Warped $AdS_{2}$ and $SU(1,1|4)$ symmetry in Type IIB

We investigate the existence of solutions with 16 supersymmetries to Type IIB supergravity on a spacetime of the form $AdS_{2}\times S^{5}\times S^{1}$ warped over a two-dimensional Riemann surface $\Sigma$. The existence of the Lie superalgebra $SU(1,1|4) \subset PSU(2,2|4)$, whose maximal bosonic subalgebra is $SO(2,1)\oplus SO(6)\oplus SO(2)$, motivates the search for half-BPS solutions with this isometry that are asymptotic to $AdS_{5} \times S^{5}$. We reduce the BPS equations to the Ansatz for the bosonic fields and supersymmetry generators compatible with these symmetries, then show that the only non-trivial solution is the maximally supersymmetric solution $AdS_{5}\times S^{5}$. We argue that this implies that no solutions exist for fully back-reacted D7 probe or D7/D3 intersecting branes whose near-horizon limit is of the form $AdS_{2}\times S^{5}\times S^{1}\times \Sigma$.


Introduction
In the study of half-BPS solutions to Type IIB supergravity, recent progress was made on spacetimes with the following factors warped over a two-dimensional Riemann surface Σ: AdS 6 × S 2 [1][2][3][4] and AdS 2 × S 6 [5,6]. In the former case, globally regular and geodesically complete solutions were obtained which provide the near-horizon geometry of (p, q) five-brane webs [7,8]. Such solutions are holographic duals to five-dimensional superconformal field theories, with the SO(2, 5) ⊕ SO(3) isometry extending to invariance under the exceptional Lie superalgebra F (4). In the latter case, the isometry SO(2, 1)⊕SO(7) extends to a different real form of the Lie superalgebra F (4). While solutions were obtained which locally match those of (p, q)-strings [9] in the near-horizon limit, some questions remain regarding their geodesic completeness. Additional families of non-compact globally regular and geodesically complete solutions were obtained independent from any string junction interpretation.
These two cases provide the most recent example of half-BPS solutions to supergravity on pairs of spacetimes with internal factors related by "double analytic continuation", e.g. AdS p × S q and AdS q × S p . While the Minkowski signature of the overall spacetime precludes the possibility of having more than one AdS factor, one may in general have multiple internal spaces of Euclidean signature, provided the bosonic symmetries of the half-BPS configuration are appropriately realized. An earlier example where such pairs of half-BPS solutions to Type IIB supergravity have been constructed are the spacetimes AdS 4 × S 2 × S 2 × Σ [10] and AdS 2 × S 2 × S 4 × Σ [11], providing the holographic duals to interface solutions and Wilson loops (respectively). In both cases, the solutions are asymptotic to AdS 5 × S 5 , and the respective isometries extend to invariance under Lie superalgebras that are both subalgebras with 16 fermionic generators of P SU (2, 2|4).
In [12], a general correspondence was proposed between certain Lie superalgebras with 16 fermionic generators and half-BPS solutions to either Type IIB supergravity or M-theory. In the case of Type IIB, the semi-simple Lie superalgebras H are subalgebras of P SU (2, 2|4), and the corresponding half-BPS solutions are invariant under H and locally asymptotic to the maximally supersymmetric solution AdS 5 × S 5 . It is shown that there exist a finite number of such subalgebras H, and thus one obtains a classification of half-BPS solutions with the above asymptotics. Among these are the special classes of exact solutions previously found in [10] and [11], while those of [1][2][3][4] and [5,6] are absent since neither F (4) nor any of its real forms are subalgebras of P SU (2, 2|4).
Half-BPS solutions related to D7 branes in Type IIB supergravity are of particular interest. The near-horizon limits of D7 probe or D7/D3 intersecting branes seem to support the existence of corresponding half-BPS solutions. However, D7 branes also produce flavor multiplets which ultimately break exact conformal invariance, and by the arguments of [13] and [14,15] (which follows earlier work on D7 branes in [16,17]) no fully backreacted near-horizon limit solutions corresponding to D7 branes should exist. The classifi-2 AdS 2 × S 5 × S 1 × Σ Ansatz in Type IIB supergravity In this section, we review key aspects of Type IIB supergravity, then obtain the Ansatz for bosonic supergravity fields and susy generators with SO(2, 1) ⊕ SO(6) ⊕ SO(2) symmetry.

Type IIB supergravity review
The bosonic fields of Type IIB supergravity consist of the metric g M N , the complex-valued axion-dilaton field B, a complex-valued two-form potential C (2) and a real-valued four-form field C (4) . The field strengths of the potentials C (2) and C (4) are given as follows, The field strength F (5) satisfies the well-known self-duality condition F (5) = * F (5) . Instead of the scalar field B and the 3-form F (3) , the fields that actually enter the BPS equations are composite fields, namely the one-forms P, Q representing B, and the complex 3-form G representing F (3) , given in terms of the fields defined above by the following relations, Under the SU (1, 1) ∼ SL(2, R) global symmetry of Type IIB supergravity, the Einsteinframe metric g M N and the four-form C (4) are invariant, while B and C (2) transform as, where SU (1, 1) is parametrized by u, v ∈ C with |u| 2 − |v| 2 = 1. The field B takes values in the coset SU (1, 1)/U (1) q and Q plays the role of a composite U (1) q gauge field. The transformation laws for the composite fields are as follows [19], Equivalently, one may formulate Type IIB supergravity directly in terms of g M N , F (5) , P, Q and G provided these fields are subject to the Bianchi identities [19,20], The fermion fields of Type IIB supergravity are the dilatino λ and the gravitino ψ M . The conditions that these fields and their variations δλ, δψ M vanish yield the BPS equations, 1 where ε is the supersymmetry generator transforming under the minus chirality Weyl spinor representation of SO (1,9) and ∇ M is the covariant derivative acting on this representation.

SO(2, 1)⊕SO(6)⊕SO(2)-invariant Ansatz for supergravity fields
We construct a general Ansatz for the bosonic fields of Type IIB supergravity consistent with the SO(2, 1) ⊕ SO(6) ⊕ SO(2) symmetry algebra. A natural realization is a spacetime geometry of the form AdS 2 × S 5 × S 1 warped over a two-dimensional Riemann surface Σ. The SO(2, 1) ⊕ SO(6) ⊕ SO(2)-invariant Ansatz for the metric is then of the following form, where the radii f 2 , f 5 , f 1 and ds 2 Σ are functions of Σ. We define an orthonormal frame, whereê m ,ê i , andê 9 respectively refer to orthonormal frames for the spaces AdS 2 , S 5 , and S 1 with unit radius. Here, e a is an orthonormal frame on Σ only, so that we have, The axion-dilaton field B is a function of Σ only, so the 1-forms P and Q can be written as, where the components p a , q a are complex and depend on Σ only. Finally, the complex 3-form G and self-dual 5-form field strength F (5) = * F (5) are given as follows, where the indicesā run over the values 7, 8, 9. The coefficients are constrained by SO(2, 1) ⊕ SO(6) ⊕ SO(2) invariance, so that both the real-valued functions f , q a and complex-valued functions p a , h, g a , g 9 depend only on Σ. This completes the Ansatz for the bosonic fields.
2.3 SO(2, 1) ⊕ SO(6) ⊕ SO(2)-invariant Ansatz for susy generators We decompose the supersymmetry generator ε onto the Killing spinors of the various components of AdS 5 × S 5 × S 1 . The Killing spinor equations on AdS 2 and on S 5 were derived in the appendices of [11] and [21], and are given (respectively) by, Here,∇ m and∇ i are the covariant spinor derivatives on the respective spaces, and integrability requires that η 2 1 = η 2 2 = 1. The action of the chirality matrices is given by, while under charge conjugation we have, The components are found by first choosing (χ c ) ++ ≡ χ −− , then using the chirality matrix γ (1) and charge conjugation matrices B (1) , B (2) to obtain the following relations for all η 1 , η 2 : Killing spinors χ η 3 on S 1 are single functions for each value of η 3 which solve the equation, As explained in [18], we may set (χ η 3 ) * = χ −η 3 , with the values η 3 = ±1 corresponding to a double-valued representation for the spinors. The most general 32-component complex spinor ε that can be decomposed onto the Killing spinors of AdS 2 × S 5 × S 1 , and which is consistent with the 10-dimensional chirality condition Γ 11 ε = −ε, is of the following form, where we have defined the constant spinor, Finally, the charge conjugate spinor is given by, This completes the construction of the SO(2, 1) ⊕ SO(6) ⊕ SO(2)-invariant Ansatz.

Reducing the BPS equations
For purely bosonic Type IIB supergravity fields, half-BPS configurations are those for which the BPS equations yield 16 independent solutions. In this section, we reduce the BPS equations to the Ansatz, employing the same strategy and methods as those used in [18].

The reduced BPS equations
As in [1,5] we use the τ matrix notation introduced originally in [22] to compactly express the action of the various γ matrices on ζ. Defining τ (ijk) = τ i ⊗ τ j ⊗ τ k with i, j, k = 0, 1, 2, 3, τ 0 the identity matrix and τ i for i = 1, 2, 3 the standard Pauli matrices, we can write, The reduction of the BPS equations (2.9) using the decomposition of ε (2.20) onto the Killing spinors (2.15) is discussed in Appendix B. The reduced dilatino equation is given by, while the various components of the reduced gravitino equations are as follows, The derivative D a is defined with respect to the frame e a of Σ, so that the total differential d Σ takes the form d Σ = e a D a , while the U (1)-connection with respect to frame indices isω a .

Symmetries of the reduced BPS equations
Upon reduction to the Ansatz, the U (1) q gauge transformations (2.4) are now given by, In addition to the continuous symmetries, there are linear discrete symmetries which leave the reduced supergravity fields unchanged and act on the supersymmetry generator as follows, Finally, composing complex conjugation with an arbitrary U (1) q transformation, we have,

Further reduction and chiral form of the BPS equations
We now derive the restrictions to one of the linear discrete symmetries which are implied by the reduced BPS equations, following the same procedure that was used for [18]. From (3.5), we see that only τ (033) ∈ S 0 commutes with the BPS differential operator and admits real eigenvalues, so we analyze the restriction of the BPS equations to the two eigenspaces, The non-zero components of the ζ are redefined in terms of a new ζ spinor with two indices, The remaining elements iτ (030) , iτ (003) ∈ S 0 (3.5) map between identical ν, and along with the complex conjugations symmetries (3.6) reduce under (3.8) to the following transformations, We then decompose the spinors ζ η 1 ,η 2 in terms of complex frame basis e a = (e z , ez) on Σ, with a metric δ zz = δz z = 2 and Clifford algebra generators γ a = (γ z , γz) defined as follows, Similar relations hold for p a , q a , g a , e.g. p z = p 7 − ip 8 and pz = p 7 − ip 8 . In this same 2dimensional spinor basis, we decompose the two-index spinor ζ into the chirality components, where ξ * η 1 ,η 2 , ψ η 1 ,η 2 are 1-component spinors. In this basis, the reduced dilatino equation is, The components of the reduced gravitino equation along AdS 2 , S 5 , S 1 are given by, together with the components along Σ, The action of the complex conjugation symmetry (3.9) is given by, with the transformations on the bosonic fields (3.6) translated as follows in the chiral basis, Finally, we note that shifting the metric factor f 1 → νf 1 removes all explicit dependence on ν from the reduced BPS equations, which is irrelevant since the supergravity fields only ever depend on the square f 2 1 . Thus, for every solution to the reduced BPS equations with ν = +1, there exists another solution with ν = −1 so that a systematic doubling of the total number of spinor solutions is produced. Together with the counting of components for the basis of Killing spinors in (2.20), this implies that any solution with ν = +1 produces 16 linearly independent solutions to the BPS equations, thereby generating a half-BPS solution.

Metric factors in terms of spinor bilinears
In this section, we relate the metric factors f 1 , f 2 , f 5 to Hermitian forms of the spinors ψ, ξ. The reality properties of the metric factors impose the conditions that the spinor bilinears be real and invariant under U (1) q transformations. The only combinations that satisfy these requirements are those of the form ψ † τ (αβ) ψ, ξ † τ (αβ) ξ. We seek relations that hold for generic values of the supergravity fields f 1 , f 2 , f 5 , f, g z , gz, h and g 9 . Following the same procedure that was used for [18], we will use combinations of the differential equations (±) in (3.14), and of the algebraic gravitino BPS equations (3.13) to find relations of the following type, 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.

Vanishing Hermitian forms
We can use the reality properties of various combinations of the BPS equations to show that certain Hermitian forms vanish automatically. We consider the following Hermitian forms, h− are purely imaginary. In the following sections, we consider three particular combinations. Then in Section (5.4), separating out the real and imaginary parts yields the full sets of vanishing and non-trivial Hermitian relations.

First set of Hermitian relations
We consider the linear combination (m) + 2(i) + (9) of the BPS equations (3.13). Note that all the terms containing f , g z , gz, h, h * are cancelled. Multiplying the first equation by ξ t τ (αβ) , the second by −ψ t τ (αβ)t , then adding them and taking the transpose, we obtain,

Second set of Hermitian relations
We eliminate the D z f i , g z , g * z terms in each set of equations (m), (i), or (9). We calculate only the (m) and (i) equations, since the relations for the (9) equation can be obtained from a linear combination of the (m) and (i) equations, together with the first set of relations.
For each pair of the f 2 and f 5 equations in the BPS equations (3.13), we multiply the first by ξ t τ (αβ) and the second by ψ t τ (αβ) . The g z , g * z terms then vanish automatically if τ (αβ) t = −τ (αβ) . Adding both to cancel the D z f i terms, then taking the transpose, we have,

Third set of Hermitian relations
Finally, we consider the combination (i) − (9). We multiply the first equation by ξ t τ (αβ) and the second by ψ t τ (αβ) , with τ (αβ) t = +τ (αβ) . Taking the difference and then the transpose,

Summary of all Hermitian relations
The full set of vanishing Hermitian relations is given by, The remaining non-trivial Hermitian relations are as follows. We have the first set, and finally the third set,

Implications for the metric factors
Together with (4.10), the above relations imply the vanishing of the following constants,

General solutions to the reduced BPS equations
In this section, we use the vanishing Hermitian forms to solve the reduced BPS equations. We follow the same procedure and reach the same conclusion as in [18], namely that the only solution to the reduced BPS equations is the maximally supersymmetric solution AdS 5 × S 5 .

First type of solution
For the first type of solutions (6.2), the following Hermitian forms vanish automatically, The remaining vanishing Hermitian forms (5.5) yield two sets of conditions for H (αβ) From the relations (5.5), we also have a set of conditions for the Hermitian forms H = 0 and thus all r η 1 ,η 2 = 0. Non-trivial solutions correspond to G + G − = 0, and one can show that for either choice G ± = 0 and G ∓ = 0, the conditions (6.8) and (6.9) plus the non-trivial relations from Section (5.4), imply that all r η 1 ,η 2 = 0. For example, if G + = 0 and G − = 0 then (6.8) is automatically satisfied, while (6.9) and the second (21) relation in (5.7) imply, The Hermitian forms that vanish under r t τ (30) r = r t τ (33) r = 0 cause a number of relations in Section (5.4) to become trivial, which in turn produce conditions that can only be satisfied if all r η 1 ,η 2 = 0. The only remaining possibility is g 9 = 0, which yields extra vanishing forms: 0 = H The top line of (6.12) plus the original vanishing Hermitian forms imply the conditions, r 1 r 4 = r 2 r 3 = 0 r 1 r 2 − r 3 r 4 = 0 (6.13) Without loss of generality, we choose r 4 = 0, so that either r 1 = r 3 = 0 or r 2 = 0, and examine the dilatino equation (3.12). If r 1 , r 3 = 0 and r 2 = r 4 = 0, then we must have, for non-vanishing spinor solutions. But in order to have non-trivial solutions while satisfying both the original conditions (6.10) and the bottom line of (6.12), we must set h = 0 so that, On the other hand, if we take r 1 = r 3 = r 4 = 0 and r 3 = 0, then this result is automatic.  , we obtain the conditions, e i(Λ 1 −Λ 3 ) + e −i(Λ 1 −Λ 3 ) e iθ g 9 ± e −iθ g * 9 r 1 r 3 = 0 e iθ g 9 ± e −iθ g * 9 r 2 1 − r 2 3 = 0 (6.18)

Second type of solution
For non-trivial solutions with g 9 = 0, we must have Re e i(Λ 1 −Λ 3 ) = 0 and r 2 1 = r 2 3 . Under this choice, the (22) relations of (5.6) and (5.8) reduce to H (00) + = 0 and thus all r η 1 ,η 2 = 0. So we again must have g 9 = 0, which then yields the additional vanishing Hermitian forms, Examining the dilatino equation (3.12), we find the same constraints as (6.14) on the supergravity fields. The extra forms in (6.17) together with (6.20) impose the following conditions: where the η i = ± for i = 1, 2, 3 independently, and we have defined the quantities, An analysis similar to the one used for the first type of solution again yields the result (6.15).

Vanishing G implies the AdS 5 × S 5 solution
When G = 0, we have g z = gz = g 9 = h = 0. For half-BPS solutions, ψ and ξ cannot both vanish, and the reduced dilatino equation (3.14) implies p z = pz = 0. By the Bianchi identities (2.5), P = 0 implies dQ = 0, and we use the U (1) q gauge symmetry to set Q = 0. Therefore, the requirements (6.15) can be obtained directly by imposing the vanishing of G.

A Clifford algebra basis adapted to the Ansatz
The Dirac-Clifford algebra is defined by {Γ A , Γ B } = 2η AB I 32 , where A, B are 10-dimensional frame indices and η AB = diag(− + · · · +). We choose a basis for the Clifford algebra which is well-adapted to the AdS 2 × S 5 × Σ × S 1 Ansatz, with the frame labeled as in (2.11), where the lower dimensional Dirac-Clifford algebra is defined as follows, The chirality matrices on the various components of AdS 2 × S 5 × Σ × S 1 are given by, which yields the following 10-dimensional chirality matrix, The complex conjugation matrices in each component 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 satisfies, and in this basis has the following form,