Non-BPS walls of marginal stability

We explore the properties of non-BPS multi-centre extremal black holes in ungauged \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathcal{N}=2 $\end{document} supergravity coupled to nv vector multiplets, as described by solutions to the composite non-BPS linear system. After setting up an explicit description that allows for arbitrary non-BPS charges to be realised at each centre, we study the structure of the resulting solutions. Using these results, we prove that the binding energy of the composite is always positive and we show explicitly the existence of walls of marginal stability for generic choices of charges. The two-centre solutions only exist on a hypersurface of dimension nv +1 in moduli space, with an nv-dimensional boundary, where the distance between the centres diverges and the binding energy vanishes.


Introduction and overview
The description of black holes in supergravity, viewed as a low energy effective description of string compactifications, has been a useful tool for understanding the structure and properties of nonperturbative features of the theory. In particular, the possible bound states of D-branes manifest themselves as multi-centre supergravity solutions at strong coupling [1]. In the BPS sector, the properties of the supergravity solutions, such as the walls of marginal stability and attractor flow trees [1,2], have been instrumental in uncovering this connection, leading to remarkable results on the description of D-brane bound states in terms of quiver quantum mechanics [3].
The purpose of this paper is to set the stage for a similar study of a particular subsector of the non-BPS spectrum. We restrict attention to zero temperature under-rotating JHEP10(2013)179 multi-centre black holes, i.e. charged and rotating extremal black holes, for which the extremality bound is saturated by the charges. 1 This class includes BPS solutions, wherein all charges involved allow for some supersymmetry to be preserved and no local rotation at the horizons is allowed, even though there is a global angular momentum generically. In this paper, we study the reverse situation, i.e. solutions where only non-BPS charges (of strictly negative quartic invariant) are allowed at the centres, as described by the composite non-BPS system [4][5][6]. The mixed situation, in which both BPS and non-BPS charges are allowed is described by the more complicated almost-BPS system [6,7], but will not be discussed here.
Using the formalism developed in [6], we are able to solve the system completely, in a general duality frame. As was already noted in [4,5], the resulting composite solutions only exist on certain hypersurfaces of the moduli space, unlike the BPS solutions whose domain of existence is of codimension zero in moduli space. The origin of this complication can be understood from the property that the phase of the central charge, which determines the BPS flow in multi-centre solutions, is somehow replaced by the n v − 1 flat directions of the individual charges 2 in the composite non-BPS system. It follows that instead of the N − 1 equations for N centres one finds in the BPS system, one now finds n v × (N − 1) equations, which not only fix the distances between the centres, but also constrain the electromagnetic charges and the asymptotic scalars in general. Nonetheless, restricting attention to the relevant hypersurface in moduli space where the solution exists, we find that the situation is essentially the same as for BPS solutions, i.e. that this hypersurface admits a co-dimension one boundary in moduli space corresponding to walls of marginal stability, where (some of) the distances between the centres diverge.
Furthermore, we study explicitly the binding energy of multi-centre solutions within the composite non-BPS system. This is based on an extension of the notion of the fake superpotential, as it has been defined for single-centre solutions [8][9][10][11][12][13]. Here, we give the general expression for the single-centre fake superpotential, for any value of the charge vector, in terms of the scalar fields and the parameters describing the flat directions mentioned above. The latter correspond precisely to the auxiliary variables introduced for the ST 2 and STU models in [13], and can be identified with them. In this paper we prove that the 'true fake superpotential' describing the single-centre flow is not only obtained as an extremum of the flat directions dependent potential as defined in [13], but is in fact always a global maximum.
Let us stress that the expression of the fake superpotential linear in the charges as defined in [13], is a rather involved function of the moduli and the flat directions parameters, already for the STU model. Proving that the extrema of the parameters describing flat directions were maxima therefore has been a technical obstacle for some time. Using a parametrisation of the moduli and the flat directions that depends explicitly on the 1 Black holes for which the extremality bound is saturated by the angular momentum are similarly called over-rotating. 2 For a single center solution of given non-BPS charge, there are exactly nv − 1 scalars that remain constant throughout the flow and are by definition determined by the nv − 1 non-compact generators of the duality group leaving the charge invariant.

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electromagnetic charges, as inspired by the structure of the general single-centre solution, we shall see that the expression of the fake superpotential simplifies drastically such that we are able to prove our results for any cubic model with symmetric Kähler space. Using the property that the energy of a composite bound state is described by the same potential linear in the charges, at non-extremum values of the flat directions parameters [4,5], we are able to prove that the binding energy is always positive. Furthermore, we also exhibit that the total energy at the location of a wall of marginal stability in moduli space is equal to the sum of the masses of the constituents that decouple, irrespectively of whether they are single-centre or composite themselves.
In fact we also find that the total mass of a composite solution is always lower than that of a single-centre solution of the same total charge, so that composite solutions are actually energetically favored, whenever they exist. This is in contrast with BPS configurations, for which the mass is entirely determined by the total electromagnetic charges and the asymptotic scalars, such that a BPS bound state always has the same mass as the singlecentre BPS black hole with identical charges. The existence and structure of the composite solutions is also shown to be connected to a notion of attractor flow tree, very similar to the corresponding one for BPS solutions [1].
This paper is organised as follows. In section 2 we introduce some notations and discuss the composite non-BPS system without any restriction on the charges, in a convenient basis. We then discuss the properties of single-centre and multi-centre non-BPS solutions in section 3, using the same basis. In particular, we present the most general single-centre solution in section 3.1, while in section 3.2 we define the fake superpotential and consider its properties. These are then used in section 3.3, where the multi-centre solutions are presented and the walls of marginal stability and the binding energy of the composites are studied. Some of our results are illustrated in an explicit two-centre example carrying D0-D6 and D0-D4-D6 charges, in section 3.4. Section 4 is devoted to the detailed derivation of several results used in the previous sections for the single-and multi-centre solutions in a frame independent formulation. We conclude in section 5, where we discuss our results and point to further directions. Finally, we recall some technicalities about T-dualities derived in [6] in appendix A, we show the appearance of space-dependent Kähler transformations to identify different sections describing the same solution in appendix B, and in appendix C we compute the stabilizer of two generic charges of negative quartic invariant.

Composite non-BPS system
In this section, we give some basic properties of the supergravity models we consider, in subsection 2.1, and define the general composite non-BPS system in a convenient basis in subsection 2.2. Using this basis, we give expressions for the general multi-centre solution, in terms of harmonic functions, referring to section 4 for the details of the derivation in a general basis.

Preliminaries
In this paper we wish to describe stationary asymptotically flat extremal black holes in the context of N = 2 supergravity coupled to n v vector multiplets. The bosonic field content JHEP10(2013)179 consists of the metric, n v complex scalar fields, t i , and n v + 1 gauge fields, A I , where i = 1 . . . n v and I = 1 . . . n v + 1. The bosonic Lagrangian then reads [14,15] (see [6] for our conventions) Here, the F I µν = ∂ µ A I ν − ∂ ν A I µ encompass the graviphoton and the gauge fields of the vector multiplets, while G µν I are the dual field strengths, defined in terms of the F I µν though the scalar dependent couplings. The explicit form of these couplings and of the Kähler metric, g i , will not be relevant in what follows, but can be computed in terms of the prepotential, which we will always consider to be cubic Here, the tensor c ijk , i = 1, . . . n v , is completely symmetric and we introduced the cubic determinant det[X] = 1 6 c ijk X i X j X k and the shorthand boldface notation for objects carrying an index i, j, . . . .
Here, we consider N = 2 supergravity theories for which the special Kähler target space, M 4 , is a symmetric space and can be obtained by Kaluza-Klein reduction from the corresponding five dimensional theories 3 defined in [16]. In this case, M 4 is a coset space, while the symmetric tensor c ijk satisfies special properties.
In order to set up the notation used throughout this paper, we define the cross product where we use boldface notation for vectors, omitting the indices i , j , . . . for brevity. Symmetric special target spaces are defined by tensors satisfying the Jordan algebra identity which will be used extensively in what follows. Note that the notation tr [ac] = a i c i denotes the contraction of two elements with two different kinds of indices. 4 Similar notation will be used for vector and scalar fields when writing components, so that we write t for the complex scalars. This notation is rather natural for the so-called magic theories, for which JHEP10(2013)179 a vector a can be represented as a three by three Hermitian matrix over a Hurwitz algebra (i.e. R, C, H, O) [16]. Throughout this work, we use objects transforming covariantly under electric/magnetic duality, in order to naturally parametrise solutions. The gauge field equations of motion and Bianchi identities can then be cast as a Bianchi identity on the symplectic vector (2.6) whose integral over any two-cycle defines the associated electromagnetic charges through where we explicitly show the decomposition of the charge vector in the n v + 1 electric and magnetic components. We use exactly the same decomposition for all other symplectic vectors. The symplectic inner product in this representation then takes the form Finally, the physical scalar fields, t, also appear through a symplectically covariant object, the so called symplectic section, V, which is uniquely determined by the physical scalar fields as up to the local U(1) phase X 0 .
Quartic invariant and charges of restricted rank. The invariance of the cubic norm deta can be used to define duality invariants and restricted charge vectors, a concept that is of central importance for the applications we consider later in this paper. First, we introduce the quartic invariant for a charge vector Γ, as where we also defined the completely symmetric tensor t M N P Q for later reference. It is also convenient to define a symplectic vector out of the first derivative, I 4 (Γ), of the quartic invariant, as In the following, all instances of I 4 (Γ 1 , Γ 2 , Γ 3 , Γ 4 ) will denote the contraction of the tensor t M N P Q in (2.10) with the four charges, without any symmetry factors, except for the case with a single argument, as in I 4 (Γ) and I 4 (Γ).
We are now in a position to introduce the concept of charge vectors of restricted rank. A generic vector leads to a nonvanishing invariant (2.10) and is also referred to as a rankfour vector, due to the quartic nature of the invariant. Similarly, a rank-three vector, Γ 3 , is a vector for which the quartic invariant vanishes, but not its derivative. An obvious example is a vector with only p = 0 and all other charges vanishing, so that the derivative I 4 (Γ 3 ) is nonzero and proportional to the cubic term detp.
There are two more classes of restricted vectors, defined analogously as rank-two (small) and rank-one (very small) vectors. A rank-two vector, Γ 2 , is defined such that both I 4 (Γ 2 ) = I 4 (Γ 2 ) = 0, and a simple example is provided by a vector with all entries vanishing except the p, with the additional constraint that detp = 0. Finally, a very small vector, Γ 1 , is defined such that for any vector Γ. Examples of very small vectors are given by vectors where only the q 0 or p 0 component is nonzero. More generally, we will use the parametrisation (2.14) for a general very small vector, where the choice of normalisation is for later convenience. Note that a general rank one vector can always be written in this way up to a possibly singular rescaling. Since the black hole solutions described in what follows do not depend on the normalisation ofR, this parametrisation is completely general, although it is singular for specific rank one vectors. In the discussion of explicit black hole solutions, we will need to define a second constant very small vector, denoted R * 0 , that does not commute withR, so that in the parametrisation (2.14), it reads 5 (2.15) 5 We use the particular notationR and R * 0 for the two vectors in order to simplify comparison with the notation introduced in [6], as well as with section 4 below, which uses the notation of that paper.

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where e * is defined such that det(e + e * ) = 0 and R , R * 0 = 4 by construction. Despite the fact that this provides a natural parametrisation for R * 0 , it turns out it is not the most convenient, as it obscures the action of T-dualities, which are central to our construction and we describe next.
T-dualities. A crucial ingredient in the description of black hole solutions in supegravity is the action of abelian isometries of the scalar manifold in the real basis. These isometries are defined as including the standard spectral flow transformations, given by (the notation exp( for this action will become clear shortly) (2.17) For the purposes of this paper, we define general T-dualities as the collection of all abelian subgroups in the duality group, obtained from the spectral flows by dualities. These can be described in terms of real vector parameters in the general case, similar to spectral flows, as shown in [6]. We refer to that work for the details of the description in the symplectic real basis and concentrate on the results for the representation of T-dualities that will be used extensively in constructing black hole solutions. It is useful to think of T-dualities as raising and lowering operators T ± on the components in (2.16)- (2.17). This is clearly the case for e.g. the spectral flow parametrised by k in (2.16), whose generators never generate p 0 , while the magnetic components, p, are only generated by the action on p 0 etc. As shown in [6], this structure is general to all T-dualities, which act on four separate eigenspaces in a similar fashion. The relevant generator is given by whereR and R * 0 are two mutually nonlocal very small vectors. One can verify that h T preserves both the symplectic product and the quartic invariant. For example, taking e → 0 and e * → ∞ in (2.14)-(2.15) leads to a pairR, R * 0 along p 0 and q 0 respectively and to the decomposition seen in the spectral flow transformations (2.16)-(2.17). In the following, we denote the four eigenspaces of (2.18) by their corresponding eigenvalue. 6

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Indeed, it is simple to show thatR and R * 0 have eigenvalues +3 and −3 respectively, while the remaining charge components are evenly split into +1 and −1 eigenvalue vectors. For the spectral flows of (2.16), the magnetic components p are of eigenvalue +1, while the electric components q are of eigenvalue −1.
In the general case, one should use the parametrisation (2.14) forR, which by using (2.16), can be written aŝ Here we used the property that the vector is invariant with respect to • T + in the second line. This way it is straightforward to write another parametrisation for R * 0 in (2.15), where such a T-duality parameter appears polynomially, as Of course this base will be rather singular when dete = 0, but this is only the case for isolated points in the moduli space of pairs of rank one vectors with a fixed symplectic product. Using the relations above, we can obtain an explicit representation for general Tdualities, denoted T ± , that will be useful in what follows, especially in section 3. As explained in appendix A, the representation (2.19) and (2.20) allows one to define the generic T + from the spectral flows and their S-dual through (A.17), or explicitly Similarly, one can define the dual T-dualities T − k − through (A.18), but these do not appear in the composite non-BPS system studied here.
We emphasise that all explicit formulae above are fully duality covariant, despite the fact that we use spectral flows as preffered transformations in order to define a representation. On the contrary, our parametrisation identifies the correct combinations of a general charge vector that transform under the simple spectral flows (2.17)- (2.16). To be precise, we record the following rewriting of the charge vector in the preffered basis, It is straightforward to verify that the action of the general T-duality (2.21) on Γ is equivalent to the action of • T + in (2.17) on the combinations P , l Γ , p + e×e dete p 0 and p 0 , in the order they appear in (2.22). Therefore, P is the charge of grade −3, l Γ is of grade −1, while p + e×e dete p 0 and p 0 are of grade +1 and +3 respectively. Similarly, we use the definition in (2.21) to act on the moduli, given the known action of • T ± . By definition, the spectral flow • T − is the T-duality shifting the axions as which is exactly the action of • T − on the physical scalar following by application of (2. 16) on the section in (2.9). Finally, the action of • T + on (2.9) leads to the transformation Here, the inverse is the Jordan inverse t −1 ≡ t×t dett and the first equality expresses the fact that • T + is related to • T − by an S-duality.

Definition of the system
We are now ready to introduce the composite non-BPS system for constructing multicentre black hole solutions. We assume stationary backgrounds and restrict ourselves to the solutions with a flat R 3 base space. We therefore introduce the standard Ansatz for the metric in terms of a scale function U(x) and the Kaluza-Klein one-form ω(x) (with spatial components only), which are both required to asymptote to zero at spatial infinity. Here and henceforth, all quantities are independent of time, so that all scalars and forms are defined on the flat three-dimensional base. For a background as in (2.26), the n v + 1 gauge fields of the theory, together with their magnetic duals, as arranged in the symplectic vector, F, in (2.6) are decomposed as Here, we defined the gauge field scalars ζ, arising as the time component of the corresponding gauge fields, and the one-forms w describing the charges. Of these components, only the vector fields w are indepedent, while the ζ can straightforwardly be constructed once the solution for the scalars is known.

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In order to describe a solution, one therefore needs to specify the spatial part of the gauge fields, dw, the scalar section V (or the physical moduli t i directly), as well as the metric components e U and ω. The composite non-BPS system can be described by introducing two constant, mutually nonlocal very small vectors,R and R * 0 , as above and two vectors of functions, denoted H 0 and K. The former contains eigenvectors of eigenvalues (−1) ⊕ (+3) with respect to the grading (A.2) and will be parameterised as (2.28) Here, L and V are the two functions parametrising the (−1) and (+3) components respectively. 7 The second vector of functions, K, appears only as a parameter of T-dualities that vary in space. We therefore do not need its explicit covariant form, but only the corresponding parameter in the chosen representation, which we denote by K. The two vectors, K and L, are harmonic on the flat R 3 base, as while the function V is specified by the Poisson equation The final dynamical equation required is the one for the angular momentum vector ω, which is given by where M is a new local function that appears explicitly in the solutions. Taking the divergence of (2.31), we obtain the Poisson equation in terms of K and L.
The solutions are then given by the above functions, as follows. The scalars are given by

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where we have used in the second line the explicit form of H 0 and the very small vectors. The physical scalars do not depend on the Kähler phase α. Note that the vector of harmonic functions, K, appears in place of the constant parameter of the basis, k, which can be viewed as the asymptotic value of K, parametrising R * 0 (cf. also the discussion below (4.2)). Similarly, the vector fields are defined from the first order equation (2.34) so that the additional harmonic functions K modify the charges explicitly. One computes the gauge fields scalars according to [6] We note that (2.33) can be solved in exactly the same way as for the BPS solutions [18], which in our basis gives Similarly, the metric scale factor is given by Regularity implies that the n v harmonic functions L must correspond to a strictly positive Jordan algebra element, so that (2.37) leads to a non-degenerate metric and the scalar fields (2.36) lie in the Kähler cone. Strictly positive means that the three eigen values of L must be strictly positive for a classical Jordan algebra, and equivalently in the STU truncation that the three functions L i are strictly positive. Explicit solutions to this system were derived in a particular frame in [4,5] while in the next section we discuss the general solution carrying arbitrary charges in the specific base above. The general manifestly duality covariant solution is derived in section 4, independently of any specific frame.

JHEP10(2013)179 3 Composite non-BPS solutions
In this section we discuss the general properties of composite non-BPS solutions, in the explicit parametrisation of the previous section. This representation is useful in studying the properties of solutions, since it provides explicit formulae for all quantities, as explained above. In particular, the natural parametrisation of the moduli in terms of integration constants in (2.36) allows us to study the behaviour of solutions as a function of the asymptotic scalars for fixed electromagnetic charges.
We find that all regular composite solutions only exist for moduli constrained to a (n v + 1)-dimensional hypersurface with an n v -dimensional boundary defining a wall of marginal stability. The solution admits a non-zero binding energy that tends to zero at the wall, while the distance between the centres diverges, in complete analogy to BPS composite solutions. Somewhat surprisingly, we find that a single-centre solution always has a greater energy compared to the total energy of a composite solution of the same total charge, at points in moduli space where it exists. Finally, we show that one can introduce a notion of attractor tree flow, similar to the existing one for BPS solutions [1].
In section 3.1 we first discuss the general single-centre solution, while in section 3.2 we give a detailed presentation of the properties of the fake superpotential for single-centre solutions, in the basis introduced in the previous section. This completes a longstanding discussion in the literature [8-13, 17, 19], and at the same time establishes various relations that are crucial in our treatment of multi-centre solutions. Indeed, as it turns out, this same function describes the total mass of the multi-centre solutions, and satisfy to a generalisation of the triangular identity that permits to prove the positivity of the binding energy. Finally, in section 3.4 we present an explicit example including two centres, for which we make all relations fully explicit, including a numerical treatment of some aspects of the solution.

Revisiting the single-centre solution
We now turn to an explicit description of the general single-centre solution in the representation introduced above. The general solution was constructed in [20], but it has not been given in a fully explicit form, while the mass formula and the properties of the relevant fake superpotential were only briefly discussed in that paper. In addition, a precise description of these properties will prove crucial in the discussion of composite solutions in what follows.
For a single-centre solution, the functions K can be consistently set a to a specific constant k Γ , 8 which depends on the charge vector Γ. We then find from (2.34) that the charge is defined by the poles of L and V i.e. of grade (−1) ⊕ (+3). This is not a constraint, but rather a choice of basis, as a single charge can be always brought to this form by choosingR and R * 0 appropriately. Indeed, consider a general charge vector and

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constrain the vector e as satisfying dete 2 √ 2 R , Γ = p 0 + tr ep − tr e × e q + dete q 0 = 0 , (3.1) which from (2.23) sets the grade (−3) charge to zero. One then finds For a single-centre solution, one may additionally choose the grade (+1) component of the charge to vanish, by choosing k appropriately. The appropriate value, k Γ , is found by setting the third row in (3.2) to zero, as One then obtains the charge which is indeed a general vector of grade (−1) ⊕ (+3) for k = k Γ . Note that we used the parametrisation I 4 (Γ) 4detl Γ for the charge of grade (+3), instead of the equivalent expression in the last line of (3.2). The general solution of (3.4) for k Γ is where the determinant of l Γ is given explicitly by dete detl Γ = detp + q 0 tr e × e p × p + dete detq + −dete q 0 + tr e × e q (q 0 p 0 + tr qp)

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It is then straightforward to define the general single-centre solution from these data. 8) and the corresponding section reads

One chooses the vector of harmonic functions
One then obtains the scaling factor and the scalar fields where we used (2.25) and (2.24). The solution will be regular provided 13) and the vector L × L is a positive Jordan algebra element everywhere. 9 For l positive, this requires that l Γ be positive, which fixes some conditions on the vector e that parametrises partially the asymptotic scalars. Note that, in principle, we should consider regularity of k Γ as well, but since the denominator of the explicit solution in (3.6) is dete detl Γ , this condition is already implied by the regularity of l Γ , e. Before concluding our discussion of the single-centre solution, let us return to the choice made above (3.1) and note that one can write the same solution with non constant K.

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This function is however quite restricted, since the requirement of regularity at the horizon implies that its poles are proportional to those of L. The relevant expressions for the various functions then follow from (2.30)-(2.32) as where γ is the proportionality constant relating the poles of L and K. The asymptotic scalars are then parametrised according to for k = k Γ + γl. The full expression for the moduli follows from (2.36) and, as it turns out, is equivalent to the one in (3.12), where all functions are harmonic. One can easily check that (3.14) is only a rewriting of the simple single centre solution, since γ can be absorbed in a redefinition of the parameters, as m → m + γdetl. The proof in the general frame independent case is given in B. This redefinition defines a different set of coordinates in moduli space (3.15), which will prove useful in various settings below.

The fake superpotential
The mass formula for single-centre solutions is crucial for the applications that follow, especially in comparing the mass of multi-centre solutions to that of their constituents. We therefore wish to rewrite the explicit expression of the mass in the representation used in this paper, in terms of the fake superpotential proposed in [8] and defined in [12,13]. Using the parametrisation (3.12) for the moduli, the non-BPS mass formula takes the rather simple form Similarly, the asymptotic central charge in this basis is Noting that the constant m is finite for regular values of the moduli, it is simple to verify that W (Γ) > |Z(Γ)|, provided I 4 (Γ) < 0. In contrast, one would have W (Γ) < |Z(Γ)| for I 4 (Γ) > 0. This proves that such a regular non-BPS extremal black hole always satisfies to the BPS bound. We emphasise that our formula (3.16) is not linear in the charge Γ, as the parametrisation (3.12) of the asymptotic scalars we use depends implicitly on the charge (through k Γ and e via the condition R , Γ = 0). In order to understand this property, it is convenient to rewrite the mass formula in a form that only depends on the asymptotic JHEP10(2013)179 scalars, the charges, and the auxiliary vector e. This vector, defined such that it satisfies to (3.1), can then be understood as parametrising the n v − 1 flat directions associated to the charge vector Γ. To this end, it is convenient to introduce some shorthand notation, that will be used in the remainder of this section. First, we define one complex and one real variable which appear in all expressions involving W . Here, the inverse is the Jordan inverse e −1 = e×e dete (and similarly for (t + e −1 ) −1 ). It is important to note that these variables only depend on the moduli t, the electromagnetic charge Γ and the parameter e. Using these objects, one computes indeed that W (Γ) can be rewritten as where l Γ is given by (3.3). Note that this expression is linear in the charge Γ. In this form, the fake superpotential reproduces the formula derived in [13], where the vector e satisfying to (3.1) parametrises the n v − 1 flat directions associated to the charge Γ. Moreover, this parametrisation of the flat directions exhibits the similarity of the fake superpotential and the central charge in this basis. The latter can be shown to take the form 20) in this basis, using the constraint Γ,R = 0. Note that the explicit dependence of (3.20) on e is due to exactly this constraint, and can be eliminated by rewriting l Γ and p e in terms of the charges. According to [13], e must be such that it extremises W (Γ, e), with respect to variations preserving (3.1). In order to check this property we compute the variation of W with respect to e −1 while keeping the charge and the moduli fixed. The variation of u following from (3.18) reads so that we obtain For the single-centre solution, one computes that this variation reduces to
in agreement with [13]. Note that it is important [13], that the flat directions parameter extremises the fake superpotential for arbitrary moduli, and so the reader may worry that we only check this variation within the solution. But note that the asymptotic scalars are completely arbitrary in this solution, and so this is perfectly consistent.
In the construction of [13] it is also important that these extrema are unique, so as to fix unambiguously the expression of the fake superpotential in terms of the charge Γ and the moduli. To check this, we can simply consider the moduli to be parametrised by (3.15) with k arbitrary and not necessarily equal to k Γ , in which case e would extremise W as we just explained. One computes that the condition that δW is proportional to tr l Γ δe gives for some arbitrary Lagrange multipliers γ. Using the property that l is positive, one can simplify this equation to which because l Γ is also positive, reduces to Since the term in γ can always be reabsorbed in a redefinition of m as in (3.14) without affecting e, we find that the unique solution for e is indeed the expression it takes for a single-centre solution.
Beyond the first order variation (3.23), it is important for the multi-centre applications that follow to consider the second variation of W as well, as it turns out to be crucial in comparing the mass of a composite to that of its constituents. In the remainder of this section, we compute explicitly the Hessian of W at its extremum, viewed as a function of e, imposing the constraint that this vector is such that Γ,R = 0. We find that the resulting quadratic form is negative definite along all directions preserving the constraint, in an open set in moduli space for general charges, so that one can extend the result to the full moduli space by duality. Because the extremum is unique, it follows that it is moreover a global maximum. The result that the extremum of W is moreover a global maximum is crucial in defining a generalisation of the triangular identity for BPS black holes, which states that the mass of a composite is always lower than the masses of its constituents. However, the details of the proof are technical and not directly relevant for the remainder of this paper, so that they can be skipped by a hasty reader.

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In order to prove that the second derivative of W is a negative definite quadratic form, we consider the explicit form of the latter, which reads Substituting the single-centre expression one obtains where in the second equality we neglected the component that vanishes assuming that the variation of e preserves R , Γ = 0. We shall prove that the above defined quadratic form is negative definite for appropriate variations of e −1 preserving this constraint. However it is generally not negative definite for arbitrary variations δe −1 , therefore it is important to take the constraint into account.

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In order to proceed, it turns out that a change of variable from e to k is useful, where k is the arbitrary vector parametrising the asymptotic scalars as in (3.15). Because we consider the variation of e −1 at fixed moduli, the variation δe −1 is determined by the corresponding variation of k such that (3.15) is kept constant. For k = k Γ , one can always find the corresponding l , e , m such that For infinitesimal variations of the parameters in the vicinity of k = k Γ , one obtains Note that the variation of k Γ is required because the correct k Γ as a function of e is evaluated at e + δe at this order. It is convenient to rewrite this equation as One can compute the variation of k Γ as such that the terms quadratic in k Γ cancel in (3.33) after substituting u = m−i detl l + k Γ . Decomposing (3.33) into its imaginary and real components one obtains where δm is itself determined such that tr l Γ δe −1 = 0. The last two expressions exhibit that δe −1 and δl are completely determined by the variation δk. The second formula in particular provides the required change of variable from δe −1 to δk. As a variation of δk proportional to l can be reabsorbed in a redefinition of m (cf. (3.14)), one can restrict attention to the variations of k linearly independent of l. This parametrisation of the flat directions in terms of k is more useful because we can forget about the constraint, which now simply determines the decomposition of (3.36) in the variations δe −1 and δm. Moreover, it is the variation δk rather than δe −1 that will appear explicitly in the twocentre solutions as will be shown shortly.

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The quadratic form written in terms of δk is a rather complicated expression in general, and we shall only consider the limit of small and large l. These limits both correspond to asymptotic scalars that have order one axions and very large dilatons. Note however that we do not consider any restriction on the electromagnetic charges, so it is enough to prove the result on an open set in moduli space to ensure that this property holds in general, given that the mass formula is duality invariant and the duality group G 4 acts transitively on the moduli space. The only restriction on our representation arises from singularities at regions where dete = 0, but the corresponding configurations define isolated points in moduli space. Therefore, showing that the extrema of W (Γ, e) are maxima on an open set in moduli space for all charges, is enough to circumvent this issue. The reader might note that the domain of large dilatons in moduli space corresponds to small volume moduli in string theory, and the supergravity approximation cannot be trusted in this limit. However this is only a technical detail at this level, since the proof extends by duality to all values of the moduli.
Small l. For very small l, the expression of δe −1 simplifies to and the quadratic form reads which can be shown to be negative for all δk. To prove this one can use the G 5 invariance of this equation to chose both l and l Γ to be diagonal Jordan algebra elements sitting in the STU truncation. Regularity of the solution then requires all components of l and l Γ to be strictly positive. This permits to rewrite which is manifestly negative. The last line shows that the non-diagonal components of δk necessarily contribute negatively to δ 2 W and (δk) 2 is the Jordan square of δk, i.e. as a matrix in an explicit basis. For completeness, we note that this region corresponds to small JHEP10(2013)179 volume moduli and finite axions Large l. In this case δe −1 reduces to After some algebra one obtains the perturbation of the mass to be Although this is not manifest in this equation, it is possible to show that (3.42) is always strictly negative, in a similar way as shown for the case of very small l, provided that δk is not proportional to l (in which case δe −1 itself vanishes). To do this we restricted ourselves to the STU truncation in a first place, requiring that all components of l, l Γ are strictly positive. Using the same argument as for the small l limit, one then shows that the non-diagonal terms of δk only contribute negatively because is a strictly positive Jordan algebra element. This region also corresponds to small volume moduli with finite axions

The multi-centre solutions
We now turn to multi-centre solutions, using the same parametrisation of charges and moduli as for the single-centre solutions above. The scalar section and the gauge fields are now given by (2.33)-(2.34), where the harmonic functions K parametrising the T-dualities are now nontrivial. In the preferred basis of the previous section, we consider a system of N centres, labeled by an index A. Then, (2.34) implies that all charges commute with the vectorR, so that all P A = 0 in the decomposition (2.22). Of the remaining components, it turns out that only the l Γ A appear in the various expressions, since we have where k, l and γ A are constants and r A = |x − x A | is the distance from centre A. We stress here that regularity imposes that the poles of K and L be linearly dependent at each centre (see the constraint (4.34) ). Given these expressions, the solutions to (2.30)-(2.32)

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are given by (4.36), (4.37) and (4.43) upon substituting the explicit expression (2.28) for H 0 , leading to In these expressions, J Ai is the intrinsic 'under-rotating' angular momentum at each centre, R AB is the distance between the centres labeled by A and B, while the function F A,BC was defined in [5] as the everywhere regular solution to (4.35). We refer to that work for the properties of this function. Using these explicit functions, one can now write down the scalar fields and the metric, using (2.36)-(2.37). In addition, one can readily understand the property that the poles of K and L must be linearly dependent at each centre, by considering the explicit expression for the scalar fields at the horizons which leads to the requirement that K − γ A L must be regular at x A . The finite values of these functions at each horizon define a set of T-duality parameters that play an important role in the definition of the electromagnetic charges.
In the multi-centre case, the charges at the various centres are allowed to have nontrivial grade (+1) components, but are still constrained to have a vanishing grade (−3) component, as (2.34) commutes withR. It follows that the most general charge allowed in each centre JHEP10(2013)179 is given by (3.2). In the explicit parametrisation of the previous section, the additional components can be computed by decomposing all charges as in (3.5), i.e. by viewing each charge as the result of a T-duality acting on the poles of L and V . This is conveniently realised in terms of the functions above, since the expression for the charges at a given centre, obtained by integrating (2.34), is indeed given as a T-duality of parameter K(x A ) − γ A L(x A ) acting on an underlying vector defined from the poles of L and V at that centre, as because the action of T-dualities is faithful on charges carrying a non-zero grad 3 component as does (3.48). Alternatively, the same result is obtained by use of the general formula derived in (4.41) below, which can be used in any other basis as well. Note that (3.49) is consistent with the property that k = k Γ for a single centre solution, due to (3.4) and (3.14), since the single-centre limit of the multi-centre solution naturally leads to a nontrivial K. We stress that although this formula is identical to the ones displayed in [4,5] in a specific duality frame, the dependence of the vectors l, k, k Γ and l Γ in terms of the charges and the asymptotic scalars is here manifest. Within the formulation of this paper, we can therefore keep the charges fixed and rather consider (3.49) as a constraint on the asymptotic scalars.
Although there is no solution for generic charge configurations with more than three centres, the problem generally admits a solution for two centres. In this case, one can easily solve (3.49) as between the two centres, denoted by R 12 . This sums up to a total of n v + 1 parameters for the 2n v asymptotic scalars (this holds for n v ≥ 3). Note that the parameter m does not count, because it can always be reabsorbed in a redefinition of k. This is a more general property that persists when adding more centres, so that composite non-BPS solutions only exist for moduli constrained to an (at most) (n v + 1)-dimensional subsurface, specified by the charges at the centres.
For the asymptotic scalar fields to be well defined, l in (3.50) must moreover define a positive Jordan algebra element. As both l Γ 1 and l Γ 2 must be positive for the solution to be well behaved at the two horizons, the positive contribution must come from k Γ 2 − k Γ 1 . Regularity therefore requires that k Γ 2 − k Γ 1 is a strictly positive Jordan algebra element (or strictly negative, depending on the sign of γ 1 − γ 2 ). Indeed, one can always find a G 5 element that rotates l Γ 1 + l Γ 2 to a Jordan algebra element proportional to the identity. The group K 5 defined as leaving the identity element invariant then permits to rotate k Γ 2 − k Γ 1 to a diagonal Jordan algebra element. For the three eigenvalues of l to all be positive, it is then clear that the three eigen values of (γ 1 − γ 2 ) −1 (k Γ 2 − k Γ 1 ) must themselves be strictly positive.
A final aspect of the multi-centre solutions worth discussing is the issue of flat directions. As is well known, the scalar fields of single-centre solutions admit n v − 1 flat directions, in the sense that n v − 1 of the 2 n v scalar fields are undetermined constants throughout the flow. In the description of the previous subsection, one can easily check that the n v − 1 parameters in e account for exactly these flat directions. These directions can be also viewed as the invariance group of the non-BPS charge vector, embedded in the duality group [21]. These are the flat directions of the individual centres, whereas the flat directions associated to the common stabilizer of the two charges are much more restricted if not trivial. The latter define the actual flat directions of the solution, and we shall discuss them latter.
The scalar fields at the horizon x A , (3.47), are still localised at the non-BPS attractor of the corresponding charge Γ A , and therefore only know about the global structure of the solution through the explicit expression of their own flat directions parametrised by e. In this case, e is not determined by the charge Γ A of the centre at hand and the asymptotic moduli only, as it would be in the single centre solution, but is determined by the property that e extremises the fake superpotential W ( A Γ A , e) with respect to the variations leaving invariant all the constraints R , Γ A = 0 for all centres A = 1, N . Through this property, e is in fact a function of the asymptotic scalars and all the charges Γ A . This vector does not depend explicitly on the distances between the centres, although the latter are eventually determined in terms of the asymptotic moduli and the individual charges themselves.
Given the above, the possibility of genuine flat directions for multi-centre solutions is not excluded. This turns out to depend on the values of the charges at the centres, as we show explicitly in section 3.4 for a two-centre example of restricted charges, while in appendix C we discuss the classification of the allowed flat directions for a two-centre configuration of generic charges.

Binding energy of composite states
One of the most important advantages of obtaining explicit general multi-centre black hole solutions, as we have done in this paper, is the possibility of studying the binding energy of the constituents. Indeed, showing that the solutions obtained are genuine bound states, rather than collections of marginally interacting objects, could be useful in the study of the non-BPS bound states at the microscopic level.
The energy of a composite solution is defined by the standard ADM expansion of the metric at infinity, which can be carried out in the general multi-centre case. For the solutions in the previous section, this involves expanding the metric scale function in (2.37) near infinity, using the expressions (3.45)-(3.46). The resulting expression, in terms of the parameters introduced above, takes the form Note that the first line of this expression is simply the sum of the single-centre mass formulae (3.16) for each centre. In the non-interacting limit, where all charges mutually commute, all γ A are equal and k = k Γ + γl, so that the second line can be reabsorbed into a redefinition of m, as in (3.14). While the structure of (3.52) is suggestive of a nontrivial binding energy, verifying this directly is rather complicated in general. Using the parametrisation (3.15), one shows that (3.52) can still be rewritten in terms of the fake superpotential (3.19), where the value of e does not however extremise each of its components in the sum. This simplifies the comparison to the mass of the constituent centres, since the single-centre mass, M A is found by extremising the fake superpotential, W (Γ A , e), with respect to e, to obtain a vector e A . As we have seen in section 3.2, the resulting value W (Γ A , e A ) is a global maximum of the fake superpotential, and therefore one finds for each centre separately. This directly implies that the binding energy of any composite solution is necessarily positive, since Moreover, by exactly the same argument, one finds that the energy of a composite solution is lower than the the energy of a single-centre solution of the same total charge, as where Γ t = A Γ A . We then conclude that single-centre non-BPS solutions are energetically disfavored over multi-centre solutions with the same total charge. Note that this comparison can only be done on the relevant hypersurface in moduli space where the composite solutions exist. This property is in contrast with the BPS case, in which the mass is uniquely determined by the total charge and the moduli. However, we should mention that single-centre black holes are generically entropically favored over multi-centre black holes, as can be computed using the area law and the properties of the quartic invariant. We shall prove that this is always the case in the two-centre configurations we consider in section 3.4. The discussion above illustrates that the parameter e, describing flat directions for non-BPS solutions, plays a role analogous to the one of the Kähler phase α of the central charge in the corresponding BPS solutions. Indeed, the linear mass formula for a BPS black hole also follows from a superpotential, given by where α is chosen such that W BPS is maximised, i.e. to be the phase of the central charge Z(Γ). The resulting mass formula is of course The energy of a two-centre solution then satisfies the triangular identity where α is the phase of Z(Γ 1 + Γ 2 ) and α A are the phases of Z(Γ A ) for A = 1, 2. Positivity of the binding energy between the two centres then follows from the property that α A is a maximum of Re[e −iα Z(Γ A )]. As seen above, the composite non-BPS mass formula obeys the same property, but with the subtle difference that, if a single-centre non-BPS solution of charge Γ 1 + Γ 2 exists, it has a mass bigger than that of the bound states of constituent charges Γ 1 and Γ 2 , due to (3.56). For a two-centre solution, one can make these properties more explicit in the regime of large separation of centres, using the expression (3.49), to obtain the variation from the single-centre solution. In this case, the term proportional to γ A l can be reabsorbed into a redefinition of m, so that the relevant small perturbation δk that determines δe in (3.36) is whenever the solution exists, provided l Γ B is linearly independent of l. This can be used in (3.29) to verify that the two-centre binding energy is indeed nontrivial whenever the JHEP10(2013)179 solution exists. Moreover, (3.52) in the two-centre case simplifies to where we used the explicit form of k in (3.60). This expression of the mass looks naively like the sum of the individual masses plus a manifestly positive quantity on the second line, which would be in contradiction with a positive binding energy. It is important to point out however that this is not the case because these expressions of the individual masses do not correspond to the individual masses at the same moduli whenever R 12 < ∞, because then k = k Γ A + γ A l. When R 12 → ∞, one gets instead that k = k Γ A + γ A l and the second line vanishes, which shows that the binding energy vanishes in the limit of large radius. This limit corresponds to a wall of marginal stability in moduli space, as we shall discuss in more detail in the next subsection. It is interesting to compare this expression of the mass to the central charge, which reads Now, for a regular solution, one has to demand both that tr l×l l Γ A detl > 0 and that where all three terms are separately positive. These formulae show explicitly that the energy is always strictly above the BPS bound, as expected. We stress that this result holds everywhere in moduli space, where the non-BPS solution exists. It is interesting to note that using (3.50) one finds that there is always a critical radius R c at which the vector l becomes degenerate (i.e. detl = 0), and the asymptotic scalars are singular, as Im(t) × Im(t) = O(R − R c ) for finite value of the other parameters. One computes that both the energy and the central charge diverge as (R − R c ) −1 in this limit, while still consistent with the BPS bound up to order O(R − R c ) (3.65) This limit should not be considered as a boundary of the (n v + 1)-dimensional surface in moduli space on which the solution exists, since it is itself at the boundary of moduli space, consistently with the property that the BPS bound is saturated in this limit.

Attractor tree and walls of marginal stability
Given the results of the previous subsection on the positivity of the binding energy, the natural next step is to consider the possibility of decay of composite solutions at regular points in moduli space, i.e. the existence of walls of marginal stability. Before turning to the corresponding analysis, we emphasise that the question at hand is in principle more subtle for non-BPS composites, which only exist on appropriate hypersurfaces in moduli space, compared to the BPS solutions, which exist in codimension zero subspaces of the moduli space. In practice, this means that BPS solutions exist in codimension zero domains in moduli space, whose boundaries define walls of marginal stability where some of the distances R AB diverge. For non-BPS composites one finds exactly the same situation, but now restricted on the relevant hypersurface where the given solution exists, as discussed above. It then follows that the walls of marginal stability are only defined on the appropriate hypersurfaces as their boundaries in moduli space and do not extend outside of them, as we discuss in more detail now. Consider a general two-centre solution, as described by (3.49)-(3.51) above. Assuming that k Γ 2 − k Γ 1 is indeed a strictly positive Jordan algebra element, the solution clearly exists for arbitrary large distance between the centres, R 12 , since the distance dependent term in these relations becomes irrelevant at this limit. Taking the limit R 12 → ∞, the moduli are regular, so that one finds a wall of marginal stability for finite moduli.
For a marginally bounded solution, one expects that the energy of the composite solution is equal to the sum of the masses of the constituent black holes. In order to verify that, we consider (3.52) in the limit of marginal stability, where (3.49) becomes simply This is identical to the value of k for which (3.52) describes non-interacting centres, as explained below that equation. It follows that the total energy decouples lim This formula generalises to any number of centres, as long as the solution exists. Suppose that we have a solution to (3.49) for N + 1 centres, such that l is strictly positive, in the limit where x A goes to spatial infinity for some A 0 , then (3.66) again applies for that value A = A 0 , while (3.49) for A, B = A 0 define the corresponding equations for N centres. Therefore, one finds that the mass formula satisfies The property that the binding energy is finite at finite radius relies on the property that e is not an extremum of each W (Γ A , e) for generic variations preserving R , Γ A = 0 separately. Nonetheless, given a total charge Γ t and a set of charges, Γ A , such that Γ t = A Γ A , one checks that e extremises W on the subvariety satisfying R , Γ A for all A.

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Indeed, one computes using (3.49) that Here one can interpret the coefficients γ A as Lagrange multipliers for the conditions The existence of a well defined extremum then constrains e and the moduli. For a regular multi-centre solution e must in fact satisfy (3.70), but for a single centre of charge A Γ A the e following from (3.69) does not extremise correctly the function W Γ t , e in one direction, and would not reproduce the same mass, as discussed in (3.56). In fact it follows that unlike for the BPS solutions, a multi-centre non-BPS solution admits an energy strictly lower than the energy of the single-centre solution with the same total charge and asymptotic scalars, so the composite configuration is energetically favored whenever it exists. Given this extremisation problem, we may now ask the reverse question and consider a top-down approach where one seeks to infer criteria for the existence of solutions from the function W (Γ A , e) alone, instead of using the explicit knowledge of solutions to derive the properties of the fake superpotential. One may wonder if the condition that e extremises W on the subvariety satisfying (3.70) is strong enough to ensure the existence of a solution. One computes in general that for e to be such an extremum, the condition must hold for some arbitrary Lagrange multipliers γ A . Using the property that l is positive, one can simplify this to where A AB is an antisymmetric matrix. Upon identifying the γ A as the proportionality constants in (3.44), this solution would reproduce equation (3.49) for However, this is not simply a particular parametrisation of A AB , since an N × N antisymmetric matrix comprises N (N −1) 2 independent components, whereas there are only 3(N − 2) independent distances R AB in three dimensions for N ≥ 3 (so that the solution is not general for N ≥ 5). Therefore the condition that e extremises W on the appropriate subvariety satisfying (3.70) does not ensure the existence of a solution in general.

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Nevertheless, this shows that there is a natural generalisation of the existence of an attractor flow tree associated to BPS composite solutions. Indeed, any solution to (3.49) with N + 1 centres admits a large radius limit in which the solution decouples in a single-centre solution and an N -centre solution to (3.49), as in (3.68). Therefore, one can solve (3.49) by adding each centre one after the other by following the inverse procedure that ensures (3.66) to be satisfied at each addition. Moreover, the existence of a limit of marginal stability implies that for any two-centre solution of charges Γ 1 and Γ 2 , there exist moduli for which the solution e extremising both W (Γ 1 , e) and W (Γ 2 , e) is the same. Then, adding a third centre requires that there exist moduli such that e extremise W (Γ 1 + Γ 2 , e) with respect to variations preserving both R , Γ 1 and R , Γ 2 , and extremise W (Γ 3 , e) with respect to variations preserving R , Γ 3 = 0. Each new centre or cluster of centres already satisfying (3.49) requires similarly the existence of a common extremising vector e for some moduli.
This construction clearly extends Denef's attractor tree, where the role of the central charge phase is now played by the Jordan algebra element e parametrising the flat directions of the individual centres. The existence of such a non-BPS attractor tree is clearly required for the solution to exist, and it is therefore natural to wonder if it provides a sufficient condition. However, the construction of such a tree is a rather difficult task in practice. An obvious obstacle is that, unlike for BPS solutions, where a closed form formula for the phase of the central charge is available, there is no such formula for the vector e extremising the fake superpotential.
It is important to point out that the restrictions on e can rapidly become overconstraining. In particular, for n v centres one finds configurations of charges such that R , Γ A determines e completely. Then there is no freedom in tuning e to accommodate a new centre anymore, and although the solution may still exist, it can only be marginally stable. For example, this is the case for solutions within the one modulus model, for which e is necessarily fixed already for a single-centre solution. In this case we have found that there exist two-centre solutions, but the total energy is independent of the distance and is equal to the sum of the constituent masses.

Explicit two-centre example
In this final subsection, we make the above considerations fully explicit for a two-centre example. While the structure and properties of the solution are exactly those discussed in the general case above, having an explicit example allows for an even more detailed description of the possible bound states. We display most of the formulae within the STU truncation, which does not constitute a restriction for the charge configurations we consider. We also perform numerical estimates of the binding energy. We consider a system of two black holes, carrying charges Γ 1 and Γ 2 respectively, which we choose as follows. Up to an electric/magnetic duality, one can always bring one of the JHEP10(2013)179 charges to be a D0-D6 with q 0 = −p 0 ≡ Q 0 > 0, so we choose for simplicity (3.75) Given that we used duality covariance to restrict one of the charges, the second charge is a priori unrestricted for the most general two-centre solution up to dualities. Here, we shall nevertheless restrict the second charge to be a D0-D4-D4-D4-D6 for simplicity. We do not expect a significant change in the physical properties of the solution by adding a D2 charge. We therefore take where we assume that I 4 (Γ 2 ) = −4q 0 detp − (p 0 q 0 ) 2 < 0, so that we are indeed dealing with two non-BPS charges. Note that the inner product Γ 1 , Γ 2 = Q 0 (q 0 + p 0 ), does not depend on the magnetic charges, p, and the interaction of the two centres vanishes in the limit q 0 = −p 0 . It is also worth mentioning that the uplift of this configuration to five dimensions describes a pair of doubly spinning extremal Kerr black holes 10 located at the two tips of a two-centre Taub-NUT geometry, while the magnetic fluxes p are threading the two-cycle between the centres. We start by considering the relevant constraints coming from the two charges on the auxiliary vector e. First, the condition R , Γ 1 = 0 simply implies that dete = 1, and (3.3), (3.6) for Γ 1 also simplify dramatically, to give The regularity condition at the horizon requires that e is a positive Jordan algebra element (i.e. all three e i are positive numbers within the STU truncation) and we shall always assume that this condition is satisfied. Using the results of subsection 3.1, it is straightforward to construct the general single center solution carrying the particular D0-D6 charge in (3.75). 11 Turning to the second charge (3.76), the equations (3.1) and (3.3) become respectively Solving for p 0 using the first relation, one finds

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as well as where we wrote the expressions within the STU truncation in the second lines. These are relevant in what follows, because we can always use the G 5 symmetry of the problem to diagonalise both e and p and solve these equations within the STU truncation without loss of generality. There are two inequivalent branches of solutions depending of the sign of γ 1 − γ 2 , since regularity requires (γ 1 − γ 2 )(k Γ 2 − k Γ 1 ) to be strictly positive.
Existence of solution. The conditions above for γ 1 − γ 2 > 0 can clearly be solved by requiring that q 0 and p are both positive, and p 0 is negative, but this is not the only solution. Without loss of generality, we can assume that One can then verify that regularity requires q 0 > 0, while p is not necessarily positive and satisfies It follows that q 0 , p 2 , p 3 must all be strictly positive, but p 1 can possibly be negative. Note that the presence of a nonzero p 0 is required for the latter possibility so that I 4 (Γ 2 ) < 0 (even though we have solved for p 0 in all equations). Indeed, we have by construction that min q 0 e 3 p 3 , e 3 p 3 q 0 ≤ 1 and therefore so that −p 0 > q 0 + e 3 p 3 . This also guarantees that the two charges are never mutually commuting, as The existence of a regular solution for e to the first of (3.78) leads to a bound on p 0 . For p 1 ≤ 0, it turns out there are solutions for −p 0 > q 0 arbitrary close to q 0 , whereas it must satisfy to 12 − p 0 > q 0 + 3 3 detp , (3.85) when p 1 > 0. The limit case −p 0 = q 0 + 3 3 √ detp is physical but degenerate, and will be discussed separately.
We shall now concentrate on the example without flat directions, for which p is strictly positive. In the general situation described by (3.85), it is simple to consider the STU JHEP10(2013)179 truncation, where one can solve for e explicitly as e 2 = −e 1 p 0 − e 2 1 p 1 − e 1 q 0 ± −4e 1 p 2 p 3 + (e 1 p 0 + e 2 1 p 1 + e 1 q 0 ) 2 2e 1 p 2 where the signs appearing in the two expressions must be opposite. Defining one obtains that in the generic situation, for which x > 1, a positive solution of (3.78) for e exists, provided that e 1 satisfies 2x + a 1 +ā where a i are the three cubic roots of 2−x 3 +2i √ .
This inequality shows explicitly that one can reach infinite radius at finite values of the moduli, and therefore exhibits the existence of a wall of marginal stability. Within the STU truncation, the six moduli are parametrised by the four free parameters R 12 , e 3 , γ 1 , γ 2 , up to the conditions (3.88), (3.89). Therefore, the wall defined at R 12 → ∞ is clearly of co-dimension one in the 4-dimensional hypersurface of allowed moduli. Note that R 12 can be arbitrarily small, but the limit R 12 → 0 is located at the boundary of moduli space. A similar analysis leads to the same conclusions for p 1 ≤ 0.
It is also interesting to compare the entropy of such a two-centre configuration with the entropy of the single-centre solution that would carry the same total electromagnetic charges and total angular momentum. For simplicity, we consider the case in which the intrinsic angular momenta of the two individual centres vanish. In this case the total angular momentum of the solution is the symplectic product of the two charges. Using the regularity conditions that Q 0 > 0, q 0 > 0 and, either detp > 0 and (3.85) is satisfied, or and therefore We conclude that for all the regular solutions we consider in this section, the entropy of the single-centre solution carrying the same total electromagnetic charges and angular momentum is always strictly greater than the entropy of the two-centre solution. We shall only very briefly discuss the branch γ 1 − γ 2 < 0. In this case one shows that for ordered values of p i e 1 p 1 ≤ e 2 p 2 ≤ e 3 p 3 , (3.92) regularity requires q 0 > 0 and
These conditions also imply that p 0 < 0. The total charge admits necessarily a strictly negative quartic invariant. The angular momentum produced by the dyons has the opposite sign There is no drastic difference between the two branches, so let us close this parenthesis and come back to the branch γ 1 − γ 2 > 0 for the rest of this section. We did several numerical checks of this solution in the parametrisation above, unfortunately excluding the region of large volume in moduli space. The reason is that reaching large imaginary values requires a severe fine tuning of the parameters, such that l is very large and at the same time An example plot of this function for p 1 < 0 is shown in figure 1, which immediately shows that it falls off monotonically as one approaches asymptotic infinity. Here, we choose not to display any further details, since this behaviour seems to be universal for all examples we considered. The numerical simulations we have been doing all manifest the behaviour that so that the binding energy is extremely small compared to both the total energy of the system and the energy above the BPS bound. In connection to this, it is interesting to note that given a two-centre non-BPS composite, generically one can define a BPS solution with the same total charges and moduli, e.g. as two centre configurations. Such a solution is then largely favored energetically and one may expect the non-BPS composite to decay into the BPS composite by tunneling relatively rapidly. Although they carry the same total charges, these solutions are however extremely different on large scales, and such a decay would require rather non-local quantum effects, and it is by no means clear that this can occur in string theory. Let us comment finally on the limiting case −p 0 = q 0 +3 3 √ detp, when there is a unique solution e = p×p (detp) 2/3 for the vector parametrisingR. In this case, the solution is always equivalent up to duality transformations to a solution of the one modulus model (or T 3 model). Since a non-BPS charge has no flat directions in this model, the binding energy always vanishes in this case, and the energy of the solution is independent of the radius. This solution is therefore only marginally stable for all values of the radius. This behaviour is peculiar to the T 3 model and it is worth mentioning that the composite non-BPS system is associated to a nilpotent orbit that in fact does not exist in g 2(2) [4], the three-dimensional duality group of this model. The existence of non-BPS composite solutions within the T 3 model despite the absence of the relevant nilpotent orbit in g 2(2) was pointed out in [17].
Flat directions. One can distinguish three cases among these examples, depending on the sign of the minimum eigen value p 1 of p. The latter determines the flat directions associated to the solution. The D0-D6 charge is by construction left invariant by a subgroup G 5 ⊂ G 4 , whereas the common stabiliser of the two charges is the stabiliser of p in G 5 . The stabiliser of p is the maximal compact subgroup K 5 of G 5 if p 1 > 0, whereas it is a noncompact real form K * 5 ⊂ G 5 if p 1 < 0. The non-compact real form K * 5 is the divisor group that would define the pseudo-Riemannian scalar manifold K * 5 \G 5 of the theory obtained by time-like reduction of a genuine six-dimensional theory, so we shall write its maximal compact subgroup as K 6 ⊂ K * 5 . 13 In the special case p 1 = 0, the stabiliser subgroup IK 6 13 We write this group K6 because it is also the maximal compact subgroup of the six-dimensional theory duality group for magic supergravity theories. For the infinite series of axion-dilaton theories, K6 ∼ = SO(n) is the compact group acting on the n vector multiplets coupled to gravity and one tensor multiplet in six dimensions.

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is the contracted form of K 5 interpolating between the two real forms K 5 and K * 5 . For example, in the exceptional theory with Kähler geometry (U(1)×E 6(−78) )\E 7(−25) , one has (3.98) In the second branch γ 1 − γ 2 < 0 we have seen that p is a strictly negative Jordan algebra element and the stabilizer K 5 is compact. Note however that these configurations are not generic, because the common stabiliser in E 7(−25) of two independent generic vectors Γ 1 and Γ 2 is either Spin (8) or Spin(1, 7) in general, as shown in appendix C. Note that when Spin(8) stabilizes both p and q, one can always use E 6(−26) to diagonalise them, such that they can be realised within the STU truncation, whereas this is not possible when their common stabilizer is Spin(1, 7). In the latter situation, both p and q are negative, and are linearly independent.
One can easily convince oneself from our analysis that such configurations indeed exist. The positivity condition on l Γ 2 and k Γ 2 − k Γ 1 are identical upon substituting p − 2e × q to p and adding to k Γ 2 − k Γ 1 a term linear in q. If one consider a situation in which q is very small compare to the other charges (q << p), it is clear that one can find solutions as deformations of the D0-D4-D6 ones. This does not require any particular property of q with respect to p apart from being very small, and so one can clearly find regular solutions for charge of common stabilizer Spin (8) or Spin(1, 7).
The two-centre solutions therefore admit drastically different sets of flat directions, from zero to sixteen in the exceptional theory, e.g.
{1} ⊂ Spin(7)\Spin(1, 7) ⊂ · · · ⊂ Spin(9)\F 4(−20) . (3.99) Although the flat directions associated to the individual charges seem to play an important role in the physical properties of the solution through their link to the vector e parametrising the fake superpotential, it is not clear at this stage if the common flat directions of the two charges carry similar properties. We did not find specific differences between the solutions carrying or not flat directions.

Derivation of composite non-BPS solutions
In this section, we present the detailed analysis of the duality covariant form of the composite non-BPS system, as defined in [6]. This leads to a characterisation of solutions in terms of harmonic functions in an arbitrary symplectic basis, leading to the results already presented in section 2.2 in a convenient basis. After a short summary of the system as defined in [6] in section 4.1, we discuss the general single-centre solution of the multi-centre system in section 4.2. This turns out to be slightly more complicated than the purely single-centre system of [20], but leads to exactly the same physical results. We then turn to the analysis of the multi-centre configurations in section 4.3, where we present the general solution of the system in an arbitrary frame and give the duality covariant constraints on the allowed charges and distances between centres.

The composite non-BPS system in a general basis
Given the ansatze for the metric and gauge fields in (2.26)-(2.27), the first order flow equation for the composite non-BPS system is given as Here, M , V , are functions to be specified below, whileR andR * are a constant and a non-constant very small vector respectively, where R ,R * = 4. The non-constantR * is related to a constant very small vector, R * 0 , bŷ which also satisfies R , R * 0 = 4. In this and all equations in this section, T + K is a generator of the T-dualities leavingR invariant, parametrised by a vector of harmonic functions, K. The vector of parameters K lies in the grade (−1) component of the vector space according to the decomposition (A.2) implied by the T-duality, i.e. a three-charge vector satisfying which indeed specifies a vector of n v degrees of freedom. Without loss of generality, we will consider the K to asymptote to zero, i.e. all the harmonic functions contained in this vector have no constant parts. This choice identifies the asymptotic value ofR * with the constant vector R * 0 and can be changed by passing to a different R * 0 by a constant T-duality. Note that this choice is convenient for discussing the general properties of the system, but not necessarily for constructing explicit solutions. Indeed, we use the freedom of reintroducing asymptotic values for K in the discussion of the explicit representation of solutions in section 3.
The solutions to the flow equation (4.1) are simplified by introducing a vector, H 0 , of grade (−1) ⊕ (+3), i.e. satisfying Note that (4.3) is trivially a solution of the last equation, found by setting the grade (+3) component, R * 0 , H 0 , to vanish. In practice, once a basis is chosen, as was done in section 2, the constraints (4.3) and (4.4) determine n v and n v + 1 allowed components for the two vectors, K and H 0 respectively (cf. (2.28)).
The equations resulting from (4.1) take the form where V is now identified with the grade (+3) component of H 0 , as V = H 0 , R * 0 . The compatibility relation for the last relations leads to the field equation for H 0 , given by

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As the right hand side of this relation is only alongR, it follows that all grade (−1) components of H 0 are harmonic, whereas V is not, leading to by taking the inner product of (4.7) with R * 0 . Note that this is a linear equation for V , since the grade (+3) component of H 0 drops out from the right hand side (cf. (2.30) in a specific basis). The final dynamical equation required is the one for the function M in (4.6) and the angular momentum vector ω, both of which are conveniently given as Taking the divergence of this equation, one obtains a Poisson equation for M , as whose solution can be used back in (4.9) to obtain the angular momentum one-form, ω. These equations can be seen to be equivalent to the formulation given in section 2, by choosing the constant vectorsR and R * 0 as in (2.19)- (2.20). Similarly, one can verify that the formulations of the composite non-BPS system given in a fixed duality frame in [4,5] can be also obtained from the above equations. The relevant choice for comparing with [4] while [5] uses the base obtained by an S-duality on the choice above. Note that (4.11) are very similar to (2.19)-(2.20), but do not include the arbitrary overall T-dualities that allow to cover all frames. It then follows that one can only describe a restricted set of charges using (4.11), contrary to the system in section 2.

Single centre flows
As a first application of the covariant system defined in this section, we now consider single-centre flows, i.e. the explicit solution when only one centre is involved. While this case was treated in detail in [20], we find it illuminating to solve the general equations in this case, since they are still nontrivial even though they lead to the same physical results as in a purely single-centre treatment. Additionally, the structure of the solution near each centre in the multi-centre case is necessarily of the type discussed here and the precise embedding of the single-centre attractor in a multicentre solution is of particular importance for later applications.
We therefore assume that all functions depend only on the coordinates relative to one point, which represents the single horizon, and which we take to be the origin of R 3 . In where h and Γ 0 are two constant vectors satisfying the same constraints as H 0 , that correspond to the harmonic parts that remain arbitrary, while d are the poles of the vector K, defined as K = d r . (4.13) As noted below (4.2), a possible constant part in K can be absorbed in R * 0 and can be disregarded. The non-harmonic terms in (4.12) arise by solving (4.8) for the only nontrivial component, V , by (4.14) The harmonic function V s is the grade (+3) component of the ones in (4.12), which are naturally decomposed as We may now relate the integration constants in H 0 to the physical charges, by using (4.6) and the definitions above, to obtain the following equation so that the charge vector at the given pole is given by Note that the presence of nontrivial T-dualities implies that the poles of H 0 are different than the charges, which explicitly involve the constant parts of H 0 , through h. Note that once a given set of charges Γ is chosen, one can use the fact that the poles of H 0 and T + d h lie in independent Lagrangian submanifolds to determine them explicitly.
The final equation to be solved is (4.9), which leads to the solution Here, M 0 stands for an dipole harmonic function describing rotation through where J is the angular momentum along the axis θ = 0, as is conventionally chosen for a single-centre solution.

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We now consider the metric starting with the expression for the scale factor, given in the standard way by 20) where H 0 and M are given by the expressions above. Firstly, the quartic invariant can be expanded in the possible combinations of the different terms in (4.12), leading to where we omitted the terms of lower order in r, that can however be straightforwardly computed. Using the expression for V n as given in (4.12), one finds poles of order higher than 4 in this expression, which lead to unphysical behaviour near the horizon and must not be present for a physical solution.
We now note that only the grade (−1) components of h and Γ 0 appear in the full expression (4.21), since one can verify that the components alongR in This still contains an unwanted pole of order 5, proportional to I 4 (Γ 0 , Γ 0 , Γ 0 ,R), whenever the angular momentum is nonzero, i.e. in the presence of a dipole harmonic term in M 0 . This term can be easily canceled by adding a dipole harmonic piece in the function V , which amounts to shifting Note that in solving all non-harmonic equations above, we did not use the freedom of adding fixed harmonic pieces in all functions, as the one in (4.24). Indeed, adding such terms not only simplifies expressions significantly, but also leads to a simpler identification of charges. We therefore modify the solution (4.12) and (4.18) to where we have computed the full expression rather than the leading term. Observe that this is identical to the corresponding single-centre expression for charges equal to Γ 0 , if the poles of H s are taken to be equal to the charges (4.17) computed above. Using this solution, it is now possible to completely fix the integration constants in terms of the charges, starting from (4.17), which is now modified to after taking into account the additional harmonic term in (4.25). Now, we can use (4.22) to rearrange the second term as where we used the linearity of T-dualities and the fact that the grade (+3) component of h and Γ 0 drops out from both sides. Combining this with (A.10), the expression (4.27) takes the rather simple form so that the charges are indeed given by Γ 0 , up to an overall finite T-duality with parameter −γ h (−1) . In order to show that the equivalence with the single-centre solutions is complete, one needs to show that not only the metric in (4.26), but also the scalar fields are driven by a single-centre flow with charges as in (4.29). This computation involves a local Kähler transformation governed by the non harmonic part of H 0 above and leaving the physical moduli invariant. Such gauge transformations were recently discussed in [22]. The interested reader can find an outline of this computation in appendix B, where we show that an appropriate Kähler transformation indeed brings the section (4.5) with H 0 as in (4.25) to exactly the single-centre form. The action on the various functions is given in (B.6)-(B.5) in the general case and in (3.14) in the basis used in sections 2-3.
A final point worthwhile discussing is the inversion of (4.29) to find the asymptotic constants −γ h (−1) in terms of the charges. Since this equation is based on a finite Tduality that leavesR invariant and acts nontrivially on R * 0 by definition, one needs to relate a combination of the charges to the vector R * 0 . This can be easily done starting from the expression by contracting with the charges. Using the manifest covariance of (4.30), we can boost Γ 0 to the physical charge Γ according to (4.29), to obtain where we also used (4.31) to express Γ 0 , R * 0 in terms of charges. The last expression gives the combination −γ h (−1) in terms of the charges in any given frame, once the vectors R and R * 0 are determined. It follows that once these vectors and the total charge are chosen, one can directly invert (4.29) to obtain the charge Γ 0 that governs the flow. Note that this does not imply that the above combination is fixed in terms of charges only, since in practice the constant vector R * 0 depends on asymptotic scalars, as in the single-centre case discussed in [20]. In terms of an explicit basis, the value −γh for the relevant T-duality parameter can be seen explicitly by the shift the harmonic functions K in (3.14).

Multi-centre flows
In view of the solutions presented in the previous section on single-centre flows, one can consider solutions involving multiple centres. In this setting, one has to superpose a set single-centre black holes, as described above, by allowing for the various functions to have poles in all allowed centres. Of central importance in this respect is the fact that all centres in a given solution must be compatible with a single pair of vectorsR and R * 0 , which poses a strong constraint on the allowed structures.
Local structure. In the composite non-BPS system, all centres carry non-BPS charge vectors and their near horizon regions are of the type described in the previous section. Considering a multi-centre flow, all but one function in H 0 continue to be harmonic, in addition to the K describing the T-dualities. Therefore, they take a form similar to (4.12), as where V n contains the non-harmonic part of the function V , to be determined below. In order to have a regular solution near the centres, we need to impose the restrictions found for the single-centre case above, and in particular (4.22), so that the poles of H s and K must be colinear at every centre, i.e.
where γ A are a set of constants.
We can now use these expressions to obtain the non-harmonic functions V n and M , by solving (4.8) and (4.10) respectively. As shown in [5], it is possible to find the exact solutions for these functions in terms of the function F AB,C , defined as the everywhere regular solution to the Poisson equation

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While the existence and regularity of this function was shown in [5], it cannot be expressed by elementary functions generically, but only in the special case when all three centres A, B, C are aligned. Additionally, in view of the discussion in the previous section, we find it convenient to include a harmonic part in each of these functions, both to impose regularity of the metric at each pole, as well as to simplify some expressions. The complete expressions for these two functions are as follows where J i A is the angular momentum vector associated to the centre A. Note that both these expressions contain harmonic parts, chosen so that the appropriate behaviour near each centre is obtained, in direct analogy with (4.25). Upon specifying to the frame (4.11), one easily recovers the results of [5], while the general choice (2.19)-(2.20) similarly leads to (3.45)-(3.46). Using these expressions, one can show that the expression for the charges at each centre takes a form very similar to the single-centre result (4.29), as Here and in the following, we use the combination   (4.41) which must hold at each centre independently. Similar to (4.32), one can write the solution as Note that this relation explicitly contains the distances between the various centres, through (4.39). This property will be used in the following, in the discussion of the global structure of solutions. Once again, writing the formal expression (4.41) in terms of an explicit basis leads to a simple identification of T-duality parameters, as in (3.49).
Global structure. Turning to the global features of multi-centre solutions, we consider the total angular momentum. Inserting (4.33) and (4.37) in (4.9), we obtain the full expression for the total angular momentum one-form as where ω AB,C is defined as the solution to In order to extract the angular momentum, we expand in the asymptotic region, using the asymptotic expansion for the function F AB,C as given in [5], along with the corresponding contribution to the angular momentum one-form through (4.9). One can verify that the resulting expression for the asymptotic total angular momentum is

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where in the second equality we used (4.40) to rewrite the second sum as the inner product of charges at all the centres, and the property that I 4 (Γ 0A , Γ 0B , Γ 0C ,R) = I 4 (Γ A , Γ B , Γ C ,R) because of the grading. Note that, while the first two terms are standard, the third term in this expression, resulting from the asymptotic expansion of ω AB,C , is rather non-standard and appears only when not all centres lie on a line. The final step is to consider the spatial structure of the solution, by fixing the distances between the centres through the interactions described above. To this end, we define the antisymmetric combination which is fixed in terms of charges by (4.41). Using (4.39), this can be written as To obtain explicit solutions to (4.47) however, one has to appreciate the fact that while the h parametrise (some of) the asymptotic scalars, all other terms are fixed in terms of charges at each centre by (4.41), so that this set of equations is overconstrained. This was shown in detail in section 3, where we saw the emergence of hypersurfaces on which the composite solutions are constrained to exist. It would be interesting to investigate whether particular contractions of (4.47) can be used to study these properties directly, i.e. without going to an explicit basis.

Conclusion
In this paper, we presented the first detailed analysis of the properties of non-BPS black hole bound states in extended supergravity, which is not attached to a specific duality frame. This permitted us to study the domain of existence of these solutions in moduli space for fixed electromagnetic charges. In particular, we showed explicitly the existence of walls of marginal stability where the binding energy vanishes. Moreover we define the notion of a non-BPS attractor flow tree as a criterion of existence for these solutions. This was done for a relatively simple subclass of non-BPS extremal solutions, corresponding to the so-called composite non-BPS system of black hole solutions. These correspond to a system of black holes each carrying a non-BPS charge (of negative quartic invariant) and an angular momentum that is bounded above by the charges.
The results derived in section 3 show that all the features familiar from the study of multi-centre BPS solutions appear for non-BPS composites as well. The only crucial difference is that, while BPS solutions a priori exist on codimension zero subspaces of moduli space, non-BPS solutions can only exist on specific hypersurfaces in moduli space, depending on the charges involved [4,5]. One of our main technical result is to prove that the composite solutions always carry a non-trivial binding energy between the constituents, JHEP10(2013)179 exhibiting that they indeed define bound states (with the exception of the solutions of the T 3 model). In order to arrive to this conclusion we showed that the fake superpotential, defined in [13] as a function of auxiliary parameters associated to the flat directions of the individual centres, defines the mass of a single-centre non-BPS black hole at its global maximum. Using the property that the mass of a composite solution is defined from the same function at different values of these auxiliary parameters, one concludes that the binding energy is always positive.
As for the BPS solutions, the distances between the centres are determined in terms of the individual charges and the moduli. We show explicitly that the distance between two centres (or two clusters of centres) diverges for finite values of the asymptotic scalars. Moreover, we prove that the corresponding binding energy vanishes in this limit, exhibiting that it defines a wall of marginal stability in moduli space. These domains indeed define codimension one boundaries of the hypersurface on which the solution exists in moduli space.
As it turns out, the auxiliary parameters associated to flat directions play a very similar role to that of the Kähler phase of the central charge for BPS solutions, leading to a natural notion of attractor flow tree. Indeed, as explained in section 3.3.2, a solution may only exist if a wall of marginal stability exists, on which the values for the auxiliary parameters for the constituents are the same as for the bound state. It then follows that any solution can be assembled in this way, so that one can associate an attractor flow tree to any such composite solution. It is therefore tempting to conjecture that the reverse would be true, i.e. that the existence of such an attractor flow tree would imply the existence of a solution, as proposed in [1] for BPS solutions.
Note that the property that these solutions only exist on a hypersurface in moduli space may be an artifact of the composite non-BPS system we are solving, rather than a physical property. The system somehow forces us to restrict ourselves to a hypersurface without boundaries in moduli space, on which we can identify a boundary carrying all physical properties of a wall of marginal stability. If we know that there is no deformation of our solutions in the normal directions to this hypersurface within the composite non-BPS system of equations, there may exist more general extremal solutions that would extend the domain of existence to a codimension zero domain in moduli space. This would require to give up some special properties of these solutions, as for example the condition that the three-dimensional Euclidean base is flat.
There are a number of future directions related to the above developments. First, it would be interesting to extend the analysis to more general systems of black hole composites. The most obvious such example is to allow for BPS charges as well, so that a BPS/non-BPS system of charges may arise. This is described by the almost-BPS system [6,7,23,24], which can be treated in a very similar fashion. Further extensions may involve solutions that do not admit a flat three-dimensional base space, which are however much less understood and there is no known system of equations to describe them systematically. Finally, it would be very interesting to understand the possible higher dimensional origin of the hypersurfaces in moduli space on which the non-BPS solutions are constrained to exist.
In the main text, we always use (A.14) in combination with (A.5), (A.6) and (A.10)-(A.13) to compute the action of a general T-duality on an arbitrary vector in a general frame.
In order to construct particular black hole solutions, it is necessary to give a representation of T ± explicitly, which is simplified by observing that the variety of very small vectors, such as R and R * , can be generated by action of any T-duality on any very small vector that is not invariant under it. 14 Therefore, we can choose any distinguished pair of T-dualities, such as the spectral flows

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More concretely, the most general transformation that brings the charge along q 0 to the most general vector R is given by exp where k ± 0 are arbitrary parameters of grade (∓1). Here, • R is the vector along q 0 and we used that all T-dualities • T + are defined as leaving • R invariant. The associated R * then takes the form R * where • R * is the vector along p 0 and we stress the fact that the new R * depends on both k ± 0 . It now follows that all T-dualities can be obtained by conjugating the simple spectral flows (2.16) by duality transformations above, so that, e.g.
The representation for the dual T-dualities T − , can be easily obtained from (A.17), as all other symmetric theories with a cubic prepotential, since the computation would go exactly the same way for the three other magic supergravity theories, and is even simpler for the infinite series of axion-dilaton theories. It is convenient to consider the following graded decomposition of E 6(−26) , which arises by viewing the five dimensional theory as the Kaluza-Klein reduction of a six dimensional theory of duality group Spin (1,9)  In the corresponding decomposition of the five dimensional vector fields in terms of the six-dimensional field components, the singlet comes from the six-dimensional metric, the spinor from the six-dimensional 1-forms, and the vector from the six-dimensional 2-forms. Let us write the electric and magnetic charges according to (C.2) as q = (q 1 , χ, q a ) , p = (p 1 , ψ, p a ) , (C.3) respectively, where q 1 , p 1 , are real numbers, χ, ψ are commuting Spin (1,9) Majorana-Weyl spinors of opposite chirality and q a , p a are vectors. One then obtains that detp = p 1 p a p a − 2 p aψ γ a ψ , detq = q 1 q a q a + 2 q aχ γ a χ , tr pq = p 1 q 1 + 4ψχ + 2 p a q a . (C.4) The action of the GL(1) × Spin (1,9) subgroup on these components is manifest, so we shall only display the transformations associated to the other generators, parameterised by spinor parameters Λ + , Λ − of opposite chirality, as δp 1 = 4Λ − ψ , δψ = p 1 Λ + − p a γ a Λ − , δp a = 2Λ + γ a ψ , δq 1 = −4Λ + χ , δχ = q 1 Λ − − q a γ a Λ + , δq a = −2Λ − γ a χ . (C.5) Using these generators, one can always set the spinor component of p to zero. The stabilizer of a generic magnetic charge with detp = 0 contains therefore the stabilizer of the non-null vector p a in Spin (1,9), and the elements generated by the spinor generators satisfying to For p a time-like, these generators are compact if p 1 p a is a positive energy vector (i.e. p 1 p 0 > 0), and non-compact otherwise. Accordingly, one finds that the stabilizer of p is decompose into 8 compact plus 8 non-compact generators, such that the stabilizer subgroup is also F 4 (−20) . This reproduces the results derived in [25]. If the stabilizer of p is compact, one can always use it to rotate the second charge q to a basis in which its spinor component vanishes as well. However, this is not always possible when the stabilizer of p is non-compact. Nevertheless, we will see that it is enough to consider an example with vanishing spinor component to get all possible stabilizers of generic charges. Assuming that the spinor component of q vanishes, the constraint that a spinor generator leaves it invariant gives (q 1 p 1 − q a γ a p b γ b )Λ − = 0 . (C.8) Consistency requires that Λ − can only be non-zero if q a q a p b p b − 2q 1 p 1 q a p a + q 2 1 q 2 1 = 0 , (C.9) which is not the case for generic charges. It follows that for generic charges (without spinor components), the stabilizer of p and q in E 6(−26) is identified with the stabilizer of p a and q a in Spin (1,9). Being generic, these vectors are linearly independent. Their common stabilizer is therefore Spin (8), unless they are both space-like and q a q a p b p b > (q a p a ) 2 , in which case it is Spin (1,7). Note that altogether with the four invariants detp , tr qp , tr q × q p × p , detq , (C.10) the angles of the homogeneous spaces Spin(8)\E 6(−26) and Spin(1, 7)\E 6(−26) , provide the 2 × 27 parameters of the two charges. The same counting applies for two four-dimensional charges in the 56 with stabilizer Spin (8) or Spin(1, 7) and their seven E 7(−25) invariants defined in [26]. Although we did not consider generic charge configurations, for which one cannot remove the spinor components of both the electric and magnetic charges q, p, the stabilizer of the two charges must also be a real form of the same complex group D 4 in this case, since it is always possible to remove the spinor component by a complex F 4 rotation. However, there is no other real form of D 4 than Spin (8) and Spin(1, 7) that one can embed in E 6(−26) , and the result above is therefore general.
A similar analysis shows that the stabilizer of two four-dimensional charges can only be Spin(8) when one charge is BPS, since Spin (8) is the only real form of D 4 inside E 6(−78) . Instead, the stabilizer of a non-BPS charge of positive quartic invariant is E 6(−14) which includes Spin (2,8) and therefore Spin(8), Spin(1, 7) and Spin (2,6). If the two charges are both of this type, the stabilizer can be any of these three groups, as can be checked explicitly in the SL(2) × SO(2, 10) truncation of the theory. These stabilizers are discussed in [26].
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