One Thousand and One Bubbles

We propose a novel strategy that permits the construction of completely general five-dimensional microstate geometries on a Gibbons-Hawking space. Our scheme is based on two steps. First, we rewrite the bubble equations as a system of linear equations that can be easily solved. Second, we conjecture that the presence or absence of closed timelike curves in the solution can be detected through the evaluation of an algebraic relation. The construction we propose is systematic and covers the whole space of parameters, so it can be applied to find all five-dimensional BPS microstate geometries on a Gibbons-Hawking base. As a first result of this approach, we find that the spectrum of scaling solutions becomes much larger when non-Abelian fields are present. We use our method to describe several smooth horizonless multicenter solutions with the asymptotic charges of three-charge (Abelian and non-Abelian) black holes. In particular, we describe solutions with the centers lying on lines and circles that can be specified with exact precision. We show the power of our method by explicitly constructing a 50-center solution. Moreover, we use it to find the first smooth five-dimensional microstate geometries with arbitrarily small angular momentum.


Introduction
Microstate geometries are solitonic solutions of the equations of motion of supergravity theories. Classical results from General Relativity established that this type of nonsingular solutions cannot be accommodated in a four-dimensional spacetime 1 , at least when non-Abelian matter fields are absent. Actually, asymptotically flat, spherically symmetric non-Abelian solitons have been known to exist in four and more dimensions for decades, [3][4][5][6][7][8][9][10], although multicenter solutions have only been discovered very recently [11]. Abelian solitons in supergravity, however, are only possible in five dimensions or more. In that case earlier no-go theorems can be circumvented due to the presence of Chern-Simons topological terms in the action; magnetic fluxes threading non-contractible 2-cycles become effective sources of electric charge, mass and angular momentum. It is for this reason that Abelian microstate geometries require the spacetime manifold to have non-trivial topology.
A set of rules to construct Abelian microstate geometries as supersymmetric solutions of five-dimensional supergravity was discovered in [12,13], where these were conjectured to be related to the classical description of black hole microstates within the context of the fuzzball proposal [14]. These works generalized earlier results [15][16][17][18][19][20][21] by making use of the solution generating technique of [22,23]. On the other hand, based on the results of the program for the study of non-Abelian black holes in string theory [24][25][26][27][28][29][30][31], this technique has been extended to include the construction of non-Abelian microstate geometries in [11]. In this article we introduce a unified framework, so our discussion can be applied to all five-dimensional supersymmetric microstate geometries on a Gibbons-Hawking base 2 .
The aforementioned solution generating technique, however, has very limited applications. In few words, this is due to the fact that it is not known how to systematically avoid the presence of Dirac-Misner string singularities or general closed timelike curves (CTCs). We refer to this fact generically as the CTCs problem, since Dirac-Misner strings, if present, can only be resolved if the time coordinate is periodic, which introduces CTCs as well. According to the technique to construct solutions, Dirac-Misner strings are absent when the bubble equations are solved, see (2.4), while the geometry is free of general CTCs when the quartic invariant function is positive everywhere, (3.7). The problem arises because it is not known how the parameters that specify the solution must be chosen in order to satisfy these two types of constraints. For this reason, the solution generating technique of [12,13] is rarely directly applied to find explicit solutions, being effective only for simplified configurations with few centers or with very special relations among the specifying parameters, see [33,34] and references therein. More general solutions have been found by making use of more sophisticated tools, like the merging of several few-center solutions, [35,36], or the link between three-charge supertube configurations and five-dimensional microstate geometries on a Gibbons-Hawking base, [37][38][39]. Superstrata solutions, which belong to a different class of smooth horizonless solutions of six-dimensional supergravity, deserve a special mention, as they might reproduce the degeneracy of microstates of general three-charge black holes, [40,41].
In order to understand the situation better, it is convenient to discuss the origin of these pathologies. Timelike supersymmetric solutions of five-dimensional supergravity have a metric of conformastationary form, (1.1) Here h mn dx m dx n is a four-dimensional hyperKähler metric (usually a Gibbons-Hawking space [42,43]) known as the base space, while f and ω are respectively a function and a 1-form defined on this base space 3 . The 1-form ω must transform as a tensor under coordinate transformations on the base space, since otherwise the hypersurfaces defined by constant values of the coordinate t would not be Cauchy surfaces. As ω is specified by a differential equation, this just means that we need to satisfy the corresponding integrability condition everywhere. This integrability condition becomes a set of algebraic relations known as the bubble equations. From a physical perspective, this phenomena is related to the frame-dragging generated by the interactions between electric and magnetic sources. These sources have Dirac string singularities, and their elimination from any influence in electromagnetic interactions requires imposing charge quantization. In a similar manner, we can think of the bubble equations as the conditions that require the frame-dragging is also invisible to those string singularities 4 . These constraints relate the charge parameters with the sizes of the non-contractible 2-cycles, and are typically interpreted as restrictions for the latter.
On the other side, the above condition is necessary but not sufficient to ensure t is a global time coordinate. Besides, it is necessary that any hypersurface defined by a constant value of t is a Riemannian manifold with timelike normal vector, so that there are no CTCs. For these solutions, this requires the quartic invariant function to be positive. In the context of BPS microstate geometries, this can be rephrased in terms of the signs of charge and energy densities at separated locations. The existence of stationary multicenter supersymmetric solutions is typically related to the cancellation of attractive and repulsive (gravitational and electromagnetic) interactions 5 . If these cancellations cease to take place the configuration is not truly supersymmetric, and this is reflected in the form of CTCs when trying to describe the solution as such. Hence, the problem is to find configurations for which all charge densities are of the same sign. However, the relation between the parameters specifying the solution and the charge densities, which arise from the interactions of magnetic fluxes threading non-contractible cycles, is rather complex. This is the reason why the solution of this problem has remained unclear.

Results and plan of the paper
In this article we propose a systematic method to solve the CTCs problem that can be used to find all five-dimensional BPS microstate geometries on a Gibbons-Hawking hyperKähler space. It can be summarized as follows: 1. The bubble equations are non-linear and hard to solve if the locations of the centers are taken as the unknowns. However, those can be rewritten as a simple system of linear equations by choosing a different set of unknown variables: the magnetic fluxes. The bubble equations become for some symmetric matrix M.
2. We conjecture that any solution satisfying the bubble equations is free of CTCs if and only if all the eigenvalues of the matrix M are positive.
When trying to build generic microstate geometries, all parameters specifying the solution can be treated in an equal footing. Therefore, there is no reason to consider the charge parameters more fundamental than the size of the bubbles. We begin with a description of the parameter space in section 2. Then, we consider the CTCs problem in section 3. In particular, in section 3.1 we rewrite the bubble equations as a linear system with the same number of equations than variables, while in section 3.2 we expose our conjecture and discuss evidence in its support. In section 3.3 we discuss the application of our method to the construction of scaling solutions, describing how the introduction of non-Abelian fields strongly enriches the spectrum of this type of solutions. Afterwards, in section 4, we put our method in practice and describe some solutions with properties and characteristics previously absent in the literature. For instance, as a striking result, we are able to find smooth horizonless five-dimensional solutions with arbitrarily small angular momentum. Those had not been discovered so far and they were even thought to be nonexistent. We also describe some solutions with the centers lying on a circle or a line whose parameters can be specified with complete accuracy (to the best of our knowledge, all the explicit microstate geometries with several centers known so far can only be obtained approximately). Last but not least, we find a solution with 50 centers that contains more than a thousand 2-cycles, which inspired the title of this article. We conclude in section 5 making some final comments and proposing future lines of research to exploit the possibilities that our method of construction offers. The appendices contain the details about the theory and the solution generating technique we use.

The parameter space and its restrictions
This section is devoted to the description of the space of parameters of supersymmetric five-dimensional microstate geometries and the restrictions that physical configurations must fulfill.
We work in the theory of five-dimensional N = 1 Super-Einstein-Yang-Mills (SEYM) supergravity 6 . The bosonic field content of this theory is given by the metric, three Abelian vector multiplets and a SU(2) triplet of non-Abelian vector multiplets. If the latter are truncated, the STU model of supergravity is recovered, the theory in which five-dimensional Abelian microstate geometries are constructed. Our scheme, therefore, is completely general and accommodates both classes of solutions, Abelian and non-Abelian. The appendices contain a workable description of the theory and how this type of solutions are constructed.
The theory of SEYM supergravity can be embedded in string theory, as recently shown in [10,29,30]. It emerges from the toroidal compactification of heterotic supergravity, followed by a convenient truncation that eliminates half of the original supercharges.
Microstate geometries are completely specified by a set of harmonic functions in three-dimensional Euclidean space 7 and some restrictions that strongly constrain the parameter space. Let us first introduce the parameter space and discuss the constraints afterwards. We distinguish the harmonic functions in the Abelian sector We refer to the n poles of the harmonic functions as centers, so r a ≡ | x − x a | is the Euclidean distance from the a th center. The first function, H, plays a special role. It determines the geometry of a four-dimensional Gibbons-Hawking ambipolar space [2,47], To describe asymptotically flat solutions we need to recover four-dimensional Euclidean geometry in the base space in that limit. Hence, we shall impose a q a = 1. On the other hand, regularity at the centers demands that the Gibbons-Hawking charges q a are integer numbers, and therefore some of them must be negative. The horizonless condition and the regularity of the metric at the centers require that the parameters l i a , m a and σ a are given by a certain combination of q a , k i a and other constants, which are specified in appendix A.3. Moreover, asymptotic flatness also fixes the value of m 0 and imposes one constraint on the product of l 0 0 , l 1 0 and l 2 0 . The two remaining degrees of freedom in these constants are related to the moduli of the solution; that is, to the asymptotic value of the two Abelian scalars of the theory. Also, as discussed in the appendix, only (n − 1) of the k i a parameters are physical, as there is one redundant degree of freedom associated to gauge transformations of the vectors.
Therefore, asymptotically flat horizonless configurations are specified by 4(n − 1) charge parameters (k i a and q a ), the 2 moduli parameters, the n non-Abelian hair parameters λ a and, of course, the coordinates of the centers, which add 3(n − 2) degrees of freedom. In total, the parameter space of the solutions is 8(n − 1)-dimensional.
Not every point in the parameter space, however, yields a physically sensible solution. Actually, it is most frequent that a random election of such point gives a solution with closed timelike curves (CTCs) and Dirac-Misner string singularities connecting some of the centers. The absence of Dirac-Misner strings is achieved by imposing the so-called bubble equations, b =a the non-Abelian gauge coupling constant is denoted by g and r ab is the distance separating the centers a and b. The combinations Π i ab are the magnetic fluxes of the i th Abelian vector threading the non-contractible 2-cycle defined by the two centers a and b. Notice that only (n − 1) equations are independent, as the sum in the index a that labels the n equations yields a trivial identity. Typically, solving the bubble equations constitutes a very hard step when constructing explicit microstate geometries. This is because, traditionally, those have been understood as equations for the variables r ab , which have to be solved in terms of the independent charge parameters k i a and q a and the moduli 8 . Then, after solving the system, usually by numerical methods, one finds that the obtained values of r ab rarely represent the distances between a collection of points, as they should all be real, positive numbers satisfying the triangle inequalities r ac ≤ r ab + r bc for all a, b, c. To construct explicit microstate geometries, one usually relies on further restrictions that reduce the number of independent parameters but make it easier for the bubble equations to admit proper solutions.
However, there seems to be no reason for considering the charge parameters more fundamental than the locations of the centers not only in the bubble equations, but also in the complete description of a particular microstate geometry. On one side, asymptotically the system looks like a black hole and its main characteristics are determined by the charge parameters and the moduli. On the other side, the existence of well-separated centers is of the utmost importance for resolving the horizon, and their locations are responsible for the distinction between different microstates associated to the same black hole. In this paper we show that the bubble equations can be solved analytically in full generality, with complete access to the whole parameter space of regular solutions, by considering the location of the centers among the independent variables.
As far as general CTCs are concerned, so far there has been no known analytically solvable restriction to guarantee their absence. Usually this has to be checked by evaluating numerically the positivity of the quartic invariant of the solution, once the bubble equations have been solved and all parameters are already specified. In the next section we propose an algebraic condition on the space of parameters that allows us to distinguish whether or not the solution has CTCs, without making use of the numerical analysis of a function.

Solving the bubble equations analytically
As we outlined in the previous section, the bubble equations have been traditionally solved for the distances between the centers using numerical methods. This, in turn, makes the task of constructing explicit microstate geometries complex and typically limits the regions of the parameter space that can be accessed. Here we use a different approach to address the problem that allows for the analytic resolution of the bubble equations in full generality.
Since the number of independent magnetic fluxes associated to a given vector is the same as the number of independent equations, (n − 1), and those appear linearly in the system, it is reasonable to take them as the unknown variables for which the system is solved. As there are three different Abelian vectors, there are three possible ways in which we can write the system. In this section we write the explicit expressions when the 2-fluxes are taken as the unknowns, although equivalent relations can be readily obtained for the 0-and the 1-fluxes. If we define and it is straightforward to see that (2.4) can be rewritten as This is a system of n equations, but the sum of all of them is trivially satisfied 9 . Therefore, we can directly eliminate one of the equations, which is chosen to be the first one. We define the variables that will play the role of unknowns in the system of equations as follows Then, the rest of the 2-fluxes can be easily written in terms of these quantities as For this variables, we get a system of (n − 1) linear equations where the components of the matrix M 2 and the vector B 2 are given by Thus, in our scheme the bubble equations can be solved by standard methods for an arbitrarily large number of centers 10 . This is, of course, if the solution exists. Let us go back some steps to understand this issue. We explained in the previous section that asymptotically flat horizonless configurations are determined by 8(n − 1) parameters. To become regular microstate geometries, these configurations need to satisfy (n − 1) independent algebraic constraints known as bubble equations. This means that there are 9 Notice that the fluxes Π i ab are antisymmetric in their indices, while the matrix α 2 ab is defined to be symmetric. Also, we have that a β 2 a = 0. 10 Provided computational resources are unlimited. at most 7(n − 1) independent parameters. In this section we have shown a way in which the independent parameters can be chosen in order to solve the bubble equations in full generality. But, still, there are special values of the independent parameters for which the bubble equations do not admit a solution: the values for which the determinant of the coefficient matrix M 2 is zero.
To get some intuition about this, let us suppose for a moment that the 7(n − 1) independent parameters are continuous variables. Then, the condition |M| = 0 defines a codimension one hypersurface in the parameter space, to which we refer as a wall. Although walls represent a very small region of the total space, we strongly believe that their presence is highly relevant. We plan to study this issue in detail in a forthcoming publication [48].

Absence of CTCs: an algebraic criterion
At this stage, there is one last restriction that physically sensible configurations must satisfy: the spacetime cannot contain closed timelike curves. As we already mentioned, this problem is translated to the positivity of a function, the quartic invariant where we use the combinations (see the appendices for more information) The parameters are chosen such that Z I and ω 5 do not diverge at the centers. Moreover, when the bubble equations are satisfied ω 5 vanishes at the centers. Asymptotically, Z i (just the Abelian sector) go to the positive constant l i 0 while ω 5 goes to zero. In short, this means that I 4 is dominated by the first factor both near the centers and far from them. Motivated by this observation, in this article we claim that the positivity of the quartic invariant is guaranteed if the first term is strictly positive, which implies that the term with ω 5 is irrelevant for the study of CTCs, even at intermediate regions, unless the first term in I 4 vanishes 11 . As Z i changes sign when H does, the inequality (3.9) can be converted into a collection of simpler inequations In terms of the parameters, these combinations of functions can be written as are the coordinates of the unit vector n a ≡ x− xa ra (recall that α = 3, 4, 5). Evaluating the Abelian functions Z i H at the centers we obtain and, from the first set of inequalities in (3.10), we find that the combination of parameters inside the brackets must be positive for all centers. Notice that in these expressions the purely Abelian limit is effectively recovered by taking g → ∞, and that in this limit the last inequality in (3.10) is trivial. At first sight, it is noteworthy that these combinations of parameters look very similar to the elements in the diagonal of the coefficient matrices But, however, what it is truly remarkable is that the positivity of the elements in the diagonal of those matrices is sufficient to ensure the positivity of the divergences of the functions Z i H at the centers 12 . This suggests that there might be a relation between the properties of the linear system of bubble equations and the absence of CTCs. To understand this relation better, it is convenient to consider simple configurations.
The study of two-center, purely Abelian microstate geometries provides a great deal of insight in this problem. In this case the bubble equation is where X 2 1 = Π 2 12 . This configuration will not contain CTCs if for i = 0, 1, 2. Since r 1 and r 2 are positive numbers, we just need to impose the positivity of the function inside the bracket. Without loss of generality, we can take q 1 to be positive and q 2 to be negative, with q 1 + q 2 = 1. Then, the function (q 1 r 2 + q 2 r 1 ) is bounded from below by the number q 2 r 12 , so we can write (3.16) In particular, for Z 2 H we need the combination (l 2 0 q 2 r 12 − q 1 q 2 Π 0 12 Π 1 12 ) to be positive. This combination looks very similar to the eigenvalue of the matrix M 2 (in this case a 1 × 1 matrix) that defines the bubble equation (3.14). Actually, as q 1 q 2 < q 2 , if the eigenvalue is positive then the above combination is positive and therefore Z 2 H > 0.
In purely Abelian configurations, the three different type of fluxes appear in the bubble equations exactly in the same manner. Then, the bubble equations can be solved for the 0-fluxes or the 1-fluxes, defining two more matrices M 0 and M 1 respectively. Following the same reasoning as in the previous paragraph, we conclude that if the eigenvalues of the three matrices M i are positive then Z i H > 0 and therefore the configuration is free of CTCs. Of course, M 0 and M 1 depend on the 2-fluxes, which play the role of unknowns in the bubble equations. So, naively, it might seem that this criterion for identifying CTCs is of little help in practice, as we would like to dispose of a set of conditions on the parameter space only. However, using the bubble equations, it is immediate to prove that the eigenvalues of the three matrices are of the same sign.
For example, for M 0 we have that its eigenvalue is positive if Π 1 12 Π 2 12 − l 0 0 r 12 > 0. This combination can be rewritten using the bubble equations (3.14) as Since the numerator on the right hand side is positive, the left hand side has the same sign as the denominator, which proves our point. At this stage we have shown that if the eigenvalue of M 2 is positive, then we have Z i H > 0 and, according to our claim at the beginning of this section, the quartic invariant is positive and the solution does not contain CTCs. But, what would happen if the eigenvalue were negative? According to the preceding discussion it might be possible to have Z 2 H > 0 even when the eigenvalue is not positive. That is, it is possible to choose the parameters such that Remarkably, it turns out that these inequalities imply that Z 0 H and Z 1 H eventually become negative! For instance, we can check it explicitly for the latter (both proofs are identical). In first place, notice that the first inequality requires Π 0 12 Π 1 12 > 0. Using the bubble equations we can write Z 1 H = l 0 0 −q 1 q 2 r 12 (Π 0 12 ) 2 + (Π 0 12 Π 1 12 − l 2 0 r 12 ) (q 1 r 2 + q 2 r 1 ) − Π 0 12 Π 1 12 q 1 q 2 r 12 r 1 r 2 (Π 0 12 Π 1 12 − l 2 0 r 12 ) . (3.19) This function is positive asymptotically, but negative at r 2 = 0, r 1 = r 12 whenever (Π 0 12 Π 1 12 − l 2 0 r 12 ) < 0. In summary, we have proved that Abelian two-center microstate geometries do not contain CTCs if and only if the eigenvalue of M 2 is positive. The same conclusion can be obtained for non-Abelian two-center microstate geometries. In this case the proof is similar, although it is more technical and not particularly illuminating. In view of this result and based on the observations exposed at the beginning of this section for multicenter configurations, we make the following proposal.
Conjecture: Five-dimensional microstate geometries on a Gibbons-Hawking base, with or without non-Abelian fields, do not contain CTCs if and only if the coefficient matrix of the bubble equations is positive-definite.
If our conjecture is true, the construction of five-dimensional microstate geometries without CTCs will no longer require the numerical evaluation of any function on R 3 , but it will be sufficient to check an algebraic property of a matrix. This would extraordinarily simplify the problem of describing and studying this type of supergravity solutions, giving rise to a new plethora of smooth geometries.
To close this section let us mention that, although we have not been able to prove our conjecture in full generality, we have tested its validity with a large number of multicenter configurations. We have analyzed more than 100,000 solutions with pseudorandom parameters, finding a perfect agreement with our proposal.

Contractible clusters and scaling solutions
Scaling microstate geometries can be defined as solutions for which the centers can be brought arbitrarily close without significantly modifying the asymptotic charges, [35]. As the centers approach each other, the geometry of the system does not only reproduce the asymptotic charges of an extremal black hole, but also starts to look like one at intermediate regions. In the zero-size limit all centers merge, a horizon is developed and the configuration becomes a black hole. But right before reaching the black hole limit, the solution is still horizonless and partially reproduces the throat that characterizes the near-horizon geometry of a black hole, capping off smoothly at some finite depth, although arbitrarily large. It is for this reason that scaling microstate geometries are expected to correspond to the classical description of individual microstates of a black hole, [34]. We now show that the formalism we have presented is extraordinarily wellsuited to describe and study scaling solutions.
We consider scaling solutions that preserve the shape of the distribution 13 . This means that we can write the distances between centers as r ab = µd ab , (3.20) where d ab remain constant in the scaling process, which is controlled by varying µ to arbitrarily small positive numbers. We can define the following quantities, which are manifestly invariant during the scaling process. Then, upon substitution of (3.20) in (3.5), the bubble equations can be written as If compared with the original equation, we have µM 2 =M 2 + µM 2 for the coefficient matrix and µB 2 =B 2 + µB 2 for the column vector. In terms of the parameters, we havē The bubble equations as written in (3.23) are well defined even in the zero-size limit µ = 0, where they cease to have a physical meaning. Scaling solutions can be identified as those for which one can take the zero-size limit through a continuous transformation and still obtain a valid solution of the bubble equations, without any of the asymptotic charges becoming zero. The existence of this limit cannot be taken for granted. Actually, for purely Abelian solutions one always hasB 2 = 0 and the bubble equations become a homogeneous system in the zero-size limit. Then, there are non-trivial solutions (that is, solutions with X 2 a = 0 for some a) only if the determinant of the corresponding coefficient matrix,M 2 , vanishes. In other words, purely Abelian scaling solutions necessarily flow to special points of the parameter space when taking the zero-size limit. It is for this reason that many Abelian solutions cannot be scaled without some of the asymptotic charges becoming zero.
The situation is completely different when non-Abelian fields are also considered. In this situation we haveB 2 = 0 and the system is still inhomogeneous in the zero-size limit. This means that non-Abelian microstate geometries can typically be scaled 14 . Actually, from our point of view this is the most important contribution that non-Abelian fields bring to the "microstate geometries program". Typically these fields enter the solutions modifying the spacetime metric, the asymptotic charges and the size of the bubbles very softly; in most cases these physical properties are practically preserved after introducing the non-Abelian distortion. A reason for it is that, from the 10-dimensional Heterotic supergravity perspective, these fields appear at first order in α ′ . However, this distortion becomes critical when we take the zero-size limit, enlarging the spectrum of scaling solutions.

Exact solutions on lines and circles
There is one issue that might worry some of the readers: the fluxes that solve the equations are in general irrational numbers. This is because the distances between a collection of points in three dimensions are usually irrational. The asymptotic charges, which are expected to be quantized when these solutions are properly interpreted within the context of string theory, are directly related to the fluxes, see equations (A.41). Therefore, we would prefer the fluxes to be given by, at least, rational quantities. This fact can be seen as a motivation for the traditional approach to solve the bubble equations, in which the fluxes are guaranteed to be rational.
However, as the method we present here is completely general, it must also describe all microstate geometries with rational fluxes. These classes of solutions are obtained if the centers are chosen such that all the relative distances between them are rational numbers. An obvious possibility is to take all centers laying on a line, so the solution is axisymmetric. This kind of microstate geometries are easier to build and study, and most of the explicit constructions known are axisymmetric. As a first application of the aforementioned procedure we show in Table 1 a first example of a 5-center solution 15 . Motivated by the recent results of [38,39], the locations of the centers present a hierarchical structure; i.e. the values of the relative distances vary between different orders of magnitude. As argued in those articles, such structure potentially favors finding solutions whose angular momentum is far from maximal. The 5 center solution described here has negligible angular momentum, see Table 3, and constitutes the first five-dimensional microstate geometry that exhibits this property. As shown in Table 1, the solution can be scaled without any problem introducing a conformal factor µ for the coordinates of the centers.
In order to go beyond axisymmetry, we now define a very interesting, arbitrarily large set of points with rational relative distances lying on a circle. The result is based on the original proof of the Erdős-Anning theorem, [49], that states that any infinite collection of points can have integral distances only if these are aligned. However, as we are about to see, it is possible to have an infinite set of points with mutual rational distances. First, pick a circle with unit diameter centered at the origin of coordinates. A primitive Pythagorean triple 16 P i is composed of three coprime natural numbers a i , b i and c i such that a 2 i + b 2 i = c 2 i . The triple P i defines a right triangle whose hypotenuse 15 Our criterion to find a solution free of CTCs is to systematically look for parameters for which the coefficient matrix M 2 is positive definite. In any case, we have also checked numerically the absence of CTCs for all the examples displayed in this article. 16 Primitive Pythagorean triples are generated through Euclid's formula,   . Then, the triangle defines three points at rational distances on the unit diameter circle.
In virtue of Ptolemy's theorem, any other point with rational distances to the pair of points (− 1 2 , 0) and ( 1 2 , 0) is necessarily separated by a rational distance from (− 1 2 + l i , h i ) as well. In particular, this means that we can use the same Pythagorean triple P i to find three more valid points: Any additional primitive triple can add up to four points more to the set in the obvious manner. Moreover, since the four points associated to a triple define two new diameters of the circle those can also be used as hypotenuses, providing new possibilities to enlarge the collection. The procedure can be prolonged without end, defining a dense set of points on the circle. Finally, the value of the radius can be set to any rational number µ. Therefore, these configurations are very well-suited to build scaling solutions. Table 1 contains a couple of examples of microstate geometries with 6 and 10 centers lying on a circle. Once again, a hierarchic structure has been imposed by making use of Pythagorean triples for which a i ≪ b i .
Some of these solutions have more bubbles than any previously known example and, furthermore, can be specified with exact accuracy. Increasing the number of centers is feasible, although computationally demanding. On one side, solving a linear system of equations is a problem of complexity P of order O(n 3 ). That is, the time required to solve the bubble equations approximately scales with the cube of the number of centers. On the other side, increasing the number of centers seems to favor the appearance of CTCs, so it is more likely that random elections of the parameters yield to unphysical solutions. According to our conjecture, this is just a natural consequence; as the coefficient matrix becomes bigger it is harder and harder to find the parameters such that all its eigenvalues are positive. Nevertheless, we have been able to describe solutions with a very large number of centers by focusing on regions of the parameter space that seem to favor the coefficient matrix is positive definite 17 . A particular example of an axisymmetric 50-center solution is given in Table 2.
Another interesting issue is that the parameters λ a that determine the non-Abelian seed functions seems to play a subleading role in the CTCs problem. In fact, once a solution without CTCs has been found, we can generate as many as we want by modifying the non-Abelian parameters 18 , as long as all of them remain positive. This for any pair of coprime integers m i > n i > 0. 17 Our main guides are to impose the presence of hierarchical structures and to take all k 0,1 a coefficients of the same sign. 18 We have checked this by taking arbitrary values of the non-Abelian parameters in a finite range. However, based on how these parameters appear in the non-Abelian seed functions, we think that one  Table 3: Asymptotic charges and angular momenta of the solutions for µ = 1.
is the reason why we have not specified any particular values in the tables. Therefore, the inclusion of non-Abelian fields not only makes it easier to find scaling solutions, but also enlarges the number of solutions with a given set of asymptotic charges, as expected [11].

General locations
It is comforting that we can use our method to describe, for the first time, many-center five-dimensional microstate geometries with an exact accuracy. Nevertheless, we are also interested in the possibility of describing more general solutions with centers at general locations, which includes the possibility of having irrational distances. In practice, this implies that the bubble equations must be solved approximately. We distinguish two possibilities: • Approximate fluxes. The first possibility is to solve the bubble equations for the fluxes. In this case these will be given by irrational numbers and, as discussed at the beginning of the preceding subsection, this can be considered inconvenient because they are related to the asymptotic charges. Then, one valid option is to round the fluxes such that the charges take valid values, and admit that the solution is only specified approximately. This can be a useful possibility when one is interested in studying generic properties of the solutions, rather than in performing a very precise analysis.
• Approximate locations. The procedure that we follow to avoid having approximate fluxes can be summarized as follows. In a first step, we choose our favorite distribution of centers and solve the bubble equations for the fluxes. Then, we round the values and solve again the equations for the distances between the centers, using the fluxes as input data now. We expect the distances not to change too much for small enough changes of the fluxes. Once we know the distances, we have to place the centers in the tridimensional space R 3 . Unfortunately, this can take any positive value for them and CTCs will not appear.
can only be done in full generality for four centers at most, so in configurations with more centers one has to impose restrictions in the locations when solving the bubble equations the second time (for example, one can consider axisymmetric configurations only).

Final comments and further directions
In this article we have presented an efficient method to construct general five-dimensional supersymmetric microstate geometries on a Gibbons-Hawking base. We have conjectured that the CTCs problem can be solved through the evaluation of a simple algebraic relation without the need to numerically evaluate the quartic invariant function. We have accompanied the exposition with a few explicit solutions, which were found making use of our method. These solutions exhibit novel properties in their class, such as arbitrarily small angular momentum or large number of centers, being some of them not axisymmetric distributions. This not only reveals that the spectrum of smooth microstate geometries on a Gibbons-Hawking base is actually very rich, but also that it is possible to find and study this type of solutions.
In particular, this method can be used to describe simple five-dimensional smooth, horizonless scaling solutions with the asymptotic charges of a D1-D5-P black hole without angular momentum. It would be interesting to study general properties of these geometries and compare them with those of a black hole; their geodesics, how they interact with incoming particles or their stability under perturbations. So far, this type of analysis has only been performed for two-charge microstate geometries or three-charge geometries with atypical asymptotic charges and angular momentum 19 [51][52][53][54].
As the procedure described is systematic, it would be very interesting to apply the tools developed in [38] to perform macroscopic explorations of the parameter space. For instance, in [39] this type of analysis has been successfully used to study generic fourcenter axisymmetric configurations, which can be constructed systematically, showing that those can only reproduce solutions with an angular momentum larger than 80% of the cosmic censorship bound when they are smooth in five dimensions, while it is possible to find solutions with arbitrarily small angular momentum if the configuration contains a supertube (which are smooth only in six dimensions or more). Making use of the method that we propose here, we can access the full space of parameters of multicenter, not necessarily axisymmetric, solutions. Work along these lines is in progress [48]. model is characterized by a constant symmetric tensor with the following non-vanishing components The first three vectors, A 0 , A 1 and A 2 are Abelian, while A 3 , A 4 and A 5 correspond to a SU(2) triplet. For convenience, we separate the range of values of the indices I, J in two sectors: the Abelian sector i, j = 0, 1, 2 and the non-Abelian sector α, β = 3, 4, 5. Therefore, if the latter sector is truncated we immediately recover the STU model of supergravity, with C ijk = |ε ijk |/6, the theory in which five-dimensional BPS microstate geometries are naturally described.
It is convenient to introduce (n v + 1) functions of the physical scalars h I (φ x ), which are subjected to the following constraint The functions h I can be interpreted as coordinates in a (n v + 1)-dimensional ambient space, so the above constraint defines a codimension 1 hypersurface parametrized by the scalars φ x known as the scalar manifold. In the ST [2, 6] model, a convenient parametrization is We also define The matrix a IJ is the metric in the ambient space, and the σ-model metric g xy in the action is given by the pullback of a IJ on the hypersurface. They are both determined by the election of C IJK as We only consider symmetric scalar manifolds, for which The field strength and covariant derivatives are defined in the usual manner, We consider the gauge group SU (2) with structure constants f IJ K = ε IJ K , with the understanding that they vanish whenever any of the indices takes values in the Abelian sector. The covariant derivatives of the functions of the scalars are

A.2 Timelike supersymmetric solutions with one isometry
Supersymmetric solutions of this theory admit a Killing vector of non-negative norm.
In adapted coordinates the metric and vectors are independent of the time coordinate, see [24], and can be written as where dŝ 2 is a hyperKähler metric. The equations of motion are reduced to the following BPS system of differential equations on this four-dimensional space, where ⋆ 4 is the Hodge dual in the hyperKähler space,F I is the field strength of the vectorÂ I andD is the covariant derivative with connectionÂ. In these equations we have introduced the functions Z I ≡ h I /f , so the metric function f is conveniently obtained as by virtue of equation (A.7).
The system of BPS equations is non-linear due to the presence of non-Abelian fields, although the three equations could be solved independently in the order they have been presented. However, it is possible to further simplify the system under the assumption that the solution admits a spacelike isometry [27], in a way that reduces the problem to a set of equations in three dimensional Euclidean space. First, consider the following decompositions where ψ is the coordinate adapted to the spatial isometry. When these expressions are substituted in the BPS system of equations, we obtain the following simplified system of differential equations and algebraic relations whereF I is the field strength of the vectorȂ I andD is the covariant derivative with connectionȂ.
The Abelian functions H, M, Φ i and L i are just harmonic functions in E 3 , and the 1-forms χ andȂ i are completely determined from those functions. In the non-Abelian sector, equations (A.21) are non-linear and must be solved simultaneously for Φ α and A α , which make their presence in (A.22). The construction of non-Abelian microstate geometries requires finding a multicenter solution to these equations. The only known example of such solution is the multicolored dyon, found by one of us in [11], which we review in appendix A.3. Last but not least, we have the differential equation (A.23), whose integrability condition will give rise to the bubble equations.
Notice that these solutions are left invariant under the following transformations of the harmonic functions generated by the parameters g i , whose sole effect is a gauge transformation of the Abelian vectors,

A.3 Microstate geometries in a nutshell
The previous section describes a procedure to find supersymmetric solutions of SEYM theories in terms of a set of three-dimensional seed functions: H, M, Φ I and L I . As we already commented, those in the Abelian sector are just multicenter harmonic functions with poles in a collection of n points located at (x 1 a , x 2 a , x This condition is solved everywhere if Q is a harmonic function with the poles at the same locations than P . Therefore, the complete non-Abelian multicolored dyon is specified by two harmonic functions 20 P = 1 + The term T ab appears due to the presence of non-Abelian fields, that alter the value of the parameters l 0 a when compared with purely Abelian configurations. The i-fluxes threading the non-contractible 2-cycles ∆ ab defined by any path connecting two centers x a and x b behave effectively as sources of electric charge and mass. When all the bubble equations are satisfied, the solutions are regular at the centers and do not present Dirac-Misner string singularities, which otherwise could only be removed by compactifying the time direction. The last restriction for the construction of physically sensible microstate geometries comes from demanding that the solution does not contain closed timelike curves (CTCs). The metric can be rewritten in the following manner where I 4 is the quartic invariant, defined as Therefore, a general restriction that must be satisfied in order to avoid CTCs is the positivity of the quartic invariant

A.4 Asymptotic charges
The electric asymptotic charge of each Abelian vector can be readily obtained from the asymptotic expansion of the associated warp factor, see [34], This can be easily seen from the fact that, asymptotically, A i t,∞ ∼ Z −1 i . The electric charges are In a similar manner, the two angular momenta can be read from the term in dψ at the asymptotic expansion of ω [34], whose contribution comes entirely from the function ω 5 , obtaining such that If all the centers were placed at the same location, the solution would describe a black hole with the same asymptotic charges (with J L = 0) and an event horizon whose area would be given by It is convenient to define the entropy parameter H of a microstate geometry as whose value indicates how far from maximal rotation the represented black hole is.