| (D1 ± iD2)φ |2= 0. (10) It can be checked that the solutions of the BPS equations belong to a subset of the full set of solutions, that satisfy the variational equation of motion. Section III - Construction of the NC CP(1) model 4 Let us now enter the noncommutative spacetime. The first task is to generalize the scalar gauge theory (3) to its NC version, keeping in mind that the latter must be ∗-gauge invariant. The NC action is, Ŝ = ∫ d3x (D̂µφ̂)† ∗ D̂µφ̂ = ∫ d3x (D̂µφ̂)†D̂µφ̂, (11) where the NC covariant derivative is defined as D̂µφ̂ = ∂µφ̂ − iµ ∗ φ̂. Depending on the positioning of µ and φ̂, the covariant derivative can act in three ways , D̂µφ̂ = ∂µφ̂ − iµ ∗ φ̂ = ∂µφ̂ + iφ̂ ∗ µ = ∂µφ̂ − i(φ̂ ∗ µ − µ ∗ φ̂) (12) which are termed respectively as fundamental, anti-fundamental and adjoint representations. We have chosen the fundamental one. 1 Notice that for the time being we have not considered the target space (CP(1)) constraint. We will return to this important point later. The NC action (11) is invariant under the ∗-gauge transformations, δ̂φ̂† = −iλ̂ ∗ φ̂† ; δ̂φ̂ = iλ̂ ∗ φ̂ ; δ̂µ = ∂µλ̂ + i[λ̂, µ]∗. (13) We now exploit the Seiberg-Witten Map [1, 16] to revert back to the ordinary spacetime degrees of freedom. The explicit identifications between NC and ordinary spacetime counterparts of the fields, to the lowest non-trivial order in θ are, µ = Aµ + θ σρAρ(∂σAµ − 1 2 ∂µAσ) φ̂ = φ − 1 2 θρσAρ∂σφ ; λ̂ = λ − 1 2 θρσAρ∂σλ. (14) As stated before, the ”‘hatted”’ variables on the left are NC degrees of freedom and gauge transformation parameter. The higher order terms in θ are kept out of contention as there are certain non-uniqueness involved in the O(θ2) mapping. The significance of the Seiberg-Witten map is that under an NC or ∗-gauge transformation of µ by, δ̂µ = ∂µλ̂ + i[λ̂, µ]∗, Aµ will undergo the transformation δAµ = ∂µλ. Subsequently, under this mapping, a gauge invariant object in conventional spacetime will be mapped to its NC counterpart, which will be ∗-gauge invariant. This is crucial as it ensures that the ordinary spacetime action that we recover from the NC action (11) by applying the 1This is the first difference between our model and [7] who use the anti-fundamental representation. In fact, in [7], it is difficult to proceed with the fundamental definition [14]. On the other hand, in the present work, the choice between the first and second definition is not very important as it affects the overall sign of θ only. Similar type of situation prevails in [15, 13]. 5 Seiberg-Witen Map will be gauge invariant. Thus the NC action (11) in ordinary spacetime variables reads, Ŝ = ∫ d3x[(Dµφ)†Dµφ + 1 2 θαβ{Fαµ((Dβφ)†Dµφ + (Dµφ)†Dβφ) − 1 2 Fαβ (D µφ)†Dµφ}] (15) The above action is manifestly gauge invariant. The equation of motion now satisfied by Aµ is, i(−2iAµφ†φ + φ†∂µφ − ∂µφ†φ)(1 − 1 2 θαβFαβ) + 1 2 θαµ[∂ α{(Dβφ)†Dβφ} − ∂β{(Dαφ)†Dβφ + (Dβφ)†Dαφ}] −1 2 θαβ [∂ α{(Dβφ)†Dµφ + (Dµφ)†Dβφ) + iF αµ(φ†Dβφ − (Dβφ)†φ}] = 0 (16) Remember that so far we have not introduced the CP 1 target space constraint in the NC spacetime setup. Let us assume the constraint to be identical to the ordinary spacetime one, i.e., 2 φ†φ = 1. (17) The reasoning is as follows. Primarily, after utilizing the Seiberg-Witten Map, we have returned to the ordinary spacetime and its associated dynamical variables and the effects of noncommu- tativity appears only as additional interaction terms in the action. Hence it is natural to keep the CP 1 constraint unchanged. Alternatively, the above assumption can also be motivated in a roundabout way. Remember that the the CP(1) constraint has to be introduced in a ∗-gauge invariant way. In order to introduce the CP 1 constraint directly in the NC action (11) or (15), the constraint term ∫ λ(φ†φ − 1) has to be generalized to a ∗-gauge invariant one by the application of the (inverse) Seiberg-Witten Map. This is quite straightforward but needless because as soon as we apply the Seiberg-Witten Map to the ∗-gauge invariant constraint term, we recover the earlier ordinary spacetime constraint. This allows us to write, Aµ = −iφ†∂µφ + aµ(θ) (18) with aµ denoting the O(θ) correction, obtained from (16,17). For θ = 0. Aµ reduces to its original form. Note that aµ is gauge invariant. Thus the U(1) gauge transformation of Aµ remains intact, at least to O(θ). Keeping in mind the constraint φ †φ = 1, let us now substitute(18) in the NC action (15). Since we are concerned only with the O(θ) correction, in the θ-term of the action, we can use Aµ = −iφ†∂µφ. However, in the first term in the action, we must incorporate the full expression for Aµ given in (18). Remarkably, the constraint condition conspire to cancel the effect of the O(θ) correction term aµ. Finally it boils down to the following: the action for the NC CP 1 model to O(θ) is given by (15) with the identification Aµ = −iφ†∂µφ and φ†φ = 1. 2This is the second difference between our model and that of [7], where θ-correction terms are present in the CP (1) constraint. This is a serious difference as it drastically alters the structures of the model in [7] from ours. Apart from the conceptual reasoning given above, there also appears a technical compulsion. We would like to obtain perturbative θ- corrections to the ordinary spacetime CP (1) model. Incorprtating θ- corrections in the CP (1) constraint as in [7] in our system will lead to a differential equation for the multiplier λ, instead of an algebraic one as in the ordinary spacetime case. This will change the φ- equation of motion in a qualitative way. We stress that our model is a perfectly well defined NC theory which, incidentally, is distinct from the existing NC CP (1) model [7]. 6 Section IV - Energy-momentum tensor for the NC CP(1) model Our aim is to study the possibility of soliton solutions for the action (15). Let us try to derive the Bogolmony bound and BPS equations in the present case. The first task is to compute the EM tensor. We follow [10] in computing the symmetric form of the EM tensor by coupling the model with a weak gravitational field and get, T Sµν = (1− 1 4 θαβFαβ )[(Dµφ) †Dνφ+(Dνφ) †Dµφ−gµν (Dσφ)†Dσφ]+ 1 2 θαβ [Fαµ((Dβφ) †Dνφ+(Dνφ) †Dβφ) +Fαν ((Dβφ) †Dµφ + (Dµφ) †Dβφ) − gµνFασ((Dβφ)†Dσφ + (Dσφ)†Dβφ)]. (19) The T Sµν stands for the symmetric form of the EM tensor. In the static situation, φ̇ = 0 → A0 = F0i = D0φ = 0 and the static energy density simplifies to, T S00 = (1 + 1 2 θ12F12)(D iφ)†Diφ, (20) where Fµν is also expressible in the form Fµν = −i[(Dµφ)†Dνφ − (Dνφ)†Dµφ] = −i[(∂µφ)†∂νφ − (∂νφ)†∂µφ]. Now we discuss the canonical form of the EM tensor. Remembering that the indices of adjacent φ’s are summed, the expanded form of the Lagrangian is, L̂ = (Dµφ)†Dµφ + λ(φ †φ − 1) − i 2 θαβ [2∂µφ †∂βφ∂αφ †∂µφ +2∂µφ †∂βφφ †∂αφφ †∂µφ−2∂βφ†∂µφφ†∂αφφ†∂µφ−∂αφ†∂βφ∂µφ†∂µφ−∂αφ†∂βφφ†∂µφφ†∂µφ]. (21) The canonical energy-momentum tensor is, Tµν = δL̂ δ(∂µφ†) ∂νφ † + δL̂ δ(∂µφ) ∂νφ − gµνL̂ = (1 − 1 4 θαβFαβ )(Dµφ †Dνφ + (µ ↔ ν)) −iθαβ [∂νφ†∂βφ(∂αφ†∂µφ + φ†∂αφφ†∂µφ) + (µ ↔ ν) − ∂βφ†∂νφφ†∂αφφ†∂µφ − (µ ↔ ν)] −iθµα[(− 1 2 ∂νφ †∂αφ + 1 2 ∂αφ†∂νφ)(D µφ)†Dµφ − ∂σφ†∂νφ(∂αφ†∂σφ + φ†∂αφφ†∂σφ) +∂σφ †∂αφ(∂νφ †∂σφ+φ†∂νφφ †∂σφ) +∂νφ †∂σφφ †∂αφφ†∂σφ−∂αφ†∂σφφ†∂νφφ†∂σφ]−gµνL̂. (22) Note that θαβ part is symmetric. In the energy density T00 the contribution coming from the non-symmetric parts in the θ-contribution drop out if only space-space noncommutativity 7 is assumed, i.e. θ0i = 0. For this special case, in the static limit, the above θ-contribution completely drops out and the energy density reduces to T N00 = −L̂ = (1 + 1 2 θ12F12)(D iφ)†Diφ. (23) Clearly this is identical to the static energy (20) obtained from the the symmetric form T Sµν . Indeed, it is satisfying that in this particular case, both the canonical and symmetric forms of the EM tensor lead to the same expression of the static energy, which is manifestly gauge invariant and conserved (as it comes from the canonical form). Interestingly to O(θ), the noncommutativity effect factors out from the ordinary spacetime result. Also notice that in the two spatial dimensions that we are considering, the θ-term in the energy density is proportional to the topological charge density n(x) in (8). This is because the expression for the topological current remains unchanged since the dynamical variables as well as the CP(1) constraint is unaltered in our model. We specialize to only space-space noncommutativity, θij = θ�ij , θ0i = 0, and find, T (θ) 00 = πθn(x)(D iφ)†Diφ, (24) where the superscript S or N is dropped. Section V - Analysis of the Bogomolny bound In order to obtain the Bogomolny bound, we follow the same procedure as that of the CP 1 model in ordinary spacetime and rewrite the static energy functional in the following form, E = ∫ d2x (1 + πθn)[| (D1 ± iD2)φ |2 ±2πn] = ∫ d2x [(1 + πθn) | (D1 ± iD2)φ |2] ± 2πn ± 2π2θn2] = ∫ d2x (1 + 1 2 πθn)2 | (D1 ± iD2)φ |2 ±2πN ± 2π2θ ∫ d2x n2(x) + O(θ2). (25) Now individually all the terms in the energy expression are positive definite. Hence we obtain the Bogomolny bound to be E ≥ N + 2π2θ ∫ d2x n2(x) (26) and the saturation condition is (1 + 1 2 πθn)2 | (D1 ± iD2)φ |2= 0. (27) The BPS equation turns out to be, D1φ = ±iD2φ. (28) Thus we find that the BPS equation remains unchanged and there is a O(θ) correction in the static energy of the soliton. Note that both of the above results do not agree with [7, 12]. But this is not unexpected since as we have mentioned before, the defining conditions of the NC CP(1) models are different. However, we repeat that a priori there is nothing inconsistent in our NC model. 8 Finally, we are ready to discuss the curiosity. It appears that the BPS equations (28) do not satisfy the equation of motion for φ. A straightforward computation yields the dynamical equation for φ, Dµ[(1 − 1 4 θαβF αβ)Dµφ] + 1 2 θαβ [iDα{(Dσφ)†DσφDβφ} + Dβ{FαµDµφ} + Dµ{FαµDβφ} −iDα{(Dβφ)†Dµφ+ (Dµφ)†Dβφ)Dµφ}+iDµ{(Dβφ)†Dµφ+ (Dµφ)†Dβφ)Dαφ}]−λφ = 0. (29) (Details of the derivation are provided in the appendix.) To get λ, contract by φ† and use φ†φ = 1. For the time being, instead of writing the full equation of motion, we want to check the consistency of the programme only, that is whether the solution of the BPS equation satisfies the equation of motion, which they should. Since the BPS equation remains unchanged here, the θ-term in the equation of motion should vanish for those solutions that satisfy the BPS equation as well. So we consider the equation of motion in a simplified setting where the BPS equation is satisfied and only θ12 is non- zero and obtain for λ λ = φ†DiDiφ + 1 2 θ12φ†Di(F12Diφ). (30) Putting λ back in the equation of motion, we get DiDiφ − φ†DiDiφφ + 1 2 θ12[Di(F12Diφ) − φ†Di(F12Diφ)φ] = 0. (31) This equation can be rewritten as DiDiφ − φ†DiDiφφ + 1 2 θ12[∂iF12Diφ − φ†∂iF12Diφφ + F12(DiDiφ − φ†DiDiφφ)] = 0. (32) Clearly the last term, that is 1 2 θ12F12(D iDiφ − φ†DiDiφφ) ≈ O(θ2) and can be dropped. The term θ12φ†∂iF12Diφφ = θ12∂iF12φ†Diφφ = 0. However the remaining O(θ)-term, 1 2 θ12∂iF12Diφ does not vanish. This is the purported mismatch between the BPS equations and the full equation of motion. This brings us to the last part - the collective coordinate analysis, where we can check explicitly the above conclusions in a simplified setup. Section VI - Collective coordinate analysis We consider the topological charge N = 1 sector. As we have discussed before, expression for the topological current and subsequently the charge remains same (in our NC CP(1) model) as that of the ordinary spacetime CP(1) model. This means that we can use the same parameterizations as before [8] to introduce the collective coordinates. As a first approximation, only the zero mode arising from the global U(1) invariance is being quantized. In the O(3) nonlinear sigma model, the N = 1 sector is characterized by [8] na = {r̂sin(g(r)),cos(g(r))} ; a = 1, 2, 3 with the constraint nana = 1 and the boundary conditions g(0) = 0; g(∞) = π. Keeping in mind the O(3)−CP(1) duality and the Hopf map na = φ†σaφ, the soliton profile in the CP(1) variables is of the form, φ ≡ ( φ1 φ2 ) = ( cos( g 2 ) sin( g 2 )ei(ϕ+α(t)) ) (33) 9 where the gauge (φ1)∗ = φ1 has been used and r,ϕ refer to the plane polar coordinates. α(t) is the collective coordinate. Substituting the above choice (33) in the static energy expression in (20) leads to, E(r) = (1 + θ sin(g)g′ 2r )[(g′)2 + sin2(g) r2 ], (34) where g′ = dg dr . In figure (1) the effect of the θ- correction is shown where the following simple form of g(r) is considered, g(r) ≈ π(1 − e−µr). (35) One can clearly see that with typical values of the parameters, (θ = 1 , µ = 1) the energy density for the NC case is more sharply peaked. (In reality, θ should be smaller.) This assures us of the rationale of our previous Bogolmony analysis. Next we look in to the equation of motion. It is straightforward check that the profile (33) satisfies the BPS equation as well as the equation of motion for the ordinary spacetime situation, θ = 0. For θ 6= 0, the BPS equations are once again satisfied since they remain unaltered. So we concentrate only on the problem term 1 2 θ12∂iF12Diφ in the equation of motion (32). With the particular form of g(r) in (35) we obtain 1 2 θ12∂iF12Diφ = θg′ 4 [ g′′ 2r sin( g 2 )− g ′ 2r2 sin( g 2 ) + (g′)2 4r cos( g 2 )] ( sin( g 2 ) −cos( g 2 )eiϕ ) ≡ ( X(r) Y (r) ) . (36) In Figure 2 we again use the g(r) given in (35) and compare the magnitude of the above expression with a typical term, ∂i∂iφ ≡ ( S(r) T(r) ) , occurring in the equation of motion (32). For consistency, the expressions in (36) should have vanished. However, even for θ = 1, (which is quite a large value), the mismatch term is small in an absolute sense. Section VII - Conclusions In this paper, we have attempted to recover the soliton solutions in the non- commutative CP(1) model, discovered earlier [7]. The U(1) and NC U(1) gauge invariances in the CP(1) and NC CP(1) models respectively, requires the use of the Seiberg-Witten map, to convert the NC action to an action comprising of ordinary spacetime dynamical variables. The effects of noncommutativity are manifested as interaction terms. For theoretical as well as technical reasons, we found it convenient to keep the CP(1) constraint unchanged, i.e. without any θ correction. From the above action, we construct both the symmetric and canonical forms of the energy momentum tensor. For only spatial noncommutativity, both the above forms reduce to an identical (gauge invariant) expression for the static energy. The Bogomolny analysis yields a lower bound in the energy, hinting at the presence of a new type of soliton. However, the resulting BPS equations do not match completely with the full variational equation of motion. The present model is otherwise a perfectly well defined NC field theory with the expected features. Hence we conclude that inadequacy in the definitions of the energy momentum tensor 10 in an NC field theory is responsible for this failure. The above phenomena are nicely visualized in a collective coordinate framework. The above awkward situation clearly demands further study. Finally, as a future work we mention that inclusion of the Hopf term, (in the form of Chern- Simons term in CP(1) variables), in the NC theory would indeed be interesting. The Hopf term was introduced [8, 17] to impart anyonic behavior to the CP(1) solitons. The exact form of the NC version of the Chern Simons term is known [15] - it is a ”non-abelian” generalization of the Chern Simons term. In our formalism, application of the Seiberg-Witten map will reduce it to the ordinary Chern Simons term [15] but there will appear O(θ) correction terms since the gauge field of the Chern Simons term is actually a non-linear combination of the CP(1) variables. Another interesting problem is the reconstruction of the NC CP(1) model of [7] in our framework. Appendix: To get the φ-equation of motion, we consider variation of φ†. and exploit the relations, (Dµφ)†φ = φ†Dµφ = 0, (37) δ(Dµφ)† = δ(∂µφ† + (φ†∂µφ)φ†) = ∂µ(δφ†) + (δφ†∂µφ)φ† + (φ†∂µφ)δφ† δ(Dµφ) = δ(∂µφ − (φ†∂µφ)φ) = −(δφ†∂µφ)φ). (38) In the action the terms are products of the generic form ∫ (Dµφ)†(Dνφ)X(x). The variation of (Dµφ) will reproduce ∫ (Dµφ)†δ(Dνφ)X = − ∫ (Dµφ)†φ(δφ†∂µφ)X = 0, by using (37). Similarly, the variation of (Dµφ)† will yield ∫ δ(Dµφ)†(Dνφ)X = ∫ (∂µ(δφ†) + (δφ†∂µφ)φ† + (φ†∂µφ)δφ†)DνφX = − ∫ δφ†Dµ(DνφX) by partial integration and using (37). the above identities simplifies the computations consid- erably and leads to the equation (29). Acknowledgements: It is a pleasure to thank Professor Hyun Seok Yang for fruitful correspondence. Also I thank Professor Avinash Khare for a helpful discussion. Lastly I am indebted to Bishwajit Chakraborty for a free access to his notes on CP(1) model. References [1] N.Seiberg and E.Witten, JHEP 9909(1999)032. [2] For reviews see for example M.R.Douglas and N.A.Nekrasov, Rev.Mod.Phys. 73(2001)977; R.J.Szabo, Quantum Field Theory on Noncommutative Spaces, hep-th/0109162. [3] J.Polchinski, Phys.Rev.Lett. 75 (1995)4724. 11 [4] C.S.Chu and P.-M.Ho, Nucl.Phys. B550 (1999)151; V.Schomerus, JHEP 06 (1999)030; R.Banerjee, B.Chakraborty and S.Ghosh, Phys.Lett. B537 (2002)340. [5] S.Minwalla, M. Van Raamsdonk and N.Seiberg, Noncommutative perturbative dynamics, hep-th/9912072. [6] R.Gopakumar, S.Minwalla and A.Strominger, JHEP 0005 (2000)020. [7] B.-H.Lee, K.Lee and H.S.Yang, Phys.Lett. B498 (2001)277; K.Furuta et al., Low energy dy- namics of Noncommutative CP 1 Solitons in 2+1 Dimensions, hep-th/0203125; H.Otsu et al., New BPS Solitons in 2+1 Dimensional Noncommutative CP 1 Model, hep-th/0303090. [8] M.Bowick, D.Karabali and L.C.R.Vijewardhana, Nucl.Phys. B271 (1986)417. [9] J.M.Grimstrup et. al., hep-th/0210288. [10] A.Das and J.Frenkel, Phys.Rev.D 67 (2003)067701 (hep-th/0212122). [11] H.S.Snyder, Phys.Rev. 71 (1947)38. For some later works, see for example S.Doplicher, K.Fredenhagen and J.E.Roberts, Phys.Lett. 331B(1994)39; K.Morita Lorentz-invariant noncommutative QED, hep-th/0209234; S.Ghosh, Phys.Rev. D66 045031(2002). [12] J.Murugan and R.Adams, Comments on Noncommutative Sigma Models, hep-th/0211171. [13] S.Ghosh, Phys.Lett. B558 (2003)245; ibid 563 (2003)112. [14] H.S.Yang, private communications. [15] N.Grandi and G.A Silva, Phys.Lett. 507B(2001)345. [16] B.Jurco et al., hep-th/0104153. [17] See also, B.Chakraborty, S.Ghosh and R.P.Malik, Nucl.Phys. B600 (2001)351; B.Chakraborty, Mod.Phys.Lett. A17 (2002) 115. 12