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Three-dimensional (p,q) AdS superspaces and matter couplings

We introduce N-extended (p,q) AdS superspaces in three space-time dimensions, with p+q=N and p>=q, and analyse their geometry. We show that all (p,q) AdS superspaces with X^{IJKL}=0 are conformally flat. Nonlinear sigma-models with (p,q) AdS supersymmetry exist for p+q<=4 (for N>4 the target space geometries are highly restricted). Here we concentrate on studying off-shell N=3 supersymmetric sigma-models in AdS_3. For each of the cases (3,0) and (2,1), we give three different realisations of the supersymmetric action. We show that (3,0) AdS supersymmetry requires the sigma-model to be superconformal, and hence the corresponding target space is a hyperkahler cone. In the case of (2,1) AdS supersymmetry, the sigma-model target space must be a non-compact hyperkahler manifold endowed with a Killing vector field which generates an SO(2) group of rotations of the two-sphere of complex structures.

For arbitrary values of p and q allowed, the pure (p, q) AdS supergravity was constructed in [1] as a Chern-Simons theory with the gauge group OSp(p|2; R) × OSp(q|2; R). Similar ideas can readily be used, e.g., to construct 3D higher-spin (p, q) AdS supergravity [3]. However, this Chern-Simons construction appears to become less powerful when it comes to coupling AdS supergravity to supersymmetric matter, especially in the important cases N = 3 and N = 4. In order to describe general supergravity-matter systems in these cases, superspace approaches appear to be the most useful ones.
As is well known, a universal approach to engineering supergravity theories in diverse dimensions is to realise them as conformal supergravity coupled to certain compensating supermultiplet(s) [4]. Making use of the conformal supergravity constraints on the superspace torsion proposed in [5], in our recent work [6] the superspace geometry of 3D N -extended conformal supergravity was developed 2 and then applied (building on the structure of off-shell superconformal σ-models in three dimensions [12]) to construct general off-shell supergravity-matter couplings for the cases N ≤ 4. In order to illustrate how the formalism of [6] can be used to describe matter-coupled AdS supergravity theories, the cases p + q = 2 were studied in detail in [13]. In particular, Ref. [13] provided two dual off-shell formulations for (1,1) AdS supergravity and one off-shell formulation for (2,0) AdS supergravity. The most general σ-model couplings to (1,1) and (2,0) AdS supergravity theories were constructed in [13] from first principles. These results generalise those obtained earlier [14,15] within the Chern-Simons approach [1]. 3 The present paper is devoted to new applications of the formalism developed in [6]. First of all, here we introduce (p, q) AdS superspaces and study their geometric properties. Secondly, we develop an off-shell formalism for constructing rigid supersymmetric theories in AdS with p + q = 3, and specifically concentrate on describing general supersymmetric nonlinear σ-models.
Within the framework of [6], (p, q) AdS superspace originates as a maximally symmetric supergeometry with covariantly constant torsion and curvature generated by a symmetric torsion S IJ = S JI , with the structure-group indices I, J taking values from 1 to N . It turns out that S IJ is nonsingular, and the parameters p and q = N − p determine its signature. The ordinary AdS space is the bosonic body of AdS (3|p,q) . The curvature of AdS 3 is proportional to S 2 = S IJ S IJ /N (with the structure-group indices being raised and lowered using δ IJ and δ IJ ). The Killing 2 The special cases of N = 8 and N = 16 conformal supergravity theories were independently worked out in [7,8] and [9] respectively. Recently, new results on N = 8 conformal supergravity and its applications have appeared [10,11]. 3 Ref. [14] constructed only those σ-model couplings to (2,0) AdS supergravity in which the scalar fields are neutral under the gauged U(1) R-symmetry group. Ref. [15] studied locally supersymmetric σmodels on homogeneous spaces of the form G/H × U(1) in which the scalar fields are charged under the gauged U(1) R-symmetry group. Such σ-model couplings to (2,0) AdS supergravity are special cases of those constructed in [13].
In fact, starting from the superspace geometry of N -extended conformal supergravity [6] and restricting the torsion to be covariantly constant and Lorentz invariant, a general AdS superspace solution for N ≥ 4 includes not only the torsion S IJ described above but also a completely antisymmetric torsion X IJKL = X [IJKL] . It turns out that the latter may be non-zero only if S IJ = Sδ IJ , which means p = N and q = 0. Such solutions define new AdS superspaces, AdS 3|2N S,X , for which the isometry group is, in general, a subgroup of OSp(N |2; R) × SL(2, R).
Why bother to study supersymmetric nonlinear σ-models in AdS (3|p,q) ? Part of our motivation comes from four dimensions. Recently, it has been realised that rigid supersymmetric field theories in AdS 4 have drastically different properties compared to their counterparts in Minkowski space [16,17,18,19,20,21,22]. Analogous results apply in five dimensions [23,24]. It is therefore natural to study the specific features of rigid supersymmetric field theories in AdS 3 . 4 And then we can immediately see that the 3D story is much richer than the 4D one, for in 3D there exist several versions of N -extended AdS superspace. These superspaces have different isometry groups, and therefore they should allow different matter couplings. This paper is organised as follows. In section 2 we introduce the (p, q) AdS superspaces and study their geometrical properties. In section 3 we prove that all (p, q) AdS superspaces with X IJKL = 0 are conformally flat. The specific features of the (p, q) AdS superspaces with p + q ≤ 4 are studied in section 4. In section 5 we develop a general setup to construct (3,0) and (2,1) supersymmetric theories in AdS. Specifically, we define a family of covariant projective supermultiplets to describe supersymmetric matter, and then present a manifestly supersymmetric action. We also give an expression for the action obtained by integrating out the superspace Grassmann variables. In section 6 we demonstrate how to reformulate any (3,0) and (2,1) supersymmetric field theory in AdS as a dynamical system in a certain N = 2 AdS superspace. In section 7 we construct general off-shell supersymmetric σ-models in AdS. Section 8 contains a brief discussion of the results obtained. Some details on the derivation of the component action (5.37) are collected in the appendix.
2 Three-dimensional (p, q) AdS superspaces In this section, we develop the differential geometry of three-dimensional N -extended (p, q) AdS superspaces.

Superspace geometry of N -extended conformal supergravity
All (p, q) AdS superspaces can be realised as special background configurations within the 3D N -extended conformal supergravity that was originally sketched in [5] and then fully developed in [6]. In this subsection we recall those results of [6] which are necessary for our subsequent analysis.
In three dimensions, N -extended conformal supergravity can be described using a curved superspace which is parametrized by real bosonic (x m ) and real fermionic (θ µ I ) coordinates, z M = (x m , θ µ I ) , m = 0, 1, 2 , µ = 1, 2 , I = 1, · · · , N , (2.1) and is characterised by the structure group SL(2, R)×SO(N ). The superspace differential geometry is encoded in covariant derivatives of the form where the tangent space indices take the values α = 1, 2, a = 0, 1, 2, I = 1, · · · , N . In eq. (2.2), is the Lorentz connection; and is the SO(N )-connection. The Lorentz generators with two vector indices (M ab ), with one vector index (M a ) and with two spinor indices (M αβ ) are related to each other by the rules: M a = 1 2 ε abc M bc and M αβ = (γ a ) αβ M a . The generators of the group SO(N ) are denoted by N IJ . For more details on our notation and conventions see Appendix A of [6]. The generators of SL(2,R)×SO(N ) act on the covariant derivatives as follows: 5 To describe conformal supergravity, the covariant derivatives have to obey certain constrains [5]. With the constraints imposed, the Bianchi identities lead to the following (anti) commutation relations 6 derived in [6]: This algebra is given in terms of three dimension-1 tensor superfields, X IJKL , S IJ and C a IJ , which are real and have the symmetry properties The dimension-3/2 components of the curvature in (2.6b), R αβ K γ de and R αβ K γ P Q , are known algebraic functions [6] of first spinor covariant derivatives of the dimension-1 tensor superfields (2.7). It is useful to represent S IJ as a sum of its irreducible components 7 The Bianchi identities imply the following set of differential equations [6] where the dimension-3/2 superfields satisfy: The supergravity gauge group is generated by local transformations of the form 10) 6 For the purposes of this paper, we only need the explicit expressions for the dimension-1 components of the torsion and the curvature. 7 In this paper, we make use of S IJ and S IJ , as well as S and S := δ IK δ JL S IJ S KL /N . We hope our imperfect notation will not lead to any confusion.
with all the gauge parameters obeying natural reality conditions but otherwise arbitrary. Given a tensor superfield T , it transforms as follows: The conformal supergravity constraints proposed in [5] are invariant under super-Weyl transformations. This invariance plays a key role in the discussion of the multiplet structure of N -extended conformal supergravity in [5]. The super-Weyl transformations were not given explicitly in [5]. For N = 8 conformal supergravity, the finite form of super-Weyl transformations first appeared in [7]. In the case of N -extended conformal supergravity, the infinitesimal form of these transformations was described in [6]. Here we present for the first time the finite form of the N -extended super-Weyl transformations [6]. This result is essential for the analysis in section 3.
The super-Weyl transformation of the covariant derivatives is while the dimension-1 torsion and curvature tensors transform as For later use, we rewrite the super-Weyl transformations of S IJ and C a IJ in the following equivalent form: This concludes our summary of the superspace geometry of N -extended conformal supergravity [6].

Definition of (p, q) AdS superspaces
We are now prepared to introduce AdS superspaces. By definition, they correspond to those conformal supergravity backgrounds which satisfy the following requirements: (i) the torsion and curvature tensors are Lorentz invariant; (ii) the torsion and curvature tensors are covariantly constant.
Requirement (ii) has a series of implications. First of all, the conditions imply that all the dimension-3/2 curvatures in the second line of (2.6b) are identically zero. Moreover, the integrability conditions for the constraints (2.16), 17) are equivalent to the following algebraic constraints on S IJ and X IJKL : We now have to analyse all the implications of the Bianchi identities [ABC) [D A , [D B , D C }} = 0 (2.19) in the case that the covariant derivatives obeying the (anti) commutation relations (2.6a)-(2.6b) are further subject to the constraints (i) and (ii). Solving the Bianchi identities is straightforward albeit somewhat tedious and we omit the details. By analysing (2.19) we obtain a new crucial constraint on the torsion tensor S IJ : This shows thatŜ is a nonsingular symmetric N × N matrix if S 2 > 0; in this caseŜ/S is an orthogonal matrix. Moreover, by solving the Bianchi identities one readily derives a commutator of two vector covariant derivatives. The complete algebra of covariant derivatives turns out to be We recall that the structure-group indices are raised and lowered using δ IJ and δ IJ . Due to (2.20), the constraints (2.18a)-(2.18b) become In accordance with (2.20), there are two conceptually different cases: (a) S > 0; and (b) S = 0 and hence S IJ = 0. In the former case, eq. (2.21c) tells us that the commutator of two vector covariant derivatives is exactly that of 3D AdS space. The latter case corresponds to a flat superspace with the following algebra of covariant derivatives: This superspace is of Minkowski type for N = 1, 2, 3. However, for N ≥ 4 there may exist a non-zero constant antisymmetric tensor X IJKL constrained by so that the resulting superspace is a deformation of N -extended Minkowski superspace.
In what follows, our analysis will be restricted to the AdS case, S > 0.

Analysis of the (p, q) AdS constraints
We have seen that the (anti) commutation relations (2.21) require the algebraic constraints (2.20) and (2.22) as the consistency conditions. Let us analyse the implications of these equations. The most important equation to study is (2.20).
The torsionŜ = (S IJ ) is a real symmetric N × N matrix. A local SO(N ) transformation can be performed to diagonaliseŜ. Then, without loss of generality, the general solution of the constraint (2.20) is where S = (S IJ S IJ )/N > 0 is a positive parameter of unit dimension. In the 'diagonal frame' (2.25), we are left with an unbroken local group SO(p) × SO(q). In what follows, we assume p ≥ q. Such a superspace should originate as a maximally symmetric solution of the (p, q) AdS supergravity. The integers p and q determine the signature of S IJ .
For our subsequent analysis, it is handy to introduce a special notation associated with the diagonal frame (2.25). All the isovector indices running from 1 to p will be overlined, while those taking values from p + 1 to N will be underlined. With this notation, the components of S IJ in the diagonal frame are The diagonal frame is especially useful for solving the constraints obeyed by X IJKL .
Making use of (2.26), it is easy to see that the constraint (2.22a) is equivalent to This means that the only non-zero components of X IJKL are those which have all the indices of the same type, i.e. X IJKL and X IJKL .
Using (2.26) and (2.27), the second constraint on X IJKL , eq. (2.22b), dramatically simplifies. The strongest condition arises when one chooses the index I overlined and the index J underlined. In this case, due to (2.26) and (2.27), the expression (2.22b) is non-trivial only if, among the indices K, L, P and Q, one is overlined and the other three are underlined or vice versa. We then get the following equations It is clear that these equations can have nontrivial solutions only in the (N , 0) case. We have thus proved that the curvature X IJKL can only consistently appear in the AdS algebra if S IJ = S δ IJ . In this case, the equation (2.22b) simplifies to This is the same algebraic equation which emerges in the case of deformed Minkowski superspace, eq. (2.24).
The first case where X IJKL can appear in the algebra is N = 4. Here where X is a real constant parameter, and ε IJKL the completely antisymmetric Levi-Civita tensor (normalised by ε 1234 = 1) which is invariant under SO(4). The constraint (2.29) is automatically satisfied due to the identity Here we do not give a general solution of eq. (2.29) for N > 4. We just mention that a particular solution of eq. (2.29) for any N > 4 is obtained by choosing X IJKL = Xε IJKL 5 6··· N , with ε I 1 ...I N the appropriate Levi-Civita tensor. This solution is invariant under a subgroup SO(4) × SO(N − 4) of the gauged R-symmetry group SO(N ).
We conclude by rewriting the algebra of covariant derivatives (2.21a)-(2.21c) in the diagonal frame (2.25) for general (p, q) with q > 0 (in the case (N , 0) the algebra of covariant derivatives is given by eqs.
Note that the R-symmetry group of this superspace is SO(p) × SO(q).

The Killing vector fields of (p, q) AdS superspace
To describe rigid supersymmetric field theories in (p, q) AdS superspaces, we need to develop a superfield description of the corresponding isometry transformations. Here we use the diagonal frame where the results become more transparent, and consider only the cases X IJKL = 0. The isometry transformations are generated by (p, q) AdS Killing vector fields, which by definition obey the equation for some parameters Λ IJ , Λ IJ and Λ ab . This equation is equivalent to the relations which imply the standard Killing vector equation In accordance with (2.35), the parameters ξ α I , ξ α I , Λ IJ , Λ IJ and Λ ab are uniquely determined in terms of ξ a . It can be shown that the (p, q) AdS Killing vector fields generate the AdS supergroup OSp(p|2; R) × OSp(q|2; R).
It was demonstrated in [13] that AdS (3|2,0) and AdS (3|1,1) are conformally flat superspaces. Here we generalise this result to the case of arbitrary (p, q) AdS superspaces with X IJKL = 0. All superspaces AdS (3|p,q) are demonstrated to be conformally flat. Since the super-Weyl transformation of X IJKL is homogeneous, any AdS superspace with X IJKL = 0 is not conformally flat.
The super-Weyl transformations in N -extended conformal supergravity are given by eqs. (2.12a)-(2.12b). Our goal is to show that there exists a local parametrisation of the superspace AdS (3|p,q) such that the covariant derivatives D A take the form for some real scalar σ. Here D A = (∂ a , D I α ) are the covariant derivatives of N -extended 3D Minkowski superspace, obeying the (anti) commutation relations Under the super-Weyl transformations, the dimension-1 torsion and curvature superfields transform according to the equations (2.13a)-(2.13c). The superspace AdS (3|p,q) is characterised by the conditions C a KL = X IJKL = 0. Hence the parameter e σ in (3.1) must satisfy the equations Moreover, in accordance with the analysis of the previous section, the superfield S IJ has to be covariantly constant, and obey the algebraic constraint (2.20) which we rewrite as The equations (3.4) and (3.5) have to be obeyed by e σ in addition to the condition (3.3b).
To find a solution of the above equations, we make a Lorentz invariant ansatz for the super-Weyl parameter where The constant parameters a, b, c and d in (3.6) are real and dimensionless. As to the constant tensor s IJ , it is also real and has unit mass dimension.
The ansatz (3.6) has been shown to be the correct one in the case of the (2,0) and (1,1) AdS superspace [13]. It is also reminiscent of the conformally flat parametrisation of the 4D N -extended AdS superspace derived in [27] using group-theoretic techniques (direct proofs based on the use of super-Weyl transformations in 4D N = 1 and N = 2 supergravity theories can be found, e.g., in [28] and [29] respectively).
Let us turn to solving the conditions for conformal flatness using the ansatz introduced. As a first step, we observe from eq. (3.3a) that (3.8) By considering the θ-independent part of (3.5), we see that s IJ has to be constrained by Next, we impose the equation (3.3b). After some algebra, one sees that (3.3b) is satisfied provided To fix the value of a, it suffices to use again the equation (3.5) which so far has been solved at the θ = 0 order only. This equation tells us that a = −1 . (3.11) We end up with the following expression for the super-Weyl parameter: The geometry is characterized by the torsion S IJ that, by using (3.3a), can be computed to be It is an instructive exercise to prove the important relations To complete the analysis, we need to prove that the condition (3.4) holds, with the covariant derivatives defined by eqs. (3.1). This is equivalent to proving that D I α S JK = 0. We can simplify such a check by making a series of simple considerations. First of all, due to the very nature of the super-Weyl transformations, the covariant derivatives (3.1) define some conformal supergravity background with the additional conditions: the corresponding torsion and curvature tensors satisfy the equations (2.9a)-(2.9c) with X IJKL = C a KL = 0. This means that the torsion S IJ = S IJ + δ IJ S in eq. (3.13) satisfies It follows that for N = 4, a sufficient condition to have D I α S JK = 0 is D I α S = 0. This is indeed the case in accordance with (3.14a).
In the N = 4 case, the condition D I α S = 0 follows from (3.14a) and (3.15b). However we need to independently check whether the requirement S α I = 0 holds indeed. Using Here the left-hand side is zero due to (3.14b), and hence Since S I J is invertible, this equation gives S α I = 0 in the case that S = 0. On the other hand, choosing S = 0 in (3.17) gives Ultimately this equation tells us that S α I = 0. Therefore, we have demonstrated that S IJ is covariantly constant.
We conclude with a comment about the space-time geometry associated with the superspace AdS (3|p,q) . Given the expression for e σ , eq. (3.12), and the explicit form of the vector covariant derivative D a , eq. (3.1b), we can read off the space-time metric This coincides with a standard expression for the metric of AdS 3 computed using the stereographic projection for an AdS hyperboloid. 8 4 Elaborating on the AdS superspaces with p + q ≤ 4 In this section we would like to reformulate the algebra of covariant derivatives, which corresponds to a given (p, q) AdS superspace with p + q ≤ 4, in a form that is more suitable for describing matter couplings within the supergravity formulation of [6].

N = 1
In the N = 1 case, only the (1, 0) AdS superspace is available. Its geometry is determined by the (anti) commutation relations In the N = 2 case, there are two AdS superspaces: (2,0) and (1,1). They have already been studied in [13]. Here we would like to re-derive the main results of [13] using the analysis of the previous section.

(2,0) AdS superspace
The (2,0) covariant derivatives satisfy the (anti) commutation relations where ε 12 = ε 12 = 1 and we have introduced the U(1) generator J following [6] It is useful to switch to a complex basis for the spinor covariant derivatives, D I α → (D α ,D α ), such that D α andD α possess definite U(1) charges With the definition D a = D a , the algebra of covariant derivatives becomes together with their complex conjugates. Upon a redefinition of the AdS parameter, S → ρ/4, these (anti) commutation relations become identical to those which define the (2,0) AdS geometry introduced in [13].

(3,0) AdS superspace
Let us start with the (3,0) AdS geometry described by As shown in [6], in order to define important off-shell supermultiplets and matter couplings in N = 3 conformal supergravity, it is useful to introduce a new basis for the spinor covariant derivatives, D I α → D ij α , defined by the rule that any SO(3) isovector index is replaced by a pair of symmetric SU(2) isospinor indices. Specifically, the new covariant derivative D ij α is defined as 9 D ij α : where J kl = J lk is the SU(2) generator. In isospinor notations the (3,0) algebra takes the form

(2,1) AdS superspace
In the diagonal frame, the (2,1) algebra is We want to rewrite the previous algebra in isospinor notations. To do that, we first observe that the algebra is constructed from the AdS algebra in the general frame (2.21a)-(2.21c) with the choice S IJ = S δ IJ − (w 3 ) I (w 3 ) J with the vector (w 3 ) I = (0, 0, √ 2) in the third direction. It is clear that with an SO(3) rotation we can move to a general frame where S IJ = S (δ IJ − w I w J ) and w I such that w I w I = 2. Clearly, the structure group is still SO(2) since, for example, the algebra admits a central extension with constant central charge field strength given by 10 w IJ = ε IJK w K , D A w K = 0. The algebra (4.14a)-(4.14d) can be seen to become (4.15c) 9 We refer the reader to section 5 and Appendix A of [6] for details on our N = 3 isospinor notations including the properties and explicit definition of the (τ I ) ij matrices. From this form it is easy to move to isospinor notations in a general frame. We obtain Note that in the algebra the R-symmetry group is generated by the U(1) operator w pq J pq .

(4,0) AdS superspace
We start with the (4,0) case. This is particularly interesting being the first geometry where the covariantly constant X IJKL curvature can be used to deform the AdS geometry. Since X IJKL = Xε IJKL for N = 4, the (4,0) algebra is Note that the scalar X is a free parameter that does not affect the curvature of the body of AdS. In particular, we can freely add it also to the N = 4 Minkowski superspace. Its role is to deform the SO(4) part of the structure group. To see in details how the X field affects the algebra we change notations for the SO(4) isovector indices and move to pairs of SU(2) L ×SU(2) R isospinor indices making use of the isomorphism SO(4) ∼ = SU(2) L × SU(2) R /Z 2 . We define new covariant derivatives D iī α as 11 The SO(4) generator N KL in isospinor notation takes the form where L kl and Rkl are respectively the left and right SU(2) generators. Finally, the (4,0) algebra becomes It is interesting to note that for generic value of X the entire SO(4) group has non-trivial curvature in the algebra. But there are two points in which either the SU(2) R or the SU(2) L curvatures are zero and the structure group is reduced. These are given by

(2,2) AdS superspace
The next case we consider is the (2,2) geometry. In the diagonal frame this takes exactly the form (2.32c)-(2.32e) where the N IJ rotates the directions I = 1, 2 and the N IJ rotates the directions I = 3, 4 in the isovector space. The torsion S IJ = S diag(1, 1, −1, −1) is traceless δ IJ S IJ = 0. We can use this information to derive the (2,2) geometry in isospinor notations. The traceless condition tells us that which can be easily seen by remembering that [6] The constraint (2.20) in isospinor notation gives the condition In a general frame, in isospinor notations, the (2,2) algebra then takes the following form where we have defined the U(1) L and U(1) R generators L := l kl L kl , R := rklRkl . We are left with the (3,1) case. In the diagonal frame the geometry is with w I satisfying w I w I = 2 but otherwise an arbitrary isovector. The operatorN IJ generates an SO(3) algebra inside SO(4). This can be easily seen by observing that w I is left invariant,N KL w I = 0 and thenN IJ generates rotations orthogonal to w I . Note that the previous representation of the (3,1) algebra is the same in describing the general (N − 1,1) cases. By using (4.28a)-(4.28c) we can derive a representation of the (3,1) algebra in isospinor notations. This takes the form Note that the spinor covariant derivatives algebra can be rewritten as 30) or equivalently as where we have defined J kl := L kl + w kk w ll Rkl , (4.32a) The operator J kl = w kk w ll Jkl, or equivalently Jkl = w kk w ll J kl , generates the residual SU(2) algebra of the (3,1) AdS geometry and leaves w iī invariant.
5 Rigid N = 3 supersymmetric field theories in AdS:

Off-shell multiplets and invariant actions
In this and the next sections, our goal is to apply the supergravity techniques of [6] to describe general nonlinear σ-models in AdS 3 possessing N = 3 supersymmetry. We recall that the case of N = 2 AdS supersymmetry has already been studied in [13]. Similar in some aspects to N = 3, the case of N = 4 AdS supersymmetry nevertheless requires a separate analysis that will be given elsewhere.
In discussing off-shell supermultiplets and supersymmetric actions, we first give a unified presentation that applies equally well to the (3,0) and (2,1) AdS supersymmetry types. After that, we spell out those technical aspects of N = 3 supersymmetric theories in AdS 3 which look essentially different for the cases (3,0) and (2,1).
For our subsequent consideration, it is useful to rewrite the (anti) commutation relations for the (3,0) and (2,1) covariant derivatives in a unified form (which is inspired by the algebra of covariant derivatives in N = 3 conformal supergravity [13]): (2,1) AdS : where the covariantly constant tensor w ij = w (ij) is normalised by w ij w ij = 2.
The N = 3 Killing equations These relations imply, in particular, the following equations which will be important for our subsequent consideration. We recall that the parameter Λ ij is real, Λ ij = Λ ij .

Covariant projective supermultiplets
In complete analogy with matter couplings in N = 3 supergravity [6], a large class of rigid supersymmetric theories in (3,0) and (2,1) AdS superspaces can be formulated in terms of covariant projective supermultiplets. Before introducing these supermultiplet, a few words are in order regarding the so-called projective superspace approach.
A covariant projective supermultiplet of weight n, Q (n) (z M , v i ), is defined to be a Lorentz-scalar superfield that lives on the appropriate N = 3 AdS superspace M 3|6 (which is AdS (3|3,0) or AdS (3|2,1) ), is holomorphic with respect to isospinor variables v i on an open domain of C 2 \ {0}, and is characterised by the following conditions: is a homogeneous function of v of degree n, that is, Under the appropriate AdS isometry supergroup, which is OSp(3|2; R) × Sp(2, R) or OSp(2|2; R) × OSp(1|2; R), Q (n) transforms as follows: where ξ denotes an arbitrary AdS Killing vector field, eq. (2.33), and Λ ij the associated SU(2) parameter defined by (2.34). 13 In eq. (5.9), we have introduced The transformation law (5.9) involves an additional isotwistor u i , which is only subject to the condition (v, u) = 0, and otherwise is completely arbitrary. Both Q (n) and δ ξ Q (n) are independent of u i .
(iii) Q (n) obeys the analyticity constraint The analyticity constraint (5.11) and the homogeneity condition (5.8) are consistent with the interpretation that the isospinor v i ∈ C 2 \ {0} is defined modulo the equivalence relation v i ∼ c v i , with c ∈ C * , hence it parametrizes CP 1 . Therefore, the projective multiplets live in M 3|6 × CP 1 .
Two comments are in order. Firstly, it follows from eq. (5.9) that Secondly, the constraints (5.11) are fully consistent due to the facts that Q (n) is a Lorentz scalar, and the operators D α obey the anti-commutation relations (5.14) A more general family of off-shell supermultiplets is obtained by removing the condition (iii) in the above definition, while keeping intact the conditions (i) and (ii). Such supermultiplets are called isotwistor. These superfields can be used to construct projective ones with the aid of the so-called analytic projection operator There exists a real structure on the space of projective multiplets known as the smile conjugation. 14 Given a weight-n projective multiplet Q (n) (v i ), its smile conjugate, Q (n) (v i ), is defined by is a weight-n projective multiplet. In particular,Q (n) (v) obeys the analyticity constraint D One can also check thatQ Therefore, if n is even, one can define real projective multiplets,Q (2n) = Q (2n) . Note that geometrically, the smile-conjugation is complex conjugation composed with the antipodal map on the projective space CP 1 .
We now list several projective multiplets that can be used to describe superfield dynamical variables. A complex O(m) multiplet, with m = 1, 2, . . . , is described by a weight-m projective superfield H (m) (v) of the form: The analyticity constraint (5.11) is equivalent to If m is even, m = 2n, we can define a real O(2n) multiplet 15 obeying the reality condition H (2n) = H (2n) , or equivalently The field strength of an Abelian vector multiplet is a real O(2) multiplet [6]. For n > 1, the real O(2n) multiplet can be used to describe an off-shell (neutral) hypermultiplet.
An off-shell (charged) hypermultiplet can be described in term of the so-called arctic weight-n multiplet Υ (n) (v) which is defined to be holomorphic in the north chart C, of the projective space CP 1 = C ∪ {∞}: and its smile-conjugate antarctic multipletΥ (n) (v), Here we have introduced the inhomogeneous complex coordinate ζ = v 2 /v 1 on the north chart of CP 1 . The pair consisting of Υ [n] (ζ) andΥ [n] (ζ) constitutes the so-called polar weight-n multiplet.

Supersymmetric action
In order to formulate the dynamics of rigid N = 3 supersymmetric field theories in AdS 3 , a manifestly supersymmetric action principle is required. It can be readily constructed by restricting the locally supersymmetric action introduced in [6] to the appropriate AdS superspace. The action is generated by a Lagrangian L (2) (z, v), which is a covariant weight-2 real projective multiplet, and has the form: Here the line integral is carried out over a closed contour γ = {v i (t)} in CP 1 . The action involves an isotwistor superfield C (−4) (z, v) defined by for some isotwistor multiplet U (n) such that 1/∆ (4) U (n) is well defined. The superfield C (−4) is required to write the action as an integral over the full AdS superspace. It is actually a purely gauge degree of freedom in the sense that (5.24) is independent of the explicit choice of U (n) . Indeed, varying U (n) gives In the contribution to δS[L (2) ] which comes from the second term, we can integrate by parts, to strip δU (n) of ∆ (4) , and make use of the fact that L (2) and ∆ (4) U (n) are covariant projective multiplets. As a result, we obtain δS[L (2) ] = 0.
In the case of (2,1) AdS supersymmetry, there is a simple choice for C (−4) :

Supersymmetric action: Integrating out all the fermionic directions
The action (5.24) is manifestly invariant under arbitrary isometry transformations of the appropriate AdS superspace, AdS (3|3,0) or AdS (3|2,1) . The price to pay for this is twofold: (i) the action involves the superfield C (−4) which is a purely gauge degree of freedom; (ii) the action is given by an integral over six Grassmann variables while the Lagrangian L (2) depends only on four of these coordinates. Both drawbacks can be eliminated, at the cost of losing the manifest invariance under the AdS isometry supergroup, if one integrates out two or all of the six fermionic directions. To achieve this, one could use the powerful method of normal coordinates around a submanifold of curved superspace [48]. Here we are going to use an alternative technique which was first developed to derive the N = 1 supersymmetric action in AdS 5 [49].
Our point of departure is the N = 3 projective superspace action in three-dimensional Minkowski space which was introduced in [12]. It has the form where the Lagrangian L (2) (z, v) is a real weight-two projective multiplet, and the operators D These operators depend not only on the isotwistor v i (t), which varies along the integration contour, but also on a constant (t-independent) isotwistor u i chosen in such a way that v i (t) and u i are linearly independent at each point of the contour γ, that is v(t), u = 0. The action (5.27) is actually independent of u i , since it proves to be invariant under arbitrary projective transformations of the form where the matrix elements a(t) and b(t) obey the first-order differential equations f := df (t)/dt for any function f (t). This invariance follows from the following properties of the Lagrangian: (i) L (2) (v) is a homogeneous function of v i of degree two; and (ii) L (2) (v) obeys the analyticity condition It turns out that the property (ii) suffices to prove that the action (5.27) is invariant under the standard N = 3 super-Poincaré transformations in three dimensions [12].
We now try to generalise the above construction to the AdS case. Let z M = (x m , θ µ ı ) be local coordinates of the AdS superspace. Given a tensor superfield U (x, θ), we define its restriction to the body of the superspace, θ µ ı = 0, specifically 16 We also define the double-bar projection of the covariant derivatives Since for both the (3,0) and (2,1) AdS geometries it holds that [D a , D b ] = −4 S 2 M ab , we can use the freedom to perform general coordinate and local structure group transformations to choose a (Wess-Zumino) gauge in which where ∇ a stands for the covariant derivative of anti-de Sitter space AdS 3 , We are interested in constructing an AdS generalisation of the action (5.27). On general grounds, it should have the form with e := det −1 (e m a ). Note that in (5.36) the dots stand for curvature dependent corrections which are necessary for the action to be invariant under the symmetries of its parent action (5.24). It is interesting to note that there is one symmetry which is shared by the flat action (5.27) and the parent curved full superspace action (5.24): both are manifestly projective invariant (5.29). On the other hand S 0 is not projective invariant. As discussed in [49,42,48], one can actually exploit projective invariance as a tool to iteratively find the completion of S 0 to S[L (2) ] in (5.36). In Appendix A we sketch how to describe this approach for the (3,0) and (2,1) AdS cases. Let us now write down the form of the full N = 3 AdS projective action principle in components Here we have used the definitions The actions corresponding to the (3,0) and (2,1) AdS superspaces are obtained from (5.37) by choosing the curvature as follows: 6 Supersymmetric action: Reduction to N = 2 superspace The representation (5.37) obtained in the previous subsection, corresponds to the situation when all the Grassmann integrals in the action (5.24) have been done. Here we take a different course and reduce the superspace integral in (5.24) to that over a certain N = 2 subspace of the full N = 3 AdS superspace under consideration. Such a procedure cannot be carried out in a unified way for the cases (3,0) and (2,1), and thus a separate consideration should be given in each case.

AdS superspace reduction: (3,0) to (2,0)
To identify an N = 2 subspace of the N = 3 AdS superspace, we need a subset of four spinor covariant derivatives which, together with D a , lead to a closed set of (anti) commutation relations.
The notion of N = 2 projection is especially useful when dealing with projective multiplets. Given a covariant weight−n projective multiplet Q (n) (v), it can always be described in terms of a related superfield Q [n] (ζ) which depends on ζ and is proportional to the original superfield, Q [n] (ζ) ∝ Q (n) (v). The precise definition of Q [n] (ζ) depends upon the specific projective multiplet under consideration. Using Q [n] (ζ), the analyticity constraint (5.11) becomes 14) or equivalently This equation shows that the dependence of Q [n] (x, θ ı , ζ) on the Grassmann coordinates θ µ 12 is completely determined in terms of its dependence on the other Grassmann coordinates θ µ 11 and θ µ 22 . In other words, all information about the projective multiplet Q [n] (ζ) is encoded in its N = 2 projection Q [n] (ζ)|.
We now list the transformation laws of several projective multiplets under the (3,0) AdS isometry group, OSp(3|2; R) × Sp(2, R). All multiplets will be projected to (2,0) AdS superspace, however will will not indicate explicitly the bar-projection.
We recall that a weight-n projective superfield Q (n) transforms under the isometry group OSp(3|2; R) × Sp(2, R) as which follows from eq. (5.9). Given an arctic weight-n multiplet Υ (n) (v), it can be conveniently represented as Then Υ [n] (ζ) transforms as follows: Given an antarctic weight-n multipletΥ (n) (v), it is represented in the form The transformation law ofΥ [n] (ζ) is Given a real weight-(2n) multiplet G (2n) (v),G (2n) = G (2n) , it is represented as The transformation law of G [2n] (ζ) is To conclude the analysis of this subsection, we present the (3,0) supersymmetric action reduced to (2,0) superspace. In accordance with (6.21), associated with the Lagrangian L (2) (v) is the superfield L [2] (ζ) defined by the rule L (2) (v) = i(v 1 ) 2 ζL [2] (ζ). It turns out that the (3,0) supersymmetric action (5.24) takes the following form in (2,0) AdS superspace To prove that (6.23) is the (2,0) reduction of the (3,0) action (5.24) we check explicitly that it is invariant under the full isometry group of (3,0) AdS superspace, OSp(3|2; R) × Sp(2, R). Making use of (6.22), the variation of (6.23) is The expression in the first line corresponds to the variation of L [2] under an infinitesimal isometry transformation of (2,0) AdS superspace. Since the action is manifestly invariant under the (2,0) AdS isometry group, this variation vanishes. For the remaining variation, upon integration by parts, we obtain which is identically zero due to the identities We conclude by noticing that the auxiliary superfield C (−4) (5.24) has dropped out upon reduction to (2,0) AdS superspace.
6.2 AdS superspace reduction: (2,1) to (2,0) We now turn to developing N = 2 reduction schemes for the projective multiplets in (2,1) AdS superspace. The main difference between the (3,0) and (2,1) AdS superspaces is that the latter possesses the covariantly constant tensor w ij (which can be interpreted as the field strength of a frozen N = 3 vector multiplet). As follows from the algebra of (2,1) AdS covariant derivatives, eq. (4.16), the R-symmetry part of the holonomy group of this superspace is no longer SU(2) R , as in the (3,0) case; instead it is the group U(1) R which is associated with the generator by J = − i 2 w ij J ij . Therefore, the local SU(2) R group can be used to choose the SU(2) R connection to be In this gauge the tensor w ij becomes strictly constant, w ij = const, and turn into an invariant tensor of the (2,1) AdS isometry group OSp(2|2; R) × OSp(1|2, R). It turns out that different numeric choices for w ij correspond to the possibility to perform reduction either to the (2,0) AdS superspace or to the (1,1) one. In other words, the (2,1) AdS superspace allows two inequivalent N = 2 reduction schemes.

N = 3 supersymmetric sigma models in AdS
We are now prepared to apply the formalism developed above to construct general (3,0) and (2,1) supersymmetric nonlinear σ-models in AdS 3 .

Sigma models with (3,0) AdS supersymmetry
By analogy with the rigid supersymmetric case, it is natural to expect that a general nonlinear σ-model with (3,0) AdS supersymmetry can be realised in terms of covariant weight-one arctic multiplets Υ (1) I (v) and their smile-conjugatesΥ (1)Ī (v), with I = 1, . . . , n. What can be said about the Lagrangian L (2) of such a theory? The specific feature of the (3,0) AdS superspace is that there are no background projective multiplets which are invariant under the isometry supergroup OSp(3|2; R) × Sp(2, R). 18 In order for L (2) to be a covariant weight-two projective multiplet, it cannot depend explicitly on v i . It must be a function of the dynamical superfields only, where K(Φ I ,ΦJ ) is a homogeneous function of its arguments of degree one, In order for L (2) to be real with respect to the smile-conjugation it suffices to subject K(Φ,Φ) to additional conditions [40,41] This condition means that K(Φ,Φ) can be interpreted as the Kähler potential of a Kähler cone, see e.g. [50]. By definition, this is a Kähler manifold (M, g IJ ) possessing a homothetic conformal Killing vector χ with the property In particular, χ is holomorphic. Its properties include: with K the Kähler potential. Local complex coordinates for M can always be chosen such that which correspond to our specific case, eq. (7.3).
Consider the σ-model where At the moment we assume only the homogeneity condition (7.2). The transformation law of L [2] must be This should be induced by the variations of Υ [1] andΥ [1] in (7.9), which are It is a short calculation to show that L [2] given by eq. (7.9) transforms as in (7.10) if eq. (7.2) holds. On the other hand, the Lagrangian (7.9) is real under the smile conjugation provided the stronger conditions (7.3) hold.
In (2,1) AdS superspace, general nonlinear σ-models can be described in terms of covariant weight-zero arctic multiplets Υ I (v) and their smile-conjugatesΥĪ(v) using the Lagrangian where K(Φ I ,ΦJ ) is the Kähler potential of a real analytic Kähler manifold X . The interpretation of K as a Kähler potential is consistent, since the action generated by (7.13) turns out to be invariant under Kähler transformations of the form with Λ(Φ I ) a holomorphic function. The target space M of this σ-model proves to be an open domain of the zero section of the cotangent bundle of X , M ⊂ T * X . This can be shown by generalizing the flat-superspace considerations of [36,52].
In general, K(Φ,Φ) in (7.13) is an arbitrary real analytic function of n complex variables. In the case that K(Φ,Φ) obeys the homogeneity condition (7.3), the Lagrangian (7.13) proves to define an N = 3 superconformal σ-model. Such a theory can be reformulated entirely in terms of covariant weight-one arctic multiplets, Υ (1) I (v), and their smile-conjugates in complete analogy with the four-dimensional N = 2 supersymmetric σ-models in AdS [22]. This requires to make use of an intrinsic hypermultiplet, q i , associated with the (2,1) AdS superspace. This hypermultiplet is defined in complete analogy with the 4D consideration given in section 2.2 of [22].
It has been shown in the previous section that the (2,1) AdS superspace allows two types of N = 2 reduction, depending on the choice of w ij made. Any field theory in AdS (3|2,1) can be reformulated as a dynamical system in AdS (3|2,0) or in AdS (3|1,1) . Upon reduction to the (2,0) AdS superspace, the supersymmetric σ-model (7.13) proves to be described by the action while the other components Υ I 2 , Υ I 3 , . . . , are unconstrained complex N = 2 superfields. It is known that (2,0) AdS supersymmetry allows only R-invariant σ-model couplings [13]. As concerns the (2,1) supersymmetric σ-model (7.15), it possesses the following U(1) symmetry: compare with [53]. This symmetry is a special case of the transformation law (6.35) obtained by setting t = α = const and switching off the other parameters.

Conclusion
In conclusion, we briefly summarise our main results and list some open problems. In this paper we introduced the three-dimensional (p, q) AdS superspaces, studied their geometric properties and proved their conformal flatness when X IJKL = 0. Building on the results of [6], we then developed the fully-fledged projective-superspace formalism to construct off-shell N = 3 rigid supersymmetric field theories in AdS 3 . There are two types of such theories, with (3,0) and (2,1) AdS supersymmetry respectively. We are especially interested in theories possessing (p, q) AdS supersymmetry with N = p + q ≤ 4 because nonlinear σ-models exist only in these cases. We recall that the σ-models with N = p+q = 2 were studied earlier in [14,15,13]. The explicit construction of (p, q) supersymmetric σ-models with p+q = 3 was the subject of the present work. An open interesting problem is to extend our analysis given in this paper to the cases N = p+q = 4. Conceptually, this should be similar to the N = 3 case studied above, however some nontrivial new aspects will emerge. In particular, of special interest are those (4,0) supersymmetric σ-models which correspond to the extremal case (4.21).
In this paper we constructed the general (3,0) and (2,1) supersymmetric σ-models described by off-shell polar hypermultiplets defined on the (3,0) and (2,1) AdS superspaces respectively. We then reduced these σ-models to certain N = 2 AdS superspaces. An interesting open problem is to reformulate the σ-models obtained in terms of N = 2 chiral superfields in AdS 3 . (The importance of such a formulation is that it should provide a direct access to the hyperkähler geometry of the target space [54,19,20].) This can be achieved by generalising the approaches developed in [55,56,22].
As shown in this paper, the N = 3 AdS supersymmetry imposes nontrivial restrictions on the σ-model hyperkähler target spaces. The most unexpected outcome is that (3,0) AdS supersymmetry requires the σ-model target spaces to be hyperkähler cones. Nevertheless, this result has a natural geometric origin. The main difference between the two types of N = 3 supersymmetric σ-models in AdS 3 is encoded in the corresponding R-symmetry groups: SO(3) in the (3,0) case and SO(2) in the (2,1) case. It can be shown that any one-dimensional subgroup H = SO(2) of the R-symmetry group acts faithfully by rotations on the two-sphere of complex structures of the hyperkähler target space (M, g, J A ). 20 Here g µν is the hyperkähler metric, and (J A ) µ ν is the complex structures of M, J A = (J 1 , J 2 , J 3 ), obeying the quaternionic algebra J A J B = −δ AB 1 + ε ABC J C .
Suppose that J 3 is invariant under the action of the subgroup H, and let V µ be the Killing vector V µ associated with H. Without loss of generality, we have The Killing vector V µ is holomorphic with respect to J 3 , and we can introduce the corresponding Killing potential K defined by As shown in [22], K is a globally defined function over M, and is the Kähler potential with respect to J 1 and J 2 and indeed any complex structure J ⊥ which is perpendicular to J 3 . In other words, It follows that the Kähler forms associated with J 1 and J 2 are exact, and thus M is non-compact [19,20,22]. As shown in [22], the Kähler potential K with respect to J 3 can be chosen such that So far, we have taken into account only the fact that the R-symmetry group contain a subgroup SO (2). In the case that the R-symmetry group coincides with SO(3), the above consideration implies that K = K, and hence We further deduce that χ µ = g µν ∇ ν K is a homothetic conformal Killing vector, and therefore M is a hyperkähler cone [50,51]. In regard to the above discussion, we should also mention an interesting work [58] in which it was shown that a sufficient condition for a 4D N = 2 σ-model in projective superspace to be superconformal is that its R-symmetry is SO(3).
The supergravity techniques of [6] can straightforwardly be applied to construct offshell σ-models in the deformed N = 4 Minkowski superspace described by covariant derivatives obeying the (anti) commutation relations {D iī α , D jj β } = 2iε ij ε¯ijD αβ + 2iε αβ ε¯ijXL ij − 2iε αβ ε ij XR¯ij , The integrand can be considerably simplified. Using the algebra of covariant derivatives, (5.1a)-(5.1b), it is not difficult to derive the following relation which has to be plugged in eq. (A.10). Next, we evaluate the anti-commutators in (A.10) and iteratively move all the Lorentz and SU(2) generators to the right. Once they hit L (2) we use the identities M αβ L (2) = v i v j J ij L (2) = 0 and v i u j J ij L (2) = −(v, u)L (2) . To compute the contributions coming from u i u j J ij L (2) one has to use the following formula This can be obtained using the results of [42,48], and it holds for any operator T These results give the desired action (5.37).