Warped AdS6 × S2 in Type IIB supergravity I: local solutions

We investigate the existence of solutions with 16 residual supersymmetries to Type IIB supergravity on a space-time of the formc AdS6×S2 warped over a two-dimensional Riemann surface Σ. The SO(2, 5) × SO(3) isometry extends to invariance under the exceptional Lie superalgebra F (4). In the present paper, we construct the general Ansatz compatible with these symmetries, derive the corresponding reduced BPS equations, and obtain their complete local solution in terms of two locally holomorphic functions A±\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathcal{A}}_{\pm } $$\end{document} on Σ, subject to certain positivity and regularity conditions. Globally, (A+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathcal{A}}_{+} $$\end{document},A−\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathcal{A}}_{-} $$\end{document}) are allowed to be multiple-valued on Σ and be holomorphic sections of a holomorphic bundle over Σ with structure group contained in SU11×ℂ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathrm{S}\mathrm{U}\left(1,1\right)\times \mathbb{C} $$\end{document}. Globally regular solutions are expected to provide the near-horizon geometry of (p, q) 5-brane and 7-brane webs which are holographic duals to five-dimensional conformal field theories. A preliminary analysis of the positivity and regularity conditions will be presented here, leaving the construction of globally regular solutions to a subsequent paper.


Introduction
Gauge-gravity duality, namely the equivalence between a quantum field theory in d dimensions and a gravitational theory in d + 1 dimensions via holography, has become one of the cornerstones of modern theoretical physics. The best understood examples are provided by the AdS/CFT correspondence and involve conformal field theories (CFTs) with a large number of fields and their anti-de Sitter (AdS) dual space-times with large degrees of supersymmetry. Maldacena's original paper [1] contained examples of such equivalences in d = 2, 3, 4 and 6 and many more dual pairs have been uncovered in these dimensions since then. However, what is almost entirely missing to date is any well understood example of the AdS/CFT correspondence in dimension d = 5, namely between a 4+1 dimensional CFT and its 5+1 dimensional dual AdS space-time.
One reason that d = 5 is special is the absence of maximally supersymmetric theories, in which the 16 Poincaré supersymmetries of a 4+1 dimensional CFT would be enhanced by 16 conformal supersymmetries to the maximum allowed number of 32 supersymmetries. The complete classification of superconformal algebras [2] indeed shows that d = 5 is singled out and does not support a maximally supersymmetric CFT. Unlike its lower dimensional cousins, supersymmetric Yang-Mills theory with maximal Poincaré supersymmetry in 4+1 dimensions is not a conformal field theory at the origin of its moduli space. Instead, the theory is believed to flow in the ultraviolet (UV) to the 5+1 dimensional (2, 0) theory, a theory which itself features prominently in AdS/CFT (see for example [3]).

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What does exist, however, are supersymmetric CFTs (SCFTs) in 4+1 dimensions with 8 Poincaré supersymmetries which are enhanced by 8 conformal supersymmetries to a total of 16 supersymmetries. The superconformal algebra in this case is based on the exceptional Lie superalgebra F (4). This Lie superalgebra has a real form whose maximal bosonic subalgebra is SO(2, 5) ⊕ SO(3) [4], which may be viewed as the direct sum of the conformal algebra in 4+1 dimensions [4] and an R-symmetry. Explicit field theory examples realizing this algebra have been uncovered in [5,6]. They are based on supersymmetric gauge theories which flow to a strongly coupled CFT in the UV, as long as the number of matter fields coupled to the gauge field remains sufficiently small. The upper bound on the number of matter fields depends on the gauge group and on the representations of the matter content and can generically be determined from the 1-loop running of the gauge coupling [6]. Since these supersymmetric theories allow for a large N limit, one may expect that they will possess a holographic AdS 6 dual.
String theory realizations of these 4+1 dimensional CFTs via brane embeddings are known and can serve as a natural starting point to construct their AdS 6 duals. For example, 4+1 dimensional SCFTs based on the gauge group Sp(N ), with hypermultiplets in the fundamental and anti-symmetric tensor representations of Sp(N ), were realized in [5] via a Type IIA string theory construction involving a stack of N D4 branes near a collection of O8 planes and D8 branes. A Type IIA supergravity solution based on this construction was given in [7] and generalized to quiver gauge groups in [8]. While some interesting questions can be addressed in this geometry, its construction leads to singularities, because the presence of the O8 planes forces the dilaton to blow up.
A much more general brane realization of these 4+1 dimensional CFTs can be given via (p, q) brane webs [9,10] in Type IIB string theory. In these brane webs, D5 branes are suspended between NS5 branes, giving a 4+1 dimensional version of the construction pioneered in [11] for 2+1 dimensional gauge theories. The brane webs can realize the gauge theory at any point on its moduli space, as well as in the presence of relevant deformations such as mass terms or a finite bare coupling. In the limit when all the branes essentially lie on top of one another the webs do realize large classes of the 4+1 dimensional SCFTs of [6], most naturally those based on SU(N ) gauge theories, possibly with couplings to matter in the fundamental representation. 1 D7-branes can be added to the (p, q) webs [13], thereby slightly expanding the set of SCFTs that can be realized. As we will see, the most general supergravity Ansatz respecting the symmetries of the (p, q) web does allow for a nontrivial axion winding which is required to accommodate D7-branes. When no D7-branes are present, the (p, q) webs form a special sub-class for which the axion winding vanishes.
It is our goal to construct Type IIB supergravity solutions which are holographically dual to the 4+1-dimensional CFTs realized via (p, q) webs in the large N limit. This will require solving the difficult problem of obtaining fully localized solutions for the corresponding intersecting branes. Examples where such solutions can give rise to warped AdS JHEP08(2016)046 spaces were given in [14,15]. Our approach will be to use the general Ansatz for the Type IIB fields consistent with the symmetries, reduce the BPS equations to this Ansatz, and then solve the BPS equations explicitly.
Earlier attempts at using the BPS equations appeared in [16][17][18], where reduced BPS equations were obtained, but not generally solved. A Type IIB T-dual configuration of the D4/D8 solution in Type IIA [7] was used to test these equations. This T-dual solution is even more troublesome than the original Type IIA solution. In addition to the singularity caused by the presence of the O8 plane on the IIA side, one now has an additional singularity due to T-dualizing a U(1) subgroup of the SO(4) isometry associated with an internal sphere. The circle that is being T-dualized shrinks to zero size at the poles of the sphere, giving rise to a new singularity in the Type IIB solution. Nevertheless, this T-dual to the D4/D8 system provides a useful check on the BPS equations obtained in [16][17][18] and we will use it for a similar purpose.
In the present paper, we will construct the general Ansatz in Type IIB consistent with the SO(2, 5) × SO(3) symmetries, derive the reduced BPS equations, and construct their general local solutions in terms of two locally holomorphic 2 functions A ± on a Riemann surface Σ. To connect these local supergravity solutions to the CFTs originating from the (p, q) brane system requires that we impose the necessary physical regularity conditions on the supergravity fields of the solutions. For this purpose, Σ must be compact, with or without boundary, and geodesically complete. The conditions on A ± needed to guarantee the proper Minkowskian signature of the metric are local on Σ and given by the following inequalities, where ∂ w is the derivative on Σ with respect to a local holomorphic coordinate w, and B is defined, up to an additive constant, by the relation ∂ w B = A + ∂ w A − − A − ∂ w A + . As a result, B is also locally holomorphic. If Σ has a boundary then the two inequalities of (1.1) must hold strictly in the interior of Σ, and become equalities on the boundary of Σ. The group SU(1, 1) acting linearly on the doublet (A + , A − ) leaves the conditions (1.1) invariant and induces the standard SU(1, 1) = SL(2, R) duality transformations on the supergravity fields. Additionally, the supergravity solutions are invariant under constant shifts A ± → A ± + a ± for constants satisfying a − =ā + , which form the additive group isomorphic to C. The SL(2, Z) duality symmetry of Type IIB string theory allows us to consider supergravity solutions with identifications under SL(2, Z), namely with non-trivial axion winding number. Mathematically, the problem then becomes to obtain holomorphic sections (A + , A − ) of a holomorphic bundle over Σ with structure group contained in SL(2, Z) × C, subject to the positivity conditions of (1.1).
In the simplest case where the Riemann surface Σ is compact and has no boundary, the second inequality in (1.1) becomes trivial since the arbitrary constant in B can always be chosen to satisfy the inequality. The associated mathematical problem then also simplifies,

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and may be formulated directly in terms of a holomorphic bundle of one-forms ∂ w A ± satisfying the first relation in (1.1). In this paper, we will provide a preliminary analysis into the existence of such global solutions but leave a detailed investigation for future work.
We close the introduction with some remarks on the relation of this work to other investigations into half-BPS solutions to Type IIB, M-theory, and six-dimensional supergravities on space-times built as products of an AdS space and one or several spheres warped over a Riemann surface Σ. In each case, the isometries of the AdS space and the sphere factors are used to reduce the BPS equations to a complicated set of non-linear partial differential equations on Σ, which can be solved exactly in terms of harmonic or holomorphic data on Σ. This strategy was employed successfully to the construction of a large variety of novel supergravity solutions in different contexts. Type IIB supergravity duals to four-dimensional N = 4 super-Yang Mills theory were found in the presence of a planar interface [19] giving supersymmetric Janus field theories [20,21] and in the presence of Wilson loops [22]. M-theory duals to field theories in three and six dimensions were found in the presence of various defect operators in [23][24][25]. Finally, six-dimensional supergravity duals were found to two-dimensional conformal field theories with string junctions in [26]. 3 The unifying principle of this strategy was explained in [32] as follows. The integrability conditions on the BPS equations produce Bianchi and field equations. With enough supersymmetry, all the Bianchi and field equations may be obtained as integrability conditions on the system of BPS equations, which therefore play a role somewhat analogous to that of a Lax pair for integrable systems. Upon reduction to AdS 6 × S 2 warped over Σ, the BPS equations reduce to equations for functions on Σ and genuinely become a set of Lax equations for a system that must therefore be integrable in the classic sense. The conformal invariance of these systems lies at the origin of their solvability in terms of harmonic and holomorphic data on a Riemann surface Σ.
We note that close cousins to the half-BPS solutions to Type IIB supergravity obtained here are the half-BPS solutions on a space-time of the form AdS 2 × S 6 warped over a Riemann surface Σ. The superconformal algebra is now a different real form of the exceptional Lie superalgebra F (4), this time with maximal bosonic subalgebra SO(2, 1) × SO(7), and 16 supersymmetries as well. The two problems are related just as duals to Wilson loops in [22] are related to the planar interface solutions in [19].

Organization
The outline of this paper is as follows. In section 2 we will review the basics of Type IIB supergravity and introduce a suitable Ansatz. In section 3 we use the decomposition of the supersymmetry generators onto Killing spinors of AdS 6 and S 2 to reduce the BPS equations for the reduced supergravity fields to a system of algebraic and partial differential equations on the surface Σ, which we partially solve. In section 4 we solve the reduced BPS equations completely and obtain the most general local solutions to the BPS equation in terms of holomorphic data. In section 5 we summarize the expressions for all the supergravity fields in terms of the holomorphic data, analyze their behavior under SU(1, 1) symmetry JHEP08(2016)046 of Type IIB supergravity, obtain the regularity conditions, recover the singular T-dual of the D4/D8 system, and conclude the section with arguments in favor of monodromy of the holomorphic data. We conclude with a discussion in section 6. In appendix A, a basis for the Dirac-Clifford algebra adapted to our Ansatz is presented, while the geometry of Killing spinors is reviewed in appendix B. Details of the derivation of the BPS equations are in appendix C, of the Bianchi identities in appendix D, and of the expressions for the supergravity fields in terms of holomorphic data are in appendix E.
2 Type IIB supergravity and AdS 6 × S 2 × Σ Ansatz In this section, we provide a brief review of the Type IIB supergravity fields, Bianchi identities, field equations and BPS equations, and their SU(1, 1) duality symmetry, and go on to construct the general Ansatz for the bosonic fields of the solutions we seek to construct. As laid out in the introduction, the bosonic symmetries of the Ansatz are completely determined by the superconformal algebra.

Type IIB supergravity review
The bosonic fields of Type IIB supergravity [33,34] consist of the metric g M N , a one-form P and gauge connection Q representing the axion-dilaton field strengths, a complex threeform field strength G, and a self-dual five-form field strength F 5 . The fields satisfy the following Bianchi identities, The field strength F (5) is required to be self-dual, The field equations are given by, The fermionic fields are the dilatino λ and the gravitino ψ M , both of which are complex Weyl spinors with opposite 10-dimensional chiralities, given by Γ 11 λ = λ, and Γ 11 ψ M = JHEP08(2016)046 −ψ M . The supersymmetry variations of the fermions are, where B is the charge conjugation matrix of the Dirac-Clifford algebra. 4 The BPS equations are obtained by setting δλ = δψ M = 0. The Bianchi identities (2.1) for the field strengths P, Q, G, F (5) can be solved in terms of a complex scalar B; a complex 2-form potential C (2) , and a real 4-form potential C (4) . The fields P and Q are expressed as follows, while the fields G and F (5) are conveniently expressed in terms of C (2) and C (4) with the help of the complex field strength F (3) = dC (2) , The scalar field B is related to the complex scalar τ and the axion χ, and dilaton φ by, The expectation value of e 2φ is related to the string coupling constant.

SU(1, 1) duality symmetry
Type IIB supergravity is invariant under SU(1, 1) = SL(2, R) symmetry. This symmetry leaves the Einstein frame metric g M N as well as the 4-form C (4) invariant, acts on the field B by Möbius transformations, and acts on the 2-form C (2) and its complex conjugateC (2) by a linear transformation, with u, v ∈ C and |u| 2 − |v| 2 = 1. In this non-linear realization of SU(1, 1) on B, the field B takes values in the coset SU(1, 1)/U(1) q , and the fermions λ and ψ µ transform linearly 4 Our convention for the signature of the 10-dimensional space-time metric is η = diag(− + · · · +); the Dirac-Clifford algebra is defined by the relations {Γ M , Γ N } = 2η M N I32; and the charge conjugation matrix B is defined by the relations BB * = I and BΓ M B −1 = (Γ M ) * . We will use the convention that repeated indices are to be summed; complex conjugation will be denoted by bar for functions and by star for spinors; and we will use the notation Γ · T ≡ Γ M 1 ···Mp TM 1 ···Mp for the contraction of any antisymmetric tensor field T of rank p and the Γ-matrix of the same rank.

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under the isotropy gauge group U(1) q with composite gauge field Q. The transformation rules for the field strengths are [33], where the phase θ is defined by, The SU(1, 1) = SL(2, R) symmetry will serve as a useful guide to organize the holomorphic data in our local solution. As is well-known, the invariance of Type IIB supergravity under the continuous group SL(2, R) is reduced in Type IIB string theory to invariance under the discrete SL(2, Z) S-duality symmetry, due to the charge quantization of non-perturbative one-branes, five-branes and D-instantons. In the construction of supergravity solutions, we will always allow for the continuous symmetry.

The AdS 6 × S 2 × Σ Ansatz
We seek a general Ansatz in Type IIB supergravity with the following symmetry group, The factor SO(2, 5) requires the geometry to contain AdS 6 , while the factor SO(3) requires S 2 , so that our space-time is given by, Here Σ stands for the remaining two-dimensional space over which the product AdS 6 ×S 2 is warped. In order for the above space to be a Type IIB supergravity geometry, Σ must carry an orientation as well as a Riemannian metric, and is therefore a Riemann surface, possibly with boundary. This SO(2, 5) × SO(3)-invariant Ansatz for the metric can be written as, where f 6 , f 2 and ds 2 Σ are functions of Σ only. We introduce an orthonormal frame, e a a = 8, 9 (2.14) whereê m andê i respectively refer to the orthonormal frames for the spaces AdS 6 and S 2 with unit radius, and e a is an orthonormal frame for the metric on Σ. In particular, we have, mnê m ⊗ê n η (6) = diag(− + + + ++)

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By SO(2, 5) × SO(3)-invariance, the fields P, Q, G and F (5) are given as follows, P = p a e a G = g a e a ∧ e 67 Q = q a e a F (5) = 0 (2.16) where e 67 = e 6 ∧ e 7 . The components p a , q a , and g a are complex. Note that the Bianchi identity for the five-form field (2.1) is automatically satisfied with this Ansatz.

Reducing the BPS equations
The residual supersymmetries, if any, of a configuration of purely bosonic Type IIB supergravity fields are governed by the BPS equations of (2.4). Our interest is in purely bosonic field configurations which preserve 16 independent supersymmetries given by the AdS 6 × S 2 × Σ Ansatz of the preceding section. It will turn out that any such configuration automatically solves the Bianchi and field equations, and thus automatically provides a half-BPS solution to Type IIB supergravity.
In this section, we will reduce the BPS equations to the AdS 6 × S 2 × Σ Ansatz by decomposing the supersymmetry parameter ε of (2.4) onto the Killing spinors of AdS 6 ×S 2 . We will expose the residual symmetries of the reduced BPS equations, and solve those reduced equations which are purely algebraic in the supersymmetry spinor components. This will produce simple algebraic expressions for the metric factors f 2 , f 6 in terms of the spinors. The remaining reduced BPS equations will then gradually be solved for the remaining bosonic fields as well as for the residual supersymmetries in subsequent sections. The strategy employed here is very similar to the one used in [19] and so our discussion will closely follow that work.

Killing spinors
The Killing spinor equations on AdS 6 × S 2 are, 5 where∇ m and∇ i stand for the covariant spinor derivatives respectively on the spaces AdS 6 and S 2 with unit radius. Recall that m, i, and a are all frame indices. The spinors χ η 1 ,η 2 α are 16-dimensional, and the parameters η 1 and η 2 can take the values ±1. The solutions to these equation are 4-fold degenerate for each value of η 1 , η 2 , and this degeneracy will be labeled by the index α = 1, 2, 3, 4. The chirality matrices act as follows, The decomposition of the 10-dimensional Dirac-Clifford matrices under the reduction to the AdS6×S 2 × Σ Ansatz, and the details of the Killing spinor equations, are relegated to appendices A and B respectively.

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The way these equations should be understood is as follows. We begin with η 1 = η 2 = +, and pick a basis χ ++ α for the four-dimensional vector space of spinors for fixed η 1 , η 2 such that the action γ (1) and γ (2) is diagonal. Then, we can simply define the basis for χ η 1 ,η 2 α for the remaining three values of η 1 , η 2 by the action of the chirality matrices above.
Since ε * appears in the fermion variations of (2.4), we also need to understand how to express the complex conjugate spinor in this basis. If χ η 1 ,η 2 α satisfies (3.1), then by complex conjugating the entire first equation and using (γ m ) * = B (1) ) * satisfies the same equation, with the same values of η 1 , η 2 . Proceeding analogously for the second equation, we conclude that I 6 ⊗ B −1 (2) (χ η 1 ,η 2 α ) * also satisfies the same equation, with the same values of η 1 , η 2 . As a result, we must have the following linear relation, for some matrix M αβ for each pair η 1 , η 2 .
We will now show that one may choose a basis for the Killing spinors χ η 1 ,η 2 α in which M = I. Iterating the complex conjugation, we conclude that (M η 1 ,η 2 ) * (M η 1 ,η 2 ) = I, for all values of η 1 , η 2 . Specializing first to η 1 = η 2 = +, we have a single matrix M +,+ satisfying (M +,+ ) * (M +,+ ) = I. Now every such matrix may be rotated to the identity by a general linear complex-valued 4 × 4 matrix U , using the relation M +,+ = (U * ) −1 U . An easy way to construct U is as follows. An arbitrary invertible complex matrix M in GL(4, C) may be written as an exponential, M = exp(iH) of a complex matrix H. Given M , the matrix H is not unique. The condition M * M = I requires H to be real-valued. Thus, we choose the solution U = exp i 2 H to the equation M +,+ = (U * ) −1 U , and the relation for η 1 = η 2 = + may be diagonalized as follows, for α = 1, 2, 3, 4. For the other values of η 1 , η 2 we use (3.2) to express χ η 1 ,η 2 α in terms of χ +,+ α , Using the fact that γ (1) commutes with B (1) , while γ (2) anti-commutes with B (2) , we find, for all values of η 1 , η 2 , α. Since this decomposition is now canonical in terms of the degeneracy index α, we will no longer indicate it explicitly.

Decomposing onto Killing spinors
An arbitrary 32-component complex spinor ε may be decomposed onto the above Killing spinors as follows,

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where ζ η 1 ,η 2 is a complex 2-component spinor for each η 1 , η 2 , and the 4-fold degeneracy has not been indicated explicitly. As a supersymmetry generator in Type IIB, the spinor ε must be of definite chirality Γ 11 ε = −ε, which places the following chirality requirements on ζ, The charge conjugate spinor is given by, Since γ (2) anticommutes with B (2) , we obtain, It will be convenient to denote the result as follows, As in [19] we will use the τ matrix notation introduced originally in [29] in order to compactly express the action of the various γ matrices on ζ. Defining τ (ij) = τ i ⊗ τ j with i, j = 0, 1, 2, 3, τ 0 the identity matrix and τ i with i = 1, 2, 3 the standard Pauli matrices, we can write,

Symmetries of the reduced BPS equations
Using the decomposition of ε into Killing spinors and the streamlined notation of the τ matrices we can finally write down the BPS equations in a reduced form. The reduced dilatino equation is, while the reduced gravitino equations take the following form, The derivative D a is defined with respect to the frame e a , so that the total differential d takes the form d = e a D a , while the U(1)-connection with respect to frame indices isω a . Before we move on to solving these equations, let us briefly look at their symmetries. The axion/dilaton field B transforms non-linearly under SU(1, 1) of Type IIB supergravity and takes values in the coset SU(1, 1)/U(1) q . Global SU(1, 1) transformations on the fields JHEP08(2016)046 are accompanied by local U(1) q gauge transformations, given in (2.9), and which induce the following symmetry transformations on the fields of the reduced BPS equations, The reduced BPS equations are also invariant under the following discrete symmetries, which leave all the bosonic fields invariant. Both I and J commute with U(1) q . Finally, complex conjugation is naturally combined with U(1) q , and we have, The chirality requirement of Type IIB restricts the spinor ζ to the subspace, In the next subsection, we will investigate the restrictions to the eigen-spaces of J and K imposed by the reduced BPS equations. The symmetries I, J , K commute with one another, so that we may diagonalize them simultaneously, and restrict to any one of their common subspaces.

Restricting to a single subspace of J
We will assume that p a does not vanish identically. We now use the dilatino equation to derive a first set of bilinear relations. Multiply the dilatino equation to the left by ζ t T σ 2 and choose the τ -matrix T so that the flux part vanishes, Since σ 2 γ a is symmetric for a = 8, 9, this condition will require T to satisfy the condition that the product T τ (03) is anti-symmetric, which has the following solutions, (11) , τ (31) , τ (20) , τ (22) , τ (23) } (3.20) The equation implied on p a is then given by the complex conjugate of,

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When p a = 0, we can draw from this equation only an orthogonality relation. To obtain a full vanishing condition, we make further use of the chirality condition, and obtain, We obtain such a relation when both T and T τ (11) belong to T , which is the case for only a single pair, namely T = τ (20) , τ (31) . As a result, we have the equivalent relations, Next, we analyze the gravitino equations. We multiply equations (m) and (i) of (3.14) on the left by ζ † T σ p for p = 0, 3, and obtain cancellation of the last term when T τ (03) is antisymmetric (which is the same condition we had for the dilatino equation), In view of (3.23), the second term will cancel when T = τ (20) and T = τ (31) , so that we obtain the following relations from the remaining cancellation of the first term, and their chiral conjugates, obtained by using the chirality condition τ (11) Next, we use the general result of [19] that the bilinear equation ζ † M ζ = 0 is solved by projecting ζ onto a subspace with the help of a projection matrix P that anti-commutes with M . This result was established for the case of 2-dimensional M , which is in fact the case also here by reduction. Thus, we must find a projector P which commutes with I and with the following properties, The solutions to these conditions are τ (32) and τ (23) , possibly multiplied by a factor of σ 3 . These four possibilities are pairwise equivalent under the chirality relation. Now the projector P = τ (32) precisely corresponds to the symmetry J , so imposing a restriction on the spinor space by this operator is the only consistent restriction. Therefore, we will impose, which solves all the above bilinear relations. One must pick one value of ν or the other in the projection.

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Imposing the chirality relation (3.18), τ (11) σ 3 ζ = −ζ, as well as the projector (3.28) we just derived, we may solve for the relations between the components of ζ. Denoting the components by ζ abc , which take values ±, the a, b components labels the τ -matrix basis, while c labels the chirality basis in which σ 3 is diagonal. We then have two independent complex-valued components which we denote by α and β, and which are defined as follows, To reduce the equations to a basis of complex frame indices z,z, we will use the following conventions, In particular, we have, It will also be convenient to have the following results of γ a b σ 2 ,

The reduced BPS equations in component form
Using the solution (3.29) we found for the projection condition on the preserved supersymmetries, the reduced dilatino equations become, The algebraic gravitino equations are,

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Finally, the component decomposition of the differential equations is as follows, In addition, we have the complex conjugate equations to all of the equations above. Note that since G and P are complex-valued, we have in general (g z ) * = gz and (p z ) * = pz.

3.6
Determining the radii f 6 , f 2 in terms of the spinors We begin with the two equations that involve D z f 6 , To eliminate the contributions from the first term in each equation, we add the first line timesβ to the second line timesᾱ, and we obtain, Proceeding analogously for D z f 2 , we have, To eliminate the contributions from the first term in each equation, we add the first line times −β to the second line timesᾱ, and we obtain, These combinations suggest that we should evaluate the covariant derivatives D z (αᾱ ± ββ) out of the differential equations (3.35) for α, β,ᾱ,β, and we find,

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We may now eliminate all flux dependences between (3.37), (3.39) and (3.40), and we find, Given that the arguments of the derivatives are real functions, we conclude, for constant c 2 , c 6 . In obtaining the equation for f 2 from the last equation in (3.41), we have assumed that |α| 2 − |β| 2 does not vanish identically.

Solving the remaining algebraic gravitino equations
To obtain the results of the previous subsection, we have taken only pairwise linear combinations of the algebraic gravitino equations. Here, we take the orthogonally conjugate pairwise linear combinations, multiplying the first equation byᾱ and the second by −β.
The determinant of the two linear combinations is αᾱ + ββ = 0, so that the four resulting bilinear equations are guaranteed to be equivalent to the original four algebraic gravitino equations. The terms in D z f 6 and D z f 2 cancel out, and we are left with, The last equation may be simplified with the help of the first and yields, Recall that ν can take the values ν = ±1.

Summary of remaining equations
We may summarize the remaining equations as follows. The dilatino equations are the only ones involving p z , and may be viewed as determining p z , Next, we have the radii in terms of the spinors,

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and the algebraic relation between the spinors and the fluxes, and finally the differential equations, along with their complex conjugates.

Local solutions to the BPS equations
In the previous section the BPS equations were reduced and solved for the radii f 6 , f 2 .
The remaining equations are all complex-valued and are organized as follows: we have one algebraic relation (3.47) and four differential equations (3.48) for the spinor components α, β, the reduced flux fields g z , g * z and their complex conjugates. Two further algebraic equations give the axion-dilaton field p a in terms of the spinors and the fluxes. In this section we will solve completely for the local solutions to this system. Specifically, we will derive expressions for all supergravity fields that satisfy the BPS equations, the Bianchi identities, and the supergravity field equations, in terms of two locally holomorphic functions A + , A − on Σ. We will discuss the conditions on the local solutions imposed by the proper Minkowski signature of the metric and the absence of singularities in section 5.
The construction of the local solution is involved and proceeds in a number of steps which we will now outline, and carry out in this section in detail. First, we will eliminate the reduced flux fields g z and gz in favor of the reduced axion-dilaton fields p z , pz and similarly for their complex conjugates. Second, we will use the expression for the reduced axiondilaton fields p z , pz, q z , qz in terms of ρ and B to decouple and integrate one pair of the differential equations, and obtain the spinor componentsᾱ,β in terms of two holomorphic oneforms κ ± as well as B and ρ. Third, we will eliminate the spinor components α, β in favor of κ ± , ρ, and B as well, and thereby produce three nonlinear partial differential equations for the complex field B and the real field ρ. Being non-linear, these equations are not easy to solve, and give rise to a situation reminiscent of [19]. However, in a fourth step we will identify a sequence of two changes of variables which decouples these non-linear differential equations. In a final fifth step, we will solve all decoupled equations in terms of two holomorphic functions A ± which are related to κ ± by κ ± = ∂ w A ± . For the reader who wishes to skip this entire derivation we have summarized the final result in the introduction to section 5.2.

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4.1 Eliminating the reduced flux fields (step 1/5) First, we eliminate the reduced flux fields g z , gz and their complex conjugates in favor of p z , pz and their complex conjugates, using the dilatino BPS equations (3.45), The algebraic relation (3.47) becomes, The differential equations (3.48) take the following form, Equations (4.2) and (4.3) are the remaining relations to be solved. The integrability conditions on the differential equations reproduce the Bianchi identities for the fields P and Q.

4.2
Integrating the first pair of differential equations (step 2/5) In the second step, we show that the first two equations of (4.3) can be solved in terms of holomorphic functions. Multiplying the first equation of (4.3) by α and the second equation of (4.3) by β, we get equivalently, We switch to conformally flat complex coordinates (w,w) on Σ, such that the metric reads ds 2 Σ = 4ρ 2 dwdw and we have, The extra factor of ρ −1 in the derivatives D z , Dz is due to z,z being frame indices. We then express p z and q z in terms of the complex field B using (2.5), to recast (4.4) as follows,

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By taking suitable linear combinations it is straightforward to verify that the following two equations are equivalent to (4.6), These equations are solved in terms of two independent holomorphic 1-forms κ ± , as follows, Inverting the relation (4.8), we get expressions for the spinor components α, β, and their complex conjugatesᾱ,β, The right side of all four equations involves only the holomorphic data κ ± and the B-field and their complex conjugates. It remains to solve for the ρ and B-fields.

Solving the second pair of differential equations (step 3/5)
In the third step we express the third and fourth equation of (4.3) in terms of B, ρ and the local complex coordinates giving, Next, we compute the derivatives ofᾱ andβ using (4.3). After taking suitable linear combinations, with coefficients (B, 1) and (1, B) of the resulting equations, we find the following equivalent system, One now eliminates the spinor components using (4.9) in terms of B, f and ρ, and we obtain, after some simplifications,

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Here we have used the following abbreviation for the phase angle ϑ, Note that by subtracting the two equations (4.12) the dependence on the metric factor ρ can be eliminated and one arrives at an equation for B andB only in terms of κ ± , The equations (4.12) are of course supplemented by their complex conjugates. While eq. (4.14) seemingly depends on both holomorphic one-forms κ + and κ − , in fact it depends only on their ratio. Assuming that κ − does not vanish identically, we define the holomorphic or meromorphic function λ by, it is straightforward to show that (4.14) depends on λ alone and takes the form, where the phase of (4.13) is now given by the expression, The last equation we have to deal with is the algebraic relation (4.2). As before, it may be expressed in terms of B, ρ and the local complex coordinates giving, Eliminating ρα 2 and ρβ 2 using (4.9), we obtain, Its dependence on κ ± cannot be reduced to a dependence solely on λ since the one-form κ − is needed to combine with ρ 2 to produce an equation in which all terms transform as (1, 0) forms, and we obtain, In summary, we have reduced the remaining BPS equations and expressed them in terms of complex differential equations given by (4.12) and (4.19) along with their complex conjugate equations. We will show in appendix D that these equations imply that the Bianchi identities are satisfied, and that there are hence no more constraints to take into account. Integrating these equations will give the complex scalar field B and the metric field ρ in terms of the holomorphic one forms κ ± . In the next section we will perform several variable redefinitions which bring the equations into a form where they can be decoupled and integrated.

Decoupling by changing variables (step 4/5)
In this subsection, we will perform two consecutive changes of variables to decouple the remaining equations. The corresponding choices will be motivated first and then carried out on the equations.

First change of variables, from B to Z
A first change of variables replaces B by a complex field Z and is designed to parametrize the phase e iϑ in (4.16) without the square root required from its definition in (4.17). The following rational change of variables eliminates B in terms of a complex function Z by, and will allow us to express e iϑ and f 2 as rational functions of Z and its complex conjugate, The derivatives of B andB take the following form, Implementing this change of variables on equation (4.16) produces the following form, Equation (4.16) was just one linear combination of the two equations in (4.12), given by the difference of the two equations (4.12). We will take the first equation of (4.12) as the complimentary independent equation, and eliminate its B-dependence in favor of Z, Finally, eliminating B in favor of Z in the algebraic flux equation (4.20) as well, we obtain, It remains to solve the system of equations (4.24), (4.25) and (4.26).

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4.4.2 Second change of variables, from Z to R, ψ A second change of variables is inspired by the form of equation (4.25), in which the norm of Z and its phase enter in distinct parts of the equation. We express the complex field Z in terms of two real variables, the absolute value R of Z, and its phase ψ, It will also be useful to change variables from ρ toρ in the following way, In terms of these variables (4.25) takes the form, while (4.24) becomes, and (4.26) becomes, Adding (4.30) and (4.31) cancels the terms proportional to ∂ w R. Remarkably, and the secret to decoupling the equations, is now that the entire R-dependence of this sum resides in an overall multiplicative factor of (1 + R) 2 . Omitting this factor, the sum becomes, Together with (4.29), which we repeat here for convenience, equation (4.32) forms a system of equations for only two of the three real fields of the system, namely ψ andρ 2 . This system is very similar to the one solved in [19], and we will approach it with similar methods. We note in passing that the integrability condition on (4.33) viewed as an equation forρ 2 as a function of ψ and given λ is given by, 1 − |λ| 2 e +iψ = 0 (4.34)

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which is a conformal invariant field equation for ψ of the sine-Gordon Liouville type [35], very similar to equation (1.3) of [19]. The equation for R will be dealt with in section 4.5.2. Adding twice (4.33) to (4.32) eliminates the term proportional to e −iψ , and we obtain, Clearly, this equation involves only the following specific complex combination ofρ 2 and ψ, ξ = 1 ρ 2 e iψ/2 (4.36) in terms of which we may express (4.35) as follows, where we have used the relation κ + = λκ − . With the help of this sequence of changes of variables, the integrable structure of the system of equations (4.24), (4.25) and (4.26) has been brought out clearly. Indeed, equation (4.37) involves only the field ξ, which is the particular combination ofρ and ψ used in (4.36). Having solved for ξ, either equation (4.33) or (4.35) becomes an equation for only a single variable, eitherρ or ψ, and may be solved. Finally, havingρ and ψ, equation (4.30) becomes an equation for R only, and we will see below that it may be solved as well.
4.5 Solving for ψ,ρ 2 and R in terms of A ± (step 5/5) Having decoupled the reduced BPS equations in the preceding subsection, we will solve the decoupled equations in the present section. To do so, we begin by solving equation (4.37) for ξ, and then obtain ψ,ρ 2 and R. Introducing locally holomorphic functions A ± such that, the function λ may be expressed in terms of A ± by, Given the one-forms κ ± , the functions A ± are unique up to an additive constant for each function. Viewed as equations for the supergravity fields in terms of A ± , the reduced BPS equations are therefore invariant under shifting the holomorphic functions A ± by arbitrary complex constants.

Solving for ξ
In terms of A ± equation (4.37) may be recast in the following form, Equation (4.40) is solved in terms of an arbitrary locally holomorphic function A 0 by,

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which provides the general solution to (4.37). To determine A 0 in terms of A ± , we enforce (4.33) on the result (4.41). Upon eliminating ξ using (4.41), we find, To proceed we change variables from the holomorphic function A 0 to a new holomorphic function ϕ, related as follows, A 0 = −A + + λ(A − + ϕ). The equation for ϕ then becomes, If λ is constant, then ϕ must be constant as well. Assuming henceforth that ∂ w λ = 0, we take the derivative of the entire equation with respect tow, and regroup terms according to their holomorphicity properties, The left side is holomorphic, while the right side is anti-holomorphic, and hence both sides must equal a complex constant α, so that we find, Eliminating the derivative ∂ w ϕ between both equations gives, Assembling all these results gives, Recalling that the functions A ± were defined only up to additive constant shifts, we may absorb the constant α into A ± , so that our final expression for the solution becomes, where L is given by, Note thatρ and ψ are directly determined by ξ using equation (4.36).

Solving for R
To solve for R, we start from equation (4.30) and eliminate the term proportional to e −iψ using (4.29). We then divide the resulting equation by R, and find,

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Changing variables from R to W defined by, renders equation (4.50) linear in W , with an inhomogeneous part, The homogeneous equation is solved straightforwardly by, To solve the homogeneous equation, W 0 is an arbitrary constant which is required to be real for real W . To find a particular solution to the inhomogeneous equation, we let W 0 be a function, which then must satisfy, Recasting the equation in terms of the variable ξ =ρ −2 e iψ/2 , and then expressing ξ in terms of L and λ, using (4.48), we find, The first two terms on the right side may be evaluated using the expression for L of (4.49).
Putting all together, we find the following equation for W 0 , Using the explicit formula forL, and the fact that λ∂ w A − = ∂ w A + , we find, Now the right side is a holomorphic 1-form, and so locally there exists a holomorphic function B, defined up to the addition of an arbitrary complex constant such that, 6 The general solution is then given by, and an arbitrary integration constant, which parametrizes the admixture of the solution to the homogeneous equation, has been absorbed into B. Note that the entire solution is real, as is required by the nature of W and R. This completes the solution of the decoupled reduced BPS equations for the fields ψ,ρ and R.

JHEP08(2016)046 5 Local solution to type IIB supergravity
In this section we summarize the complete local solution for the supergravity fields, which is parametrized by two holomorphic functions A ± and various constants. The doublet (A + , A − ) transforms linearly under the group SU(1, 1), which is isomorphic to the group SL(2,R). The transformation properties of the supergravity fields under SU(1, 1) can be made transparent by expressing the supergravity fields with the help of natural invariants. The SU(1, 1) transformation properties of the holomorphic data and its invariants will be spelled out in section 5.1, before we give the supergravity fields in 5.2 and discuss the SU(1, 1) transformations induced on them in section 5.3. We discuss positivity and regularity conditions respectively in sections 5.4 and 5.5. In 5.6 we show how the T-dual of the D4/D8 solution in massive type IIA supergravity can be recovered as a special case of our general solution, while in 5.7 we give a local solution which satisfies the positivity and regularity conditions in a finite but local region near a boundary segment. Finally, in section 5.8, we discuss the conditions on the holomorphic data A ± and on the supergravity fields under which solutions with monodromy can exist.

SU(1, 1) transformations of the holomorphic data
The basic data parametrizing the general local solution are two holomorphic functions A ± , which transform linearly under SU(1, 1). Parametrizing the elements of SU(1, 1) by u, v ∈ C subject to |u| 2 − |v| 2 = 1, the functions A ± transform as, Note that theĀ ∓ and κ ± transform in the same fashion as A ± does. The functions A ± determine the holomorphic function B introduced in (4.58), up to an additive constant, by, The right hand side is invariant under the SU(1, 1) transformations (5.1) and consequently B transforms at most by a constant shift. Derived quantities which will copiously appear in the expressions for the supergravity fields are κ ± and λ defined by, Since κ ± transform as A ± , we are led to the following natural invariant, Finally, a combination which was already encountered earlier in (4.59) and given by, also has simple transformation properties, since the first and second terms on the right side combine into an SU(1, 1) invariant. Since B transforms at most by shifts under SU(1, 1), so does G. Moreover, G is the only place where B shows up, and we see that of the generally complex constant of integration hiding B, only the real part is relevant. For later use, we note the following relation between G and κ 2 given by κ 2 = −∂ w ∂wG.

Supergravity fields in terms of holomorphic data
The general local solution to Type IIB supergravity with SO(2, 5) × SO(3) symmetry can now be expressed in terms of the holomorphic data introduced above. Translating the local solution to the reduced BPS equations obtained in section 4 back to the supergravity fields is straightforward, except in the case of the flux field for which the derivation of the flux potential is quite involved, and detailed calculations are relegated to appendix E. Here we will summarize the results and discuss some of the immediate properties of the solutions.
Recall that the symmetries of the problem dictate the Ansatz for the bosonic supergravity fields, while the fermionic fields vanish. The five-form field strength vanishes, while the Ansatz for the metric and three-form flux fields are as follows, where f 6 , f 2 , ρ, C and the dilaton-axion field B are all functions on Σ, andê 67 is the volume form on an S 2 of unit radius defined in (2.15).
The metric functions f 2 , f 6 and ρ can be expressed solely in terms of G, and for notational convenience we introduce the composite quantities R and W defined in terms of G through, The right side is real and so are R and W . The metric functions are then given by, The axion-dilaton field B of (2.7) is given by, Finally, the flux potential function C for the three-form field strength F (3) is derived in appendix E and given by, Here, K 0 is a complex integration constant which represents the residual gauge transformation degree of freedom in C (2) restricted to our Ansatz, and does not affect the gauge-invariant field strength F (3) .

SU(1, 1) transformations induced on the supergravity fields
The expressions for the supergravity fields in terms of the holomorphic data, obtained in section 5.2 allow us to specify more precisely which transformations of the holomorphic data leave the supergravity fields invariant, or transform them according to SU(1, 1)-duality.

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The radii f 6 and f 2 must be invariant since they parametrize the metric in Einstein frame. Therefore, R and the combination κ 2 /ρ 2 must be invariant, and hence |∂ w G| 2 /κ 2 and as a result also G and κ 2 themselves must be invariant by (5.7). We will now implement these invariance requirements on the holomorphic data A ± themselves. Invariance of κ 2 requires that the holomorphic one-forms transform under SU(1, 1) as, where we have parametrized SU(1, 1) by u, v ∈ C with |u| 2 − |v| 2 = 1. Integrating the above transformation laws to obtain the holomorphic functions A ± we get, where a ± are complex constants. The addition of the constants a ± leaves κ ± unchanged but transforms ∂ w B as follows, Integrating this relation, we find, where b 0 is a complex constant. Therefore, the transformation law for G is as follows, Invariance of G requires, Since A − andĀ + transform under SU(1, 1) by the same formula, the restriction a − =ā + is automatically SU(1, 1)-invariant. To analyze the transformation properties of the dilatonaxion field B, we use the solution (5.9) and again appeal to the fact that G, ∂ w G and R are invariant under SU(1, 1). The transformation property of B under (5.12) is as follows, Finally, the transformation law for the flux potential C (2) = Cê 67 under (5.12) is given by, The last term is constant and amounts to a gauge transformation on the field C (2) . Therefore, up to gauge transformations, we recover (2.8), as required. We see that the transformations of (5.1) indeed induce the appropriate SU(1, 1) transformations on the supergravity fields. But at the level of the holomorphic data we have an additional shift symmetry, which leaves the supergravity fields invariant and results from the fact that A ± are determined from κ ± only up to additive constants.

Positivity conditions
By construction in (5.8), the metric fields f 2 2 , f 2 6 and ρ 2 are real, but they are not necessarily positive, as is required by the Minkowski signature of ten-dimensional space-time. In this subsection, we investigate the requirements on the holomorphic data implied by the positivity of f 2 2 , f 2 6 and ρ 2 , and the condition |B| ≤ 1. There are no reality or positivity requirements derived from the flux field C (2) since it is complex. The further conditions needed to produce regular solutions will be investigated in the next section.
Positivity of the expressions for f 2 2 and f 2 6 in (5.8) requires, The expression for ρ 2 in (5.8) is then automatically positive. By definition R is an absolute value, so that we must have R ≥ 0 and therefore W ≥ 2 by (5.7), which implies, Finally, we verify that |B| ≤ 1 holds, or equivalently f 2 ≥ 1, using (4.22), It is manifest that Actually, these two branches are mapped into one another under a complex conjugation, which includes the reversal of the complex structure on Σ. Specifically, this transformation reverses the sign of κ 2 and G, maps R to R −1 , and interchanges w andw, and may be realized on the holomorphic functions as follows, and thus on holomorphic forms by, One verifies that these transformations have the desired action on κ 2 , G, and R, and therefore leave the metric functions f 2 2 , f 2 6 , ρ 2 invariant, while complex conjugating the fields B and C combined with a reversal of the complex structure on Σ,

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Therefore, we may restrict to considering just a single branch of the solutions, specified by, the other branch being related by complex conjugation.

Regularity conditions
To describe holographic duals to 4 + 1 dimensional CFTs, we are mainly interested in solutions where the AdS 6 factor governs the entire non-compact part of the geometry. Therefore, we will assume that Σ is compact, with or without boundary. It will be convenient to examine the regularity conditions required in each one of these two cases separately. When the Riemann surface Σ is compact and without boundary, a regular supergravity solution requires the metric functions f 2 2 , f 2 6 and ρ 2 to remain strictly positive, and the axion-dilaton field to satisfy the strict inequality |B| < 1 throughout Σ. As a result, the corresponding conditions on the holomorphic data are given by the strict inequalities, throughout Σ. The inequality κ 2 > 0 implies the strict inequality |λ| < 1. When the Riemann surface Σ has a non-empty boundary ∂Σ, a regular supergravity solution may be obtained when the function f 2 2 vanishes on ∂Σ, provided such behavior corresponds to the shrinking of a sphere S 2 as part of a regular three-dimensional sub-manifold. It is clear from the explicit solutions in (5.8) that 9f 2 2 < f 2 6 and therefore the vanishing of f 2 6 will force f 2 2 to vanish as well producing a space-time geometry with a short-distance singularity. To avoid such physically unacceptable singularities, we will assume henceforth that the regular part of the boundary of Σ is characterized by f 2 2 = 0 and f 2 6 > 0. In particular, a topologically non-trivial three-cycle, or three-sphere, may arise as part of the space-time manifold of a regular solution from fibering the sphere S 2 over a line segment on Σ which is spanned between two points on ∂Σ, and which cannot be continuously contracted to a point.
We will now investigate the behavior of a regular supergravity solution, and its associated holomorphic data, near a boundary point or segment of Σ. To analyze this behavior, we express f 2 and f 6 as directly as possible in terms of holomorphic data, and we find, The neighborhood of a regular boundary point is realized by letting f 2 2 → 0 while keeping f 2 6 finite. On the variables R and κ 2 , this requires the following limiting behavior, The first condition ensures that f 2 /f 6 → 0, the second that f 6 stays finite. As a result, we have κ 2 → 0 and R → 1 at a regular boundary point, while their ratio stays finite.

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Furthermore, the limiting behavior R → 1 implies that W → 2 + O((R − 1) 2 ), which imposes a condition on the behavior of G in view of the equation (5.7) for R. To derive this condition, we recast (5.7) as follows, Near a regular boundary point, the left side remains finite. Assuming that ∂ w G also remains finite, we are led to the following limiting behavior for G, Finally, we examine the regularity condition on the axion-dilaton field, namely |B| < 1 or equivalently f 2 ≥ 1 and f 2 remains finite near a regular boundary point of ∂Σ. From equation (5.21) these conditions will be realized provided the following ratio has a finite limit, which gives us information on the behavior of the relative phase between λ and Z 2 . To analyze this condition, we make use of the definition Z 2 = Re iψ and the relation e iψ =ξ/ξ = L/L, with L defined in (4.49). Expressing the result further with the help of A ± we obtain, This condition is automatically fulfilled as long as (Ā − − A + )/L remains finite. In summary, the behavior of the supergravity fields near a boundary point on ∂Σ is regular if and only if κ 2 → 0 and the following two ratios have a finite and non-zero limit, When these conditions are obeyed, all supergravity fields are regular in the neighborhood of the corresponding regular boundary point. Of course, one may wish to consider supergravity solutions with sufficiently mild singularities, such as a diverging dilaton field at isolated points on the boundary. In this case, the condition G/κ 2 being finite may have to be relaxed.

Recovering the T-dual of D4/D8
In this subsection, we will recover the T-dual of the D4/D8 solution given in [36] and show that this solution, even though singular, solves the BPS equations derived in the present paper. To find the T-dual of D4/D8, we consider a local coordinate system w,w on a Riemann surface Σ, and make the following Ansatz for the holomorphic data, with a, b ∈ R. As a result, we find the following auxiliary quantities,

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The combinations G and R, which are also required to construct the supergravity fields of the solutions, are given as follows, The positivity conditions of (5.26) require κ 2 ≥ 0, G ≥ 0 and R ≤ 1, and thus, Note that neither w +w = 0, nor w +w = 1 satisfy the regularity conditions of (5.34), so that we must expect the resulting supergravity solution to have singularities there.
The metric functions are found as follows, For the axion and dilaton we use the formulas in appendix E, eqs. (E.11) and (E.12), to get them separately right away. This yields, To match to [36], we change from w,w to real coordinates θ, φ 3 defined by, and fix the parameters a, b, c 6 by the following choice, a = 27 16 m 1/3 b = 9 8m 1/3 c 6 = 1 (5.42) With the notation W = (m cos θ) −1/6 , the axion and dilaton fields become, This reproduces the results of (A.1) of [36], noting that the definition of the dilaton in (2.7) differs from that used in [36] by a factor 2. The metric functions become, Our metric is in Einstein frame, that of [36] in string frame. The metric functions in string frame are simply given by dropping the dilaton factors in the above expressions, i.e. (f 2 6 ) string = e φ f 2 6 etc. Upon this change to string frame, we have exactly the metric in (A.1) of [36]. This just leaves the flux field C (2) , which we obtain from (5.10) as, 6aww − a(w +w) 2 (3 + Y −2 ) + 6b(w −w) − K 0 ê 67 (5.45)

Satisfying positivity and regularity conditions locally near ∂Σ
In this subsection, we shall show that, at least locally in a finite neighborhood of part of the boundary of Σ, it is possible to satisfy both positivity and regularity conditions. By a conformal transformation, we map a boundary component of Σ to the real axis, and take the interior of Σ to be part of the upper half plane parametrized by complex coordinates w,w. To realize the positivity condition κ 2 = 0, or equivalently |λ| = 1, on ∂Σ we choose λ = (1 + iw)/(1 − iw), so that the functions A ± are related by the following equation, We restrict attention to local solutions for which ∂ w A ± are rational functions of w, and can be decomposed into a sum of simple poles plus a constant additive term, The positions of the poles x n are chosen to be real. We guarantee the absence of zeros for ∂ w A − in the upper half plane by requiring a n e −iφ ∈ R for all n and some n-independent phase φ. The integrals A ± are as follows, where α ± are complex constants. For N ≥ 2, the calculation of B reveals the presence of dilogarithms. To avoid this complication, we shall examine only the simplest case N = 1 where no dilogarithms appear in B. Setting a 1 = a and x 1 = x, we then obtain,

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which indeed vanishes on R and is strictly positive in the upper half plane. Integrating these equations as well as the one for B, we find, where B 0 is an integration constant. We use these ingredients to compute G. The vanishing of G on the real axis requires B 0 +B 0 = 0 as well as a 2 = −|a| 2 and α − =ᾱ + , and we obtain, In the upper half plane the pre-factor −2i(w −w) is positive, so that we have G > 0 inside a semicircle centered at x with Im (w) > 0 and ln |w − x| 2 < 2, while G = 0 on the real axis. Therefore, we have established that it is possible, locally in a finite region near a boundary component, to satisfy both the positivity and regularity conditions on κ 2 and G. On the semi-circle defined by ln |w − x| 2 = 2 the behavior fails to be regular, since we have G = 0 but κ 2 > 0 reminiscent of the singularity of the T-dual to D4/D8 at w +w = 1. It remains to find supergravity solutions which obey the positivity and regularity conditions globally, an investigation that we shall reserve for future work.

Supergravity solutions with monodromy
When Σ is a compact Riemann surface without boundary, and the locally holomorphic functions A ± are assumed to be single-valued on Σ, then they must be constant, which does not produce any supergravity solutions. Therefore, on any compact Σ without boundary, the existence of regular supergravity solutions will require the locally holomorphic functions A ± to have non-trivial monodromy, or poles which conspire in such a way that the supergravity fields remain finite.
Under the weaker assumption that the locally holomorphic function λ is single-valued on a compact Riemann surface Σ without boundary, but A ± are allowed a non-trivial monodromy, it follows from the condition |λ| < 1 that λ must be constant. The relation between κ ± may then be integrated explicitly, and we have, where a 0 is an arbitrary complex constant. The monodromies of (5.12), which we repeat here for convenience, will be compatible with the relation A + = λA − + a 0 provided A ′ + = λA ′ − + a 0 as well. This condition in turn imposes two complex-valued conditions on the three complex-valued monodromy parameters u, v, a + =ā − , given as follows,

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Manifestly, the only solution to these equations with trivial SU(1, 1) monodromy, namely u = 1, v = 0 are solutions with no monodromy at all since a + − λā + = 0 implies a + = 0 in view of |λ| < 1. Thus, solutions with constant λ must necessarily involve a non-trivial SU(1, 1) monodromy on A ± . The existence of such solutions will be investigated in detail in a subsequent publication.
Allowing all three locally holomorphic functions A ± and λ to have non-trivial monodromy under SU(1, 1)×C, the mathematical problem becomes quite interesting, and quite involved. The pair (A + , A − ) may then be viewed as a holomorphic section of a holomorphic bundle over Σ with a structure group which is a subgroup of SU(1, 1) × C. The fibers of this bundle are subject to the regularity conditions κ 2 > 0 and G > 0.
The simplest situation is when Σ is compact and without boundary, and we shall attempt to give a plausible mathematical context for this case. The SU(1, 1) × C invariance of G then guarantees the continuity of G as a function on Σ and therefore its boundedness. Using the freedom to shift B by an arbitrary constant, we may always adjust the shift to make G > 0, and therefore to render this condition trivially satisfied. The only remaining condition on the holomorphic sections (A + , A − ) of the bundle is then a condition on its associated one-forms, The construction of this bundle should be expected to parallel the construction of vector bundles over Σ with structure group contained in SU(1, 1), with the important difference that in our case the space of one-forms ∂ w A ± is subject to the above inequality which makes each fiber into a solid cone in C 2 rather than a vector space. Holomorphic vector bundles over compact Riemann surfaces were considered early on in [37] and classified when their structure group is a subgroup of SU(n) and the bundle satisfies certain stability conditions [38,39]. 7 The equivalence classes of rank n holomorphic vector bundles are then in one-to-one correspondence with the irreducible representations of the fundamental homotopy group π 1 (Σ) into SU(n). The concrete realization of these representations on a compact surface of genus g ≥ 2 is constructed as follows. We begin by introducing a basis of A i and B i cycles for the first homology group H 1 (Σ, Z), with canonical normalization of their intersection matrix #(A i , A j ) = #(B i , B j ) = 0 and #(A i , B j ) = δ ij for i, j = 1, · · · , g. The representation γ of π 1 (Σ) into SU(n) may then be concretely described by assigning elements γ(A i ) and γ(B i ) in SU(n) to the homology cycles A i and B i respectively, subject to the standard closure condition on the commutators, The product in (5.59) is ordered in the index i, and I stands for the identity matrix in SU(n). The discrete group Γ is then freely generated by the elements γ(A i ) and γ(B i ) for i = 1, · · · , g, subject to the closure relation (5.59).

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Holomorphic vector bundles over Σ whose structure group Γ is a subgroup of SL(2, R) = SU(1, 1) instead should admit an analogous construction. We should assign to each homology generator A i , B i a transformation γ(A i ), γ(B i ) in Γ subject to the closure relations of (5.59). The corresponding group Γ is then freely generated by these elements, just as was the case for the unitary groups. In the simplest case where the structure group is Z 2 , the corresponding differential forms span the space of Prym differentials [41] on the surface Σ, familiar from Z 2 orbifold constructions in string theory. The case of a more general structure group Γ may be viewed as a generalization of Prym differentials to the case of un-ramified covers of Σ with higher structure group Γ. Finally, for our set-up the fibers are solid cones of one-forms in C 2 rather than vector spaces, but the construction of such bundles is expected to follow in parallel. A detailed investigation into these possibilities will be relegated to future work.

Discussion
Supersymmetric solutions to Type IIB supergravity with an AdS 6 factor and the corresponding isometries are of considerable interest to describe holographically the still mysterious 4+1 dimensional CFTs discovered in [5,6] and realized in string theory via (p, q) brane webs. As quantum field theories these CFTs are defined only indirectly, as the non-trivial UV fixed points of certain 4+1 dimensional super-Yang-Mills theories. The description as Yang-Mills theories, however, is non-renormalizable and only captures the IR effective action of a relevant deformation of the CFT. Having supergravity duals at our disposal would allow for extensive quantitative studies of the actual UV CFTs, which from the field theory side are hard to do even in principle without a known Lagrangian description.
In this work we have constructed the general local solution to Type IIB supergravity with SO(2, 5)×SO(3) isometry and 16 supersymmetries, which are precisely the symmetries expected for holographic duals to these 4+1 dimensional SCFTs. The local solutions are constructed from AdS 6 and S 2 spaces, which realize the desired isometries and are fibered over a Riemann surface Σ. With the supergravity fields spelled out in section 2, the local solutions are given in terms of two locally holomorphic functions A ± , as summarized in section 5.1 and 5.2. The solutions transform properly under SL(2, R), and this symmetry serves as an organizing guide throughout the derivations. Finally, we recover the (singular) T-dual of the D4/D8 solution in Type IIA as a special case.
A crucial ingredient which has allowed us to go beyond earlier works, where the BPS equations were reduced to a set of coupled PDEs [16][17][18], was keeping the freedom to choose convenient coordinates on the Riemann surface Σ. Separating holomorphic and anti-holomorphic dependences in conformally flat coordinates on Σ featured prominently in the derivations and allowed us to actually solve the reduced BPS equations. The complete local solution provides the basic building blocks for constructing globally regular solutions with the aforementioned isometries, and is a significant step towards understanding and eventually classifying supergravity duals for the 4+1 dimensional CFTs realized by (p, q) brane webs.

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The natural next question is whether or not there are globally regular solutions with the symmetries discussed above. While the existence of a large-N limit on the CFT side suggests that there should be dual supergravity solutions, the difficulties in finding such solutions so far call for a more careful perspective. Indeed, there are known examples where a superconformal field theory with a large-N limit does not admit a dual description which reduces to supergravity alone, even at large N and strong coupling [42]. The existence of a supergravity dual is obstructed in that case by the absence of a gap in the spectrum to isolate a small number of states with low scaling dimension. As a result, the dual description always involves stringy states. In the example of [42], however, realizing a superconformal theory at large N needs a similarly large number of flavors, of the same order as N . This Veneziano limit is different from the usual 't Hooft limit and from the case we are looking at here, where superconformal theories exist also with small numbers of flavors. There is, quite on the contrary, an upper limit on the number of flavor multiplets in the CFTs we are attempting to find a dual description for. The large-N limit should therefore indeed be a 't Hooft limit, and these arguments suggest that finding a dual supergravity description should be possible. We discussed regularity conditions in section 5, but constructing globally regular solutions is still a non-trivial task, and we plan to come back to it in the future.

Acknowledgments
We are happy to acknowledge useful conversations with Oren Bergman and David Gieseker.

A Clifford algebra basis adapted to the Ansatz
The signature of the space-time metric is chosen to be (− + · · · +). The Dirac-Clifford algebra is defined by {Γ M , Γ N } = 2η M N I 32 . We choose a basis for the Clifford algebra which is well-adapted to the AdS 6 × S 2 × Σ Ansatz, with the frame labeled as in (2.14), Γ m = γ m ⊗ I 2 ⊗ I 2 m = 0, 1, 2, 3, 4, 5

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where a convenient basis for the lower dimensional Dirac-Clifford algebras is as follows, We will also need the chirality matrices on the various components of AdS 6 × S 2 × Σ, and they are chosen as follows, so that, The 10-dimensional chirality matrix in this basis is given by, 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 is given by, The geometry of Killing spinors On the sphere S 2 with unit radius, the Killing spinor equation is given by, where σ 1 , σ 2 , σ 3 are the standard Pauli matrices, and ∇ i is the spin connection on the sphere S 2 with unit radius, and the derivative is expressed with respect to frame indices i. An equivalent equation is obtained by letting σ i σ 3 → iσ i and ε ′ = e −iπσ 3 /4 ε. Integrability of this system of equations requires η = ±1. The system may be solved by restricting the canonical flat connection on SU(2), as was reviewed in an appendix of [19]. Here, we will take a more direct approach to obtaining an explicit solution.
We parametrize the sphere S 2 with unit radius by angles θ 1 , θ 2 and take the following conventions for the round metric, The frame and the spin connection are given by, and satisfy the vanishing torsion condition, The Killing spinor equation on the sphere S 2 of unit radius is of the form, with η = ±1 required by integrability. With the Dirac matrices defined in appendix A we have γ 1 = σ 1 and γ 2 = σ 2 , and this equation becomes, and is solved by, Here, ε 0 is an arbitrary constant spinor. Hence the space of Killing spinors is twodimensional. We can verify the following statements for a Killing spinor ε(η) which satisfies the Killing equation with a given η = ±1, 1. γ (2) ε(η) = σ 3 ε(η) satisfies the Killing equation with η → −η; 2. σ 1 ε(η) * satisfies the Killing equation with η → −η; 3. σ 2 ε(η) * satisfies the Killing equation with η → +η.

B.2.1 Explicit form of the Killing spinors
We parametrize AdS 6 with unit radius by a radial coordinate r and x i ∈ R for i = 1, 2, 3, 4, and use the SO(2, 5)-invariant metric,

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The Killing spinor equation is given by, with η = ±1. The frame and spin connection are given by, Hence the Killing spinor equation becomes, The general solution for the Killing spinor equation is, where ε 0 is a constant spinor, hence the space of Killing spinors of AdS 6 is eight dimensional. We can label the basis vectors by η = ±1 which each have a four dimensional degeneracy which will play no role in the following other than leading to the correct number of preserved supersymmetries in the end. We can verify the following statements for an AdS 6 Killing spinor ǫ(η) which satisfies the Killing equation with a given η = ±1, 2. B (1) ǫ(η) * = 1 2 ⊗ σ 2 ⊗ σ 1 ǫ(η) * satisfies the Killing equation with η → η; 3.B (1) ǫ(η) * = σ 3 ⊗ σ 1 ⊗ σ 2 ǫ(η) * satisfies the Killing equation with η → −η.
Note that we choose the complex conjugation matrix to be B (1) in appendix A

C Derivation of the BPS equations
We begin by collecting some identities that will be useful during the reduction of the BPS equations. We will also need the following decompositions of ε and B −1 ε * , where we use the abbreviations, in τ -matrix notation. In addition, we have the chirality relations,

C.1 The dilatino equation
The dilatino equation is, Reduced to the Ansatz of subsection 2.3, we have the following simplifications, The dilatino equation now becomes, from which we extract the equation satisfied by the ζ-spinors, Using the explicit expression for ⋆ζ and reversing the sign of η 2 , we find, Recasting this equation in terms of the τ -notations, we have, 0 = −4p a τ (03) γ a σ 2 ζ * + g a γ a ζ (C.9)

C.2 The gravitino equation
The gravitino equation is, where A, B are the 10-dimensional frame indices.

C.2.1 The calculation of (d + ω)ε
The spin connection components are ω a b , whose explicit form we will not need and,

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The hats refer to the canonical connections on AdS 4 and S 2 respectively. Projecting the spin-connection along the various directions we have, where the prime on the covariant derivative indicates that only the connection along AdS 4 and S 2 respectively is included. Using the Killing spinor equations (3.1) we can eliminate the primed covariant derivatives, which yields, where we have pulled a factor of Γ M out front. It will turn out that all terms in the gravitino equation contain Γ M χ η 1 ,η 2 ,η 3 , and we will require the coefficients to vanish independently, just as we did for the dilatino equation. The coefficient of Γ M χ η 1 ,η 2 ,η 3 can be expressed in the τ -matrix notation as, The relevant expression is as follows, A few useful equations are as follows,

D Verifying the Bianchi identities
In this appendix we show that the reduced BPS equations summarized in section 4.3 imply that the Bianchi identities (2.1) are satisfied. In the complex coordinates introduced in (4.5), the Bianchi identities for P and Q take the following form, They are satisfied automatically if P and Q are expressed in terms of B as in (2.5). The Bianchi identity that needs to be checked is the one for G. We will find that (4.12) alone implies that it is satisfied. In terms of the lower-case expansion coefficients p, q, g defined in (2.16), the Bianchi identity (2.1) becomes, ∂ w ρf 2 2 gz − ∂ w ρf 2 2 g z + ρ 2 f 2 2 − iq z gz + iqzg z + p z (g z ) ⋆ − pz(gz) ⋆ (D.2) Using (4.1) and (2.5), this can be rewritten in terms of B, ρ and the κ ± alone as, where △ = ∂ w ∂ w and,ā are used as shorthands for the corresponding combinations of κ ± . The only place where ρ 2 shows up is in the log derivatives, where it can be eliminated using (4.12), leaving an expression solely in terms of B andB. To show that (4.12) implies that the Bianchi identity is satisfied, we first eliminate the two-derivative term. To this end, we act on (4.14) with ∂ w and on the complex conjugate of (4.14) with ∂ w . The result are two equations linear in △B and △B, namely, where X = b 4 + 2a 2 b 2 e iθ and Y = a 4 + 2a 2 b 2 e −iθ . Eliminating △B yields an equation for △B in terms of first derivatives, namely, Substituting this expression in (D.3) leaves only first derivatives. We can now use (4.14) and its complex conjugate to eliminate the derivatives ofB. After this step the left hand side of (D.3) collapses to zero, showing that the Bianchi identity is satisfied.

JHEP08(2016)046 E Supergravity fields in terms of holomorphic data
In this appendix we derive the expression for the supergravity fields from the local solution to the BPS equations derived in section 4. The results are summarized in section 5. As a prerequisite, we note that (5.7) arises from (4.59) with,

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The explicit expression for f 2 can be obtained from (4.22) as, For τ we find To get to the axion and dilaton we separate the real and imaginary parts and use τ = χ + ie −2φ , which yields The axion becomes

E.2 Flux field
The complex 3-from field G (3) is better expressed in terms of the complex 3-form field F (3) by the first relation of (2.6). Since F (3) is a closed 3-form it may locally be written in terms of a complex flux potential 2-form field C (2) by F (3) = dC (2) . Inverting this relation, we have, The symmetries of the problem force C (2) and G (3) to be of the following form, C (2) = Cê 67 G (3) = g a e a ∧ e 67 = f 2 2 g a e a ∧ê 67 (E.14) so that, ∂ w C = ρf 2 2 f (g z + Bḡ z ) ∂wC = ρf 2 2 f (gz + Bḡz) (E.15) Using the conversion of G into P and then into derivatives of B using (2.5) and (4.1), 16) we obtain the following expressions, We will now work towards expressing the right side in terms of the solutions to the BPS equations, and then integrating the equations to obtain C and its complex conjugate.

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Changing variables from Z to Z 2 = Re iψ , we find, Expressing this combination in terms of W = R + R −1 , we find, Using equation (4.52), we eliminateρ in favor of ξ, and eliminate ξ in favor of L and λ, ∂ w W = −(W + 1)∂ w ln LL + (W − 2)∂ w ln(1 − |λ| 2 ) + 3i∂ w ψ (E.28) Putting all together, we have, where we have used the relation e iψ =L/L to express the derivative i∂ w ψ in terms of L.

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Proceeding analogously forC, and using the equations for ∂ w ξ and ∂ wξ , we find, for some holomorphic function K 2 . Equating now the two different expressions forC +λC, we get after some simplifications which eliminate the dependence on W completely, Separating holomorphic and anti-holomorphic dependences, we find, where K 0 is an arbitrary complex constant. Thus, we have, Evaluating that further gives Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.