N=6 superconformal gravity in three dimensions from superspace

A unique feature of N=6 conformal supergravity in three dimensions is that the super Cotton tensor W^{IJKL} can equivalently be viewed, via the Hodge duality, as the field strength of an Abelian vector multiplet, W^{IJ}. Using this observation and the conformal superspace techniques developed in arXiv:1305.3132 and arXiv:1306.1205, we construct the off-shell action for N=6 conformal supergravity. The complete component action is also worked out.

we make use of a unique property of the N = 6 case to construct the corresponding off-shell action from superspace. Upon reduction to components, our action coincides with that found a week ago by Nishimura and Tanii [9] who used completely different techniques.
This paper is organized as follows. In section 2 we briefly review N = 6 conformal supergravity in superspace and then describe the unique feature of N = 6 conformal supergravity. In section 3 we make use of this property to construct the off-shell action for N = 6 conformal supergravity, both in superspace and in terms of the component fields. The main results and their implications are discussed in section 4.

N = 6 conformal supergravity
We begin with a brief review of N = 6 conformal superspace following [2]. After that we describe the unique feature of N = 6 conformal supergravity mentioned in the introduction.
We consider a curved three-dimensional N = 6 superspace M 3|12 , parametrized by local bosonic (x m ) and fermionic coordinates (θ µ I ), z M = (x m , θ µ I ), where m = 0, 1, 2, µ = 1, 2 and I = 1, · · · , 6. The N = 6 conformal superspace [2] is obtained by gauging the N = 6 superconformal algebra osp(6|4, R) and then imposing appropriate constraints. The covariant derivatives have the form Here E A = E A M (z)∂ M is the inverse supervielbein, M ab are the Lorentz generators, N IJ are generators of the SO(6) group, D is the dilatation generator and K A = (K a , S I α ) are the special superconformal generators. 2 The complete set of the generators of osp(6|4, R) consists of Xã = (P A , X a ), where P A = (P a , Q I α ) are the super-Poincaré generators.
The Lorentz generators obey The SO (6) and dilatation generators obey The special conformal generators K A transform under Lorentz and SO (6) transformations as while under dilatations as Among themselves, the generators K A obey the algebra Finally, the algebra of K A with ∇ A is given by The remaining (anti-)commutators are not essential here and may be found in [2].
Under the supergravity gauge group, the covariant derivatives transform as where K denotes the first-order differential operator Covariant (or tensor) superfields transform as The covariant derivatives obey the (anti-)commutation relations of the form where T is the torsion, and R(M), R(N), R(D), R(K) are the curvatures.
The algebra of covariant derivatives corresponding to N = 6 conformal supergravity is is the super Cotton tensor, which is a completely antisymmetric primary superfield of dimension 1, It satisfies the Bianchi identity 14) The N = 6 case is special because it has the important property that the Hodge dual of the Cotton tensor satisfies the Bianchi identity for the field strength of an Abelian N = 6 vector multiplet 3 Therefore, associated with the N = 6 Weyl multiplet is a uniquely defined Abelian N = 6 vector multiplet. 3 The N -extended vector multiplet in conformal superspace is described in [2].
As a result we can use W IJ to define a closed two-form It is the field strength of the vector multiplet. We associate with the field strength F a gauge one-form A, It is the existence of this gauge one-form which distinguishes the N = 6 case from the N < 6 cases.

Conformal supergravity action
In this section, we start by recalling the method to construct the N < 6 conformal supergravity actions employed in [1]. After that, we present a generalization of the method that is suitable in the N = 6 case.

Construction of N < 6 conformal supergravity actions
The idea of the method employed in [1] is to look for two solutions, Σ CS and Σ R , to the superform equation and Γãb denotes a properly normalized Cartan-Killing metric of the N -extended superconformal algebra; the explicit form of R 2 is One solution is always the Chern-Simons form Σ CS which exists for any N . Its explicit form is found to be correspond to the Riemann and SO(N ) curvature tensors. The other solution is the so-called curvature induced form Σ R such that its components are constructed in terms of the super Cotton tensor and its covariant derivatives. It turns out that Σ R exists for N < 6 (in the cases N = 1 and N = 2, Σ R vanishes), see [1] for more details.
The difference is a closed three-form which may be used to construct a locally supersymmetric action This action principle is not applicable for N ≥ 6 since Σ R does not exist. However, in the N = 6 case there is a way out and that is to make use of the closed two-form (2.17) constructed in terms of the vector multiplet field strength W IJ , eq. (2.15).

Modified N = 6 curvature induced form
In the N = 6 case, we can modify the superform equation where A is some constant we will determine. Here F is the closed two-form (2.17). The Chern-Simons solution is now modified to We may now attempt to find a covariant solution to eq. (3.8) which will also be called the curvature induced form and denoted Σ R . We make the ansatz for the lowest components 5 with A and B some constants to be determined, and turn to analyzing the superform equation (3.8) by increasing dimension.
At the lowest dimension we find that we must set A = −2 and The higher dimension components are found to be To derive these results, we have made use of the following consequences of the Bianchi identity (2.16): Now, the closed three-form J = Σ CS − Σ R generates a locally supersymmetric action according to the rule (3.7).

The component action
The complete component analysis of the N -extended Weyl multiplet was given in [1]. Here we specialize to the N = 6 case where the auxiliary fields coming from the 5 When referring to components of the curvature induced form we denote Σ R by Σ to avoid awkward notation.
super Cotton tensor are defined as: These definitions agree with [10,11]. There is also an additional component field X αβ I 1 ···I 6 := i∇ However, this field turns out to be a composite object as it is the component U(1) field strength F ab up to contributions involving the gravitino: As the action is invariant with respect to the gauge transformations (2.8) up to a total derivative, it follows that the dependence on b m must drop out. Equivalently, we can simply adopt the K-gauge b m = 0. Using the action (3.7) and the Chern-Simons form (3.9), we find the Chern-Simons contribution to be where the component curvatures R ab cd and R ab IJ are defined as for the component projection of a three-form along with the explicit expressions for the components of Σ ABC , we find where we have used the relations Combining this result with the Chern-Simons contribution gives the full action Our choice of normalization for the auxiliary fields allows a simple truncation to lower values of N . From the above action one can truncate the auxiliary fields to N = 5 by taking (with I, J, K = 1, 2, 3, 4, 5) For the gauge fields one must switch off the U(1) gauge field A b −→ 0, while truncation is obvious for the other gauge fields. The N < 5 cases can be obtained via the truncation procedure given in [1].

Discussion
Our component action for N = 6 conformal supergravity (3.20) agrees with that derived recently in [9], where alternative techniques were used involving the consistent truncation of the off-shell multiplet of N = 8 conformal supergravity [12]. It also correctly reduces to the action for N = 5 conformal supergravity [1] via the truncation procedure (3.21). Eliminating the auxiliary fields is equivalent to removing the last three lines in (3.20). The resulting on-shell action for N = 6 conformal supergravity does not agree with that obtained in [13,14] by gauging the N = 6 superconformal algebra in x-space (the action given in [13,14] does not contain the U(1) Chern-Simons term). Instead it coincides with the action given in [15].
In conclusion, we comment on the structure of a supercurrent multiplet associated with a superconformal matter theory coupled to N -extended conformal supergravity (see also [2]). In general, the supergravity-matter system is described by an action of the form where S CSG denotes the conformal supergravity action and S matter the matter action. The conformal supergravity equation is Here W is the N -extended super Cotton tensor (with its indices suppressed) and T the matter supercurrent multiplet. The supercurrent has the same algebraic type as W and obeys the same differential constraints W is subject to. For any N , the super Cotton tensor (and also the supercurrent) is a conformal primary superfield, with ∆ W the dimension of W . We now recall the structure of W for various values of N following [2].
The N = 1 super Cotton tensor [16] is a completely symmetric spinor W αβγ of dimension 5/2. It obeys the conformally invariant constraint (4.4) In the N = 2 case, the super Cotton tensor [17,18] is a completely symmetric spinor W αβ of dimension 2. The corresponding conformally invariant constraint is In the N = 3 case, the super Cotton tensor is a symmetric spinor W α of dimension 3/2. It obeys the conformally invariant constraint For N > 5, the super Cotton tensor [5,6] is a completely antisymmetric tensor W IJKL of dimension 1. It obeys the conformally invariant constraint In the N = 4 case, the super Cotton tensor is equivalently described by a scalar primary dimension-1 superfield W IJKL := ε IJKL W . The corresponding conformally invariant constraint is As we have demonstrated, the specific feature of the N = 6 case is that the super Cotton tensor is equivalent to the U(1) vector multiplet field strength (2.15). Therefore, the N = 6 supercurrent T IJ has the same multiplet structure. This agrees with the Nishimura-Tanii analysis [9] of the supercurrent of the ABJM model [19] coupled to conformal supergravity.

A The supersymmetry transformations
In this appendix we present the complete Q-and S-supersymmetry transformations for the component fields of the Weyl multiplet for N = 6. The component action (3.20) is manifestly supersymmetric by virtue of our superspace construction. We refer the reader to [1] for details on the component projection rules and the precise definition of the gauge component fields.