Embedding formalism for N -extended AdS superspace in four dimensions

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Introduction
The simplest AdS superspace in four dimensions, AdS 4|4 , was introduced in the early years of supersymmetry by Keck [2] and Zumino [3] as the coset superspace 1 and the thorough study of general superfield representations on AdS 4|4 was given by Ivanov and Sorin [4].It was also realised that AdS 4|4 originates as a maximally supersymmetric solution in the following off-shell formulations for N = 1 supergravity: (i) the old minimal supergravity [5][6][7][8] with a cosmological term [9], see [10,11] for a review; and (ii) the nonminimal AdS supergravity [12].
The group-theoretic realisation (1.1) of N = 1 AdS superspace has a natural extension to the N > 1 case (see, e.g., [13]) The description of AdS 4|8 as a maximally supersymmetric solution in the minimal off-shell formulation for N = 2 supergravity with a cosmological term, developed by de Wit, Philippe and Van Proeyen [14], was given in [15][16][17]. 2  The conformal flatness of AdS 4|4 was first established in [4], and it was later re-derived in textbooks [10,11] within the supergravity framework.The superconformal flatness of AdS 4|4N  was demonstrated in [20] for arbitrary N .Alternative proofs of the conformal flatness of AdS 4|8  were given in [16,21] using the off-shell N = 2 supergravity framework.Ref. [22] described different conformally flat realisations for AdS 4|4 and AdS 4|8 which are based on the use of Poincaré coordinates.
In the non-supersymmetric case, there exist two different but equivalent realisations of AdS d : (i) as the coset space O(d − 1, 2)/O(d − 1, 1); and (ii) as a hypersurface in R d−1,2 Both realisations of AdS d have found numerous applications in the literature.As regards AdS 4|4N , only the coset superspace realisation (1.2) had existed for many years.The supertwistor and bi-supertwistor formulations for AdS 4|4N have recently been developed [1].Analogous results in three dimensions have been derived in [1,23].In this paper we elaborate on the superembedding formalism 3 for AdS 4|4N .
Since the work by Ferber [26], supertwistors have found numerous applications in theoretical and mathematical physics.In particular, supertwistor realisations of compactified N -extended Minkowski superspaces have been developed in four [27,28] and three [29,30] dimensions, and their harmonic/projective extensions have been derived [29][30][31][32][33][34][35][36][37]. 4 Recently, supertwistor formulations for conformal supergravity theories in diverse dimensions have been proposed [39,40].To the best of our knowledge, the supertwistor realisations of AdS superspaces in three and four dimensions have been given only in [1,23], although (super)twistor descriptions of (super)particles in AdS spaces had been studied in the literature earlier [41][42][43][44][45][46][47][48][49][50] (see also [51]). 5  This paper is organised as follows.In section 2 we give a brief review of the (bi)supertwistor description of AdS 4|4N and present a manifestly OSp(N |4; R) invariant model for a superparticle in AdS 4|4N .Section 3 is devoted to presenting a novel realisation of the AdS supergroup, which is then used in section 4 to develop a coset construction for AdS 4|4N .The coset construction is applied in section 5 to work out the differential geometry of AdS 4|4N .In section 6, by employing the framework of conformal superspace, we describe the most general conformally flat supergeometry and then specialise our construction to the case of AdS 4|4N .The main body of the paper is accompanied by several technical appendices.Appendix A includes essential definitions concerning the supergroup OSp(N |4; R) and corresponding supertwistors.Appendix B provides a review of the conformally flat atlas for AdS d .In appendix C, we spell out the N -extended superconformal algebra.
2 The (bi)supertwistor description of AdS 4|4N   In this section we give a brief review of the construction of [1].The reader is referred to appendix A for the technical details concerning the supergroup OSp(N |4; R) and supertwistors.
Associated with the space of even complex supertwistors, C 4|N , is a Grassmannian of even two-planes.Given such a two-plane P, it is spanned by two even supertwistors T µ , (2.1) The property of P being a two-plane means that the bosonic bodies of T 1 and T 2 are linearly independent complex four-vectors.An arbitrary element Q ∈ P is a linear combination Q = T µ q µ , with the coefficients q µ being even elements of the Grassmann algebra.By construction, the supertwistors (2.1) are defined modulo the equivalence relation since both T µ and T ′µ define the same two-plane P.
We restrict our attention to the subset of those two-planes which satisfy the constraints det P sT JP = 0 , (2.3a) P † JP ≡ ( * P) sT JP = 0 . (2.3b) Here (2.3a) refers to the body of the 2 × 2 supermatrix P sT JP, and * T denotes the conjugate of a pure supertwistor T , see eq. (A.14).The conditions (2.3) imply that the bodies of the four even supertwistors T μ = (T µ , * T μ) form a basis for C 4 , in particular the supertwistors (2.1) generates a two-plane.We emphasise that the conditions (2.3) are invariant under the equivalence transformations (2.2).In what follows, the supertwistor * T will be denoted T .
We say that any pair of even supertwistors P, eq.(2.1), constrained by the conditions (2.3) constitutes a frame.The space of frames will be denoted F N .The supergroup OSp(N |4; R) acts on F N by the rule This group action is naturally extended to the quotient space F N / ∼.The latter proves to be a homogeneous space of OSp(N |4; R), which was identified in [1] with the AdS superspace, Given two frames T μ, T μ ∈ F N , one can construct the following OSp(N |4; R)-invariant two-point functions: with ℓ a fixed positive parameter.They do not change if T and T are replaced by their equivalent frames (2.2), and therefore these OSp(N |4; R)-invariant two-point functions are well defined on AdS 4|4N .In the non-supersymmetric case, N = 0, the three two-point functions (2.6) coincide.
Given a point in F N , we associate with it the graded antisymmetric matrices ) (2.7b) These supermatrices are invariant under arbitrary equivalence transformations (2.2), and therefore they may be used to parametrise AdS 4|4N .The bi-supertwistors (2.7) have the following properties: where X [AB X CD} denotes the graded antisymmetric part of X AB X CD .Using the results of [36], the bi-supertwistor description of AdS 4|4N defined by (2.8) may be shown to be equivalent to the supertwistor one described earlier.
Restricting the above bi-supertwistor realisation of AdS 4|4N to the N = 0 case gives the bi-twistor formulation of AdS 4 , which in turn leads to a standard embedding formalism for AdS 4 .Building on the analysis given in section 3.3 of [1], it may be used to derive the reality condition (2.9) Here X α β and X α β denote the J-traceless parts of X α β and Xα β , respectively, Associated with X α β is a real 5-vector Here Γ â = (Γ â) α β are real 4 × 4 matrices which obey the anti-commutation relations and are characterised by the property The explicit realisation of Γ â is given, e.g., in [30].Making use of the completeness relation we obtain It may be shown that the bi-twistor description of AdS 4 is equivalent to the bi-spinor formalism introduced in [53].
Since the two-point functions (2.6) are invariant under arbitrary equivalence transformations (2.2), they can be expressed in terms of the bi-supertwistors (2.7).In terms of the supermatrices X = (X A B ) and X = ( XA B ) defined by these expressions have the form: ) ) We point out that the OSp(N |4; R) transformation (2.4) acts on X and X as follows The bi-supertwistor realisation described above facilitates the construction of manifestly OSp(N |4; R) invariant models.Indeed, let us consider the following worldline action for a superparticle on AdS 4|4N where τ parametrises the world line, e denotes the einbein, κ is a real dimensionless parameter, and m is a mass parameter.We can see that in the non-supersymmetric case, N = 0, the κ-term is absent, since the three two-point functions (2.6) coincide.

Isomorphic realisation of the AdS supergroup
The supergroup OSp(N |4; R) possesses an alternative realisation, which we introduce below and which turns out to be useful for applications.There is a simple motivation to look for such a realisation.To explain it, we consider the non-supersymmetric case, N = 0.It follows from (2.3a) that for every frame the 2 × 2 matrices F and G are non-zero.In the framework of the coset construction, however, it would be useful to deal with an isomorphic realisation of Sp(4, R) that would allow a frame such that either F = 0 or G = 0 .
The supergroup OSp(N |4; R) C proves to be isomorphic to OSp(N |4; R).The proof is based on considering the following supermatrix correspondence: in conjunction with the supertwistor transformation for every supertwistor T .Here the supermatrix U is defined as It obeys the useful properties: and These conditions imply that Associated with OSp(N |4; R) C are two invariant inner products defined as for arbitrary pure supertwistors T and S. The conditions (3.2) impose restrictions on the blocks of g.For these are: and In the original realisation of OSp(N |4; R) the reality condition could be realised as the coincidence of the supertranspose and the Hermitian conjugate, eq.(A.13b).For our new realisation of the supergroup, (A.13b) is replaced with the following condition From this we have the following conditions (3.15d) We will now discuss involution for the supertwistors T .Since the transformation (3.5b) applies to every supertwistor T , we can also consider it applied to * T .We have This acts explicitly on a supertwistor T as The components of * T are given by * T A = (−1) Let us introduce a new operation, denoted by ⋆, by removing the factor of −i in (3.17): The components of ⋆T are given by We therefore have the following reality condition with respect to the map ⋆ The map ⋆ is an involution, since it satisfies the property We also observe that which, in conjunction with the properties (3.8), yields the following It is useful to express the constraints (2.3), the two-point functions (2.6), and the bisupertwistors (2.7) in terms of the new realisation of the supergroup.The constraints can be expressed as For the two-point functions we find The bi-supertwistors (2.7) can be expressed in terms of transformed supertwistors T as follows They satisfy the following properties For the case N = 0, the ε-traceless parts of the bi-supertwistors take the form As before, we can express the two-point functions (3.26) in terms of the supermatrices X A B and XA B defined by They then take the form )

Coset construction
The alternative realisation of the AdS supergroup described in the previous section is ideal for developing a coset construction for AdS 4|4N .To start with, it is worth recalling some basic definitions, see e.g.[54] for more details.Consider a homogeneous space X = G/H x 0 , where G is a Lie group and H x 0 is the isotropy subgroup (or stabiliser) of some point x 0 ∈ X.A global coset representative is a bijective map S : X → G such that π • S = id X , where π denotes the natural (canonical) projection π : G → G/H x 0 .For many homogeneous spaces, no global coset representative exists.In such a case, local coset representatives S A : U A → G with the property π • S A = id U A can be introduced on open charts {U A } that provide an atlas for X.In the intersection of two charts U A and U B , U A ∩ U B = ∅, the corresponding coset representatives S A and S B are related by a little group transformation, S B (x) = S A (x)h AB (x), with h AB (x) ∈ H x 0 .

Isotropy subgroup
As a marked (preferred) point P (0) of AdS 4|4N , we choose The stabiliser H of P (0) consists of those elements h of the AdS supergroup OSp(N |4; R) C which satisfy the conditions These conditions imply that Thus the stability subgroup H is isomorphic to The bi-supertwistors (3.27) corresponding to the preferred point P (0) take the form

Generalised coset representative
The freedom to perform arbitrary equivalence transformations (2.2) can be used to fine-tune the conditions (3.25) to for a fixed positive parameter ℓ.Such a frame is said to be normalised.Under the condition (4.6a), the equivalence relation (2.2) turns into The space of normalised frames will be denoted F N .Along with the definition (2.5) given earlier, the N -extended AdS superspace can equivalently be defined as AdS 4|4N where the equivalence relation is given by (4.7).
The conditions (4.6) can be recast in terms of the two-plane and imply the following constraints: Relation (4.9a) tells us that at least one of the 2 × 2 matrices F and G is nonsingular.
Associated with the normalised two-plane P is the following group element The fundamental property of S(P) is that S(P)P (0) = P, for any normalised two-plane P ∈ F (ℓ) N .We point out that D is symmetric, D = D T .The functional forms of the matrices A and C are fixed through the condition S(P)P (0) = P and the reality conditions (3.15a) and (3.15c).The remaining blocks are then fixed by the group requirements (3.12) and (3.13).It is possible to obtain alternate expressions for the blocks D and B, which may be more suited to performing calculations.They take the following form These expressions can be seen to coincide with (4.10d) and (4.10e) by using the group requirements and the general form for the inverse of a supermatrix.
The group element S(P) is characterised by the property with N ∈ SL(2, C).This relation means that S(P) is not a genuine coset representative that is used in the coset construction.However, S(P) will allow us to obtain a coset representative if we pick a single two-plane in each equivalence class.This may be readily done in coordinate charts for AdS 4|4N .

AdS space (N = 0)
As noted above, at least one of the 2 × 2 matrices F and G, see eq. (4.8), is nonsingular.Therefore we can naturally introduce two coordinate charts for F (ℓ) N that provide an atlas.We define the north chart to consist of all normalised two-planes with det F = 0. Similarly, the south chart is defined to consist of all normalised two-planes with det G = 0.
In the north chart, we can use the freedom (4.7) to choose F ∝ ½ 2 , and then where λ = 0 is a parameter, and σ are the Pauli matrices.The constraints (4.9a) and (4.9b) give, respectively, x † = x .(4.14b) It follows that x m is real and x 2 := η mn x m x n = ℓ 2 .We also observe that λ = λ .Since there is still a remnant of the equivalence relation (4.7), T µ ∼ −T µ , it can be used to fix λ > 0. Then we observe that the coordinate chart is specified by and the parameter λ is given by (4.16) The real coordinates x m parametrise AdS 4 in the north chart.Direct calculation of the twopoint function (3.26a) in this chart yields In the south chart, the gauge freedom (4.7) can be used to choose G ∝ ½ 2 , and then for some parameter γ = 0. Now, repeating the north-chart analysis tells us that the local coordinates y m are real, and the following relations hold: The two-point function (3.26a) in the south chart is In the intersection of the two charts, the transition functions are It follows that x 2 < 0 ⇐⇒ y 2 < 0. Comparing the above relations with those described in appendix B, we find complete agreement except for the sign difference (4.21) and (B.10).

N = 0
The analysis of the previous subsection can be extended to the supersymmetric case in a similar fashion.Let us consider the north chart in which the matrix F in (4.8) is nonsingular.
The equivalence relation (4.7) can once again be used to choose F ∝ ½ 2 , and then Making use of (4.6) leads to the relations The former is solved by We see that the two-planes (4.22) are parametrised by the chiral coordinates x m + and θ I µ , with The coset representative in the north chart is given by The two-point function (3.26a) computed in the north chart yields where In the non-supersymmetric case, N = 0, this reduces to (4.17).
In the south chart, the gauge freedom (4.7) can be used to fix G ∝ ½ 2 .Repeating the analysis of the north chart leads to with The former is solved by We see that the two-planes in the south chart (4.28) are parametrised by the chiral coordinates y m + and ξ I µ , with y m + = y m + iξ I σm ξI .
The coset representative in the south chart is given by The two-point function (3.26a) computed in the south chart yields where In the intersection of the two charts, the transition functions are given by In addition, the two coset representatives (4.25a) and (4.31a) are related in the intersection by the point-dependent little group transformation Explicitly, h −1 is given by ) We see that n −1 is chiral, through the transition functions (4.34).
So far we have only considered the form of the two-planes in the north and south charts.It is also useful to describe the form of the bi-supertwistors (3.27) in an explicit coordinate system.In the north chart they take the form It is of interest to compare this supermatrix with a similar result for compactified N -extended Minkowski superspace, see eq. (3.17) in [36].

Superspace geometry
In this section we give explicit expressions for the vierbein, connection, torsion tensor and curvature tensor.From these expressions the graded commutation relations of the covariant derivatives can be derived.

Geometric structures in AdS 4|4N
Let us denote by G the superalgebra of the AdS supergroup OSp(N |4; R) C , and by H the algebra of the stability group (4.4).Let W be the complement of H in G, G = H ⊕ W. The superalgebra G consists of even supermatrices ) Additionally, the elements w ∈ W take the form With the following row-vector definition the elements (5.3) take the form We may uniquely decompose the Maurer-Cartan one-form ω = S −1 dS as a sum ω = E + Ω, where E = S −1 dS| W is the vierbein taking its values in W. The Maurer-Cartan one-form is where the blocks are given by ) One can make use of the group conditions (3.12) to recast E in an equivalent form In the above, E Θ is given by The Maurer-Cartan one-form (5.6) can be decomposed into supermatrices of the form (5.2) and (5.5) to obtain the vierbein and connection.The connection is where It is possible that these expressions may be simplified by using an explicit form for A −1 , with A given by (4.10b), however the above expressions appear most convenient for proving the required properties (5.12) The vierbein is where and E is defined as in (5.7b) or (5.8).It is straightforward to show that (5.14) is Hermitian, using (4.9b).
Using the above expressions we can now compute the torsion T and curvature R. In accordance with the coset construction, they are defined as follows: There exists another simple expression for both T and R, given by Following (5.15), the torsion is given by where (5.18e) The curvature is given by where )

Covariant derivatives
The vierbein and connection (as well as curvature and torsion) can be decomposed into the bases corresponding to the superalgebra W and the algebra H. Accordingly, we must introduce a basis W A = (P a , q Iα , qI α) for the superalgebra W and a basis H Î = (M ab , J IJ ) for the algebra H.The elements h of H and w of W, given by (5.2) and (5.3), may be written as a linear combination of generators w = v a P a + ψ Iα q Iα + ψI α qI α .
The vierbein and the torsion two-form, as elements of W, can be decomposed with respect to the basis as ) to obtain the one-form E A = (E a , E Iα , ĒI α) and the the torsion T A = (T a , T Iα , T I α).A similar procedure follows for the curvature.We may further decompose the torsion and curvature components as (5.24b) Building on the approach used in [55], we can use (5.16) and the graded commutation relations (5.22) to determine the non-vanishing components of the torsion and curvature to be These components can be used to construct the graded commutation relations of the covariant derivatives (5.26) The algebra of covariant derivatives is thus given by ) ) (5.27f)

N = 1 AdS superspace
Many of the expressions in subsection 5.1 contain A −1 and D. These are, in principle, expressible in terms of F , G, and Θ.These expressions are, however, N -dependent.Below, we will discuss both of these in the N = 1 case.

Using the group requirements (3.12a) we can rearrange for (A
(5.28) which in the N = 1 case yields the following expression where Furthermore, D has the explicit solution We can use these expressions to compute E from the vierbein (5.7b).For N = 1 it is where This expression coincides with (5.9) when considered in the N = 1 case.

North and south charts
The results of subsections 5.1, 5.2 and 5.3 did not make use of the freedom (4.7) to fix a coordinate system.In this section we will use these results to describe the geometry in the N = 1 case for the north and south charts, given by two-planes of the form (4.22) and (4.28).
In the north chart, the vierbein (5.13) reads where η θ is computed using (5.33) as In the above, dθ and ẽ = e a (σ a ) are the flat N = 1 superspace vielbeins.The general forms for the vielbeins of a superspace with superconformally flat geometry are ) where σ (σ) is chiral (antichiral).In our case it is straightforward to compute the coefficients in (5.36), which yields the following expression λ = e − 1 2 σ .
(5.37) Indeed, (5.34) can be shown to take the form with Ẽ = E a (σ a ) and (E θ ) given by (5.36).The connection is given by where the components of the connection read (5.40) We introduce the inverse Let us then define the vector fields (5.42) Here D M := (∂ m , D µ , D μ) are the N = 1 flat superspace covariant derivatives.We find (5.43c) The components of the connection Ω north were given with respect to the basis ε M in (5.40).Using the inverse vierbein defined by (5.41), the connection can be decomposed into the basis {E A }, with which we can then construct explicit expressions for the covariant derivatives (5.44) They take the following form (5.45c) The expressions (5.45) can be seen to coincide with the general form for the covariant derivatives of a conformally flat superspace.
In the south chart, the vierbein (5.13) is given by where η ξ is We showed in section 4.4 that the coset representatives in the north and south charts were related by a little group transformation, see (4.35).Under such a transformation, the vierbein and connection transform as follows ) (5.48b) We can see then that the vierbein supermatrix in the north chart is related to that in the south chart by which yields with n −1 given by (4.36b).The vector fields E A are also related in the intersection of the two charts.We find (5.51b)

Conformally flat supergeometry
This section is devoted to a description of the most general four-dimensional conformally flat supergeometry.Our approach will be to begin with a general conformally flat superspace whose local structure group is the superconformal group. 6Then, by performing a series of gauge fixings, and passing through the conventional U(N ) and SU(N ) superspaces, we realise the AdS supergeometry within this framework.

Conformal superspace: conformally flat geometry
We consider a conformally flat N -extended superspace M 4|4N , parametrised by local coordinates z M = (x m , θ µ ı , θı μ), where m = 0, 1, 2, 3, µ = 1, 2, μ = 1, 2 and ı = 1, . . ., N .The structure group is chosen to be SU(2, 2|N ), the N -extended superconformal group.Its corresponding Lie superalgebra, su(2, 2|N ), is spanned by the translation ), Lorentz M ab , R-symmetry Y and J i j , dilatation D, and the special conformal K A = (K a , S α i , Si α) generators, see appendix C for more details.The geometry of this superspace is encoded within the conformally covariant derivatives , which take the form: where E A M denotes the inverse supervielbein and the remaining superfields are connections associated with the non-translational generators of the superconformal group.
By definition, the gauge group of conformal supergravity is generated by local transformations of the form where the gauge parameters satisfy natural reality conditions.Given a conformally covariant tensor superfield U (with its indices suppressed), it transforms under such transformations as follows: In general, the algebra of covariant derivatives [∇ A , ∇ B } should be constrained such that it: (i) has a super Yang-Mills structure; and (ii) is expressed solely in terms of a single superfield, the super-Weyl tensor.In this section, we will restrict our attention to conformally flat backgrounds, which are characterised by vanishing super-Weyl tensor.As a result, the only non-vanishing sector of [∇ A , ∇ B } is

Degauging (i): U(N ) superspace
According to eq. ( 6.2), under an infinitesimal special superconformal gauge transformation K = Λ B K B , the dilatation connection transforms as follows Thus, it is possible to impose the gauge B A = 0, which completely fixes the special superconformal gauge freedom. 7As a result, the corresponding connection is no longer required for the covariance of ∇ A under the residual gauge freedom and may be extracted from ∇ A , Here the operator D A involves only the Lorentz and R-symmetry connections The next step is to relate the special superconformal connection F AB to the torsion tensor associated with D A .To do this, one can make use of the relation In conjunction with (6.4), this relation leads to a set of consistency conditions that are equivalent to the Bianchi identities of (conformally flat) U(N ) superspace [58].Their solution expresses the components of F AB in terms of the torsion tensor of U(N ) superspace and completely determines the algebra [D A , D B }.

N = 1 case
We begin by solving eq.(6.8) in the N = 1 case.The outcome of this analysis is: ) ) Here R is a chiral scalar superfield while X α is the chiral field strength of a U(1) vector multiplet and G α α is a real vector superfield.These are related via We now pause and comment on the geometry described by D A .In particular, by employing (6.8) one arrives at the following anti-commutation relation It follows that if one performs the shift then the G-dependent terms in (6.11) vanish.The resulting algebra of covariant derivatives, up to dimension-3/2, takes the form which describes a U(1) superspace [10,58] with vanishing super-Weyl tensor.
Above we made use of the special conformal gauge freedom to degauge from conformal to U(1) superspace.Now, we will show that the residual dilatation symmetry manifests in the latter as super-Weyl transformations.To preserve the gauge B A = 0, every local dilatation transformation should be accompanied by a compensating special conformal one This is the case only if the special conformal parameter is We now determine what transformation of D A and the torsions of U(1) superspace this induces.They may be determined by making use of the following relation Specifically, one finds that the super-Weyl transformations of the degauged geometry are: which are in agreement with the ones presented in [59].Additionally, for infinitesimal Σ, these transformations may be obtained from the ones presented in [58].

N > 1 case
We now extend the analysis presented above to the N > 1 case.A routine calculation leads to the following expressions for the degauged special conformal connection: The dimension-1 superfields introduced above have the following symmetry properties: and satisfy the reality conditions The U(1) R charges of the complex fields are: Now, by employing (6.8), we find that the anti-commutation relations for the spinor covariant derivatives are: At the same time, the consistency conditions arising from solving (6.8) lead to the Bianchi identities: Now, in complete analogy with the N = 1 story described above, we show how the residual dilatation symmetry of conformal superspace manifests in the present geometry as super-Weyl transformations.It may be shown that the following combined dilatation and special conformal transformation, parametrised by a dimensionless real scalar superfield Σ = Σ, preserves the gauge B A = 0: At the level of the degauged geometry, this induces the following super-Weyl transformations where we have made the definitions: In the infinitesimal case, these transformations are a special case of the ones presented in [58]. 8urther, for N = 2, these may be read off from the finite super-Weyl transformations presented in [60].

Degauging (ii): SU(N ) superspace
In the preceeding subsection we have shown that the degauging of the N -extended conformally flat supergeometry described in section 6.1 leads to (conformally flat) U(N ) superspace.
The latter is characterised by the property that its local structure group is SL(2, C) × U(N ) R .In the present section we will further degauge this geometry by breaking the local R-symmetry group down to SU(N ) R .This procedure consists of the following steps.First, one must eliminate the U(1) R curvature.This involves redefining D A to absorb such terms in the algebra of covariant derivatives and employing super-Weyl transformations to set the remaining contributions, which describe purely gauge degrees of freedom, to zero.For N = 1, this role is played by the chiral spinor X α , while in the N > 1 case, G α αi j should be gauged away.Next, by performing some local U(1) R transformation one may always set Φ A = 0, and so the local R-symmetry group has been reduced to SU(N ) R .Finally, one must identify the class of residual combined super-Weyl and local U(1) R transformations preserving this geometry.As will be shown below, such transformations are parametrised by a dimensionless chiral scalar Σ (and its conjugate).

N = 1 case
As pointed out above, the spinor X α is the chiral field strength of an Abelian vector multiplet and describes purely gauge degrees of freedom.By employing the super-Weyl transformatons (6.17f) it is possible to fix the gauge By inspecting the algebra of covariant derivatives (6.13), it is clear this leads to vanishing U(1) R curvature.Hence, in this gauge the U(1) R connection Φ A may also be gauged away Then, the algebra of covariant derivatives (6.13) reduces to ) which describes a conformally flat GWZ geometry [61].This algebra should be accompanied by the constraints (6.10), provided one sets X α = 0.
Equation (6.17f) tells us that imposing the condition X α = 0 does not fix completely the super-Weyl freedom.The residual transformations are generated by parameters of the form However, in order to preserve the U(1) R gauge Φ A = 0, every residual super-Weyl transformation (6.30) must be accompanied by the following compensating U(1) R transformation (6.31)This leads to the transformations: .32e) In the infinitesimal limit, these transformations may be obtained from the ones given in [62].

N > 1 case
As discussed above, in the N > 1 case, the torsion G α αi j describes purely gauge degrees of freedom.Thus, by employing the super-Weyl freedom described by eq. ( 6.25), it may be gauged away .33)In this gauge, it is natural to shift D a as follows: Then, by making use of (6.22), we find that these covariant derivatives obey the algebra: ) In the N = 2 case this algebra of covariant derivatives coincides with conformally flat limit of the one derived by Grimm [63].It should be pointed out, however, that no discussion of super-Weyl transformations was given in [63].As a result, the setup of [63] is insufficient to describe conformal supergravity.These transformations were later computed in [15].
The geometric superfields appearing above obey the Bianchi identities (6.23) (upon imposing (6.33)).Now, by examining equations (6.35), we see that the U(1) R curvature has been eliminated and therefore the corresponding connection is flat.Consequently, it may be set to zero via an appropriate local U(1) R transformation; Φ A = 0.As a result, the gauge group reduces to SL(2, C) × SU(N ) R .Hence, we will refer to this supergeometry as conformally flat SU(N ) superspace.
It turns out that the gauge conditions (6.33) and Φ A = 0 allow for residual super-Weyl transformations, which are described by a parameter σ constrained by The general solution of this condition is where the parameter σ is covariantly chiral, with zero U(1) R charge, but otherwise arbitrary.
To preserve the gauge condition Φ A = 0, every super-Weyl transformation, eq. ( 6.25), must be accompanied by the following compensating U(1) R transformation As a result, the algebra of covariant derivatives of (conformally flat) SU(N ) superspace is preserved by the following set of super-Weyl transformations: For N = 2 case these transformations are a special case of the ones given in [15].It is important to point out that for N = 4 the chiral parameter σ and its conjugate σ appear in (6.39)only in the real combination σ + σ.
In the case that , the covariant derivatives of N -extended Minkowski superspace M 4|4N , the relations (6.39) provide a conformally flat realisation for an arbitrary conformally flat superspace.

Degauging (iii): N -extended AdS superspace
As an application of the superspace geometries sketched above, we now show how the N -extended AdS supergeometry may be described within SU(N ) superspace.Such a supergeometry is characterised by the following conditions: (i) the torsion and curvature tensors are Lorentz invariant; (ii) the torsion and curvature tensors are covariantly constant.
These conditions imply the following relations: Keeping in mind these constraints, the algebra obeyed by D A reduces to the following: with the identification R = −S when N = 1.Additionally, one may impose the reality condition S ij = S ij by performing some rigid U(1) phase transformation D i α → e iφ D i α .In the N = 2 case the resulting geometry coincides with the one of [16].We will not impose this reality condition below.
When N > 1, the constraint D A S jk = 0 implies the following integrability condition As δ ij is the SO(N ) invariant tensor, it follows that the R-symmetry group reduces to SO(N ) R .
The former may then be utilised to raise and lower indices in accordance with the rule Further, upon inspection of (6.41), the R-symmetry generators only appear in the algebra of covariant derivatives via the combination The SO(N ) R generator J ij may be shown to act on isospinors as follows The resulting algebra of covariant derivatives is as follows: ) This algebra coincides with the one presented in eq. ( 5.27) provided one fixes S = −2.
By definition, a conformally flat supergeometry may be related to a flat one by performing some super-Weyl transformation.In the case of AdS superspace, this means that the curved covariant derivatives D A are related to those of Minkowski superspace D A = (∂ a , D i α , D α i ), see eq. (6.39), as follows: while the AdS superspace curvature S takes the form Here the chiral parameter σ is required to obey the following constraints:

.47e)
As compared with [20], our work provides an alternative proof of the conformal flatness of N -extended AdS superspace.It should also be pointed out that the logarithm of the chiral parameter λ, which was defined in equations (4.22) and (4.23), is proportional to σ; ln λ ∝ σ.Further, in N = 1 case, they are related via eq.(5.37).

Conclusion
This work has completed the construction of the embedding formalism for AdS 4|4 initiated in [1].In the original realisation [1], superspace Poincaré coordinates for AdS 4|4N are naturally introduced, and therefore that realisation is well suited for AdS/CFT calculations in the spirit of [53].The novel realisation of the N -extended AdS supergroup OSp(N |4; R), which has been introduced in this paper, is more suitable for the coset construction, The AdS superparticle model (2.19) is one of the main results of this paper.Setting κ = 0 in (2.19) gives a unique AdS extension of the model for a massive superparticle in Minkowski superspace.In terms of the local coordinates in the north chart described in subsection 4.4, the kinetic terms have the form where the one-form e m is defined in (4.27).In the non-supersymmetric case, N = 0, the κ-term is absent.Therefore, for N > 0 the κ-term does not contain purely bosonic contributions.
It may be checked that the κ-term contains a higher-derivative contribution proportional to iℓ( θ2 − θ2 ).Thus our superparticle model (2.19) may be viewed as an AdS analogue of the Volkov-Pashnev model [64]. 9In N -extended Minkowski superspace, for N > 1 it was possible to add a fermionic WZ-like term to the superparticle action [65].Such structures are more difficult to generate in the AdS case.
In this paper we have also provided descriptions of the most general conformally flat Nextended supergeometry in four dimensions.Specifically, we have realised this geometry in three different superspace frameworks: (i) conformal superspace; (ii) U(N ) superspace; and (iii) SU(N ) superspace.Additionally, we computed the finite super-Weyl transformations within the U(N ) and SU(N ) superspaces.As an application of this construction, we utilised it to obtain a new realisation for AdS 4|4N and describe the specific super-Weyl transformation (6.47) required to 'boost' to this superspace from a flat one.
coordinates y a will be chosen to correspond to the intersection of the hyperplane Z d = 0 with the straight line Γ A S (t) connecting Z A and the "south pole" Z A south = (0, . . ., 0, ℓ).In the north chart, the straight line Γ A N (t) can be parametrised as Γ A N (t) = (1 − t)Z A north + tx A , x A = (x a , 0) , (B.2) and Z A ∈ AdS d corresponds to some value t ′ of the evolution parameter, Γ A N (t ′ ) = Z A .We then derive 3) The embedding coordinates Z A can be expressed in terms of the local ones, For the induced metric we obtain (B.7) The embedding coordinates Z A are expressed in terms of the local ones y a as follows: The induced metric has the form It may also be seen that x 2 < 0 ⇐⇒ y 2 < 0 in the intersection of the charts.
C The N -extended superconformal algebra In this appendix, we spell out our conventions for the N -extended superconformal algebra of Minkowski superspace, su(2, 2|N ).It was initially described in the literature by Park [66], see also [67].We emphasise that the appropriate relations differ by an overall sign as compared with those of eq.(5.22).This distinction arises from our adoption of the convention where generators act on fields and operators in a consistent manner.The R-symmetry group U(N ) R is generated by the U(1) R (Y) and SU(N ) R (J i j ) generators, which commute with all elements of the conformal algebra.Amongst themselves, they obey the commutation relations The superconformal algebra is then obtained by extending the translation generator to P A = (P a , Q i α , Q α i ) and the special conformal generator to K A = (K a , S α i , Si α).The commutation relations involving the Q-supersymmetry generators with the bosonic ones are: We emphasise that all (anti-)commutators not listed above vanish identically.