Simple variables for AdS$_5 \times S^5$ superspace

We introduce simple variables for describing the AdS$_5\times S^5$ superspace, i. e. $\frac{PSU(2,2|4)}{SO(4,1)\times SO(5)}$. The idea is to embed the coset superspace into a space described by variables which are in linear (ray) representations of the supergroup $PSU(2,2|4)$ by imposing certain supersymmetric quadratic constraints (up to two overall U(1) factors). The construction can be considered as a supersymmetric generalisation of the elementary realisations of the $AdS_5$ and the $S^5$ spaces by the SO(4,2) and SO(6) invariant quadratic constraints on two six-dimensional flat spaces.

String theory in AdS 5 × S 5 has been studied extensively in recent years, because of the AdS/CFT correspondence [1]. The theory is also a prime example of string theories with nonzero Ramond-Ramond fields in their backgrounds, and has a very high degree of symmetry, in particular, the maximal supersymmetry P SU (2, 2|4). The classical action of the theory [2] in the Green-Schwarz formalism [3,4] describes the propagation of the strings in the target superspace P SU (2,2|4) SO(4,1)×SO (5) , which is expressed as a coset space. It is the purpose of the present note to point out that this superspace has a simple realisation in which the supersymmetry and the fermionic variables are represented in a particularly clear manner. Fermionic and bosonic variables are treated on an equal footing in this formalism.
The realisation can be considered as a supersymmetric generalisation of the standard definition of the AdS 5 and the S 5 spaces (with radii R) by embedding them into two sixdimensional flat spaces, Here Xİ 's and Y I ′ 's are six-dimensional real vectors. The indicesİ,J = 0, 1, . . . , 5 refer to SO(4, 2) vector indices and the metric is given by ηİJ = diag(−1, 1, 1, 1, 1, −1); I ′ , J ′ = 1, 2, . . . , 6 are SO(6) vector indices, η I ′ J ′ = diag(1, . . . , 1). The manifolds defined by these equations are equivalent to the coset spaces SO(4, 2)/SO(4, 1) and SO(6)/SO(5) respectively up to global issues which we shall ignore throughout this note. This type of embedding by quadratic constraints is often useful, in particular, to make the symmetry properties more transparent, as was originally pointed out by Dirac [5]. In our construction we will use linear representations (more precisely linear ray representations) of P SU (2, 2|4) and introduce certain quadratic constraints on the representation spaces. The supermanifold defined by these constraints will be shown to be equivalent to Representations As the supersymmetrisations of Xİ and Y I ′ , we use two sets of variables X AB and Y AB . They belong to the (super-)anti-symmetric and symmetric products of the fundamental ray representations of P SU (2, 2|4), Our notation is as follows. Indices A, B, . . . are those for the P SU (2, 2|4) fundamental ray representation and take eight values. They consist of four SU (2, 2) "bosonic" componentṡ a,ḃ, . . . and four SU (4) "fermionic" components a ′ , b ′ , . . .. 1 The A, B, . . . indices on the exponent of (−1) should be understood as either 0 for the bosonic components or 1 for the fermionic components. More explicitly we have Theȧḃ and a ′ b ′ components of X's and Y 's are commuting and theȧb ′ and a ′ḃ components are anti-commuting. We use two irreducible representations of P SU (2, 2|4), rather than one irreducible representation. At first sight, it might seem that the use of the two variables (X's and Y 's) would make the superspace a direct product of two superspaces. Actually, the constraints we introduce below intertwine the two variables so that the final superspace cannot be written as a direct product of two spaces. This is consistent with the fact that while the bosonic part of the superspace AdS 5 × S 5 is written as a direct product, the full superspace P SU (2,2|4) SO(4,1)×SO (5) is not.
In order to formulate the constraints we introduce further conventions and notations on the supersymmetric tensor calculus. We use the standard "left derivative" convention for supersymmetric tensor indices, such that is a scalar: the indices A should be contracted in this manner. In this convention, Kronecker's delta has the index structure δ B A .
They can be used to lower and raise the indices. An element of the fundamental ray representation of the P SU (2, 2|4) supergroup transforms as where Uȧ˙b, U a ′ b ′ 's are commuting and Uȧ b ′ , U a ′ḃ are anti-commuting. The variables X, Y 's transform under P SU (2, 2|4) transformations by the transformation rule, The condition defines the U (2, 2|4) supergroup. A further constraint defines the SU (2, 2|4) supergroup [6]. Finally, by identifying two U 's related by an overall we obtain the P SU (2, 2|4) supergroup. This identification implies that the fundamental representation should be considered as a ray (or projective) representation, namely elements of the representation space should be identified as follows As a consequence, the spaces described by the variables X, Y also have natural identifications Alternatively, we may also speak about linear representations of SU (2, 2|4) or U (2, 2|4), without introducing the identification, though P SU (2, 2|4) is the physically interesting case. We denote the complex conjugate of X AB by We define X with lower indices by (The sign factor (−1) (B+C )A above equals 1 because η is diagonal.) Similarly, we define Constraints On the space described by the variables X AB and Y AB , we introduce the following quadratic constraints, The factor (−1) AB in (24) is necessary to make the index structures of the LHS and the RHS match. By construction, the LHS and the RHS of the constraints have the same transformation properties under P SU (2, 2|4) transformations, which can also be verified directly using (13)-(15). Hence these constraints have invariant meanings under P SU (2, 2|4) transformations.
The constraints are invariant also under the two overall U (1) transformations Hence the constraints (23), (24) are consistent with the identifications (19): the constraints are correctly defined on the ray representations. 2 Equivalence to P SU (2,2|4)

SO(4,1)×SO(5)
We will now show that the supermanifold defined by the constraints (23), (24) is equivalent to the coset superspace P SU (2,2|4) SO(4,1)×SO (5) . Any two points on the supermanifold which are related by a P SU (2, 2|4) transformation are equivalent. It is therefore natural to start by choosing a representative point on the manifold and study the manifold in the vicinity of the point.
We first choose a pair of vectors Xİ (0) , Y I ′ (0) satisfying the constraints (1), (2). The representative point is constructed from Xİ (0) , Y I ′ (0) using the Clebsch-Gordan coefficients relating SO(4, 2) and SU (2, 2), Γİȧ˙b, and SO(6) and SU (4), Γ I ′ a ′ b ′ , This point in the superspace satisfies the constraints (23),(24), 3 which can be checked using These formulae follow from properties of the Clebsch-Gordan coefficients summarised in the appendix. It is sometimes useful to specify further the point by choosing Xİ (0) = (0, . . . , 0, 1), Y I ′ (0) = (0, . . . , 0, 1). We next consider the orbit of this representative point under all possible P SU (2, 2|4) transformations. The equivalence of the constrained superspace to the coset superspace will be shown in two steps. First we will show the equivalence of the coset space and the orbit space and then the equivalence of the orbit space and the constrained superspace.
The orbit and the coset are equivalent if the subgroup of P SU (2, 2|4) which leaves the representative point fixed is precisely SO(4, 1) × SO (5).
It is sufficient to consider the infinitesimal transformations of the representative point specified by (X AB (0) , Y AB (0) ). An infinitesimal transformation U A B = δ A B + δU A B satisfies, from (15), The infinitesimal transformation rules of X's and Y 's are derived from (13), (14), The bosonic transformations consist of SU (2, 2) transformations acting onȧ,ḃ indices and SU (4) transformations acting on a ′ , b ′ indices. 4 The only non-zero components on which a SU (2, 2) transformation can act are Xȧ˙b (0) . By the standard property of the Clebsch-Gordan coefficients, this action is equivalent to the action of the corresponding SO(4, 2) transformation on Xİ (0) . The SO(4, 2) transformations which leave Xİ (0) (satisfying (1)) invariant are precisely those forming SO(4, 1). Similarly, the subset of SU (4) transformations which leave X AB (0) 's and Y AB (0) 's invariant are equivalent to SO(5) transformations which leave Y I ′ (0) invariant. It therefore remains to be shown that under any fermionic transformations (with parameters δUȧ b ′ , δU a ′ḃ, satisfying (33)), the representative point is not fixed. The representative point transforms under the fermionic transformations as, Since Xȧ˙b (0) and Y a ′ b ′ (0) are invertible (see (32)), it follows that the representative point is not fixed under any fermionic transformations. Thus the equivalence between the orbit and the coset is established.
From the covariance of the constraints (23), (24), it follows that all points on the orbit space will satisfy the constraints. Therefore the orbit space is contained in the space defined by the constraints.
Hence, in order to show that the orbit space and the constrained manifold are equivalent, it is sufficient to check that the constrained manifold does not contain "extra directions". Hence establishing that the constrained manifold contains the correct number of bosonic and fermionic dimensions is enough to ensure the equivalence of the constrained superspace and the orbit space.
Since all points on the manifold will be equivalent, it suffices to check this property in the vicinity of the representative point, The constraint (23) can be linearised to yield, and (24) gives The formulae (39) and (41) mean that the unwanted components belonging to the tendimensional symmetric representations of SU (4) (in δX) and of SU (2, 2) (in δY ) are actually eliminated by the constraints. In order to understand the meaning of (42), (45), we write using the fact that each of Γİȧ˙b, Γ I ′ a ′ b ′ spans a basis of 4 × 4 anti-symmetric matrices. In terms of this notation (42), (45) imply δXİ ΓİȧċXJ (0) ΓJċ˙b + Xİ (0) ΓİȧċδXJ ΓJċ˙b = 0, (47) Here δXİ and δY I ′ are six-dimensional complex vectors. By decomposing them into real and imaginary parts we obtain, Thus, the imaginary parts of the vectors δXİ and δY I ′ are proportional to X (0)İ and Y (0)I ′ respectively; they are related to the original representative point by infinitesimal U (1) transformations (25) and therefore should be neglected in the ray representations. The real parts of the vectors δXİ and δY I ′ are orthogonal to X (0)İ and Y (0)I ′ respectively; they are nothing but the tangent spaces of AdS 5 and S 5 at the representative point. Thus the bosonic tangent space of the constrained manifold is just as it should be.
The constraints for the fermionic components (40), (43), (44) are actually all equivalent to Before imposing the constraints, the independent fermionic fluctuations δXȧ b ′ and δYȧ b ′ have 32 complex components. The above constraint imposes a certain reality condition on them. Because of this we have 32 real components, which is the correct number for the superspace under consideration. Hence the constrained supermanifold captures correctly fermionic directions of the orbit, or equivalently the coset space. 5 Thus finally the equivalence of P SU (2,2|4) SO(4,1)×SO (5) and the supermanifold defined by the constraints (23), (24) is established. 6 Discussion It should be possible to write down the superstring Green-Schwarz action using the variables X AB , Y AB as fields defined on the string worldsheet. The constraints should be imposed by introducing δ-functionals associated with the constraints, in the path integral in terms of the fields X AB and Y AB . It may also be possible to take a linear sigma model type approach, in which one first studies unconstrained X, Y fields with various coupling constants, and take an appropriate limit of these coupling constants to realise the constraints. One should take into account of the U (1) × U (1) identifications (19) in order to ensure that no extra degrees of freedom enter. It may also be possible to (partially) eliminate the U (1) degrees of freedom by introducing non-quadratic constraints constructed using the superdeterminant. 5 One can also check this more directly. The supersymmetry variation of the representative point (36), (37) satisfies (51) under the condition (33). Conversely, for any variation δX and δY satisfying (51), one can find the fermionic infinitesimal parameters satisfying (33) which produce the variation by (36), (37). 6 Alternatively, if one does not introduce the identification (19), the same argument presented here establishes the equivalence of the constrained space to U (2,2|4) SO(4,1)×SO (5) .
It is interesting to study variables similar to the ones presented in this note for other AdS superspaces, in particular those associated with the supermembrane theory on the AdS 4 × S 7 and AdS 7 × S 4 spaces [7].
We hope that the present formulation may provide a point of view which simplifies and clarifies the structure of supersymmetric theories on AdS spacetimes. The formalism may also be useful for study of quantities controlled by the P SU (2, 2|4) symmetry such as observables in N = 4 Super-Yang Mills theory in four-dimension.
This note presents results of work done several years ago. I was stimulated to write up the present results by two very recent papers [8,9] which develop a new formulation of superstring theory on AdS 5 × S 5 using a parametrisation of the superspace built along similar directions to the approach proposed in this note. The bosonic degrees of freedom in [8,9] are represented in a similar way as done in (26), (27), where we specify a part of the bosonic coordinates of the representative point. The fermionic degrees of freedom are however introduced differently in our formalism compared to that of [8,9]. The supersymmetry is realised on the (constrained) coordinates of our superspace in a linear fashion, whereas in [8,9] a non-linear realisation of the supersymmetry is used. The formalism presented here may be advantageous for some applications, as in particular in quantum field theories linearly realised symmetries can often be more straightforwardly dealt with compared to non-linearly realised symmetries.
An explicit representation is,