Bubbling geometries for AdS_2 x S^2

We construct BPS geometries describing normalizable excitations of AdS_2 x S^2. All regular horizon-free solutions are parameterized by two harmonic functions in R^3 with sources along closed curves. This local structure is reminiscent of the"bubbling solutions"for the other AdS cases, however, due to peculiar asymptotic properties of AdS_2, one copy of R^3 does not cover the entire space, and we discuss the procedure for analytic continuation, which leads to a nontrivial topological structure of the new geometries. We also study supersymmetric brane probes on the new geometries, which represent the AdS_2 x S^2 counterparts of the giant gravitons.


Introduction
AdS 2 ×S 2 has a peculiar status in string theory: while being especially interesting as a near horizon limit of the extremal black holes in four dimensions, this space still evades a holographic description available for its counterparts in higher dimensions [1,2,3]. Although a significant progress towards formulation of the AdS 2 /CFT 1 correspondence has been made over the last two decades [4], the complete understanding of the 'field theory side' is still missing. Nevertheless, one can apply the methods used to study the bulk side of the AdS/CFT correspondence to AdS 2 ×S 2 , and remarkably this space shares some of the nice analytic properties with its higher dimensional counterparts. In particular, integrability of strings, which was discovered for AdS 5 ×S 5 and AdS 3 ×S 3 [5,6], persists for AdS 2 ×S 2 as well [7,8] 1 .
In this article we will study supersymmetric excitations of AdS 2 ×S 2 , which come in several varieties. Very light perturbations correspond to the gravity multiplet, which is included in the analysis of [7]. As the energy of a supersymmetric excitation increases, a better semiclassical description is given in terms of probe branes on AdS 2 ×S 2 , and the counterparts of such branes in higher dimensions are known as giant gravitons [11,12]. When the energy increases even further 2 , the gravitational backreaction of branes becomes important, and one needs to find deviations from the AdS p ×S q geometry. In the past this problem has been analyzed for p = 3, 4, 5, 7, where all supersymmetric branes have been classified [13,14], all regular geometries preserving half of the supersymmetries have been constructed [15,16,17], and a progress towards finding 1/4-BPS geometries has been made [18,19,20]. The goal of this article is to extend these successes to supersymmetric excitations of AdS 2 ×S 2 .
In string theory Anti-de-Sitter spaces naturally appear as near-horizon limits of D branes, but in this context one recovers only a part of the AdS geometry known as a Poincare patch. Since this patch is geodesically incomplete, and any point of this geometry is separated from the center of AdS by an infinite distance, such near-horizon limits look somewhat singular. On the other hand, a completion of this space gives the global AdS space, which is smooth everywhere. While the geodesic completion of the bosonic sector is straightforward, this procedure modifies the boundary conditions for fermions, so the relation between string theories on the Poincare patch and on the global AdS is not trivial. Large classes of BPS excitations of the Poincare patch can be constructed by considering several parallel stacks of D branes, and such configurations correspond to the Coulomb branch of the dual field theory [1,21]. Not surprisingly, all such solutions have singularities at the locations of the branes, where explicit sources are introduced, but these singularities are infinitely far from any point on the Poincare patch. Such singularity seems unavoidable if one starts from the probe branes and surrounds them by a large sphere: since such sphere carries a quantized flux of an appropriate field strength 3 , making the sphere smaller and smaller, one arrives at a singular point. In contrast to the Poincare patch, the geometry of the global AdS is completely smooth, and we will now review the mechanism of such regularization in various dimensions.
We begin with the global AdS 5 ×S 5 and its embedding into the 'bubbling ansatz' [17]. 1 In fact, intergrability persists even for the eta-deformation of AdS 2 ×S 2 [9], which is analogous to a similar modification of the higher dimensional spaces [10]. 2 The transition between graviton, brane, and geometry regimes is governed by the scaling of energy with the string coupling constant. This scaling can also be rewritten in terms of number of branes which created the AdS geometry: gravitons have finite energy, branes have energy that scales as N , and energy of classical geometries scales as N 2 .
3 For example, D3 branes in ten non-compact dimensions can be surrounded by S 5 which carries a flux of F 5 , D1-D5 bound states in six non-compact dimensions can be surrounded by S 3 carrying a flux of F 3 , and so on.
The metric contains two explicit S 3 factors, the time, and a three-dimensional base, then the quantized flux is carried by S 5 , which is constricted by fibering one of the S 3 over a two-dimensional surface on the base. Naively, the arguments presented in the last paragraph suggest that a singularity becomes unavoidable since S 5 can contract to zero size, but in the bubbling geometries such contraction is avoided since the second sphereS 3 collapses causing the space to end while the size of S 5 is still finite. By taking a singular limit of the bubbling geometries, one can make the regions whereS 3 collapses arbitrarily small, this leads to recovery of the Poincare patch and to singular geometries produced by multiple stacks [21]. To summarize, the ten-dimensional bubbling geometries are regularized by a topological mechanism based on termination of space by collapsing one of the spheres.
While a similar mechanism can be applied to some D1-D5 geometries [22], in general regularization of the D1-D5 system happens for a different reason. At infinity there is a clear separation between the three dimensional sphere S 3 , which supports the flux, and one of the compact directions. If such separation persisted everywhere, then a singularity would be unavoidable, but, as demonstrated in [23], it is possible to mix the sphere and a compact direction so that the sphere which collapses at the 'location of the branes' is not the same S 3 that carries the flux. As the result of this construction, the space can end in a smooth fashion without violating the conservation of flux. The 'location of the branes' has a smooth geometry of the KK-monopole [23], so the sources are completely dissolved in geometry. In [16] this construction had been extended to all 1/2-BPS geometries with AdS 3 ×S 3 asymptotics. Interestingly, in the AdS 3 ×S 3 case, one can continuously connect the flat space with global AdS coordinates [23,15], and this fact has inspired a very promising approach to the resolution of the black hole information paradox known as the fuzzball proposal [15,24]. Regularization of 1/4-BPS geometries is slightly different, and it comes from simultaneous change in signature of the base space and the harmonic functions. The complete picture that works for all such geometries is still missing, and we refer to [18,20] for further discussion.
So far we have encountered two mechanisms for resolving singularities and making the brane sources geometric: the first one relies on termination of space via the 'bubbling mechanism', and the second one is based on dissolving the brane charges in the geometry of the KK-monopole. It turns out that AdS 2 ×S 2 geometries are regularized in yet another way, which is based on a peculiar property of the global AdS 2 : in contrast to its higherdimensional counterparts, this space has a disconnected boundary. In this article we will demonstrate that a bubbling picture with flat base, which worked for AdS 5 ×S 5 and AdS 3 ×S 3 , can be extended to the AdS 2 ×S 2 , but to cover this space completely, one needs two copies of the base R 3 , and these copies are connected through a branch cut 4 . This feature is not very surprising since R 3 has a connected boundary, while the boundary of AdS 2 ×S 2 contains two disconnected pieces. Notice that by taking a near horizon limit of a brane configuration, one can obtain only a half of the space, so the Poincare patch is singular. As we will demonstrate in section 3.3, the branch cut introduces a new mechanism which makes all bubbling geometries with AdS 2 ×S 2 asymptotics regular without compromising the conservation of flux. The branch cuts also lead to very interesting topological structures, which are discussed in section 3.5. This paper has the following organization. In section 2 we analyze the dynamics supersymmetric branes on AdS 2 ×S 2 , which can be viewed as lower-dimensional counterparts of the giant gravitons. In particular, we will find that, unlike giant gravitons in AdS 5 ×S 5 , which can expand only on AdS or on a sphere, the branes on AdS 2 ×S 2 can be placed at any point on a three dimensional space while stretching in the time direction. A similar feature is exhibited by the giant gravitons in AdS 3 ×S 3 and in all 1/2-BPS fuzzball geometries, and such branes are discussed in Appendix A. In section 3 we construct all BPS geometries with AdS 2 ×S 2 ×T 6 asymptotics which can be viewed as backreactions of the configurations discussed in section 2. We will demonstrate that regular BPS geometries (3.5)- (3.6) are parameterized by one complex harmonic function, which has a form (3.23), and to recover a geodesically-complete space, one must connect at least two copies of the three-dimensional base through two-dimensional branch cuts. The details of such analytic continuation and the interesting topological structures originating from it are discussed in sections 3.4 and 3.5. While we demonstrate that the task of finding the BPS geometries reduces to solving a Laplace equation with some mixed Dirichlet/Neumann boundary conditions, this linear problem is still rather nontrivial, and in section 4 we present several explicit examples of regular solutions. Finally, in section 5 we add probe branes to the new geometries, solve their equations of motion, and prove supersymmetry by analyzing the kappa-projection. The resulting picture is very reminiscent of the one found for branes on AdS 3 ×S 3 . Some technical calculations and supplementary material are presented in the appendices.
2 Brane probes on AdS 2 ×S 2 According to the standard AdS/CFT dictionary, AdS 2 ×S 2 must correspond to a vacuum of some quantum mechanics living on the boundary of the AdS space, and excitations of this quantum mechanics are mapped into normalizable modes on the bulk side. The light excitations are mapped into strings moving on AdS 2 ×S 2 , and as demonstrated in [7], such strings are integrable. Heavier excitations correspond to probe D branes, which will be discussed in this section, and when many such branes are put together, they produce normalizable deviations from the AdS 2 ×S 2 geometry, and the resulting solutions of supergravity will be constructed in section 3.
We begin with reviewing some known properties of AdS 2 ×S 2 . There are several ways of embedding this space in string theory, and we will mostly focus on the implementation based on D3 brane sources [26]: Here bullets denote the directions wrapped by the branes and tildes denote the coordinates in which branes are smeared. We also assume that directions (X 1 , are compactified on a torus with finite volume V . Four stacks of D3 branes (2.1) produce an asymptotically flat geometry constructed in [26] which describes a BPS black hole with an area The near horizon limit of (2.2) is the AdS 2 ×S 2 ×T 6 , where the AdS and the sphere have the same radius In this article we will focus on configurations with Q 1 = Q 2 = Q 3 = Q 4 , then the near-horizon geometry can be written in global coordinates as Since the main goal of this article is construction of supersymmetric geometries, we use supergravity normalization of fluxes [27] throughout the paper and set κ = 1. To write the action for the probe branes, such as (2.8), one should recall that fluxes in string theory have different normalization, in particular, F (string) 5 = 4κ gs F 5 . This is the origin of the additional factor of 4 in (2.8) and in other actions for the brane probes. See [28] for the detailed discussion of the map between string and supergravity normalizations for various fluxes.
Supersymmetric excitations of the metric (2.5) fall into several categories: perturbative gravitons, D-branes, and topologically nontrivial deformation of the metric, and one moves between these three cases as a mass of the excitation grows. The excitations whose energy scales as N 0 appear as perturbative gravitons, and they were studied in [29] using the standard analysis of spectrum which had previously been applied to other AdS×S spaces [30]. Semiclassical excitations with energies of order N 1 behave as supersymmetric D branes, which are studied in this section. Once the energy reaches N 2 , gravitational backreaction of branes becomes important, and the resulting solutions of supergravity are constructed in the next section.
Supersymmetric branes on AdS p ×S p are known as 'giant gravitons' [11,12]: they expand on contractible cycles on AdS or the sphere, and they are prevented from collapsing by angular momentum. If p > 3, then the size of the giant graviton is fixed by the angular momentum, while for p = 3 the giant gravitons may expand to an arbitrary size. Moreover, on AdS 3 ×S 3 one can generalize giant gravitons to branes wrapping cycles on both AdS and the sphere, and this construction is presented in Appendix A. The AdS 2 ×S 2 case is very similar with a small caveat: the 'giant gravitons' are pointlike. As we will see in the next section, this similarity between p = 3 and p = 2 gives rise to a similarity in classifying gravity solutions for AdS 3 ×S 3 and AdS 2 ×S 2 .
To study supersymmetric D3 branes on (2.5), we impose the static gauge for the worldvolume: and assume that (ρ, θ, φ) are functions of τ . Then the action for the D brane 6 , Equations of motion are solved by constant (ρ, θ,φ) as long as two relations are satisfied: Combining these relations, we conclude thaṫ φ = 1, c θ = ρ tan β. (2.10) In particular, β = π 2 corresponds to ρ = 0 and an arbitrary θ, giving the AdS 2 ×S 2 counterpart of the giant graviton present in the higher dimensions [11]. Another special case, β = 0, gives θ = π 2 and an arbitrary ρ, which corresponds to the dual giant [12]. Interpolating values of β have counterparts only in the AdS 3 ×S 3 case, which is discussed in the Appendix A, and they correspond to arbitrary points in the (ρ, θ) space. For completeness we also give the expressions for the angular momentum and the energy densities of the branes: As we will see in the next section, supersymmetric geometries preserve Killing spinors which are proportional to ǫ 0 as well, and the prefactor varies in space. Moreover, in section 5 we will demonstrate that Killing spinors for the solutions corresponding to rotating branes reduce to (2.17) in the vicinity of a probe brane.
We conclude this section by observing that the ansatz (2.7) describing supersymmetric branes can be generalized by applying an SO(3) rotation on two vectors (X 1 , X 2 , X 3 ) and (Y 1 , Y 2 , Y 3 ). The reduced action (2.9) remains unchanged, and the supersymmetry analysis is modified by the appropriate rotation matrices.

Supergravity solutions 3.1 Local structure
Our goal is to construct a family of supersymmetric solutions of ten-dimensional supergravity which approach at infinity. The 'vacuum' (3.1) can be lifted to a supersymmetric ten-dimensional geometry in several ways. We will mostly focus on the embedding (2.5) into type IIB SUGRA, and some alternative options will be discussed in subsection 3.6. We are looking for supersymmetric excitations of (2.5) which preserve the torus and the structure of F 5 : Here g mn , F , andF are undetermined ingredients, which can be found by requiring the state (3.2) to be supersymmetric. This implies that the gravitino equation, must have a nontrivial solution, and combining (3.3) with the equation of motion for F 5 , one can determine all functions appearing in (3.2). The details of this analysis are presented in the Appendix B, and here we only mention one interesting feature: the tendimensional equation (3.3) reduces to equation (B.16) for an effective four-component spinorη in four dimensions spanned by x m : The complete solution of the gravitino equation and equations of motion solution is derived in Appendix B, and it reads (see (B.47), (B.48), (B.57), (B.72)) This geometry is parameterized by two functions h and α, which in turn can be extracted from two harmonic functions H 1 and H 2 : Equations (3.5)-(3.6) completely specify the local structure of the solution at regular points of (H 1 , H 2 ), but the harmonic functions must have sources, where solution may become singular. One encounters a similar problem in the AdS 3 ×S 3 case, where it was shown that the singularity is absent for the sources allowed in string theory [15,16]. Unfortunately in the present case the condition on the sources is less intuitive, so we begin with studying it for the vacuum (3.1) before extending it to a general state in subsection 3.3.

An example of a regular solution: AdS 2 ×S 2
The easiest way to recover AdS 2 ×S 2 from (3.5)-(3.6) is to set but the resulting geometry covers only the Poincare patch, so it is not geodesically complete. Since relations (3.5)-(3.6) describe all supersymmetric solutions which have the form (3.2), they must include the global AdS 2 ×S 2 as well, and in this subsection we will discuss such embedding and analyze the mechanism that makes AdS 2 ×S 2 regular in spite of singularities in (H 1 , H 2 ). In the next subsection we will use the intuition acquired from the AdS 2 ×S 2 solution to classify all regular geometries covered by (3.5)-(3.6). The geometry of the global AdS 2 ×S 2 is given by and to put it in the form (3.5) we shift the angular coordinate asφ = φ +t. For convenience we also rescale time,t = t/L, to make the harmonic functions dimensionless. This gives the metric Geometry (3.9) has the form (3.5), in particular, the metric on the base, is flat, as can be seen by going to cylindrical coordinates (r, φ, y): does not affect the new coordinates, so every point in (r, y) half-plane, with an exception of (r, y) = (L, 0), corresponds to two points in the (ρ, θ) space. As we will see, this double cover plays an important role in ensuring regularity of the solution.
The harmonic functions and the vector field V corresponding to the solution (3.9) can be extracted by a direct comparison with (3.5): Expressions for H 1 and H 2 can be encoded in terms of a complex valued harmonic function which for (3.13) has a very simple form: We will use the complex function (3.14) to parameterize the general solution (3.6) as well.
Expressions (3.13) become singular on the r = L circle in the y = 0 plane, and this curve coincides with the set of fixed points of the transformation (3.12). To analyze the behavior of (H 1 , H 2 ) in the vicinity of the circle, we introduce polar coordinates (R, ζ) in the plane orthogonal to the singular curve: The leading order near R = 0 gives the harmonic functions (3.17) and the metric Naively expressions (3.16) suggest that ζ ∈ [0, 2π), then the metric (3.18) appears to have a conical singularity at R = 0. However, we recall that a point in the (r, y) plane corresponds to two points in the global AdS (see (3.11)), and this double cover breaks down precisely at R = 0. For small values R we find approximate expressions 19) so to cover the full vicinity of ρ = 0, coordinate ζ must vary between zero and 4π: R > 0, 0 ≤ ζ < 4π. (3.20) This range ensures regularity of (3.18) at R = 0 and provides a double cover of the (r, y) plane (3.16). As ζ changes from 0 to 2π a point goes from one copy (r, y) to another, so a branch cut must be introduced on the way. As in the case of multivalued functions in a complex plane, the location of such branch cut is ambiguous, but it has to be a surface bounded by the singular curve 8 . An example of such branch cut and images of one closed loop on two copies of R 3 are depicted in figure 1. We conclude this section by summarizing the mechanism of regularization for AdS 2 ×S 2 : • Geometry (3.5)-(3.6) is regular in the Cartesian coordinates x a everywhere away from the singular points of H. To cover the entire AdS 2 ×S 2 we need two copies of the flat base, as depicted in figure 1. • On the curve where H has sources, the geometry remains regular, but the Cartesian coordinates break down, and they should be replaced by (R, ζ) defined by (3.16). The range (3.20) covers the vicinity of the singular curve on both copies, but the curve itself (R = 0) is covered only once, so two copies are glued along this curve.
• By going around the singular curve, a point moves from one copy to another, so a "branch cut surface" must be introduced. As in the case of Riemann surfaces, the precise location of this branch cut is ambiguous, as long as it is bounded by the singular curve.
In the next subsection we will use the insights from the AdS 2 ×S 2 geometry to formulate the regularity conditions for an arbitrary closed curve.

Regularity conditions
Solutions presented in section 3.1 provide a local description of supersymmetric geometries, but a generic harmonic function H gives rise to a singular metric (3.5). The simplest example of such singular solution is the Poincare patch of the AdS space, which corresponds to harmonic functions (3.7). Introducing several point-like sources for H 1 while keeping H 2 = 0, one would describe a geometry produced by several stacks of D3 branes, which corresponds to the AdS 2 counterpart of the Coulomb branch discussed in [21]. In this paper we are interested in regular solutions, which are analogous to the bubbling geometries of [17], and as we will show in this subsection, the requirement of regularity imposes severe constraints on the allowed sources of H 1 and H 2 . We will demonstrate that once such constraints are satisfied, the geometries (3.5)-(3.6) are guaranteed to be regular, as long as the appropriate analytic continuation is performed.
To preserve the AdS 2 ×S 2 asymptotics, harmonic functions H 1 and H 2 must vanish at infinity, this implies that they must have sources at finite points in R 3 , rendering the coordinate system (3.5) singular. Generic sources lead to curvature singularities in (3.5), but for some special configurations geometry may remain regular, as we saw in the last subsection. A similar situation has been encountered in the AdS 3 ×S 3 case, where sources parameterized by string profiles on the base space led to regular solutions [15,16], but in the present case there is an important caveat: while the geometry may remain regular, the patch covered by the coordinate system (3.5) cannot be geodesically complete. We have already encountered this phenomenon in the last subsection, where coordinates (3.5) covered only a half of the AdS 2 ×S 2 parameterized by (ρ, θ), and a second copy of R 3 had to be attached to describe the full geometry. This is not very surprising since the global AdS 2 space is known to have two boundaries, and the asymptotic region of R 3 described by large (x n x n ) can only describe a vicinity of one boundary. In this subsection we will identify the sources of H 1 and H 2 that lead to regular geometries and describe the procedure for extending a patch (3.5) to a geodesically complete space.
First we recall that in the AdS 3 ×S 3 case all 1/2-BPS solutions are parameterized by several harmonic functions defined on a flat four-dimensional base [15], and all regular geometries share the same mechanism for resolving singularities at the location of the sources [16]. Using that case as a guide, we expect the mechanism of regularization described in the last subsection to be generic for all metrics (3.5). Specifically, we focus on complex harmonic functions H (3.14) which satisfy four conditions: (a) H can have sources only on closed curves 9 , and the space (3.5) develops a conical defect π in the vicinity of every point on the curve, so branch cuts have to be introduced.
(b) Once an analytic continuation to the second sheet is performed, the metric remains regular in a vicinity of the curve.
(c) To ensure that the metric is regular away from the curve, the harmonic function H cannot vanish at finite points in R 3 .
(d) For asymptotically AdS 2 ×S 2 geometries the harmonic function approaches 3.21) at infinity.
We will now present a procedure for constructing the functions satisfying conditions (a)-(d) and demonstrate that they lead to regular solutions after an appropriate analytic continuation. The regular geometry will be parameterized by a closed contour, by a charge density, and by an additional vector field. Figure 2: An example of a singular curve and the corresponding vectors F and S, which lead to a regular geometry.

S F
As demonstrated in Appendix C, requirement (a) determines the leading contribution to the metric in a vicinity of a curve. Selecting an arbitrary point on a curve and introducing cylindrical coordinates (R, ζ, x 3 ) with an origin at that point and with x 3 axis pointing along the curve, we find with a complex parameter a which can vary along the curve. Clearly, the space (3.17), (3.18) has this form. Additional analysis presented in Appendix C demonstrates that the most general harmonic function with properties (3.22) in the vicinity of the sources has the form Here F(v) is the location of the profile, σ(v) is the 'charge density', H reg is a harmonic function that remains regular everywhere, and A(v) is a complex vector field subject to two constraints: Such field can be expressed in terms of one real vector S, and in the natural parameterization of the curve, where (Ḟ) 2 = 1, (3.25) the answer becomes especially simple: Figure 3: Discussion after equation (3.27) requires the cuts to approach the singular curve from a particular direction, but this still leaves some ambiguity depicted in figure (a). This is analogous to an ambiguity for the cuts in a complex plane that remains after imposing a particular direction for the cut at the branching points, as shown in figure (b).
A pictorial representation of vectors F and S in shown in figure 2. Expression (3.23) can be used for several closed curves as well, but the integral should be understood as integration over every connected piece and summation over such pieces. Harmonic function (3.23) satisfies the condition (a), but to ensure the regularity condition (b) one needs subleading contributions to (3.22). Moreover, one has to impose the requirement (c) since vanishing of H at any finite point leads to singularities in the metric. Conditions (b) and (c) are enforced by a specific choice of the branch cuts and the regular function H reg , which we will now describe.
In a vicinity of every point on a singular curve function H behaves as (3.22): For a given complex a we can choose the range ζ 0 < ζ < ζ 0 + 2π where the real part of H remains positive and introduce a branch cut at ζ = ζ 0 , where the first term in (3.27) is purely imaginary. This determines the direction of the branch cut in the vicinity of every point on the singular curve, but still leaves an ambiguity in the complete location of the cut, which will not affect our discussion. One encounters an analogous ambiguity for the holomorphic function f (z) = √ z 2 − 1 by requiring that the branch cut goes in the real direction from z = ±1 (see figure 3). Our choice of the branch cut guarantees that the real part of the function (3.23) remains finite on the cut, then we can determine the harmonic function H reg by requiring Re H| cut = 0. (3.28) We will now demonstrate that this construction leads to regular solutions satisfying conditions (a)-(d). Moreover, once functions (F, A, σ) and the location of the branch cut are chosen, function H exists, and its real part is unique. The analysis contains two ingredients: 1. Regularity in the vicinity of the curve 2. Regularity away from the curve and we will now present the relevant arguments.

Regularity in the vicinity of the curve
To prove regularity of the metric at the location of the singular curve, we should analyze the subleading contributions to H. Let us pick a point on a curve and introduce local Cartesian coordinates (x, y, z) by choosing x direction alongḞ, z direction along F, and y direction alongF ×Ḟ. According to (3.27), the leading contribution to the harmonic function is 29) and the next order can be written as In this approximation the curve is contained in the (x, z) plane, and the branch cut is given by an open surface In particular, equation f = 0 describes the curve in the y = 0 plane, so coefficients (e 1 , e 2 , e 3 ) must be real. Laplace equation for function H determines (e 4 , e 5 , e 6 ) and leads to the final expression The gauge field can be found by integrating the defining relation and in a convenient gauge V y = 0 the result is Substituting the harmonic functions (3.30), (3.32), (3.34) into the metric (3.5) and removing the cross terms between dx and other coordinates on the base by shifting the z coordinate as 35) we arrive at the final expression for the metric in the vicinity of the singular curve: In the leading order and metric (3.36) becomes Metric (3.38) remains regular for arbitrary values of (c 1 , c 2 ), as long as angle ζ is identified with periodicity 4π. As in the AdS 2 ×S 2 example, we observe that the branch cut and introduction of a second copy of R 3 plays a crucial role in making the solution regular.
To summarize, we have demonstrated that the prescription (3.23), (3.28) ensures regularity of the solution in the vicinity of the singular curve. We will now show that the geometry (3.5) does not develop singularities elsewhere.

Regularity away from the curve
Since the complex harmonic function H remains finite and differentiable away from the singular curve, the metric (3.5) can become singular if and only if H vanishes at some point. This can only happen when the real and imaginary parts of this function vanish at the same point. Since condition (3.28) imposes a restriction only on the real part of the regular harmonic function H reg (see (3.23)), one can always shift the imaginary part of this object to ensure that Im[H] never vanishes on the branch cut: This does not fix Im[H reg ] completely, and in section 3.4 we will impose additional restrictions which lead to a convenient analytic continuation. For regularity it is sufficient to require (3.40) and to prove that Re[H] > 0 away from the cut. This would guarantee that |H| 2 never vanishes.
To demonstrate positivity of the real part of H, we introduce additional cuts shaped as thin tubes around singular curves, as depicted in figure 4. These tubes begin and end on the cuts introduced earlier. We also remove the infinity by focusing on the interior of a very large sphere. The construction presented after equation (3.27)  (c) |H 1 | < ǫ on the large sphere, and ǫ can be made arbitrarily small by increasing the radius of the sphere.
(d) H 1 is harmonic and finite in the region bounded by the cuts.
These conditions imply that function H 1 is non-negative on the boundary of a finite region surrounded by the cuts, and application of the strong maximum principle for harmonic functions leads to the conclusion that H 1 must be positive away from the cuts. Hence H cannot vanish anywhere, and the metric (3.5) cannot have singularities away from the curves r = F. As we have already demonstrated, the solution remain regular near such curves as well.
As a byproduct of the analysis presented above, we also conclude that functions (F, S, σ) and the choice of the branch cuts lead to the unique harmonic function H 1 . Indeed, if two such functions were possible, their difference ∆H 1 would remain finite on all cuts, and by making the tubes sufficiently small and the sphere sufficiently large, one can ensure that |∆H 1 | < ǫ on all cuts. Then using the maximum principle, one concludes that ∆H 1 = 0, proving uniqueness of H 1 . Existence of H 1 and H 2 follows from the standard arguments for the Dirichlet problem for the Laplace equation. Notice that the construction presented here does not lead to a unique function H 2 , and the freedom in selecting this function will be fixed by performing an analytic continuation and by requiring regularity on the additional sheets. This will be discussed further in section 3.5.
To summarize, we have demonstrated that a regular solution can be constructed by performing the following steps: (1) Starting with functions (F, S, σ) parameterizing the profile, construct the harmonic function (3.23) with undetermined H reg .
(2) Select branch cuts terminating on the singular curve and ensure that Re[H] ≥ 0 in the vicinity of the singular curves on one of the sheets (see the discussion following equation (3.27)).
(3) Determine the regular part of the harmonic function H reg by enforcing (3.28) and (3.40), as well as the asymptotic behavior (3.21).
This construction guaranties that the resulting solution remains regular in the vicinity of the singular curve and at all points on the selected sheet. In addition, one has to perform an analytic continuation and to enforce regularity and an appropriate asymptotic behavior on the second sheet, but conditions (3.28) and (3.40) are not sufficient to guarantee uniqueness of the analytic continuation through the branch cut. Moreover, it might be possible to have more than two sheet, and such solutions would have several asymptotic AdS 2 ×S 2 regions. The detailed discussion of such interesting geometries is beyond the scope of this article, and in the next subsection we will focus on describing the simplest analytic continuation for a large class of regular solutions.

Special case: planar curves
While geometries with several AdS 2 ×S 2 regions are very interesting, in this article we are focusing on solutions describing the backreaction of supersymmetric branes discussed in section 2. In particular, such branes are not expected to introduce drastic changes far away from the sources, so we expect to have only two sheets with asymptotic AdS 2 ×S 2 regions, as happened for the vacuum solution (3.8), (3.13). In principle, this does not eliminate a possibility of having 'handles' 10 , such as one depicted in figure 5(b), but we will focus on the simplest case of two sheets connected through a series of branch cuts as in figure 5(c). To find the explicit expression for the full geometry, we will also require all curves to be in the (x 1 , x 2 ) plane. Let us introduce Cartesian coordinates (x 1 , x 2 , y) in R 3 and assume that all profiles are drawn in the (x 1 , x 2 ) plane: F = (F 1 , F 2 , 0). SinceḞ andF belong to the same plane, vector S parameterizing the profile through (3.23) and (3.26) must point along y direction. Then approximation (3.27) for the integral (3.23) implies that in a small vicinity of the curve, Re [H] can vanish only at y = 0, so one can choose the branch cuts to be in the (x 1 , x 2 , y) plane. To perform an analytic continuation, we remove the space with y < 0 and introduce a boundary at y = 0. Part of this boundary (black regions in figure 6) is formed by the branch cuts where Re H| black = 0, (3.41) and another part (white regions) extends to infinity. According to (3.27), Im H remains finite in the white region, so one can always choose function H reg by requiring Since now we have a space with a boundary at y = 0, conditions ( Figure 6: An example of a droplet configuration in the (x 1 , x 2 ) plane. Black and white regions correspond to boundary conditions (3.43) for the complex harmonic function H.
the analysis presented in the last subsection guarantees that the harmonic function (3.23) with boundary conditions (3.43) is regular at y ≥ 0, and that the resulting geometry (3.5) has a conical singularity with a deficit angle 3π 2 at the location of the singular curve. Thus gluing three more sheets along such curves would produce regular geometries.
Conditions (3.43) make the analytic continuation rather simple. Starting with a harmonic function H defined at y ≥ 0, we introduce four sheets: Then the gluing across the cuts is performed using the following rules: white : A pictorial representation of the continuation (3.45) is shown in figure 7. All four sheets converge at the location of the profile F, and since each sheet describes a wedge of a flat space with an opening angle π 2 , the total angle around the curve adds to 2π, so the arguments presented in section 3.3 guarantee regularity in the vicinity of the curve. Conditions (3.45) ensure that function H and its derivatives are continuous across all branch cuts, so the geometry remains regular on the cuts as well. Finally, to demonstrate regularity at a generic point we have to show that |H| never vanishes. To do so, we combine sheets A and B to produce an R 3 with branch cuts along the black disks. Then H approaches (3.21) at infinity, and arguments presented in the last subsection prove that |H| does not vanish on A or B sheets. Then the explicit analytic continuation (3.45) ensures that |H| never vanishes, and the geometry is regular everywhere.
To analyze the asymptotic behavior of the geometry, it is convenient to construct two copies of R 3 by combining (A, B) and (C, D) sheets. These two copies are glued through the black region, as expected from the general analysis presented in section 3.3. At infinity functions (H A , H B ) approach H 0 given by (3.21), while functions (H C , H D ) approach (−H 0 ). In both cases the geometry approaches AdS 2 ×S 2 , but the two asymptotic regions are disconnected. We have already encountered this situation in section 3.2, and now we see that backreaction of the branes modified the structure of the black regions, but it preserves the asymptotic behavior, as expected for the normalizable excitations.
To summarize, in this subsection we have focused on planar curves and we found an explicit construction for the global geometry which preserves the asymptotic structure of AdS 2 ×S 2 . Starting from the general solution (3.23), (3.26) with planar curves, one should divide the y = 0 plane into black and white regions and impose the 'bubbling boundary conditions' (3.43) along with asymptotic behavior (3.21) to determine the unique harmonic function H reg . Then analytic continuation (3.44), (3.45) leads to the harmonic function which describes the global geometry, and the resulting metric is regular. In the next subsection we will discuss the topological structure of the new solutions.

Topology and fluxes
The branch cuts and analytic continuations, which make solutions constructed in section 3.1 regular, also introduce some interesting topological structures. In particular, the fourdimensional part of the geometry (3.5) acquires some non-contractible two-cycles, which can support quantized fluxes of F 2 . Upon lifting to ten dimensions these fluxes can be interpreted as dissolved D3 branes 12 . In this subsection we will analyze the topological structure of (3.5) and the associated fluxes. We will first focus on planar curves, for which the explicit analytic continuation is known, and then extend the discussion to more general solutions.
Let us consider a plane divided into black and white regions and draw a curve C (not to  b)). The resulting surface can be deformed into two disks above and below the y = 0 plane (figure (c)) to make evaluation of (3.47) easier.
be confused with a 'singular curve' separating the regions) that lies entirely in the white area. Then we attach a cap ending on this curve and approaching the curve vertically, as shown in figure 8(a). Next we construct a smooth closed surface D by combining the cap on the sheet A and its image under (3.44) on the sheet B (see figure 8(b)). If the curve C can be contracted without leaving the white region, then D is contractible. On the other hand, if one tries to contract the curve C by moving it through a black region, then D would develop a cusp when C approaches a singular curve 13 , and smooth continuation beyond this point would not be possible. This implies that curve C circling a black region gives rise to a non-contractible surface D, which is topologically equivalent to S 2 , and this surface lies entirely on sheets A and B. There is a 'mirror image' of this sphere on sheets C and D, and the two surfaces go into each other by passing through the the cut, but they never collapse. We have already encountered this phenomenon for the AdS 2 ×S 2 example in section 3.2, where the surface can be taken to be (3.46) in parameterization (3.8). For positive values of ρ this sphere remains on A and B sheets, for negative values of ρ it belongs to C and D sheets, and at ρ = 0 the surface goes through the branch cut without collapsing. Similarly, a curve in black region that circles around a white droplet gives rise to a non-contractible surface, which lies either on A and C sheets or on B and D sheets. Every non-contractible surface D based on a contour C in a white region carries a flux of the fieldF from (3.5). To evaluate DF , we deform the surface into two disks on sheets A and B located very close to the y = 0 plane (see figure 8(c)): (3.47) We will now demonstrate that the integrals in the right hand side receive contribution only from the parts of the disk immediately above or below the black droplet, so the left hand side does not change is one varies the contour C within the white region.
To treat the while and black regions symmetrically, we define a complex two-form Using relations which follow from equations (3.5)- (3.6), and boundary conditions (3.43), we conclude that the integral is real in the white region and pure imaginary in the black region, so expression (3.47) receives contributions only from integration over the black droplets. The right hand side of (3.47) can be viewed as a jump of a relevant function across the branch cuts going through the black droplets, and this interpretation leads to the final expression Notice that the second term in (3.51) picks up only ∆(∂ y H), so the boundary conditions (3.43) guarantee reality of the last equation. Similarly, starting with contour C in a black region and attaching caps to it, one finds a manifold the has a topology of a two-sphere, which is spanned by the flux where cut ′ denotes a cut along a white region encompassed by D. Notice that integral in (3.51) involves sheets A and B, while integral in (3.52) involves sheets A and C. For the symmetric analytic continuation (3.44), all integrals can be expressed in terms of the sheet A: The integrations are performed only over the interior of a defining curve C. Nontrivial integrals (3.53) of F andF give rise to fluxes of the five-form F 5 over the relevant five-cycles. For instance, starting with a surface D with a non-vanishing integral of F and combining it with various circles on the torus, one can construct several closed five-cycles D 5 with where l T is a linear size of the torus T 6 . Since the last integral must be quantized in the units of 2π 2 l 4 p , the natural unit for fluxes (3.53) is (2π 2 l 4 p )/l 3 T . Some examples of D 5 are given by where X a and Y a are defined in (2.6). Nontrivial integrals ofF give rise to similar fluxes of F 5 .
To summarize, we have demonstrated that planar curves give rise to a rich topological structure of bubbling geometries (3.5)-(3.6) through connections between different branches. Any non-contractible curve C in a white or a black region gives rise to a nontrivial S 2 , which is supported by fluxes (3.53). It is also possible to construct surfaces with more interesting topology (for example, figure 9 depicts a non-contractible torus), but the fluxes are always given by (3.53).
This construction can be extended to non-planar curved discussed in section 3.3, although in this case the situation is slightly less symmetric due to the absence of white regions and a lack of explicit formulas for the analytic continuation. Let us consider a  Figure 10: An example of a closed surface for a genetic singular curve constructed from a contour C on the branch cut.
collection of branch cuts in R 3 associated with some number of singular curves and draw a surface D that does not touch the cuts. Two such surfaces are homotopic if they can be transformed into each other without crossing the cuts. On the other hand, a surface that cannot be collapsed to a point without crossing a cut has a nontrivial topology, and it is supported by the flux (3.51). Notice that the integrals in (3.51) involve only one copy of R 3 (previously they were written in terms of sheets A and B which form this copy), so the details of the analytic continuation are not important. The second type of surfaces is constructed by choosing contours C in the branch cuts and attaching two caps to them. One of this caps extends to R 3 , and the other cap goes to the second branch, as shown in figure 10, so the details of the analytic continuation are important for constructing such surfaces. If one focuses only on the first copy of R 3 , as we did in section 3.3, then the non-contractible surfaces of the the second type look open (see figure 10). Such surfaces are supported by the flux ofF . A better understanding of the analytic continuation for arbitrary branch cuts would shed more light on structure of such non-contractible surfaces and fluxes supported by them. It would also make the treatment of F andF more symmetric, as in the case of the planar droplets.
We conclude this subsection with a brief comment concerning angular momentum of the bubbling solutions 14 . Although geometries (3.5) are not static, they do not give rise to a nontrivial ADM angular momentum if one insists on preserving the AdS 2 ×S 2 asymptotics. To see this, we recall that the ADM charges on AdS p contain a multiplicative factor (p − 2) (as discussed, for example, in [32]), so such charges always vanish for the AdS 2 . This absence of angular momentum plays a very important role in the counting of states for the four-dimensional black holes [33]. Vanishing of the angular momentum is consistent with the statement that geometries (3.5) describe the backreaction of the giant gravitons discussed in section 2: as in the AdS 3 ×S 3 case, the angular momentum of such objects comes from the flat connection associated with the spectral flow operation [23,34] rather than with ADM construction. In particular, writing AdS 2 ×S 2 in coordinates (3.5), one finds that the probe branes have vanishing angular momentum 15 , and so do the geometries (3.5) produced by them.

Embeddings into type IIA supergravity
In this article we are focusing on AdS 2 ×S 2 solutions in type IIB supergravity, but solutions (3.5)- (3.6) can also be lifted to type IIA SUGRA and to M theory. In this subsection we will briefly discuss such embeddings following the duality chains described in [35,26].
We begin with defining real coordinates (X a , Y a ) on the torus, and rewriting the field strength appearing in (3.5) in terms of them: The system (3.5)-(3.6) can be mapped to type IIA theory in several ways, and we will focus on three of them: 1. T dualities along (Y 1 , Y 2 , X 2 ) directions lead to a D4-D4-D2-D2 system, which lifts to M theory as an M5-M5-M2-M2 configuration.
2. T dualities along (Y 1 , Y 2 , Y 3 ) directions lead to a D6-D2-D2-D2 system, which lifts to M theory as a set of three orthogonal stacks of M2 branes on a background of a KK-monopole.
3. T dualities along (Y 1 , Y 2 , X 3 ) directions lead to a D4-D4-D4-D0 system, which lifts to M theory as three stacks of M5 branes on a plane wave background.
None of the T dualities affect the nontrivial part of the metric (3.5). Let us briefly discuss all three options.
15 See section 5 and Appendix A for the detailed discussion of this issue. 16 An extra factor of two comes from combining the T duality rules with supergravity normalization of F 5 . See [28] for the detailed discussion of the relation between normalization of fluxes in string theory and in SUGRA. This is a mixture of 4-and 6-forms, and electromagnetic duality leads to the final solution in terms of F 4 only: All ingredients of (3.59) can be written in terms of one complex harmonic function H: Geometry (3.59) corresponds to a brane configuration, which can be obtained from (2.1) by application of the T dualities: This picture in terms of branes becomes useful only if one focuses on a real harmonic function H: in this case the geometry does have sources. For a complex harmonic function satisfying regularity conditions, the metric is source-free, so the representation (3.61) is rather schematic. Configuration (3.59) trivially lifts to eleven dimensions by adding one more flat direction to the metric and identifying F 4 with a four-form in M theory. The special case of (3.59) with real harmonic function H (which corresponds to a singular near horizon limit of four stacks (3.61)) was discussed in [35,26].

D6-D2-D2-D2 intersection
Next we apply T dualities along (Y 1 , Y 2 , Y 3 ) to (3.57), this leads to a solution of type IIA supergravity with fluxes Application of the electromagnetic duality to this mixture of 2, 4, 6, 8-forms leads to the final solution in terms of F 2 and F 4 only: Once again, all ingredients can be expressed in terms of the harmonic function H using (3.60). The brane picture corresponding to (3.63) is a T-dual version of (2.1): Although the geometry (3.63) can be lifted to M theory using the standard embedding to do this explicitly, one needs to determine the one-form C by solving the defining equation (3.66) Unfortunately we were not able to find a nice expression for C for the general solution (3.63). In a special (albeit singular) case Re[H] = 0, we find Another example is AdS 2 ×S 2 solution (3.8), which has In this case, the coordinate y corresponds to an S 1 Hopf fibration over S 2 , and three coordinates (θ,φ, y) combine into S 3 in eleven dimensions [35,7].

D6-D2-D2-D2 intersection
Finally, application of T dualities along (Y 1 , Y 2 , X 3 ) to (3.57) leads to the fluxes and electromagnetic duality gives the final answer: Interestingly, solution (3.70) can be obtained from (3.63) by swapping F andF , and since these fields appear on the same footing in (3.60), our formalism does not distinguish between embeddings (3.63) and (3.70). The situation becomes rather different if one insists on using a real harmonic function H: as we already saw such restriction leads to a Poincare patch of the AdS space, making F purely electric andF purely magnetic. For such solutions (3.63) and (3.70) are interpreted as rather different brane configurations: (3.63) corresponds to (3.64), while (3.70) is produced by These special cases were discussed in [35,26]. From our perspective, (3.63) and (3.70) should be viewed as the same embedding of two different regular bubbling solutions into type IIA supergravity.
To summarize, in this subsection we presented three alternative embeddings of the regular AdS 2 ×S 2 solutions into type IIA supergravity and discussed lifts to eleven dimensions. The rest of this paper is focused on the type IIB solutions (3.5)-(3.6), but all results extend trivially to the embedding (3.59), (3.63), (3.70).

Examples
In this section we will consider several examples of regular geometries (3.5)-(3.6). To have complete solutions with all asymptotic regions, we will focus on planar curves, for which the analytic continuation is well understood.

AdS 2 ×S 2 and its pp-wave limit
Embedding of AdS 2 ×S 2 into the general solution (3.5)-(3.6) has been already discussed in section 3.2, and here we will briefly mention some additional aspects of this embedding. Recall that the AdS 2 ×S 2 geometry (3.8) can be expressed in the form (3.5) by defining new coordinates (t, φ) as φ =φ −t, t = Lt (4.1) and rewriting (3.8) in terms of them: The flat metric on the base, ds 2 base is given by (3.10), and function h is Extracting A t andÃ t from the time components of (4.2), we can verify the expressions for the magnetic components of F 5 : and for the harmonic function Translation to cylindrical coordinates is given by (3.11): Notice that the standard coordinates (ρ, θ) of AdS 2 ×S 2 can be viewed as oblate spheroidal coordinates on the flat base (3.10), and this seems to be a generic feature of all AdS spaces. As demonstrated in [36], in all known cases where AdS×S space can be written as a fibration over a flat base, the standard parameterization of the global AdS is associated with the oblate spheroidal coordinates on the base. Moreover, supersymmetric geometries can have integrable geodesics if and only if the Hamilton-Jacobi equation separates in ellipsoidal coordinates [36], and the oblate spheroidal parameterization is a special case. As discussed in section 3.2, parameterization (3.5) of AdS 2 ×S 2 has a branch cut at ρ = 0, which corresponds to a disk of radius L in the y = 0 plane. Expressions (3.13) on the sheet A, which corresponds to y > 0, satisfy the boundary conditions (3.43), and the analytic continuation (3.44) gives the full AdS 2 ×S 2 . Any two-dimensional surface surrounding the branch cut is non-contractible, and the flux through it is given by the first expression in (3.53). The integrand, can be interpreted as an area form for the black droplet, and the integral (3.53),  Figure 11: Relation between the droplet picture for AdS 2 ×S 2 and the pp-wave: a graphical representation of the limit (4.10).
must be quantized in the units of (2π 2 l 4 p )/l 3 T , where l p is a ten-dimensional Planck length, and l T is a linear size of T 6 . Since the AdS 2 ×S 2 geometry (4.2) does not have compact white droplets, it is impossible to form a topologically nontrivial manifold on the base that carries a nontrivial flux of F .
The pp-wave limit of the geometry (4.2) is obtained in a standard way [37] by zooming in on a vicinity of the singular curve. For the harmonic function (3.15) this implies a limit which leads to 11) and the relevant coloring of the y = 0 plane is depicted in figure 11. The resulting geometry is ds 2 = −h −2 (dt +Ṽ ) 2 +h 2 dx a dx a + dz˙adzȧ = 2dtdφ − [ρ 2 +θ 2 ]dt 2 + dρ 2 + dθ 2 + dz a dz a (4.12) and as demonstrated in section 3.3, this is a generic behavior of the metric in a vicinity of the singular curve (see equation (3.38)).
In the next subsection we will construct regular geometries corresponding to small perturbations of AdS 2 ×S 2 , and light excitations of the pp-wave can be obtained from them by taking the limit (4.10).

Perturbative solution
After discussing the AdS 2 ×S 2 solution corresponding to the ground state of the system with a given amount of flux, we consider perturbations of this geometry. The light excitations are describes by the 'gravitons', i.e., by combinations of the metric and fluxes, and coupled equations for such degrees of freedom have been extensively discussed in the literature for various AdS spaces and spheres [30,29]. Although one can perform a similar analysis for AdS 2 ×S 2 [29], here we are interested in supersymmetric excitations, which are guaranteed to be covered by our ansatz (3.5), so the study of 'gravitons' reduces to the analysis of small perturbations in the complex harmonic function H parameterizing the bubbling solution (3.5). In this subsection we will expand H around H 0 corresponding to AdS 2 ×S 2 and construct the solutions describing small regular perturbations.
We will focus on the sector corresponding to planar curves, where 'gravitons' correspond to small changes in the shape of the circles. Such ripples have been studied for the AdS spaces in higher dimensions, where geometries can be written explicitly in terms of functions parameterizing the curves [15,17]. While in the present case it is difficult to solve the Laplace equation with arbitrary boundary conditions (3.43), small perturbations around AdS 2 ×S 2 can be found explicitly. First we note that an arbitrary ripple on the circular shape with radius L can be parameterized in polar coordinates as where the sum is assumed to be infinitesimal in comparison to the leading contribution. Every profile (4.13) generates a solution with harmonic function where H 0 is given by (3.15), (4.6), 15) and every set of amplitudes a m in (4.13) translates into a particular mode expansion in H ′ : We will now determine the functional form of h m by solving the Laplace equation for H ′ and imposing regularity conditions on the geometry (3.5).
Writing H = H 0 + H ′ and expanding the metric (3.5) to the first order in H ′ , we find Here ds 2 0 is the metric (3.5) for the AdS 2 ×S 2 space, and V ′ is the vector field corresponding to H ′ . To ensure regularity of (4.17) in the vicinity of the singular curve, it is sufficient to require for small ρ and cos θ. This implies that H ′ should vanish at least as ρ or as cos θ.
To construct the relevant solutions, we observe that the Laplace equation for function H ′ on the flat base (3.10) is equivalent to the wave equation on the AdS 2 ×S 2 (3.9), ds 2 = −(ρ 2 + cos 2 θ) dt − L sin 2 θdφ ρ 2 + cos 2 θ 2 + L 2 dρ 2 ρ 2 + 1 + dθ 2 + sin 2 θ(ρ 2 + a 2 )dφ 2 ρ 2 + cos 2 θ , (4.19) with an additional assumption of t-independence. Going to the standard coordinates of AdS 2 ×S 2 by shifting and rescaling coordinates as 20) we conclude that H ′ can depend only on three coordinates (ρ, θ,φ−t). The wave equation on the AdS 2 ×S 2 separates between two subspaces, and, to ensure the t-independence of H ′ , we are looking for solutions which have the form Function R(ρ) must vanish at infinity to preserve the AdS 2 ×S 2 asymptotics, and this implies that R(0) = 0. Then to ensure regularity at the singular curve (see (4.18)), function Θ(θ) must vanish at θ = π 2 . Since the wave equation separates between the AdS space and the sphere, the angular part of the function (4.22) can be written as a superposition of spherical harmonics, Θ(θ)e imφ = Y l,m (θ,φ), (4.23) and the only harmonics that vanish at θ = π 2 are Y |m|+1,m (θ,φ) ∝ e imφ [sin θ] |m| cos θ (4.24) Substituting (4.23) with l = |m| + 1 into (4.22) and writing the wave equation for H ′ in the metric (4.21), we arrive at an ordinary differential equation for R(ρ), which can be solved in terms of the associated Legendre functions. In particular, the solution that vanishes at ρ = ∞ is 26) and it approaches zero as (4.27) As expected, all radial functions R l vanish faster than H 0 ∼ 1 ρ . The first few cases of (4.26) are given by Rewriting the function (4.22) in the original coordinates used in (4.19), we arrive at the final expression: Harmonic function (4.29) gives rise to regular perturbations of the AdS 2 ×S 2 geometry via (3.5)- (3.6), and it corresponds to exciting the (l − 1)-st harmonic on a circle. Notice that (4.29) does not satisfy the boundary conditions (3.43) since it corresponds to an infinitesimal perturbation, but a condensate of such modes would obey (3.43). Some particular condensate deforms the circle into an ellipse, and an explicit solution for this case will be constructed in the next subsection.

Elliptical droplet
Although finding solutions with mixed boundary conditions (3.43) is not easy, some examples can be constructed using separation of variables. It is well-known that Laplace equation in three dimensions separates only in ellipsoidal coordinates and in their degenerate cases [38], and the standard coordinates of AdS 2 ×S 2 defined by (4.7) correspond to such a degenerate case. We will now consider a more general situation involving generic ellipsoidal coordinates and use them to construct a harmonic function H satisfying conditions (3.43) with an elliptical droplet. We begin with recalling the ellipsoidal coordinates on R 3 using the notation of [39]. Starting from the Cartesian coordinates (x 1 , x 2 , y), one defines the ellipsoidal coordinates as solutions of a cubic equation for u: (4.30) where (a, b, c) are some positive constants. Without loss of generality, we assume that Denoting the solutions of (4.30) by (ξ, η, ζ), one can find the explicit formulas for the Cartesian coordinates: In the ellipsoidal coordinates the metric of the flat space becomes In section 4.1 we used the oblate spheroidal coordinates, which are obtained by taking the limit b → a while keeping ξ, η and ζ + a 2 a 2 − b 2 ≡ cos 2 φ fixed. Then defining coordinates (ρ, θ) by 33) we recover the transformation (4.7): x 1 + ix 2 = L ρ 2 + 1 sin θe iφ , y = Lρ cos θ. (4.34) In particular, the disk ρ = 0 corresponds to ξ = −c 2 , and the rest of the y = 0 plane corresponds to η = −c 2 . This pattern persists for the elliptical droplet as well: as we will see, the interior of the ellipse corresponds to ξ = −c 2 , its exterior corresponds to η = −c 2 , and the ratio a/b determines the eccentricity of the ellipse. Going back to the general ellipsoidal coordinates (4.31) and setting ξ = −c 2 we find The range of ζ allows us to define a new angular coordinate φ by 36) then (4.37) and for the allowed values of η the coordinates in the plane satisfy an inequality Thus (4.35) describes the interior of an ellipse. Similarly, the region η = −c 2 can be parameterized by ξ and φ as (4.39) and this describes the exterior of the same ellipse.
To determine the harmonic functions (H 1 , H 2 ), we have to solve the Laplace equation on the flat base and impose the boundary conditions in the interior and exterior of the ellipse. The functions have the correct behavior in the y = 0 plane, and a direct calculation shows that H 1 and H 2 are harmonic in the flat space with the metric (4.32). At large values of ξ, which correspond to infinity of R 3 , we also find the correct behavior: 4.41) Recall that at large values of ξ, it is √ ξ that plays the role of the radial coordinate of R 3 (see (4.31)). An explicit expression for V corresponding to the harmonic functions (4.40) can be found, but it is not very illuminating.
We conclude this subsection by writing the approximate expressions for H 1 and H 2 in the vicinity of the singular curve. To do so, we introduce the counterparts of coordinates (ρ, θ, φ) used for the circular droplet: (4.42) Here we defined two convenient constants: 4.43) which control the size of the ellipse and its eccentricity e. Specifically, the relation (4.38) for the interior of the ellipse can be written as (4.44) so the eccentricity of the ellipse is given by (4.45) In the vicinity of the singular curve we find the approximate expressions for the Cartesian coordinates, 4.46) and for the harmonic functions As before, the singular curve is located at ρ = 0, θ = π 2 . To summarize, in this subsection we have presented an interesting example of an explicit solution that goes beyond the circular droplet. The success in constructing this example is based on our ability to solve the boundary problem (3.43) using separation of variables. Unfortunately, the boundary conditions (3.43) for a generic droplet are not amenable to an analytical treatment, but for every shape the solution exists, and it is unique.

Asymptotically-flat solution
So far we have been focusing on regular geometries which approach AdS 2 ×S 2 at infinity, and it might be interesting to look for asymptotically flat solutions as well. One example of such solution is given by (2.2), but this geometry has a singularity at r = 0. This is an example of a general situation for AdS p with p > 3: the regular solutions are described by bubbling geometries, which cannot be connected to flat space, while the asymptotically flat configurations of branes can only produce a singular Poincare patch of the AdS space. This dichotomy stems from different boundary conditions for the fermions on global AdS and on its Poincare patch, and only the latter can be glued to flat space. The situation is rather different in the AdS 3 case, where the global AdS can be connected to flat space via the spectral flow procedure developed in [23]. Although the fermions on the global AdS and on the Poincare patch still have different boundary conditions (they correspond to the Neveu-Schwarz and to the Ramond sectors of the dual field theory), one can go from one description to another by performing a spectral flow on the boundary [40], which corresponds to a diffeomorphism in the bulk 17 . Specifically, starting from the NS vacuum described by the AdS 3 ×S 3 geometry (A.1), one can go to one of the Ramond vacua by mixing the sphere and AdS coordinates as 4.48) and using (θ,φ,ψ) rather than (θ, φ, ψ) to parameterize the sphere at infinity. As demonstrated in [23], coordinates (ρ, θ,φ,ψ) can be extended to the asymptotically flat region, where they parameterize R 4 . We will now demonstrate that solutions (3.5) can accommodate a similar interpolation between a regular interior of the global AdS 2 and the flat space.
To connect the global AdS 2 ×S 2 (3.8) and the flat space, it is convenient to write the metric on the base in terms of the oblate spheroidal coordinates (3.10): The infinities of the two (x 1 , x 2 , x 3 ) sheets correspond to ρ = ±∞, and the harmonic function (4.6) describing AdS 2 ×S 2 approaches zero in both regions. For the flat space function H should approach a constant, and since at large values of ρ the real part of (4.6) dominates, it is natural to look for flat region where H 1 approaches a constant and adjust H 2 accordingly 18 . Writing the harmonic functions as (4.50) we conclude that function h 1 should vanish at ρ = 0 to satisfy the boundary condition (3.43) with a black circle in the y = 0 plane, and it must approach a constant when ρ goes to infinity. The easiest way to satisfy these requirements is to assume that h 1 depends only on one variable ρ, then the Laplace equation for h 1 has a unique solution with the desired properties: Here c 1 is the asymptotic value of h 1 . To determine the function h 2 , one can go to the Cartesian coordinate and ensure regularity by requiring that the complex harmonic function H has the form (3.30) with function f given by (3.32). A simpler way of ensuring regularity is to notice that function is harmonic, it satisfies the correct boundary conditions in the y = 0 plane (ρ = 0 or θ = π 2 ), and it vanishes at infinity. Enforcement of regularity on a circle, which amounts to imposing (3.32), leads to a relation between c 1 and c 2 . Specifically, near the singularity we find and this becomes a function of ρ − i cos θ (an analog of z + iy in (3.32)) if For this value the definition (3.6) of the V field gives The leading term corresponds to the AdS 2 ×S 2 space, and the expressions in the curly brackets remain finite in the vicinity of the circle ρ 2 + cos 2 θ = 0, so the metric remains regular. To see this, it is sufficient to look at the (t, φ) sector: If the relation (4.54) is not imposed, then V φ has logarithmic singularities, e.g., c 2 = 0 gives V φ = − L sin 2 θ ρ 2 + cos 2 θ 1 + 4c 1 π ρ arctan ρ − 2Lc 1 π ln 1 + ρ 2 ρ 2 + cos 2 θ (4.57) Such logarithms lead to geometries with shock waves [41]. Similar singularities have been encountered in the AdS 3 ×S 3 case [42], where it was shown that shock waves can be removed by perturbing the sources [43]. A similar resolution for solutions (4.51), (4.52) violating (4.54) might also be possible, but we will not discuss this further.
To summarize, we have constructed an example of an asymptotically flat regular geometry, and it is given by (3.5), (3.6) with and V φ from (4.55). The resulting metric has two length scales: one determined the AdS radius, and the other one defines the scale of the transition between the AdS and flat regions. At sufficiently large values of |ρ|, the second term in (4.58) dominates, and the geometry can be approximated by a flat metric. As expected, there are two such regions: they come from two copies of R 3 in (3.5), and they correspond to positive and negative values of ρ. The transition to the near horizon regime happens when 1 ρ 2 + cos 2 θ ∼ 2c 1 π arctan(ρ) (4.59) and the size of the AdS region, ρ trans depends on the value of c 1 through (4.59). On the other hand, the radius of the AdS space is L (or one, if measured in ρ coordinate), so a meaningful AdS region exists only if ρ trans ≫ 1, in other words, if parameter c 1 is small. In the AdS 3 case an extension of geometries from the AdS region to flat asymptotics was accomplished by adding one to the harmonic function [15], but now such procedure is more complicated even for the simplest state (4.58). It would be interesting to find the general algorithm for extending all solutions (3.5) from the near horizon geometry to an asymptotically flat space.

Brane probes on bubbling geometries
In section 2 we analyzed supersymmetric branes on AdS 2 ×S 2 , and in section 3 we constructed geometries produced by such objects. Gravitational backreaction becomes important only when many branes are put on top of each other, and it might be interesting to study dynamics of one additional brane on the geometry produced by such stacks. This dynamics is governed by the DBI action for the probes placed on (3.5)- (3.6), and in this section we will analyze the behavior of such probes.
Supersymmetric D3 branes on (3.5)-(3.6) must wrap three directions on T 6 , and it is convenient to introduce real coordinates X a , Y a instead of complex z a used in (3.5) by writing z a = X a + iY a (5.1) We begin with discussing a brane that wraps directions (X a , Y a ) is a specific way, and we will comment on the general situation in the end of this section. Let us assume that a brane appears as a point in the (Y 2 , Y 3 ) subspace and wraps (X 2 , X 3 ) as well as a line in the (X 1 , Y 1 ) plane. Then the following static gauge can be imposed Assuming that (x 1 , x 2 , x 3 ) are functions of τ , we find the action for the D3 brane 19 : Equations of motion are solved by constant (x 1 , x 2 , x 3 ), as long as the following constraints are satisfied:

Solution (3.5) has
so relation (5.4) can be rewritten as The easiest way to satisfy this constraint is to make β a coordinate dependent quantity and to set Such configurations solve all equations of motion, moreover, the action (5.3) vanishes on the solutions, and this property often indicates an unbroken supersymmetry.
To identify the supersymmetries preserved by the rotating branes, we recall the kappasymmetry projection associated with a D3 brane [31]: The origin of the factor of four in the Chern-Simons term is explained in the footnote 5 on page 6.
Rotating brane (5.2) in the geometry (3.5) has 20 The last relation implies that the brane preserves supersymmetry satisfying a projection that depends on β: The brane is supersymmetric if and only if the last relation is consistent with the projection imposed in (3.5),η = h −1/2 e iαΓ 5 /2ǫ , Γ t Γ 5ǫ =ǫ, (5.10) in particular, the coordinate dependences of the projectors (5.9) and (5.10) must match. Notice that relations (5.9) are written for a spinor in ten dimensions, while (5.10) are formulated in terms of a reduced four-dimensional object, and the relation between the two is described in Appendix B.1: Writing similar relations for ǫ andǫ, we find Here we used the explicit expressions (B.6) for the gamma matrices ΓX 1 and ΓX 2 and for their product: To summarize, we found that a ten-dimensional spinor η for the solution (3.5) can be expressed in terms of a constant spinor ǫ as 5.14) 20 As in section 2, Γm denote the gamma matrices with flat indices (see equation (2.15)).
Here we used the relation (5.7) to express α in terms of β. Comparing the projection (5.9) coming from the brane and projection (5.14) coming from the geometry, we find a perfect match in the functional dependence of the two spinors 21 , and the two coordinateindependent restrictions on ǫ and η 0 are also consistent. We conclude that the brane (5.2) does not break any supersymmetry of the background, as long as it is placed at an appropriate point, i.e., as long as relation (5.7) is satisfied. In other words, orientation of the D branes on the torus (angle β) must be adjusted to match the known function of coordinates α, which comes from (3.5).
Although our argument were made for the brane (5.2) that does not stretch in (Y 2 , Y 3 ) directions, it can be easily generalized to branes with generic orientation on the torus. We conclude this section by presenting such generalization for the action (5.3), and extension of the supersymmetry analysis is straightforward, although the notation becomes cumbersome.
Any supersymmetric D3 brane wrapping three directions of T 6 can be described in a static gauge that generalizes (5.2): (5.15) where M is a 3 × 3 complex matrix with a non-zero determinant. The induced metric on the brane is where index µ goes over four non-compact directions, including time. The determinant of this metric is det g ind = g µνẋ µẋν det(M † M) . (5.17) We can always normalize coordinate ξ m to ensure that det(M † M) = 1, then the DBI action coming from (5.17) is identical to the first term in (5.3). Next we look at the pullback of the gauge potential that appears in the Chern-Simons term: Here we defined angle β by An extra normalization factor h −1/2 in (5.14) is irrelevant since projection is a linear relation.
The consistency condition, (5.20) is satisfied due to normalization of M. It is clear that the pullback (5.18) gives rise to a Chern-Simons term, which is identical to the one used in (5.3), so the entire action (5.3) is recovered for an arbitrary complex matrix M in (5.2). The angle β, which translates into the location of a brane in the non-compact direction and into the kappa projection via (5.5) and (5.7) is determined for every normalized matrix M by (5.19).

Discussion
In this paper we have constructed regular BPS geometries with AdS 2 ×S 2 ×T 6 asymptotics and demonstrated that such solutions of supergravity are parameterized by one complex harmonic function on R 3 with sources distributed along arbitrary curves. To construct a geodesically complete space, one has to glue several copies of R 3 through a series of branch cuts, and we have presented the explicit procedure for the analytic continuation in the case when all curves belong to one plane. Although the geometric data paramaterizing the new solutions is analogous to its conterparts for the bubbling geometries in ten dimensions (where one specifies the white and black regions in a plane) and for the sixdimensional 1/2-BPS fuzzballs (where one specifies contours in a 4-dimensional base), the mechanism of resolving the singularity in the AdS 2 ×S 2 case is very peculiar, and it is based on existence of several copies of the base and on analytic continuation. Another peculiar feature of the new solutions is the lack of a clear connection between the gravity picture and a theory on the boundary, which was present in the six-and tendimensional cases. For example, the 1/2-BPS D1-D5 geometries of [15] corresponded to chiral primaries in the dual field theory, and this connection could be visualized via an effective multiwound string [44]. The ten-dimensional bubbling solutions of [17] were mapped to a quantum mechanics of a matrix model on the boundary [45] via a very explicit correspondence. Unfortunately, the field theory dual to AdS 2 ×S 2 ×T 6 is not wellunderstood, and this impedes the construction of an explicit map between the boundary and the bulk, but perhaps one can use the gravity side to get some insights into the dynamics of fields theory using the methods developed in [46]. This may also allow one to count the bubbling states in supergravity and extend the fuzzball proposal [24] to the four-dimensional black holes constructed from intersecting D3 branes.

A Giant gravitons on AdS 3 ×S 3 and on fuzzballs
In this article we study supersymmetric branes on AdS 2 ×S 2 and their gravitational backreaction, and we find that such branes are rather different from their counterparts on AdS 5 ×S 5 . It turns out that branes on AdS 3 ×S 3 share some of these peculiar properties, but to see this one has to go beyond the standard giant gravitons discussed in [11,12]. In AdS 5 ×S 5 giant gravitons exhaust all 1/2-BPS configurations, and their counterparts with lower supersymmetry have also been classified in [13]. In this section we will analyze the probe branes on AdS 3 ×S 3 and on its supersymmetric excitations.
We begin with discussing branes on AdS 3 ×S 3 : The standard giant graviton [11] and the dual giant [12] are obtained by imposing the following ansatz for the worldvolume of the D1 brane: However, in the AdS 3 ×S 3 case, one can introduce a more general ansatz, which leads to the following combination of the DBI action and the Chern-Simons term: Equations of motion for cyclic variables (φ, χ, ψ) are satisfied automatically, while equations for ρ and θ give These equations are solved bẏ .5) and arbitrary (ρ, θ). Moreover, configurations (A.5) have vanishing Lagrangian density so their energy and angular momentum are equal up to a sign, 6) and they also preserve supersymmetries, as we will see below.
Giant gravitons (A.2) wrapping the sphere or the AdS space have counterparts in higher dimensions [11,12], but their "mixed" generalization (A.2) exists only in AdS 3 ×S 3 and AdS 2 ×S 2 . This is related to another peculiar property of giant gravitons observed in [12]: in sharp contrast to higher dimensional cases, where the size of the giant graviton is fixed by its angular momentum, the branes (A.2) on AdS 3 ×S 3 can wrap arbitrary cycles on AdS or on a sphere since the potential for their size is flat. Now we see that not only the size of a cycle is arbitrary, but a mixture between AdS and sphere is also allowed.
We will now demonstrate that configurations (A.5) are supersymmetric. Rather than proving this only for giant gravitons on AdS 3 ×S 3 , we will show that counterparts of (A.5) on any 1/2-BPS geometry preserve SUSY. First we recall that all 1/2-BPS geometries with AdS 3 ×S 3 asymptotics are known explicitly [15] 22 , and they are given by Here dx i dx i denotes the metric on a flat four-dimensional base, and dz a dz a denotes a metric on T 4 . All functions can depend on (x 1 , x 2 , x 3 , x 4 ). Harmonic functions (H 1 , H 5 ) and the gauge field A are determined from microscopic analysis, and expressions give rise to regular solutions [15,16]. To recover the AdS 3 ×S 3 (A.1) from this construction, one has to choose a circular profile in the four dimensional space (x 1 , x 2 , x 3 , x 4 ) and to perform a spectral flow [23,48,15]. Specifically, the relevant harmonic functions are given by [48] and combining this with flat metric on the base, we find This relation reduces to (A.1) after identification Q = L 2 and a change of coordinates .9) In particular, the shift of the angular coordinates corresponds to a spectral flow from the Ramond to the NS sector in the dual CFT [23]. Rewriting this spectral flow as we conclude that configurations (A.5) correspond to constant values of (r, θ,φ,ψ), i.e., to one point on the base. This observation suggests a simple ansatz generalizing (A.5) to branes on an arbitrary supersymmetric geometry (A.7): one should put a D1-brane at one point on the base, while stretching it alongt and y: .11) Assuming that coordinates x i on the base depend only on time, we find the action governing the dynamics of D1 branes on (A.7): It is clear that all equations of motion are solved by constant x i , moreover, the action vanishes on such solutions. To verify that configuration (A.11) preserve supersymmetry, we recall the expression for the kappa-symmetry projection associated with a D1 brane [31]: For configurations (A.11) this expression reduces to a very simple projection Γǫ = ǫ, Γ = iσ 3 σ 2 ⊗ ΓtΓŷ, (A.13) which is consistent with supersymmetries preserved by the background geometry (A.7). An analogous projection for the D5 branes wrapping (t, y, z a ) is Backreaction of such D1 and D5 branes modifies the geometry, while leaving it in the general class (A.7). To summarize, in this appendix we reviewed some properties of supersymmetric branes on AdS 3 ×S 3 and on fuzzball geometries constructed in [15,16]. We demonstrated that such branes are much more general than the giant gravitons in higher dimensions [11,12], but they are very similar to the branes on AdS 2 ×S 2 discussed in sections 2.

B Derivation of the solution
In this appendix we derive the geometry (3.5) by solving equations for the Killing spinors and self-duality conditions for F 5 after imposing the ansatz (3.2): Here index m refers to four non-compact directions, and index a runs from one to three.

B.1 Reduction to four dimensions
Supersymmetry of the geometry (B.1) implies an existence of a Killing spinor η. Since only the five-form is excited, the variations of dilatino under supersymmetry transformations vanish trivially, and we only have to solve the gravitino equations [27] ∇ It is convenient to separate the gravitino equation into its torus components, and the remaining projections, To proceed we choose a convenient basis of gamma matrices 23 : Noticing that in this basis we can compute several useful products: Recall that the Killing spinor must satisfy the chiral projection of type IIB SUGRA: Duality between F andF implies a relation 24 and equations (B.3), (B.4) become Next we decompose the spinor η into eight components η ±±± : The first equation in (B.10) with a = 1 and (+ − −) projection of (B.11) give These two equations appear to be inconsistent for nontrivial F and η +−− , and since we are looking for solution with flux, we will set η +−− = 0. Other mixed components of η vanish for the same reason, and we end up with two non-vanishing projections, η +++ and η −−− , which trivially satisfy (B.10) and mix in equation (B.11).
To factorize the torus, we define two four-component objectsη, (B.13) which are subject to constraints due to the chirality condition (B.8). Equation (B.11) reduces to a system 24 To verify the signs, one can look at a particular component in frames, whereǫ 0123 = 1, e.g., which can be written in compact form using (non-chiral) spinor The final equation for the four dimensional spinorη is (B.16) and it will be analyzed in the remaining part of this appendix.

B.2 Spinor bilinears
Existence of Killing spinors severely constrains the geometry, and a powerful technique for extracting the constraints is based on analyzing spinor bilinears [49]. We will now use this technique to explore the consequences of equation (B.16). To simplify the notation, we will drop tildes in (B.16), then equations for the Killing spinor and its conjugate become From now on Γ A = e A m γ m and η denote four dimensional gamma matrices and Killing spinor 25 . Combining the duality relations (B.9), i˜ F = − F γ 5 , and expressing F in terms ofF , we obtain an alternative form of (B.17) : Using an identity 19) we find equations for the spinor bilinears: The last two equations imply an existence of a Killing vector K and an exact form L: Using the Fierz identities, we conclude that K m cannot be space-like 26 . In this paper we will focus on time-like K m and choose coordinate t along this vector. The exact form L m dx m selects a second coordinate y, and we can choose the remaining two coordinates (x 1 , x 2 ) to be orthogonal to y. Notice that due to the second Fierz identity (B.25), t is also orthogonal to y, so we arrive at the most general metric consistent with (B.24) and (B.25): Furthermore, equations for the bilinears ensure that we can choose a gauge wherē We will now determine the functional form of η by combining the last two equations with relations following from definitions of K and L: After solving equation (B.28) by introducing a real angle α, we define a new spinor ǫ by Substitution of this expression into (B.27), (B.29) gives relations for the bilinears involving ǫ: Taking a difference of the first two relations, 33) and observing that 1 2 [1 − Γ t Γ 5 ] is a projector, we conclude that Then the first and the last relations in (B.32) imply that We conclude that a four-component spinor ǫ satisfies two independent projections: so it effectively reduces to a one-component complex object. Moreover, the normalization condition determines ǫ up to a pure phase.
To restrict the form of the two-dimensional metricq αβ in (B.26), we consider equation for a new bilinear that does not involve complex conjugation of spinors 27 : This equation implies a relation, which can be written either in terms of η or in terms of ǫ: Choosing the frames for the metric (B.26): B.40) and using projectors (B.36), the one-form entering (B.39) can be simplified: then the resulting equation reads Using this relation to define a complex coordinate w: 27 We work in the representation where Γ t , Γ y and Γ 1 are symmetric, while Γ 2 is antisymmetric. Then Γ 2 γ T m Γ 2 = −γ m and Γ T 5 = −Γ 5 .
we find In other words, the two dimensional metricq αβ must be flat.
To summarize, we have used some spinor bilinears to determine the bosonic fields: and the Killing spinor: In the next two subsections we will analyze the remaining equations for the Killing spinor and Bianchy identities for F 5 to find the relations between h, α, and vector V .

B.3 Remaining equations for the Killing spinor
Although introduction of y coordinate was helpful in deriving the metric (B.45), from now on it is convenient to treat all three coordinate on a flat base of (B.45) uniformly 28 : and impose only one of the projections from (B.46) (B.48) In this subsection we will verify the equation (B.17) for the Killing spinor. To do so, we introduce the frames and simplify the expression for F : We used a relation ǫ abc Γ ab = 2Γ c Γ 123 = 2Γ ct Γ t123 = 2iΓ tc γ 5 .
Substitution of (B.50) into (B.17) gives Using expressions (B.45) for A t andÃ t , the last equation can be rewritten as Further rewriting of η in terms of ǫ (see (B.46)), we arrive at the equation for the ǫ, which is equivalent to (B.17): The rest of this subsection will be devoted to proving that this equation is satisfied, as long as the vector field V obeys a duality relation (B.57).
To verify equation (B.52), we need to compute the spin connection ω p,mn : we find the nontrivial components of ω This leads to the following expressions for the derivatives in the orthonormal frame: Using projection (B.48) and relations the t component of (B.52) can be simplified: Since three spinors Γ c ǫ are independent, the last equation is equivalent to the relation Spacial components of (B.52) give We used the relation Substitution of the duality relation (B.57) into (B.58) leads to the conclusion that ǫ is a constant spinor.
To summarize, we have demonstrated that the geometry (B.47) admits a Killing spinor η given by (B.46) with constant ǫ, as long as the duality relation (B.57) is satisfied. In the next subsection we will show that the self-duality condition for F 5 leads to additional relations between h and α.

B.4 Equations for the field strength
To find supersymmetric solutions of type IIB supergravity it is sufficient to solve the equations for the Killing spinors and the Bianchy identities for various fluxes [49]. Equation (B.57) gives the condition for existence of a Killing spinor in the geometry (B.47), and now we will analyze the Bianchi identity for F 5 .
To determine the F ab components in (B.47), it is convenient to go back to (B.50), F = −2(Γ ta ∂ a A t − iΓ 5 Γ ta ∂ aÃt ), (B.59) and use a "duality relation" for gamma matrices: Removing Γ 5 from (B.59), (B.61) and reading off various components of F , we arrive at the final expression for the field strength F : where H 1 and H 2 are harmonic functions on the three-dimensional base. The full expressions for the fields strength are given by The results of this appendix are summarized in equations (3.5)-(3.6).
C Regularity analysis: lessons from AdS 2 ×S 2 As we saw in section 3.2, AdS 2 ×S 2 geometry remains regular in spite of singularities in the harmonic functions. Of course, generic sources in H would lead to singular geometries, and in this appendix we will use the insights from the AdS 2 ×S 2 example to identify the allowed sources. The result of this heuristic analysis is summarized in equation (C.22) and in section 3.3 we demonstrate that this setup indeed leads to regular solutions. We begin this appendix with recalling the harmonic function (3.15) for AdS 2 ×S 2 , rewriting it in Cartesian coordinates: 1) and writing this function as an integral over the sources of H. It is clear that such sources are located on the circle of radius L in the y = 0 plane, and we introduce a parameter v along this circle. Then the singular curve can be written as To simplify further analysis, we fixed the freedom in selecting the parameter v by choosing the natural parameterization of the curve, wherė Direct calculation demonstrates that function (C.1) can be written as We will now generalize (C.4) and (C.5) to an arbitrary profile F(v) and find the conditions on A that make the solution (3.5)-(3.6) regular. Using the intuition from the AdS 3 ×S 3 case [15,16], where harmonic functions for all 1/2-BPS states were given in terms of the same integrals involving string profiles, we will look for a similar construction here. Any one-dimensional curve in three dimensions (x 1 , x 2 , x 3 ) can be parameterized by a profile F(v) that satisfies the normalization condition (C.3) 29 . Motivated by the AdS 2 ×S 2 example, we associate such a curve with a (complex) harmonic function where the complex vector A(v) is yet to be determined. At large values of r the harmonic function approaches so it is natural to interpret an arbitrary function σ(v) as a dimensionless charge density. In the AdS 2 ×S 2 example this function was equal to one. The Laplace equation for function the H given by (C.6) implies a relation which is satisfied if and only if (A · A) = 0, (C.8) so vector A must be complex. Regularity of the geometry (3.5) imposes additional constraints on this vector. Let us consider a small vicinity of a point F(v 0 ) on the profile. To simplify notation, we shift parameter v to set v 0 = 0 and choose a new coordinate system with the origin at F(0) and x 3 axis pointing alongḞ(0). Further, we introduce polar coordinates (R, ζ) in the (x 1 , x 2 ) plane: F(0) = 0,Ḟ 1 (0) =Ḟ 2 (0) = 0, x 1 + ix 2 = Re iζ . (C.9) The four-dimensional part of the metric (3.5) becomes ds 2 = −h −2 (dt + V ) 2 + h 2 dR 2 + R 2 dζ 2 + dx 2 3 . h 2 = HH (C.10) To reproduce the regularization mechanism encountered for AdS 2 ×S 2 , we require the leading contributions to V and h to have the form: where (Q, P,P ) approach constants as R → 0. Clearly this requiresP = −1. Expression for h 2 determines H up to a phase, and since ζ-dependence of Φ must be linear, the Laplace equation completely fixes this phase: H ≃ Q √ R e i(ζ−ζ 0 )/2 (C.12) We will now relate the complex parameter Qe −iζ 0 /2 (which generically depends on a point on the singular curve) with σ and A which enter (C.6).
To analyze the behavior of function (C.6) near the singular curve, we first focus on the x 3 = 0 plane, wherẽ To extract the leading contribution to the last expression, one is tempted to replace the profile by a straight line and integrate over v from minus infinity to infinity. Unfortunately such replacement leads to unphysical divergences at large values of |v|. To cure this problem, we take a derivative of (C.13) with respect to R, [2R − 2(Fn)]dv (C.14) before replacing the profile by a straight line. Introducing expansions .15) and recalling thatḞ 2 = 1, we find the leading contribution to (C.14) at small values of R by extending the integral over v from minus infinity to infinity: The fact that all additional terms in the expansion (C.15) lead to subleading contributions becomes especially obvious if one introduces a new integration variable u = v/R. Equation (C.16) has to match the R-derivative of (C.12): 17) and this matching leads to a constraint on A 0 . Indeed, when n changes sign, ζ shifts by π, and expression (C.17) is multiplied by i. Equation (C.16) reproduces this transformation law only if (Ḟ 0 A 0 ) = 0, (C. 18) i.e., if vector A belongs to a plane transverse to the profile. In this case the integral (C.16) simplifies, and we recover (C.17) with a particular value of Q. Substitution into (C.12) gives the leading contribution to H in terms of A: We conclude that complex vector A is subject to two conditions (C.8) and (C.18): (AḞ) = 0, (AA) = 0. (C. 21) Notice that these are the necessary conditions for regularity which come only from the analysis of the leading divergence, so we have proved that function H can give rise to a regular geometry (3.5)- (3.6) only if it has the form (AḞ) = 0, (AA) = 0.
Here H reg is a regular part of the harmonic function, which is not fixed by our analysis of the divergent terms in a vicinity of the singular curve. In section 3.3 we will start with imposing (C.22), derive some additional constraints on H, and prove that the resulting harmonic functions always produce regular geometries via (3.5)-(3.6). We conclude this appendix by rewriting the constraints (C.21) on a complex vector field A in terms of a real vector. Breaking A into a real and imaginary part as Introducing ζ as an angle between S and n, we can rewrite (C.20) in a form that matches (C.12): 26)