Rigidity of SU(2,2|2)-symmetric solutions in Type IIB

We investigate the existence of half-BPS solutions in Type IIB supergravity which are invariant under the superalgebra SU(2,2|2) realized on either AdS_5 x S^2 x S^1 or AdS_5 x S^3 warped over a Riemann surface \Sigma with boundary. We prove that, in both cases, the only solution is AdS_5 x S^5 itself. We argue that this result provides evidence for the non-existence of fully back-reacted intersecting D3/D7 branes with either AdS_5 x S^2 x S^1 x \Sigma or AdS_5 x S^3 x \Sigma near-horizon limits.

in 4 dimensions [28,29,30]. All known exact half-BPS solutions have known brane realizations. The brane picture suggests cases, however, of half-BPS near horizon limits of brane intersection where no corresponding supergravity solutions have been found yet.
Perhaps the most important case where the superalgebra and brane pictures appear to be at odds with the absence of any known solutions is for Type IIB supergravity solutions which are half-BPS, locally asymptotic to AdS 5 × S 5 , and invariant under SU(2, 2|2). The duals to such solutions would be 4-dimensional N = 2 super-Yang-Mills theories with exact conformal invariance. Their brane picture would be in terms of the near-horizon limit of stacks of D3-and D7-branes, intersecting along a 4-dimensional space-time, or equivalently in terms of probe D7-branes inside AdS 5 .
The conflict is as follows. The existence of the subalgebra SU(2, 2|2) ⊂ P SU(2, 2|4) clearly makes it possible to have asymptotically AdS 5 × S 5 half-BPS solutions with this symmetry. The existence and stability of probe D7-branes in AdS 5 with SU(2, 2|2) × SU (2) symmetry also supports the possibility of having such solutions, though at the expense of including an extra bosonic SU(2) symmetry group not mandated by superconformal invariance. The question is whether such probe brane solutions extend to fully back-reacted supergravity solutions where the D7-branes dissolve into regular flux solutions.
Several arguments have been presented [31,19,32] (following earlier work [33,34]) that no such fully back-reacted solutions corresponding to D7-branes should exist. A first argument against is that D7-branes produce flavor multiplets [35,36] transforming under the fundamental representation of the gauge group in the dual gauge theory. Their contribution to the renormalization group β-function for the gauge coupling is non-vanishing, and thus breaks the exact conformal invariance needed to maintain the AdS 5 factor on the gravity side. A second argument [37] makes use of local 1/2 BPS-operators: the invariance superalgebra H for half-BPS solutions which are genuinely asymptotic to AdS 5 × S 5 must have rank strictly less than the rank 6 of P SU (2, 2|4). This argument leads us to conclude that solutions with SU(2, 2|2) × SU(2) symmetry cannot be asymptotic to AdS 5 × S 5 , though there appears to be no obstruction to the existence of locally asymptotic solutions.
We shall prove here that the only solution with at least 16 supersymmetries for spacetimes of the form AdS 5 ×S 2 ×S 1 ×Σ and AdS 5 ×S 3 ×Σ, invariant respectively under SO(2, 4)× SO(3)×SO (2) and SO(2, 4)×SO (4), is just the maximally supersymmetric solution AdS 5 × S 5 . Here, the products AdS 5 × S 2 × S 1 and AdS 5 × S 3 are warped over the Riemann surface Σ. It then follows that no near-horizon limit can exist (in the supergravity limit) for fully back-reacted intersection solutions of D3-and D7-branes with those symmetries and spacetime geometries. It is in this sense that the SU(2, 2|2)-invariant solutions are rigid. Note that rigidity of half-BPS solutions has already been encountered in analyzing analogous problems in M-theory [38].
Finally, we note that there is one possible interesting near-horizon limit of D3-and D7brane fully back-reacted solutions that is not yet excluded by our study. One could imagine a space-time of the form AdS 5 × (S 2 × k S 1 ) × Σ where the product S 2 × k S 1 is twisted by the Hopf fibration, with k ∈ Z being the monopole number. For k = 0, we recover the case AdS 5 × S 2 × S 1 × Σ studied here. For k = 0 one recovers S 3 /Z k topologically, but the metric may be deformed to that of a squashed S 3 . For k = 1, and no squashing, we recover the case AdS 5 × S 3 × Σ studied here. In general, however, the squashing parameter (the ratio of the radius of S 1 to that of S 2 ) may be non-trivial, and actually be a function of Σ. Such cases with non-trivial fibration and squashing require a generalization of the construction methods proposed in [23], and are clearly of physical interest. They are considerably more complicated than the cases treated here; their study will be postponed to future work.

Organization
The remainder of this paper is organized as follows. In section 2, Type IIB supergravity is briefly reviewed, mostly to fix notations, and the AdS 5 ×S 2 ×S 1 ×Σ Ansatz is implemented on the Type IIB supergravity fields. In section 3, the BPS equations are reduced on this Ansatz; they are the starting point of the exact solutions. In section 4, we show that the vanishing of the 3-form field strength implies the maximally supersymmetric solutions AdS 5 × S 5 . In section 5, we solve for the metric factors of the AdS 5 × S 2 × S 1 spaces, and express them in terms of bilinears in the supersymmetry spinors. In section 6, the vanishing of several types of Hermitian forms in the supersymmetry spinors are derived from the reduced BPS equations. In section 7, with the help of these vanishing Hermitian forms, it is shown that the only viable solution for the AdS 5 × S 2 × S 1 × Σ Ansatz is AdS 5 × S 5 . In section 8, the analysis is extended to the AdS 5 × S 3 × Σ Ansatz. It is shown that also here the only viable half-BPS solution is AdS 5 × S 5 . In Appendix A, we give Clifford algebra representations adapted to the two different Ansätze. In Appendices B and C, we provide the details of reducing BPS equations for the two different Ansätze. In Appendix D and E, technical details of solving metric factors and vanishing Hermitian forms are presented.

Type IIB supergravity
The canonical fields of Type IIB supergravity are the metric g M N , and the 4-form C (4) , both of which are real fields, the axion/dilaton scalar field B, and the 2-form B (2) , both of which are complex fields, as well as the dilatino λ and gravitino ψ M spinor fields. Our conventions are those of [39,40] (see also [41]). It is convenient to formulate Type IIB supergravity in terms of the field strengths, Bianchi identities, and field equations. For the axion/dilaton, we introduce the 1-form fields P, Q, while for the 2-and 4-form fields we introduce the field strengths, The scalar field B is related to the complex string coupling τ , the axion χ, and dilaton Φ. In terms of the composite fields P, Q, and G, there are Bianchi identities given as follows, The field strength F (5) is required to be self-dual, The field equations are given by, The BPS equations for bosonic solutions are obtained by setting the supersymmetry variations δλ and δψ M of the fermionic fields to zero, in the presence of vanishing fermi fields, 2 Here, B is the complex conjugation matrix of the Clifford algebra. 3 Type IIB supergravity is invariant under SU(1, 1) ∼ SL(2, R) symmetry, which leaves g M N and C (4) invariant, acts by Möbius transformation on the field B, and linearly on B (2) , with u, v ∈ C andūu −vv = 1. The transformation rules for the composite fields are [39], where the phase θ is defined by e 2iθ = vB + ū vB +ū (2.14) In this form, the transformation rules clearly exhibit the U(1) q gauge transformation that accompanies the global SU(1, 1) transformations.

The SO(2, 4) × SO(3) × SO(2)-invariant Ansatz
We seek the most general Ansatz in Type IIB supergravity with SO(2, 4) × SO(3) × SO(2) symmetry. The SO(2, 4)-factor requires the geometry to contain an AdS 5 -factor, while the SO(3)-and SO(2)-factors respectively require S 2 -and S 1 -factors, all warped over a 2-dimensional surface Σ. The total space-time then has the structure, Following the general constructing as in [10], the Ansatz will be given in terms of SO(2, 4) × SO(3) × SO(2) invariant supergravity fields, which are functions on the surface Σ. The Ansatz for the metric is given by, where f 5 , f 2 and f 1 are real functions on Σ. Orthonormal frames are defined by, Here, the metrics ds 2 AdS 5 , ds 2 S 2 , and ds 2 S 1 , as well as the orthonormal framesê m ,ê i , andê 9 refer to the spaces AdS 5 , S 2 , and S 1 with unit radius. In particular, we have Similarly, the dilaton/axion field B is a function of Σ only. As a result, the 1-forms P and Q have no component along e 9 , so that p 9 = q 9 = 0 and their structures is simply given by, Finally, the most general Ansätze for the 3-and 5-form fields G and F (5) consistent with SO(2, 4) × SO(3) × SO(2) invariance, are given by, 4 G = igãe 56ã + h e 789 = ig a e 56a + ig 9 e 569 + h e 789 F (5) = f e 01234 − e 56789 (2.20) Note that the components of G involving any of the directions 0, 1, 2, 3, 4 must vanish by SO(2, 4) symmetry, and that the remaining terms are the only allowed ones by SO(3)invariance. By SO(2, 4) × SO(3) × SO(2)-invariance, the coefficient functions p a , q a , g a , g 9 , h, and f depend only on Σ. The functions f, q a are real, while p a , h, g a , g 9 are complex-valued.

Immediate constraints from the Bianchi identities
There are two immediate consequences of the Bianchi identities and field definitions for the coefficient functions of the Ansatz.
• The coefficient g 9 is determined by the field definitions. This may be shown directly by observing that the g 9 -term in G can arise only from a contribution b 0ê 569 to F (3) where b 0 is a complex function of Σ. Closure of F (3) requires b 0 to be constant, so that g 9 is given by, with b 0 an arbitrary complex constant. Note that the corresponding B (2) -field is given by b 0 x 9ê56 . This form is invariant up to a gauge transformation under the action of SO (2), namely under a constant shift in the coordinate x 9 of the circle S 1 .
• The Bianchi identity for F (5) reduces to dF (5) = 0, since G ∧Ḡ vanishes identically on the Ansatz. Using the closure ofê 01234 , we find that f f 5 5 must be constant, where c 0 is an arbitrary real constant.

The reduced BPS equations
Half-BPS configurations and solutions in Type IIB supergravity with vanishing Fermi fields are such that the BPS equations (2.11) for the spinor ε admit 16 linearly independent solutions. The spinor ε is covariant under SO(2, 4) × SO(3) × SO(2), and may be built from the Killing spinors on the space AdS 5 × S 2 × S 1 , where each factor space has unit radius. Once this structure is in place, the reduction of the full BPS equations of (2.11) is standard.
The chirality matrix γ (2) of the γ i -Dirac algebra produces a reversal of the sign of η 2 , allowing us to make the following identification, reverses the signs of both η 1 and η 2 , so that (χ c ) η 1 ,η 2 is proportional to χ −η 1 ,−η 2 . The proportionality factor depends on η 1 , η 2 , and may be determined once a sign convention has been chosen for one of the four components of χ η 1 ,η 2 . It will be convenient to make the choice χ c++ = χ −− . The remaining components are then related by using the γ (2) -chirality matrix, and the properties of the charge conjugation matrices B (1) and B (2) . For example, we have χ c+− = (B (1) ⊗ B (2) ) −1 γ (2) χ * ++ by the definition of χ c , and the action of γ (2) . Commuting the γ-matrix past the B-matrices, we have χ c+− = −γ (2) (B (1) ⊗ B (2) ) −1 χ * ++ , which in turn equals −χ −+ . The remaining relations may be similarly obtained, and we find, a relation which is now valid for all four assignments of η 1 , η 2 . The Killing spinors on S 1 are solutions to the equation, Here, we have η 3 = ±, and χ η 3 consists of just a single function, since the irreducible spinor representation of SO(2) on S 1 are 1-dimensional. By choosing a proper overall phase convention, we may set (χ η 3 ) * = χ −η 3 . Note that the values of η 3 are not required by integrability, since the above equation is integrable for all values of η 3 . Rather, the values η 3 = ±1 naturally correspond to a double-valued representation, suitable for spinors. Other odd values of η 3 are allowed as well, but will be equivalent to our choice, upon redefinition of the metric function f 1 of the circle S 1 .

The reduced BPS equations
Details of the reduction of the BPS-equations of (2.11) have been deferred to Appendix B. The final results are as follows. The dilatino BPS equation is given by, while the gravitino equation decomposes into a system of 4 equations,

Symmetries of the reduced BPS equations
The symmetries of the reduced BPS equations will play a fundamental role in the analysis of half-BPS solutions. It will be natural to separate the symmetries into three groups: continuous symmetries; linear discrete symmetries which map ζ to a linear function of ζ; and complex conjugations which map ζ to a linear function of ζ * .

Continuous symmetries
One manifest continuous symmetry consists of the U(1) frame rotations on the orthogonal frame e a on Σ. The axion/dilaton field B transforms non-linearly under the continuous S-duality group SU(1, 1) of Type IIB supergravity, and the field B takes values in the coset SU(1, 1)/U(1) q . Thus, SU(1, 1) transformations on the B-field and on other fields are accompanied by local U(1) q gauge transformations, given in (2.13) and (2.14). This U(1) q survives the reduction to our Ansatz, and induces the following transformations, ζ → e iθ/2 ζ g a → e iθ g a q a → q a + D a θ g 9 → e iθ g 9 p a → e 2iθ p a h → e iθ h (3.15) The real function θ depends on the SU(1, 1) transformation, as well as on the field B.

Linear discrete symmetries
The following linear transformations on ζ are symmetries of all the BPS equations, upon leaving the metric factors f 1 , f 2 , f 5 , the flux fields f, g a , g 9 , h, p a , q a unchanged, The set S 0 forms an Abelian group. The BPS equations have further linear symmetries ζ → ζ ′ = Sζ, given by, The transformations in S 1 leave f 1 , f 2 , f 5 , and f, g a , h, p a , q a unchanged but, in contrast with the transformations in S 0 , must be accompanied by a sign reversal g 9 → g ′ 9 = −g 9 . Thus, the generators of S 1 are genuine symmetries of the bosonic supergravity fields only if g 9 = 0. The union S 0 ∪ S 1 forms a non-Abelian group, whose generators may be chosen to be

Complex conjugation (or anti-linear) symmetries
The BPS equations are invariant under 8 discrete complex conjugations, which may be viewed as the compositions of the 8 transformations of S 0 ∪ S 1 with a single complex conjugation. It will be useful to record a more general transformation, in which we compose the discrete complex conjugation with an arbitrary continuous U(1) q transformation, which gives, The pure complex conjugation corresponds to the special case where θ = 0.
3.4 Further reduction using ζ → τ (303) ζ symmetry The transformations in S 1 are genuine symmetries of the BPS equations only if g 9 = 0, while the complex conjugations are symmetries only if the dilaton/axion and flux fields obey reality conditions p * a = p a , q a = 0, g * a = g a , g * 9 = −g 9 , and h * = h, up to U(1) q . For general bosonic fields, only the transformations in S 0 are genuine symmetries of the BPS equations. Only the transformation ζ → τ (303) ζ commutes with the BPS differential operator acting on real linear combinations, so that this is the only symmetry that may be diagonalized simultaneously with the BPS operator. (Note that the other two non-trivial symmetries of S 0 square to −I, and admit no real eigenvalues.) Thus, we may now analyze separately the restrictions of the BPS equations to the two eigenspaces of τ (303) , given ν = ±1, Redefining the non-vanishing component of the spinor ζ in terms of a spinor with only two indices (which, by abuse of notation, we denote again by ζ), we have, In terms of this two-index spinor ζ, the BPS equations become, The matrices τ (ij) act on the two-index spinor ζ in the obvious manner.
Remarkably, the dependence on ν of the reduced BPS equations is entirely concentrated in the first term of equation (9) in (3.21), and may be undone by changing the sign of the metric factor f 1 . As only the square f 2 1 enters into any of the supergravity fields, this sign redefinition is immaterial, and the projections ν = ±1 represent the same bosonic supergravity fields. A solution to the reduced BPS equations for ν = +1 thus automatically produces another solution with the same bosonic fields but with ν = −1. In this manner, a systematic doubling of the number of spinor solutions is produced.
As a corollary, this doubling up of the number of spinor solutions implies that any solution to the BPS equations with ν = +1 effectively produces 16 linearly independent solutions of the full BPS equations, and thus generates a half-BPS solution.

Residual symmetries of the reduced form
Having projected ζ onto the two eigenspaces τ (303) ζ = νζ, any residual symmetry of (3.21) must map these eigenspace into themselves, requiring the transformation to commute with τ (303) . The transformations U(1) q thus continue to be symmetries of (3.21). The remaining elements in S 0 , namely iτ (300) and iτ (003) map between identical ν, and reduce to the following transformations on two-index ζ variables, both of which are symmetries of the further reduced BPS equations (3.21). The transformations in S 1 all anti-commute with τ (303) , thus map the system with ν to the system with −ν, and do not yield symmetries of the further reduced BPS equations (3.21). Amongst the complex conjugation symmetries of the original reduced BPS equations (3.13) and (3.14), only the transformation of (3.19), and its images under S 0 , commute with τ (303) . The transformation of (3.19) acting on 2-index ζ variables is given by, and is a residual symmetry of the further reduced BPS equations (3.21).

Chiral form of the reduced BPS equations
It will be convenient to separate the two components of the 2-dimensional spinors ζ η 1 ,η 2 ,η 3 , or of their two-index reductions ζ η 1 ,η 2 . This decomposition goes hand in hand with the complex structure decomposition on the Riemann surface Σ, and may be achieved by introducing the standard complex frame basis e a = (e z , ez), with metric δ zz = δz z = 2. The explicit expressions for the frame and flux fields in this basis may be deduced from Similar relations hold for the components of the fields p a , q a , g a , so that, for example, p z = p 7 − ip 8 , and pz = p 7 + ip 8 . The Pauli matrices in this basis take the form, We proceed to decompose the two-index spinor ζ into its two chirality components in this same 2-dimensional spinor basis, where ξ * η 1 ,η 2 and ψ η 1 ,η 2 are 1-component spinors. In this basis, the reduced BPS equations take the following form, The action of the complex conjugation symmetry (3.23) is given by It acts by interchanging ξ and ψ, together with the transformations on the bosonic fields, listed already in (3.19), and translated as follows in the chiral basis, This concludes the full reduction of the BPS equations onto the AdS 5 × S 2 × S 1 × Σ Ansatz.

Vanishing G implies the AdS 5 × S 5 solution
In this section, we shall prove that, whenever the 3-form flux field G vanishes, the solution to the BPS equations is AdS 5 × S 5 itself. We shall also spell out the properties of this solution.
When G = 0, we have g z = gz = g 9 = h = 0. For solutions with at least 16 supersymmetries, ψ and ξ cannot both vanish. The reduced dilatino equations (d) in (3.27) imply p z = pz = 0, so that P = 0. By the Bianchi identities, we then have dQ = 0. Using the U(1) q gauge symmetry (3.15), we may now choose a gauge in which Q = 0, so that, The gauge choice leaves open the symmetry of performing constant gauge transformations, which we shall fix later.

Using the discrete symmetries of the reduced BPS equations
Given the vanishing of the components of G, P, Q in (4.1), all four generators T 1 , T 2 , T 3 , and T 4 of (3.18) and (3.19) produce genuine discrete symmetries of (3.27). The generators T 1 , T 2 , and T 4 mutually commute and may be simultaneously diagonalized as follows, where ν, γ, µ take on independently the values ±1. The projection of T 1 has already been carried out in the previous section. Clearly, the sign µ in the T 3 projection can always be gauged away with a constant gauge transformation in U(1) q . Henceforth, we shall choose µe iθ = 1. In terms of ψ and ξ, these projections become, Solutions with opposite values of γ are equivalent. Henceforth, we set γ = +, and shall treat ψ and ξ as two-component spinors, omitting their fixed η 1 indices.
After these simplifications, the remaining reduced BPS equations of (3.27) take the form,
• Using equation (4.9) to eliminate f = f −1 5 from the (+) equation in (4.5), we find, Multiplying the first byᾱ, the second byβ, and adding gives an equation that is automatically satisfied. Thus, we may retain just the first equation. After some simplifications, it reduces to a holomorphic equation, which may be easily solved, and gives the solution with z 0 being the arbitrary integration constant. In fact, it will be convenient to express both α and β solely in terms of the function A, namely, To determine metric factors in terms of spinor components 5 we make use of both algebraic and differential equations. On the one hand, from the (i) equations in (4.5), we have On the other hand, using differential equations (+) and (−), we obtain the relation Combining the above equations gives, where c 2 is a constant. Similarly, from (9) equations, we may derive the following relations, Combining these in turn implies, where c 1 is a constant.
• Using the expression for metric factors, α and β, it can be checked that (i) and (9) equations are all satisfied, if c 1 = c 2 = 1. So all the reduced BPS equations are completely solved, and all fields are determined by a holomorphic function A(z).

Solution of AdS 5 × S 5
Choosing A(z) = −e −2z , f 5 = 1, we recover the AdS 5 × S 5 solution, To exhibit the metric in standard form, we change parametrization, e x 7 = tan( θ 2 ), which is indeed AdS 5 ×S 5 . The solution to the spinor ζ is characterized by three projections, where we can have, independently, ν = ±1, γ = ±1. In terms of functions ζ 1 , ζ 2 defined below, the non-vanishing components of spinor solutions are (with z = (x 7 + ix 8 )/2): With 8 independent solutions to killing spinors χ and 4 independent solutions to ζ, we indeed recover 32 supersymmetries as expected.

Example of metric factor calculation
The detailed calculation is relegated to appendix D, here, we shall illustrate the procedure by giving an specific example of finding spinor bilinears of f 5 .
We shall use combinations of the differential equations (±) in (3.28) and of the algebraic gravitino BPS equations to bring out the corresponding relations. To this end, we compute, Then, left-multiplying equation (m 1 ) by ψ † , and left-multiplying the complex conjugate of equation (m 2 ) by ξ † , we find, With appropriate combinations of the above equations, all terms would cancel out. We may get the following relation, , so that f 5 may be expressed as the product of constant C (00) 5 and spinor bilinears,

Summary of metric factor expressions
As a summary, we list all the generic relations, valid for arbitrary values of all the fields of the supergravity Ansatz. For notational simplicity of this and the remaining sections, we shall use the following definition for Hermitian forms.

Vanishing Hermitian forms
In the preceding section, it was shown that certain Hermitian forms of the supersymmetry spinors are related to the metric factors f 1 , f 2 , f 5 , in terms of unknown constants. In the present section, we shall show that certain Hermitian forms vanish automatically as a result of the BPS equations. These relations will be obtained using the reality properties of various combinations. Their derivations will involve combinations in which the quantities p z , p * z , g z , and g * z are algebraically eliminated, since these functions do not, generically, obey definite reality conditions. In this section, we shall use the following definition for Hermtian forms, where the first line reproduces the definition already used in (5.5

Example of Hermitian relation calculation
We leave the detailed calculation in appendix E, and quote the results below. There are three different ways to compute vanishing hermitian forms, each will provide additional relations among Hermitian forms, which are also summarized below.
Before giving the complete list of all Hermitian relations, we illustrate one example about computing Hermitian forms.
Considering a combination of the BPS equations of (3.27), given by 2(m) + (i) + (9). All f, g z , gz, h, h * terms cancel in these combinations, and the resulting equations are, Multiplying the first equation by ξ t , the second by −ψ t , adding both to cancel the differential terms, and taking the transpose, we obtain, Each term in ξ † τ ξ or ψ † τ ψ, and each combination of terms involving g 9 and g * 9 , is either real or purely imaginary. Separating real and imaginary parts of the above equations, we have, So we obtained one vanishing Hermitian form, and a relation among them.

Summary of all Hermitian relations
As a summary, we list all the vanishing Hermitian relations below, The remaining equations from the first set are as follows, g− = 0 (6.6) Those of the second set are as follows, Finally, those of the third set are given by, h− = 0 (6.8)

Implications for the metric factors
The relations between the metric factors f 5 , f 2 , f 1 and Hermitian forms, derived in (5.6), together with the vanishing Hermitian forms of the preceding section, produce vanishing value for the following constants, The remaining non-vanishing Hermitian forms f 1 , f 2 , f 5 are given as follows, Here, we have used the normalization C  by c 1 , c 2 , and c 5 respectively, all three of which are real constants. A series of additional constant Hermitian forms were derived in (5.7) and (5.8) respectively when g 9 = 0 and h = 0 or both. These relations are readily seen to follow directly from the equations (6.6), (6.7), and (6.8), and may thus be omitted.

General solution to the reduced BPS equations
In this section, we shall exploit the vanishing hermitian forms to solve the BPS equations, and conclude that the only solution to the reduced BPS equations is AdS 5 × S 5 .

Solving the Hermitian relations H
We begin by solving the Hermitian relations of the form H (αβ) ± = 0, namely those relations given in the first two lines of (6.5). Once we shall have solved for those, we shall solve the remaining Hermitian equations of (6.5) as well as the remaining BPS equations.
Every set of four Hermitian relations can be grouped together to derive an equivalent set of simplified relations between spinor components, From the first group (7.11), we deduce that |ξ +± | = |ψ −± |, |ξ −± | = |ψ +± |. Thus, the corresponding functions are related by phases, which we introduce as follows, Substituting these results into the second group (7.12), we obtain, while from the third and fourth groups (7.13) and (7.14), we obtain, The solution depends upon whether the spinor components ψ ±± generically vanish or not.

The generic case for which
When the components ψ ++ , ψ +− , ψ −+ , and ψ −− are all generic and non-vanishing, then equations (7.16) require, We introduce the angles Θ and Θ ′ , defined by, Note that we also have 4Θ = θ 2 − θ 3 + π = θ 1 − θ 4 + π = θ 2 − θ 4 + π in view of (7.18). The equations of (7.17) amount to relations between the phases of the components of ψ which may all be expressed in terms of the angle Θ as follows, The solutions to these relations may be parametrized in terms of 4 real arbitrary functions r ++ , r +− , r −+ , r −− , the angle Θ, and a further arbitrary angle Λ, The solutions for ξ now follow immediately, where Λ ′ = 2Θ ′ + Λ. (Using the U(1) gauge freedom, we could remove the overall relative phase factor between ψ and ξ, namely Λ ′ − Λ, and set this angle to zero so that Λ ′ = Λ. To keep symmetry manifest, and to remain as general as possible, we shall not make this gauge choice here.) One is then left with 4 real functions r, and two angles Θ, and Λ.

Non-generic cases with vanishing spinor components
It is clear from the above generic solution that a non-generic solution could be viewed as the limit of the generic case. For example, if ψ −− = 0, then this case can be viewed as the limit in which r −− → 0. Tracking this case back to the original equations, it is clear that one retains the first and third equations of (7.21), whose solution simply amounts to the generic solution in which we set r −− = 0. One can check that the only exceptions to the generic solution is when either ψ −+ = ψ −− = 0, ψ ++ ψ +− = 0 or ψ ++ = ψ +− = 0, ψ −+ ψ −− = 0, which has equivalent derivations. We shall focus on the first possibility then, which may be parameterized by four real functions r ++ , r +− , Λ 1 , Λ 2 , and equations (7.16), (7.17) are automatically satisfied.
We shall call this the second type of solution, the difference from the generic solution being ψ ++ and ψ +− may have different phases.

Two types of solutions
The solutions of (7.22) and (7.23) constitute the first type of solution. They may be expressed in matrix form, as follows, where r is given by, We can readily check that this form of ψ and ξ indeed reproduces all the vanishing Hermitian forms of the first two lines of (6.5). In checking this, the dependences on the angles Λ and Λ ′ clearly drop out, but the dependence on Θ may or may not drop out. We find, This combination will vanish for a given assignment of (αβ) and all values of r and Θ provided either (i) the terms in the square brackets cancel; or (ii) τ (αβ) is anti-symmetric and commutes with τ (30) (which is the case precisely only for τ (02) and τ (32) ). These two vanishing conditions reproduce all the vanishing Hermitian forms of the first two lines of (6.5), as well as two additional relations, These relations do not figure amongst the first two lines of (6.5), and remarkably never enter the additional relations either. Equation (7.24) gives the second type of solution, with non-vanishing components,

Solving the Hermitian relations H
(αβ) g±

For the first type of solution
We shall focus on the first type of generic solution (7.25) in the section. Expressing H (αβ) g± in terms of the above parametrization gives This form may be simplified according to whether τ (αβ) commutes or anti-commutes with τ (30) , so that the formula may be split into two cases, (30) τ (20) τ (αβ) r (7.31) It follows from the vanishing of r t τ (γδ) r, when τ (γδ) is anti-symmetric, that we have the following automatic cancelations, The remaining cases of the middle two lines of (6.5), impose the following conditions. For H (30) g− , H where G ± ≡ g 9 e i(Λ ′ −Λ) ± g * 9 e −i(Λ ′ −Λ) . Since r t r = 0, this equation must admit non-trivial solutions, requiring that the determinant of the above 2 × 2 matrix vanish, i.e. G + G − = 0. It is manifest that G + = 0, with G − = 0, cannot be a viable solution. Hence, we must have which then automatically solves also the equations (7.33). Now if g 9 = 0, we have G + = 0, and also the relation r t τ (30) r = 0. r 2 1 + r 2 2 = r 2 3 + r 2 4 (7.36)

Vanishing of g 9 for second type of solution
The computation of H

First type of solution
Without loss of generality, let's choose r 4 = 0 in equations (7.41) and (7.42). It then follows that either r 1 = r 2 = 0 or r 3 = 0. We now examine the dilatino equations in (3.27), If r 1 = r 2 = r 4 = 0, to have a non-vanishing value for r 3 , we must have From the analysis in section 4, we know that this leads to the AdS 5 × S 5 solution. On the other hand, if r 3 = r 4 = 0, the first dilatino equation reads in component form as, In order to have non-vanishing spinor solutions, we should require p z = pz = 0 |h| 2 + |g z | 2 = 0 |h| 2 + |gz| 2 = 0 (7.47) The second dilatino equation provides similar results, requiring p z = pz = g z = gz = h = 0. The solution is still limited to AdS 5 × S 5 .

Second type of solution
The analysis is identical to the one used for the first type of solution, and we again reach the conclusion that the only possible solution is AdS 5 × S 5 .

Ansatz with SO(2, 4) × SO(4) symmetry
Another realization of the superalgebra SU(2, 2|2) in Type IIB supergravity is motivated by the system of a stack of D3-branes in the presence of probe D7-branes (see the Introduction). This realization requires extending the invariance by adjoining a purely bosonic SU(2) group, so that we have full SU(2, 2|2) × SU(2) symmetry [23]. Its bosonic symmetry is SO(2, 4) × SO(4) × SO (2). In this section, we shall use a less restrictive bosonic symmetry SO(2, 4) × SO(4), thereby increasing the generality of our results. We shall show that the only solution is AdS 5 × S 5 for this realization of the symmetry.

Construction of the Ansatz
We seek the most general Ansatz in Type IIB supergravity with SO(2, 4) ×SO(4) symmetry.
The SO(2, 4)-factor requires the geometry to contain an AdS 5 -factor, while the SO(4)-factor requires an S 3 -factor, both of which are warped over a 2-dimensional surface Σ with boundary. The total space-time then has the structure AdS 5 × S 3 × Σ. Following a construction similar to the one given in section 2, the Ansatz for the metric is found to be, where f 3 and f 5 are functions on Σ only. Orthonormal frames are defined by. Here, the metrics ds 2 AdS 5 and ds 2 S 3 , as well as the orthonormal framesê m andê i , refer to the spaces AdS 5 and S 3 with unit radius. In particular, we have The dilaton/axion field B is a function of Σ only, and the associated composites P, Q of (2.1) are decomposed as follows, P = p a e a , and Q = q a e a , where p a and q a are again functions of Σ only. Finally, the most general Ansätze for the 3-and 5-form fields G and F (5) consistent with SO(2, 4) × SO(4) invariance, are given by By SO(2, 4) × SO(4)-invariance, the coefficient functions p a , q a , g and f depend only on Σ. The functions f, q a are real, while p a , g are complex-valued.

Reducing the BPS equations
The supersymmetry spinor ε must be covariant under SO(2, 4) × SO(4), and may be built from the Killing spinors on the space AdS 5 × S 3 , where each factor space has unit radius.

The reduced BPS equations
The detailed calculations for the reduction of the BPS equations are in Appendix C. Here, we summarize the final results. The dilatino BPS equation is given by, Any solution with non-vanishing ζ requires g = 0. The second equation implies det(p a σ a ) = 0, from which we derive either p 8 = ip 9 = 0 or p 8 = −ip 9 = 0 or p 8 = p 9 = 0. The gravitino equation decomposes into a system of 3 equations,

The chiral form of the reduced BPS equations
Using the complex basis defined in (3.24), we proceed to decompose the two-index spinor ζ into its two chirality components in this same 2-dimensional spinor basis, The reduced gravitino BPS equations now becomes

Solving the reduced BPS equations
From the argument given below (8.57), we have three possible solutions for p a . We shall now show that only the p a = 0 solution is viable. Indeed, suppose that we had a solution with p 8 = ip 9 = 0. From (8.57), this implies that ψ = 0. Then from (− 2 ) in (8.60), we find f = 0 in order to have non-zero ξ. But now the (m 1 ) equation becomes 1 2f 5 τ (30) ξ = 0, which in turn implies ξ = 0, making the resulting solution trivial. The same arguments eliminates the case p 8 = −ip 9 = 0. Thus we are left with the only remaining solution p 8 = p 9 = 0. Now when p a = 0, the Bianchi identity dQ = 0 implies that q a are constants. With the help of the U(1) q gauge symmetry, we may choose the gauge q a = 0.

Using discrete symmetries
There are two commuting discrete symmetries for the reduced BPS equations, so we can project the solution to the eigenspace of τ (30) and τ (03) , It turns out that different projections will give the same solution for supergravity fields. We restrict to the projection with ν = γ = 1, so that only the ζ ++ component can be non-zero; we shall abbreviate this component simply as ζ.

Hermitian forms for metric factors
The calculations of Hermitian forms for metric factors proceed in the same way as for SO(2, 4) × SO(3) × SO(2) case, and we will omit some details in the calculation. Using differential equations (−)(+) in equation (8.60), we find, D z (ψψ +ξξ) = 0 (8.63) Soψψ +ξξ is a constant, which may be normalized to f 5 , since f 5 is a constant too, which is derived from (m) equation in the next subsection. Using (+) and (−) and (i) equations, we find D z (f −1 3 (ψψ −ξξ)) = 0, whose general solution is given by, f 3 = c 3 (ψψ −ξξ). Using the fact that f f 5 = 1 and the (i) equation, the normalization factor c 3 is determined to be 1. So the expression for f 3 becomes,

Generic Solution to reduced BPS equations
From equations (m 1 ) and (m 2 ), we have, the solution to which is, which imply that f and f 5 are constants. The solution to (− 1 ) and (+ 2 ) is given by, Where α(z), β(z) are holomorphic functions. Using the above expression, the (− 2 ) and (+ 1 ) equations now become, From the Hermitian form for f 5 in equation (8.64), we derive ρ = f 5 /(|α| 2 + |β| 2 ), and the differential equations become These equations may be compared to equations (4.10), which differ only by a factor of i. Let A(z) be an arbitrary holomorphic function, the solution is

Uniqueness of the AdS 5 × S 5
The above solution coincides with AdS 5 × S 5 . This may be seen by a convenient choice of the coordinate z, for which A(z) = (e z − ie −z )/(e z + ie −z ), for which the metric manifestly factorizes and agrees with the unit radius metric on AdS 5 ×S 5 . Because there are 8 solutions to the Killing spinor χ η 1 η 2 , as well as 4 solutions for spinor ζ from the different projections, we recover indeed 32 supersymmetries.

A Clifford algebras
A.1 Clifford algebra basis for AdS 5 × S 2 × Σ × S 1 space We choose a basis for the Clifford algebra which is well-adapted to the AdS 5 × S 2 × Σ × S 1 , with the frame labeled as in (2.17), where a convenient basis for the lower dimensional Clifford algebras is as follows, We also find the following matrix The 10-dimensional chirality matrix in this basis is given by The complex conjugation matrices in each subspace are defined by where in the last column we have also listed the form of these matrices in our particular basis. The 10-dimensional complex conjugation matrix B is defined by (Γ M ) * = BΓ M B −1 and BB * = I, and in this basis is given by A.2 Clifford algebra basis for AdS 5 × S 3 × Σ space We choose another basis of the Clifford algebra for this ansatz, which is well-adapted to the AdS 5 × S 3 , with the frame labeled as in (8.49), where a convenient basis for the lower dimensional Clifford algebras is as follows, We also find the following matrix The 10-dimensional chirality matrix in this basis is given by The complex conjugation matrices in each subspace are defined by where in the last column we have also listed the form of these matrices in our particular basis. The 10-dimensional complex conjugation matrix B is defined by (Γ M ) * = BΓ M B −1 and BB * = I, and in this basis is given by B Reducing the BPS equations: In this appendix, we shall present the detailed calculations required for the reduction of the BPS equations in AdS 5 × S 2 × Σ × S 1 case. The reduction of the BPS equations (2.11) to our Ansatz requires the following combinations, The corresponding Γ-matrices, in the basis given in Appendix A.1, are given by, We shall also need the decompositions of B −1 ε * , where we use the abbreviation, * ζ η 1 ,η 2 ,η 3 = −σ 2 η 2 ζ * −η 1 ,−η 2 ,−η 3 * ζ = −iτ (121) σ 2 ζ * (B.4) in τ -matrix notation.

B.2 The gravitino equation
The gravitino equation is The non-vanishing spin connection components are as follows, where D a = ρ −1 ∂ a . The hats refers to the canonical connections on AdS 5 , S 2 respectively. Projecting the spin-connection along the various directions we have the expression for dε+ωε where the prime on the covariant derivative indicates that only the connection along AdS 5 , S 2 respectively is included. Using the Killing spinor equations (3.1) we eliminate the primed covariant derivatives, and bring out overall factors of Γ M , which yields All terms in the gravitino equations (m), (i), and (9), will contain Γ M χ η 1 ,η 2 χ η 3 , and we must require the coefficients to vanish independently. The coefficients of Γ M χ η 1 ,η 2 χ η 3 can be expressed in the τ -matrix notation as The F (5) part in the calculation of φ is as follows, Projecting along the various directions, we have Using the τ -matrix notation, we write the coefficients of Γ M χ η 1 ,η 2 χ η 3 or χ η 1 ,η 2 χ η 3 in the form The relevant expression for the calculation of g is as follows, A few useful equations are as follows, Projecting along the various directions we obtain Using the τ -matrix notation, we write the coefficient of Γ M χ η 1 ,η 2 χ η 3 or χ η 1 ,η 2 χ η 3 in the form (3g a τ (010) * ζ − gbτ (010) σb a * ζ − 3ihσ a * ζ) (9) 1 16 (3g 9 σ 3 * ζ − g a σ a * ζ − 3ihτ (010) * ζ)

B.2.1 Assembling the complete gravitino BPS equation
Now we combine the three equations (B.12), (B.15), and (B.19) to obtain the reduced gravitino equations. We again argue that the Γ M χ η 1 ,η 2 or χ η 1 ,η 2 are linearly independent which leads to the equations (3.14). For equations (m), (i) and (9), we have dropped an overall factor of Γ M . In equation (a), ω a = (ω 78 ) a is the spin connection along Σ, and we have used the connection formula (B.9) for the covariant derivative and the fact γ 78 = i σ 3 .
C Reducing the BPS equations: In this appendix, we shall present the detailed calculations required for the reduction of the BPS equations in AdS 5 × S 3 × Σ case. The corresponding Γ-matrices, in the basis given in appendix A.2, are given by, We shall also need the decompositions of B −1 ε * ,

C.1 The dilatino equation
The dilatino equation becomes, Using the above form of the matrices, we obtain, 0 = ip a σ a σ 2 ζ * −η 1 ,η 2 + The hats refers to the canonical connections on AdS 5 , S 3 respectively. Projecting the spinconnection along the various directions we have the expression for dε + ωε where the prime on the covariant derivative indicates that only the connection along AdS 5 , S 3 respectively is included. Using the Killing spinor equations (8.52) we can eliminate the primed covariant derivatives, which yields combining the terms, we have where we have pulled a factor of Γ M out front for equation (m), (i). It will turn out that all terms in the gravitino equation (m) (i) will contain Γ M χ η 1 ,η 2 , and we require the coefficients to vanish independently, just as we did for the dilatino equation. The coefficient of Γ M χ η 1 ,η 2 , χ η 3 can be expressed in the τ -matrix notation as In the subsequent subsections, we shall combine these results with corresponding combinations from the algebraic gravitino BPS equations, and seek relations of the following type, D z r 1 ψ † τ (αβ) ψ + r 2 ξ † τ (αβ) ξ + D z f i f i r 3 ψ † τ (αβ) ψ + r 4 ξ † τ (αβ) ξ = 0 (D.14) where i = 1, 2, 5 and the coefficients r 1 , r 2 , r 3 , r 4 may depend on i, and α, β, but not on Σ.
We shall seek such relations for generic fields f 1 , f 2 , f 5 , f, g z , gz, h, and g 9 , though we shall find that when special relations hold (such as g 9 = 0), additional relations of type (D.14) exist.
E Vanishing Hermitian forms for AdS 5 × S 2 × S 1 × Σ In this Appendix, we derive three sets of vanishing Hermitian forms.

E.1 First set of Hermitan relations
The first set of Hermitian relations is obtained by considering the combination of the BPS