Non-BPS Floating Branes and Bubbling Geometries

We derive a non-BPS linear ansatz using the charged Weyl formalism in string and M-theory backgrounds. Generic solutions are static and axially-symmetric with an arbitrary number of non-BPS sources corresponding to various brane, momentum and KKm charges. Regular sources are either four-charge non-extremal black holes or smooth non-BPS bubbles. We construct several families such as chains of non-extremal black holes or smooth non-BPS bubbling geometries and study their physics. The smooth horizonless geometries can have the same mass and charges as non-extremal black holes. Furthermore, we find examples that scale towards the four-charge BPS black hole when the non-BPS parameters are taken to be small, but the horizon is smoothly resolved by adding a small amount of non-extremality.

In [1][2][3], the Weyl formalism has been generalized to a "charged" Weyl formalism for Einstein-Maxwell theories. Solutions are still determined by D−3 harmonic functions. They are sourced by rods that can be now charged under the gauge field(s) and still correspond to either black hole horizon or smooth bubbles. In this system, the sources are set to have the same charge-to-mass ratio. Moreover, as these are non-BPS systems, this ratio can be generically different from 1. In [2,3], a "bottom-up" approach has been adopted to highlight the physics of the new non-BPS solitons, such as chains of smooth charged bubbles in six dimensions or "bubble bag ends". In the present paper, we adopt a top-down perspective by applying the same formalism to string theory backgrounds to derive and explore a non-BPS linear ansatz.

Summary of the results
Our initial construction takes place in M-theory on T 6 ×S 1 . We consider that the gauge field is sourced by three sets of M2 branes that wrap three orthogonal 2-tori inside T 6 . Moreover, we allow for a magnetic KKm vector associated to the S 1 . 1 By applying the charged Weyl formalism, we derive what we call a "non-BPS floating brane ansatz" for the M2-M2-M2-KKm system in M-theory. The solutions are determined in terms of eight harmonic functions. Four are related to the deformations of the T 6 while the other four are associated with the gauge potentials. In this paper, we focus on solutions that are asymptotic to R 1,3 ×T 6 ×S 1 or R 1,4 ×T 6 .
The asymptotically-R 1,3 solutions are static, axisymmetric and induced by an arbitrary number of non-BPS sources on the symmetry axis. A physical source corresponds to a regular coordinate degeneracy of either the time or one of the T 6 ×S 1 directions. If the time degenerates, it corresponds to the horizon of a static non-extremal M2-M2-M2-KKm black holes [38][39][40]. If one of the T 6 directions or the S 1 shrinks, it defines the locus of a M2-KKm bubble or a KKm bubble respectively. The sources are separated by strings with negative tension, or struts, which disappear if they are connected [2]. We explicitly construct solutions that are chains of non-extremal static black holes, separated by struts or by KKm bubbles. In addition, we construct non-BPS bubbling geometries of M2-KKm bubbles on a line. They can have the same charges and mass as static non-extremal M2-M2-M2-KKm black holes, but they are horizonless and smooth. Moreover, we show that they have a BPS limit where they converge to the BPS black hole. Therefore, when they are slightly non-BPS, they are almost indistinguishable from the BPS black hole, develop an AdS 2 throat, and resolve its horizon into non-BPS bubbles by adding a small amount of non-extremality. This is the first example of such a resolution in the non-BPS regime.
We also consider the non-BPS floating brane ansatz in different duality frames such as the D1-D5-P-KKm frame in type IIB. We show that there are types of bubbles that are smooth only in a specific duality frame, and we construct bubbling geometries in the D1-D5 framework.
Solutions that are asymptotic to five-dimensional flat space plus compactification circles can be obtained by considering the S 1 to be the Hopf fiber of a four-dimensional base. The non-BPS sources correspond to either three-charge non-extremal black holes [41][42][43], smooth M2 bubbles or vacuum bubbles. We explicitly construct and study black hole chains and smooth bubbling geometries that are asymptotically-flat in five dimensions.
In section 2, we derive the non-BPS floating brane ansatz by solving Einstein-Maxwell equations with the charged Weyl formalism. We discuss the different regimes and the reduction in five and four dimensions. In section 3, we classify the asymptotically-flat solutions and the regular sources. In the sections 4 and 5, we construct explicit families of non-BPS four-dimensional and five-dimensional solutions, and we discuss their physics.

Non-BPS floating branes in M-theory
We consider M-theory solutions on T 6 =T 2 ×T 2 ×T 2 for which each two-torus can be wrapped by M2 branes. Moreover, we allow one of the other four spatial directions to be an extra S 1 that can carry KKm charges. The dynamics in the bosonic sector of Mtheory is given by where G 11 is the eleven-dimensional Newton constant and F 4 = dA 3 is the four-form field strength of the three-form gauge field. The Einstein-Maxwell equations are (2.2) We parametrize the T 6 by {y a } a=1,.., 6 , and each direction is 2πR ya periodic. The remaining four-dimensional spatial directions will be decomposed into a three-dimensional basis in cylindrical Weyl coordinates (ρ, z, φ) and a U(1) fiber, y 7 . This fiber will correspond either to a compact circle with 2πR y 7 periodicity for asymptotically R 1,3 ×S 1 ×T 6 solutions, or to an angle with 2π periodicity for solutions that asymptote to R 1,4 ×T 6 .

Equations of motion
We derive the equations of motion, obtained from (2.2), for solutions that are static, axisymmetric, and sourced by a Kaluza-Klein magnetic vector and three stacks of M2 branes wrapping the two-tori. Without restriction, we consider the following ansatz that fits the equations: , F 4 =d [T 1 dt ∧ dy 1 ∧ dy 2 + T 2 dt ∧ dy 3 ∧ dy 4 + T 3 dt ∧ dy 5 ∧ dy 6 ] . There are three electric gauge potentials T I induced by the stacks of M2 branes and a magnetic gauge potential H 0 for the KKm vector. We have introduced four warp factors, Z Λ , which couple naturally with each gauge potential. In addition, we have four independent warp factors, W Λ , which are associated with torus deformations. Specifically, W 0 corresponds to a Kähler structure deformation of the T 6 since det(T 6 ) = W 2 0 , while the other W I define complex structure deformations of the internal T 2 . Finally, e 2ν determines the nature of the three-dimensional base. All functions depend only on ρ and z by symmetry. More generically, we use capital greek letters for the label Λ = 0, 1, 2, 3 and capital latin letter for the label I = 1, 2, 3. We introduce the cylindrical Laplacian operator of a flat three-dimensional base for axisymmetric functions: The equations of motion (2.2) can be decomposed into three almost-linear sectors: Torus sector: ∆ log W Λ = 0 , Maxwell sector: Base sector: where S are source functions that depend non-trivially on (W Λ , Z Λ , T I , H 0 ) (A.1). The equations can be treated linearly except the Maxwell sector which is a set of non-linear coupled differential equations.

Non-BPS floating brane ansatz
In [1], a procedure has been found to extract linear closed-form solutions from the above equations without reducing to BPS equations. In the present context, charged Weyl solutions are determined by eight harmonic functions The pairs of scalars (Z I , T I ) and (Z 0 , H 0 ) are given by 2 The integrability is guaranteed by the harmonicity of the functions on the right-hand sides. These equations can be integrated in a case-by-case manner depending on the choice of sources for the harmonic functions. It is important to point out that W Λ are "vacuum" warp factors, as they are not coupled to any gauge potentials, whereas Z Λ are generated and induced by the gauge potentials. However, one can take a neutral limit by sending all gauge potentials to zero while keeping Z Λ non-trivial, which then satisfy the vacuum equation ∆ log Z Λ = 0. The neutral limit is given by with e b Λ /a Λ and a Λ L Λ held fixed. (2.11) It shows that the present ansatz is generically in the non-BPS regime of the M2-M2-M2-KKm system. Moreover, at the other side of the parameter space, (2.12) the solutions converge to the linear branch Z Λ = L Λ + b Λ /a Λ . As we will see in the next section, this leads to the BPS equations for the M2-M2-M2-KKm system. Therefore, the ansatz admits a BPS limit considering the above transformation.
It is remarkable that one can deviate significantly from the BPS regime while maintaining the linearity of the Einstein-Maxwell equations. Moreover, it only changes the warp factors from a linear function (BPS regime) to a sinh function (non-BPS regime) in terms of harmonic functions. As we will see later, the price to pay for having a linear ansatz will be to use non-BPS sources that have a fixed charge-to-mass ratio, but not necessarily 1.

BPS regime
We consider that the two-tori and six-torus are rigid, W Λ = 1. Moreover, we take the limit (2.12) for the scalars (Z Λ , H 0 , T I ) (2.7), which leads to L Λ = Z Λ − b Λ /a Λ . Therefore, the solutions can be better written such that Z Λ are harmonic functions and the gauge potentials are determined by More generically, the scalars are given by where G (Λ) can be freely chosen among the following five generating functions of one variable: However, we did not find regular solutions for the branches = 2, 4, 5 (see [1][2][3] for more details). Therefore, we consider only the branch = 1 from which the function = 3 can be obtained with (2.12).
We recognize the BPS equations of motion one can obtain from supersymmetric static M2-M2-M2-KKm solutions. With such a choice, ν = 0 (2.8), and the base is flat R 3 as expected.
Physical solutions are necessarily sourced by point particles on the z-axis: 14) The sources at (ρ = 0, z = z i ) are either static extremal four-charge black holes if q Therefore, the linear non-BPS floating-brane ansatz contains the BPS floating brane ansatz for the static M2-M2-M2-KKm system.

Non-BPS regime
Generic non-BPS solutions are constructed for non-zero (a Λ , b Λ ) in (2.7). One can check that sourcing the harmonic functions by point sources lead to naked singularities. We therefore consider that the harmonic functions are sourced by rods, that is segment sources.  In this section, we establish our formalism and highlight the basic ingredients that can be added linearly through the ansatz. We save the construction of the solutions for later.
We consider n distinct rods of length M i > 0 along the z-axis centered around z = z i . Without loss of generality, we can order them as z i < z j for i < j. Our conventions are illustrated in Fig.1. The coordinates of the rod endpoints on the z-axis and the distance to them are given by Moreover, we introduce the following generating functions They satisfy the equations The functions log can be used as the building blocks for the eight harmonic functions: where the constants, G 19) and ν is induced by the generic functions ν ij : (2.20) As we will see precisely later, the weights of the torus warp factors, G i , are not involved in any of the conserved charges of the solutions as they are deformations of the T 6 . However, the four P (Λ) i are associated to the four charges carried by the i th rod, 3 We have normalized the charges in units of volume: where T 4 I is the four-torus orthogonal to the I th stack of M2 branes and S 1 is the y7-fiber.
Moreover, we will have to add a semi-infinite rod to construct asymptotically-R 1,4 ×T 6 solutions as we will see in detail in the section 3.2. It can be obtained from the above expressions by sending one of the endpoints to infinity.
Finally, the non-BPS solutions have a very large phase space determined by the geometry of the rod sources, the eight weights associated with each and the four pairs of gauge field parameters (a Λ , b Λ ). However, the parameters will be constrained to have physical solutions as we will discuss in section 3.

Profile in five and four dimensions
In this section, we describe the ansatz in five and four dimensions after reduction along the T 6 and the S 1 . We will keep the least number of degrees of freedom that encompasses the ansatz. Note that we do not expect to obtain the Lagrangian of the STU model as it is usually the case for BPS multicenter solutions [18][19][20][21][22][23][24] and some of their non-BPS extensions [44][45][46][47]. Indeed, we are not compactifying along a rigid T 6 since it has nontrivial deformations due to the W Λ (2.3).
We will therefore apply a series of KK reductions from M-theory using the generic rules of [48,49]. The reduction rules and the derivation are detailed in the appendix B.1.

Reduction to five dimensions
We first perform a series of KK reductions to five dimensions by compactifying along {y i } i=1,..,6 . For that purpose, we suitably define seven scalars (X I , Φ Λ ) and three one-form gauge fields such that the M-theory metric and gauge field (2.3) is now given as (2.23) where F (I) = dA (I) are two-form field strengths. The reduced five-dimensional action is (see the appendix B.1) where G 5 = G 11 (2π) 6 6 a=1 Ry a is the five-dimensional Newton's constant, is the Hodge star operator in five dimensions, and L 5D ST U is the STU Lagrangian of five-dimensional N = 2 supergravity: where repeated indices are summed over, IJK is the rank-3 Levi-Civita tensor, and we have defined the scalar kinetic term In this framework, the ansatz (2.3) defines solutions of the above action such that Remarkably, the torus warp factors in M-theory, W Λ , decouple entirely and the fivedimensional metric is independent of these factors. Therefore, from a five-dimensional point of view, these functions are simple harmonic scalar decorations on an STU background. For solutions that are asymptotic to five-dimensional flat space, the ADM mass is obtained from the following asymptotic expansion 4 where r is the five-dimensional radial coordinate, related to the cylindrical coordinates such as (ρ, z) = 1 2 (r 2 sin 2θ, r 2 cos 2θ). Note that the sources in W Λ does not indeed induce mass.

Reduction to four dimensions
We further reduce to four dimensions after compactification along y 7 . One can simply use known results about the reduction of the STU model from five to four dimensions (see for instance [45,51,52]). Therefore, the reduced four-dimensional action is given by is the four-dimensional Newton's constant, is now the Hodge star in four dimensions, and L 4D ST U is the STU Lagrangian of four-dimensional N = 2 supergravity. This Lagrangian is better written in terms of three independent complex scalars z I and four two-form field strengths,F Λ = dĀ Λ , We refer the reader to the appendix B.1.2 for the scalar and gauge field couplings, g IJ , I ΛΣ and R ΛΣ (B.12), (B.13) and (B.14). The four-dimensional metric, scalar and gauge fields are derived from five-dimensional fields such as In this context, our ansatz (2.3) consists of a metric, three purely imaginary complex scalars, four real scalars and four gauge fields given by 4 We use the conventions of [50].
For solutions that are asymptotic to four-dimensional flat space, the ADM mass is obtained from the following asymptotic expansion where r is the radial coordinate in four dimensions, (ρ, z) = (r cos θ, r sin θ).

Classification of asymptotically-flat solutions
In this section, we construct families of non-BPS solutions introduced in section 2.4, and discuss their regularity. We first discuss solutions that are asymptotic to a four-dimensional flat space plus compactification circles and then build solutions asymptotic to five-dimensional flat space. For the self-consistency of this section, we remind the ansatz of metric and gauge field in M-theory (2.3):

Four-dimensional solutions
We consider that y 7 is an extra S 1 of 2πR y 7 periodicity. Generic non-BPS solutions are given by (2.7), and are sourced by n rods as depicted in Fig.1. The warp factors and gauge potentials are given by eight weights at each rod, (P i ), and four pairs of gauge field parameters, (a Λ , b Λ ), such that where we remind the main functions and variables 3) , and the exponents We have defined a family of non-BPS solutions given by 10n + 8 parameters. We now have to discuss the regularity of the solutions that constrains the parameter space.

Regularity constraints
Potential constraints come from coordinate singularities on the z-axis, regularity of the spacetime elsewhere, and conditions on the asymptotic. The details of the analysis can be found in the appendix C.1.
• Condition on the asymptotics: We consider the asymptotic spherical coordinates (ρ, z) = (r sin θ, r cos θ). At large r, we have Therefore, the four-dimensional metric of the solutions (2.31) is asymptotically flat if • Regularity out of the z-axis: To avoid extra zeroes out of the z-axis, one needs With this condition, the solutions are regular out of the z-axis.
• Regularity at the rods on the z-axis: We consider the local spherical coordinates centered around the i th rod, At the rod, r i → 0, we have while all other quantities are well-behaved. From (3.2), we have 5 and therefore, the metric (3.1) is singular at the rod, except for specific choices of weights that will characterize the physical rods of the non-BPS floating brane ansatz in M-theory. It will be convenient to use the following aspect ratios, d i , The regularity at each rod requires that the sources are in one of these categories (see appendix C.1.3 for more details): -Black rods: the y a -fibers and the φ-circle have a finite size at the i th rod if One can check that α ii = 1 and the timelike Killing vector ∂ t shrinks at the rod which therefore defines the locus of a horizon. The topology of the horizon is either T 7 ×S 2 or T 6 ×S 3 depending on the close environment around the rod (see Fig.2 and 3). 6 Moreover, the surface gravity and horizon area associated to the black hole give 7 (3.13) The rod on its own corresponds to the static limit of a four-charge non-extremal black hole [38][39][40]. 8 However, in the present ansatz, one can stack linearly multiple black holes on a line and their interactions can be studied.
-Bubble rods: One of the T 6 directions, for instance y 1 or y 2 , shrinks at the i th rod while all other fibers are finite by taking 14) where the "+" corresponds to a shrinking y 1 -circle and the "−" corresponds to a shrinking y 2 -circle. The weights for which another T 6 direction degenerates can be obtained by permuting the I = 1, 2, 3 indexes. Similarly, the y 7 circle shrinks while all other fibers remain finite by considering For each possibility, the (r i , y a )-subspace corresponds to an origin of an R 2 defining the locus of a bubble with a T 6 ×S 2 or T 5 ×S 3 (see Fig.2 and 3). 9 It corresponds to a smooth coordinate degeneracy if an algebraic equation, which we can denote as a bubble equation, is satisfied. For a rod where the y 1 or y 2 circle 6 The horizon topology depends on the close environment of the black rod. If the rod is connected on one side as in Fig.3, it corresponds to a T 6 ×S 3 horizon. Otherwise, it is a T 7 ×S 2 horizon as in Fig.2. 7 The surface gravity can be read from the local geometry such as ds 2 ∝ dρ 2 i − ρ 2 i κ 2 i dt 2 + . . . with ρi → 0, and is associated to the temperature of the black hole, Ti = κi/(2π). Moreover, note that if the rod is connected to another one, for instance z are non-zero, the rod carries non-zero M2-M2-M2-KKm charges given by (2.22). 9 As for a black rod, the local topology of the bubble depends if the bubble rod is disconnected (Fig.2) or connected (Fig.3).
degenerates smoothly, the bubble equation relates the asymptotic radius to the internal parameters as follows 10 where k i is an orbifold parameter k i ∈ N that corresponds to the order of the conical deficit on the R 2 . The constraint corresponding to the degeneracy of another T 6 direction or the y 7 -circle is obtained by permuting e b 1 a 1 → e b I a I or e b 1 a 1 → e b 0 a 0 respectively. Moreover, one can check that the gauge field is regular at the rod (see Appendix C.1.3). The smooth bubble where one of the T 6 directions degenerates has nonzero P i . Therefore, it carries only a M2 brane charge and a KKm charge, given by (2.22). Similarly, the smooth bubble that shrinks the y 7 direction carries only a KKm charge. Moreover, these charges can be freely reduced to zero without changing the topology of the bubble by taking the neutral limit (2.11), and the resulting objects will be smooth vacuum bubbles in M-theory. Therefore, seven types of physical rod sources correspond to a smooth coordinate degeneracy of one the T 6 ×S 1 directions defined by an origin of an R 2 where a smooth bubble is sitting.
There are 8 categories of sources that can be used to generate physical asymptotically-  otherwise, (3.17) which means that the three-dimensional base does not depend on the specific nature of the sources.
Finally, one can think of other types of sources that are singular in M-theory but regular in other duality frameworks. This will be clarified when discussing the ansatz in other string theory frameworks in the section 3.3.
The effective mass induced by the i th rod, M i , can be derived from (2.32) by isolating the contribution of the rod. We find where the brane charges carried by the rod are given in (2.22). Therefore, the chargeto-mass ratios induced by the branes are determined by a Λ coth b Λ , and are the same for all the rods. It is a consequence for having a linear ansatz. If it still allows to remarkably deviate from the BPS regime, one cannot have two rods with opposite charges for instance. • Regularity on the z-axis out of the rods: ±± are finite and nonzero. Therefore, the warp factors are well-defined and the φ-circle degenerates as the cylindrical coordinate singularity. The analysis reduces to the regularity of the three-dimensional base, ds 2 3 = e 2ν dρ 2 + dz 2 + ρ 2 dφ 2 , and the warp factor e 2ν can induce conical singularities. More precisely, we have where we remind that d i are the aspect ratios (3.11). Therefore, the base metric above and below the rod configuration, z < z − 1 and z > z + n , has no conical singularity and is smooth there. However, in between the (i − 1) th and i th rods, we have ds 2 , the segment has a conical excess and describes a strut, i.e. a string with a negative tension (see Fig.2). The strut is a singularity that accounts for the lack of repulsion between the (i − 1) th and i th sources. It can be associated with a curvature singularity and an energy as done in [53] and reviewed in [2].
The existence of a strut between two rods requires that they are disconnected. Therefore, by connecting the rods, which implies that they are of different nature, the solutions are free of struts and conical excesses [2,3,33]. These solutions consist in connecting bubbles and possibly non-extremal black holes on a line (see Fig.3). For such configurations, the sources do not require a singular mechanism to be prevented from collapsing and are kept apart by topological bubbles [2]. This provides a new paradigm in the microstate geometry program for constructing smooth bubbling geometries without the aid of supersymmetry. The standard lore of supersymmetry is that gravitational attraction is counterbalanced by electromagnetic repulsion. In the present ansatz, the solutions are generically non-BPS with arbitrarily low electromagnetic potentials, and the lack of repulsion is balanced by the inherent pressure of squeezed bubbles [2].
In conclusion, the non-BPS floating brane ansatz contains a large phase space of axisymmetric regular static solutions that are asymptotic to four-dimensional flat space plus compactification circles. Typical solutions consist of non-extremal four-charge black holes and 7 types of smooth bubbles on a line, which are kept apart by struts if disconnected. In addition, smooth bubbling geometries can be constructed by considering chains of connected bubbles of different species. We will construct explicit solutions in section 4.

Five-dimensional solutions
To construct five-dimensional non-BPS solutions, the four-dimensional base, (ρ, z, φ, y 7 ), must be considered as a four-dimensional infinite space. That is, we take y 7 to be an angle of periodicity 2π. As detailed in the appendix C.2, one must also consider the neutral limit (2.11) for the pair (Z 0 , H 0 ) and source Z 0 by a semi-infinite rod.
Generic five-dimensional solutions are therefore given by (3.1), sourced by n finite rod of length M i and a (n + 1) th semi-infinite rod (see Fig.4). We will consider that the n th and (n + 1) th rods are connected to avoid a conical excess between them, z + n = z − n+1 . The warp factors and gauge potentials are similar to the ones given for four-dimensional solutions (3.2) with the neutral limit (2.11) for (Z 0 , H 0 ) and additional terms corresponding to the semi-infinite rod, that is where the main functions are given in (3.3) and the exponents, α ij , are now given by 11 Note that the solutions have no KKm charges along y 7 because H 0 = 0. We have therefore defined a family of non-BPS M2-M2-M2 solutions given by 10n + 8 parameters.

Regularity constraints
We will be brief in the regularity analysis since it is similar to the discussion for fourdimensional solutions. We refer to the appendix C.2 for more details.
The five-dimensional metric (2.26) is asymptotically flat if The solution is regular everywhere out of the finite rods, especially at the semi-infinite rod where the y 7 degenerates smoothly as an origin of a R 2 if and only if Moreover, the solution is regular at the finite i th rod if its weights fall into one of these eight categories (see appendix C.2.3 for more details): -Black rods: We consider that and the rod corresponds to a horizon of a black hole where the timelike Killing vector ∂ t shrinks. The surface gravity, temperature and area of the black hole are given by where d i is still given by (3.11). Because the three P (I) i are finite, the black hole carries generically three M2 charges given by (2.22). Each charge can be taken to be zero by considering the neutral limit (2.11), which corresponds to a I = sinh b I → ∞.
More precisely, the rod corresponds to the horizon of a three-charge static nonextremal black hole for which the single-center solutions have been derived in [41][42][43].
-Bubble rods: The y 1 or y 2 circle shrinks at the i th rod while all other fibers are finite if we consider where the "+" corresponds to a shrinking y 1 -circle and the "−" corresponds to a shrinking y 2 -circle. One can obtain the weights for another degenerating T 6 direction by permuting the I = 1, 2, 3 indexes. Similarly, on can make the y 7 circle degenerate by considering For each possibility, the local geometry corresponds to a bubble sitting at a smooth origin of a R 2 if a bubble equation is satisfied. For a rod that makes the circle y 1 or y 2 degenerate smoothly, the bubble equation constrains the internal parameters as a function of the extra-dimension radius such as (3.28) where k i is an orbifold parameter, k i ∈ N, that corresponds to the order of the conical deficit on the R 2 . The bubble equation corresponding to another degenerating T 6 direction or the y 7 -circle is obtained by permuting e b 1 a 1 → e b I a I or e b 1 a 1 → 1 respectively. The smooth bubble where one of the T 6 direction degenerates carries only a M2 brane charge, given by (2.22), because only one P (I) i is nonzero (3.26). Similarly, the smooth bubble where the y 7 direction shrinks is purely topological and carries no charge. We have therefore six types of M2 bubbles and one type of vacuum bubble that can smoothly source the solutions.
The mass induced by each rod, M i , can be derived from (2.27) and related to the brane charges (2.22) such as The charge-to-mass ratios for each stack of M2 branes are fixed by a I coth b I for all rods.
As for four-dimensional solutions, a strut is separating each pair of disconnected rods. More precisely, if the (i − 1) th and i th rods are not connected, the φ-circle degenerates in between them with a conical excess of order d −1 i > 1 (3.11). This singularity can be bypassed by considering connected rods. For such configurations, the struts disappear and the non-BPS sources are held apart by pure topology.
To conclude, the family of solutions, given by (3.20), contains a large phase space of regular non-BPS axisymmetric static solutions that are asymptotic to five-dimensional flat space plus compactification circles. Typical solutions consist of non-BPS non-extremal three-charge black holes and 7 types of smooth bubbles on a line, which are held apart by struts if disconnected. Moreover, smooth bubbling geometries can be constructed by considering chains of connected bubbles of different species. We will construct explicit solutions in section 5.

Other duality frames
So far we have limited the discussion to the M-theory frame, but the construction can be dualized to different string theory frames through dimensional reduction and T-dualities. Each frame will have new types of regular rod sources that are singular in M-theory. To illustrate this property, we will focus on dualization in the D1-D5-P-KKm frame, but we refer the interested reader to the appendix B.2 where we have dualized the solutions in six different frames, from the D2-D2-D2-D6 to the P-M5-M2-KKm frame. In addition, the study of the D1-D5-P-KKm frame will provide a link to the author's previous constructions of smooth bubbling geometries [1][2][3].
After reduction along y 5 and a series of T-dualities along y 1 , y 2 and y 6 , the M2-M2-M2-KKm solutions (3.1) transform to D1-D5-P-KKm solutions where the common direction for the momentum P charge, the D1 and D5 branes is y 6 . The metric, the R-R gauge fields C (p) , the NS-NS two-form gauge field B 2 and the dilaton in type IIB are given by where H 2 is the electromagnetic dual of T 2 , and is given by the same expression as H 0 in 2). One can recognize the type IIB frame of topological stars and five-dimensional charged Weyl solutions of [1] by taking 31) Taking T 3 = 0 while keeping a non-trivial Z 3 = W is possible by considering the neutral limit for the pair (Z 3 , T 3 ) (2.11), that is no P-charges. If we now consider Z 0 to be different from Z 1 = Z 2 , we retrieve the framework of [2,3] where six-dimensional smooth bubbling solutions, as bubble bag ends [3], have been constructed.
The warp factors and gauge potentials are governed by the linear system of equations obtained from the non-BPS floating brane ansatz of section 2.2.
For the non-BPS solutions, the physical rod sources are different from the eighth found in sections 3.1.1 and 3.2.1. For instance, the y 6 -circle degenerates at the i th rod for a solutions given by (3.2) defining the locus of a bubble if we consider its associated weights to be As a black rod (3.12), they do not require to turn on the weights of the torus warp factors W Λ , and therefore the T 4 is rigid as for BPS solutions. The only difference is that a 3 P negative is compatible with the regularity condition (3.7) only if one takes the neutral limit, (a 3 , b 3 ) → ∞ (2.11). It requires to take the gauge potential for the P charge to be zero and the solutions have no momentum charges as in [1][2][3]. A rod given by (3.32) corresponds to a smooth D1-D5-KKm bubble in type IIB. 12 However, it is singular in the M-theory frame (3.1). Therefore, there are specific species of bubble rods that correspond to smooth loci in a unique string theory frame.

Explicit four-dimensional solutions
In this section, we derive explicit solutions that are asymptotic to R 1,3 with extra compact dimensions using the non-BPS floating brane ansatz in different string theory frames. There is a very large number of configurations that one can think about. First, we will construct chain of non-extremal four-charge black holes, either separated by struts or smooth bubbles, and discuss their physics. Second, we will construct smooth bubbling geometries in the manner of [2,3], that have the same mass and charges as non-extremal four-charge black holes.

Chain of static non-extremal black holes
We first study solutions for which the main ingredients are non-extremal four-charge black holes on a line.

Black holes separated by struts
We consider n finite rods of length M i on the z-axis with weights corresponding to black rods given by (3.12) (see Fig.5). It is convenient to divide the warp factors, Z Λ (3.2), into two parts: one will encode changes in topology and the other will correspond to a well-behaved flux decoration such as The remaining functions (3.2) give where the generating functions R The function U t behaves as a product of "Schwarzschild factors" that vanishes at each rod and makes the timelike Killing vector shrink. It induces the topology of the solutions and forms the horizons. The Z Λ depend only on the gauge field parameters, and are always positive and finite. In the neutral limit (2.11), all Z λ converge to 1 while U t remains unchanged. We retrieved the solutions of Schwarzschild black holes on a line [54,55]. At large distance r, with (ρ, z) = (r sin θ, r cos θ), the geometry is asymptotically flat if (3.6) is satisfied, and one can read the ADM mass from (2.32): Moreover, the solutions carry four charges that are M2-M2-M2-KKm charges in M-theory. They are given by the sum of the rod charges carried by each rod (2.22): • A unique rod: map to the four-charge non-extremal black hole [38][39][40].
We consider a unique rod, n = 1, and the spherical coordinates The four-dimensional metric, gauge fields and scalars (4.2) simplify to 13 We retrieve the static solutions of a fourcharge non-extremal black hole [38][39][40] by relating the boost parameters in these papers, δ Λ , to our parameters, b Λ , such as cosh 2δ Λ = coth b Λ .
• Multiple four-charge black holes on a line.
By considering n > 1, we perform the multicentric generalization of the static solutions in [38][39][40]. In the IR, the geometry is made of a chain of four-charge static non-extremal black holes on the z-axis held apart by struts (see Fig.5). At each rod, the timelike Killing vector ∂ t vanishes defining the locus of a horizon with area (3.13). If one wants the black holes to be in thermal equilibrium, their surface gravity must be equal (3.13), which non-trivially constrains their positions and sizes. Moreover, in between each rod the φ-circle shrinks with a conical excess of order d −1 i > 1 (3.11) defining the loci of the struts. The struts carry an energy that counterbalances the attraction between the black holes. This energy corresponds to the interaction between the black holes in string theory. It can be derived using the method of [2,53]. More precisely, the energy of the strut that separates the (i − 1) th and i th black holes on the chain carries an energy given by For a configuration of two identical black holes separated by a distance , the energy of the strut in between them is given by  where M is the length of each rod. When the separation is large, M , the energy of the strut approximates the Newtonian potential between two particles of mass, M 2G 4 , in four dimensions. From this perspective, the strut measures the binding energy between the two black holes, or rather the potential energy needed to keep the two black holes from collapsing on each other. An important observation is that the effective ADM masses of the black holes (4.3), from the Newtonian point of view, depend on b Λ , which are associated to the four charges of the black holes (4.5) with the regularity condition (3.6). This implies that the binding energy as measured by the strut also accounts for effects due to the electromagnetic fields of the non-extremal black holes.
For finite separation between the two black holes, the gravitational potential between them deviates significantly from that of the Newtonian limit. In particular, the gravitational potential between two black holes vanish when → 0. In this limit, the two rod sources merge and the two-body configuration becomes a single non-extremal black hole.

Black holes separated by smooth bubbles
Struts are singularities that do not have a consistent UV description in string theory as they correspond to cosmic strings with negative tension. Therefore, it is crucial to find a mechanism to resolve them and to obtain more relevant configurations in string theory. The non-BPS regime will require a new mechanism for this task. In the analysis of [53] and continued in [2], it was shown that struts can be classically resolved into smooth bubbles. More precisely, one can consider the same black hole systems as before but sourcing each segment between them by a bubble rod. The resulting configuration will consist in a chain of connected rods which would be a succession of black holes and bubbles without struts.
The condition for having connected rods fixes their centers in terms of their size: One can also freely choose the origin of the z-axis so that z + n = 0. For simplicity, we consider the bubble rods to be smooth KKm bubbles given by the weights (3.15), but one could a priori take any of the seven species of bubble rods detailed in section 3. Therefore, we consider n = 2N + 1 rods of lengths M i where the N + 1 odd rods correspond to black rods (3.12) and the N even rods correspond to KKm bubble rods (3.15) (see Fig.6). As in the previous section, we divide the warp factors Z Λ (3.2) in meaningful pieces such as where R (i) ±± are given in (3.3). The metric and gauge potential in M-theory are given in (3.1), while the five-dimensional reduction along T 6 is given by (2.26) The warp factors U t and U y 7 force the degeneracy of the t and y 7 fibers at the rods. In the regions where U y 7 vanishes, that is at the even rods, the y 7 direction shrinks forming a smooth origin of an R 2 (with potential conical defects) if we have (3.16) 12) where we remind that z ± j are the rod endpoint coordinates on the z-axis (3.3) and d j are the aspect ratios (3.11). These N "bubble equations" fix the lengths of the bubble rods, M 2i , according to the other parameters. Moreover, if one wants the black holes to be in thermal equilibrium, one needs in addition to impose that their surface gravities (3.13) are equal, which also constrains the length of the black rods.
The φ-circle does not shrink anymore along the rod configuration and the four-charge non-extremal black holes are held apart by KKm bubbles. The struts have been then successfully replaced by smooth bubbles. This mechanism has been explored in [2,53]. In one word, vacuum bubbles are reluctant to be squeezed as it wants to expand [34], and provide the necessary pressure between two non-BPS objects [53]. If the electromagnetic fluxes stabilize the bubbles [35], their reluctance to be squeezed remains [2].

Smooth bubbling geometries
We now seek to construct smooth non-BPS bubbling solutions without black hole sources and struts. This consists in building a chain of connected bubble rods of different species. In M-theory alone, there are 7 species of smooth bubbles that can be used, and there are several others in different duality frameworks as described in section 3.3. This induces a wide variety of configurations possible. In this section, we will construct an explicit M2-M2-M2-KKm configuration and a D1-D5-KKm configuration in type IIB.

In M-theory
As described in section 3.1.1, among the seven species of physical bubble rods in M-theory, six correspond to the smooth degeneracy of one of the T 6 direction and carry a unique M2 charge and a KKm charge. To construct a smooth configuration with M2-M2-M2-KKm charges, one needs at least three species of these kinds.
Therefore, we will consider a chain of n = 3N connected bubble rods of length M i which will successively make the y 1 , y 3 and y 5 circles shrink (see Fig.7). Having connected rods constrains the rod centers in terms of their sizes as in (4.8). The expression of the warp factors and gauge potentials are given by (3.2) with the following choices of weights, obtained from (3.14), 3i = 0. (4.15) As in the previous constructions, it is convenient to split the warp factors Z Λ such as The functions U I are products of "Schwarzschild factors" that vanish at all the (3i + I) th rods while the Z Λ are non-zero and finite. The remaining functions (3.2) give ±± are given in (3.3). The metric and gauge potentials in M-theory, obtained from (3.1), are given by The solutions are regular out of the rods since U I and Z Λ are finite and positive there. At the rods, the y 1 , y 3 and y 5 fibers shrink to zero size where U 1 , U 2 and U 3 vanish respectively (see Fig.7). These loci end the spacetime as smooth origin of R 2 (with potential conical defects) if 3N bubble equations are satisfied. These bubble equations arise from the regularity condition (3.16) at each rod. They are non-trivial multivariate polynomials that will fix all rod lengths M i in terms of the independent parameters: the extra-dimension radii R ya , the gauge-field parameters (a Λ , b Λ ), the number of bubbles n = 3N and the orbifold parameters k i ∈ N. These equations are a priori not solvable analytically if N is not small, but can be solved numerically. Moreover, interesting approximations can be performed in the large N limit, that is when the number of bubbles is large, and analytic solutions can be found. In this limit, the bubbles highly deform the spacetime as a "bubble bag end" geometry (see [3] for such an example with a configuration of two species of bubbles).
Each species of bubbles carries a M2 charge and a KKm charge as discussed in section 3.1.1. Therefore, by combining the three species, the solutions have M2-M2-M2-KKm charges. They can be derived by summing the individual rod charges (2.22), and we find In the four-dimensional frame obtained after reduction along the T 6 ×S 1 (2.31), the solutions are given by Furthermore, since the extra scalars, Φ Λ , are turned on, the four-dimensional solutions are not solutions of the STU Lagrangian as detailed in section 2.5.2. However, they have the same conserved charges as non-extremal four-charge static black holes given by (4.5), but they are horizonless and terminate the spacetime as a chain of non-trivial bubbles wrapped by fluxes. We leave the comparison of the two geometries for a later project. More precisely, it will be interesting to study, in the manner of [3], the compactness of the bubble structure with respect to the size of the horizon of the corresponding black hole.
• BPS limit: We have seen in (2.12) that generic non-BPS solutions approach the BPS regime by taking (b Λ , a Λ ) → 0. One can derive the BPS limit of our specific non-BPS smooth bubbling solutions by considering (b Λ , a Λ ) small.
We consider the specific flow where b Λ = a Λ = λ with λ → 0. First, from the bubble equations (3.16), we have where q i are finite constants as λ → 0. Therefore, the full rod configuration shrinks to a point. However, it is clear that the charges (4.19) and mass (4.21) do not vanish such as  where (r, θ) are the spherical coordinates (ρ, z) = (r sin θ, r cos θ). Therefore, the solutions approach a four-charge static BPS black hole, or static BMPV black hole. At small (b Λ , a Λ ), the bubbling geometries are almost indistinguishable from the BPS black hole. They must develop an AdS 2 throat that caps off smoothly at r = O(λ) as non-BPS bubbles. This is the first example of such a resolution in the non-BPS regime. It also shows that the geometries can be as compact as a black hole, unlike the previous ones constructed by the author [2,3].
It would be interesting to push the comparison further. Specifically, it would be to analyze how the BMPV black hole horizon at r = 0 is smoothly resolved by adding a small amount of non-extremity when moving in the non-BPS regime, and how the geometries develop an AdS 2 throat.

In type IIB
We now work in the D1-D5-P-KKm framework given by (3.30). The advantage of constructing smooth non-BPS bubbling geometries in type IIB is that there are two species of bubbles that do not require to turn on the warp factors W Λ : a KKm bubble given by the weights (3.15) and a D1-D5-KKm bubble given by (3.32). Such configurations are solutions of the STU Lagrangian when reduced to four dimensions (2.28). However, the regularity of the D1-D5-KKm bubble requires that the solutions have no momentum P charge.
We will therefore consider a chain of n = 2N + 1 connected rods of lengths M i that consists of a succession of D1-D5-KKm and KKm bubbles (see Fig.8). 14 Having connected rods constrains the rod centers in terms of their sizes as in (4.8).
The warp factors and gauge potentials (3.2) give 15 14 We are considering an odd number of rods to make the link with the solutions of [2,3], but one could have taken any other configuration. 15 As discussed in section 3.3, one needs to consider the neutral limit (2.11) for the pair (Z3, T3). That is we have considered a3 = e b 3 /2 = ∞ and a3P where we have defined (4.26) The metric, the dilaton and gauge fields in type IIB (3.30) lead to The solutions carry D1, D5 and KKm charges given by (4.28) The solutions are asymptotic to R 1,3 when reduced to four dimensions if (3.6) is satisfied. Moreover, the ADM mass (2.32) is given by If we assume (a 1 , b 1 ) = (a 2 , b 2 ), we retrieve the non-BPS smooth bubbling solutions constructed in [2,3]. Therefore, we refer the reader to [2,3] for an exhaustive analysis of these backgrounds, their regularity and their depiction as "bubble bag ends" [3]. In short, the geometries are strongly distorted and have a S 2 that suddenly opens up near the bubble loci like a bag of smooth bubbles. However, in the absence of P charge, their BPS limit is not a BPS black hole but a Taub-NUT space. Thus, they cannot develop an AdS throat and cannot be as compact as a black hole.

Explicit five-dimensional solutions
In this section, we derive explicit solutions that are asymptotic to R 1,4 plus extra-compact dimensions. We will be more brief than in the previous section. Indeed, one can transform any asymptotically-R 1,3 solution into an asymptotically-R 1,4 geometry while preserving the brane structure of the solution by simply taking a neutral limit (2.11) for the pair (Z 0 , H 0 ) and adding a semi-infinite rod in Z 0 : 16 All other warp factors and gauge potentials can be left unchanged. Five-dimensional solutions can therefore be easily extracted from the four-dimensional solutions constructed before. However, the regularity constraints at the sources are modified by the presence of the semi-infinite rod as discussed in section 3.2.1.
We will first derive solutions consisting of a chain of three-charge non-extremal black holes before constructing smooth bubbling geometries. The generic solutions are given in section 3.2, and the warp factors and gauge potentials are given in (3.20).

Chain of static non-extremal black holes
We consider that the n finite rods of length M i on the z-axis correspond to black rods given by the weights (3.24). As for four-dimensional solutions, it is more convenient to divide the warp factors into meaningful parts: 16 The neutral limit for the pair (Z0, H0) has sent a0 to infinity while keeping a0P The remaining functions (3.20) give ±± are given in (3.3). The metric and gauge potentials in M-theory are given in (3.1), while the five-dimensional reduction along T 6 is given by (2.26): The warp factors U t vanishes at each rod as a product of "Schwarzschild factors," thereby inducing the horizons. The Z I depend only on the gauge field parameters, and are always positive and finite. The condition for having an asymptotically flat five-dimensional space requires (3.22), and one can read the ADM mass from (2.27): If we consider a unique rod, n = 1, one retrieves the static limit of the single threecharge black hole constructed in [41][42][43]. The map can be done by changing coordinates and identifying the boost parameters δ I used in [41][42][43] to the gauge field parameter b I such as cosh 2δ I = coth b I . Taking n > 1 consists therefore in a chain of static three-charge black holes in five dimensions. The black holes are separated by struts, that are segments where the φ-circle degenerates with a conical excess of order d −1 i > 1 (3.11). The struts account for the lack of repulsion between the non-extremal black holes and encodes the binding energy of the system.
The singular struts can be replaced by regular bubbles in a manner similar to that used for the chain of four-dimensional black holes in the section 4.1.2. Specifically, one can source the M-theory solutions with bubble rods at each segment between the black holes. The bubble rods can either be vacuum bubbles that make the y 7 circle degenerate smoothly (3.27), or M2 bubbles that correspond to the degeneracy of one of the T 6 directions (3.26).

Smooth bubbling geometries
We now construct smooth non-BPS bubbling solutions that consist in a chain of connected bubble rods of different species. We will restrict to the five-dimensional analogs of the M-theory solutions constructed in section 4.2.1. One could have also built asymptotically- We consider a chain of n = 3N connected bubble rods of length M i where the circles y 1 , y 3 and y 5 successively degenerate. Having connected rods constrains the rod centers in terms of their sizes as in (4.8), and we also assume that the origin of the z-axis is at z + n = 0. The warp factors and gauge potential are given by (3.20) with the following choices of weights, obtained from (3.26), It is also convenient to divide the warp factors Z Λ such as The remaining functions (3.20) give where R The solutions are regular out of the finite and semi-infinite rods since U I , Z I and r The semi-infinite rod, ρ = 0 and z > z + n = 0, corresponds to a coordinate degeneracy since r (n) + − z = 0. The y 7 -circle reduces to zero size as a smooth origin of R 2 (see appendix C.2.3 for more details).
At the finite rods, the y 1 , y 3 and y 5 fibers shrink successively to zero size since U 1 , U 2 and U 3 vanish alternatively. These loci correspond to smooth origin of R 2 with potential conical defects if 3N bubble equations are satisfied (see Appendix C.2.3 for more details). These bubble equations arise from the regularity condition (3.28) at each rod. They are non-trivial multivariate polynomials that fix all rod lengths, M i .
Each species of bubble rod carries a M2 charge as discussed in section 3.2.1. Therefore, by combining the three species, the solutions have M2-M2-M2 charges. They are given by In the five-dimensional frame after reduction on the T 6 (2.26), the solutions are Therefore, they are singular at the rods in five dimensions, and these singularities are resolved in M-theory as the degeneracy of the extra dimensions. Moreover, the solutions are asymptotic to R 1,4 if (3.22) is satisfied. The ADM mass is given by (2.27) Furthermore, since the extra scalars, Φ Λ , are turned on, the five-dimensional solutions are not solutions of the STU Lagrangian as detailed in section 2.5.1. Nevertheless, they have the same conserved charges as a non-extremal three-charge static black hole in five dimensions, but they are horizonless and terminate the spacetime as a chain of non-BPS bubbles in M-theory.
• BPS limit: As for the four-dimensional bubbling geometries constructed in section 4.2.1, one can derive the BPS limit of the present five-dimensional solutions. This consists of taking the gauge field parameters to zero, (b I , a I ) → 0. For simplicity, we consider b I = a I = λ with λ → 0. First, from the bubble equations (3.28), we have where q i are finite constants as λ → 0, and the whole structure reduces to a point. However, it is clear that the charges (5.11) and the mass (5.13) do not vanish: The warp factors and gauge potentials, (5.8) and (5.9), behave as where (r, θ) are the five-dimensional spherical coordinates (ρ, z) = 1 2 (r 2 sin 2θ, r 2 cos 2θ). Therefore, the solutions approach a three-charge static BPS black hole in five dimensions.
The BPS black hole horizon is thus smoothly resolved into smooth non-BPS bubbles wrapped by flux by adding a small amount of non-extremity. At small (a I , b I ), the solutions are almost indistinguishable from the BPS black hole. They must develop an AdS 2 throat that does not end in a horizon but as non-BPS bubbles.

Discussion
In this paper, we have shown that non-BPS linear ansatz that allow non-trivial matter fields and topology can be directly derived from Maxwell-Einstein equations in string theory. In M-theory on T 6 ×S 1 , our ansatz enables four gauge potentials corresponding to three stacks of M2 branes and a KKm vector, and it can be dualized to other string frames such as the D1-D5-P-KKm frame. Focusing on solutions that are asymptotic to four-and fivedimensional flat space, we derived families of four-charge non-extreme black holes on a line and non-BPS bubbling geometries. The latter can have the same charges and mass as nonextremal black holes but are horizonless and smooth. We have highlighted examples that are almost indistinguishable from the BPS black hole when they are slightly non-BPS but terminate the spacetime smoothly as a chain of non-BPS bubbles.
While the present non-BPS floating brane ansatz opens a new door to the study of non-BPS solitons in string theory, several questions need to be explored in future work. First, although the smooth charged bubble is known to be a meta-stable vacuum [35,36], the stability of chains of such objects remains to be studied. Second, one can think of constructing new families of solutions with different boundaries than the flat asymptotic. Third, it will also be interesting to add brane degrees of freedom to the ansatz in M-theory, such as M5 or P brane charges. This will enable Chern-Simons interactions to be turned on as in the BPS floating brane ansatz for multicenter solutions. In addition, to construct more astrophysically-interesting solutions, a rotational degree of freedom will need to be added.
Furthermore, understanding the origin of the solutions as bound states of strings and branes will require careful analysis of the geometric transition that occurs. For BPS solutions, for example, the geometric transition that gives rise to smooth geometries is well understood [13]. The present one uses very different mechanisms because our solutions are far from being the non-BPS extensions of known BPS smooth bubbling solutions as their BPS limit suggests, and they are based on non-trivial topology changes of compact tori. They should therefore be very illuminating on constructions of string and brane boundary states in the non-supersymmetric regime, but one should not expect these bound states to use transitions similar to those of BPS bound states.
Finally, it would be interesting to have a better geometrical and physical understanding of the smooth bubbling geometries. One can compare them with the non-extremal black hole with the same mass and charges as in [3]. Moreover, one can derive their multipole moments, probe them by light geodesics or compute their quasi-normal modes using technologies developed for similar geometries [52,[56][57][58][59][60][61]. One could also being interested in performing M2-brane probe computation in the smooth bubbling solutions in M-theory. If we do not expect the branes to feel an entirely flat potential that allows them to "float" everywhere in space, our non-linear ansatz suggest special flat regions where probe can be trapped.

A Equations for ν
The equations for the base warp factor, ν, obtained from (2.2) with the ansatz of metric and gauge field given by (2.3), are

B The non-BPS floating-brane ansatz in other frames
In this section, we derive several frames dual to our M-theory ansatz. We first detail the four-and five-dimensional reduction, before discussing the ansatz in different type IIB and IIA frameworks.

B.1 Profile in five and four dimensions
We reduce the ansatz (2.3) to five and four dimensions after compactification along the T 6 and the S 1 . We apply a series of KK reductions from M-theory using the generic rules of [48,49]. We briefly summarize them here by truncating the degrees of freedom that are not present in our ansatz.
We aim to describe the reduction of the eleven-dimensional action (2.1) to D = 4, 5 dimensions for solutions of the following type (B.1) where φ is a vector of 11 − D dilatons, A (i) and A (I) are one-form gauge fields that have components only along the D-dimensional directions and g and γ i are constant vectors that we will make precise. We assume that all y i directions are U(1) isometries.
After the series of reduction along the y i , we obtain an Einstein-Maxwell-dilaton theory with the following action in D = 4 or 5 dimensions where F (I) = dA (I) and F (i) = dA (i) , a I and b i are also constant vectors, and the Newton constant is given according to the eleven-dimensional one by The constant vectors are all obtained from g ≡ 3 (s 1 , s 2 , . . . , s 11−D ) , where s i ≡ 2/((10 − i)(9 − i)), such as In D dimensions, our solutions are defined by a metric, given by ds 2 D , 11 − D scalars and KK gauge fields and 11−D 2 one-form gauge fields that arise from F 4 .

B.1.1 Reduction to five dimensions
We first perform a series of KK reductions to D = 5 dimensions by compactifying along {y i } i=1,..,6 . For the ansatz (2.3), there are no KK gauge fields along those fibers, and therefore we take A (i) = F (i) = 0. Moreover, it is convenient to change the basis of six scalars φ = (φ i ) i=1,..,6 to a more suitable one that allows a simpler final Lagrangian and a better comparison with the STU model in five dimensions: where we have defined seven scalars with the constraint X 1 X 2 X 3 = 1. With such a definition, the metric and gauge field (B.1) for D = 5 are given by We apply (B.2), and the reduced five-dimensional action is then where L 5D ST U is the STU Lagrangian of five-dimensional N = 2 supergravity: where repeated indices are summed over, and we have defined the scalar kinetic term Note that the Cherns-Simons term can be dropped because it is trivially zero for our ansatz. Nevertheless, we have kept it for a clearer identification with the STU Lagrangian.
In this framework, the ansatz (2.3) defines solutions of the five-dimensional action such as

B.1.2 Reduction to four dimensions
We further reduce to four dimensions by compactifying along y 7 for solutions where y 7 corresponds to a compact direction. One could reapply the generic reduction rules summarized before, but, thanks to the identification (2.24), it is simpler to directly use known results about the reduction of the STU model from five to four dimensions [45,51,52]. The reduced four-dimensional action is given by where L 4D ST U is the STU Lagrangian of four-dimensional N = 2 supergravity and G 4 = . This Lagrangian is better written in terms of three independent complex scalars z I and the field strengths of four one-form gauge fieldsF Λ =Ā Λ , and is given by Relabelling the scalar fields as the metric of the scalar σ-model g IJ follows from the Kähler potential the gauge kinetic couplings are and the axionic couplings are (B.14) The four-dimensional metric, scalars and gauge fields arise from the five dimensional ones such as (B.15) In this context, our ansatz is composed of a metric, three purely-imaginary complex scalars, four real scalars and four one-form gauge fields given by

B.2 Other string theory frames
In this section, we present the non-BPS floating brane ansatz (2.3) in different duality frames in type IIA and type IIB theories. We will make a great use of T-duality rules and compactification from M-theory, which we summarize here following [62]. The set of bosonic fields in M-theory is given by the metric G µν and the three-form gauge field A 3 µνρ . After the compactification along y i we are left with type IIA supergravity with the fields g µν , which are related to the eleven-dimensional fields as follows (note that we are working in string frame): Solutions in Einstein frame are obtained by transforming g Eµν = e Φ 2 g µν . T-duality transformations act on the supergravity fields mixing them according to Buscher's rules [63]. We assume that y j is the direction along which one performs the T-duality transformation, and we decompose the string metric, B-fields and RR gauge fields such as where the forms, B (2) , C (p−1) y and C (p) , do not have legs along y j and are functions only of the x µ coordinates, the dualized fields are By reducing the M-theory ansatz (2.3) along y 7 , the solutions correspond to D2-D2-D2-D6 solutions given by (in string frame) This framework requires that the y 7 direction in M-theory is compact and is ill-defined for the family of asymptotically-R 1,4 solutions constructed in the sections 5 and 3.2.

B.2.3 The D1-D3-F1-KKm frame
By applying a T-duality along y 1 , the solutions correspond to D1-D3-F1-KKm solutions given by The ". . ." in C (4) corresponds to the term arising from C (5) in type IIA. It is appropriate to use the self-duality of F (5) to derive this term with We find where H 2 has the same form as H 0 in (3.2) but with P Note that the electromagnetic dual of T 2 dt ∧ dy 1 ∧ dy 3 ∧ dy 4 only couples with Z 2 since we have 10 (dx a ∧ dt ∧ dy 1 ∧ dy 3 ∧ dy 4 ) = −ρZ 2 2 b a dx b ∧dφ∧dy 2 ∧dy 6 ∧dy 7 where x a = (ρ, z).

C Regularity analysis
In this section, we detail the regularity of the non-BPS M-theory solutions constructed in section 2.4. We first discuss solutions that are asymptotic to R 1,3 ×S 1 ×T 6 before the asymptotically-R 1,4 ×T 6 solutions.

C.1 Four-dimensional solutions
Generic non-BPS Weyl solutions in M-theory, that are asymptotic to R 1,3 plus compactification circles, are sourced by n rods of length M i , and are obtained from the ansatz (2.3), with the warp factors and gauge potentials given in (3.2).

C.1.1 Regularity out of the z-axis
At large distance, (ρ, z) = (r sin θ, r cos θ) and r large, the warp factors and gauge potentials (3.2) behave as At finite distance but out of the z-axis, the warp factors and gauge potentials are finite. However, the warp factors Z Λ can change sign due to their sinh form. This will necessarily induce closed timelike curves. By noting that R (i) − → 0 at the i th rod and R is enough to guarantee that all Z Λ are positive and that the solutions are regular out of the z-axis.

C.1.2 Regularity on the z-axis
We first discuss the regularity at the rods before the regularity elsewhere on the z axis.

C.1.3 At the i th rod
The local spherical coordinates around the i th rod are given by The two-dimensional base behaves as Moreover, Thus, As for e 2ν , we have Therefore, where we have defined the constants where the expansion of Z I can take two forms if P (i) i is zero or not given by (C.6), IJK is the rank-3 Levi-Civita tensor and we have used the convenient notation (y 1 , y 2 , y 3 , y 4 , y 5 , y 6 ) = (x (1) (C.11) Therefore, the Killing vector along the extra dimensions, ∂ ya , or the timelike Killing vector, ∂ t , can vanish at the rod while all other have a finite norm by fixing the 8 weights (G i ). We then have 8 possible choices of regular rods. Because all T 6 directions are similar by permutation, we will treat three categories: a black rod where ∂ t vanishes defining the horizon of a M2-M2-M2-KKm non-extremal black hole, a bubble rod where ∂ y 1 or ∂ y 2 vanishes defining the locus of a smooth M2-KKm bubble and another bubble rod where ∂ y 7 vanishes defining the locus of a smooth KKm bubble.
We first assume that the rod is disconnected from the others.
• A four-charge non-extremal static black hole.
We consider G The metric components along the compact dimensions and the φ-circle are finite while the time component vanishes as g tt = O(r i ). More concretely, the i th rod corresponds to a horizon where the timelike Killing vector ∂ t shrinks. Indeed, we have which implies that the θ i -dependent factors in g tt and g r i r i are remarkably the same. The local metric around the i th rod is then where ρ 2 i ≡ 4r i , the g xx (θ i ) are all finite and non-zero for 0 ≤ θ i ≤ π (C.10) and the surface gravity, κ i , is given by (C.14) The metric corresponds to the horizon of a black hole with a S 2 ×T 7 topology. One can relate the surface gravity to the temperature of the black hole by requiring smoothness of the Euclideanized solution. We find Note that if we study axisymmetric solutions with multiple black holes in thermal equilibrium the temperature associated to each black rod must be fixed to be equal.
(θ i ) is remarkably independent of θ i , the area of the horizon is simple to derive. We find are finite, the black hole carries generically three M2 charges and a KKm charge given by (2.22). Each charge can be taken to be zero by considering the neutral limit (2.11), which corresponds to a Λ = sinh b Λ → ∞. Moreover, since we have T I ∼ −a I 1 + O(ρ 2 i ) (C.10), the field strength is vanishing at the rod which guarantees its regularity.
We consider that one of the directions of the first T 2 shrinks at the rod while all other directions have finite size, that is where the "±" imposes the degeneracy of the x ± fiber. One can obtain any other directions of the T 6 by permuting the I = 1, 2, 3 indexes.
For these weights, the spacelike Killing vector ∂ x (1) ± shrinks on the i th rod corresponding to a coordinate singularity of an origin of R 2 space. We have which implies that the θ i -dependent factors in g x (1) ± x (1) ± and g r i r i are the same. The local metric around the i th rod is then where all the g xx (θ i ) can be obtained from (C.10) and are finite and non-zero for 0 ≤ θ i ≤ π. Moreover, we have defined ρ 2 i ≡ 4r i and the constant, C i , is given by The two-dimensional subspace (ρ i , x ± ) describes a smooth origin of R 2 or a smooth discrete quotient R 2 /Z k i if the parameters are fixed according to the radius of the x (1) ± -circle as To conclude, the time slices of the eleven-dimensional space at the i th rod is a bolt described by a warped S 2 ×T 6 fibration over an origin of a R 2 /Z k i space.
Moreover, one can check that metric determinant of the S 2 ×T 6 is remarkably independent of θ i if there are no black rods in the configuration. Therefore, one can easily derive the area of the S 2 ×T 6 bubble for such configurations and we find Finally, since only P (1) i and P (0) i are finite, the bubble carries a unique M2 charge and a KKm charge given by (2.22). Each charge can be taken to be zero by considering the neutral limit (2.11), which corresponds to a Λ = sinh b Λ → ∞. Moreover, since we have T 1 ∼ −a 1 1 + O(ρ 2 i ) (C.10), the component of the field strength along x (1) ± is vanishing at the rod which guarantees its regularity.
The y 7 fiber shrinks to zero size at the rod while all others are finite if the rod has the following weights The analysis is similar to the M2-KKm bubble. We have and the local metric takes the same form as in (C.18) by replacing x (1) ± by y 7 and C i is now given by The two-dimensional subspace (ρ i , y 7 ) describes a smooth origin of R 2 or a discrete quotient R 2 /Z k i if the parameters are fixed according to the radius of the y 7 -circle Therefore, the time slices of the eleven-dimensional space at the i th rod is a bolt described by a warped S 2 ×T 6 fibration over an origin of a R 2 /Z k i space. Because all P i , the bubble carries only a KKm charge given by (2.22). Finally, the area of the S 2 ×T 6 KKm bubble is also derivable when there are no black rods and give the same formula as the M2-KKm bubbles (C.21).
If we assume now that the rod is connected, we still have the eight same choices of weights but the local topology might change. For instance, we consider that the i th rod is a black rod. If the rod is not connected from below and connected from above to a M2-KKm bubble rod where x (1) + = y 1 shrinks, then the local metric at the horizon is still given (C.13). However, g x (1) + x (1) + (θ i ) andḡ φφ (θ i ) are not finite for 0 ≤ θ i ≤ π anymore. More precisely, we have g x (1) + x (1) + (θ i ) ∼ 0 andḡ φφ (θ i ) sin 2 θ i finite around θ i → 0. Therefore, the y 1 -circle pinches off at the north pole and the horizon has a S 3 ×T 6 topology. Note that the surface gravity computed in (C.14) is still the same and still well-defined. If the rod is now connected from above and below to two M2-KKm bubble rods where y 1 shrinks, we have a S 2 ×T 7 horizon again but the S 2 is now described by (θ i , y 1 ). Similar scenarios happen if the i th rod is a bubble rod: we can have either an S 2 ×T 6 bubble or a S 3 ×T 5 bubble depending on what is surrounding the rod.
To conclude, we have eight types of rods that can source physically our solutions on the axis. For each type of rods the φ-circle has a finite size. The different rods and their physics has been depicted in Fig.2 and Fig.3.
Moreover, note that the exponents α ij drastically simplify for the physical rods: if the i th and j th rods are of the same nature, 1 2 otherwise. (C.26) Therefore, the three-dimensional base, determined by the warp factor e 2ν (3.2), does not depend on the specific nature of the rods.

C.1.4 On the z-axis and out of the rods
We now study the behavior of the solutions on the z-axis, ρ → 0, and out of the rods where the φ-circle can shrink to zero size. On these segments, each R At ρ = 0 and out of the rods, we want this space to correspond to the cylindrical coordinate degeneracy. First we have Therefore, we get e 2ν ∼ 1 if z < a 1 − M 1 2 and z > a n + Mn where d i is given in (C.9). First we notice that, asymptotically, z > z + n and z < z − 1 , the base space is directly flat R 3 without conical singularity. However, in between two rods, we have two possibilities if they are connected or disconnected.

• Disconnected rods:
We consider the segment in between the (i − 1) th rod and i th rods with z + i−1 < z − i . The three-dimensional base is then given by the metric The segment corresponds to a R 3 base with the local cylindrical angle φ i ≡ φ d i . Moreover, we have Thus, we necessarily have d i < 1. Thus, the segment has a conical excess and the period of the local angle, φ i = φ d i , is 2π d i > 2π. This manifests itself as a string with negative tension, or strut, between the two rods. The strut exerts the necessary repulsion so that the whole structure does not collapse. We can calculate the stress tensor and the energy of the strut using the method described in [2,53], and we will similarly find that the energy is given by (C.33) • Connected rods: We consider the intersection between the (i − 1) th rod and i th rods with z + i−1 = z − i . The intersection then consists of a point with coordinates (ρ, z) = (0, z + i−1 ) = (0, z − i ) . We first define local spherical coordinates as follows The two-dimensional base transforms to dρ 2 + dz 2 = dr + At r i → 0 we have Thus, the φ-circle keeps a finite size unlike the disconnected case. This means that the intersection is protected from the conical excess associated to the degeneracy of the φ-circle in between two rods. In order to determine the local topology, a distinction must be made between different types of rods. Being connected, the i th and (i − 1) th rods are necessarily of a different nature. 18 Consequently, we have two possible scenarios: an intersection between a black rod and a bubble rod and between two different bubble rods.
For the first scenario we consider that the (i − 1) th rod is a M2-KKm bubble where y 1 shrinks, but all other choices would have led to the same results. From the above expressions, we find that the local metric is given by where α i and β (a) i are irrelevant finite constants and the surface gravity of the black rod κ i and the orbifold parameter of the M2-KKm bubble rod k i−1 are given in (C.14) and (C.20). This corresponds to the usual metric at the north pole of a horizon. At this type of loci, a circle composing the surface of the horizon is degenerating which is here y 1 . In the present case, it degenerates with a conical defect, parametrized by k i−1 , related to the M2-KKm bubble connected to the black hole.
We now analyze the scenario of two connected bubble rods. For simplicity, we assume that they are two M2-KKm bubbles where the y 1 and y 2 circles shrink respectively. The time slices of the metric are locally given by where α i and β (a) i are irrelevant finite constants and (k i−1 , k i ) are the orbifold parameters of the connected bubble rods. This corresponds to the metric of the origin of an orbifolded R 4 parametrized by (r i , τ i , y 1 , y 2 ). The two angles have the same conical defects as the connected bubbles but the local topology is free from struts and conical excess. Moreover, if k i−1 = k i = 1, the time slices of the metric corresponds to the origin of a R 4 with a T 5 fibration and the local spacetime is entirely smooth.

C.2 Five-dimensional solutions
To construct five-dimensional solutions, one needs to consider the four-dimensional spatial part of the eleven-dimensional spacetime (2.3), ds 2 4 = 1 Z 0 (dy 7 + H 0 dφ) 2 + Z 0 e 2ν dρ 2 + dz 2 + ρ 2 dφ 2 , (C.37) as a four-dimensional base. First, we consider y 7 to be an angle with 2π periodicity. Moreover, as suggested in [32], one also needs Z 0 to be sourced by a semi-infinite rod and to consider the neutral limit for the pair (Z 0 , H 0 ), which implies H 0 = 0. Generic five-dimensional Weyl solutions in M-theory are therefore given by the ansatz (2.3), sourced by n finite rods of length M i and a (n + 1) th semi-infinite rod above the rod configuration. We will consider that the n th and (n + 1) th rods are connected to avoid a conical excess between them, z + n = z − n+1 . The warp factors and gauge potentials are given by (3.20).

C.2.1 Regularity out of the z-axis
We first discuss the condition on the asymptotics by considering (ρ, z) = 1 2 (r 2 sin 2θ, r 2 cos 2θ) and r large. The warp factors and gauge potentials (3.20) behave as Therefore, the solutions, when reduced to five dimensions after KK reduction on the T 6 (2.26), are asymptotically: dr 2 + r 2 dθ 2 + sin 2 θdy 2 7 + cos 2 θdφ 2 . which makes the solutions regular out of the z-axis.

C.2.2 Regularity on the z-axis
We will first discuss the regularity at the rods before the regularity elsewhere on the z axis.

C.2.3 At the rod
The regularity at the n finite rods is very similar to the one performed for four-dimensional solutions in section C.1.3. However, the regularity at the semi-infinite rod will require a specific analysis.
At ρ = 0 and z > z + n , we have (C.44) The rod corresponds to a regular coordinate singularity where the y 7 -circle pinches off defining a smooth origin of a R 2 .
• At the i th rod: ρ = 0 and z − i < z < z + i .
The analysis is very close to the one led for four-dimensional solutions in section C.1.3, and we refer the reader to this section for most of the details of the expansion. The only difference is that there are new terms arising from the semi-infinite rod in Z 0 and e 2ν that must be carefully expanded.
We also use the local spherical coordinate (r i , θ i ) (C.3), and derive the local geometry at r i → 0. The new terms behave such that r (n) where the two possible developments of Z I are given in (C.46), IJK is the rank-3 Levi-Civita tensor and we have also used the convenient notation (y 1 , y 2 , y 3 , y 4 , y 5 , y 6 ) = (x (1) − , x − , x i ). Moreover, the physical rods are similar to the ones for fourdimensional solutions, and they are given by the same weights knowing that G i = a 0 P (0) i . We refer then to the analysis led in section C.1.3 for the description of the local geometry and topology, and to the summary in section 3.2.1.
In a word, we have eight types of rods that can source physically our solutions on the axis. Moreover, as for four-dimensional solutions, the exponents α ij (3.21) drastically simplify for the physical rods and give (C.26).
Finally, the regularity analysis on the z-axis but out of the rods, that is where the φ-circle pinches off, and at the intersections of connected rods are identical to the analysis led for the four-dimensional solutions. We refer the reader to the appendix C.1.4.