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Physics Letters B 560 (2003) 239–244

www.elsevier.com/locate/np

Noncommutative Maxwell–Chern–Simons theory, duality ana new noncommutative Chern–Simons theory ind = 3

Ömer F. Dayia,b,c

a The Abdus Salam ICTP, Strada Costiera 11, 34014 Trieste, Italyb Physics Department, Faculty of Science and Letters, Istanbul Technical University, 80626 Maslak–Istanbul, Turkey

c Feza Gürsey Institute, P.O. Box 6, 81220 Çengelköy–Istanbul, Turkey

Received 11 February 2003; received in revised form 12 March 2003; accepted 21 March 2003

Editor: M. Cvetic

Abstract

Noncommutative Maxwell–Chern–Simons theory in 3 dimensions is defined in terms of star product and noncomfields. The Seiberg–Witten map is employed to write it in terms of ordinary fields. A parent action is introduced adual action is derived. For spatial noncommutativity it is studied up to second order in the noncommutativity paramθ .A new noncommutative Chern–Simons action is defined in terms of ordinary fields, inspired by the dual action. Mortransformation between noncommuting and ordinary fields is proposed. 2003 Published by Elsevier Science B.V.

1. Introduction

An equivalence of “ordinary” (commutative) and noncommutative gauge fields leads to a transforbetween them which is known as the Seiberg–Witten (SW) map [1]. This permits one to study noncommgauge theories in terms of ordinary fields. In fact, in [2] (S) duality is incorporated into noncommuMaxwell theory action in terms of ordinary fields after performing the SW map. In 4 dimensionsoriginal theory is noncommutative Maxwell theory where noncommutativity is spatial, its dual theorynoncommutative gauge theory whose time variable is effectively noncommuting with the other coordinaThis interesting phenomenon is a consequence of the fact that the duality transformation includes 4-dimtotally antisymmetric tensor.

In 3 dimensions the most extensively studied duality is between Maxwell–Chern–Simons (MCS) theoself dual theory [3]. It leads to two equivalent descriptions of the dynamics of massive spin-1 field. Onemost known applications is bosonization in 3 dimensions [4]. We wonder what would be the consequegeneralization of this duality to noncommutative MCS theory. In [5] and [6] some generalizations of the menduality to noncommutative theories are investigated in terms of noncommuting fields. However, duality c

E-mail address: dayi@itu.edu.tr (Ö.F. Dayi).

0370-2693/03/$ – see front matter 2003 Published by Elsevier Science B.V.doi:10.1016/S0370-2693(03)00413-1

240 Ö.F. Dayi / Physics Letters B 560 (2003) 239–244

due toterms of

map.l actioncee fields.ms ofrmation

uct

he usual

alues in

nd

be studied employing ordinary fields in the spirit of [2]. Although at first sight this can appear to be trivialthe fact that 3-dimensional noncommutative Chern–Simons (CS) action becomes the usual CS action inthe SW map [7–10], we will show that it gives nontrivial results.

We write 3-dimensional noncommutative MCS action in terms of ordinary gauge fields utilizing the SWWe introduce a parent action in terms of ordinary fields to obtain the dual description. We study the duaup to the second order in the noncommutativity parameterθ , when we let only spatial noncommutativity. Onthe dual description is obtained it inspires a new noncommutative CS theory in terms of ordinary gaugWe discuss equations of motion following from this new action. Moreover, we propose to write it in ternoncommuting fields as the simplest generalization of Abelian CS action. This leads to an explicit transfobetween noncommutative and ordinary fields.

2. Duality and noncommutative MCS theory

It is well known that noncommutativity between coordinates can be introduced in terms of the star prod

(1)∗ ≡ expiθµν

2

(←∂ µ

→∂ ν −

←∂ ν

→∂ µ

),

where θµν are antisymmetric constant parameters. Thus, one retains coordinates commuting under tproduct.

Noncommutative MCS theory in 3 dimensions can be defined as

(2)S = SM + SCS,

in terms of the noncommutative CS action

(3)SCS= m

2εµνρ

∫d3x

(Aµ∂νAρ + 2

3Aµ ∗ Aν ∗ Aρ

)and the noncommutative Maxwell theory

(4)SM =−1

4

∫d3x Fµν F

µν.

We employed the noncommutative field strength:

(5)Fµν = ∂µAν − ∂νAµ − iAµ ∗ Aν + iAν ∗ Aµ.

Aµ are not operators but they are called noncommutative gauge fields in the sense that they take vnoncommutative space.

One can show that the noncommutative MCS action (2) is invariant under the gauge transformations

(6)δλAµ = ∂µλ+ iλ ∗ Aµ − iAµ ∗ λ.

The equivalence relation between the noncommutingAµ, λ and the ordinary (commuting) gauge fields agauge parameterAµ, λ:

(7)Aµ(A)+ δλAµ(A)= Aµ(A+ δλA),

leads to the SW map [1]. To the first order inθ it is written explicitly as

(8)Aµ =Aµ − θρν(Aρ∂νAµ − 1

2Aρ∂µAν

).

Ö.F. Dayi / Physics Letters B 560 (2003) 239–244 241

comes

n be

When the change of variables which follows from (7) is performed the noncommutative CS action (3) bethe usual action [7]

(9)SCS= m

2εµνρ

∫d3x Aµ∂νAρ.

Thus, in terms of the SW map the action (2) can be expressed as

(10)S =∫

d3x

{−1

4

[FµνF

µν +L(θ,F )]+ m

2εµνρAµ∂νAρ

},

where theθ dependent part can be written as

L(θ,F )≡ Lθ (F )+Lθ2(F )+ · · · .Lθn is at thenth order inθ . In the following we will use only the first and the second order terms, which cawritten as [2]

(11)Lθ(F )= 2θµνF νρFρσFσµ − 12θµνFµνFρσF

ρσ ,

(12)

Lθ2(F )= 2θνµF νρθρσFσδFδξF

ξµ + θµνFνρFρσ θ

σδFδξFξµ + θµνF

νµθρσFσξFξδF

δρ

− 1

8(θµνF

µν)2FρσFσρ + 1

4θµνF

νρθρσFσµFδξF

ξδ.

Let us introduce the parent action

(13)S =∫

d3x

{−εµνρBµ∂νAρ + 1

2BµB

µ + m

2εµνρAµ∂νAρ − 1

4

[Lθ(F )+Lθ2(F )+ · · ·]}.

Equations of motion with respect toBµ are

Bµ = εµνρ∂νAρ.

When we substituteBµ with this in (13) the noncommutative MCS action (10) follows.On the other hand the equations of motion with respect toAµ

(14)∂ν

[εµνρ(Bρ −mAρ)− 1

2

δLδFνµ

]= 0,

can be solved forAµ as

(15)Aµ = 1

mBµ + 1

mbµ(θ,B).

We definedbµ(θ,B) in terms of the equation

(16)bµ(θ,B)+ 1

4εµνρKνρ

θ

(H

m+ h(θ,B)

m

)= 0,

whereH = dB, h(θ,B)= db (θ,B) and

Kµνθ (F )≡ δL(θ,F )

δFµν

.

Obviously,bµ(θ,B) can be expanded in powers ofθ as

bµ(θ,B)= bµθ + b

µ

θ2 + · · · .

242 Ö.F. Dayi / Physics Letters B 560 (2003) 239–244

r,choose

r

hat theyy oblige

(9) [7].yan

ian CS

When we plug the solution (15) into (13) the dual of noncommutative MCS action follows:

(17)SD =∫

d3x

{1

2BµB

µ + 1

2mεµνρ

[Bµ∂νBρ + bµ(θ,B)∂νbρ(θ,B)

]− 1

4L

(θ,

H

m+ h(θ,B)

m

)}.

In the most general case we can add to the solution (15) the term∂µκ , whereκ is an arbitrary function. Howevethis alters the dual action (17) only up to a surface term which we can drop. Obviously, this is equivalent toa vanishingκ .

When the noncommutativity is along the spatial coordinates:

(18)θ ij = θεij , θ0i = 0,

the first order term (11) can be written as

(19)Lθ(F )= θF12FµνFµν.

Now, we can solve (16) to obtain

b0θ =

θ

m2

[HµνH

µν + 2H12H12], b1

θ =2θ

m2H12H02,

(20)b2θ =

2θ

m2H12H10.

When we use these in (17) explicit form of the dual action to the second order inθ follows. To the second ordein θ (17) can be written as

SD,(2) ≡∫

d3x

{1

2BµB

µ + 1

2mεµνρB

µ∂νBρ − 1

4Lθ

(H

m

)+ 1

2mεµνρb

µθ ∂

νbρθ −

1

4Lθ2

(H

m

)

+ 2

m2θµν(HνρhθρσH

σµ + 2HνρHρσhσµθ

)− 12

m2θµν(hµνθ HρσH

ρσ + 2HµνhθρσHρσ

)},

wherehµνθ = ∂µbνθ − ∂νbµθ .

Although, the dual actions (10) and (17) are obtained from the parent action (13), it is not guaranteed tyield the same partition function. Indeed, we deal only with the classical aspects. Quantum corrections maus to regulate the action (13) [2], which is not addressed in this work.

3. A new noncommutative CS theory

When the SW map is employed the noncommutative CS action (3) becomes the ordinary CS actionThus, a noncommutative CS theory formulated in terms of the ordinary gauge fieldsAµ and the noncommutativitparameterθ is not available. However, we can utilize the actionSD (17) to define a new noncommutative AbeliCS theory in terms of the ordinary gauge fieldsBµ and the noncommutativity parameterθµν .

Settingθµν = 0 and dropping the ordinary mass termB2 in SD (17) lead to the ordinary Abelian CS action:

(21)SCS[B] = M

2

∫d3x εµνρB

µ∂νBρ,

whereM ≡ 1/m. We would like to take advantage of this observation to define a new noncommutative Abeltheory in terms of the ordinary gauge fieldsBµ as

(22)SNCS=∫

d3x

{M

2εµνρ

[Bµ∂νBρ + bµ(θ,B)∂νbρ(θ,B)

]− 1

4L

(θ,MH +Mh(θ,B)

)},

Ö.F. Dayi / Physics Letters B 560 (2003) 239–244 243

tions

also being someaction.

S

patial

rms ofto yield

.eories.

by dropping theB2 term in (17). Obviously, this action is invariant under the Abelian gauge transformaδBµ = ∂µλ and yields the ordinary CS theory (21) when one setsθ = 0.

Equations of motion are

(23)εµνρ∂ν(Bρ + bρ(θ,B)

)− 4εσνρ∂κ[Gσ

κµ(H)∂νbρ]= 0,

where we defined

(24)δbµ(θ,B(y))

δHνρ(x)=Gµ

νρ(H)δ3(x − y).

Observe that the simplest solution of (23) is

Hµν = 0,

which is independent ofθ . To the first order inθ the equations of motion (23) get the simple form

(25)εµνρ∂ν(Bρ + b

ρθ

)= 0.

The SW map (7) expresses the noncommutative gauge fieldsAµ in terms of the ordinary gauge fieldsAµ

utilizing the equivalence relation (8). A transformation between noncommutative and ordinary fields canderived by assuming an equivalence relation between the action (22) and another one written by introducfieldsB(B, θ) taking values in noncommutative space. However, there is no unique choice for the latterOne should make an assumption about the form of the action in terms of the noncommuting fieldsB(B, θ). Letus suppose that the action in terms of the noncommutative fieldsB(B, θ), is in the same form as the Abelian Ctheory:

(26)SNCS≡ M

2

∫d3x εµνρBµ(B, θ)∂νBρ(B, θ).

One can show that there exists a transformation betweenBµ(B, θ) and Bµ. Indeed, one can solve forBperturbatively inθ . At the first order inθ one should solve

(27)εµνρBµθ H

ρν = Lθ (MH),

whereBµθ = ∂Bµ/∂θ |θ=0. There is not a unique solution. For instance, when the noncommutativity is only s

(18), a solution is

Bµθ =

Mθ

2H21ε

µνρHνρ.

Although the assumption (26) is very plausible, in principle one may define some other actions in tefields taking values in noncommutative space. Nevertheless, the assumed form of the action (26) is showna map between the noncommutative gauge fieldsBµ(B, θ) and the ordinary onesBµ which is not the SW map (8)Moreover, the form of the action (26) can be useful to generalize this construction to non-Abelian gauge th

Acknowledgement

I thank the referee for useful comments.

References

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