Minimal $\cal N=4$ topologically massive supergravity

Using the superconformal framework, we construct a new off-shell model for $\cal N=4$ topologically massive supergravity which is minimal in the sense that it makes use of a single compensating vector multiplet and involves no free parameter. As such, it provides a counterexample to the common lore that two compensating multiplets are required within the conformal approach to supergravity with eight supercharges in diverse dimensions. This theory is an off-shell $\cal N=4$ supersymmetric extension of chiral gravity. All of its solutions correspond to non-conformally flat superspaces. Its maximally supersymmetric solutions include the so-called critical (4,0) anti-de Sitter superspace introduced in arXiv:1205.4622, and well as warped critical (4,0) anti-de Sitter superspaces. We also propose a dual formulation for the theory in which the vector multiplet is replaced with an off-shell hypermultiplet. Upon elimination of the auxiliary fields belonging to the hypermultiplet and imposing certain gauge conditions, the dual action reduces to the one introduced in arXiv:1605.00103.


Introduction
A unique feature of three spacetime dimensions (3D) is the existence of topologically massive Yang-Mills and gravity theories. They are obtained by augmenting the usual Yang-Mills action or the gravitational action by a gauge-invariant topological mass term. Such a mass term coincides with a non-Abelian Chern-Simons action in the Yang-Mills case [1,2,3,4] and with a Lorentzian Chern-Simons term in the case of gravity [3,4]. Without adding the Lorentzian Chern-Simons term, the pure gravity action propagates no local degrees of freedom. The Lorentzian Chern-Simons term can be interpreted as the action for conformal gravity in three dimensions [3,5,6]. 1 Topologically massive theories of gravity possess supersymmetric extensions. In particular, N = 1 topologically massive supergravity was introduced in [9] and its cosmological extension followed in [10]. The off-shell formulations for N -extended topologically massive supergravity theories were presented in [11] for N = 2 and in [12] for N = 3 and N = 4. In all of these theories, the action functional is a sum of two terms, one of which is the action for pure N -extended supergravity (Poincaré or anti-de Sitter) and the other is the action for N -extended conformal supergravity. The off-shell actions for N -extended supergravity theories in three dimensions were given in [13] for N = 1, [14,15] for N = 2, and [14] for the cases N = 3, 4. The off-shell actions for N -extended conformal supergravity were given in [5] for N = 1, [16] for N = 2, and [17] for N = 3, 4. The latter work made use of the formulation for N -extended conformal supergravity presented in [18].
The off-shell structure of 3D N = 4 supergravity [14] is analogous to that of 4D N = 2 supergravity (see, e.g., [19] for a pedagogical review) in the sense that two superconformal compensators are required (for instance, two off-shell vector multiplets, one of which is self-dual and the other anti-self-dual) in order to realise pure Poincaré or anti-de Sitter (AdS) supergravity theories. We recall that the equations of motion for pure N = 4 Poincaré or AdS supergravity are inconsistent if one makes use of a single compensator [12]. By construction, the off-shell N = 4 topologically massive supergravity theory of [12] makes use of two compensators. However, in [20] the consistent system of dynamical equations was proposed for N = 4 topologically massive AdS supergravity with a single compensating hypermultiplet, following earlier work in [21,22,23] on ABJ(M) models. A peculiar feature of this model, like those considered in [21,22,23], is that it has no free parameter. Consequently the dimensionless combination, µℓ, of mass µ and AdS radius ℓ takes a fixed value, µℓ = 1, as in chiral gravity [24]. In [24] it was argued that µℓ = 1 is the only value for the quantum theory to have a chance to be free of ghosts. It is thus interesting that the N = 4 theory of [20] picks precisely this value. 2 In [20] a supergravity action functional was also postulated to generate the dynamical equations given. This action was claimed to be off-shell without giving technical details. In this paper we propose a new off-shell model for N = 4 topologically massive supergravity which is minimal in the sense that it makes use of a single compensating vector multiplet. The theory is consistent only if the term corresponding to N = 4 conformal supergravity is turned on. An important maximally supersymmetric solution for this theory is the so-called critical (4,0) AdS superspace introduced in [25]. Our supergravity theory is first presented in a manifestly supersymmetric form, and then its action functional is reduced to components. By choosing appropriate gauge conditions at the component level and performing a duality transformation, we show how to reduce our off-shell supergravity action to the one postulated in [20]. This paper is organised as follows. In section 2 we recall the superspace geometry of the two N = 4 vector multiplets and the corresponding locally supersymmetric actions. In section 3 we present two models for minimal N = 4 topologically massive supergravity, analyse their equations of motion and give a brief discussion of the maximally supersymmetric solutions. Section 4 is devoted to the component structure of minimal N = 4 topologically massive supergravity. Concluding comments are given in section 5. The main body of the paper is accompanied with three technical appendices. The essential details of the known superspace formulations for N = 4 conformal supergravity are collected in Appendices A and B. Some useful super-Weyl gauge conditions in SO(4) superspace and their implications are given in Appendix C.

The N = 4 vector multiplets
There are two inequivalent irreducible N = 4 vector multiplets in three dimensions, self-dual and anti-self-dual ones, as discovered by Brooks and Gates [26]. In this section we review the superspace geometry of these supermultiplets in the presence of N = 4 conformal supergravity [14,18] and the corresponding locally supersymmetric actions [14].
Throughout this paper we make use of both the SO(4) superspace formulation of conformal supergravity, which was sketched in [27] and fully developed in [14], and the conformal superspace formulation presented in [18]. These formulations are related to each other since SO(4) superspace may be viewed as a gauge fixed version of the N = 4 conformal superspace [18]. Due to this reason, we will first start by formulating vector multiplets in conformal superspace. We refer the reader to Appendix A for the salient details of the conformal superspace formulation. The geometry of SO(4) superspace in briefly reviewed in Appendix B.

Kinematics
To describe an Abelian vector multiplet in a curved superspace M 3|8 parametrised by coordinates z M = (x m , θ µ I ), we introduce gauge covariant derivatives where the gauge parameter τ (z) is an arbitrary real scalar superfield.
The algebra of gauge covariant derivatives is where the torsion and curvatures are those of conformal superspace but with F AB corresponding to the gauge covariant field strength and must be subject to covariant constraints to describe an irreducible vector multiplet.
In order to describe an N = 4 vector multiplet, the superform F is subject to the constraint (see [14] for more details) and then the Bianchi identity fixes the remaining components of F to be where G IJ is primary and of dimension 1, Moreover, the field strength G IJ is constrained by the dimension-3/2 Bianchi identity It is well known (see [14] and references therein) that the constraint (2.7) defines a reducible off-shell supermultiplet. 3 The point is that the Hodge-dual of G IJ , obeys the same constraint as G IJ does, where ε IJKL is the Levi-Civita tensor. As a result one may constrain the field strength G IJ to be self-dual,G IJ = G IJ or anti-self-dual,G IJ = −G IJ . These choices correspond to two different irreducible off-shell N = 4 vector multiplets, which we denote by G IJ + and G IJ − , respectively. In what follows we will make use of an (anti-)selfdual Abelian vector multiplet such that its field strength G IJ ± is nowhere vanishing, G 2 ± := 1 2 G IJ ± G ±IJ = 0. When working with N = 4 supersymmetric theories, a powerful technical tool is the isospinor notation based on the isomorphism SO(4) ∼ = SU(2) L × SU(2) R /Z 2 , which allows one to replace each SO(4) vector index with a pair of isospinor ones. In defining the isospinor notation, we follow [14] and associate with a real SO(4) vector V I a second-rank isospinor V iī defined as where we have introduced the τ -matrices (τ I ) iī = (½, iσ 1 , iσ 2 , iσ 3 ) , I = 1, · · · , 4 , i = 1, 2 ,ī =1,2 . (2.11) The isospinor indices of SU(2) L and SU(2) R spinors ψ i and χī, respectively, are raised and lowered using the antisymmetric tensors ε ij , ε ij and ε¯ij, εīj (normalised by ε 12 = ε 21 = ε12 = ε21 = 1) according to We then have the following dictionary: where V I and U I are SO(4) vectors, A IJ and B IJ are anti-symmetric second-rank SO(4) tensors. The left-hand side of (2.13d) is the Levi-Civita tensor in the isospinor notation.
In the isospinor notation, the self-dual (G IJ + ) and anti-self-dual (G IJ − ) vector multiplets take the form 14) and the Bianchi identity (2.7) turns into At this stage it is useful to introduce left and right isospinor variables v L := v i ∈ C 2 \ {0} and v R := v¯i ∈ C 2 \ {0}, which can be used to package the anti-self-dual field strength G ij and the self-dual field strength G¯ij into fields without isospinor indices, G R (v R ) := Gījv¯ivj, respectively. The same isospinor variables can be used to define two different subsets, ∇ (1)ī α and ∇ (1)i α , in the set of spinor covariant derivatives ∇ iī α by the rule It follows from (A.17) that the operators ∇ (1)ī α obey the anti-commutation relations: where The rationale for the definitions given is that the constraints (2.15) now become the analyticity conditions which tell us that each of G (2) L and G (2) R depends on half the Grassmann coordinates. The constraints (2.18) do not change under re-scalings v i → c L v i and v¯i → c R v¯i, with c L , c R ∈ C \ {0}, with respect to which G (2) L (v L ) and G (2) R (v R ) are homogeneous polynomials of degree two. We see that the isospinor variables v L and v R are defined modulo the equivalence relations v i ∼ c L v i and v¯i ∼ c R v¯i, and therefore they parametrise identical complex projective spaces CP 1 L and CP 1 R . The superfields G . The field strengths G   [14] in SO(4) superspace and later reformulated in [12] within the conformal superspace setting. There are two types of covariant projective multiplets, the left and right ones. A left projective multiplet of weight n, Q Secondly, it is subject to the constraint Thirdly, it is a holomorphic function of v L . Fourthly, it is homogeneous function of v L of degree n, Q Every left projective multiplet is inert with respect to SU(2) R and transforms under SU(2) L as and made use of the differential operator Here we have also introduced a second left isospinor variable u L := u i which is restricted to be linearly independent of v L , that is (v L , u L ) = 0. One may see that As shown in [14] the self-dual vector multiplet, G R (v R ), can be described in terms of a gauge prepotential V L (v L ), which is a left weight-0 tropical multiplet and is real with respect to the analyticity preserving conjugation called the smile conjugation. The interested reader is referred to [14] for the technical details. Similar properties hold for the anti-self-dual vector multiplet except all 'left' objects have to be replaced by 'right' ones and vice versa.

Dynamics
General off-shell matter couplings in N = 4 supergravity were constructed in [14].
The action for such a supergravity-matter system may be represented as a sum of two terms (one of which may be absent), (2.25) The left S L and right S R actions, are naturally formulated in curved N = 4 projective superspace. The left action has the form where the Lagrangian L . It is defined to be real with respect to the smile-conjugation defined in [14] and obeys the differential equation Here ∆ L denotes the following fourth-order operator 4 ∆ (4) ij , (2.28) 4 The operator ∆ L is a covariant projection operator. Given a covariant left projective multiplet Q  There are two equivalent action functionals to describe the dynamics of a single self-dual Abelian vector multiplet coupled to conformal supergravity. One of them is a right action formulated in terms of a right O(2) multiplet G (2) R (v R ) = vīvjG¯ij, which is associated with the superfield strength G¯ij of the vector multiplet. This action, has the form 5 [14] S (+) where the weight-one arctic multiplet Υ R and its smile conjugateΥ (1) R are pure gauge degrees of freedom. The action (2.29) is the 3D N = 4 counterpart of the projectivesuperspace action [29] for the 4D N = 2 improved tensor multiplet [30]. The other representation for S (+) VM makes use of a left tropical prepotential V L (v L ) of the self-dual vector multiplet with gauge transformations The gauge parameter λ L is an arbitrary left arctic multiplet of weight zero. The gauge invariant field strength, G¯ij, is related to V L through Here u L = u i is a constant isospinor such that (v L , u L ) = 0 along the closed integration contour. 6 The action (2.29) can be recast as a left BF -type action [12] S (+) (2.33) 5 We should emphasise that in this paper we have defined the vector multiplet actions with "wrong" sign, because in our approach they correspond to superconformal compensators. 6 One may show that the right-hand side of (2.31) is independent of u L .
The composite left superfield G ij can be equivalently realised as the anti-self-dual SO(4) bivector G IJ − . Similarly, the action for the anti-self-dual vector multiplet [14] can be recast as the right BF -type action [12] S (−) is the tropical prepotential of the anti-self-dual vector multiplet. The composite right superfield (2.35) can be equivalently realised as the self-dual SO (4) bivector G IJ + . The composite O(2) multiplets can be expressed in terms of SO(4) vector indices as follows [12] where we have defined (2.37) To show that G IJ ± is primary and satisfies the Bianchi identity, the following identities prove useful It is worth mentioning that the two N = 4 linear multiplet actions (2.32) and (2.34) are universal [12] in the sense that all known off-shell supergravity-matter systems (with the exception of pure conformal supergravity) may be described using such actions with appropriately engineered composite O(2) multiplets G

Minimal topologically massive supergravity
In this section we present two new supergravity-matter systems as models for minimal topologically massive supergravity.

Action principle and equations of motion
Our models for minimal topologically massive supergravity are described by N = 4 conformal supergravity coupled to a vector multiplet, either self-dual or anti-self-dual, via the following supergravity-matter actions: where S CSG denotes the conformal supergravity action [17]. We will refer to the theories with actions S + and S − as the self-dual and anti-self-dual topologically massive supergravity (TMSG) theories, respectively.
As shown in [12], the equation of motion for the vector multiplet derived from the action (3.1) is equivalent to while the equation of motion for the conformal supergravity multiplet (that is, the N = 4 Weyl supermultiplet) is Here T ± is the supercurrent, which corresponds to the action S (±) VM , One can check that the supercurrent T ± obeys the conservation equation [31] ∇ α(I ∇ J) when the matter equation of motion (3.2) is satisfied.
Making use of the Bianchi identity (2.7) as well as the equations of motion (3.2)-(3.4), one finds the following equations on G ± : We now turn to an analysis of the consequences of the equations of motion (3.6).

Analysing the equations of motion
To analyse the equations of motion corresponding to the action (3.1) we need to fix the gauge freedom. Firstly, we use the special conformal transformations to make the dilatation connection vanish, B A = 0. This corresponds to degauging of conformal superspace to SO(4) superspace [14] and gives rise to new torsion terms 7 which can be expressed in terms of superfields S IJ , S, C a IJ and their covariant derivatives. We refer the reader to [14] for details and provide a summary of the salient details of SO(4) superspace in Appendix B.
Upon imposing the gauge B A = 0 one can show that (3.6) is equivalent to where D I α is the SO(4) superspace covariant derivative [14,27] (see also [18]). In isospinor index notation, for the self-dual vector multiplet one obtains while for the anti-self-dual vector multiplet one finds One should keep in mind that the equations of motion for G + and G − derived from the actions S + and S − , respectively, were used in the above results.
Under super-Weyl transformations the SO(4)-covariant derivatives and the torsion terms transform as 8 where σ is a real unconstrained superfield. Within the superconformal framework, all supergravity-matter actions are required to be super-Weyl invariant.
The super-Weyl gauge freedom may be used to impose useful gauge conditions. For instance, one can make use of the super-Weyl transformations to gauge away the self-dual or anti-self-dual part of C a IJ such that the remaining torsion components are expressed directly in terms of the matter fields. For concreteness, let us consider the theory described by the action S + , with corresponding equations of motion (3.8), and gauge away C aīj via a super-Weyl transformation. We then find In this gauge, we see that the geometry is determined in terms of a single superfield, which is chosen to be the scalar G + . After imposing this super-Weyl gauge condition it is possible to show that there is enough super-Weyl freedom left to impose the additional condition 2S + W = 0 , (3.12) see Appendix C for the derivation. This condition proves to lead to the following nonlinear equation for G + : The main virtue of the super-Weyl gauge conditions imposed is that all the torsion and curvature tensors are descendants of the single scalar superfield G + . However, this gauge choice is not particularly useful from the point of view of studying (maximally) supersymmetric backgrounds. A more convenient super-Weyl gauge fixing is G + = const. We spell out the implications of such a gauge condition below.
Given a vector multiplet with a superfield strength G IJ such that G is nowhere vanishing, one can always make use of the super-Weyl transformations to choose a gauge where Such a gauge condition has slightly different consequences on the superspace geometry for the two vector multiplets G IJ + and G IJ − satisfying the equations of motion (3.2) and (3.3). In both cases the super-Cotton tensor is constant, while the constraints on the remaining torsion components differ. For the on-shell self-dual vector multiplet one finds the following consistency conditions while for the on-shell anti-self-dual vector multiplet one finds In the case where C IJ a vanishes, the algebra of covariant derivatives coincides with that of (4, 0) AdS superspace in the critical case where 2S ∓ W = 0, see [25]. 9 In general, however, C a IJ does not vanish and instead satisfies some differential conditions implied by the Bianchi identities To analyse the Bianchi identities in detail it will be useful to convert to isospinor notation.
We consider in detail the self-dual TMSG theory. In the isospinor notation, the covariant derivative algebra which follows from the equations of motion is The N = 4 super-Cotton tensor is denoted by X in [14,25] .
Analysing the Bianchi identities (3.18) determines the remainder of the covariant derivative algebra: as well as the following differential constraint on C aīj The above constraint implies, in turn, Since the SU(2) R curvature vanishes, we can completely gauge away the corresponding connection. Such a gauge condition is assumed in what follows. In this gauge, the field strength G¯ij becomes a constant symmetric isospinor subject to the normalisation condition G¯ijGīj = 1. It is invariant under a U(1) subgroup of SU(2) R .
We are now in a position to describe all maximally supersymmetric solutions of the theory. In accordance with the general superspace analysis of supersymmetric backgrounds in diverse dimensions [32,33,34], such superspaces have to comply with the additional constraint which leads to the integrability conditions The general solution of (3.23b) is where Cīj is a constant symmetric rank-2 isospinor. Without loss of generality, Cīj can be normalised as CījCīj = 1. The covariant constancy conditions (3.22) and (3.23a) now amount to We recall that the Lorentz generator with a vector index, M a , acts on a three-vector by the rule M a C b = ε abc C c . The second condition in (3.25) implies that C b is a Killing vector of constant norm, Thus there are three types of backgrounds depending on whether the Killing vector C a is chosen to be time-like, space-like or null. The algebra of covariant derivatives for such a background is One may think of this algebra as a Lie superalgebra. 10 By construction, the theory involves the constant symmetric isospinor G¯ij being invariant under a U(1) subgroup of the group SU(2) R . If Cīj does not coincide with G¯ij, then the group SU(2) R is completely broken. This indicates that Cīj = G¯ij.
The simplest maximally supersymmetric solution of the theory is characterised by (see also [20]) (3.28) It corresponds to the critical (4,0) AdS superspace introduced in [25]. Its algebra of covariant derivatives is as follows: The last relation shows that the cosmological constant is Λ = −W 2 = −ℓ −2 , in agreement with [20,25]. Here ℓ is the radius of curvature in AdS 3 . The latter relation is equivalent to µℓ = 1, which corresponds to chiral gravity [24].
More generally, the (p, q) AdS superspaces, p + q = N , in three dimensions were classified in [25]. 11 In the N = 4 case, the (3,1) and (2,2) AdS superspaces are necessarily conformally flat, W = 0. The distinguished feature of (4,0) AdS supersymmetry is that the super-Cotton scalar W may have a non-zero value. The algebra of covariant derivatives is given by [25] where the positive constant S determines the curvature of AdS 3 . For a generic value of W the entire SO(4) R-symmetry group belongs to the superspace holonomy group. But there are two special values of W for which either the SU(2) R or the SU(2) L curvature vanishes and the structure group is reduced. These are given by and correspond to the critical (4,0) AdS superspaces. As briefly discussed in [35], the isometry group of (4,0) AdS superspace is isomorphic to D(2, 1; α) × SL(2, R) in the non-critical case W = ±2S, where D(2, 1; α) is one of the exceptional simple supergroups, with the real number α = −1, 0, see e.g. [36,37] for reviews. The supergroup parameter α is related to the (4,0) AdS parameter q = 1 + W 2S introduced in [35]. If the values of α are restricted to the range 12 −1 < α ≤ − 1 2 , then we can identify −2α = 1+ W 2S . The case α = − 1 2 corresponds to the conformally flat (4,0) AdS superspace, for which W = 0. Its isometry group is OSp(4|2)×SL(2, R). The limiting choice α = −1 corresponds to one of the two critical (4,0) AdS cases, W = 2S. 13 The isometry group of this (4,0) AdS superspace is SU(1, 1|2) ⋊ SU(2) × SL(2, R), see also the discussion in [38].
It is worth giving a few general comments about maximally supersymmetric warped AdS backgrounds in N -extended supergravity theories. Such backgrounds do not exist in the case of N = 1 supergravity. This result was first demonstrated by Gibbons, Pope and Sezgin [44], and it follows trivially from the general superspace analysis of supersymmetric backgrounds in diverse dimensions [32,33,34]. 14 However, maximally supersymmetric warped AdS backgrounds do exist in extended supergravity, N > 1, if the structure group includes not only the Lorentz group SL(2, R) but also a nontrivial R-symmetry group. For instance, the structure group for N = (2, 0) AdS supergravity is SL(2, R) × U(1) R , and thus this theory possesses maximally supersymmetric warped AdS backgrounds, which were described in [11,34] using the superspace techniques, and some time later in [41,42] using the component approach. On the other hand, the structure group for N = (1, 1) AdS supergravity coincides with the Lorentz group, and therefore this theory possesses no maximally supersymmetric warped AdS backgrounds, see [11,34] for more details. One can also derive further equations on descendants of δC αβīj using the constraint (3.20). In particular, one finds where D iī α denotes the spinor covariant derivative of the critical (4, 0) AdS superspace. The component projection of δC αβγ iī is proportional to the linearised gravitino field 14 Indeed, the superspace geometry of N = 1 supergravity is determined by two torsion tensors, a scalar S and a symmetric spinor C αβγ = C (αβγ) , see [13,14] for more details. According to [33,34], every maximally supersymmetric background is characterised by the conditions C αβγ = 0 and S = const, see also [45]. The resulting algebra of covariant derivatives corresponds to N = 1 AdS superspace for S = 0, or Minkowski superspace for S = 0.
In the above we worked with the self-dual TMSG theory, however the analysis of the equations of motion corresponding to the action S − is completely analogous. There one finds the covariant derivative algebra is where C a ij satisfies the Bianchi identity Using the above equation one finds The solution C a ij = 0 corresponds to (4, 0) AdS superspace in the critical case 2S = −W . We now linearise around the (4, 0) AdS superspace and set C a ij = δC a ij where δC a ij is a small disturbance. It can be seen that δC a ij obeys the equation where D a corresponds to the vector covariant derivative of the (4, 0) AdS superspace. After applying another vector derivative one finds

Component actions
In this section we give the component results corresponding to the minimal N = 4 topologically massive supergravity action (3.1).

The component conformal supergravity action
The complete component analysis of the N -extended Weyl multiplet was given in [17]. Here we specialise to the N = 4 case where the auxiliary fields coming from the super-Cotton tensor are defined as: The full N = 4 conformal supergravity action was given in [17] and is

The component vector multiplet actions
The component N = 4 linear multiplet actions were given in [12]. Making use of the results there, one can construct the left and right vector multiplet actions.
The component fields of the vector multiplets are defined as The component gauge one-forms v (±)a are defined as where V ± is the superspace gauge one-form associated with the field strength G IJ ± . It is useful to replace h (±) IJ by the fieldŝ which proves to be (anti-)self-dual The component self-dual vector multiplet action is where the bolded component fields correspond to those of the composite vector multiplet, The component anti-self dual vector multiplet action is Plugging in the superspace expressions for G ± IJ one one can construct the component fields of the composite vector multiplets. The component fields are found to be (4.14) Here we have introduced the following:

N = 4 topologically massive supergravity in components
To simplify our results it is useful to make use of the gauge freedom to impose some gauge condition. One can always choose a gauge condition where (4.17) At the component level these require The first gauge condition fixes the dilatation transformations, the second fixes the S-supersymmetry transformations and the third fixes the conformal boosts. For a right G ij and left G¯ij vector multiplet we can use the respective SU(2) symmetry to fix their lowest components to a constant. This then gives With the above gauge conditons we find Using the above conditions one finds (upon integrating by parts) the self-dual vector multiplet action is while the anti-self-dual vector multiplet action is The complete component action for minimal N = 4 topologically massive supergravity (3.1) is then given by where S CSG is the component action (4.2). As a simple check one can readily verify that the equation of motion on the field y gives which is consistent with the supergravity equation of motion being W = ∓µG ± in the presence of the vector multiplet compensator.
For completeness we will also give the component action in isospinor notation. The N = 4 conformal supergravity action (4.2) becomes where the component SU(2) curvatures R ab ij and R abīj are The self-dual vector multiplet action in isospinor notation is while the anti-self-dual vector multiplet action is Having derived the component actions for minimal N = 4 topologically massive supergravity, it is worth elaborating on these results further. For instance, if we consider just one of the vector multiplet actions without the conformal supergravity action, one can see that the equation of motion for y leads to an inconsistency. This is equivalent to the fact that the superfield equations of motion for the N = 4 gravitational superfield 16 derived from the actions S which is completely consistent. Moreover, this equation is consistent with our gauge conditions because imposing the gauge G + = 1 implies G − = 1, which in turn implies that the auxiliary field y cancels. Furthermore, the fields w andĥ IJ become auxiliary and their equation of motion is the requirement that they vanish. The equations of 16 The N = 4 gravitational superfield is a scalar prepotential describing the multiplet of N = 4 conformal supergravity. It is the 3D N = 4 counterpart of the N = 2 gravitational superfield in four dimensions [48]. In the presence of the conformal supergravity action the gauge conditions G + = G − = 1 are no longer consistent [12] and instead one has to use the results in subsection 4.2 in the general gauge. If one also adds to (4.29) the supersymmetric cosmological term [14], the resulting theory corresponds to (2,2) AdS supergravity as was described in detail in [12,14].
It is worth mentioning some simplifications that can be made to the N = 4 topologically massive supergravity actions upon using the equations of motion. To illustrate this let us consider the theory with a self-dual vector multiplet. In this case the equation of motion for the SU(2) L gauge field is which tells us that the SU(2) L gauge field can be completely gauged away. The equation of motion for the auxiliary fieldĥ ij sets the auxiliary field to zero and removes it from the action. The equation of motion on y just sets w = −µ and gives rise to a cosmological term. The resulting action is Similar simplifications can be made for the anti-self dual vector multiplet action.
We can now show how to derive the supergravity action postulated in [20] from our theory S − . The crucial observation is that the U(1) gauge field appears in the action (4.28) only via its field strength f (−)ab , and therefore it may be dualised into a scalar field. To implement this, we replace (4.28) with an equivalent first-order action Plugging this back into (4.34) gives the dual action If we impose a Weyl gauge ϕ = 1 and make use of the equation of motion for the auxiliary fieldĥ¯ij + , which isĥ¯ij + = 0, we recover the bosonic matter sector of the topologically massive supergravity action in [20] up to conventions and fermion terms.
Since the auxiliary fieldĥ¯ij + has been integrated out, the action given in [20] does not appear to be off-shell.

Discussion
In this paper we constructed minimal N = 4 topologically massive supergravity. It has several unique features that we summarise here.
• Unlike the other N -extended TMSG theories with N ≤ 4 [9,10,11,12], its action cannot be viewed as the supergravity action (with or without a supersymmetric cosmological term) augmented by the conformal supergravity action playing the role of a topological mass term. The point is that the theory becomes inconsistent upon removing the conformal supergravity action, as was explained in section 4.3.
• Our theory makes use of a single superconformal compensator. We recall that all known Poincaré or AdS supergravity theories with eight supercharges in diverse dimensions require, in general, two such compensators in order for the corresponding dynamics to be consistent. One known exception is the off-shell formulation for 4D N = 2 AdS supergravity given in [49], which makes use a single massive tensor compensator (described by an unconstrained chiral scalar prepotential) and no compensating vector multiplet. 17 In the case of higher derivative theories, two compensators are no longer required. This was observed in four dimensions for models involving the N = 2 supersymmetric R 2 term [57], and in three dimensions for N = 4 topologically massive supergravity [20].
• Our minimal TMSG theory does not allow any supersymmetric cosmological term. However, a cosmological term gets generated at the component level upon integrating out the auxiliary fields. This is manifested in the fact that the critical (4,0) AdS superspace [25] is a maximally supersymmetric solution of the theory.
• The theory has only one coupling constant.
• Our minimal TMSG theory is the first off-shell N = 4 supergravity theory in three dimensions with the property that the critical (4,0) AdS superspace [25] is a solution of the theory. Upon integrating out the auxiliary fields we recover the model discussed in [20].
• Our theory is an off-shell N = 4 supersymmetric extension of chiral gravity [24]. It is obvious that such an extension, which has never been constructed before, must involve a single conformal compensator.
The above features demonstrate the physical relevance of the theory proposed.
As mentioned in section 1, there exist N = 6 and N = 8 supersymmetric extensions of chiral gravity [24]. Unlike our theory, these TMSG theories are necessarily on-shell. The off-shell structure of our N = 4 theory is indispensable for at least two reasons: (i) it allows for the general coupling to matter supermultiplets; and (ii) at the quantum level, it allows one to derive supersymmetric power-counting rules through the use of supergraph techniques.
In the on-shell construction of topologically gauged N = 6 and N = 8 ABJM type theories [21,22,23], a crucial role is played by a sixth order scalar potential. In the off-shell approach, such a scalar potential is automatically generated upon elimination of the auxiliary fields, as was demonstrated in [58] where the N = 6 and N = 8 ABJM models were realised in N = 3 harmonic superspace. There is an analogous feature in our actions. Specifically, before imposing any gauge condition there is a term w 2 g ± in our actions and upon eliminating the auxiliary fields the term µ 2 g 3 ± is generated. This term plays a similar role as the sixth order polynomial in [21,22,23] in the sense that its coefficient is fixed by the equations of motion (in terms of the coupling coefficient of the conformal supergravity action) and the conformal coupling between the Einstein-Hilbert term and the O(2) multiplet. In this respect our model is akin to those of [21,22,23].
Both models for minimal N = 4 topologically massive supergravity constructed in this paper possess dual formulations. They are obtained by replacing the vector multiplet actions S and similarly for the left hypermultiplet action S (−) HM . In the dual formulation, its compensating multiplet is the so-called polar hypermultiplet described by the weightone arctic multiplet Υ  We begin with a curved three-dimensional N = 4 superspace M 3|8 parametrized by local bosonic (x m ) and fermionic coordinates (θ µ I ): where m = 0, 1, 2, µ = 1, 2 and I = 1, · · · , 4. The structure group is chosen to be OSp(4|4, R) and the covariant derivatives are postulated to have the form Here E A = E A M ∂ M is the inverse vielbein, M ab are the Lorentz generators, N IJ are generators of the SO(4) group, D is the dilatation generator and K A = (K a , S I α ) are the special superconformal generators.
The Lorentz generators obey The SO(4) and dilatation generators obey The Lorentz and SO(4) 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(4), 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(4), 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 [18], 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 [18] it was shown how to constrain W IJ entirely in terms of the super Cotton tensor (or scalar for N = 4). The super Cotton scalar W , is a primary superfield of dimension 1, The algebra of covariant derivatives is where the super Cotton scalar W satisfies the following dimension 2 Bianchi identity 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 Here L kl is the SU(2) L generator and Rkl is the SU(2) R generator. They are related to the SO(4) generators N KL as N KL → N kkll = εklL kl + ε kl Rkl . (A.14) 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 {∇ iī α , ∇ jj β } = 2iε ij ε¯ij∇ αβ + 2iε αβ ε¯ijW L ij − 2iε αβ ε ij W R¯ij − iε αβ ε ij ∇ γ kī W S kj γ + iε αβ ε¯ij∇ γ ik W S jk γ + 1 4 ε αβ ε ij ∇ γ kī ∇ kj δ W − ε¯ij∇ γ jk ∇ ik δ W K γδ (A. 17) and the action of the S-supersymmetry generator on ∇ iī α is {S iī α , ∇ jj β } = 2ε αβ ε ij ε¯ijD − 2ε ij ε¯ijM αβ + 2ε αβ ε¯ijL ij + 2ε αβ ε ij R¯ij . (A.18)

B The geometry of SO(4) superspace
For many applications it is useful to work with a superspace formulation with a smaller structure group than that of conformal superspace. The superspace formulation of [14,27], known as SO (4) superspace, provides such a formulation and may be obtained from conformal superspace via a degauging procedure [18]. For the N = 4 case one chooses the structure group to be SO(4). The SO(4) superspace formulation for N = 4 conformal supergravity has been used to construct general off-shell supergravity-matter couplings [14].
The covariant derivatives have the form: with T AB C the torsion, R AB cd the Lorentz curvature and R AB KL the SO(4) curvature.
The algebra of covariant derivatives must be constrained to describe conformal supergravity. The appropriate constraints [27] lead to the following anti-commutation relation [14]: It is often useful to make use of the isomorphism SO(4) ∼ = SU(2) L × SU(2) R /Z 2 and make use of isospinor notation, D I α → D iī α , by replacing each SO(4) vector index by a pair of isospinor ones. For our notation and conventions we refer the reader to [14].

C Super-Weyl gauge conditions
In this appendix we show how one can use the super-Weyl freedom to impose certain gauge conditions in SO(4) superspace. In particular, within the SO(4) superspace formulation we will show that one can impose either We begin by introducing, within the SO(4) superspace geometry, an off-shell selfdual vector multiplet G¯ij and an anti-self-dual vector multiplet G ij . They are constrained by the differential constraints for O(2) multiplets Using these constraints it is possible to build some of the components of the torsion in terms of these multiplets. In particular, one finds where G 2 + = G¯ijGīj and G 2 − = G ij G ij . The vector multiplets transform homogeneously under super-Weyl transformations G¯ij → e σ G¯ij , G ij → e σ G ij , (C.5) which tells us that the super-Weyl freedom can be completely fixed by imposing the gauge condition G + = 1 or G − = 1. If we impose G + = 1 we find the conditions (C.1), while if we impose G − = 1 we find the conditions (C.2). Therefore, these conditions can always be imposed by an appropriate super-Weyl transformation.