Non-Abelian Chern-Simons Actions in Three-dimensional Projective Superspaces

We construct an action for the superconformal Chern-Simons theory with non-Abelian gauge groups in three-dimensional N=3 projective superspace. We propose a Lagrangian given by the product of the function of the tropical multiplet, that represents the N=3 vector multiplet, and the O(-1,1) multiplet. We show how the tropical multiplet is embedded into the O(-1,1) multiplet by comparing our Lagrangian with the Chern- Simons Lagrangian in the N=2 superspace. We also discuss N=4 generalization of the action.


Introduction
Off-shell superfield formalisms of supersymmetric gauge theories with eight supercharges have been 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 superspace formalism [4,5].
Recently, the projective superspace formalism that keeps manifest N = 3 and N = 4 superconformal symmetries in three dimensions has been developed [6]. This results open up a window 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-Simons actions [7,18] while it had not been constructed in three-dimensional projective superspace formalisms. In [9], we have constructed the N = 3 and N = 4 superconformal Chern-Simons actions with Abelian gauge group in the projective superspace. We have shown that the N = 3 Abelian Chern-Simons Lagrangian is written in the product of the tropical multiplet representing an Abelian vector multiplet V [0] with weight 0 and an O(−1, 1) multiplet G [2] with weight 2 corresponding to the field strength of the vector multiplet 1 . The N = 3 vector multiplet consists of a (anti)chiral superfield Φ (Φ) and a vector superfield V in the N = 2 standard superfield 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 multiplet into the tropical multiplet. This is the key point to obtain the N = 3 superconformal Chern-Simons actions in the projective superspace formalism since the action is constructed so that 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 vector multiplets in terms of N = 2 superfields have not been studied in detail. This is because the relation between them becomes non-linear and complicated. Formal treatments of non-Abelian vector 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 hypermultiplets that couple with the non-Abelian vector multiplet and found the relations among the non-Abelian vector multiplet (V, Φ,Φ) and the N = 3 tropical multiplet V [0] . We have explicitly 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. We also constructed the actions of hypermultiplets coupled with the non-Abelian vector multiplet in the three and four-dimensional projective superspaces.
The purpose of this paper is to construct the non-Abelian Chern-Simons action in the three dimensional N = 3 projective superspace. Although the action has been discussed in the harmonic superspace formalism, our study provides a complimentary analysis of the non-Abelian Chern-Simons theory in the projective superspace formalism. The Lagrangian is written 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 to the O(−1, 1) multiplet. We will show that G [2] actually satisfies the condition of O(−1, 1) multiplet. We also find the explicit form of the function f which is consistent with the Abelian limit. Finally we will show that the action proposed in this letter completely reproduces the known action for the non-Abelian Chern-Simons theory in the N = 2 standard superfield 1 A brief introduction of the projective superfield formalism is found in Appendix B.

formalism.
The organization of this letter is as follows. In the next section, we give a brief review about the embedding of the N = 3 non-Abelian vector multiplet into the tropical multiplet in the projective superspace formalism. We work in the formalism that the superconformal symmetry is manifest [6]. In Section 3, we propose the non-Abelian Chern-Simons action in the N = 3 projective superspace. We show that the action in the projective superspace reproduces that in the N = 2 superspace. We also discuss the N = 4 generalization of the action. Section 4 is conclusion and discussions. Notations and conventions of the three-dimensional N = 2, 3, 4 ordinary superspaces are given in Appendix A. A brief summary of the three-dimensional N = 3 and the N = 4 projective superspace formalisms are found in Appendix B.

Non-Abelian vector multiplet in projective superspace
In this section, we give a brief overview about non-Abelian vector multiplets in the projective superspace formalism. A summary of the three-dimensional N = 3 and N = 4 projective superspace 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] with weight 0. In the Lindström-Roček gauge, V [0] is expanded by the projective coordinate ζ parameterizing CP 1 as [10] where z is the N = 3 standard superspace coordinate defined in Appendix A. The component superfields V −1 , V 0 , V 1 are adjoint representation of a non-Abelian gauge group. The tropical multiplet satisfies the projective superfield constraint D [2] α V [0] = 0. This is equivalent to the following constraints on the component superfields: Following the analysis in four dimensions [5], we decompose the tropical multiplet as follows: whereV + (V − ) contains only positive (negative) powers of ζ andV 0 contains terms with ζ 0 . In general, they are expanded by ζ aŝ whereV ±n andV 0 are functions of V ±1 , V 0 . Finding the closed expressions of the functionŝ V ±n ,V 0 in terms of the components in the tropical multiplet V [0] is difficult in the projective superspace formalism. However they are obtained perturbatively. Up to O(V 4 ),V ±n ,V 0 are found to be,V , We note that in the Abelian limit, these components becomeV ±1 = V ±1 ,V 0 = V 0 andV ±n = 0 (n > 2). As we will see in below, even for the non-Abelian gauge group, the componentŝ V n (n > 2),V −n (n > 3) do not appear in the Lagrangian and they are not relevant to our discussion.
In order to express the N = 3 vector multiplet (V, Φ,Φ) in terms of the component su- , we consider the action for hypermultiplets that couples to tropical multiplets. The hypermultiplets charged under the non-Abelian gauge group are embedded in the (ant)arctic multiplets Υ [1] (Ῡ [1] ) with weight 1, which are expressed as They also satisfy the projective superfield constraints D [2] α Υ [1] = D [2] αῩ [1] = 0. The Lagrangian for the hypermultiplet in the fundamental representation of the non-Abelian gauge group is originally written in the N = 3 superspace. After choosing a frame where the isospinor u is fixed, the Lagrangian becomes (see (B.8)): where the contour γ is chosen such that it does not through the north pole of CP 1 . We can reduce this Lagrangian to that in terms of N = 2 superfields. Below we briefly explain how to reduce it and how to find the relation between the component superfields and the N = 2 superfields (V, Φ,Φ). First we define the following new fields which satisfy the gauge-covariantized projective superfield constraints: Here D [2] α is the gauge covariant derivative defined to be where the gauge connections are defined as [13] Γ The algebra that the gauge covariant derivatives D α , D 12 α ,D α satisfy was calculated in [13]. With the use of (2.8), (2.7) is simplified to be This is a convenient form to write down the Lagrangian in terms of the N = 2 superfields sinceΥ [1] andῩ [1] can be expanded as the same way in (2.6): One finds the expanded forms by replacing Υ n withΥ n and so on. Substituting their expanded forms into (2.13) and integrating over ζ, we are left withΥ 0 andΥ 1 in the Lagrangian. The other fieldsΥ n (n ≥ 2) are integrated out since they are non-dynamical. Equation (2.9) gives a constraint forΥ 1 and it is incorporated into the Lagrangian with the Lagrange multiplier N = 2 superfield. Then integratingΥ 1 and going back to the original fields without tilde, one obtains the Lagrangian in terms of Υ 0 , the Lagrange multiplier which is denoted as Y 0 ,V −1 ,V 0 andV 1 . Now we consider the N = 3 gauge interacting Lagrangian in terms of the N = 2 superfield language. The charged hypermultiplet consists of two chiral superfields (S, T ) and they couple with the non-Abelian vector multiplet (V, Φ,Φ). The Lagrangian is given by (2.14) We can directly compare this with the Lagrangian in terms of Υ 0 , Y 0 ,V −1 ,V 0 andV 1 coming from (2.7). In [13] we found that S = Υ 0 , T =D 2Ȳ 0 eV 0 are chiral superfields andȲ 0 =Ȳ 0 eV − and In [13] we have checked that Φ (Φ) defined in the above actually satisfies the (anti)chirality conditionD α Φ = D αΦ = 0. We note that in the Abelian limit, we reproduce the correct expression found in [9] In the next section, we propose the supersymmetric N = 3 and N = 4 Chern-Simons Lagrangians by utilizing the above results.

Chern-Simons Lagrangian
In this section, we construct a Lagrangian for the non-Abelian Chern-Simons theory in the projective 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] Here k is the Chern-Simons level and t is an auxiliary integration variable. The first term can be rewritten as We first start from the N = 3 Abelian Chern-Simons theory. When the gauge group is Abelian, the Lagrangian (3.1) is reduced to the following form: We have constructed the Lagrangian in the projective superspace that reproduces the expression where the gauge invariant O(−1, 1) multiplet G [2] with weight 2 is a function of the tropical multiplet V [0] and is expanded as Each component should satisfy the projective superfield condition D [2] α G [2] = 0, namely, Taking account of the gauge invariance of G [2] , these constraints are solved by This is a gauge-fixed form of the relation found in [15]. It is easy to check that Φ 0 (Φ 0 ) 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 = 3 projective superspace. For non-Abelian gauge groups, the embedding of the tropical multiplet into the O(−1, 1) multiplet (3.7) becomes non-linear. We assume that the Lagrangian is given by the product of a function of V [0] and an O(−1, 1) multiplet G [2] even for non-Abelian gauge groups. Since the Lagrangian should reproduce the result (3.4) in the Abelian limit, we propose the following non-Abelian Chern-Simons Lagrangian in the N = 3 projective superspace: where f is a projective superfield with weight 0 which is a function of the tropical multiplet V [0] and G [2] = i ζ Φ 0 + L + iζΦ 0 is an O(−1, 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 gauge invariance of the action is not manifest, the component expression of the action will ensure it. In the Lindström-Roček gauge, the ζ expansion of the function f is generically given by 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 to that in the N = 2 superspace, We look for the explicit forms of the components (f −1 , f 0 , f 1 ) and (Φ 0 , L,Φ 0 ) that reproduce the action (3.1) in the N = 2 superspace. First, we identify Φ 0 ,Φ 0 with the non-Abelian adjoint (anti)chiral superfields Φ,Φ defined in (2.15) (with extra overall factors ±i): Then, using the constraints (2.2), we can show that the above definition satisfies a part of the projective superfield constraints of G [2] , D αΦ0 =D α Φ 0 = 0 [13]. Next, we rewrite parts of the D-terms in the Lagrangian (3.10) to F-terms: where we have used the fact thatD α Φ = D αΦ = 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 the components L as and f 0 as When we employ the expression (3.2) instead of (3.1), we have The two expressions of L, (3.14) and (3.16), are physically equivalent. Since L is a component of the O (−1, 1) multiplet G [2] , it should satisfy the constraint D 2 L =D 2 L = 0. We examine this 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) 2 . The expression (3.14) is manifestlȳ D-exact form. Therefore we find that the first constraint is satisfied trivially: On the other hand, the second condition D 2 L = 0 is not manifest. We examine the second condition by perturbative calculations. We concentrate on the next leading order in V where the non-Abelian property begins to appear for the first time 3 . Up to O(V 3 ), we have For non-Abelian gauge groups, the N = 2 vector superfield V =V 0 is perturbatively expressed asV Using the constraints for the tropical multiplet V [0] , we find (3.20) Then up to O(V 3 ), we find that the expression (3.14) satisfies the constraints D 2 L =D 2 L = 0. Therefore, all the components (Φ 0 , L,Φ 0 ) are correctly embedded into the O(−1, 1) multiplet G [2] . Finally, we look for expressions of the functions f 1 , f −1 . Comparing the second and the third terms in (3.13) with the F-terms in the Lagrangian (3.1), we find the following relations, We need to solve f −1 , f 1 in the above relations. Since the anti-chiral superfieldΦ (3.12) is shown to be D 2 -exact form [13] On the other hand, the chiral superfield Φ (3.11) is not manifestlyD 2 -exact. Again, we calculate the function f 1 by perturbation in V . From (3.11) we have Using the projective superspace constraints of the tropical multiplet, we find that the chiral superfield is rewritten as the D 2 -exact form up to O(V 3 ) calculation, Then, the function f 1 is determined to be R ], (3.27) where the function f is the same one found in the N = 3 model. The O(−1, 1) multiplet of the left sector G [2] L is the function of the right tropical multiplet V [0] R and vice versa: In order to be consistent with the Abelian limit [9], we take the each component in G [2] L,R as the same one in the N = 3 case. Here the superfields ( R as in the N = 3 case. Performing the ζ L , ζ R integration, we find This Lagrangian contains two gauge fields and mixing interactions between the left and right multiplets. This kind of theory is known as the BF-theory [16,17]. The N = 4 supersymmetric BF-theory is discussed in the harmonic superspace formalism [18].

Conclusion and discussions
In this letter we have studied the N = 3 and N = 4 non-Abelian Chern-Simons actions in the three-dimensional projective superspaces. We work in the projective superspaces where the superconformal symmetry is manifest. The N = 3 and N = 4 vector multiplets are defined by the tropical multiplet V [0] with weight 0. The relations among the component superfields and the vector multiplet (V, Φ,Φ) in the N = 2 superspace are quite nonlinear for non-Abelian gauge groups. The explicit relations among them are found in our previous paper [13]. In this letter, using the explicit relations of the component superfields, we propose the Lagrangian . The Lagrangian (3.8) has the correct Abelian limit [9]. We demonstrated that the proposed Lagrangian (3.8) successfully reproduces the N = 3 non-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 on the perturbation, they are not trivial even in the next leading order in V . Moreover, the very suggestive 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] in this letter. For an Abelian gauge group, the O(−1, 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 is challenging 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 = 6 ABJM action [19] in the projective superspaces 4 . The gauge field part of the ABJM model is the U(N) × U(N) Chern-Simons model with opposite Chern-Simons level (k, −k). We can easily 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 rewritten as the sum of the two Chern-Simons terms constructed by the two vector superfields V and V ′ with opposite Chern-Simons level (k, −k).
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 and right 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 superspace is also interesting. We will come back to these issues in the future works.
The Grassmann measure of integration in the N = 2 superspace is defined by They are normalized such that, For an N = 2 superfield F (x, θ,θ), the following relation holds within the spacetime integration, The chiral and anti-chiral coordinates are defined by The N = 3 superspace coordinates are defined by z = (x m , θ α ij ) where i = 1, 2 is the SU(2) R Rsymmetry spinor index and the Grassmann coordinate satisfies the reality condition θ α ij = θ αij . The SU(2) R spinor indices and the SO(3) R vector indices are intertwined by the relation θ α ij = (τ I ) ij θ α I . 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 (A.6) The N = 4 superspace coordinates are defined by z ′ = (x m , θ α ij ) where i = 1, 2 andj = 1, 2 are indices for the SU(2) L × SU(2) R subgroup of SO(4) R R-symmetry and the Grassmann coordinate satisfies the reality condition θ α ij = θ αij . The supercovariant derivatives in the N = 4 superspace are defined by We use the following relations among the N = 2, N = 3 and N = 4 superspaces [6]:

B Projective superspace formalisms
In this section, we summarize conventions and notations of the projective superspaces in three dimensions. 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 completeness relation, The supercovariant derivative in the projective superspace is defined as A projective superfield Q (n) with weight n is defined by 3) The N = 3 superconformal invariant action is where L (2) is a real superconformal projective superfield with weight 2. The line integral is evaluated over a closed contour γ in CP 1 . Since the action (B.4) is independent of u, we can choose a frame where u i = (1, 0). We take the contour γ in (B.4) such that it does not pass through the north pole v i = (0, 1). We then introduce a complex inhomogeneous coordinate ζ ∈ C in the upper hemisphere of CP 1 , Then the supercovariant derivative D (2) α is rewritten as where Q k (z) are standard N = 3 superfields subject to the constraints. Then the action (B.4) reduces to the following form, where we have used (B.6) and the constraint (B.7). Here the symbol | θ 12 =0 means that the superfields 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 CP 1 [6]. The complex projective spaces CP 1 L × CP 1 R are parametrized by the homogeneous complex coordinates v L = (v i ), v R = (vk) and u L = (u i ), u R = (uk). They satisfy the completeness relation (B.1) independently. The N = 4 supercovariant derivatives are defined by (B.9) In the N = 4 case, one introduces the left and right projective superfields with weight n independently. They are defined by Since the left and right parts have almost the same property, we focus on the left part in the following. We introduce the complex inhomogeneous coordinate ζ L by Then the supercovariant derivative becomes As for the N = 3 case, the v 1 dependencies of the projective superfields can be factored out and one can define a new field Q 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 where L The contour γ L (γ R ) is chosen such that the path goes the outside of the north pole in CP 1 L (CP 1 R ). After fixing u i = (1, 0), uk = (1, 0) in CP 1 L and CP 1 R , the action (B.14) is rewritten in the N = 2 superspace: where the symbol | θ ⊥ =0 means that the superfields in the Lagrangian are projected on the N = 2 superspace.