Remarks on Gauge Fixing and BRST Quantization of Noncommutative Gauge Theories

Brazilian Journal of Physics, vol. 35, no. 3A, September, 2005 645

Ricardo Amorim, Henrique Boschi-Filho, and Nelson R. F. Braga Instituto de F´ısica, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, RJ 21941-972 – Brazil Received on 12 May, 2005 We consider the BRST gauge fixing procedure of the noncommutative Yang-Mills theory and of the gauged U(N) Proca model. An extended Seiberg-Witten map involving ghosts, antighosts and auxiliary fields for non- Abelian gauge theories is studied. We find that the extended map behaves differently for these models. For the Yang-Mills theory in the Lorentz gauge it was not possible to find a map that relates the gauge conditions in the noncommutative and ordinary theories. For the gauged Proca model we found a particular map relating the unitary gauge fixings in both formulations.

I. INTRODUCTION to note that in this BRST approach the closure relation (1.3) is naturally contained in the above condition for the ghost field. Noncommutative gauge fields Y can be described in terms This means that it is not necessary to construct a SW map for of ordinary gauge fields y by using the Seiberg-Witten (SW) the parameter. map[1] defined by Here we will investigate the extension of this map also to the gauge fixed actions. Observe that in ref.[7] the Hamil- tonian formalism was used while here we consider a La- δY = δ¯Y[y] (1.1) grangian approach. In particular we study the consistency of such maps with the gauge fixing process. Considering the δ δ¯ where and are the gauge transformations for noncommu- BRST quantization of a noncommutative theory we will find tative and ordinary theories respectively and Y[y] means that that some usual gauge choices for the noncommutative theo- we express the noncommutative fields Y in terms of ordinary ries are mapped in a non trivial way in the ordinary model. ones y. In this work we will consider gauge theories whose algebra closes as Recent results coming from string theory are motivating an increasing interest in noncommutative theories. The presence [δ1,δ2]Y = δ3 Y (1.2) of an antisymmetric tensor background along the D-brane [8] world volumes (space time region where the string endpoints without the use of equations of motion. In terms of the are located) is an important source for noncommutativity in mapped quantities this condition reads string theory[9, 10]. Actually the idea that spacetime may be noncommutative at very small length scales is not new[11]. [δ¯ ,δ¯ ]Y[y] = δ¯ Y[y] . (1.3) Originally this has been thought just as a mechanism for pro- 1 2 3 viding space with a natural cut off that would control ultra- This implies that the mapped noncommutative gauge parame- violet divergences[12], although these motivations have been ters must satisfy a composition law in such a way that they eclipsed by the success of the renormalization procedures. depend in general on the ordinary parameters and also on the Gauge theories can be defined in noncommutative spaces fields y. Usually this noncommutative parameter composition by considering actions that are invariant under gauge trans- law is the starting point for the construction of SW maps. In- formations defined in terms of the Moyal structure[1]. In this teresting properties of Yang-Mills noncommutative theories case the form of the gauge transformations imply that the al- where discussed in[2–4], where these points are considered in gebra of the generators must close under commutation and detail. Other noncommutative theories with different gauge anticommutation relations. That is why U(N) is usually cho- structures are also studied in [5, 6]. sen as the symmetry group for noncommutative extensions of Originally the Seiberg-Witten map has been introduced re- Yang-Mills theories in place of SU(N), although other sym- lating noncommutative and ordinary gauge fields and the cor- metry structures can also be considered [3][13][14]. Once one responding actions. When one considers the gauge fixing pro- has constructed a noncommutative gauge theory, it is possible cedure one enlarges the space of fields by introducing ghosts, to find the Seiberg-Witten map relating the noncommutative antighosts and auxiliary fields. In this case one can define an fields to ordinary ones[2]. The mapped Lagrangian is usu- enlarged BRST-SW map[7] ally written as a nonlocal infinite series of ordinary fields and their space-time derivatives but the noncommutative Noether identities are however kept by the Seiberg-Witten map. This sY = sY¯ [y] (1.4) assures that the mapped theory is still gauge invariant. where s and s¯ are the BRST differentials for the noncommu- In this work we will first consider (section II) the case of tative and ordinary theories respectively and Y and y here in- the noncommutative U(N) Yang-Mills theory and investigate clude ghosts, antighosts and auxiliary fields. It is interesting the BRST gauge fixing procedure in the Lorentz gauge. Then 646 Ricardo Amorim, Henrique Boschi-Filho, and Nelson R. F. Braga the construction of the SW map between the noncommutative trace normalization and ordinary Yang-Mills fields, including ghosts, antighosts 1 and auxiliary fields is discussed. We will see that imposing the tr(T AT B) = δAB (2.3) Lorentz gauge in the noncommutative theory does not imply 2 an equivalent condition in the ordinary theory. Conversely, and the (anti)commutation relations imposing the Lorentz gauge condition in the ordinary gauge fields would not correspond to the same condition in the non- [T A,T B] = i f ABCTC commutative theory. {T A,T B} = dABCTC (2.4) Another model that will present an interesting behavior is the gauged version of noncommutative non-Abelian Proca where f ABC and dABC are assumed to be completely antisym- field, discussed in sections III and IV. For this model we study metric and completely symmetric respectively. the BRST gauge fixing in the unitary gauge. The model is ob- The action (2.1) is invariant under the infinitesimal gauge tained by introducing an auxiliary field which promotes the transformations massive vector field to a gauge field. This auxiliary field can ¯ be seen as a pure gauge “compensating vector field” , defined δaµ = Dµα ≡ ∂µα − i[aµ,α] (2.5) in terms of U(N) group elements and having a null curvature [15, 16]. In this model we find that the general SW map re- which closes in the algebra lates in a non trivial way noncommutative and ordinary gauge fixing conditions. However it is possible to find a particular δ¯ δ¯ δ¯ SW map that relates the unitary gauges in noncommutative [ 1, 2]aµ = 3 aµ (2.6) and ordinary theories. Some of these points have been par- with parameter composition rule given by tially considered in [5]. Regardless of these considerations, the Fradkin-Vilkovisky theorem[17] assures that the physics described by any non α3 = i[α2,α1] (2.7) anomalous gauge theory is independent of the gauge fixing, without the necessity of having the gauge fixing functions The gauge structure displayed above leads to the definition mapped. This means that the quantization can be implemented of the BRST differential s¯ such that consistently in noncommutative and ordinary theories. Note that the SW map is defined for the gauge invariant action. This places a relation between the noncommutative and ordi- sc¯ = ic2 nary theories before any gauge fixing. Once a gauge fixing is sa¯ µ = Dµc chosen one does not necessarily expect that the theories would ≡ ∂ c − i[a ,c] still be related by the same map. In the Yang-Mills case with µ µ Lorentz gauge we could not relate the complete theories by a s¯c¯ = γ BRST-SW map after gauge fixing. However, for the gauged s¯γ = 0 (2.8) Proca theory we could find a map relating the unitary gauges in both noncommutative and ordinary theories. As s¯ is an odd derivative acting from the right, it is easy to ver- ify from the above definitions that it indeed is nilpotent. Natu- rally c and c¯ are grassmannian quantities with ghost numbers γ II. GAUGE FIXING THE NONCOMMUTATIVE U(N) respectively +1 and −1. c¯ and form a trivial pair necessary YANG-MILLS THEORY to implement the gauge fixing. The functional BRST quantization starts by defining the to- In order to establish notations and conventions that will tal action be useful later, let us start by considering the ordinary U(N) Yang-Mills action (denoting ordinary actions by the upper in- (0) (0) (0) (0) dex ) S = S0 + S1 (2.9)

(0) Z µ ¶ where S0 is given by (2.1) and (0) 4 1 µν S = tr d x − f ν f (2.1) 0 2 µ µ ¶ Z γ (0) 4 µ S = −2 tr s¯ d xc¯ − + ∂µa (2.10) where 1 β appropriated to ﬁx the (Gaussian) Lorentz condition, is BRST (0) fµν = ∂µaν − ∂νaµ − i[aµ,aν] (2.2) exact. This assures that S1 is BRST invariant, due to the nilpotency of s¯. Since s¯ fµν = i[c, fµν] according to (2.2) and (0) β is the curvature. We assume that the connection aµ takes val- (2.8), it follows that S0 is also BRST invariant. In (2.10), ues in the algebra of U(N), with generators T A satisfying the is a free parameter, as usual. Brazilian Journal of Physics, vol. 35, no. 3A, September, 2005 647

(0) In general S1 can be written as

(0) (0) (0) ? ε3 = i[ε2 , ε1] (2.17) S1 = Sgh + Sg f (2.11) with the ghost action given by The total action is given by

Z ( ) 0 4 S = S0 + S1 (2.18) Sgh = −2tr d xcMc¯ (2.12) where and the gauge fixing one by µ ¶ Z µ ¶ Z Γ γ 4 ¯ ∂ µ (0) = − 4 γ − + S1 = −2 trs d xC − + µA (2.19) Sg f 2tr d x β F (2.13) β where F = F (a) is a gauge fixing function and M = δF /δα. The BRST differential s is defined through µ The Lorentz gauge condition corresponds to F1 = ∂µa , M1 = µ ∂µD . The functional quantization is constructed by function- ally integrating the exponential of iS over all the fields, with sC = −iC ?C appropriate measure and the external source terms in order to sAµ = DµC ? generate the Green’s functions. ≡ ∂µC − i[Aµ , C] The noncommutative version of this theory comes from re- sC¯ = Γ placing y = {aµ, c, c¯, γ} by the noncommutative fields Y = sΓ = 0 (2.20) {Aµ, C, C¯, Γ} as well as the ordinary products of fields by ?- Moyal products, defined through Obviously both S0 and S1 are BRST invariant. The BRST Seiberg-Witten map is obtained from the condi- µ ¶ ← → tion (1.4). In this work we will consider only the expansion of i µν ? ≡ exp θ ∂ µ ∂ ν (2.14) noncommutative fields in terms of ordinary fields to first order 2 in the noncommutative parameter θ: where θµν is a constant and antisymmetric matrix. The Moyal (1) θ2 product is associative and cyclic under the integral sign when Y[y] = y +Y [y] + O( ), appropriate boundary conditions are adopted. We also assume where Y represents the noncommutative fields A ,C,C¯,Γ. that the group structure is deformed by this product. In this µ Then we find from (1.4) and (2.20) that way, the group elements are constructed from the exponentia- tion of elements of the algebra of U(N) by formally using ? in λA A 1 λA A λB B the series, that is, g = 1 + T + 2 T ? T + .... Gen- (1) (1) 1 αβ sC¯ + i{c,C } = θ [∂αc,∂βc] eral consequences of this deformation in field theories can be 4 θ0i seen in [1, 4]. In particular, the appearance of 6= 0 breaks (1) (1) ∂ (1) (1) the unitarity of the corresponding quantum theory. Further- sA¯ µ + i[Aµ ,c] = µC + i[C ,aµ] more in a Hamiltonian formalism this would imply higher or- 1 αβ − θ {∂α c,∂βa } der time derivatives which demand a non standard canonical 2 µ treatment. This last aspect does not show up here since we are s¯C¯(1) = Γ(1) using a Lagrangian formalism. s¯Γ(1) = 0 (2.21) The noncommutative action corresponding to (2.1) can be written as The corresponding solutions for the ghost and the gauge field are Z ³ ´ 4 1 µν S = tr d x − F ν ? F 0 2 µ (1) 1 µν © ª µν (2.15) C = θ ∂µ c,aν + λ1θ [∂µ c,aν] (2.22) 4 where now

(1) 1 αβ © ª αβ = − θ α,∂β + β + σθ αβ ? Aµ a aµ f µ Dµ f Fµν = ∂µAν − ∂νAµ − i[Aµ , Aν] (2.16) 4 λ 1 θαβ As expected, the noncommutative gauge transformations + Dµ[aα,aβ], (2.23) ? 2 δAµ = ∂µε − i[Aµ , ε] close in an algebra like (1.2) with composition rule for the parameters given by where σ and λ1 are arbitrary constants. 648 Ricardo Amorim, Henrique Boschi-Filho, and Nelson R. F. Braga

It is important to remark that when we extend the space of not compatible with the solution (2.23), and its higher order fields in order to implement BRST quantization we could in extensions, for the SW mapping. (1) (1) principle find solutions for C and Aµ depending on c¯ and Alternatively we could choose in the noncommutative the- γ. However there are two additional conditions to be satisfied, ory the non trivial non linear gauge condition (to first order in besides eqs. (2.21): the ghost number and the dimension of θ ) all first order corrections must be equal to those of the corresponding zero order field. It can be checked that there are no µ 1 αβ µ © ª F = ∂ A + θ ∂ Aα,∂βA + Fβ other possible contributions to the solutions (2.22) and (2.23) 2 µ 4 µ µ involving c¯ and γ and satisfying these criteria. λ αβ µ 1 αβ µ For the trivial pair we find − σθ ∂ DµFαβ − θ ∂ Dµ[Aα,Aβ] 2 = 0 . (2.30) ¯(1) θαβ C = Hαβ If we assume the map (2.23) to hold, this gauge fixing condi- (1) αβ Γ = θ sH¯ αβ (2.24) tion would correspond just to the ordinary Lorentz condition (2.28) to first order in the noncommutative parameter θ. We where Hαβ is a function of the fields aµ,c,c¯ and γ with ghost see that if we impose the Lorentz gauge in one of the theo- number = −1 (taking the convention that the ghost numbers ries, the SW map would lead to a somehow complicated and of c and c¯ are +1 and −1 respectively). Note that the sum of potentially inconsistent gauge in the other. the mass dimensions of c and c¯ is 2. Then the mass dimension Instead of just considering the map of the gauge fixing func- of Hαβ will be the dimension of c¯ plus 2. Considering these tions, a more general approach is to consider the behaviour of points we find that the general form for this quantity is the action S1 eq. (2.19) under the SW map. This action can be written, to first order in the noncommutative parameter, as

αβ = ω + ω ∂ + ω ∂ H 1 ca¯ [α aβ] 2 c¯ [α aβ] 3 [α ca¯ β] (2.25) (0) (1) S1 = S1 + S1 , (2.31) where ω , ω and ω are arbitrary parameters. From eqs. 1 2 3 (0) (2.8), (2.24) and (2.25) we see that where S1 is given by (2.10) and

³ Z ³ γ Γ(1) θαβ ω γ ω γ∂ ω ∂ γ S(1) = −2 tr s¯ d4x C¯(1)(− + ∂ aµ) = 1 aα aβ + 2 α aβ + 3 α aβ 1 β µ

+ ω1 c¯(Dα ca + aα D c) + ω2 c¯∂α D c Γ(1) ´ β β β (1)µ ´ + c¯ (− + ∂µA ) . (2.32) ω ∂ β + 3 α cD¯ β c . (2.26) µ Note that the condition F1 = ∂ Aµ = 0 in the noncommu- µ The usual Seiberg-Witten map is defined for the S0 (gauge tative theory would be effectively mapped into F3 = ∂ aµ = 0 invariant) part of the action. When we gauge fix by includ- (1) if S could vanish. In order to see if this is possible we intro- ing S and S we find a total action that is no more gauge 1 gh g f duce (2.23) and (2.26) in (2.32) and examine the terms with invariant but rather BRST invariant. This poses a non trivial (1) problem of whether it would still be possible to relate non- the same field content. The part of S1 independent of the commutative and ordinary gauge fixed theories by a SW map. ghost sector is Let us consider the Lorentz gauge condition appearing in Z (2.19): h³ 2γ´ (1) = − θαβ 4 ∂ µ − S1noghost 2 tr d x µa β ³ ´ µ ω γ ω γ∂ ω ∂ γ F1 = ∂ Aµ = 0 (2.27) × 1 aα aβ + 2 α aβ + 3 α aβ ³ n o If we use the Seiberg-Witten map found above we see that this γ∂µ 1 ∂ σ + − aα , β aµ + fβ + Dµ fαβ condition would correspond to 4 µ λ í + 1 D [a ,a ] . (2.33) 2 µ α β µ ∂ aµ = 0 (2.28) The terms linear in the connection aµ in the integrand are µ (1) ∂ Aµ [aµ] = 0 . (2.29) 2γ³ ´ That means, besides the ordinary Lorentz condition (2.28), we − ω γ∂ a + ω ∂ γa + 2σγ¤∂ a (2.34) β 2 [α β] 3 [α β] [α β] find the additional non linear conditions on aµ from (2.23) and (2.29). If we were adopting in our expansions terms up to which will vanish modulo total differentials only if σ = 0 and µ (n) order N in θ, we would find a set of conditions ∂ Aµ [aµ] = ω3 = 2ω2. Using these results the quadratic part in aµ of the µ 0, n = 0,1,..,N. So it seems that the condition ∂ Aµ = 0 is integrand can be written as Brazilian Journal of Physics, vol. 35, no. 3A, September, 2005 649

This action is invariant under the local transformations δaµ = ∂ α and δλ = α. The equations of motion for a and λ are 2 ³ µ µ − γ2ω a a − γ ω ( ∂ a ∂ aµ + 2 a ∂ ∂ aµ ) β 1 [α β] 2 [α β] µ [β α] µ ν ν ν ´ ∂ f µ + m2(a − ∂ λ) = 0 1∂µ ∂ ∂ λ ∂ µ + (2{a α , β aµ} + { µa α ,aβ } − 4 1 µ(a α aβ )) µ µ 4 [ ] [ ] [ ] ∂µ(a − ∂ λ) = 0 (3.6) (2.35) Note that applying ∂ν to the first equation, one reobtains modulo a total differential. The quadratic term in γ can be the second one. In this case (3.3) does not come from the equations of motion. However as now the model is gauge set to zero choosing ω1 = 0. However, there is no choice for invariant, we must impose a gauge fixing function. We may λ1 and ω2 which makes the linear terms in γ in the expres- sion above vanish or be written as a total derivative. Since the choose for instance one of the gauge fixing functions F1 = ∂ µ λ λ terms from the ghost sector can not cancel the above ones, we µa , F2 = ¤ or F3 = in order to recover the original Proca theory. conclude that S(1) can not vanish. So it is not possible to relate 1 We now consider the non-Abelian generalization of this the Lorentz gauge conditions in the ordinary and noncommu- model. Now a takes values in the algebra of U(N), exactly as tative theories by the SW map. Anyway, quantization can be µ in the Yang-Mills case of the previous section. consistently implemented in both theories as pointed out in the In place of (3.2) one finds end of section I. We will see in section IV that for the gauge invariant Proca model it is possible to find a particular BRST- µν 2 ν SW map relating the gauge fixing unitary conditions for both Dµ f + m a = 0 (3.7) sectors. Applying Dν to this equation and using the property

III. THE NON-ABELIAN PROCA MODEL [Dµ,Dν]y = i[ fµν , y] (3.8) Before considering the gauge invariant noncommutative where y is any arbitrary function with values in the algebra, U(N) Proca model, it will be useful to brieﬂy present some we ﬁnd as in the Abelian case that basic properties of the ordinary Abelian Proca theory with action µ µ Dµa = ∂µa = 0 . (3.9) Z µ ¶ However, the equations of motion present non linear terms. 1 1 (0) 4 µν 2 µ Using the ”Lorentz identity” (3.9) in eq. (3.7) we obtain the S0 = d x − fµν f + m aµa . (3.1) 4 2 equations of motion for aµ

Here aµ represents the massive Abelian vector field. Variation 2 ρ µ ρ µρ of (3.1) with respect to aµ gives the equation of motion (¤ + m )a − i[aµ,∂ a + f ] = 0 (3.10) which is no longer a Klein Gordon equation. µν 2 ν ∂µ f + m a = 0 (3.2) It is worth to mention that contrarily to the Abelian case, the non-Abelian Proca model is not renormalizable, although which, by symmetry, implies that the divergencies at one-loop level cancel in an unexpected way [18],[19]. Nonrenormalizability is, in any way, an almost general property of noncommutative field theories[4]. µ ∂µa = 0 (3.3) The next step would be to consider the gauged version of this non-Abelian model. This will be done in the noncommu- Substituting (3.3) into (3.2) we find that the vector field satis- tative context in the next section. We will also discuss there fies a massive Klein-Gordon equation, as expected: the gauge fixing procedure and the BRST formalism.

2 (¤ + m )aµ = 0 (3.4) IV. THE GAUGE INVARIANT NONCOMMUTATIVE U(N) PROCA MODEL This model can be described in a gauge invariant way by introducing a compensating field λ. This kind of mechanism is useful, for instance, when calculating anomalies[15, 16]. In The gauge invariant version of the U(N) noncommutative this case, the action (3.1) is replaced by Proca model can be written as[5] Z ³ Z µ ¶ 4 1 µν 1 1 S0 = tr d x − Fµν ? F (0) 4 µν 2 ∂ λ µ ∂µλ 2 S0 = d x − fµν f + m (aµ − µ )(a − ) . ´ 4 2 2 µ µ (3.5) + m (Aµ − Bµ) ? (A − B ) (4.1) 650 Ricardo Amorim, Henrique Boschi-Filho, and Nelson R. F. Braga where the curvature is again given by eq. (2.16) and As expected, the noncommutative gauge transformations listed above close in an algebra as (1.2), for the fields Aµ, G , Bµ or Fµν. The composition rule for the parameters is as −1 Bµ ≡ −i∂µG ? G (4.2) (2.17). The associated BRST algebra obtained again by introduc- In the above expressions G is an element of the noncom- ing the trivial pair C¯ and Γ, corresponds to the Yang-Mills mutative U(N) group and Bµ is a ”pure gauge” vector field algebra of eq. (2.20) plus the transformation of the compen- in the sense that its curvature, analogous to (2.16), vanishes sating field identically. Note that Bµ is the noncommutative U(N) version of ∂µλ discussed in the previous section. We assume that the algebra generators satisfy, as in the Yang-Mills case, the trace sG = iC ? G , (4.11) normalization and (anti)commutation relations (2.3) and (2.4). By varying action (4.1) with respect to Aµ and G, we get the that corresponds to equations of motion

? µν 2 ν ν sBµ = D¯ µC = ∂µC − i[Bµ , C] . (4.12) DµF + m (A − B ) = 0 (4.3) ¯ µ µ Dµ(A − B ) = 0 (4.4) A gauge fixed action is constructed from (4.1) by adding Sgh and Sg f from eqs. (2.12) and (2.13) as in the Yang-Mills where we have defined the two covariant derivatives case. For the Lorentz gauge we choose as in the Yang-Mills case ∂ µ ∂ µ D X = ∂ X − i[A ?, X] F1 = µA , M1 = µD and we get formally the same ghost µ µ µ and gauge fixing actions. The complete action is BRST in- ¯ ∂ ? DµX = µX − i[Bµ , X] (4.5) variant, as expected. For the unitary gauge corresponding to the choice F = G− for any quantity X with values in the algebra. By taking the 4 1, we find the ghost and gauge fixing actions covariant divergence of equation (4.3) and using the noncommutative Bianchi identity

S1 = Sgh + Sg f Z ? [Dµ,Dν]X = i[Fµν , X] (4.6) = −2tr s d4xC¯ (G − 1) (4.13) one finds which is obviously BRST invariant. Since sS0 = 0, as can be verified, it follows the invariance of the complete action. µ µ Dµ(A − B ) = 0 (4.7) Let us now build up the BRST Seiberg-Witten map for this model. We start again by imposing [7] that sY = sY¯ [y] submit- which is equivalent to equation (4.4), as can be verified. Ac- ted to the condition Y[y]|θ=0 = y that lead us again to equations tually, we can rewrite (4.4) or (4.7) in the convenient form (2.21) and also to

∂ µ µ µA = DµB . (4.8) (1) (1) 1 µν (1) sG¯ − icG = − θ ∂ c∂ν g + iC g 2 µ So we see that if we choose, among the possible gauging fix- ¯ (1) (1) (1) (1) µ δBµ + i[Bµ ,c] = ∂µC + i[C ,bµ] ing functions, F1 = ∂µA or F4 = G − 1, the compensating field is effectively eliminated and the Lorentz condition is im- 1 αβ − θ {∂α c,∂βb } (4.14) plemented. 2 µ Action (4.1) is invariant under the infinitesimal gauge trans- The solutions of these equations are again obtained by formations searching for all the possible contributions with the appropriate dimensions and Grassmaniann characters. However now we have the extra compensating field bµ. So the solutions for δA = D ε µ µ A(1) and C(1) are no longer the Yang-Mills solutions (2.22) δ ε µ G = i ? G (4.9) and (2.23) but rather which also implies that

(1) 1 µν © ª µν C = (1 − ρ)θ ∂ c,aν + λ θ [∂ c,aν] 4 µ 1 µ δBµ = D¯ µε 1 µν © ª µν δ ε ? + ρθ ∂ c,bν + λ θ [∂ c,bν] (4.15) Fµν = i[ , Fµν] (4.10) 4 µ 2 µ Brazilian Journal of Physics, vol. 35, no. 3A, September, 2005 651

we find that the conditions g = 1 and G = 1 are equivalent to first order in θ, as can be verified from (4.17). So that for this (1) 1 − ρ αβ © ª αβ Aµ = − θ aα,∂βaµ + fβµ + σθ Dµ fαβ particular solution of the map, the unitary gauge in the non- 4 commutative theory is mapped in the ordinary unitary gauge λ1 αβ ρ αβ © ª + θ D [aα,aβ] + θ bα,D bβ − 2∂βa in the mapped theory. 2 µ 4 µ µ ³ ´ Additionally, if we choose ρ = 1 implying λ1 = 0 and also αβ λ θ α (1) + 2 Dµ b bβ (4.16) σ = 0 we find that in the unitary gauge Aµ = 0. So that Aµ is mapped just in aµ. In this case we find that the non gauge In these equations λ1, λ2 , ρ and σ are arbitrary constants. invariant noncommutative U(N) Proca theory is recovered in These solutions are a generalization of those in ref. [5]. the unitary gauge and at the considered order in θ. For the compensating field we find

µ ¶ (1) 1 αβ i G = − (1 − ρ)θ aα ∂βg − aβg V. CONCLUSION 2 2 λ θαβ γθαβ + i 1 aαaβg + fαβg We have investigated the extension of the Seiberg-Witten ρ αβ 2 map to ghosts, antighosts and auxiliary fields in the BRST + i(λ2 − )θ bαbβg + O(θ ) (4.17) 4 gauge fixing procedure. Two non-Abelian gauge models that present different behaviors under the BRST-SW map of the with arbitrary γ. gauge fixing conditions have been considered. For the Yang- Now we may consider the compatibility of the BRST-SW Mills theory we found that it is not possible to map the Lorentz map and the unitary gauge fixing, in the same spirit of what gauge condition in the noncommutative and ordinary theories. was discussed in the last section. First we observe that the On the other hand, for the gauged U(N) Proca model we were condition F = G−1 is mapped in g−1 plus complicated first 4 able to find a SW map that relates the unitary gauge fixing order corrections in θ. However, if we choose the parameters in the noncommutative and ordinary theories. It would be in- of the map as teresting to investigate how the BRST-SW map act in other gauge fixing conditions and models. 1 λ = (ρ − 1) 1 4 Acknowledgment: This work is supported in part by γ = 0 (4.18) CAPES and CNPq (Brazilian research agencies).

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