Non-Abelian Chern-Simons Actions in Three-dimensional Projective Superspaces
aa r X i v : . [ h e p - t h ] J un Non-Abelian Chern-Simons Actionsin Three-dimensional Pro jective Superspaces
Masato Arai † a and Shin Sasaki ♯ b † Fukushima National College of TechnologyIwaki, Fukushima 970-8034, Japan † Institute of Experimental and Applied Physics,Czech Technical University in Prague,Horsk´a 3a/22, 128 00, Prague 2, Czech Republic ♯ Department of PhysicsKitasato UniversitySagamihara, 252-0373, Japan
Abstract
We construct an action for the superconformal Chern-Simons theory with non-Abeliangauge groups in three-dimensional N = 3 projective superspace. We propose a Lagrangiangiven by the product of the function of the tropical multiplet, that represents the N = 3vector multiplet, and the O ( − ,
1) multiplet. We show how the tropical multiplet isembedded into the O ( − ,
1) multiplet by comparing our Lagrangian with the Chern-Simons Lagrangian in the N = 2 superspace. We also discuss N = 4 generalization of theaction. a masato.arai(at)fukushima-nct.ac.jp b shin-s(at)kitasato-u.ac.jp Introduction
Off-shell superfield formalisms of supersymmetric gauge theories with eight supercharges havebeen studied in the past. There are two known off-shell formalisms with eight supercharges.One is the harmonic superspace formalism [1, 2, 3], the other is the projective superspaceformalism [4, 5].Recently, the projective superspace formalism that keeps manifest N = 3 and N = 4 su-perconformal symmetries in three dimensions has been developed [6]. This results open up awindow to construct a supersymmetric Chern-Simons actions in terms of projective superfields.The harmonic superspace provides an explicit form of the N = 3 non-Abelian Chern-Simonsactions [7, 18] while it had not been constructed in three-dimensional projective superspaceformalisms. In [9], we have constructed the N = 3 and N = 4 superconformal Chern-Simonsactions with Abelian gauge group in the projective superspace. We have shown that the N = 3Abelian Chern-Simons Lagrangian is written in the product of the tropical multiplet represent-ing an Abelian vector multiplet V [0] with weight 0 and an O ( − ,
1) multiplet G [2] with weight2 corresponding to the field strength of the vector multiplet . The N = 3 vector multipletconsists of a (anti)chiral superfield Φ ( ¯Φ) and a vector superfield V in the N = 2 standard su-perfield formalism. We have found the explicit relations among the N = 2 superfields ( V, Φ , ¯Φ)and the N = 3 projective superfield V [0] and determined the embedding of the vector multi-plet into the tropical multiplet. This is the key point to obtain the N = 3 superconformalChern-Simons actions in the projective superspace formalism since the action is constructed sothat it reproduces the known Chern-Simons action in terms of N = 2 superfields. However,generalizing this result to that with non-Abelian gauge groups is not straightforward. For non-Abelian gauge groups, the relation between the tropical multiplet V [0] and the N = 3 vectormultiplets in terms of N = 2 superfields have not been studied in detail. This is because therelation between them becomes non-linear and complicated. Formal treatments of non-Abelianvector multiplets in four dimensions have been discussed for example in [10, 11, 12]. In [13],we have studied three-dimensional N = 3, N = 4 and four-dimensional N = 2 charged hyper-multiplets that couple with the non-Abelian vector multiplet and found the relations amongthe non-Abelian vector multiplet ( V, Φ , ¯Φ) and the N = 3 tropical multiplet V [0] . We haveexplicitly written down the N = 2 superfields ( V, Φ , ¯Φ) as functions of the components in V [0] .The same analysis has been done for the N = 4 and the four-dimensional N = 2 cases. Wealso constructed the actions of hypermultiplets coupled with the non-Abelian vector multipletin the three and four-dimensional projective superspaces.The purpose of this paper is to construct the non-Abelian Chern-Simons action in thethree dimensional N = 3 projective superspace. Although the action has been discussedin the harmonic superspace formalism, our study provides a complimentary analysis of thenon-Abelian Chern-Simons theory in the projective superspace formalism. The Lagrangian iswritten in the form of the product of “a gauge field f ( V [0] )” and “a gauge field strength G [2] ”.In order to have the consistent Abelian limit, the gauge field strength G [2] should belong tothe O ( − ,
1) multiplet. We will show that G [2] actually satisfies the condition of O ( − , f which is consistent with the Abelianlimit. Finally we will show that the action proposed in this letter completely reproduces theknown action for the non-Abelian Chern-Simons theory in the N = 2 standard superfield A brief introduction of the projective superfield formalism is found in Appendix B. N = 3 non-Abelian vector multiplet into the tropical multiplet in theprojective superspace formalism. We work in the formalism that the superconformal symmetryis manifest [6]. In Section 3, we propose the non-Abelian Chern-Simons action in the N = 3projective superspace. We show that the action in the projective superspace reproduces thatin the N = 2 superspace. We also discuss the N = 4 generalization of the action. Section 4is conclusion and discussions. Notations and conventions of the three-dimensional N = 2 , , N = 3and the N = 4 projective superspace formalisms are found in Appendix B. In this section, we give a brief overview about non-Abelian vector multiplets in the projectivesuperspace formalism. A summary of the three-dimensional N = 3 and N = 4 projectivesuperspace formalism is found in Appendix B. For more detail of the formalism, see [6, 9, 13].The non-Abelian vector multiplet is represented as the real tropical multiplet V [0] withweight 0. In the Lindstr¨om-Roˇcek gauge, V [0] is expanded by the projective coordinate ζ parameterizing C P as [10] V [0] ( z, ζ ) = 1 ζ V − ( z ) + V ( z ) + ζ V ( z ) , ¯ V ( z ) = V ( z ) , ¯ V ( z ) = − V − ( z ) , (2.1)where z is the N = 3 standard superspace coordinate defined in Appendix A. The componentsuperfields V − , V , V are adjoint representation of a non-Abelian gauge group. The tropicalmultiplet satisfies the projective superfield constraint D [2] α V [0] = 0. This is equivalent to thefollowing constraints on the component superfields: D α V = 0 , D α V − D α V = 0 , D α V − − D α V − ¯ D α V = 0 , ¯ D α V + 2 D α V − = 0 , ¯ D α V − = 0 . (2.2)Following the analysis in four dimensions [5], we decompose the tropical multiplet as follows: e V [0] = e ˆ V − e ˆ V e ˆ V + , (2.3)where ˆ V + ( ˆ V − ) contains only positive (negative) powers of ζ and ˆ V contains terms with ζ . Ingeneral, they are expanded by ζ asˆ V + = ∞ X n =1 ζ n ˆ V n , ˆ V − = ∞ X n =1 ζ − n ˆ V − n , (2.4)2here ˆ V ± n and ˆ V are functions of V ± , V . Finding the closed expressions of the functionsˆ V ± n , ˆ V in terms of the components in the tropical multiplet V [0] is difficult in the projectivesuperspace formalism. However they are obtained perturbatively. Up to O ( V ), ˆ V ± n , ˆ V arefound to be, ˆ V − = 112 [ V − , [ V , V − ]] + O ( V ) , ˆ V − = V − + 12 [ V , V − ] + 16 [ V − , [ V − , V ]] + 16 [ V , [ V , V − ]] + O ( V ) , ˆ V = V + 12 [ V , V − ] −
112 [ V , [ V − , V ]] −
112 [ V − , [ V , V ]] + O ( V ) , ˆ V = V + 12 [ V , V ] + 16 [ V , [ V , V − ]] + 16 [ V , [ V , V ]] + O ( V ) , ˆ V = 112 [ V , [ V , V ]] + O ( V ) . (2.5)We note that in the Abelian limit, these components become ˆ V ± = V ± , ˆ V = V and ˆ V ± n =0 ( n > V n ( n > V − n ( n >
3) do not appear in the Lagrangian and they are not relevant to ourdiscussion.In order to express the N = 3 vector multiplet ( V, Φ , ¯Φ) in terms of the component su-perfields V − , V , V in the tropical multiplet V [0] , we consider the action for hypermultipletsthat couples to tropical multiplets. The hypermultiplets charged under the non-Abelian gaugegroup are embedded in the (ant)arctic multiplets Υ [1] ( ¯Υ [1] ) with weight 1, which are expressedas Υ [1] ( z, ζ ) = ∞ X n =0 ζ n Υ n ( z ) , ¯Υ [1] ( z, ζ ) = ∞ X n =0 (cid:18) − ζ (cid:19) n ¯Υ n ( z ) . (2.6)They also satisfy the projective superfield constraints D [2] α Υ [1] = D [2] α ¯Υ [1] = 0.The Lagrangian for the hypermultiplet in the fundamental representation of the non-Abeliangauge group is originally written in the N = 3 superspace. After choosing a frame where theisospinor u is fixed, the Lagrangian becomes (see (B.8)): L = 12 πi I γ dζζ d θ ¯Υ [1] e V [0] Υ [1] , (2.7)where the contour γ is chosen such that it does not through the north pole of C P . We canreduce this Lagrangian to that in terms of N = 2 superfields. Below we briefly explain howto reduce it and how to find the relation between the component superfields V − , V , V in V [0] and the N = 2 superfields ( V, Φ , ¯Φ). First we define the following new fields¯˜Υ [1] ≡ ¯Υ [1] e ˆ V − , ˜Υ [1] ≡ e ˆ V e ˆ V + Υ [1] , (2.8)which satisfy the gauge-covariantized projective superfield constraints: D [2] α ˜Υ = 0 . (2.9)3ere D [2] α is the gauge covariant derivative defined to be D [2] α = − ¯ D α − ζ D α + ζ D α , (2.10) D α = D α + Γ ( − ) − α , D α = D α −
12 Γ ( − ) − α , ¯ D α = ¯ D α − Γ ( − )0 α , (2.11)where the gauge connections are defined as [13]Γ ( − ) − α = 0 , Γ ( − ) − α = D α ˆ V − , Γ ( − )0 α = D α ˆ V − − D α ˆ V − + 12 [ D α ˆ V − , ˆ V − ] . (2.12)The algebra that the gauge covariant derivatives D α , D α , ¯ D α satisfy was calculated in [13].With the use of (2.8), (2.7) is simplified to be L = 12 πi I γ dζζ d θ ¯˜Υ [1] ˜Υ [1] . (2.13)This is a convenient form to write down the Lagrangian in terms of the N = 2 superfieldssince ˜Υ [1] and ¯˜Υ [1] can be expanded as the same way in (2.6): One finds the expanded forms byreplacing Υ n with ˜Υ n and so on. Substituting their expanded forms into (2.13) and integratingover ζ , we are left with ˜Υ and ˜Υ in the Lagrangian. The other fields ˜Υ n ( n ≥
2) are integratedout since they are non-dynamical. Equation (2.9) gives a constraint for ˜Υ and it is incorporatedinto the Lagrangian with the Lagrange multiplier N = 2 superfield. Then integrating ˜Υ andgoing back to the original fields without tilde, one obtains the Lagrangian in terms of Υ , theLagrange multiplier which is denoted as Y , ˆ V − , ˆ V and ˆ V .Now we consider the N = 3 gauge interacting Lagrangian in terms of the N = 2 superfieldlanguage. The charged hypermultiplet consists of two chiral superfields ( S, T ) and they couplewith the non-Abelian vector multiplet ( V, Φ , ¯Φ). The Lagrangian is given by L = Z d θ (cid:0) ¯ Se V S + T e − V ¯ T (cid:1) + (cid:20) Z d θ T Φ S − Z d ¯ θ ¯ T ¯Φ ¯ S (cid:21) . (2.14)We can directly compare this with the Lagrangian in terms of Υ , Y , ˆ V − , ˆ V and ˆ V comingfrom (2.7). In [13] we found that S = Υ , T = ¯ D ¯˜ Y e ˆ V are chiral superfields and ¯˜ Y = ¯ Y e ˆ V − and V = ˆ V , Φ = 18 e − ˆ V (cid:16) − D α Γ ( − )0 α + ¯ D α Γ ( − ) − α + { Γ ( − ) α − , Γ ( − )0 α } (cid:17) e ˆ V , ¯Φ = 18 (cid:16) − D α Γ ( − ) − α − D α Γ ( − ) − α + { Γ ( − ) α − , Γ ( − ) − α } (cid:17) . (2.15)In [13] we have checked that Φ ( ¯Φ) defined in the above actually satisfies the (anti)chiralitycondition ¯ D α Φ = D α ¯Φ = 0. We note that in the Abelian limit, we reproduce the correctexpression found in [9] V = V , Φ = −
18 ¯ D V , ¯Φ = − D V − . (2.16)In the next section, we propose the supersymmetric N = 3 and N = 4 Chern-SimonsLagrangians by utilizing the above results. 4 Chern-Simons Lagrangian
In this section, we construct a Lagrangian for the non-Abelian Chern-Simons theory in theprojective superspace. We also generalize the result to the N = 4 case. The three-dimensional N = 3 Chern-Simons Lagrangian in the N = 2 superfield formalism is [14] L N =3CS = − ik π Z d θ Z dt Tr (cid:2) V ¯ D α ( e tV D α e − tV ) (cid:3) − k π Z d θ TrΦ + k π Z d ¯ θ Tr ¯Φ . (3.1)Here k is the Chern-Simons level and t is an auxiliary integration variable. The first term canbe rewritten as − ik π Z d θ Z dt Tr (cid:2) V ¯ D α ( e tV D α e − tV ) (cid:3) = ik π Z d θ Z dt Tr (cid:2) V D α ( e − tV ¯ D α e tV ) (cid:3) , (3.2)where we have used the fact that A ( M BM − ) = M B ( M − AM ) M − for variables A, B and M that satisfy { A, B } = 0.We first start from the N = 3 Abelian Chern-Simons theory. When the gauge group isAbelian, the Lagrangian (3.1) is reduced to the following form: L = ik π Z d θ V ¯ D α D α V − k π Z d θ Φ + k π Z d ¯ θ ¯Φ . (3.3)We have constructed the Lagrangian in the projective superspace that reproduces the expression(3.3). The Lagrangian in the N = 3 projective superspace is [9] L = k π I γ dζ πiζ Z d θ V [0] G [2] , (3.4)where the gauge invariant O ( − ,
1) multiplet G [2] with weight 2 is a function of the tropicalmultiplet V [0] and is expanded as G [2] = iζ Φ + L + iζ ¯Φ . (3.5)Each component should satisfy the projective superfield condition D [2] α G [2] = 0, namely, D α ¯Φ = 0 , D L = ¯ D L = 0 , ¯ D α Φ = 0 . (3.6)Taking account of the gauge invariance of G [2] , these constraints are solved by L = i ¯ D α D α V , Φ = −
18 ¯ D V , ¯Φ = − D V − . (3.7)This is a gauge-fixed form of the relation found in [15]. It is easy to check that Φ ( ¯Φ ) is the(anti)chiral superfield appearing in (3.3) by substituting (3.7) into (3.4).Now we construct a Lagrangian for the non-Abelian Chern-Simons theory in the N = 3projective superspace. For non-Abelian gauge groups, the embedding of the tropical multipletinto the O ( − ,
1) multiplet (3.7) becomes non-linear. We assume that the Lagrangian is given5y the product of a function of V [0] and an O ( − ,
1) multiplet G [2] even for non-Abelian gaugegroups. Since the Lagrangian should reproduce the result (3.4) in the Abelian limit, we proposethe following non-Abelian Chern-Simons Lagrangian in the N = 3 projective superspace: L = k π I γ dζ πiζ Z d θ Tr (cid:2) f ( V [0] ) G [2] (cid:3) , (3.8)where f is a projective superfield with weight 0 which is a function of the tropical multiplet V [0] and G [2] = iζ Φ + L + iζ ¯Φ is an O ( − ,
1) multiplet with weight 2 which satisfies the constraints(3.6). The function f should satisfy f ( V [0] ) → V [0] in the Abelian limit. Although the gaugeinvariance of the action is not manifest, the component expression of the action will ensure it.In the Lindstr¨om-Roˇcek gauge, the ζ expansion of the function f is generically given by f ( V [0] ) = 1 ζ f − ( z ) + f ( z ) + ζ f ( z ) + · · · , (3.9)where · · · are terms that contain ζ n , ( n = 0 , ±
1) which are irrelevant in the Lagrangian (3.8)and vanish in the Abelian limit. Performing the ζ integration, the Lagrangian is reduced tothat in the N = 2 superspace, L = ik π Z d θ Tr (cid:2) f Φ − if L + f − ¯Φ (cid:3) . (3.10)We look for the explicit forms of the components ( f − , f , f ) and (Φ , L, ¯Φ ) that reproduce theaction (3.1) in the N = 2 superspace. First, we identify Φ , ¯Φ with the non-Abelian adjoint(anti)chiral superfields Φ , ¯Φ defined in (2.15) (with extra overall factors ± i ):Φ = i Φ = i e − ˆ V (cid:16) − D α Γ ( − )0 α + ¯ D α Γ ( − ) − α + { Γ ( − ) α − , Γ ( − )0 α } (cid:17) e ˆ V , (3.11)¯Φ = − i ¯Φ = − i (cid:16) − D α Γ ( − ) − α − D α Γ ( − ) − α + { Γ ( − ) α − , Γ ( − ) − α } (cid:17) . (3.12)Then, using the constraints (2.2), we can show that the above definition satisfies a part of theprojective superfield constraints of G [2] , D α ¯Φ = ¯ D α Φ = 0 [13]. Next, we rewrite parts of theD-terms in the Lagrangian (3.10) to F-terms: L = ik π Z d θ Tr [ − if L ] − k π Z d θ Tr (cid:20) −
14 ¯ D f Φ (cid:21) + k π Z d ¯ θ Tr (cid:20) − D f − ¯Φ (cid:21) , (3.13)where we have used the fact that ¯ D α Φ = D α ¯Φ = 0 and dropped the total derivative terms.Comparing the first term in (3.13) with the D-term in the Lagrangian (3.1), we determine thecomponents L as L = − D α Z dt (cid:16) e t ˆ V D α e − t ˆ V (cid:17) , (3.14)and f as f = ˆ V . (3.15)6hen we employ the expression (3.2) instead of (3.1), we have L = 2 D α Z dt (cid:16) e − t ˆ V ¯ D α e t ˆ V (cid:17) . (3.16)The two expressions of L , (3.14) and (3.16), are physically equivalent. Since L is a componentof the O ( − ,
1) multiplet G [2] , it should satisfy the constraint D L = ¯ D L = 0. We examinethis condition in the following. Because the calculations for the expressions (3.14) and (3.16)are essentially the same, we focus on the expression (3.14) . The expression (3.14) is manifestly¯ D -exact form. Therefore we find that the first constraint is satisfied trivially:¯ D L = 0 . (3.17)On the other hand, the second condition D L = 0 is not manifest. We examine the secondcondition by perturbative calculations. We concentrate on the next leading order in V wherethe non-Abelian property begins to appear for the first time . Up to O ( V ), we have D h − D α (cid:16) e t ˆ V D α e − ˆ V (cid:17)i = − t (cid:16) [ D D α ˆ V , ˆ V ] + { D β D α ¯ D α ˆ V , D β ˆ V } + { D β D α ¯ D α ˆ V , D β ˆ V } + [ D α ¯ D α ˆ V , D ˆ V ] −{ D D α ˆ V , ¯ D α ˆ V } + [ D β D α ˆ V , D β ¯ D α ˆ V ] − [ D β D α ˆ V , D β ¯ D α ˆ V ] + { D α ˆ V , D ¯ D α ˆ V } (cid:17) + O ( V ) . (3.18)For non-Abelian gauge groups, the N = 2 vector superfield V = ˆ V is perturbatively expressedas ˆ V = V + 12 [ V , V − ] + O ( V ) . (3.19)Using the constraints for the tropical multiplet V [0] , we find D h − D α (cid:16) e t ˆ V D α e − t ˆ V (cid:17)i = − t (cid:0) −{ D α V , D ¯ D α V } + { D α V , D ¯ D α V } (cid:1) + O ( V )=0 + O ( V ) . (3.20)Then up to O ( V ), we find that the expression (3.14) satisfies the constraints D L = ¯ D L = 0.Therefore, all the components (Φ , L, ¯Φ ) are correctly embedded into the O ( − ,
1) multiplet G [2] .Finally, we look for expressions of the functions f , f − . Comparing the second and thethird terms in (3.13) with the F-terms in the Lagrangian (3.1), we find the following relations,¯ D f = − , D f − = − . (3.21)We need to solve f − , f in the above relations. Since the anti-chiral superfield ¯Φ (3.12) isshown to be D -exact form [13] ¯Φ = − D ˆ V − , (3.22) When we employ the other expression (3.16), the following calculations hold if D α and ¯ D α are interchanged. The leading order O ( V ) corresponds to the Abelian case. Here V represents the components V − , V , V in the tropical multiplet V [0] . f − is determined to be f − = ˆ V − . (3.23)On the other hand, the chiral superfield Φ (3.11) is not manifestly ¯ D -exact. Again, we calculatethe function f by perturbation in V . From (3.11) we haveΦ = −
18 ¯ D V + 116 [ V , ¯ D V ] − { ¯ D α V , ¯ D α V } + O ( V ) . (3.24)Using the projective superspace constraints of the tropical multiplet, we find that the chiralsuperfield is rewritten as the D -exact form up to O ( V ) calculation,Φ = 18 ¯ D (cid:18) − V + 12 [ V , V ] (cid:19) + O ( V ) . (3.25)Then, the function f is determined to be f = (cid:18) V + 12 [ V , V ] (cid:19) + O ( V ) . (3.26)Therefore all the functions f ± , f have been determined. We stress that the expression (3.26)is nothing but the first two terms in the perturbative expansion of ˆ V in (2.5). Although ourcalculations are limited to the perturbative regime, this result suggests that f is naturallygiven by f = ˆ V for the full order in V . All the expressions f ± = ˆ V ± , f = ˆ V have thecorrect Abelian limit f ± → V ± , f → V . In summary we have found explicit forms ofthe functions f − , f , f and the O ( − ,
1) multiplet in the Lagrangian (3.8) in the N = 3projective superspace. We note that only ˆ V − , ˆ V , ˆ V , ˆ V in the decomposition (2.3) appear inthe Lagrangian.We generalize this result to the N = 4 model. For the N = 4 theory, a pair of the projectivemultiplets associated with the two C P s (see Appendix B) is introduced. We propose thefollowing Lagrangian for the N = 4 generalization of the N = 3 model (3.8): L = k π I dζ L πiζ L d θ Tr[ f ( V [0] L ) G [2] L ] + k π I dζ R πiζ R d θ Tr[ f ( V [0] R ) G [2] R ] , (3.27)where the function f is the same one found in the N = 3 model. The O ( − ,
1) multiplet ofthe left sector G [2] L is the function of the right tropical multiplet V [0] R and vice versa: G [2] L,R = iζ L,R Φ R,L + L R,L + iζ L,R ¯Φ R,L . (3.28)In order to be consistent with the Abelian limit [9], we take the each component in G [2] L,R as thesame one in the N = 3 case. Here the superfields ( V L , Φ L , ¯Φ L ), ( V R , Φ R , ¯Φ R ) are defined by thecomponents in V [0] L and V [0] R as in the N = 3 case. Performing the ζ L , ζ R integration, we find L = − ik π Z d θ Z dt Tr (cid:2) V L ¯ D α ( e tV R D α e − tV R ) (cid:3) − k π Z d θ TrΦ L Φ R + k π Z d ¯ θ Tr ¯Φ L ¯Φ R + ( L ↔ R ) . (3.29)This Lagrangian contains two gauge fields and mixing interactions between the left and rightmultiplets. This kind of theory is known as the BF-theory [16, 17]. The N = 4 supersymmetricBF-theory is discussed in the harmonic superspace formalism [18].8 Conclusion and discussions
In this letter we have studied the N = 3 and N = 4 non-Abelian Chern-Simons actions inthe three-dimensional projective superspaces. We work in the projective superspaces where thesuperconformal symmetry is manifest. The N = 3 and N = 4 vector multiplets are definedby the tropical multiplet V [0] with weight 0. The relations among the component superfields( V − , V , V ) in V [0] and the vector multiplet ( V, Φ , ¯Φ) in the N = 2 superspace are quite non-linear for non-Abelian gauge groups. The explicit relations among them are found in ourprevious paper [13].In this letter, using the explicit relations of the component superfields, we propose theLagrangian (3.8) for the superconformal non-Abelian Chern-Simons theory in the N = 3 pro-jective superspace. Although we have a little principle to determine the function f of thetropical multiplet V [0] , we have found the explicit form of the function by the help of the ac-tion in the N = 2 superspace. The ζ ± , ζ components of the function f consist of ˆ V ± , ˆ V which appeared in the decomposition (2.3) of the non-Abelian tropical multiplet V [0] . We alsofound the explicit embedding of the non-Abelian tropical multiplet V [0] into the O ( − ,
1) mul-tiplet G [2] with weight 2. The O ( − ,
1) multiplet plays the role of the gauge field strengthassociated with the gauge potential V [0] . The Lagrangian (3.8) has the correct Abelian limit[9]. We demonstrated that the proposed Lagrangian (3.8) successfully reproduces the N = 3non-Abelian Chern-Simons Lagrangian in the N = 2 superspace [14]. We also discussed the N = 4 generalization of our Lagrangian. We stress that although our calculations are based onthe perturbation, they are not trivial even in the next leading order in V . Moreover, the verysuggestive expression (3.26) implies that our analysis holds true even for the full order in V .We found the functions f and G [2] in the language of the component superfields of V [0] inthis letter. For an Abelian gauge group, the O ( − ,
1) multiplet G [2] is a linear function of V [0] [9, 15]. Since for a non-Abelian case, this would become non-linear and complicated, it ischallenging to write down the Lagrangian in terms of the projective superfield V [0] .For an application of the present formalism, it is interesting to write down the N = 6ABJM action [19] in the projective superspaces . The gauge field part of the ABJM modelis the U ( N ) × U ( N ) Chern-Simons model with opposite Chern-Simons level ( k, − k ). We caneasily construct the N = 6 ABJM action in the N = 3 projective superspace. However, in the N = 4 projective superspace, the first term in the Lagrangian (3.29) is the mixing term of V L and V R . Then it is not the standard Chern-Simons term discussed in [14] but is the BF-theory.At least in the component level in the Abelian limit, we found that the first term is rewrittenas the sum of the two Chern-Simons terms constructed by the two vector superfields V and V ′ with opposite Chern-Simons level ( k, − k ). Here V = √ ( V L + V R ) and V ′ = √ ( V L − V R ).For the non-Abelian case, V and V ′ would become highly non-linear functions of V L and V R .Moreover, in order to incorporate with the bi-fundamental matters which couples to left andright parts of the gauge potentials, one may need the hybrid projective multiplet [15]. Non-Abelian gauge interactions of the hybrid projective multiplet in the N = 4 projective superspaceis also interesting. We will come back to these issues in the future works. The N = 6 ABJM action has been constructed in the harmonic superspace formalism [20]. The Chern-Simons action based on the 3-algebra has been discussed in [21]. cknowledgments The work of M. A. is supported by Grant-in-Aid for Scientific Research from the Ministry ofEducation, Culture, Sports, Science and Technology, Japan (No.25400280) and in part by theResearch Program MSM6840770029 and by the project of International Cooperation ATLAS-CERN of the Ministry of Education, Youth and Sports of the Czech Republic. The work of S. S.is supported in part by Sasakawa Scientific Research Grant from The Japan Science Societyand Kitasato University Research Grant for Young Researchers.
A Conventions and notations of ordinary superspacesin three dimensions
In this section, we provide the basic conventions and notations of the N = 2, N = 3 and N = 4 superspaces in three dimensions. The three-dimensional metric is given by η mn =diag( − , +1 , +1). The three-dimensional N = 2 superspace is represented by the coordinates( x m , θ α , ¯ θ α ) where θ, ¯ θ are two component spinors. The index α = 1 , SO (1 , ∼ SL (2 , R ) Lorentz spinors. The spinor indices are raised and lowered by the anti-symmetric epsilon symbol ε = − ε = 1. The gamma matrices which satisfy the Cliffordalgebra { γ m , γ n } = 2 η mn are defined by ( γ m ) αβ = ( iτ , τ , τ ). Here τ I ( I = 1 , ,
3) are thePauli matrices and I = 1 , , SO (3) R ∼ SU (2) R R-symmetry. Thesupercovariant derivatives in the N = 2 superspace are defined by D α = ∂ α + i ( γ m ¯ θ ) α ∂ m , ¯ D α = − ¯ ∂ α − i ( θγ m ) α ∂ m , { D α , ¯ D β } = − iγ mαβ ∂ m , { D α , D β } = { ¯ D α , ¯ D β } = 0 . (A.1)The Grassmann measure of integration in the N = 2 superspace is defined by d θ = − dθ α dθ α , d ¯ θ = − d ¯ θ α d ¯ θ α , d θ = d θd ¯ θ. (A.2)They are normalized such that, Z d θ θ = 1 , Z d ¯ θ ¯ θ = 1 , Z d θ θ ¯ θ = 1 . (A.3)For an N = 2 superfield F ( x, θ, ¯ θ ), the following relation holds within the spacetime integration, Z d θ F ( x, θ, ¯ θ ) = 116 ( D ¯ D F ( x, θ, ¯ θ )) (cid:12)(cid:12)(cid:12)(cid:12) θ =¯ θ =0 . (A.4)The chiral and anti-chiral coordinates are defined by x mL = x m + iθγ m ¯ θ, x mR = x m − iθγ m ¯ θ. (A.5)The N = 3 superspace coordinates are defined by z = ( x m , θ αij ) where i = 1 , SU (2) R R-symmetry spinor index and the Grassmann coordinate satisfies the reality condition θ αij = θ αij .The SU (2) R spinor indices and the SO (3) R vector indices are intertwined by the relation10 αij = ( τ I ) ij θ αI . The SU (2) R indices are raised and lowered by the anti-symmetric symbols ε ij , ε ij . The supercovariant derivatives in the N = 3 superspace are defined by D ijα = ∂∂θ αij + iθ βij ∂ αβ , ∂ αβ = γ mαβ ∂ m , { D ijα , D klβ } = − iε i ( k ε l ) j ∂ αβ . (A.6)The N = 4 superspace coordinates are defined by z ′ = ( x m , θ αi ¯ j ) where i = 1 , j = 1 , SU (2) L × SU (2) R subgroup of SO (4) R R-symmetry and the Grassmanncoordinate satisfies the reality condition θ αi ¯ j = θ αi ¯ j . The supercovariant derivatives in the N = 4 superspace are defined by D i ¯ jα = ∂∂θ αi ¯ j + iθ βi ¯ j ∂ αβ , { D i ¯ jα , D k ¯ lβ } = 2 iε ik ε ¯ j ¯ l ∂ αβ . (A.7)We use the following relations among the N = 2, N = 3 and N = 4 superspaces [6]: θ α = θ α = θ α , ¯ θ α = θ α = θ α , (A.8) D α = D α = D α , ¯ D α = − D α = − D α . (A.9) B Pro jective superspace formalisms
In this section, we summarize conventions and notations of the projective superspaces in threedimensions. For more detail, see [6, 9, 13].
B.1 N = 3 projective superspace We introduce the SU (2) R complex isospinors v i , u i ( i = 1 ,
2) which satisfy the following com-pleteness relation, δ ij = 1( v, u ) ( v i u j − v j u i ) , ( v, u ) ≡ v i u i = 0 . (B.1)The supercovariant derivative in the projective superspace is defined as D (2) α = v i v j D ijα , D (0) α = 1( v, u ) v i u j D ijα , D ( − α = 1( v, u ) u i u j D ijα . (B.2)A projective superfield Q ( n ) with weight n is defined by D (2) α Q ( n ) = 0 , Q ( n ) ( z, cv ) = c n Q ( n ) ( z, v ) , c ∈ C ∗ . (B.3)The N = 3 superconformal invariant action is S = 18 π I γ ( v, dv ) Z d x ( D ( − ) ( D (0) ) L (2) ( z, v ) (cid:12)(cid:12) θ =0 , (B.4)11here L (2) is a real superconformal projective superfield with weight 2. The line integral isevaluated over a closed contour γ in C P . Since the action (B.4) is independent of u , we canchoose a frame where u i = (1 , γ in (B.4) such that it does not passthrough the north pole v i = (0 , ζ ∈ C in the upper hemisphere of C P , v i = v (1 , ζ ) , ζ ≡ v v , i = 1 , . (B.5)Then the supercovariant derivative D (2) α is rewritten as D (2) α = ( v ) D [2] α , D [2] α ( ζ ) ≡ − ¯ D α − ζ D α + ζ D α . (B.6)By factoring out the v dependence in Q ( n ) ( z, v ), a new superfield Q [ n ] ( z, v ) ∝ Q ( n ) ( z, v ) isdefined as D [2] α ( ζ ) Q [ n ] ( z, ζ ) = 0 , Q [ n ] ( z, ζ ) = X k ζ k Q k ( z ) , (B.7)where Q k ( z ) are standard N = 3 superfields subject to the constraints. Then the action (B.4)reduces to the following form, S = 12 πi I γ dζζ Z d xd θ L [2] ( z, ζ ) (cid:12)(cid:12) θ =0 , (B.8)where we have used (B.6) and the constraint (B.7). Here the symbol | θ =0 means that thesuperfields in the Lagrangian are projected on the N = 2 superspace. Performing the ζ integration, we obtain the action in the standard N = 2 superspace. B.2 N = 4 projective superspace For the N = 4 projective superspace, we introduce a pair of C P [6]. The complex projectivespaces C P L × C P R are parametrized by the homogeneous complex coordinates v L = ( v i ) , v R =( v ¯ k ) and u L = ( u i ) , u R = ( u ¯ k ). They satisfy the completeness relation (B.1) independently.The N = 4 supercovariant derivatives are defined by D (1)¯ kα = v i D i ¯ kα , D ( − kα = 1( v L , u L ) u i D i ¯ kα ,D (1) iα = v ¯ k D i ¯ kα , D ( − iα = 1( v R , u R ) u ¯ k D i ¯ kα . (B.9)In the N = 4 case, one introduces the left and right projective superfields with weight n independently. They are defined by D (1)¯ kα Q ( n ) L ( v L ) = 0 , Q ( n ) L ( cv ) = c n Q ( n ) L ( cv ) ,D (1) iα Q ( n ) R ( v R ) = 0 , Q ( n ) R ( cv ) = c n Q ( n ) R ( cv ) , c ∈ C ∗ . (B.10)12ince the left and right parts have almost the same property, we focus on the left part in thefollowing. We introduce the complex inhomogeneous coordinate ζ L by v i = v (1 , ζ L ) , ζ L = v v . (B.11)Then the supercovariant derivative becomes D (1)¯ kα = v D [1]¯ kα , D [1]¯ kα = D kα − ζ L D kα . (B.12)As for the N = 3 case, the v dependencies of the projective superfields can be factored outand one can define a new field Q [ n ] L ∝ Q ( n ) L which satisfies the following condition, D [1]¯ kα ( ζ ) Q [ n ] L = 0 , Q [ n ] L ( z ′ , ζ L ) = X k ζ kL Q k ( z ′ ) , (B.13)where Q k ( z ′ ) are the standard N = 4 superfields subject to the constraint (B.10).The manifestly N = 4 superconformal invariant action is given by S = 12 π I γ L ( v L , dv L ) Z d x D ( − L L (2) L ( z ′ , v L ) | θ =0 + 12 π I γ R ( v R , dv R ) Z d x D ( − R L (2) R ( z ′ , v R ) | θ =0 , (B.14)where L (2) L ( L (2) R ) is a left (right) projective superfield with weight 2. The integration measuresare defined by D ( − L = 148 D ( − k ¯ l D ( − k ¯ l , D ( − k ¯ l = D ( − α ¯ k D ( − α ¯ l ,D ( − R = 148 D ( − ij D ( − ij , D ( − ij = D ( − αi D ( − αj . (B.15)The contour γ L ( γ R ) is chosen such that the path goes the outside of the north pole in C P L ( C P R ). After fixing u i = (1 , u ¯ k = (1 ,
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