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Symmetries of the Boundary of AdS_5 x S^5 and Harmonic Superspace

We study the boundary limit of the bulk isometries of AdS x S. The superconformal symmetry is realized on the coordinates of the AdS boundary, the fermionic superspace coordinates, and the harmonics on the sphere. We show how these may be related to the coordinates of an off-shell harmonic superspace of SCFT living on the boundary. In the special case of d=4, N=2 super Yang-Mills theory, a truncation of the N=4 SYM dual to Type IIB string theory compactified on AdS_5 x S^5, we identify the bosonic space SU(2)/U(1) of the N=2 harmonic superspace (known before as auxiliary) with the S^2 submanifold of the S^5.

There is a growing interest in the AdS/CFT correspondence conjecture [1]. One of the basic arguments in favor of this conjecture concerning N = 4 Yang-Mills theory is the fact that the superisometry of the AdS superspace forms the same SU(2, 2|4) supergroup as the symmetries of the N = 4 Yang-Mills field theory, which supposedly lives on the boundary of the AdS space. The superspaces with the bosonic AdS p+2 × S d−p−2 geometry and the proper form field were recently constructed in [2], and their superisometries were found in closed form in [3]. The purpose of this note is to motivate a connection between the limit to the boundary (ρ → ∞) of the bulk AdS 5 × S 5 superspace with the harmonic superspace [4] of the off-shell 4-dimensional N = 2 Yang-Mills theory.
Here, x are the coordinates of the 4-dimensional space parallel to the boundary of the AdS 5 space, and ρ is the AdS 5 coordinate orthogonal to the boundary, which is located at ρ = ∞. The 32 fermionic coordinates θ are the fermionic coordinates of type IIB string theory/supergravity, and finally u are harmonics of the five-sphere, which may be given as functions of the angles parametrizing the sphere, i.e. u(φ (where Z denotes all superspace coordinates) are rather complicated, and they are explicitly constructed in [3]. In short, they are given in terms of components of a covariantly constant Super-Killing field Σ(Z) and the inverse of the supervielbein At θ = 0, the supervielbein reduces to the standard vielbein, the gravitino, and a fermion-fermion sector. The Killing field Σ depends linearly on the Killing vectors and Killing spinors Σ 0 (x, ρ, u(φ)) of the background geometry (and the corresponding compensating transformations). Naturally, both E and Σ are also functions of the SU(2, 2|4) structure constants and θ.
At first glance the expressions for the isometries do not have a well defined limit at ρ → ∞. For example, the metric, and hence the vielbein, has terms with positive and negative powers of ρ: where R is the radius of curvature of the AdS 5 space as well as of the sphere S 5 .
Also, the translations Σ m 0 parallel to the boundary and the compensating Lorentz transformations Σ mρ 0 which rotate in the bulk-boundary plane (x m , ρ) have a nontrivial ρ dependence. They are of the form In addition, we see the dependence on ρ in the Killing spinors of the AdS 5 × S 5 space, which depend on the harmonics 2 of the five sphere u i I (φ): where and where ǫ andη are the parameters of the Q-supersymmetry and S-supersymmetry resp. We have found that this irregular dependence of the bulk isometries on ρ can be perfectly organized if we use two important technical tools.
• The superisometries in the bulk were found in the AdS basis of the superalgebra which treats the bulk direction on equal footing with the boundary directions since there is a manifest SO(4, 1) × SO(5) symmetry, mixing bulk and boundary directions. The limit to the boundary, however, should be performed in the superconformal basis of the algebra which has only manifest SO(1, 3) symmetry in the AdS 5 sector, reflecting the split of the AdS coordinates into x and ρ.
• We use the Killing gauge in the superspace [2], since in this gauge the gravitino vanishes not only at θ = 0 for the bosonic background but also to all orders in θ which makes various calculations doable and easy. In this gauge we find the limit to the boundary ρ → ∞ in which the radial direction ρ decouples.
The SU(2, 2|4) superalgebra with generators T Λ and structure constants f ΛΣ is defined up to linear redefinitions of the generators. The most useful basis for our purposes is the superconformal basis, whose most important property is that 2 We use a two-component notation, which means that we work with a representation of γ-matrices More details on Killing spinors in terms of the harmonics will be given in a future publication [6].
there exists a grading operator D, under which every generator T Λ has a particular conformal weight: In fact, the translation in the ρ-direction becomes the dilatation operator D, transla- Now, the matrix R Λ Λ (ρ) carries all dependence on the radial variable and the fields Σ Λ (x m , u(φ)) depend only on the bosonic coordinates of the boundary, and the harmonics of the sphere.
The dependence of Σ and E on θ is such that use of the Killing spinor gauge [2] together with (13) allows all non-trivial ρ dependence in δZ to be absorbed into new ρ-dependent structure "constants" via the matrix R(ρ) defined in (13). The crucial observation at this point is that the structure constants of the algebra have the property that for any component f ∆ holds. This identity shows then that in the 'underlined' basis the structure "constants" f ∆ ΛΣ (ρ) are numerically nothing but the old constants f ∆ ΛΣ , since i.e. the ρ-dependence cancels out. This gives us control over the limit ρ → ∞ in all expressions for superisometries. We find that the bulk-boundary isometry correspondence has the following form: δu| bulk = δu| boundary (u, θ) .
The isometry in the radial direction blows up in the limit ρ → ∞: but decouples from the boundary isometries.
We do not specify here the functions of (x, u, θ) introduced above as they are still rather complicated, but they can be read off from the generic expressions for the isometries in the bulk, given explicitly in [3]. We derive from the superisometries in the bulk [3] that after truncation the remaining symmetries at the boundary are: The symmetries of the sphere are given in terms of the coset representative (i.e. the harmonics u(φ)) as follows where, g is an element of SO(6), h(φ) is an element of SO(5), and φ ′ are the transformed angles of the sphere. Explicitly, the infinitesimal transformations take the where Λ i j are the parameters of the transformation from the SO(6) group in spinorial 3 Here we use the Killing Spinor gauge [2], which means that we decompose the 32 component fermionic coordinates θ in the same way as the Killing spinor (9), i.e.
representation and H L K (φ) are the parameters of the transformation from the stability group SO(5).
The truncated system of coordinates at the boundary has a symmetry realizing the truncated SU(2, 2|4) algebra without S, D, K, which is the simply the extended The symmetries of the reduced system then are:  4 The new bosonic coordinates relevant for extended supersymmetries were added to the superspace in the form of supertwistors and/or coset coordinates in [7]. But the origin of these additional coordinates which were used in [4] to construct an off-shell SYM theory was not clear so far to the best of our understanding. Now we seem to shed some light also on the physical origin of the supertwistors which may be related to the harmonic superspace.
They also appear explicitly in the Killing spinors on the AdS × S space [3]. Their main purpose is to allow for a change of fermionic variables from those transforming under SU(2) to those transforming under the U(1) where we took I = (+, −). It follows that The last term shows that θ ± transforms under the U(1) symmetry. This term came from the cancellation of the SU(2) transformation from the θ i by the shift of angular variables and the remaining compensating U(1) transformation is the one with the parameter Λ.
The ABC of the analytic N = 2 superspace (which is reminiscent of the chiral subspace of N = 1 susy) follows as in [4]. One can shift the real 4-dimensional coordinates x by the bilinear combination of spinors so that there is an analytic subspace x ′ , θ + ,θ + , u. Then one can construct unconstrained superfields which depend in this basis only on θ + . The complete N = 2 SYM theory with manifest off-shell supersymmetry is then easily constructed. The SYM prepotential V ++ is an unconstrained analytic superfield of U(1) charge +2, and the hypermultiplets q + are given by analytic superfields of charge +1.
The basic difference between [4] and present derivation lies in the fact that we have the coordinates of S 2 as part of the real boundary of the curved AdS 5 × S 5 space.
Also, the somewhat tedious integration rules for the integration over u, du·1 = 1 and du · u + (i u − j) = 0, find a simple explanation in the AdS 5 × S 5 picture: they all can be derived from i.e. from the integration over the physical sphere part of our boundary space.
This shows that the S 2 part of S 5 on which we compactify the superstring/supergravity theory plays a fundamental role in the existence of the off-shell hypermultiplet action and supergraph Feynman rules of N = 2 SYM theory developed in [4]. We find here the long-anticipated presence of higher dimensions, however, they are not coming from the d = 10 SYM theory, but from string theory/supergravity on AdS 5 × S 5 through the AdS/SCF T correspondence. The harmonics of [4] find their place as the harmonics of the sphere which is part of the boundary of the AdS 5 × S 5 space.
It would be very interesting to see whether a new (off-shell?) N = 4 harmonic superspace can be constructed from the untruncated boundary isometries when the harmonics on the the full S 5 sphere are used.