Supergravity-matter actions in three dimensions and Chern-Simons terms

We study off-shell N-extended Yang-Mills multiplets coupled to conformal supergravity in three spacetime dimensions. Superform formulations are presented for the non-Abelian Chern-Simons actions in the cases N=1, 2, 3, and the corresponding component actions are explicitly worked out. Such a Chern-Simons action does not exist for N=4. In the latter case, a superform formulation is given for the BF term that describes the coupling of two Abelian vector multiplets with self-dual and anti-self-dual superfield strengths respectively. The superform results obtained are used to construct linear multiplet action principles in the cases N=2, 3, 4. The N=3 and N=4 actions are demonstrated to be universal in the sense that all known off-shell supergravity-matter systems (with the exception of pure conformal supergravity) may be described using such an action. Starting from the N=3 and N=4 Abelian vector multiplets, we also construct composite O(2) multiplets which are analogues of the four-dimensional construction of an N=2 reduced chiral scalar engineered from the improved tensor multiplet. Using these composites, we present the superfield equations of motion for N=3 and N=4 anti-de Sitter and topologically massive supergravity theories. We also sketch the construction of a large family of higher derivative couplings for N=3 and N=4 vector multiplets.


Introduction
The linear multiplet 1 plays an important role in the context of matter-coupled supergravity theories with eight supercharges in four, five and six dimensions. There are two 1 In four-dimensional N = 2 Poincaré supersymmetry, the linear multiplet was introduced by Sohnius [1] as a superfield Lagrangian for the matter hypermultiplet [2] coupled to the Yang-Mills vector multiplet [3]. The linear multiplet action was generalized to N = 2 supergravity by Breitenlohner and Sohnius [4], and then reformulated by de Wit, van Holten and Van Proeyen [5] within the N = 2 superconfomal tensor calculus [6,7,8], see [9] for a recent review. The linear multiplet actions, and their use, in fivedimensional N = 1 and six-dimensional N = (1, 0) supergravity theories were described in [10,11,12] and [13] respectively. reasons for the significance of this representation of supersymmetry that can be attributed to its possible realizations as: (i) a dynamical multiplet; or (ii) a composite multiplet. In the first realization, the linear multiplet without central charge [14] (nowadays, often called the O(2) multiplet [15,16]) provides a dual off-shell formulation for the massless hypermultiplet, in which one of the four physical scalars of the hypermultiplet is dualized into a gauge (d − 2)-form in d dimensions. In the d = 4 case, the O(2) multiplet describes the field strength of the massless N = 2 tensor multiplet [17,8]. In the second realization, which is most relevant for this paper, the linear multiplet takes on the role of a Lagrangian for a locally supersymmetric action [4,5]. This action principle turns out to be universal in the sense that it supports general off-shell supergravity-matter theories. 2 Different theories correspond to different composite linear multiplets. In this paper we present three-dimensional (3D) analogues of the linear multiplet action.
The linear multiplet action actually involves two building blocks: an Abelian vector multiplet and a linear multiplet, the latter with or without central charge (no central charge is possible in six dimensions). The vector multiplet is dynamical and modelindependent. The linear multiplet is composite and contains all the information about the dynamical system under consideration. Within the superconformal tensor calculus, the action is formulated in terms of the component fields [5], which is useful for many applications. However, this component approach obscures a geometric origin of the action. On the other hand, the action acquires a simple geometric interpretation as a supersymmetric BF term when formulated in curved 4D N = 2 harmonic superspace [20] (as an extension of the rigid supersymmetric construction given in [21]) or, in the case of the linear multiplet without central charge, in curved 4D N = 2, 5D N = 1 and 6D N = (1, 0) projective superspaces [18,22,23]. 3 From the viewpoint of x-space practitioners, a disadvantage of these superspace approaches is that some work is required in order to reduce the action to components. Recently, there has appeared a new formulation for the 4D N = 2 linear multiplet action [28] that combines the advantages of both the superconformal tensor calculus and the powerful superspace techniques. It made use of 4D N = 2 conformal superspace [29] in conjunction with the superform approach to the construction of supersymmetric invariants [30,31,32,33].
The superform formulation given in [28], and its extension to describe 3D N = 1 2 Its universality may be readily justified in the case of 4D N = 2 supergravity. Within the off-shell and N = 4 topologically massive supergravity. Some implications of our results and open problems are briefly discussed in section 9.
We have included a couple of technical appendices. Appendix A includes some salient facts about the conformal superspace of [40]. In Appendix B we give the supersymmetry transformations for vector multiplets with N ≤ 4. In Appendix C we briefly review covariant projective N = 3 supermultiplets and demonstrate universality of the N = 3 linear multiplet action. In Appendix D we sketch the structure of left and right covariant projective N = 4 supermultiplets and demonstrate universality of the two N = 4 linear multiplet actions.

Vector multiplets in conformal superspace
In this section we show how to describe Yang-Mills multiplets within the superspace formulation of [40], known as conformal superspace. Conformal superspace is based on gauging the entire superconformal algebra. Its essential aspects are summarized in Appendix A.
To describe a Yang-Mills multiplet in the 3D N -extended conformal superspace M 3|2N of [40], parametrized by coordinates z M = (x m , θ µ I ), we introduce gauge covariant derivatives where the gauge parameter τ (z) takes its values in the Lie algebra of G YM .
The gauge covariant derivative algebra is where the torsion and curvatures are those of conformal superspace but with F AB corresponding to the gauge covariant field strength F = 1 2 E B ∧ E A F AB . The field strength F AB satisfies the Bianchi identity (2.4) and must be subject to covariant constraints to describe an irreducible vector multiplet. The structure of the constraints and their consequence is different for N = 1 and for N > 1. Below we describe the various cases.

The N = 1 case
In the N = 1 case, one imposes the covariant constraint [43,44] F αβ = 0 . (2.5) Then one derives from the Bianchi identities the remaining components together with the dimension-2 differential constraint on the spinor field strength Furthermore, the Jacobi identities require G α to be primary and of dimension-3/2: (2.8)

The N > 1 case
For N > 1 one imposes the following dimension-1 covariant constraint [53,45,47] where G IJ is antisymmetric, primary and of dimension-1 G IJ = −G JI , S I α G JK = 0 , K a G IJ = 0 , DG IJ = G IJ . (2.10) These constraints are a natural generalization of the N > 1 constraints in four dimensions [3,54]. The Bianchi identities then give the remaining field strength components:

11a)
(2.11b) The N = 2 case is special because G IJ becomes proportional to the antisymmetric tensor ε IJ G IJ = ε IJ G . (2.12) The components of F AB then become The Bianchi identities imply a constraint on G at dimension-2 (2.14) Unlike for N = 2, in the case N > 2 the field strength G IJ is constrained by the dimension-3/2 Bianchi identity This constraint may be shown to define an off-shell supermultiplet, see e.g. [55,40].

Chern-Simons and curvature induced three-forms
In this section our method to construct supersymmetric Chern-Simons actions is outlined. This method heavily builds on the superform formalism for the construction of supersymmetric invariants [30,31,32,33]. First of all, we sketch its salient points in the framework of 3D N -extended conformal superspace. The formalism makes use of a closed three-form Under an infinitesimal coordinate transformation generated by a vector field ξ = ξ M ∂ M = ξ A E A , the three-form varies as We note that δ ξ J = δ gct J, where δ gct stands for the general coordinate transformation associated with ξ. As discussed in Appendix A, the gauge group of conformal supergravity, G, is generated by two types of transformations: (i) covariant general coordinate transformations, δ cgct , associated with a parameter ξ A ; and (ii) standard superconformal transformations, δ H , associated with a parameter Λ a . The covariant diffeomorphism δ cgct (ξ) is related to the ordinary one δ gct (ξ) by the rule [40] The closed three-form J is required to transform by an exact three-form under the standard superconformal transformations, If we assume the components ξ M and Λ a vanish at spacetime infinity, then we have the supersymmetric invariant Here M 3 denotes the bosonic body of the curved superspace M 3|2N and i : Suitable actions must also be gauge invariant for any additional gauge symmetries of the theory under consideration. If the closed three-form J transforms by an exact three-form under the gauge transformations, then the functional (3.5) is a suitable candidate for an action.
Our method to construct Chern-Simons actions is analogous to the one used in [35,36] to derive the conformal supergravity actions for N ≤ 6. In the super Yang-Mills case, following [35], we will construct a closed three-form J by finding two solutions to the superform equation The first of these solutions is the Chern-Simons three-form Σ CS , It changes by an exact three-form under the Yang-Mills gauge transformation (2.2), It is invariant under the standard superconformal transformations, The other solution, the so-called curvature induced form Σ R , is defined to be such that its components are constructed in terms of the field strength F AB and its covariant derivatives. This three-form is required to be invariant under the Yang-Mills gauge transformations (2.2) and under the the standard superconformal ones, The existence of Σ R is not guaranteed for arbitrary N and crucially depends on the explicit structure of the constraints obeyed by the field strength. If Σ R exists, the properties of Σ CS and Σ R imply that their difference is an appropriate closed three-form that constitutes a supersymmetric action.
We would like to emphasize that the three-form Σ R is required to be conformally invariant, eq. (3.11b). Actually, it turns out that the only non-trivial invariance condition on Σ R is with respect to the special conformal generators K A . It is equivalent to the condition [35] S J β Σ a 1 ···an The above scheme is an example of a known construction where an invariant derived from a closed super d-form can be generated from a closed, gauge-invariant super (d + 1)form provided that the latter is Weil trivial, i.e. exact in invariant cohomology (a concept introduced by Bonora, Pasti and Tonin [56] in the context of anomalies in supersymmetric theories). Examples of this include higher-order invariants in other supersymmetric theories which were studied, e.g., in [57,58].

Non-Abelian curvature induced three-form
We introduce the curvature induced form By covariant we mean that the components Σ ABC are directly expressible in terms of F AB and their covariant derivatives. It should be emphasized that the curvature induced form 7 When referring to the components of the curvature induced form we will use Σ instead of Σ R to avoid awkward notation.
can only exist if the field strength F is constrained in a such a way that eq. (4.1) can be satisfied.
To see this, consider the N > 1 case where one finds at the lowest dimension the condition On dimensional grounds, the most general ansatz to take for Σ R is 8 3) It will turn out that the curvature induced three-form, based on the ansatz (4.3), can only be found for N < 4. It is in these cases that we have for some A and B.
Below we give the solution to eq. (4.1) on a case by case basis.

The N = 2 case
In the N = 2 case, we can replace G IJ with its Hodge-dual: Then we have Using the constraint (2.14) one finds the solution The curvature induced three-form can be shown to obey equation (3.11b).
It is often advantageous to make use of the complex basis for the N = 2 covariant derivatives, see [42,40] for details. In this basis, the field strength is given by where its components are The corresponding curvature induced form (4.8) may be expressed as where the components have the explicit form

The N = 3 case
In the N = 3 case we define the Hodge-dual of G IJ as which implies (4.14) Furthermore, the Bianchi identity (2.15) gives (4.15c) Using the above identities one finds the solution Conformal invariance follows since Σ R obeys equation (3.11b).

The special case of N = 4
In the previous subsections, we found that our approach struck an obstacle at the N = 4 case. In particular, eq. (4.4) no longer holds. Actually, the N = 4 case requires some additional consideration. It is well known that the constraint (2.15) does not define an irreducible off-shell supermultiplet for N = 4. In this case, the Hodge-dual of G IJ , obeys the same constraint as G IJ does, As a result, one may constrain the field strength G IJ to be self-dual, 19a) or anti-self-dual,G These choices correspond to two different off-shell N = 4 vector multiplets, the left and right ones, see [42] for more details. Now, if we consider an irreducible N = 4 vector multiplet obeying either (4.19a) or (4.19b), it may be seen that eq. (4.4) still does not hold. A possible way out is to consider two vector multiplets and a generalization of eq. (3.7) of the form dΣ = tr F 1 ∧ F 2 . However, this poses a problem for non-Abelian vector multiplets, since the two-form field strengths F 1 and F 2 are not gauge invariant; instead, they transform covariantly under the two different gauge groups. 10 In this section, we therefore restrict ourselves to Abelian vector multiplets.
We will consider the general case of two Abelian vector multiplets G IJ + and G IJ − with the two-form field strengths F + and F − respectively. In this case the superform equation (3.7) is replaced by The Chern-Simons solution Σ CS to the above is where V ± are gauge one-forms associated with the two-form field strengths, F ± = dV ± .
The curvature induced three-form Σ R is the covariant solution to the superform equation (4.20) (when it exists). For N = 4 one finds at the lowest dimension of (4.20) the condition The most general ansatz to take for Σ R is which will lead to a solution if 10 It should be mentioned that there is an alternative approach to the problem of constructing the N = 4 Chern-Simons action [60]. It is based on dualizing two scalars in the vector multiplet into vector fields and constructing a theory involving three different vectors! However, as mentioned in [48], such an approach is on-shell and cannot be used to construct matter couplings.
It is easy to see that if we let both G IJ ± be (anti-)self-dual then we cannot satisfy eq. (4.24) for any A and B. However, imposing opposite duality conditions gives us a way out. Taking G IJ + to be self-dual and G IJ − to be anti-self-dual, Using the Bianchi identity (2.15) and the (anti-)self-duality conditions (4.25), one finds the curvature induced form to be One can check that eq. (3.13) holds.
As is known, the group isomorphism SO(4) ∼ = SU(2) L × SU(2) R /Z 2 allows us to convert each SO(4) vector index into a pair of SU(2) spinor ones, for instance ∇ I α → ∇ iī α , see [42] for more details. It is instructive to look at some of the above results in the isospinor notation. The SO(4) bivector G IJ = −G JI is equivalently described by two symmetric second-rank isospinors, G ij and G¯ij, which are defined as and transform under the local groups SU(2) L and SU(2) R , respectively. For the Hodgedual SO(4) bivectorG IJ defined by (4.17), we get The Bianchi identity (4.18a) is equivalent to the two analyticity constraints [42] Thus the field strengths G ij and G¯ij are independent of each other. The (anti-)self-duality conditions (4.25) are equivalent to (4.31b) 11 It is clear that G IJ + G −IJ = 0.

Component actions
In the previous sections we have given a complete superspace description of the Chern-Simons actions for non-Abelian vector multiplets with N < 4 and of the BF action for Abelian vector multiplets in the N = 4 case. In this section we will derive the corresponding component action. To do so we will need to elaborate on the component structure of the theory. For a complete description of the component fields of the Weyl multiplet including their supersymmetry transformations we refer the reader to [35]. Here we outline some of the salient details. These connections turn out to be composite and their expressions are given in [35].
The Weyl multiplet also contains some auxiliary fields for N > 2. In the N = 3 case, there is a single fermionic auxiliary field defined by In the N = 4 case, the Weyl multiplet contains both bosonic and fermionic auxiliary fields, where W denotes the Hodge-dual of W IJKL ,

Vector multiplets in components
The component fields of vector multiplets may be extracted from the field strength G IJ . For N > 1, we define the matter fields as follows 12 The covariant field strength may be constructed from the bar-projection of the two- Making use of the identity and performing a component projection, we find The component fields of the vector multiplet form the following tower [55]: 13 12 The coefficients are chosen so that the N = 1 case may be derived via the higher N cases. 13 The tower is analogous to the one for the N -extended super Cotton tensor [55,35].
The coefficients chosen in eq. (5.6) allow for a straightforward truncation of higher N cases to the lower N ones via a procedure analogous to the one described in [35]. For the N = 1 case we have to switch off all matter fields except with the field strength G α defined in (2.6a).
The N = 4 case is special, since it allows for two inequivalent off-shell vector multiplets with field strengths G IJ + and G IJ − obeying the self-duality condition (4.19a) and the antiself-duality condition (4.19b) respectively. In this case we define the the component fields of the vector multiplets as 14) The component one-forms are given by where V ± is the gauge one-form associated with the field strength G IJ ± . The (anti-)self-duality property of G IJ ± , eq. (4.25), reduces the degrees of freedom for each vector multiplet by half. To see this, it is useful to replace h (±) IJ by the fieldŝ which prove to be (anti-)self-dual The other implications of (anti-)self-duality are Diagrammatically, this means that the components on the left hand side of Figure 1 are related to those on the right hand side via (anti-)self-duality. One can see that each vector multiplet constitutes 8+8 degrees of freedom.

Off-shell component actions
Now we have all the ingredients to construct the component actions corresponding to the closed forms found in the previous sections. To do so we just need to apply the action principle (3.5), and make use of the formula Here we present the resulting actions on a case by case basis.
Although all our actions are automatically supersymmetric, we give the supersymmetry transformations of the component fields in Appendix B.

The non-Abelian N = 1 case
The action is constructed using eqs. (3.8) and (4.5). From eq. (5.20) we find Combining this with the contribution coming from the Chern-Simons form (3.8) gives Using eqs. (5.20) and (4.8) we find Using the above result and incorporating the Chern-Simons form (3.8) gives the action Expressing the above action in terms of the complex basis gives where we have made use of the component fields in the complex basis:

The non-Abelian N = 3 case
Using eqs. (5.20) and (4.16) we find where we have defined Combining this with the contribution coming from the Chern-Simons form (3.8) gives As in [35], our choice of normalization for the component fields allows for a simple truncation to the actions for lower values of N . For example, from the above action one can truncate the auxiliary fields to N = 2 by taking (with I, J = 1, 2) For the fields of the Weyl multiplet one performs a similar truncation, which is given in [35].

The Abelian N = 4 case
Using eqs. (5.20) and (4.27) we find Combining this with the contribution coming from the Chern-Simons form (4.21) gives The results of sections 4 and 5 may be used to generate locally supersymmetric actions. This idea can be illustrated, in a simple and transparent way, by considering the N = 2 case which we discuss below. Unlike in section 5, here we use the complex basis for the N = 2 covariant derivatives, see [42] for details.
Let us consider a locally supersymmetric BF term described by the action Here V =V is the gauge prepotential of an Abelian vector multiplet, and G =Ḡ a real linear superfield, ∇ 2 G =∇ 2 G = 0. 14 The action (6.1) is invariant under gauge transformations with the gauge parameter λ being an arbitrary covariantly chiral dimensionless scalar. Eq. (6.1) defines the N = 2 linear multiplet action.
It turns out that the action (6.1) may be recast in terms of a closed three-form that involves three building blocks. First of all, F = 1 2 E B ∧ E A F AB is a closed two-form, dF = 0, associated with G. Its components are defined as in eqs. (4.9) and (4.10) by and are explicitly given as follows: The second building block, V = E A V A , is the gauge one-form describing the vector multiplet associated with V. Modulo an exact one-form, we can choose the components of V as follows: The corresponding gauge-invariant field strength F = dV has the explicit structure given by eqs. (6.4) and (6.5) with F and G replaced with F and G respectively, where G denotes the gauge-invariant field strength associated with the prepotential V. Finally, the three-form Σ is chosen to obey the equation Its components are defined by and have the following explicit form: The components of Σ are symmetric under the interchange G ↔ G. When G = G we have agreement with eqs. (4.11) and (4.12) in the Abelian case.
Associated with the closed three-form J, eq. (6.3), is the component action where the component fields are defined as in section 5 (in the complex basis): Eq. (6.11) is exactly the component form of the action (6.1).
Let us recall that the most general N = 2 supergravity-matter system (see [42,63] for more details) is described by an action of the form for some real scalar L and covariantly chiral scalar L c Lagrangians,∇ α L c = 0. Here E denotes the chiral density. 15 We assume that the dynamical supermultiplets include an Abelian vector multiplet described by prepotential V with nowhere vanishing field strength G = i∇ α∇ α V. This is the case for Type II minimal supergravity [42,63]. Then, the first term in (6.13) may be represented in the BF -form (6.1), specifically: We see that the linear multiplet action (6.1) allows us to describe a broad class of locally supersymmetric models. However, this action principle is not universal for, in general, it cannot be used to describe the chiral term in (6.13) and its conjugated antichiral one. On the other hand, the (anti)chiral action is truly universal in N = 2 supersymmetry, due to the identity [42] As demonstrated in [65], this action can equivalently be described in terms of a closed three-form Ξ, dΞ = 0, such that its components are expressed viaL c as follows: In summary, the N = 2 linear multiplet action (6.1) is useful but not universal. As will be shown in the next section, the situation in N = 3 supersymmetry is conceptually different.

Matter-coupled N = 3 supergravity
General off-shell matter couplings in N = 3 supergravity were constructed in [42]. Given a supergravity-matter system, its dynamics can be described by a Lagrangian L (2) (v) which is a real weight-two projective supermultiplet, 16 with v i the homogeneous coordinates for CP 1 . The corresponding action is given by eq. (C.14). We assume that the dynamical supermultiplets include an Abelian vector multiplet such that its gauge invariant field strength G ij is nowhere vanishing, that is G := G ij G ij = 0. As shown in Appendix C, the action functional (C.14) can be rewritten as a BF term where V(v) is the tropical prepotential for the vector multiplet, ∇ α V = 0, and is a composite real O(2) or linear multiplet. The explicit expression for G (2) (v) in terms of the superfield Lagrangian L (2) is given by eq. (C.20b). Different theories correspond to different choices of the composite linear multiplet G ij . The action (7.1a) is invariant under gauge transformations where the gauge parameter λ is an arbitrary weight-0 arctic multiplet, andλ its smileconjugate, see [42] for more details. Eq. (7.1a) defines the N = 3 linear multiplet action.
Instead of dealing with the symmetric spinors G ij and G ij , we can equivalently work with the isovectors where the sigma-matrices are defined by

Linear multiplet action
It turns out that the action (7.1a) may be recast in terms of a closed three-form: where F is the two-form field strength associated with The component action generated by J is where the component fields are defined as in section 5. They are explicitly given by To prove that the N = 3 linear multiplet action (7.1a) has the component form (7.7), it suffices to redo, in a 3D setting, the 4D N = 2 analysis given in [66].

Composite O(2) multiplet
We now present a special example of the composite O(2) multiplet defined by (C.20b). We consider a vector multiplet Lagrangian of the form where Υ (1) (v) is a weight-1 arctic multiplet andΥ (1) (v) its smile-conjugated antarctic multiplet. The superfields Υ (1) andΥ (1) are pure gauge degrees of freedom [42]. In the rigid supersymmetric limit, the Lagrangian (7.9) describes a superconformal vector multiplet, which is the 3D N = 3 analogue of the 4D N = 2 improved tensor multiplet [67,68]. 17 With the Lagrangian (7.9), the contour integral in (C.20b) can be evaluated using the techniques of [70]. Alternatively, one may look for a dimension-1 primary superfield that obeys the Bianchi identity (4.15a). The resulting composite O(2) multiplet is where is required to be nowhere vanishing. The O(2) multiplet may be expressed in terms of SU(2) indices as follows

Supercurrent
Before turning to a consideration of specific supergravity models, it is worth giving a few remarks concerning matter couplings to N = 3 conformal supergravity (see also [40,36]). In general, matter-coupled conformal supergravity is described by an action of the form Here S CSG denotes the N = 3 conformal supergravity action [35] and S matter the matter action. The equation of motion for the Weyl multiplet is 18 1 µ W α + T α = 0 , (7.14) 17 The 4D N = 1 improved tensor multiplet was introduced in [69]. The N = 2 construction of [67,68] is a natural extension of the one given in [69]. 18 The coupling constants µ and µ differ from each other by some numerical coefficient.
where T α is the matter supercurrent. As a result the supercurrent T α must have the same properties as the super Cotton tensor W α . Specifically, T α must be a primary superfield of dimension 3/2, and obey the conservation equation The latter holds provided the matter equations of motion are satisfied.
Matter-coupled Poincaré or anti-de Sitter supergravities can also be described by actions of the type (7.13) with 1/ µ = 0. The matter supermultiplets have to include a conformal compensator. In what follows, the latter is assumed to be the vector multiplet described by the field strength G I . The supergravity equation of motion is As an example, consider N = 3 AdS supergravity. It can be described by the Lagrangian [42] L (2) with κ and ξ the gravitational and cosmological constants respectively. The cosmological term is a U(1) Chern-Simons term. The choice ξ = 0 corresponds to Poincaré supergravity. The corresponding supercurrent is One can show that if G I satisfies the equation of motion for V, G I + ξG I = 0, the supercurrent does obey eq. (7.16).

(2,1) anti-de Sitter supergravity
It was discovered by Achúcarro and Townsend [71] that three-dimensional N -extended anti-de Sitter (AdS) supergravity exists in [N /2] + 1 different versions, with [N /2] the integer part of N /2. These were called the (p, q) supergravity theories where the nonnegative integers p ≥ q are such that N = p + q.
We wish to demonstrate that the Lagrangian (7.18) describes (2,1) AdS supergravity. To see this we will degauge, following the procedure described in [40], the corresponding equations of motion, to SO(3) superspace [41,42]. As in [40], the covariant derivatives of SO(3) superspace are denoted D A = (D a , D I α ). Using the results of [40] we can degauge our expression for G I , eq. (7.10), to The covariant derivatives D A no longer contain the dilatation and special conformal generators, The original local dilatation symmetry is now realized in terms of the super-Weyl transformations. In SO(3) superspace, there are two dimension-1 real torsion tensors, S IJ = S JI and C a IJ = −C a JI . The super Cotton tensor W α becomes a descendant of C a IJ , We refer the reader to [40] for more details about the degauging procedure.
Using the super-Weyl transformation of G [42], we can impose the gauge condition G = 1 . Taking a spinor derivative of G then gives Note that the Bianchi identity now simplifies to Since the supercurrent vanishes, eq. (7.20b), we must also have On the other hand, using eq. (7.26) we find Contracting the above with G I and implementing eq. (7.27) tells us that G I is covariantly constant, The fact that G I is covariantly constant strongly constrains the superspace geometry. In particular, we have which fixes the form of the curvature components as The composite vector multiplet now reduces to Due to the equation of motion (7.20a), S is seen to be constant, As a result, the covariant derivatives corresponds to (2, 1) AdS superspace [72]. Therefore the theory (7.18) indeed describes (2,1) AdS supergravity.
Without a cosmological constant, ξ = 0, we find 36) and the resulting geometry corresponds to Minkowski superspace.

Topologically massive supergravity
Topologically massive N = 3 supergravity 19 can be described by the action where S SG corresponds to the supergravity Lagrangian (7.18).
The equation of motion for the Weyl multiplet now becomes compare with (7.20b). The equation of motion for V coincides with (7.20a).
We would like to degauge the equations of motion to SO(3) superspace. Eq. (7.38) tells us that i κG where upon degauging the Cotton tensor is Using the gauge condition G = 1 we find where we have defined the constantμ = µ κ . (7.42) Therefore the Bianchi identity becomes Next, we notice that on the one hand, due to eq. (7.44), we have while on the other we have Combining the two results gives In addition to (7.49a), the Cotton superfield must obey the Bianchi identity Due to the conditions (7.47), the composite vector multiplet may be expressed as follows Furthermore, from the equation of motion (7.20a) we see that S can be expressed in terms of the Cotton tensor as For ξ = 0, a solution of the equations of motion for topologically massive supergravity is obtained by setting W α = 0 in the above relations. This solution describes a flat superspace. Linearizing the equations (7.49) around Minkowski superspace, it may be shown that W α obeys the Klein-Gordon equation with ✷ := ∂ a ∂ a . For ξ = 0, the equations of motion for topologically massive supergravity are solved by setting W α = 0. Locally it describes (2,1) AdS superspace [72].
The off-shell matter couplings in N = 4 supergravity were constructed in [42]. In general, the action for a supergravity-matter system may be represented as a sum of two terms, S = S L + S R , the left S L and right S R actions, which are naturally formulated in curved N = 4 projective superspace M 3|8 × CP 1 L × CP 1 R . The left action is given by eq. (D.12), where the Lagrangian L (2) L (v L ) is a real left projective multiplet of weight two, with v L = v i the homogeneous coordinates for CP 1 L . The structure of S R is analogous. We assume that the dynamical supermultiplets include two Abelian vector multiplets such that their field strengths G IJ + and G IJ − are self-dual and anti-self-dual, respectively, and nowhere vanishing,

Left linear multiplet action
The left linear multiplet action is given by where G L (v L ), can be equivalently realized as the anti-self-dual SO(4) bivector G IJ − . It turns out that the action S LLM may be reformulated in terms of the closed three-form where F is the super two-form associated with G IJ − and V = E A V A is the gauge one-form associated with the field strength G IJ + . The three-form Σ = 1 The corresponding component action is where the component fields are defined as in section 5: The component fields are defined so thatĥ IJ − is anti-self-dual andĥ IJ + is self-dual, see eq. (5.17b).

Right linear multiplet action
The right linear multiplet action is given by where G R (v R ), can be equivalently realized as the self-dual SO(4) bivector G IJ + .
The action S RLM may be reformulated in terms of the closed three-form where F is the two-form field strength associated with G IJ + and V = E A V A is the gauge one-form associated with the field strength G IJ − . The three-form Σ = 1 3! E C ∧E B ∧E A Σ ABC is given by The corresponding component action is where the component fields are defined as in section 5: The component fields are defined so thatĥ IJ + is self-dual andĥ IJ − is anti-self-dual, see eq. (5.17b). and make use of the representation (D.30). 20 The contour integral in (D.30) may be evaluated using the technique developed in [42]. A similar analysis may be used to construct G

Composite O(2) multiplets
L (v L ) may also be found by looking for primary superfields which obey the Bianchi identity The resulting composite O(2) multiplets are given by where we have defined To show that G IJ ± is primary and satisfies the Bianchi identity, the following identities prove useful (8.14b) In the isospinor notation, the composite O(2) multiplets constructed read These expressions may be compared with the 4D N = 2 results in [70]. 20 The arctic weight-1 hypermultiplet Υ (1) L and its smile conjugateΥ For completeness we give the expressions for the composite O(2) multiplets in SO(4) superspace: Here S ijīj = S (ij)(īj) is one of the two irreducible components of the torsion superfield S iī,jj := S IJ (τ I ) iī (τ I ) jj , defined by

Supercurrent
The remainder of this section is devoted to a study of specific supergravity theories. To start with, we would like to discuss the structure of the N = 4 supercurrent (see also [40,36].) Our consideration below is similar to the N = 3 analysis of section 7.3. Consider a dynamical system describing N = 4 conformal supergravity coupled to matter supermultiplets. In general, the supergravity-matter action has the form S = 1 µ S CSG + S matter , (8.18) where S CSG denotes the N = 4 conformal supergravity action [35] and S matter the matter action. The equation of motion for conformal supergravity reads where W is the N = 4 Cotton superfield and T is the matter supercurrent. It follows from this equation that the supercurrent must have the same properties as W . Specifically, T must be a primary superfield of dimension 1, 20) and obey the conservation equation Of course, the latter holds provided the matter equations of motion are satisfied.
Matter-coupled Poincaré or anti-de Sitter supergravities can also be described by an action of the type (8.18) with 1/ µ = 0. The matter supermultiplets have to include two conformal compensators. As before, these are chosen to be two Abelian vector multiplets such that their field strengths G IJ + and G IJ − are self-dual and anti-self-dual, respectively, and nowhere vanishing, G 2 ± := 1 2 G IJ ± G ±IJ = 0. The supergravity equation of motion is As an example, let us consider N = 4 AdS supergravity. it can be described by two Lagrangians, left and right ones, which were chosen in [42] as where κ is the gravitational coupling constant and the parameters ξ L and ξ R determine a cosmological constant. We recall that G L (v L ) and G (2) R (v R ) are the gauge invariant field strengths for V R (v R ) and V L (v L ) respectively. The cosmological term is described by the left and right BF terms in (8.23). It is known [42] that the action does not change if the BF coupling constants are modified as for any real constant a. For the action to be mirror invariant, we have to choose [42] ξ L = ξ R ≡ ξ/2 . With the left and right Lagrangians given by (8.23), the supercurrent is It may be shown that the equations of motion for V L and V R are equivalent to where the composite superfields G IJ ± are defined according to (8.13). Using these equations of motion, one can show that the supercurrent satisfies the conservation equation (8.21).

(2,2) anti-de Sitter supergravity
It turns out that the model (8.23) describes the (2,2) AdS supergravity. We will show this by degauging the equations of motion for the compensators, eq. (8.27), and the equation of motion for the Weyl multiplet, Using the results of [40] we degauge G IJ ± to SO(4) superspace We then use the super-Weyl transformations to impose the gauge condition Taking a spinor derivative of G + gives Then using the (anti-)self-duality condition The above tells us that G IJ + is covariantly constant Since the supercurrent vanishes (eq. (8.28)) we have Similarly we deduce that G IJ − is covariantly constant The covariant constancy of G ± IJ has immediate consequences on the superspace geometry. In particular, we have which gives The above leads to the constraints Eq. (8.14a) and eq. (8.39a) tells us that S IJ takes the form The composite O(2) multiplets now reduce to Combining the above result with the equation of motion (8.27) gives Then making use of eq. (8.14a) fixes the form of S IJ as follows Therefore S IJ must be covariantly constant The above geometry corresponds to (2, 2) AdS superspace [72]. To see this, we rewrite S IJ in the isospinor notation As a result, the algebra of covariant derivative coincides with that for (2, 2) AdS superspace [72]. 21 When ξ = 0 the covariant derivative algebra corresponds to that of N = 4 Minkowski superspace.

Topologically massive supergravity
Topologically massive N = 4 supergravity can be described by the action where the left S SG, L and right S SG, R actions correspond to the supergravity Lagrangians (8.23). Now, the supercurrent is non-zero, since the equation of motion for the Weyl multiplet is whereμ = µ/κ. We choose again the super-Weyl gauge condition (8.30), G + = 1. Then using the (anti-)self-duality condition (8.32) we find that G IJ + is covariantly constant Following similar reasoning as in the last subsection, we derive the constraints The anti-self-dual composite O(2) multiplet, G IJ − , becomes Using the equation of motion we find the form of S IJ to be Taking a spinor derivative of G − and using eq. (8.48) and the anti-self-duality condition (8.32) gives which, due to the Bianchi identity (8.12), lead to Upon degauging to SO(4) superspace and using eq. (8.54), we find that the composite vector multiplet G IJ + may be expressed as Then the equation of motion G IJ + + ξG IJ It follows that the equation of motion on W is with ✷ := ∂ a ∂ a .

Concluding comments
In this paper we have worked out the linear multiplet action principles in N = 3 and N = 4 conformal supergravities. At the component level, the N = 3 action is given by eq. (7.7), while the N = 4 action is a sum of the left and right sectors, given by eqs. (8.4) and (8.9) respectively. Using these locally supersymmetric actions, it is not difficult to construct the component off-shell actions for the (2,1) and (2,2) AdS supergravities and their topologically massive extensions. For instance, the composite O(2) multiplet, which has to be used in the action (7.7) in order to describe the (2,1) AdS supergravity, proves to be G I + 1 2 ξG I , where G I is given by eq. (7.10). The derivation of the component actions will be given elsewhere.
In superspace, the off-shell formulations for (2,1) and (2,2) AdS supergravities were given in [42]. The specific feature of (2,1) AdS supergravity is that its conformal compensator is a vector multiplet that can be described in terms of the tropical prepotential V(v). 22 The specific feature of (2,2) AdS supergravity is that its conformal compensators are two vector multiplets that can be described using the left and right tropical prepotentials, V L (v L ) and V R (v R ). 23 As concerns the (3,0), (3,1) and (4,0) AdS supergravity theories, the structure of the corresponding conformal compensators is not yet known, which is an interesting open problem.
Our procedure of constructing composite O(2) multiplets can be used to generate higher derivative couplings for vector multiplets, similar to the known results in 4D N = 2 supersymmetry [70,75]. 24 To illustrate the idea, let us fix N = 3 and consider the composite O(2) multiplet defined by (7.10). Choosing in (C.18) a composite prepotential of the form leads to a family of composite real O(2) multiplets Here the contour integral can be computed using the technique of [70]. Now, we have two types of composite O(2) multiplets, G (2) (v) and H n (2) (v), which differ by the number of spinor derivatives involved. Both of them can be used to generate new composite O(2) multiplets with p and q non-negative integers. The above composite O(2) multiplets are expected to appear in low-energy effective actions for quantum N = 3 supersymmetric gauge theories.
In the N = 4 case, we need two vector multiplets, G L (v L ) and G R (v R ), in order to generate higher derivative composite O(2) multiplets.
In the rigid supersymmetric case, Zupnik has derived, building on the earlier work by Howe and Leeming [78], harmonic superspace formulations for the N = 5 vector multiplet and corresponding Chern-Simons actions [79,80]. In this setting, the off-shell vector multiplet involves an infinite number of bosonic and fermionic degrees of freedom, which makes possible the construction of Chern-Simons actions.
It is known that the harmonic superspace approach is the most elaborated scheme to do supergraph calculations in off-shell theories with six and eight supercharges. It would be interesting to see how quantum corrections of the type (9.4) are generated within the background field formulation for quantum 3D N = 3 super Yang-Mills theories [81].
Here E A = E A M ∂ M is the inverse vielbein, M ab are the Lorentz generators, N IJ are generators of the SO(N ) group, D is the dilatation generator and K A = (K a , S I α ) are the special superconformal generators. 25 The Lorentz generators obey The SO(N ) and dilatation generators obey The Lorentz and SO(N ) generators act on the special conformal generators K A as while the dilatation generator acts on K A as Among themselves, the generators K A obey the algebra Finally, the algebra of K A with ∇ A is given by The covariant derivatives obey the (anti-)commutation relations of the form where T AB C is the torsion, and R(M) AB cd , R(N) AB P Q , R(D) AB , R(S) AB γ K and R(K) AB c are the curvatures corresponding to the Lorentz, SO(N ), dilatation, S-supersymmetry and special conformal boosts respectively.
The full gauge group of conformal supergravity, G, is generated by covariant general coordinate transformations, δ cgct , associated with a parameter ξ A and standard superconformal transformations, δ H , associated with a parameter Λ a . The latter include the dilatation, Lorentz, SO(N ), and special conformal (bosonic and fermionic) transformations. The covariant derivatives transform as where K denotes the first-order differential operator Covariant (or tensor) superfields transform as In order to describe the Weyl multiplet of conformal supergravity, some of the components of the torsion and curvatures must be constrained. Following [40], the spinor derivative torsion and curvatures are chosen to resemble super-Yang Mills where W IJ is some operator that takes values in the superconformal algebra, with P A replaced by ∇ A . In [40] it was shown how to constrain W IJ entirely in terms of the super Cotton tensor (for each value of N ). Remarkably, for all N the torsion tensor takes its constant flat space value, while the Lorentz and dilatation curvatures always vanish: We now summarize the resulting covariant derivative algebra for all values of N .

A.1 The N = 1 case
The N = 1 super Cotton tensor W αβγ is a symmetric primary superfield of dimension-5/2 The algebra of covariant derivatives is given by The Bianchi identities imply an additional constraint on W αβγ , the vanishing of its spinor divergence, The N = 2 super Cotton tensor W αβ is a symmetric primary superfield of dimension-2 As in the N = 1 case, its spinor divergence vanishes, 14) The algebra of covariant derivatives is where the U(1) generator J obeys The N = 3 super Cotton tensor W α is a primary superfield of dimension-3/2, with vanishing spinor divergence, The algebra of covariant derivatives is In order to define a large class of matter multiplets coupled to supergravity, it is often useful to switch to an isospinor notation using the isomorphism SO(3) ∼ = SU(2)/Z 2 . As usual, this is achieved by replacing any SO(3) vector index by a symmetric pair of SU(2) spinor indices, ∇ I α → ∇ ij α = ∇ ji α . The details of this correspondence are available in [42]. Here we only give the results essential for our discussion. Converting the indices of the SO(3) generator N KL into isospinor indices gives where the SU(2) generator J kl acts on the spinor covariant derivatives as We also have For all values of N > 3, we introduce the super Cotton tensor W IJKL , which is a totally antisymmetric primary superfield of dimension-1, The algebra of covariant derivatives is 26 where W IJKL satisfies the Bianchi identity For N = 4, the equation (A.26) is trivially satisfied, and instead a fundamental Bianchi identity occurs at dimension-2. Rewriting the super Cotton tensor as a scalar superfield, as in (5.5), the Bianchi identity is We now we turn to a discussion of special features of the N = 4 case.

A.5 The N = 4 case
For each SO(4) vector V I we can associate a second-rank isospinor V iī The original SO(4) connection turns into a sum of two SU(2) connections The left and right operators act on the covariant derivatives as In the isospinor notation, the Bianchi identity on W becomes The algebra of spinor covariant derivatives becomes

B Supersymmetry transformations
In this appendix we give the supersymmetry transformations of the component fields for vector multiplets with N < 5. For the supersymmetry transformations of the Weyl multiplet we refer the reader to [35]. In general there are additional auxiliary fields coming from the super Cotton tensor W IJKL . These are defined for N > 3 as follows [35] where I(n) is defined by eq. (5.7). Expressions involving the component fields for lower values of N may be derived via the truncation procedure given in [35].
One readily finds the Q-supersymmetry and S-supersymmetry transformations of v m to be The S-supersymmetry transformations of the non-gauge fields are Their Q-supersymmetry transformations are (B.6c) In the above we have derived the supersymmetry transformations of the component fields for general N . However, we are still missing the supersymmetry transformations of These fields only appear for N > 3, while for N = 4 χ αβ IJKL is composite once one imposes the (anti-)self-dual condition (4.25), see eq. (5.17b). Keeping in mind the definition of the component fields, eqs. (5.29) and (5.13), and the truncation procedure, we see that all the supersymmetry transformations for N < 5 are specified.

C Action principle in N = 3 supergravity
As demonstrated in [42], general off-shell N = 3 supergravity-matter systems are naturally formulated in curved N = 3 projective superspace M 3|6 × CP 1 in terms of covariant projective multiplets. These multiplets were defined in [42] in SO(3) superspace.
Here we briefly extend those definitions to N = 3 conformal superspace.
Let v i ∈ C 2 \ {0} be homogeneous coordinates for CP 1 . We use these variables to define a subset of spinor covariant derivatives It follows from (A.22) that the operators ∇ α anticommute with each other, Here we have defined and introduced the differential operator These relations involve a fixed isospinor u i which is subject to the condition (v, u) = 0, but otherwise completely arbitrary. For the covariant projective multiplet, one can define the operation of smile conjugation which takes Q (n) (v) to its smile-conjugateQ (n) (v), which is also a covariant weight-n projective multiplet, see [42] for the details. Its property is Therefore, if n is even, one can define real projective multiplets,Q (2n) = Q (2n) .
A weight-n isotwistor superfield U (n) (z, v) is defined to share with Q (n) (z, v) all its properties except the analyticity constraint (C.3).
In this paper, all covariant projective multiplets are assumed to be primary, and hence {S ij α , ∇ (2) }Q (n) = 0. Then it follows from (A.23) that the dimension of Q (n) is equal to n/2, An important example of covariant projective multiplets is a real O(2n) multiplet, with n = 1, 2, . . . It is described by a real weight-2n projective superfield H (2n) (v) of the form: (C.11) The analyticity constraint (C.3) is equivalent to while the reality conditionH (2n) = H (2n) is equivalent to The field strength of an Abelian vector multiplet, G (2) , is a real O(2) multiplet.
To describe the dynamics of a supergravity-matter system, one has to specify a Lagrangian, L (2) (v), which is postulated to be a real weight-two projective multiplet. Associated with L (2) is the supersymmetric action [42] (C.14) Here the model-independent isotwistor superfield C (−4) (v) of weight −4 is required to be conformally primary and of dimension −1, and obey the condition As shown in [42], the action (C.14) does not change under an arbitrary infinitesimal variation of C (−4) , and thus (C.14) is actually independent of C (−4) .
The second-order operator (C.17) allows us to engineer covariant projective multiplets. The point is that the superfield ∆ (4) U (n−4) (v) is a covariant weight-n projective multiplet, for any primary isotwistor superfield U (n−4) of dimension n/2 − 1.
We now derive a new representation for the action (C.14) that is valid under the assumption that there is an Abelian vector multiplet such that its gauge invariant field strength G ij is nowhere vanishing, G := G ij G ij = 0. Let V(v) be the tropical prepotential for this multiplet. By definition, V(v) is a real weight-zero projective multiplet. The superfield G ij is a real O(2) multiplet which is related to V(v) as follows: The right-hand side on (C.18) is invariant under the gauge transformations (7.2).
In the action (C.14), we first replace L (2) → G (2) [L (2) /G (2) ] and make use of the representation (C.18) for the first multiplier. As a next step, we can integrate by parts in order to let ∆ (4) hit C (−4) and then use (C.16). Finally, we can change the order of contour integrations to result in .
(C. 19) In this functional, we first re-label v ↔v, then insert the unity 1 = ∆ (4) C (−4) (v) and finally integrate ∆ (4) by parts. Since V(v) obeys the constraint (C.3), the projection operator ∆ (4) commutes with V, and we end up with the representation , is a composite real O(2) multiplet. Eq. (C.20) is our new representation for the action (C.14). It is the main result of this section.
We conclude with an example that provides evidence of the universality of the projective superspace action (C.14). Let us consider the conventional N = 3 locally supersymmetric action where the Lagrangian L is a dimensionless primary scalar superfield. It turns out that this action can be recast in the form (C.14) if we define This may be proved using the the contour integration techniques of [70].

D Action principle in N = 4 supergravity
Within the approach [42], off-shell N = 4 supergravity-matter systems are formulated in curved N = 4 projective superspace M 3|8 ×CP 1 L ×CP 1 R in terms of covariant projective multiplets. These multiplets were defined in [42] in SO(4) superspace. Here we briefly extend those definitions to N = 4 conformal superspace. Our presentation is similar to the N = 3 story of the previous section.
Let v L := v i ∈ C 2 \ {0} and v R := v¯i ∈ C 2 \ {0} be homogeneous coordinates for CP 1 L and CP 1 R respectively. We use these variables to define two different subsets, ∇ The right projective multiplets are defined similarly.
In N = 4 conformal superspace, we can also introduce hybrid projective multiplets and isotwistor projective multiplets. The corresponding definitions are completely analogous to those given [42].
All left and right projective multiplets are assumed to be primary, in particular In general, the N = 4 supersymmetric action may be represented as a sum of two terms, the left S L and right S R ones, 28 S = S L + S R . (D.11) The left action has the form L , E −1 = Ber(E A M ) , (D. 12) where the Lagrangian L where the gauge parameter λ L is an arbitrary left arctic multiplet of weight zero. The corresponding gauge invariant field strength, G¯ij, is a right O(2) multiplet related to V L as follows: (D.17) 28 There exist different action principles, in particular the one with a hybrid Lagrangian [42]. However, they may be always reduced to the form (D.11). 29 The operator ∆ L is a covariant projection operator. Given a covariant left projective multiplet Q (v L ), see [42] for the details.
Here u L = u i is a constant isospinor such that (v L , u L ) = 0 along the closed integration contour. One may show that the right-hand side of (D.17) is independent of u L .
The left O(2) multiplet G ij is associated with a right tropical prepotential V R (v R ) according to the rule: The prepotential V R can always be represented as for some isotwistor superfield T (−4) R (v R ). 30 Here ∆ R is defined similar to ∆ We now obtain an alternative representation for the left action (D.12). The idea is to insert the unity 1 = G  where we have changed the order of contour integrals. The next step is to insert the unity ∆ R C (−4) R = 1 into the integrand and then integrate by parts.
Taking the explicit form of ∆ R into account, this expression can be replaced with the following: Here the operator ∆ (−2,2) is defined by for an isospinor u i such that (v L , u L ) = 0. The operator ∆ (2,2) in (D.26) is obtained from ∆ (2,2) , eq. (D.23), by replacingv i → v i . Now, one may notice that ∆ (2,2) ∆ (2,2) in (D.26) is equivalent to (v L ,v L ) 2 ∆ R , and therefore the action turns into Since V R is a right projective multiplet, it commutes with the operator ∆ (−2,2) , and we end up with the following representation for S L : R , it will be referred to as the right linear multiplet action.