b

tial

Physics Letters B 593 (2004) 203–208

www.elsevier.com/locate/physlet

The non-commutativeU(N) Kalb–Ramond theory

Ricardo Amorima, Cristine N. Ferreirab, Cresus F.L. Godinhoc

a Instituto de Física, Universidade Federal do Rio de Janeiro PO Box 68528, 21945-910, Rio de Janeiro, RJ, Brazilb Grupo de Física Teórica José Leite Lopes, Petrópolis, RJ, Brazil

c Centro Brasileiro de Pesquisas Físicas (CBPF-CCP), Rua Dr. Xavier Sigaud 150, Urca 22.290-180, Rio de Janeiro, RJ, Brazil

Received 8 March 2004; accepted 15 April 2004

Available online 31 May 2004

Editor: Y. Yanagida

Abstract

We present the non-commutative extension of theU(N) Cremmer–Scherk–Kalb–Ramond theory, displaying its differenform and gauge structures. The Seiberg–Witten map of the model is also constructed up to 0(θ2). 2004 Elsevier B.V. All rights reserved.

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1. Introduction

The first ideas about space–time non-commutatiwere formulated by Heisenberg in the thirties[1], al-though the first published work on the subject apeared in 1947[2], introducing a possible frameworfor avoiding the characteristic singularities of quatum field theories. Although the original motivatiohave been eclipsed by the renormalization progrrecently the interest on non-commutative theories hagrown up in large scale, mainly associating the spatime non-commutativity with results coming frostring theory. In this approach non-commutativity cbe found in the study of perturbative strings in tpresence of branes in a constant background magfield [3–5]. For a review on the subject, where seeral interesting features of non-commutative field t

E-mail addresses: amorim@if.ufrj.br (R. Amorim),crgodinho@globo.com (C.N. Ferreira), crisnfer@if.ufrj.br,crisnfer@cbpf.br (C.F.L. Godinho).

0370-2693/$ – see front matter 2004 Elsevier B.V. All rights reserveddoi:10.1016/j.physletb.2004.04.053

ories can be commented, see, for instance,[6] and ref-erences therein. One of the important ingredientsnon-commutative gauge field theories is the SeibeWitten map connecting field variables which tranform under a non-commutative gauge structure wordinary field variables transforming under an ornary gauge structure[7–10]. This map seems to be esential for the construction of a phenomenologicalable non-commutative description of Nature[11,12].

Interesting models of gauge theories that havehad their non-commutative extentions very exploare those constructed with the aid of antisymmetensor fields. The antisymmetric Kalb–Ramond tsor field has been first introduced within a stritheory [13–15], but the so-called Cremmer–ScherKalb–Ramond (CSKR) model appears in a greatriety of scenarios, includingsupersymmetric theorie[16–18], cosmology[19] and cosmic strings[20].Other interesting points are related to the rank ofCSKR model gauge structure and its consequenunder quantization, including the possibility of mas

.

204 R. Amorim et al. / Physics Letters B 593 (2004) 203–208

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for gauge theories without spoiling gauge invarian[21–23].

In the present Letter, we propose a non-commtive generalization of the CSKR theory. We shothat its covariant description, with the aid of diffeential forms, can be extended in order to incorporMoyal products, characteristic of non-commutatfield theories. It is also possible to show that thereits an underlying commutative gauge invariant theand a suitable Seiberg–Witten map linking the ncommutative model and its ordinary counterpart.

The outline of this Letter is as follows: inSec-tion 2, we start by presenting the ordinaryU(N)

CSKR model in terms of differential forms. After thwe show that it is possible to deform the form structin order to incorporate Moyal products. This esstially permits the construction of the non-commutatextention of the model. InSection 3, the appropri-ate Seiberg–Witten map is derived. It takes intocount not only the usual Yang–Mills sector but alsogauge sector which arises when one considers thvariance associated with the 1-form gauge parameWe reserveSection 4for some concluding remarks.

2. The non-commutative U(N) CSKR model

To fix notations and conventions, let us start wa brief review of the non-Abelian commutative KalRamond theory. We will follow a notation close to thone found in Ref.[23]. After that we will study thecorresponding non-commutative theory.

Leta = aaµT a dxµ represent a one-form connectio

taking values in theU(N) algebra in the fundamentarepresentation. We assume that[T a,T b

] = if abcT c,T a,T b

= dabcT c,

(2.1)tr(T aT b

) = 1

2δab.

On any u(N) valued p-form α it is possible todefine the exterior covariant derivative

(2.2)Dα = dα − ia ∧ α + i(−1)pα ∧ a.

It follows the first Bianchi identity

(2.3)DDα = i[α,f ],

.

where we have defined the curvature two-form

(2.4)f = da − ia ∧ a

and the wedge product is implicit in the commutatFor completeness, we also note the second Biaidentity

(2.5)Df = 0.

The gauge sector of theU(N) Yang–Mills theory,with this notation, is described by the action

(2.6)S = Tr∫

f ∧ f,

where the symbol here denotes the space–time duAction (2.6) is invariant under the gauge transformtion

(2.7)δa = Dα

since under(2.7)f transforms as

(2.8)δf = i[α,f ]and the invariance of(2.6)is achieved due to the cycliproperty of the trace operation. In the above expsions we have usedδ to represent ordinary gauge vaations. We will letδ represent the corresponding nocommutative gauge variations.

To describe the CSKR model, besides the cnection a, we need a two-form gauge fieldb =12bµν dxµ ∧ dxν and a compensating one-form fieω = ωµ dxµ, both taking values inu(N). The 3-formfield strength associated withb is defined as

(2.9)g = Db.

Now the CSKR action

(2.10)S = Tr∫ [

f ∧ f − g ∧ g + 2mf ∧ b]

shows itself to be invariant under the set of gautransformations

δa = Dα,

δb = Dξ + i[α,b],(2.11)δω = i[α,ω] − ξ,

where we have used in(2.10)the collective two-form

(2.12)b = b + Dω

R. Amorim et al. / Physics Letters B 593 (2004) 203–208 205

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and the modified field strength

(2.13)g = Db.

We observe from(2.11)and the definitions abovthatb andω transform not only as Yang–Mills tensobut also present an additional transformation relato the one-form gauge parameterξ = ξµ dxµ. Thequantitiesb andg, however, transform only as YangMills tensors, in the same way asf in (2.8). Thisfact permits the gauge invariance of(2.10). Actually,in the Abelian case, the compensating one-formω isnot necessary, and the corresponding theory is gainvariant, although reducible[21–23].

All of the transformations defined above close inalgebra, defined by the parameters composition ru

α3 = i[α2, α1],(2.14)ξ3 = i[ξ2, α1] − i[ξ1, α2],

when the commutation of two successive gauge traformations is applied to any one of the fields appearin the theory, here generically represented byy:

(2.15)[δ1, δ2]y = δ3y.

The field–antifield quantization of the model dscribed above was studied in Ref.[23].

Let us now pass to consider the non-commutaversion of this theory. As already commented, thesic procedure to construct the non-commutative exsion of some theory consists in deforming ordinaproducts to non-commutative Moyal products. For atwo fields Φ1(x) and Φ2(x), we define their Moyaproduct as

(2.16)

Φ1(x) Φ2(x) = exp

(i

2θµν∂x

µ∂yν

)Φ1(x)Φ2(y)|x=y,

where θµν is assumed to be a real, constant aantisymmetric quantity which characterizes the noncommutativity of the theory. These products aresociative and cyclic under the integral sign, if adquate boundary conditions are assumed. As it is wknown [7,9], the U(N) group elements are also dformed by such a product in the sense that their cstruction by exponentiation involves Moyal producAlso the group multiplication is defined as a Moyproduct. In this way the symmetry structure of tnon-commutativeU(N) theory is not the same as th

corresponding commutative one and the groupsure property is only achieved if the algebra generaclose not only under commutations but also underticommutation. This essentially constitutes the reafor choosingU(N) in place ofSU(N) as a symmetrygroup of this non-commutative gauge theory, althouother possibilities can be considered[11]. Similar de-formations can also be implemented in the differtial forms structure. In this way, the exterior produis modified in order to accommodate the Moyal str

ture with the formal replacement∧ → ∧. In a coordi-nate basis, this modification is trivial and consistsintroducing Moyal star products in place of the ornary ones between the forms components, keepingwedge product between the form basis. Here werestrict ourselves to this situation. Similar proceducan also be implemented in the definition of theterior covariant derivative. The non-commutative vsion of the exterior covariant derivative of ap-form Λ

is given by

(2.17)DΛ = dΛ − iA∧ Λ + i(−1)pΛ

∧ A.

Definition (2.4) is in the same way trivially deformeto

F = dA − iA∧ A

= 1

2

((∂µAa

ν − ∂νAaµ

)T a

− i[Aa

µT a , AbνT

b])

dxµ ∧ dxν

(2.18)= 1

2Fµν dxµ ∧ dxν,

whereA = AaµT a dxµ now represents the non-com

mutative 1-form connection. The above expressshows the rule played by the non-commutative we

product∧. As can be observed,F involves both

structure functions defined in(2.1). Actually, Faµν =

∂µAaν − ∂νA

aµ + 1

2f abcAbµ

, Acν − i

2dabc[Abµ

, Acν].

Other expressions follow the same rules. The Bianidentities are now written as

DDΛ = i[Λ , F ],(2.19)DF = 0.

To construct the non-commutative version of tmodel described above, in place of the ordinary qutities b andω we define the non-commutative formB = 1

2Bµν dxµ ∧dxν andΩ = Ωµ dxµ. Also in place

206 R. Amorim et al. / Physics Letters B 593 (2004) 203–208

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he

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s

of g, b and g defined in(2.9), (2.12)and(2.13), weintroduce the corresponding non-commutative form

G = DB,

B = B + DΩ,

(2.20)G = DB.

These quantities present the following set of gatransformations:

δA = Dε,

δB = DΞ + i[ε , B],δG = i[Ξ , F ] + i[ε , G],δΩ = i[ε , Ω] − Ξ,

δF = i[ε , F ],δB = i[ε , B],

(2.21)δG = i[ε , G]and, as can be verified, the non-commutative extenof action(2.10),

(2.22)S = Tr∫ [

F ∧ F − G ∧ G + 2mF ∧ B]

is gauge invariant under(2.21)due to cyclic propertiesof the Moyal product under the integral sign. Dto the boundary conditions we are adopting, it

irrelevant to use in the above expression∧ or∧.

The transformations(2.21)close in an algebra

(2.23)[δ1, δ2]Y = δ3Y

with the composition rule given by

ε3 = i[ε2, ε1],

(2.24)Ξ3 = i[Ξ2, ε1] − i[Ξ1

, ε2]in place of(2.14)and(2.15).

As one can observe, the non-commutative extsion of the CSKR theory has been constructed wout difficulties. Of course its quantum version woushow all those characteristic points associated wnon-commutative field theories and their non-pladiagramatic expansions[6]. We will not consider thesepoints in this work. In what follows let us study thSeiberg–Witten map of the model, which presesome interesting features.

3. The Seiberg–Witten map

Accordingly to what we have been discussing inlast section, let us represent the non-commutativevariables by capital letters, here generically denobyY and the corresponding ordinary ones by smallters generically written asy. Also accordingly to thenotations of the last section, their gauge transfortions are respectively represented byδY and δy. Thebasic idea in the construction of the Seiberg–Witmap is to obtain the gauge transformationsδY of thenon-commutative variables departing from the gastructure of the ordinary theory, with variables traforming accordingly toδy. This is equivalent to solvethe equation

(3.1)δY = δY [y]which is done by using expansions in powers ofnon-commutative parameterθ and assuming that up t0(θ) terms,Y → y. This map is non-trivial when thnon-commutative parametersε andΞ are consideredas functions of the commutative parametersα and ξ

as well as of the ordinary fieldsy. In this case, thefundamental expressions implicit in(2.23)reduce to

[δ1, δ2]A[y] = Dε3[y],[δ1, δ2]Ω[y] = i

[ε3[y] , Ω

] − Ξ3[y],(3.2)[δ1, δ2]B[y] = i

[ε3[y] , B

] + DΞ3[y],where now, in place of(2.24), we get

ε3[y] = δ1ε2[y] − δ2ε1[y] + i[ε2[y] , ε1[y]],

(3.3)Ξ3[y] = δ1Ξ2 − δ2Ξ1 + i[ε2, Ξ1] − i[ε1

, Ξ2]due to the dependence of the parameters in the fiIndices 1, 2 and 3 represent the dependence ofε andΞ in αi and ξi , i = 1,2,3. For instance,ε3[y] ≡ε[α3, ξ3, y]. Related quantities such asF , B or G

also follow similar rules related to the closure of talgebra.

The first of Eq. (3.2) is not new in the literature[7,8,11], what does not occur with the second onefar as we know. They will be important for the resuthat we will derive. As in the pure Yang–Mills casthe gauge transformation 0-form parameterε can beexpanded to first order inθµν asε[y] = α + ε(1)[y]. Inthe same way, the 1-form parameterΞ is expanded aΞ [y] = ξ +Ξ(1)[y], to first order inθ . From(3.3)and

R. Amorim et al. / Physics Letters B 593 (2004) 203–208 207

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the above expansions we deduce that

δ1ε(1)2 − δ2ε

(1)1 − i

[α1, ε

(1)2

] + i[α2, ε

(1)1

] − ε(1)3

(3.4)= −1

2θµν∂µα1, ∂να2

and also that

δ1Ξ(1)2 − δ2Ξ

(1)1 + i

[α2,Ξ

(1)1

] − i[α1,Ξ

(1)2

] − Ξ(1)3

= −i[ε(1)2 , ξ1

] + i[ε(1)1 , ξ2

](3.5)+ 1

2θαβ

(∂αα2, ∂βξ1 − ∂αα1, ∂βξ2).

The general solution of(3.4)given by[8]

(3.6)ε(1) = θαβ

4∂αα, aβ + λθαβ [∂αα, aβ ],

where the term inλ is the solution of the homogeneopart of(3.4). Now it is possible to show that

Ξ(1) = 1

2θαβ

aα, ∂βξ + i

2[aβ, ξ ]

(3.7)+ iλθαβ[aα[aβ, ξ ]]

solves(3.5) when one uses(3.6) for ε(1). Once weexpand the fields to first order inθ , this is: A = a +A(1), Ω = ω + Ω(1) andB = b + B(1), we can rewritethe corresponding gauge transformations appearin(2.21) as

δA(1) − i[α,A(1)

]= −1

2θαβ∂α α, ∂βa + Dε(1),

δΩ(1) − i[α,Ω(1)

]= −1

2θαβ∂αα, ∂βω + i

[ε(1),ω

] − Ξ(1),

δB(1) − i[α,B(1)

]= −1

2θαβ

(∂αα, ∂βb − [∂αa, ∂βξ ])(3.8)+ i

[ε(1), b

] − iξ,A(1)

+ DΞ(1),

whereDε(1) and DΞ(1) represent now ordinary covariant derivatives as defined in(2.2). After inserting(3.6) into the first ofEq. (3.8) it is possible to findthe general solution of the Seiberg–Witten map forconnectionA as[8]

A = a − 1

4θαβaα,2∂βa − Daβ

(3.9)+ θαβD

(σfαβ + λ

2[aα, aβ]

)+ O

(θ2),

whereλ appears in(3.6)andσ is a second parameteassociated with the homogeneous part of the firsEq. (3.8). Observe that the indices associated wthe non-commutativity appear explicitly, what can beexpected since the Lorentz invariance is brokenthe Moyal structure. The covariance associated wthe form structure is however kept. This means tit is not necessary to write the forms componentsequations(3.4)–(3.9)since they do not mix with thenon-commutative structure. Theses features will aappear in the following.

It is possible to show from(3.6), (3.7) and thesecond one ofEq. (3.8)that

Ω(1)[y] = −1

4θαβ

aα, (∂β + Dβ)ω

(3.10)+ i

2λθαβ

[aα[aβ,ω]]

is the desired solution for the compensating 1-frfield. To solve the third equation in(3.8) for B(1), weneed to consider the already derived expressionsε(1), Ξ(1) and A(1) given above. As can be inferrefrom them, it is not an easy task to achieve a compsolution forB(1) following this route.

However, if we consider the equation defining tgauge variation forB in (3.8), we find a much simplemapping equation given by

(3.11)

δB(1) − i[α, B(1)

] = −1

2θαβ∂α α, ∂β b + i

[ε(1), b

]

whose general solution, when one keeps the fcovariance in the sense discussed above, is given

B(1) = −1

4θαβ

aα, (∂β + Dβ)b

(3.12)+ θαβ

(ρ[b, fαβ ] − i

λ

2

[b, [aα, aβ]]

),

where ρ is a new parameter associated withhomogeneous part of(3.11). Now remembering thaB = B + DΩ from (2.21), one can verify that

B(1) = B(1) − DΩ(1) + iA(1),ω

(3.13)− 1

2θαβ [∂αa, ∂βω].

Inserting(3.12)and the expressions forΩ(1) andA(1)

given in(3.10)and(3.9) in the above expression, itsimple to obtain the complete expression forB(1).

208 R. Amorim et al. / Physics Letters B 593 (2004) 203–208

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;inldsonNY,

For completeness, we note also that[8]

F (1) = 1

2θαβ

(Daα − ∂αa,Daβ − ∂βa

− 1

2

aα, (∂β + Dβ)f

+ iσ [fαβ,f ]

(3.14)− i

2λ[f, [aα, aβ ]]

)

and

(3.15)G(1) = DB(1) − i[A(1), b

] + 1

2θαβ∂αb, ∂βa.

Now the action(2.22)is mapped to

S = 2 Tr∫

d4x

(1

2f ∧ f − 1

2g ∧ g + mf ∧ b

+ f ∧ F (1) − g ∧ G(1)

(3.16)+ mf ∧ B(1) + mF(1) ∧ b

)

up to O(θ2) terms, whereF (1), B(1) and G(1) aregiven above. It can be verified that action(3.16)is in-variant under the set of ordinary gauge transformat(2.11), since the Noether identities are kept underSeiberg–Witten map.

4. Conclusions

We have studied in this work a non-commutatformulation of the U(N) Cremmer–Scherk–KalbRamond theory. The gauge structure has been conered in detail, as well as a non-commutative differtial form structure appropriated to describe the modIt was also constructed the Seiberg–Witten maptween this non-commutative gauge theory and its onary counterpart, in first order in the non-commutatparameterθ . In this last subject, not only the vectgauge sector has been considered, but also thetwo gauge algebra associated with the antisymmric tensor gauge fields. We observe that it hasbeen explored here some characteristic featuresally associated with the ordinary CSKR model suas the effective topological mass generation forvectorial sector, since they constitute direct geneizations of the ordinary case. Specific features assocated with non-commutative quantum field theories[6]such as renormalizability, broken of unitarity, presen

-

-

of anomalies or mixing of ultraviolet and infrared dvergencies have been left for consideration in a futwork.

Acknowledgements

This work is supported in part by CAPES aCNPq (Brazilian research agencies).

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