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On the Seiberg–Witten map of N=2 SYM theory in non(anti)commutative harmonic superspace
Physics Letters B 601 (2004) 81–87 www.elsevier.com/locate/physletb
On the Seiberg–Witten map of N = 2 SYM theory in non(anti)commutative harmonic superspace Batool Safarzadeh Department of Physics, School of Sciences, Tarbiat Modares University, PO Box 14155-4838, Tehran, Iran Received 14 July 2004; received in revised form 23 August 2004; accepted 6 September 2004 Available online 18 September 2004 Editor: T. Yanagida
Abstract We consider N = 2 supersymmetric U (1) gauge theory in the N = 2 non(anti)commutative harmonic superspace with the singlet deformation. We generalize analytic superfield and gauge parameter to the non(anti)commutative theory so that gauge transformations act on the component fields in a canonical form (Seiberg–Witten map). This superfield, upon a field redefinition transforms under supersymmetry in a standard way. 2004 Elsevier B.V. All rights reserved.
1. Introduction The deformation of superspace has been studied intensively [1–4]. Similar to the deformations of bosonic space time which result in noncommutative field theories [5], fermionic deformations of superspace ({θ, θ } = 0), give rise to non(anti)commutative theories [6]. In the latter case, for N = 1, half of supersymmetry is broken and the unbroken Q supersymmetry is known as N = 12 [6]. It is an interesting problem to study the deformation of the extended superspace (N = 2). Depending on whether one chooses the supercovariant derivatives Dα or the supersymmetry generators Qα as the differential operators defining the Poisson brackets, one obtains the full N = 2 supersymmetry [9] or a partial N = 2 supersymmetry [7,10], respectively. The singlet deformation of N = 2 supersymmetric U (1) gauge theory in the harmonic superspace which breaks half of N = 2 supersymmetry has been studied in [8,11]. In these works, it was shown that the gauge and supersymmetric transformations get corrections which are linear in the deformation parameter such that, with a proper redefinition of the component fields, the standard gauge transformations were recovered. It would be interesting to study the same problem by looking for a modified analytic superfield and gauge parameter such that the gauge transformations keep their usual E-mail address: [email protected] (B. Safarzadeh). 0370-2693/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2004.09.018
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form (Seiberg–Witten map) [6]. Seiberg and Witten claimed in [5] that certain noncommutative gauge theories are equivalent to commutative ones. In particular, they argued that there exists a map from a commutative gauge field to a noncommutative one which is compatible with the gauge structure of each. In this work we will apply this idea to U (1) N = 2 with a singlet deformation and obtain the deformed supersymmetry transformations of the component fields. We will redefine component fields such that the standard form of supersymmetry transformations is restored and then use them to find the action which turns out to have a simple form. Although the final Lagrangian will have the same form as the one found in [8], the present approach will prove to be much easier in the sense that the field redefinitions and supersymmetry transformations will have a simpler form. 2. Deformation of N = 2 and gauge theory To formulate supersymmetric field theories with extended supersymmetry, one necessarily has to make use of the harmonic variables [13]. So we begin by introducing the non(anti)commutative deformation of N = 2 harmonic superspace. N = 2 in the four-dimensional superspace is parameterized by (x µ , θ α , θ¯ α˙ , u+i ) where µ = 0, 1, 2, 3, α, α˙ = 1, 2 and i = 1, 2 are spacetime, spinor and SU(2)R indices, respectively.1 In this harmonic superspace, ± ¯± ¯± supersymmetry generators Q± α , Qα˙ and supercovariant Dα , Dα derivatives are defined by ± i Q± α = ui Qα ,
± ¯i Q¯ ± α = ui Qα , D¯ ± = u± D¯ i ,
(1)
and analytic superfields are defined by the differential constraint µ + + ¯ Dα+ φ = D¯ α+ ˙ φ xA , θ , θ , u = 0.
(2)
i Dα± = u± i Dα ,
α˙
i
α˙
The non(anti)commutativity in the N = 2 harmonic superspace is introduced by Moyal–Weyl star product α β 1 θi , θj ∗ = ij αβ Cs . 4 Here the ∗-product is defined by
(3)
1 →j ← − − P = − ij αβ Cs Qiα Qβ . (4) 8 The action of N = 2 supersymmetric U (1) gauge theory in this non(anti)commutative harmonic superspace is written in terms of an analytic superfield V ++ [12] ∞ V ++ (ζ1 , u1 ) ∗ · · · ∗ V ++ (ζn , un ) 1 (−i)n S= (5) , d 4 x d 8 θ du1 · · · dun + + + 2 n n (u+ 1 u2 ) · · · (un u1 ) f (θ ) ∗ g(θ ) = f (θ ) exp(P )g(θ ),
where ζi = (xA , θi+ , θ¯i+ ) and d 8 θ = d 4 θ + d 4 θ − . The action is invariant under the gauge transformation ∗ ++ V = −D ++ Λ + i Λ, V ++ ∗ . δΛ The gauge parameter Λ(ζ, u) is also analytic. D ++ = u+i
D ++
(6)
harmonic derivative is defined by
∂ ∂ ∂ ∂ − 2iθ + σ µ θ¯ + µ + θ +α −α + θ¯ +α˙ −α˙ . −i ∂u ∂θ ∂xA ∂ θ¯
(7)
Since the analytic superfield and gauge parameter contain an infinite number of (auxiliary) fields in the harmonic superspace representation, the gauge freedom can be used to set some of their components to zero [13]. In the 1 For lowering and raising spinor and SU(2) indices we use antisymmetric tensors ij , αβ , respectively. R
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Wess–Zumino (WZ) gauge we then find √ 2 √ 2 2 + 2 + i − ++ VWZ θ¯ ψ¯ ui (ξ, u) = −i 2 θ + φ¯ + i 2 θ¯ + φ − 2i θ + σ µ θ¯ + Aµ + 4 θ¯ + θ + ψ i u− i −4 θ + 2 + 2 ij − − µ θ¯ +3 θ D ui uj , and Λ = λ xA . ++ is calculated as In the case of singlet deformation, the gauge variation of VWZ √ i ˙ 2 1 µ ˙ µ ++ δλ∗ VWZ = +2i θ + σ µ θ¯ + ∂µ λ − αβ ij Cs 4i 2 θα+ σβ β˙ θ¯ β ∂µ λφ¯ − 4σα α˙ σβνβ˙ α˙ β θ¯ + ∂µ λAν 8 2
β˙ µ +j −i +i −j + 16σβ β˙ θ¯ + θ¯ α˙ θα+ ψ¯ i u− i ∂µ λ u u − u u
2 1 = −2i θ + σ µ θ¯ + −∂µ λ − √ Cs ∂µ λφ¯ + i θ¯ + Cs ∂µ λAµ 2 + 2 + β −1 µ i ∂µ λ σ ψ¯ β Cs u− θ + 4 θ¯ i . 2
Deformed gauge transformations for the component fields in the case of the singlet deformation read as 1 1 ∗ ∗ δΛ Aµ = − 1 + √ Cs φ¯ ∂µ λ, δΛ φ = √ Cs Aµ ∂ µ λ, 2 2 1 ∗ i ∗ ¯ ∗ ¯i ∗ ij δΛ φ = δΛ ψα˙ = δΛ ψα = − Cs ∂µ λ σ µ ψ¯ i α , δΛ D = 0. 2
83
(8)
(9)
(10)
3. Modified superfield and gauge parameter We may also consider another point of view in which we change the definition of the superfields and the supergauge parameter such that the gauge transformations of the component fields get the standard form. In comparisons with the previous situation, this point of view could then be interpreted as the Seiberg–Witten map for the non(anti)commutative N = 2 SYM theory [14]. ++ as follows To be precise let us consider a deformed analytic superfield VWZ √ √ 2 2 2 ++ VWZ (ξ, u) = −i 2 θ + φ¯ + i 2 θ¯ + φ − 2i θ + σ µ θ¯ + Aµ + 4 θ¯ + θ + ψ i u− i i α˙ −j + 2 + i − i αβ ¯ + 2 + µ θ¯ ψ¯ ui − ij θ θβ σα α˙ ψ¯ , Aµ u − 4 θ 2
2 2 Cs − + 3 θ + θ¯ + D ij − √ ij ∂µ Aµ , φ¯ u− (11) i uj . 3 2 Accordingly we will also change the gauge parameter Λ as the following 2 i i ¯ Λ = λ + ij u−i u−j Cs θ¯ + {∂µ λ, Aµ } + √ ij u−i u−j Cs θ + σ µ θ¯ + {∂µ λ, φ}. 4 2 2
(12)
In order to find the gauge transformation of the component fields one needs to compute the effect of D ++ on the gauge parameter Λ 2 i 2i ¯ D ++ Λ = −2i θ + σ µ θ¯ + ∂µ λ + ij u+i u−j Cs θ¯ + ∂µ λ, Aµ + √ ij u+i u−j Cs θ + σ µ θ¯ + {∂µ λ, φ} 4 2 2 2 1 ¯ + √ ij u−i u−j Cs θ + θ¯ + ∂ µ {∂µ λ, φ}. (13) 2
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Moreover we find
2 −i µ α˙ ++ Cs ij αβ θ¯ + θβ+ σα α˙ ψ¯ i , Aµ u−j = λ, λ, VWZ 2 −i − −j αβ ¯ + 2 + µ ¯ i α˙ ij kl u+ = θβ σα α˙ λ ψ , Aµ θ k ul , u 2 2 µ α˙ = −iCs θ¯ + θ +α σα α˙ λ ψ¯ i , Aµ u− i .
(14)
Here we have used the fact that the gauge group is U (1) and therefore all other terms in Λ will commute with V ++ . Plugging (13) and (14) into (6), the gauge transformation of the deformed analytic superfield (11) reads
+ µ + 2 1 ∗ ++ ¯ ¯ −∂µ λ − √ Cs ∂µ λφ + i θ¯ + Cs ∂µ λAµ δλ VWZ = −2i θ σ θ 2 + 2 + β −1 µ i 2i 2 − ¯ ¯ ∂µ λ σ ψ β Cs ui − Cs θ¯ + ∂µ λ, Aµ +4 θ θ 2 4 + 2 + 2 µ 1 i ¯ −i u−j − √ Cs θ + σ µ θ¯ + {∂µ λ, φ} ¯ − √ Cs θ θ¯ ∂ {∂µ λ, φ}u 2 2 2 µ α˙ + Cs θ¯ + θ +α σα α˙ λ ψ¯ i , Aµ u− (15) i . From this relation one can read the gauge transformations of the components of the superfield in the case of singlet deformation as follows ∗ δΛ Aµ = −∂µ λ,
∗ ∗ i ∗ ¯ ∗ ¯i ∗ ij δΛ φ = δΛ ψα = δΛ D = 0, φ = δΛ ψα˙ = δΛ
(16)
which is the same as the ordinary field theory. Therefore by making use of the deformed analytic superfield (11) the gauge transformations of the component fields (10) reduce to the canonical form. As compared to [8], here we do not need any redefinition of the component fields to get this result. Let us now write down the corresponding Lagrangian using the deformed superfield (11). By making use of the same method as in [10], the Lagrangian up to the first order of (Cs ) can be computed. The result is 1 1 1 L = Fµν F µν + F˜ µν − iψ i σ µ ∂µ ψ¯ i − ∂ µ φ∂µ φ¯ + Dij D ij + √ Cs Aν ∂µ φ¯ F µν + F˜ µν 4 4 2 √ i i i 2 Cs Aµ Aµ ∂ 2 φ¯ + √ Cs φ¯ ψ k σ ν ∂ν ψ¯ k + √ Cs ψ k σ ν ψ¯ k ∂ν φ¯ + Cs ψ¯ i ψ¯ j Dij + 2 4 2 2 √ 2 ¯ ij − Cs φD Dij , 4 where F˜ µν = i µνρσ Fρσ .
(17)
2
4. Supersymmetry transformation of modified superfield The next step would be to check how the supersymmetry transformations δξ of the component fields work for this deformed superfield. The supersymmetry transformation in the WZ gauge and for the non(anti)commutative case has been studied in [11] which has the following form ++ ++ ∗ ++ = δ˜ξ VWZ + δΛ VWZ , δξ VWZ
++ ++ δ˜ξ VWZ ≡ ξiα Qiα VWZ ,
(18)
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where in the analytic basis one has −α + ξiα Qiα = −ξ +α Q− Qα , α +ξ
with Q+ α =
∂ ∂ µ − 2iσα α˙ θ¯ +α˙ µ , −α ∂θ ∂xA
Q− α =−
∂ . ∂θ +α
To preserve the WZ-gauge we must use the most general analytic gauge parameter Λ(ζ, u): 2 2 (0,1)α˙ Λ(ζ, u) = λ(0,0)(xA , u) + θ¯α+ (xA , u) + θ +α λα(1,0)(xA , u) + θ¯ + λ(0,2) (xA , u) + θ + λ(2,0) (xA , u) ˙ λ 2 2 (1,1) (2,1)α˙ + θ + σ µ θ¯ + λµ (xA , u) + θ¯ + θ +α λ(1,2)(xA , u) + θ + θ¯α+ (xA , u) ˙ λ + 2 + 2 (2,2) θ¯ + θ (19) λ (xA , u). Using the appropriate gauge parameter and the deformed analytic superfield (11), it is easy to see how the various fields transform √ i δξ φ = − 2iξ i ψi − √ Cs ξ i σ µ {ψ¯ i , Aµ }, 2 2 δξ φ¯ = 0, δξ Aµ = iξ i σµ ψ¯ i , 1 δξ ψαi = 1 + √ Cs φ¯ σ µν ξ i α Fµν − D ij ξαj , 2 √ α˙ 1 ˙ ¯ δξ ψ¯ αi = − 2 σ¯ µ ξ i 1 + √ Cs φ¯ ∂µ φ, 2 1 kl k µ l l µ k ¯ ¯ ¯ δξ D = −iξ σ ∂µ ψ − iξ σ ∂µ ψ 1 + √ Cs φ . 2
(20)
It is straightforward to find a series of field redefinitions which bring these deformed supersymmetry transfor˙ ,D ˜ ij ) as mations to the standard form. We introduce the multiplet (aµ , ϕ, ϕ, ¯ λiα , λ¯ αi 1 2 µ ¯ ¯ ¯ aµ = F (φ)Aµ , ϕ¯ = φ, ϕ = F (φ) φ + Cs √ Aµ A , 2 2 ˙ ˙ ¯ ψ¯ αi ¯ 2 ψαi , ¯ 2 D ij , λ¯ αi (21) = F (φ) , λiα = F (φ) D˜ ij = F (φ) ¯ is a function of φ¯ and is determined as where F (φ) ¯ = F (φ)
1 1+
√1 Cs φ¯ 2
.
˙ ,D ˜ ij ) transforms canonically under supersymmetry trans¯ λiα , λ¯ αi It is easy to check that the multiplet (aµ , ϕ, ϕ, formations √ δξ aµ = iξ i σµ λ¯ i , δξ ϕ = −i 2 ξ i λi , δξ ϕ¯ = 0, δξ λiα = σ µν ξ i fµν − D˜ ij ξαj , √ α˙ ˙ = − 2 σ¯ µ ξ i ∂µ ϕ, ¯ δξ D˜ ij = −i ξ i σ µ ∂µ λ¯ j + ξ j σ µ ∂µ λ¯ i . δξ λ¯ αi (22)
Here fµν = ∂µ aν − ∂ν aµ .
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5. Invariant action In the singlet deformed U (1) theory which we consider here the action can be constructed as [9] 1 1 S= d 4 xL d 4 θ du W ∗ W = d 4 xL d 4 θ du W 2 , 4 4
(23)
where xL is a chiral-analytic coordinate: xA = xL − 2iθ − σ µ θ¯ + . µ
µ
To compute the filed strength W = 14 (D¯ + )2 V −− one can obtain V −− from this differential equation D ++ V −− − D −− V ++ + i V ++ , V −− ∗ = 0,
(24)
here D −− = ∂ −− − 2iθ − σ µ θ¯ −
∂ µ. ∂xA
In terms of the deformed analytic superfield (11), W is calculated as
i + + 2 Cs 2 2Cs − ¯ i ψ¯ j + D ij u− u− W = φ + √ Aµ Aµ + 2θ + ψ i u− − ψ + θ θ u ψ i i i j 1+ √ 1 ¯ 1+ √ 1 ¯ 2 2 2 Cs φ 2 Cs φ
− 2 2Cs 1 + i ¯j ij ¯ θ ψ ψ + D u+ + i uj 1+ √ 1 ¯ 1+ √ 1 ¯ 2 Cs φ 2 Cs φ
+ − 2Cs 2 + i ¯j ij ¯ ψ ψ + D u− − θ θ i uj 1+ √ 1 ¯ 1+ √ 1 ¯ 2 Cs φ 2 Cs φ + µν − − 2 + µ 1 ¯ i u+ ψ + θ σ θ Fµν + 2i θ θ σ ∂µ i 1+ √ 1 ¯ 2 Cs φ + 2 − + 2 − 2 2 1 1 µ i ¯ ¯ θ + 2i θ θ 1+ √ ∂ φ. (25) σ ∂µ ψ u− i − θ 1 ¯ √ 1+ 2 Cs φ ¯ 2 Cs φ
If we use redefined fields (21) to compute the Lagrangian we will see that it has this simple form 2 1 L0 , L= 1+ √ 2 Cs φ¯
(26)
where 1 1 L0 = fµν f µν + f˜µν − iλi σ µ ∂µ λ¯ i − ∂ µ ϕ∂µ ϕ¯ + D˜ ij D˜ ij . 4 4 6. Conclusions In this Letter, using the Seiberg–Witten map, we have determined the generalized analytic superfield and gauge parameter of N = 2 supersymmetric U (1) gauge theory to the non(anti)commutative harmonic superspace for which the component fields transform canonically under gauge transformations. The component fields are then redefined to preserve the standard form of supersymmetry transformations. With this redefined component fields, the Lagrangian is obtained which has the same form as the one in [8].
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Acknowledgements I would like to thank M. Alishahiha and A.E. Mosaffa for useful comments and discussions and M. Abolhasani for his sincere support.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
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On the Seiberg–Witten map of N = 2 SYM theory in non(anti)commutative harmonic superspace Batool Safarzadeh Department of Physics, School of Sciences, Tarbiat Modares University, PO Box 14155-4838, Tehran, Iran Received 14 July 2004; received in revised form 23 August 2004; accepted 6 September 2004 Available online 18 September 2004 Editor: T. Yanagida
Abstract We consider N = 2 supersymmetric U (1) gauge theory in the N = 2 non(anti)commutative harmonic superspace with the singlet deformation. We generalize analytic superfield and gauge parameter to the non(anti)commutative theory so that gauge transformations act on the component fields in a canonical form (Seiberg–Witten map). This superfield, upon a field redefinition transforms under supersymmetry in a standard way. 2004 Elsevier B.V. All rights reserved.
1. Introduction The deformation of superspace has been studied intensively [1–4]. Similar to the deformations of bosonic space time which result in noncommutative field theories [5], fermionic deformations of superspace ({θ, θ } = 0), give rise to non(anti)commutative theories [6]. In the latter case, for N = 1, half of supersymmetry is broken and the unbroken Q supersymmetry is known as N = 12 [6]. It is an interesting problem to study the deformation of the extended superspace (N = 2). Depending on whether one chooses the supercovariant derivatives Dα or the supersymmetry generators Qα as the differential operators defining the Poisson brackets, one obtains the full N = 2 supersymmetry [9] or a partial N = 2 supersymmetry [7,10], respectively. The singlet deformation of N = 2 supersymmetric U (1) gauge theory in the harmonic superspace which breaks half of N = 2 supersymmetry has been studied in [8,11]. In these works, it was shown that the gauge and supersymmetric transformations get corrections which are linear in the deformation parameter such that, with a proper redefinition of the component fields, the standard gauge transformations were recovered. It would be interesting to study the same problem by looking for a modified analytic superfield and gauge parameter such that the gauge transformations keep their usual E-mail address: [email protected] (B. Safarzadeh). 0370-2693/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2004.09.018
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form (Seiberg–Witten map) [6]. Seiberg and Witten claimed in [5] that certain noncommutative gauge theories are equivalent to commutative ones. In particular, they argued that there exists a map from a commutative gauge field to a noncommutative one which is compatible with the gauge structure of each. In this work we will apply this idea to U (1) N = 2 with a singlet deformation and obtain the deformed supersymmetry transformations of the component fields. We will redefine component fields such that the standard form of supersymmetry transformations is restored and then use them to find the action which turns out to have a simple form. Although the final Lagrangian will have the same form as the one found in [8], the present approach will prove to be much easier in the sense that the field redefinitions and supersymmetry transformations will have a simpler form. 2. Deformation of N = 2 and gauge theory To formulate supersymmetric field theories with extended supersymmetry, one necessarily has to make use of the harmonic variables [13]. So we begin by introducing the non(anti)commutative deformation of N = 2 harmonic superspace. N = 2 in the four-dimensional superspace is parameterized by (x µ , θ α , θ¯ α˙ , u+i ) where µ = 0, 1, 2, 3, α, α˙ = 1, 2 and i = 1, 2 are spacetime, spinor and SU(2)R indices, respectively.1 In this harmonic superspace, ± ¯± ¯± supersymmetry generators Q± α , Qα˙ and supercovariant Dα , Dα derivatives are defined by ± i Q± α = ui Qα ,
± ¯i Q¯ ± α = ui Qα , D¯ ± = u± D¯ i ,
(1)
and analytic superfields are defined by the differential constraint µ + + ¯ Dα+ φ = D¯ α+ ˙ φ xA , θ , θ , u = 0.
(2)
i Dα± = u± i Dα ,
α˙
i
α˙
The non(anti)commutativity in the N = 2 harmonic superspace is introduced by Moyal–Weyl star product α β 1 θi , θj ∗ = ij αβ Cs . 4 Here the ∗-product is defined by
(3)
1 →j ← − − P = − ij αβ Cs Qiα Qβ . (4) 8 The action of N = 2 supersymmetric U (1) gauge theory in this non(anti)commutative harmonic superspace is written in terms of an analytic superfield V ++ [12] ∞ V ++ (ζ1 , u1 ) ∗ · · · ∗ V ++ (ζn , un ) 1 (−i)n S= (5) , d 4 x d 8 θ du1 · · · dun + + + 2 n n (u+ 1 u2 ) · · · (un u1 ) f (θ ) ∗ g(θ ) = f (θ ) exp(P )g(θ ),
where ζi = (xA , θi+ , θ¯i+ ) and d 8 θ = d 4 θ + d 4 θ − . The action is invariant under the gauge transformation ∗ ++ V = −D ++ Λ + i Λ, V ++ ∗ . δΛ The gauge parameter Λ(ζ, u) is also analytic. D ++ = u+i
D ++
(6)
harmonic derivative is defined by
∂ ∂ ∂ ∂ − 2iθ + σ µ θ¯ + µ + θ +α −α + θ¯ +α˙ −α˙ . −i ∂u ∂θ ∂xA ∂ θ¯
(7)
Since the analytic superfield and gauge parameter contain an infinite number of (auxiliary) fields in the harmonic superspace representation, the gauge freedom can be used to set some of their components to zero [13]. In the 1 For lowering and raising spinor and SU(2) indices we use antisymmetric tensors ij , αβ , respectively. R
B. Safarzadeh / Physics Letters B 601 (2004) 81–87
Wess–Zumino (WZ) gauge we then find √ 2 √ 2 2 + 2 + i − ++ VWZ θ¯ ψ¯ ui (ξ, u) = −i 2 θ + φ¯ + i 2 θ¯ + φ − 2i θ + σ µ θ¯ + Aµ + 4 θ¯ + θ + ψ i u− i −4 θ + 2 + 2 ij − − µ θ¯ +3 θ D ui uj , and Λ = λ xA . ++ is calculated as In the case of singlet deformation, the gauge variation of VWZ √ i ˙ 2 1 µ ˙ µ ++ δλ∗ VWZ = +2i θ + σ µ θ¯ + ∂µ λ − αβ ij Cs 4i 2 θα+ σβ β˙ θ¯ β ∂µ λφ¯ − 4σα α˙ σβνβ˙ α˙ β θ¯ + ∂µ λAν 8 2
β˙ µ +j −i +i −j + 16σβ β˙ θ¯ + θ¯ α˙ θα+ ψ¯ i u− i ∂µ λ u u − u u
2 1 = −2i θ + σ µ θ¯ + −∂µ λ − √ Cs ∂µ λφ¯ + i θ¯ + Cs ∂µ λAµ 2 + 2 + β −1 µ i ∂µ λ σ ψ¯ β Cs u− θ + 4 θ¯ i . 2
Deformed gauge transformations for the component fields in the case of the singlet deformation read as 1 1 ∗ ∗ δΛ Aµ = − 1 + √ Cs φ¯ ∂µ λ, δΛ φ = √ Cs Aµ ∂ µ λ, 2 2 1 ∗ i ∗ ¯ ∗ ¯i ∗ ij δΛ φ = δΛ ψα˙ = δΛ ψα = − Cs ∂µ λ σ µ ψ¯ i α , δΛ D = 0. 2
83
(8)
(9)
(10)
3. Modified superfield and gauge parameter We may also consider another point of view in which we change the definition of the superfields and the supergauge parameter such that the gauge transformations of the component fields get the standard form. In comparisons with the previous situation, this point of view could then be interpreted as the Seiberg–Witten map for the non(anti)commutative N = 2 SYM theory [14]. ++ as follows To be precise let us consider a deformed analytic superfield VWZ √ √ 2 2 2 ++ VWZ (ξ, u) = −i 2 θ + φ¯ + i 2 θ¯ + φ − 2i θ + σ µ θ¯ + Aµ + 4 θ¯ + θ + ψ i u− i i α˙ −j + 2 + i − i αβ ¯ + 2 + µ θ¯ ψ¯ ui − ij θ θβ σα α˙ ψ¯ , Aµ u − 4 θ 2
2 2 Cs − + 3 θ + θ¯ + D ij − √ ij ∂µ Aµ , φ¯ u− (11) i uj . 3 2 Accordingly we will also change the gauge parameter Λ as the following 2 i i ¯ Λ = λ + ij u−i u−j Cs θ¯ + {∂µ λ, Aµ } + √ ij u−i u−j Cs θ + σ µ θ¯ + {∂µ λ, φ}. 4 2 2
(12)
In order to find the gauge transformation of the component fields one needs to compute the effect of D ++ on the gauge parameter Λ 2 i 2i ¯ D ++ Λ = −2i θ + σ µ θ¯ + ∂µ λ + ij u+i u−j Cs θ¯ + ∂µ λ, Aµ + √ ij u+i u−j Cs θ + σ µ θ¯ + {∂µ λ, φ} 4 2 2 2 1 ¯ + √ ij u−i u−j Cs θ + θ¯ + ∂ µ {∂µ λ, φ}. (13) 2
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Moreover we find
2 −i µ α˙ ++ Cs ij αβ θ¯ + θβ+ σα α˙ ψ¯ i , Aµ u−j = λ, λ, VWZ 2 −i − −j αβ ¯ + 2 + µ ¯ i α˙ ij kl u+ = θβ σα α˙ λ ψ , Aµ θ k ul , u 2 2 µ α˙ = −iCs θ¯ + θ +α σα α˙ λ ψ¯ i , Aµ u− i .
(14)
Here we have used the fact that the gauge group is U (1) and therefore all other terms in Λ will commute with V ++ . Plugging (13) and (14) into (6), the gauge transformation of the deformed analytic superfield (11) reads
+ µ + 2 1 ∗ ++ ¯ ¯ −∂µ λ − √ Cs ∂µ λφ + i θ¯ + Cs ∂µ λAµ δλ VWZ = −2i θ σ θ 2 + 2 + β −1 µ i 2i 2 − ¯ ¯ ∂µ λ σ ψ β Cs ui − Cs θ¯ + ∂µ λ, Aµ +4 θ θ 2 4 + 2 + 2 µ 1 i ¯ −i u−j − √ Cs θ + σ µ θ¯ + {∂µ λ, φ} ¯ − √ Cs θ θ¯ ∂ {∂µ λ, φ}u 2 2 2 µ α˙ + Cs θ¯ + θ +α σα α˙ λ ψ¯ i , Aµ u− (15) i . From this relation one can read the gauge transformations of the components of the superfield in the case of singlet deformation as follows ∗ δΛ Aµ = −∂µ λ,
∗ ∗ i ∗ ¯ ∗ ¯i ∗ ij δΛ φ = δΛ ψα = δΛ D = 0, φ = δΛ ψα˙ = δΛ
(16)
which is the same as the ordinary field theory. Therefore by making use of the deformed analytic superfield (11) the gauge transformations of the component fields (10) reduce to the canonical form. As compared to [8], here we do not need any redefinition of the component fields to get this result. Let us now write down the corresponding Lagrangian using the deformed superfield (11). By making use of the same method as in [10], the Lagrangian up to the first order of (Cs ) can be computed. The result is 1 1 1 L = Fµν F µν + F˜ µν − iψ i σ µ ∂µ ψ¯ i − ∂ µ φ∂µ φ¯ + Dij D ij + √ Cs Aν ∂µ φ¯ F µν + F˜ µν 4 4 2 √ i i i 2 Cs Aµ Aµ ∂ 2 φ¯ + √ Cs φ¯ ψ k σ ν ∂ν ψ¯ k + √ Cs ψ k σ ν ψ¯ k ∂ν φ¯ + Cs ψ¯ i ψ¯ j Dij + 2 4 2 2 √ 2 ¯ ij − Cs φD Dij , 4 where F˜ µν = i µνρσ Fρσ .
(17)
2
4. Supersymmetry transformation of modified superfield The next step would be to check how the supersymmetry transformations δξ of the component fields work for this deformed superfield. The supersymmetry transformation in the WZ gauge and for the non(anti)commutative case has been studied in [11] which has the following form ++ ++ ∗ ++ = δ˜ξ VWZ + δΛ VWZ , δξ VWZ
++ ++ δ˜ξ VWZ ≡ ξiα Qiα VWZ ,
(18)
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where in the analytic basis one has −α + ξiα Qiα = −ξ +α Q− Qα , α +ξ
with Q+ α =
∂ ∂ µ − 2iσα α˙ θ¯ +α˙ µ , −α ∂θ ∂xA
Q− α =−
∂ . ∂θ +α
To preserve the WZ-gauge we must use the most general analytic gauge parameter Λ(ζ, u): 2 2 (0,1)α˙ Λ(ζ, u) = λ(0,0)(xA , u) + θ¯α+ (xA , u) + θ +α λα(1,0)(xA , u) + θ¯ + λ(0,2) (xA , u) + θ + λ(2,0) (xA , u) ˙ λ 2 2 (1,1) (2,1)α˙ + θ + σ µ θ¯ + λµ (xA , u) + θ¯ + θ +α λ(1,2)(xA , u) + θ + θ¯α+ (xA , u) ˙ λ + 2 + 2 (2,2) θ¯ + θ (19) λ (xA , u). Using the appropriate gauge parameter and the deformed analytic superfield (11), it is easy to see how the various fields transform √ i δξ φ = − 2iξ i ψi − √ Cs ξ i σ µ {ψ¯ i , Aµ }, 2 2 δξ φ¯ = 0, δξ Aµ = iξ i σµ ψ¯ i , 1 δξ ψαi = 1 + √ Cs φ¯ σ µν ξ i α Fµν − D ij ξαj , 2 √ α˙ 1 ˙ ¯ δξ ψ¯ αi = − 2 σ¯ µ ξ i 1 + √ Cs φ¯ ∂µ φ, 2 1 kl k µ l l µ k ¯ ¯ ¯ δξ D = −iξ σ ∂µ ψ − iξ σ ∂µ ψ 1 + √ Cs φ . 2
(20)
It is straightforward to find a series of field redefinitions which bring these deformed supersymmetry transfor˙ ,D ˜ ij ) as mations to the standard form. We introduce the multiplet (aµ , ϕ, ϕ, ¯ λiα , λ¯ αi 1 2 µ ¯ ¯ ¯ aµ = F (φ)Aµ , ϕ¯ = φ, ϕ = F (φ) φ + Cs √ Aµ A , 2 2 ˙ ˙ ¯ ψ¯ αi ¯ 2 ψαi , ¯ 2 D ij , λ¯ αi (21) = F (φ) , λiα = F (φ) D˜ ij = F (φ) ¯ is a function of φ¯ and is determined as where F (φ) ¯ = F (φ)
1 1+
√1 Cs φ¯ 2
.
˙ ,D ˜ ij ) transforms canonically under supersymmetry trans¯ λiα , λ¯ αi It is easy to check that the multiplet (aµ , ϕ, ϕ, formations √ δξ aµ = iξ i σµ λ¯ i , δξ ϕ = −i 2 ξ i λi , δξ ϕ¯ = 0, δξ λiα = σ µν ξ i fµν − D˜ ij ξαj , √ α˙ ˙ = − 2 σ¯ µ ξ i ∂µ ϕ, ¯ δξ D˜ ij = −i ξ i σ µ ∂µ λ¯ j + ξ j σ µ ∂µ λ¯ i . δξ λ¯ αi (22)
Here fµν = ∂µ aν − ∂ν aµ .
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5. Invariant action In the singlet deformed U (1) theory which we consider here the action can be constructed as [9] 1 1 S= d 4 xL d 4 θ du W ∗ W = d 4 xL d 4 θ du W 2 , 4 4
(23)
where xL is a chiral-analytic coordinate: xA = xL − 2iθ − σ µ θ¯ + . µ
µ
To compute the filed strength W = 14 (D¯ + )2 V −− one can obtain V −− from this differential equation D ++ V −− − D −− V ++ + i V ++ , V −− ∗ = 0,
(24)
here D −− = ∂ −− − 2iθ − σ µ θ¯ −
∂ µ. ∂xA
In terms of the deformed analytic superfield (11), W is calculated as
i + + 2 Cs 2 2Cs − ¯ i ψ¯ j + D ij u− u− W = φ + √ Aµ Aµ + 2θ + ψ i u− − ψ + θ θ u ψ i i i j 1+ √ 1 ¯ 1+ √ 1 ¯ 2 2 2 Cs φ 2 Cs φ
− 2 2Cs 1 + i ¯j ij ¯ θ ψ ψ + D u+ + i uj 1+ √ 1 ¯ 1+ √ 1 ¯ 2 Cs φ 2 Cs φ
+ − 2Cs 2 + i ¯j ij ¯ ψ ψ + D u− − θ θ i uj 1+ √ 1 ¯ 1+ √ 1 ¯ 2 Cs φ 2 Cs φ + µν − − 2 + µ 1 ¯ i u+ ψ + θ σ θ Fµν + 2i θ θ σ ∂µ i 1+ √ 1 ¯ 2 Cs φ + 2 − + 2 − 2 2 1 1 µ i ¯ ¯ θ + 2i θ θ 1+ √ ∂ φ. (25) σ ∂µ ψ u− i − θ 1 ¯ √ 1+ 2 Cs φ ¯ 2 Cs φ
If we use redefined fields (21) to compute the Lagrangian we will see that it has this simple form 2 1 L0 , L= 1+ √ 2 Cs φ¯
(26)
where 1 1 L0 = fµν f µν + f˜µν − iλi σ µ ∂µ λ¯ i − ∂ µ ϕ∂µ ϕ¯ + D˜ ij D˜ ij . 4 4 6. Conclusions In this Letter, using the Seiberg–Witten map, we have determined the generalized analytic superfield and gauge parameter of N = 2 supersymmetric U (1) gauge theory to the non(anti)commutative harmonic superspace for which the component fields transform canonically under gauge transformations. The component fields are then redefined to preserve the standard form of supersymmetry transformations. With this redefined component fields, the Lagrangian is obtained which has the same form as the one in [8].
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Acknowledgements I would like to thank M. Alishahiha and A.E. Mosaffa for useful comments and discussions and M. Abolhasani for his sincere support.
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