Extremal black holes, nilpotent orbits and the true fake superpotential

Dimensional reduction along time offers a powerful way to study stationary solutions of 4D symmetric supergravity models via group-theoretical methods. We apply this approach systematically to extremal, BPS and non-BPS, spherically symmetric black holes, and obtain their"fake superpotential"W. The latter provides first order equations for the radial problem, governs the mass and entropy formula and gives the semi-classical approximation to the radial wave function. To achieve this goal, we note that the Noether charge for the radial evolution must lie in a certain Lagrangian submanifold of a nilpotent orbit of the 3D continuous duality group, and construct a suitable parametrization of this Lagrangian. For general non-BPS extremal black holes in N=8 supergravity, W is obtained by solving a non-standard diagonalization problem, which reduces to a sextic polynomial in $W^2$ whose coefficients are SU(8) invariant functions of the central charges. By consistent truncation we obtain W for other supergravity models with a symmetric moduli space. In particular, for the one-modulus $S^3$ model, $W^2$ is given explicitely as the root of a cubic polynomial. The STU model is investigated in detail and the nilpotency of the Noether charge is checked on explicit solutions.


Introduction
In trying to extend our string-theoretic understanding of black holes away from the supersymmetric regime, extremality is often a key simplifying assumption. Firstly, it eliminates Hawking radiation and ensures that the solution is semi-classically stable. Secondly, it implies the existence of an AdS region which may admit a dual conformal field theory description. Thirdly, it guarantees that the near-horizon solution is entirely determined by the conserved charges measurable at spatial infinity, and therefore insensitive (away from lines of marginal stability) to variations to the moduli at infinity. This attractor behavior, first discovered for supersymmetric (BPS) black holes [1,2], holds for all extremal solutions [3,4,5], and is arguably responsible for the validity of certain weakly coupled description of non-BPS black hole micro-states [6]. However, while the most general BPS solution is known explicitly [7,8], our ability to construct non-BPS extremal solutions is quite limited. Early solutions were found in [9,10] by an astute embedding of the Reissner-Nordström solution in N = 8 supergravity, while some solutions were studied numerically in [11]. More recently, it was shown how to deduce first-order equations for non-BPS extremal solutions from a "fake superpotential" [12,13,14,15], and some solutions were obtained. Unfortunately, this strategy has suffered from the lack of a general method to construct the fake superpotential. A generic 5-parameter seed solution was obtained and analyzed in [14,16,17,18,19]. Using dimensional reduction along the time direction, non-BPS solutions of the one-modulus N = 2 supergravity model were obtained in [20] via the determination of their nilpotent Noether charge. The nilpotent orbits 1 associated to extremal black holes have been constructed in [25,26].
In this paper, we build upon the insights of [12,20,25] and give a systematic method to construct the fake superpotential for non-BPS extremal solutions in supergravity models with a symmetric moduli space. In particular, we derive the fake superpotential for extremal non-BPS black holes in N = 8 supergravity in full generality, and in magic N = 2 supergravity models by consistent truncation. Before giving the outline of the paper, we start by briefly reviewing the dimensional reduction, fake superpotential and nilpotent orbit techniques which underlie our approach.

Radial evolution and geodesic motion
Stationary solutions in D = 4 supergravity are efficiently studied by reduction along the time direction [27,21,28,29] (see e.g. [30] for a review). After dualizing the oneforms in three dimensions and restricting to weakly extremal 2 solutions, one obtains a non-linear sigma model with pseudo-Riemannian target space where • R + is parametrized by the scale function U in the metric ansatz ds 2 = −e 2U (dt + ω) 2 + e −2U dr 2 + r 2 (dθ 2 + sin 2 θ dφ 2 ) , (1.2) • M 4 is the moduli space of massless scalar fields in 4 dimensions, with coordinates φ i and metric g ij , • T is a 2n V − dimensional symplectic torus parametrized by the Wilson lines ζ Λ ,ζ Λ of A Λ and its magnetic dual A Λ around the time direction, and • S 1 is the fibre of a circle bundle over T parametrized by the NUT potential σ dual to the off-diagonal metric one-form ω. Its first Chern class is proportional to the canonical symplectic form dζ Λ ∧ dζ Λ .
We shall denote the coordinates U, φ i , ζ Λ ,ζ Λ , σ on M * 3 collectively as φ µ , and U, φ i as φ a . As indicated by the ∼ sign, the metric on T × S 1 varies over the base R + × M 4 , 3 being positive definite along S 1 and negative definite along T , while the metric on the base is 1 2 dU 2 + g ij dφ i dφ j ≡ g ab dφ a dφ b , positive definite. The negative signature along T can be traced to the negative signature of the time direction along which the dimensional reduction is carried out. In the context of N = 2 supergravity, M 4 is special Kähler, and M * 3 , the "c * -map" of M 4 , is related to the "c-map" of [31] by analytic continuation (ζ,ζ, σ) → (iζ, iζ, −σ). Under this dimensional reduction, stationary solutions of 4D supergravity become harmonic maps from R 3 to M * 3 , with pointwise vanishing Lagrangian density (this latter condition follows from the restriction to weakly extremal solutions).
Assuming in addition spherical symmetry, the supergravity equations of motion become equivalent to light-like geodesic motion on M * 3 , with the affine parameter τ identified as the inverse radial distance τ = 1/r. The conserved Noether charges along the twisted torus T × S 1 are identified as the electric, magnetic and NUT charges q Λ , p Λ , k, respectively. Static solutions have zero NUT charge k = 0. In this case the Hamiltonian for light-like geodesic motion on M * 3 becomes independent of ζ Λ ,ζ Λ , and reduces to the Hamiltonian for the motion of a fiducial particle on R + × M 4 subject to the potential V , Here p a = g abφ b is the momentum conjugate to φ a , the dot denotes the derivative with respect to τ , and V ≡ −e 2U V BH encodes the (negative definite) kinetic energy along T . The latter depends quadratically on the charges p, q, and is proportional to V BH , which is sometimes called the "black hole potential".

Extremal solutions and fake superpotential
While the condition that the spatial slices be flat is necessary for extremality, it is by no means sufficient. In order that the solution be smooth, one must fine-tune the boundary conditions at spatial infinity so that the particle reaches the top of the potential hill in infinite proper time and with zero velocity, p a (τ = ∞) = 0. For fixed electromagnetic charges, this fine-tuning holds only on a Lagrangian subspace of the phase space (φ a , p a ) of initial conditions at τ = 0.
There can be different ways of performing this fine-tuning. In supergravity models with extended supersymmetry, one may impose the existence of Killing spinors to obtain first order equations which relate the momentum p a to the coordinate φ a , and guarantee that the second order equations of motion are obeyed. The resulting solution then preserves some fraction of supersymmetry and, if smooth, will also be extremal. This is most familiar in the framework of D = 4, N = 2 supergravity, where BPS black holes satisfy the"attractor flow" equations p a = −∂ φ a S BPS , (1.4) i.e. follow the gradient flow of the potential (1.5) The potential S BPS and "superpotential" W BPS satisfy such that on solutions of (1.4), the positive kinetic energy compensates the potential term V (φ a ) < 0, ensuring that the spatial slices ds 2 3 are flat. Moreover, they guarantee that a maximum of V (or minimum of V BH ) is reached at zero momentum, provided this extremum occurs at a regular attractor point |Z * | > 0 [32]. The restriction of (1.4) at τ = 0 define the "BPS" component of the Lagrangian subspace of extremal solutions, and corresponds to BPS black holes with ADM mass and Bekenstein-Hawking entropy given by 2GM = W BPS (τ = 0) , S BH = πW 2 BPS (τ = ∞) . (1.7) There may however exist other disconnected components of the Lagrangian subspace of extremal solutions corresponding to non-BPS black holes. As shown in [12,13], some non-BPS solutions can be obtained in a similar way as the BPS ones, provided there exists another function S(φ a ) = e U W (φ i ), where W (φ i ) is dubbed the "fake superpotential", such that the potential V can be written as in (1.6), (1.8) The first order equations p a = −∂ φ a S(φ a ) , (1.9) then imply, just as in the BPS case, that the second order equations of motion are satisfied, that the kinetic and potential energy compensate each other, and that the solutions reach a critical point of the potential at zero velocity. The first order equations (1.9) at τ = 0 therefore provide another component of the Lagrangian subspace of extremal solutions, and correspond to extremal non-BPS black holes with mass and Bekenstein-Hawking entropy given by (1.7) where W BPS is replaced by W . In contrast to the BPS case, the critical point of W is not guaranteed to be an isolated maximum, but could exhibit flat directions or even saddle behavior; in the presence of flat directions, some of the scalars at the horizon are determined uniquely by the conserved charges, although the entropy will be independent of the asymptotic value of the scalars [4]. Using this method, non-BPS extremal black holes for the ST U model were obtained in [12] in the axion-free case 4 . Unfortunately, there has been no systematic way of computing W without solving for the full problem (although, in some cases, one may engineer different fake superpotentials W for the same potential V using discrete symmetries). The purpose of this paper is to give a method to determine W a priori for symmetric supergravity models.

Fake superpotential and radial wave function
As a side remark, we note that the potential S(φ a ) may be identified, by virtue of (1.9), as the generating function of the Lagrangian subspace of the "small phase space" T * (R + × M 4 ) corresponding to smooth extremal solutions with fixed values of the electromagnetic charges (see [34] for a related discussion). By construction, S solves the Hamilton-Jacobi equation 5 associated to the Hamiltonian (4.13) H(∂ φ a S, φ a ) = ∂ t S = 0 . (1.10) Equivalently, both the first order equations (1.9) and the relations 6 between the charges q Λ , p Λ , k and the canonical momenta p ζ Λ , pζ Λ , p σ can be derived from the generating function on the "large phase space" T * (M * 3 ) 4 A fake superpotential for non-zero axion was postulated in [33], but it depends explicitely on the flat directions and its status is unclear to us. 5 Hamilton's principle function is usually a function of the position variables at time t, canonical momenta at initial time 0, and time t itself. The electric and magnetic charges p Λ , q Λ in (1.12) can be regarded as the values of the canonical momenta at t = 0, while the absence of explicit time dependence is a consequence of weak extremality, H = 0. 6 The numerical factors in the forthcoming relations are convention-dependent, and have been chosen consistently with the coordinates used for the ST U model in Section 4. via p φ µ = ∂ φ µS . Upon quantization of the radial evolution of the scalars by replacing [21,35], this generating function determines the semi-classical form of the radial wave function, i.e.
For BPS black holes with superpotential (1.5), one recovers the semi-classical BPS wave function found in [35,36]. In addition to the usefulness of W for determining the mass, entropy and fine-tuning at infinity, this relation to the radial wave function provides extra incentive to study the fake superpotential for non-BPS extremal black holes.

Extremal black holes and nilpotent orbits
In this note, we focus on the special case of N ≥ 2 supergravity theories with a symmetric moduli space, where group theoretical methods can be used to bear on this problem. In these cases, both M 4 and M * 3 are symmetric spaces, (1.14) Here K 4 is the maximal compact subgroup of the continuous 4D duality group G 4 , while K * 3 is a non-compact real form of the maximal compact subgroup of the continuous 3D duality group G 3 . The latter acts isometrically on M * 3 by right-multiplication, and yields a conserved Noether charge Q valued in the Lie algebra 7 g 3 . Those include not only the conserved charges for translations along the twisted torus T × S 1 , i.e. the electromagnetic and NUT charges, but also additional charges corresponding to Ehlers and Harrison transformations, as well as 4D duality rotations. The geodesic motion on M * 3 is integrable, and in fact all geodesics on M * 3 can be obtained by exponentiating a generator −P 0 τ ∈ g 3 k * 3 , where P 0 determines the momentum along the trajectory. P 0 is conjugate to the Noether charge Q via the coset representative V in G 3 , Extremal solutions correspond to special geodesics which reach the boundary U = −∞ in infinite proper time [27]. As already mentioned, it is necessary but not sufficient that the geodesic be light-like. For BPS black holes, it was observed in [21] that the Noether charge must satisfy [ad(Q)] 5 = 0, i.e. belong to a nilpotent orbit of degree 5 (See Appendix A for a summary of useful facts about nilpotent orbits). More precisely, it was shown that the Noether charge defines a 5-grading 8 of the Lie algebra g 3 = g (−4) ⊕ g (−2) ⊕ g (0) ⊕ g (2) ⊕ g (4) where the top spaces g (4) is one-dimensional. Upon quantization, the BPS phase space becomes the Hilbert space of the quaternionic discrete series of G 3 [35], closely related to the quasiconformal realization [37].
For what concerns extremal non-BPS black holes, it was later shown in the special case of the one-modulus S 3 model that extremality requires the condition [Q |7 ] 3 = 0, where 7 denotes the 7-dimensional representation of the 3D duality group G 2(2) [20]. This condition is equivalent to [ad(Q)] 5 = 0 in this particular case. G 2(2) admits two distinct nilpotent orbits of degree 5 with the same dimension 10, corresponding to extremal BPS and non-BPS black holes, respectively.
More recently, the supersymmetry and extremality conditions on the Noether charge for symmetric supergravity models were re-analyzed in [24]. It was shown in all cases where G 3 is simple that extremality requires 9 to [Q |R ] 3 = 0, where R denotes the "fundamental representation" of G 3 : for example the spinor representation if G 3 is an orthogonal group SO(2 + m, 2 + n) or SO * (2m + 4). The only exception is for G 3 = E 8 (8) or E 8(−24) , where the condition becomes [Q |3875 ] 5 = 0, with 3875 being the 3875-dimensional irreducible representation appearing in the symmetric tensor product of two adjoints.
More precisely, any generic extremal spherically symmetric black hole (i.e. with a non-zero horizon area) is characterized by a nilpotent Noether charge Q which lies inside the grade-two component l (2) 4 of g 3 with respect to the 5-grading (more appropriately, even 9-grading) which arises in the reduction from 4 to 3 dimensions: The nilpotent orbit O G 3 of Q ∈ g 3 under G 3 is characterized by the isotropy subgroup of Q in G 4 . For extremal black holes, this isotropy subgroup coincides with the isotropy subgroup of the electromagnetic charges in the four-dimensional duality group G 4 computed in [38,39]. On the other hand, the momentum P 0 is valued in the coset g 3 k * 3 , and therefore defines a K * 3 -orbit O K * 3 inside g 3 k * 3 . As explained in [25], O K * 3 is a Lagrangian submanifold of O G 3 equipped with its canonical Kirillov-Kostant symplectic form. Parametrizing this Lagrangian will be a key step towards computing the fake superpotential. 8 To match standard conventions in the mathematics literature on nilpotent orbits, we rescale the Cartan generator h by a factor of 2 compared to [21], such that the 5-grading becomes an even 9-grading; the original 5-grading with charges ranging from -2 to 2 corresponds to the minimal orbit. 9 An important assumption in [24] is that all extremal solutions can be obtained as limits of nonextremal black hole solutions. Irrespective of this, the condition [Q |R ] 3 = 0 must be supplemented by a condition on P as discussed at the end of Section A.1.

Strategy and main results
Since the coset component of the Maurer-Cartan form is conjugate to the Noether charge via P = VQV −1 , it defines a representative e ≡ P of the corresponding nilpotent orbit inside the coset component g 3 k * , and therefore defines a K * -orbit inside this coset. A general fact about nilpotent elements is that one can always find another nilpotent element f and a semi-simple generator h such that the triplet (e, f , h) defines an sl 2 subalgebra of g 3 , i.e. [ The eigenspaces of h furnish a graded decomposition of g which uniquely characterizes the complex nilpotent G C orbit [40]. Extremal solutions are such that the K * -orbit of P is characterized by a graded decomposition of k * of the same form as (1.16) [25], As we shall show explicitly in the framework of N = 8 supergravity, for static solutions (i.e. with zero NUT charge) the semi-simple element h associated to the nilpotent element P can be computed in terms of the central charges Z ij alone, and more generally, in terms of the central and matter charges which we write collectively Z I 10 . Decomposing P ∈ g k * with respect to the Ehlers U (1) and the four-dimensional R-symmetry group K 4 , where e i j is a vielbein for the metric g ij , one may recast the middle equation in (1.17) into a system of first order differential equations of the forṁ 20) where W and W i depend on the moduli φ i and electromagnetic charges Q I through the charges Z I only; moreover, we shall prove that Thus, extremal solutions attached to the given nilpotent orbit satisfy a gradient flow under the fake superpotential W . In particular, it follows from the nilpotency of P that 22) and therefore that (1.8) is obeyed. Applying this strategy to N = 8 supergravity with G 3 = E 8 (8) , we are able to determine the fake superpotentials for both BPS and non-BPS extremal black holes, and express them in terms of the SU (8) invariant combinations of the central charges: • In the BPS case, we find that W is the modulus of the largest skew eigenvalue of the central charge matrix (in particular, W 2 is largest root of a quartic polynomial whose coefficients are polynomials in Z ij ). This reproduces the result of [13].
• In the non-BPS case, we find that W = 2 , where is obtained from the nonstandard diagonalization problem (2.68). This problem is solved in Appendix B, where W 2 is expressed as a particular root of an irreducible sextic polynomial 11 . This polynomial becomes reducible at particular values of the central charges, at which points W can be computed in closed form. In particular, on the semi-line Pfaff(Z) ∈ R − we recover the result of [13].
Our expression for the fake superpotential in fact extends straightforwardly to all theories with a symmetric scalar manifold whose isometry group acts faithfully on the electromagnetic charges [27], as discussed in Section 3 below. In particular, we obtain the fake superpotential for all magic N = 2 supergravity models, and in fact for all supergravity theories with N ≥ 2 with a symmetric moduli space. In the one-modulus case, W takes a completely explicit form given in (3.33) below.

Outline
In Section 2, we apply the above strategy and find the complete fake superpotential for BPS and non-BPS extremal black holes in N = 8 supergravity. In Section 3, we extend these results to N = 4 and symmetric N = 2 supergravity models. In Section 4, we analyze the ST U model in more detail, rephrase the BPS and non-BPS, Z * = 0 solutions in terms of the para-quaternionic geometry of M * 3 , and check the nilpotency of the Noether charge on explicit solutions. Appendix A contains a detailed discussion of the real nilpotent orbits of E 8 (8) and SO(4, 4), relevant for maximal supergravity and the STU model. In Appendix B we discuss how to evaluate the fake superpotential for non-BPS, Z = 0 black holes at various loci in the space of central charges corresponding to consistent truncations. Appendix C records some extremal solutions of the STU model, which provide a useful testing bench for our analysis.
Note added: the fake superpotential (3.33) for the S 3 model was derived independently in [57], which appeared on arXiv after the present work had been completed.

Extremal black holes in N = 8 supergravity
In this section, we parametrize the Spin * (16) orbits of generic extremal spherically symmetric black holes of N = 8 supergravity, as Lagrangian submanifolds of certain nilpotent orbits of E 8 (8) in e 8(8) so * (16). For static solutions, this parametrization determines P in terms of the central charges and allows us to identify the fake superpotential. We reproduce the known result for the fake superpotential for 1/8-BPS black holes (2.42), and obtain a new expression (2.78) for the fake superpotential for non-BPS extremal black holes, valid everywhere in moduli space.

Generalities on N = 8 supergravity
We first set up our notations for N = 8 supergravity in 4 dimensions. The massless scalar fields take values in the symmetric space [42] where SU c (8) is the quotient of SU (8) by the Z 2 centre leaving invariant the representations of even rank. According to the conventions of [43] (up to normalization factors), we write the coset representative v as where little Latin letters are associated to the SU (8) gauge symmetry, whereas capital Latin letters refer to the global SU c (8) ⊂ E 7 (7) . They both run from 1 to 8, and raising or lowering indices corresponds to complex conjugation (e.g. Φ IJ = (Φ IJ ) * and Z ij = (Z ij ) * ). The invariant metric on M 4 can be written as where is the SU c (8)\E 7(7) vielbein, which is automatically a complex self-dual antisymmetric tensor by property of the e 7(7) Lie algebra. In the symmetric gauge, v can be written in terms of a complex self-dual tensor φ ijkl = 1 24 ε ijklmnpq φ mnpq , The hyperbolic functions in this expression are Taylor series in φ ijkl with SU (8) covariant contractions [43], (2.6) The SU c (8)\E 7(7) vielbein is then given by where Esh φ is the linear operator The central charge of N = 8 supergravity is a complex antisymmetric tensor Z ij . It can always be rotated via a suitable SU (8) transformation into the form such that ρ 0 ≥ ρ 1 ≥ ρ 2 ≥ ρ 3 are positive real numbers. ϕ is defined (modulo π 2 ) as the SU (8) invariant function 12) and the four SU (8) invariant functions ρ 0 2 , ρ 1 2 , ρ 2 2 , ρ 3 2 are the four roots of the polynomial The E 7(7) quartic invariant [42] can be expressed in terms of the SU (8) invariants as Being E 7(7) invariant, it is a function of the electromagnetic charges alone and independent of the moduli. In (2.68) below, we shall define a different parametrization of the central charge Z ij , which plays the same role for non-BPS black holes as (2.11) for BPS ones.

Spherically symmetric, weakly extremal solutions
As explained in [27], the dimensional reduction of N = 8 supergravity along the time direction leads to a non-linear sigma model on where Spin * c (16) is the quotient of Spin * (16) by the Z 2 subgroup that acts trivially in the chiral Weyl representation. To parametrize this space in a way suited to the dimensional reduction, recall that the Lie algebra e 8(8) admits the real five-graded decomposition such that e 7(7) is the Lie algebra of the four-dimensional duality group, and sl 2 ∼ = 1 (−2) ⊕ gl 1 (0) ⊕ 1 (2) the Lie algebra of the Ehlers duality group for stationary solutions. We write the generators of e 7(7) ∼ = su(8) ⊕ 70 as G I J , G IJKL and the ones of (2) as F, H, E, respectively. The generators of grade 1 and −1 will be written as E IJ , E IJ and F IJ , F IJ , such that they only appear in e 8 (8) through the combinations The negative weight part of the the five-graded decomposition (2.17) defines the Lie algebra of a maximal parabolic subgroup P ⊂ E 8 (8) , also known as the Heisenberg parabolic. SU c (8)\P is isomorphic to the Riemannian symmetric space 8(8) by the Iwasawa decomposition, and to a dense subset of the pseudo-Riemannian symmetric space M * 3 . A generic element of SU c (8)\P may be parametrized as where U is identified as the scale factor in the metric ansatz (1.2), v is the coset representative in (2.1), σ the NUT scalar, and Φ IJ are linear combinations of the Wilson lines ζ,ζ transforming as an antisymmetric complex tensor of SU c (8) ⊂ E 7 (7) . The associated Maurer-Cartan form decomposes into its coset and so * (16) components according to A straightforward computation gives where the e 8(8) generators with lowercase indices i, j, · · · satisfy the same commutations rules as the ones with capital indices, and are related to the latter via a fixed vielbein v, chosen to be the vielbein v 0 at spatial infinity. The equations of motion for weakly extremal spherically symmetric solutions (including those with non-zero NUT charge) then take the manifestly E 8(8) invariant form Defining the e 8(8) -valued Noether charge as we can characterize spherically symmetric weakly extremal black holes by the constant value of Q, subject to Tr Q 2 = 0, and the asymptotic value of the scalars fields v 0 ∈ E 7(7) at spatial infinity (i.e. at τ = 0). The condition of smoothness of the metric puts additional restrictions on the Noether charge discussed in the next subsection. For our purposes it will be more convenient to characterize the solutions instead in terms of the value P 0 of P at τ = 0.
P 0 transforms as a Majorana-Weyl spinor under Spin * c (16). It can be conveniently parametrized using a fermionic oscillator basis [24], where is the anti-involution defining the chiral Majorana-Weyl representation of Spin * (16), W = M + ik where M is the mass and k the NUT charge, Z ij are the supersymmetric central charges and Σ ijkl are the "scalar charges".
We will focus on the case of static solutions, such that k = 0 and the equations of the electromagnetic fields reduce to 12 These integrals of motion define the complex electromagnetic charges Q IJ ∈ 28 C . This allows to replace the 'electromagnetic component' of P by the central charges Z(v) ij , as follows Within the fermionic oscillator basis, P then reduces to the following Majorana-Weyl spinor Smoothness of the metric requires Q, and therefore P = VQV −1 , to be nilpotent of degree five when evaluated in the 3875 representation of e 8(8) [24], This condition is invariant under the adjoint action P → g −1 P g where g ∈ E 8 (8) , and therefore defines an adjoint orbit of E 8 (8) . Moreover, the Spin * c (16) ⊂ E 8(8) orbit of P defines a Lagrangian subspace of the adjoint orbit, for the Kirillov-Kostant symplectic form induced by the Killing norm [24]. The adjoint orbits of elements of e 8(8) satisfying the nilpotency conditions (2.29) are in one-to-one correspondence with the Spin * c (16) orbits of spherically symmetric black holes satisfying v 0 = 1. Other solutions with general value of v 0 can be obtained by acting further with E 7 (7) . The stratification of the moduli space of extremal black holes solutions with v 0 = 1 is identical to that of the corresponding E 8(8) nilpotent orbits [24,44], and is displayed in Figure 1. There are two E 8 (8) orbits associated to the nilpotency condition (2.29), whose union is dense in the space of solutions of this equation. They both lie in a single E 8 (C) orbit, labelled by the weighted Dynkin diagram 0 0000002 , 13 associated to the same five graded decomposition as in (2.17), up to a rescaling of the grading generator by two, A representative E of the nilpotent orbit 0 0000002 is a generic element of the grade two component 56 (2) . There are two classes of such elements which are distinguished by their isotropy subgroup inside E 7 (7) , respectively E 6(2) and E 6(6) [41,38] 14 . Each of these two E 8(8) orbits does not contain a unique Spin * c (16) orbit. Nonetheless, there is one single Spin * c (16) orbit of regular spherically symmetric black holes in each E 8(8) orbit [24]. As for the E 8(8) orbits, we can label them by the associated so * (16) weighted Dynkin diagrams. An so * (16) weighted Dynkin diagram defines the coordinates of a gl 1 generator h of a chosen Cartan subalgebra of so * (16) which defines a corresponding graded decomposition of so * (16) and its Majorana-Weyl representation 128 + such that 13 In general, a complex nilpotent orbit of G C is uniquely labelled by a weighted Dynkin diagram of G, which records the coordinates of the Cartan generator of the SL(2, C) triplet defining the orbit. Real orbits are generally uniquely labelled by a pair of weighted Dynkin diagrams for G and its maximal compact subgroup. See e.g. [40] for a thorough introduction to nilpotent orbits.
14 Note however that in the case of physical interest, the graded decomposition (2.30) is defined with respect to a gl 1 subalgebra of so * (16) such that the corresponding component e (0) 7 (7) is not the fourdimensional duality group. Nevertheless, one checks that the isotropy subgroup of the corresponding electromagnetic charges Z ij defining P are also left invariant by the same subgroups of E 7(7) . a representative of the orbit lies in the component of grade two of 128 + , in particular The two orbits of Spin * c (16) associated to generic extremal black holes (i.e. black holes with a non-vanishing horizon area) are labelled by 0 · 0000020 and 0 · 2000000 , respectively. The aim of this section is to solve explicitly Eq. (2.31) at all values of τ , re-express it as a system of first order differential equations for U and v, and read off the fake superpotential for both BPS and non-BPS extremal black holes.

1/8−BPS black holes
For a generic 1/8−BPS spherically symmetric black hole, P 0 lies in the Spin * (16) orbit [24] Spin * (16) labelled by the weighted Dynkin diagram 0 · 0000020 , associated to the five graded decomposition of so * (16) In the symmetric gauge (in which V = exp(−P 0 τ )), P = P 0 . Since P transforms as a Majorana-Weyl spinor with respect to Spin * (16), it follows that P lies in the same Spin * (16) orbit for any value of τ in the parabolic gauge.
The generator h 1 8 defining the grading (2.33) through gl 1 ≡ Rh 1 8 associated to 1/8-BPS solutions can be identified as follows. Let ω ij be an antisymmetric tensor of rank two of SU (8) such that I j i ≡ ω ik ω jk is a projector onto the two-dimensional subspace C 2 ⊂ C 8 of the preserved Killing spinors at infinity (i.e. at τ → 0), Here we have written the Killing spinor as a complex SU (2) spinor valued in the fundamental of SU (8). It transforms as a real (2,16) under SU (2) × SO * (16), where the real structure is given by the product of the pseudo-real structures on both factors. It can be then checked that the generator h 1 generates the 5-grading (2.33).
Solutions to (2.31) are Spin * (12) Majorana-Weyl spinors of the form (8), W ∈ C and Z ij ∈ 2 C 6 provide a complete parametrization of the Spin * (16) orbit (2.32). To see this, note that the only generators of so * (16) that act non-trivially on a representative of the grade two component 32 (2) associated to a gl 1 generator (2.35) are in the subalgebra of negative grade and acts transitively on the components W , Z ij associated to regular black holes. The generators of strictly negative grade can be redefined through the addition of generators of strictly positive grade such that they correspond precisely to the generators of su(8) that act non- Comparing the general form of |P (2.28) to the general representative of the relevant nilpotent orbit (2.36), we conclude that in the static case (i.e. k = 0), the field dependent h(φ) generator can be defined at any value of τ uniquely from the the central charge. Using the decomposition (2.11) of the central charge Z ij , where ρ 0 (φ) ≥ ρ n (φ), the semi-simple generator h(φ) can be written In the remainder, we shall refrain from writing the dependence on φ explicitely. Comparing (2.40) and (2.28), we see that the scalar fields U and v satisfy the first order equationsU The first equation in (2.41) identifies the fake superpotential for BPS black holes as In order to verify that (2.41) is indeed the gradient flow of W , one may use the second equation in (2.39) to write W = ω ij Z ij . One then computes (in symmetric gauge, but the proof is easily generalized to any other parametrisation of where we used the fact that the right-hand-side in the second line is an element of su (8). ω ij is invariant under an su(2) ⊕ su(6) ⊂ su (8). Moreover, the action of any generator of su(8) on ω ij gives an antisymmetric tensor with at least one index in the image of the projector I j i ≡ ω ik ω jk . Using the explicit form of Z ij (2.38), it then follows that the very last line of (2.43) can be rewritten as Besides, using the fact that W is real, we compute in the same way that Adding the two expressions of dW and canceling terms, we finally obtain It follows that the equations (2.41) can be expressed aṡ By virtue of the attractor mechanism, the scalar fields at the horizon lie at an extremum of W . Thus, P reduces to as τ goes to infinity. From (2.36) we conclude that Z ij * = 0 at the horizon, and therefore Since the E 7(7) invariant ♦(Z) depends only on the electromagnetic charges, it is constant along the flow, and equals to ρ 0 (Z ij * ) 4 at the horizon. Therefore, the value of W at the horizon can be rewritten as Using (1.7), one recovers the famous formula for the entropy of BPS black holes in N = 8 supergravity [45]. Finally, we note that the two-form ω ij associated to the central charge is invariant under SU (2) × SU (6) ⊂ SU (8) and E 6(2) ⊂ E 7(7) , respectively. Therefore the "fake superpotential" W admits flat directions homeomorphic to the symmetric space which are not determined by the attractor mechanism, but depend on the asymptotic value of the scalars [46,39]. In order to see this in our formalism, note from (2.36) that in terms of representations of the SU (2) × SU (6) isotropy subgroup of ω ij , the components of u ij IJ dv klIJ − v ijIJ du kl IJ are only non-zero along the component 15 C ⊂ 70, and are always zero along (2 ⊗ 20) R ⊂ 70. One therefore understands from the decomposition that the 40 flat directions generate a M BPS subspace.

Non-BPS extremal black holes
Non-supersymmetric extremal spherically symmetric black holes can be treated in much the same way. In this case, the Noether charge Q (and therefore the momentum P ) The associated graded decompositions of e 8 (8) and so * (16) can be read off the weighted Dynkin diagram, This is the same 5-grading as the one relevant for the minimal nilpotent orbit with weighted Dynkin diagrams ( 0 · 1000000 , 0 0000001 ) associated to 1/2-BPS black holes, after rescaling the generator h by two (see Appendix A). An orbit representative is a generic element of 28 (2) , whose stabilizer defines a U Sp(8) ⊂ SU * (8) (0) subgroup (equivalently, a generic element of the 56 (2) whose stabilizer defines an E 6(6) ⊂ E (0) 7(7) subgroup). Its associated semi-simple generator takes the form where the complex two-form Ω ij satisfies 15 Indeed, using the identities one obtains the generators of degree ±2 of so * (16) as the 'squares' of a i ± e −iα Ω ij a j , respectively, and the su * (8) generators as their commutators. One can then check that the elements of 1 (4) are of the form where N is real. The elements of grade two h 0 |C (2) = 2|C (2) can be computed to be of the form [25] |C (2) These conditions state that Ξ ij is an element of the real 27 representation of the U Sp(8) subgroup of SU (8) that leaves invariant Ω ij . Such elements |C (2) 0 correspond to special values of the central charges in the asymptotic region, whose overall phase ϕ(Z) is determined by the value of the NUT charge. The most general charge with strictly negative E 7(7) invariant ♦(Z) < 0 can be obtained with a linear combination of |C (2) 0 and |C (4) 0 , associated to the gl 1 generator In this case the general element of grade two |C (2) reads The parameters Ω ij ∈ U Sp(8)\SU (8), α, λ, N and Ξ ij ∈ 27 give a complete parametrization of the Spin * (16) orbit (2.54). To see this, we note that the elements of so * (16) that act non-trivially on a generic element of grade two of (2.55) combine into representations of its U Sp(8) ⊂ SU * (8) isotropy subgroup as The component of grade zero leaves h invariant by definition, and acts transitively on the set of parameters N, Ξ ij defining regular black holes. The generators of su(8) that act non-trivially on Ω ij are given by linear combinations in 27 (−2) ⊕ 27 (2) , while the remaining nilpotent generators 1 (−2) ⊕ 1 (2) act by shifting the two parameters α and λ.
Since we are interested in static solutions only, we must cancel the NUT charge (2.64) by choosing λ = tan(2α) (with − π 2 < α < π 2 ). The asymptotic value of P thus takes the form (2.66) where M = cos −1 (2α)N is the physical mass. The asymptotic central charge Z ij is readily obtained from the part of (2.66) bilinear in the oscillators, where ≡ M/2. Due to the conditions (2.61), the complex two-form Z ij can brought by an SU (8) rotation into the skew-diagonal form where > 0 and ξ 1 , ξ 2 , ξ 3 ∈ R can be chosen such that The condition of non-saturation of the BPS bound M > ρ n further requires that In the parametrization (2.68), the E 7(7) quartic invariant (2.14) factorizes into and is therefore strictly negative inside the tetrahedron (2.70). It vanishes whenever M = ρ m for any m, i.e. when the black hole becomes BPS with vanishing horizon area. The quantities α, , ξ 1 , ξ 2 and ξ 3 define a basis of the SU (8) invariant functions of Z ij well suited for non-BPS solutions, alternative to the basis of functions ρ n , ϕ adapted to the BPS case. The diagonalization problem (2.68) is central to the determination of the fake superpotential for non-BPS solutions, and is reduced in Appendix B to the solution of a sextic polynomial.
Comparing the general form of |P (2.28) to the general representative of the relevant nilpotent orbit (2.66), we conclude that in the static case (i.e. k = 0), the field dependent h generator is determined in terms of the central charges Z ij . Using the decomposition (2.67), with ξ n (φ) taking values in the tetrahedron (2.70), the h generator is defined by Indeed, one computes that the general solution to the equation h|P = 2|P is The non-BPS nilpotent orbit therefore determines the scalar fields U and v through the first order equationsU The first equation identifies the fake superpotential for extremal non-BPS black holes as where is the diagonal component of the central charge Z ij in the non-standard diagonalization problem (2.68). In particular, W is larger than the modulus of any of the eigenvalues ρ m of the central charge matrix, as required for a non-BPS solution. Equation (2.78) is one of the main results in this paper.
In order to check that the second line in (2.77) describes the gradient flow of W , we rewrite W = 1 2 Re Ω ij Z ij , and compute dW following the same steps as in (2.43). In this way we obtain One computes easily that and using the explicit form of the central charge (2.72) one computes that such that the second line in (2.43) cancels. The anti-hermiticity of the su(8) generators implies that With respect toΩ ij , the generators of su * (8) decompose into the generators of the usp(8) subalgebra that leaveΩ ij invariant, and the remaining 27 generators that transform as a traceless antisymmetric tensor of rank two with respect to U Sp (8). It follows that And because such that the first order equations (2.77) can be rewritten aṡ This confirms that (2.78) is indeed a fake superpotential for the non-BPS extremal solutions.
In Appendix B we discuss how W (2.78) can be evaluated in practice, and show that W 2 can be obtained in general as a specific root of a polynomial (B.14) of degree six, whose coefficients are polynomial in the central charges Z ij . At some particular loci in the ρ n , ϕ space however, this polynomial becomes reducible. This happens in particular when ϕ = 0 mod π/4, α = 0, where the sextic polynomial (B.14) becomes fully reducible. For ϕ = π/4 mod π/2, the physical root is W 0 in (B.16), i.e.
This result is in agreement with [13]. Our formula (2.78) is however more general, and does not assume any restriction on the moduli nor on the charges. Similarly, at ϕ = 0 mod π/2, the physical root is the largest of W n in (B.29), i.e.
Having obtained the fake superpotential for non-BPS black holes, one may substitute (2.86) into (2.76) and rewrite the momentum as By virtue of the attractor mechanism, the scalar fields at the horizon again lie at an extremum value of W . It follows that as τ goes to infinity. Using (2.76), one concludes that at the horizon, the central charges satisfy In terms of the SU (8) invariant functions appearing in the standard diagonalization (2.11), this can be expressed as In particular, at the horizon where we again noted that ♦(Z) depends only on the the conserved charges. Using (1.7) we recover the known entropy formula for non-BPS extremal black holes in N = 8 supergravity [47].
Similarly to the BPS case, the symplectic form Ω ij is invariant under U Sp(8) ⊂ SU (8) and E 6(6) ⊂ E 7 (7) , respectively. This implies that fake superpotential W exhibits flat directions homeomorphic to the symmetric space ( it is evident that the 42 flat directions at the horizon parametrize the symmetric space (2.94). It should be noticed that this moduli space of flat directions is isomorphic to the moduli space M 5 of the five-dimensional supergravity model obtained after decompactification [49,39]. We conclude this section by discussing the relation between the nilpotent orbits of the moment P under G 3 , and the orbits of the electromagnetic charges Q IJ under G 4 , as classified in [38,39]. To this aim, notice that, as an e 8(8) (8) is the intersection of the stabilizers of e and h in E 8 (8) . The stabilizer of h in E 8(8) is a GL(1, R) × E 7 (7) subgroup, such that the E 7(7) factor is conjugate but distinct from the four-dimensional duality group G 4 = E 7 (7) . Because any element x of a reductive algebra j e decomposes into two elements x ± ∈ j e , such that

99)
J R e must stabilize H andẐ * separately. On the other hand, the stabilizer of H is GL(1, R) × G 4 . At the horizon (i.e. at τ → +∞), J R e therefore coincides with the stabilizer of the central charge Z * ij inside E 7 (7) . J R e is then the intersection of the two E 7(7) subgroups defined by H and h, respectively. Away from the horizon however (at finite τ ), these two subgroups differ by a similarity transformation.
The above discussion is in fact completely general and applies for any extremal black hole with non-vanishing horizon area, in any theory admitting a symmetric scalar manifold whose isometry group acts faithfully on the electromagnetic charges. The stabilizers of the central charges computed in [38,39], indeed match perfectly the reductive stabilizers of the corresponding nilpotent orbits given in [41]. We have already seen in the case of maximal supergravity that we have the following correspondence between the stabilizer J P of P in E 8 (8) and the stabilizer J Q of Q IJ in E 7 (7) . In the example of the exceptional N = 2 supergravity, to be discussed in Section 3.1 below, 24) , and there are three nilpotent orbits of E 8(−24) associated to extremal black holes, characterized by the stabilizers ,

Truncations of maximal supergravity and their extensions
Our method for deriving the fake superpotential of static extremal black holes applies to any symmetric model of 4D gravity coupled to abelian vector fields and scalar fields invariant under a continuous duality group G 4 , provided G 4 acts faithfully on electromagnetic charges. Many such models can be obtained by consistent truncations of N = 8 supergravity, and all others follow by "covariantisation". In this Section we shall restrict ourselves to models that can be obtained by reduction on a circle of 5D supergravity theories. Indeed, these are the only ones which admit non-BPS black holes with no central charge being saturated [25], for which our formalism is most fruitful. All other symmetric models have only two types of generic extremal black holes, and the fake superpotential is always of the form W = ρ n .

Magic N = 2 supergravity
The maximal N = 2 truncation of N = 8 supergravity is the 'magic' supergravity theory associated to the quaternions [50]. In particular, only the components (and the ones obtained by antisymmetric permutations of the indices) are non-zero in the symmetric gauge, where φ AB is an antisymmetric tensor of U (6).
With respect to S U (2) × U (6) ⊂ SU (8), the representations of the fermions decompose as follows such that the fermionic fields of the N = 2 truncation are 2 gravitini and symplectic-Majorana spinors in the 2 ⊗ 15 representation, and the fermionic fields of the N = 6 truncation are 6 gravitini and symplectic-Majorana spinors in the6 ⊕ 20.
The SU (8) invariant functions of the central charges Z ij , ρ 0 , ρ 1 , ρ 2 , ρ 3 and ϕ then become U (6) invariant functions of Z and Z AB defined by 16 and (which are 1/6-BPS in the context of N = 6 supergravity) are controlled by the fake superpotential W = ρ 1 (Z AB ), and the non-BPS black holes with Z * = 0 0 00002 · 2 , by the fake superpotential W = 2 (Z, Z AB ), which is the same function of ρ 0 , ρ 1 , ρ 2 , ρ 3 and ϕ as for the non-BPS black holes of maximal supergravity.
In addition to (3.1),there are three additional N = 2 'magic' supergravity theories in D = 4, associated to the octonions, complex and real numbers respectively. Their moduli spaces are given by [50]  subgroup associated to the 'truncation to the quaternions', the complex 27 representation decomposes as 15 ⊕ 2 ⊗ 6, and the complex charge Z a splits this way into Z AB and Z A α . The action of the remaining generators of e 6(−78) in the (2⊗20) R of SU (2)×SU (6), is defined as follows in term of the complex self-dual parameters Λ α ABC = 1 6 ε αβ ε ABCDEF Λ DEF β . Using these generators, one computes the quadratic and the cubic invariants 17 of E 6(−78) as A generic element Z a can always be rotated to a basis in which its component Z A α = 0, and Z AB takes a block diagonal form such that a generic charge Z a is parametrized by 50 angles of Spin(8)\E 6(−78) and four invariants. The SU (6) invariant functions ρ 1 , ρ 2 , ρ 3 lift in a unique way to the roots of the E 6(−78) invariant polynomial while ρ 0 and ϕ are the U (1) × E 6(−78) invariants In terms of these invariants, the E 7(−25) quartic invariant is defined similarly to (2.15), As in magic supergravity q) there are three classes of generic extremal black holes in this model, all of which correspond to the complex nilpotent orbit of weighted Dynkin diagram superpotential W = 2 (Z, Z a ). The latter is again the same function of ρ 0 , ρ 1 , ρ 2 , ρ 3 and ϕ as in N = 8 supergravity.
We will now explain how the fake superpotential controlling the non-BPS Z * = 0 black holes can be derived from the parametrization of the nilpotent orbit of weighted Dynkin diagram 0 000002 · 2 . The latter tells us that the generator h ∈ sl 2 ⊕ e 7(−25) of the triplet associated to the nilpotent orbit defines the following five-graded decomposition of sl 2 ⊕ e 7(−25) : The corresponding decomposition of the coset component e 8(−24) sl 2 ⊕ e 7(−25) is while the orbit of a representative e ∈ 1 ⊕ 27 (2) associated to a regular black holes is 18 The generator h is parametrized in this case by two phases α and β and an element Ω a ∈ F 4(−52) \E 6(−78) satisfying 19 Let us consider for instance general charges Z and Z a , which we parametrize as As in maximal supergravity, such decomposition is unique as long as the eigenvalues of Ξ a lie in the tetrahedron (2.70). To such electromagnetic charges, one associates a generator h ∈ sl 2 ⊕ e 7(−25) that acts on the elements of P 0 as follows The solution to the equation [h, P 0 ] = 2P 0 is then As for the maximally supersymmetric case one shows that the equation [h, P ] = 2P is equivalent to a gradient flow with respect to the fake superpotential W = 2 (Z, Z a ). The only non-trivial step compared to (2.43, 2.79) is to check that the e 6(−78) generators acting on Ω a , give an element satisfying It is rather remarkable that the nilpotent orbits of the Lie algebras f 4(4) , e 6(2) , e 7(−5) and e 8(−24) associated to the real numbers, complex numbers, quaternions and octonions, respectively, satisfying the nilpotency conditions so that the corresponding moduli space of extremal black holes have the same stratified structure in term of K * 3 orbits, which is displayed in Figure 2.

Axion-dilaton N = 2 supergravity
The quaternionic magic supergravity (3.1) can also be further truncated by singling out an additional complex charge Z 1 = 2Z 34 from Z AB , such that the internal symmetry group is restricted to U (2) × U (4) ⊂ U (6). The remaining charges transform as a complex vector of SO(6) ∼ = Z 2 \SU (4) and the scalar fields parametrize the symmetric spaces More generally, one obtains this way all N = 2 supergravity theories with n + 1 vector multiplets (0 < n ≤ 6) coupled to scalar fields parametrising the symmetric space while ρ 2 and ρ 3 follow from the roots of the polynomial 20 For n > 2 there are four SL(2, R) × SL(2, R) × SO(2, n) orbits of generic extremal black holes associated to four distinct nilpotent orbits of so(4, 2 + n) of degree three in the spinor representation (i.e. such that e 3 = 0 and ad e 4 = 0). For 1/2-BPS black holes, W = |Z|; for non-BPS black holes with Z * = Z i * = 0, W = |Z 1 |; for the non-BPS black holes with Z * = Z 1 * = 0, and finally, for non-BPS black holes with Z * = 0, W = 2 with the same function as in N = 8 supergravity. Of course these results extend straightforwardly to any value of n larger than six. The STU model, to be expanded upon in Section 4, corresponds to n = 2. The STU truncation of maximal supergravity is defined by a basis such that which is preserved by a subgroup S U (2) × U (2) × U (2) × U (2) ⊂ SU (8) as well as The SU (2) subgroups leave invariant the bosonic fields and will be disregarded. Considering the whole field content of the STU model, only one SU (2) factor acts non-trivially on the fermionic fields, and corresponds to the R-symmetry group. There are five different types of extremal static black holes in this model. Four of them, including the 1/2-BPS ones, correspond to 1/8-BPS black holes within N = 8 supergravity. Their fake superpotential is then W = |Z|, |Z 1 |, |Z 2 |, |Z 3 |, respectively. The last type corresponds to the non-BPS black holes of N = 8 supergravity; its fake superpotential 20 Note that raising the SO(n) index i does not involve complex conjugation.
There are two interesting further truncations, which are obtained by restriction to the diagonal subgroup of either two SL(2, R) factors or the three of them. In the first case one obtains the S 2 T model (i.e. (3.26) with n = 1). It admits three types of extremal static black holes, the 1/2-BPS ones, for which W = |Z|; the non-BPS ones with Z * = 0, for which W = |Z 1 |; and the non-BPS ones with Z * = 0, for which W = 2 (Z, Z 1 , Z 2 ). As explained in Section B.3, the degree six polynomial (B.14) factorizes and an explicit expression of W in terms of Z, Z 1 and Z 2 can be obtained by solving the quartic polynomial (B.20).
The subsequent truncation is the S 3 model, which is the circle compactification of the minimal N = 2 supergravity theory in five dimensions.
This theory admits two types of extremal black holes, the 1/2-BPS ones for which W = |Z|, and the non-BPS ones with Z * = 0, corresponding to the two nilpotent orbits of g 2 (2) in which [e |7 ] 3 = 0 [20]. As explained in B.4, the degree six polynomial (B.14) reduces to a cubic polynomial (B.22). Identifying ρ 0 = |Z| and ρ 1 = |Z 1 | where Z 1 = −2iS 2 DZ, one obtains an algebraic expression for the fake superpotential, where L ± are given by (B.28) below, which we rewrite for convenience, (3.34) and ♦ is the moduli-independent quartic SL(2, R) invariant The fake superpotential W in (3.33) is plotted as a function of ϕ in Figure 3. Contrary to the superpotential for BPS black holes, which is ϕ independent, W is maximal at ϕ = π/2(n + 1 2 ), n ∈ Z. Note that (3.33) is only valid in the region where ♦ < 0, such that W > |Z|. In the region where ♦ > 0 it must be replaced by the BPS superpotential W = |Z|. SU (4) such that S U (4) × U (4) ⊂ SU (8). The central charges Z AB are then defined as an antisymmetric tensor of the first SU (4) that we shall identify as the R-symmetry group of the theory. For further generalization, it will be more convenient to represent the other six complex electromagnetic charges as a complex vector Z i of SO(n) ∼ = Z 2 \SU (4) for n = 6. Then, the SU (8) invariant functions become S U (4) × U (4) invariant functions defined by the two polynomials and This model admits three types of generic extremal black holes, which are in one-toone correspondence with the nilpotent orbits of so (8,8) of degree three in the spinor representation [24] (i.e. such that e 3 = 0 and ad e 4 = 0). They are: the 1/4-BPS black holes, for which the non-BPS black holes with Z * AB = 0, for which (3.39) and the non-BPS ones with Z * AB = 0, for which the fake superpotential is the same function 2 (Z AB , Z i ) as in N = 8 supergravity. These results extend straightforwardly to N = 4 supergravity coupled to any number n > 2 of vector multiplets, with moduli space Non-BPS multi-black hole solutions with Z * AB = 0 have been discussed in [26]. For n = 2, the scalar moduli space (3.41) of the N = 4 model becomes identical to that of the N = 2 model (3.26) with n = 6. The nilpotent orbit associated to the non-BPS black holes with Z * AB = 0 splits into two inequivalent orbits of the connected component of SO (8,4), giving rise to two inequivalent SO(6, 2) × SL(2, R) × SL(2, R) orbits . With only one vector multiplet, there is no non-BPS black holes with Z * AB = 0. The degree six polynomial defining the fake superpotential W = 2 associated to non-BPS black holes with Z * AB = 0 reduces to the degree four polynomial (B.20) in such a way that the explicit form of W can then be derived straightforwardly.

Axion-dilaton gravity
The non-standard diagonalization problem defining (Z) can in fact be formulated more generally for the case of Einstein gravity coupled to m + n abelian vector fields and to scalars valued in the symmetric spaces (with m ≥ 1, n ≥ 1) For m = 2 and m = 6, respectively one recovers the cases of axion-dilaton N = 2 supergravities and N = 4 supergravities discussed previously. Denoting the SO(m) and SO(n) vectors of charges by Z a and Z i , respectively, 21 the non-standard diagonalization problem can be formulated in term of two unit-norm vectors Ω a and Ω i as follows, where Ξ a and Ξ i are real vectors orthogonal to Ω a and Ω i , respectively.
We use similar notations as we did in exceptional N = 2 supergravity, i.e. W = M + ik, Σ denotes the complex scalar charge associated to sl 2 , and σ a i denote the real scalar charges associated to so(m, n). The nilpotent orbit is characterized by the generator h of so(m, 2) ⊕ so(2, n), which acts as follows The solution to the equation hP 0 = 2P 0 is then (3.44) As before, the linear equation hP = 2P is equivalent to a gradient flow with respect to the fake superpotential W = 2 (Z a , Z i ). For m ≤ 6, n ≤ 6, W is necessarily the same function of the five SO(2) × SO(m) × SO(n) invariants  (3.46) as in N = 8 supergravity, as it can be obtained by consistent truncation, and therefore remains so for any values m and n. From (3.44) it is apparent that the flat directions of W are the components of σ a i orthogonal to both Ω a and Ω i , as well as the component Ω a Ω i σ a i − Im(e −2iβ Σ), which altogether generate the expected symmetric space The subgroup GL(1, R) × SO(m − 1, n − 1) ⊂ SL(2, R) × SO(m, n) is also the stabilizer of the charges at the horizon (at τ → +∞), where they are of the form

Extremal black holes in the ST U model
In this section, we expand the previous results in the case of the ST U model of N = 2 supergravity, which generates all other symmetric models by truncation or covariantization. In particular, we interpret the BPS and non-BPS, Z * = 0 solutions in terms of the para-quaternionic structure on M * 3 , and confirm our identification of nilpotent orbits on explicit solutions.

Moduli spaces in D = 4 and D = 3
The STU model of N = 2 supergravity is governed by the prepotential The 4D moduli space consists of 3 copies of the Poincaré upper half plane, with metric 22 Upon dimensional reduction along a space-like direction, the resulting moduli space in 3 dimensions is the c-map of M 4 , (recall that ImN is definite negative, so the metric on M 3 is definite positive). Upon dimensional reduction along a time-like direction, the resulting moduli space in 3 dimensions is instead the pseudo-Riemannian space In this section, to avoid notational conflict, we rename the variable U appearing in the metric ansatz (1.2) into φ/2.
Note that in the second case, the denominator is not a maximal compact subgroup of G 3 , which accounts for the signature (8,8) of the metric on M * 3 . To check that the metrics (4.3) and (4.4) are the right-invariant metrics on the two symmetric spaces (4.5), it is convenient to choose an explicit parametrization of the Lie algebra g 3 = so(4, 4) X = E a E a , where E a are the 28 generators and E a are the 28 dual coordinates, where the four commuting SL(2, R) subgroups are generated by while the 16 real generators in the coset fit in an hypercube E a 0 ,a 1 ,a 2 ,a 3 with a i = 1, 2 (see Figure 4). Decomposing with respect to the Cartan generator H 0 leads to the real 5-grading  The symmetric space M 3 can then be parametrized in the Iwasawa gauge by the coset element The metric (4.3) on M 3 is then the right-invariant metric obtained from the Maurer-Cartan one-form θ = dV · V −1 , The Iwasawa parametrization (4.10) can also be used for the pseudo-Riemannian space M * 3 , although it is suitable only on an open subset of the full M 3 . The metric (4.4) is obtained as in (4.11) upon replacing P by where η is the quadratic form preserved by SL(2, R) 4 = SO(2, 2) × SO(2, 2) ⊂ G 3 .
The Hamiltonian associated to geodesic motion on M * 3 with Lagrangian L = ds 2 M * 3 /dρ is given by where 14) and p a = ∂L/∂φ a are the momenta conjugate to φ a . The conserved Noether charges are given by The electric and magnetic charges q Λ , p Λ , NUT charge k are proportional to the components E q Λ , E p Λ , E k of the Noether charge: Note that the charges (p Λ , q Λ ) transform as (2, 2, 2) of G 4 , so can be fit into a cube E 2,a 1 ,a 2 ,a 3 , corresponding to the front face of the hypercube in Figure 4. The Cayley hyperdeterminant of this cube, provides a quartic polynomial invariant under [SL(2, R)] 3 . This is in fact the truncation of the E 7(7) quartic invariant (2.14) in N = 8 supergravity.
For static black holes with p σ = 0, the Hamiltonian (4.13) reduces to where V BH is the "black hole potential" Note that V BH > 0, but the actual potential −e φ V BH is unbounded from below. Since M * 3 has restricted holonomy group SO(2, 2) 2 , it is possible to construct a vielbein V AA A A transforming as (2, 2, 2, 2), covariantly constant under the sl(2, R) 4valued spin connection. In order to make contact with the standard construction of the quaternionic vielbein for the c-map geometry (4.3) [31], we adapt the construction of [37] to the pseudo-Riemannian case.
For this purpose, we perform a Cayley rotation within each SL(2) factor, and define the 16 generators in the pseudo-real coset (2, 2, 2, 2) can now be fit in a complex hyper- where iA i is the eigenvalue under J 3 i and K satisfies the reality condition (4.22) The entries in K can be obtained from the real coset E a 0 ,a 1 ,a 2 ,a 3 by means of a Cayley i=0...3 (E i +F i ) , for example, The action of L 0 ≡ J 3 0 defines a complex 5-grading, and the combined action of L 0 and R 0 = J 3 1 + J 3 2 + J 3 3 projects the root system onto the G 2 root system, with multiplicity 3 for the short roots. We now perform a Weyl reflection with respect to K +−−− , in such a way that the four SU (2) subalgebras generated by J ± 0 , J ± i are rotated by an angle π/3 into the four SU (2) subalgebra generated by 23 K ±±±± and K ±(±∓∓) , with Cartan generators respectively. The algebra so(4, 4) again decomposes as [su (2)] 4 ⊕(2, 2, 2, 2) with respect to this SU (2) 4 , and the projection of the Maurer-Cartan one-form on the coset yields the desired vielbein. This may be represented in the same Figure 4, upon replacing where u, v, e (i) , E (i) denote the right-invariant one-forms It is straightforward to check that the vielbein V AA A A satisfies the reality condition In term of these one-forms, the metric takes the simple form (4.28) 23 Here the parenthesis indicates that one should include the three possible permutations.
The spin connection for each of the SU (2) factors is given by and    The constant covariance of the vielbein can be checked using the Maurer-Cartan equa- Singling out the first SU (2) as the R-symmetry group, we recover the quaternionic vielbein for the c-map geometry (4.3) [31], Finally, in order to describe the pseudo-Riemannian manifold M * 3 we conjugate the generators by exp(iπH 0 /2), whose effect is to Wick rotate The (2,2,2,2) coset V * AA A A is now obtained from Figure 4 by replacing In terms of this new vielbein V * AA A A , the pseudo-Riemannian metric (4.4) may be written as (4.35) As will become clear in Section 4.3, the two terms in bracket correspond to the kinetic and potential terms in the Hamiltonian (4.18).

Near horizon solutions
For simplicity, we focus on solutions with D4 and D0 brane charges only, and denote P i = p i , P 0 = −p 0 , Q i = q i , Q 0 = −q 0 to conform with existing literature. For extremal solutions, the near horizon geometry is AdS 2 × S 2 with constant values for the complex scalars S (i) , which have to extremize the "black hole potential" V BH . For D4 − D0, it is consistent to set the axions S (i) 1 to zero. In this case, V BH simplifies to Extremization with respect to S 2 , T 2 , U 2 leads to a unique minimum, corresponding to an entropy S BH = πV BH, * = 2π |Q 0 P 1 P 2 P 3 | . The extremum (4.37) exists for charges Q 0 , P i of any sign, but the supersymmetry properties of the solution do depend on the signs of the charges. The values of the central charges at the horizon are summarized in the table below, where z = sgn(Q 0 ) √ 2|Q 0 P 1 P 2 P 3 | 1/4 (see also [33,29,18]): Case (a) corresponds to BPS black holes of the ST U model. The next three cases (bcd) are the so-called non-BPS, Z * = 0 extremal solutions. They are non-BPS in the ST U model, but can still be lifted to 1/8-BPS black holes in maximal supergravity [48]. All these cases have ♦ > 0. In contrast, the remaining four cases (ef gh) have ♦ < 0 and are genuinely non-BPS. They are characterized by the fact that the four central charges Z, Z i at the horizon are non-vanishing, and in fact equal in modulus. We shall discuss these solutions in more detail below.

Extremal Z * = 0 solutions
In this subsection, we discuss the geodesic flow interpretation of BPS black holes, as well as so-called non-BPS, Z * = 0 black holes, as they all appear on the same footing in the context of the ST U model. Our starting point is the SL(2, R) 4 vielbein V *

AA A A
computed at the end of Section 4.1 in terms of the invariant one-forms (4.26), and the relation (4.16) between the electromagnetic and NUT charges q Λ , p Λ , k and canonical momenta conjugate to ζ Λ ,ζ Λ , σ. For a static black hole, k = −2p σ = 0, which allows to express the differentials dζ Λ , dζ Λ , dσ in terms of the electromagnetic charges q Λ , p Λ . In this way, we find where Z ≡ Z 0 and Z i are the central charge and "scalar charges", respectively, For completeness we list the central charges: It should be noted that Z i is related to Z by reversal of the sign of one of the S (i) 2 's followed by complex conjugation, e.g. (4.46) Using (4.35), we see that the black hole potential (4.19) can be rewritten as Moreover, from standard formulae in special geometry [55] ZdS (i) (4.48) or from the Maurer-Cartan equations (4.31), one can check that for any choice of W amongst [13] W = |Z| , Therefore, any W in (4.50) can be used as a fake superpotential to generate first order equationsφ = −e φ/2 W ,Ṡ (i) = −e φ/2 g S (i)S(i) ∂S(i)W (4.51) which imply the second order equations from the Hamiltonian (4.13) at zero energy, and moreover guarantee that the corresponding solution is extremal [12]. The first choice in (4.50) corresponds to BPS black holes. Its interpretation as a special kind of geodesic flow on M * 3 is well known [21,35]: indeed, the BPS attractor flow equations (4.51) with W = |Z| can be written in terms of the invariant forms (4.40) as (4.52) or equivalently, using the vielbein V AA A A , Since the index A transforms as a doublet under the R-symmetry SU (2), this is recognized as the Killing spinor equation for BPS black holes [21,35]. Note that the phase of Z varies according to d arg(Z) + A = 0 (4.54) where A is the Kähler connection, Similarly, the first order equations (4.51) with W = |Z i | can be written in terms of the invariant forms (4.40) as 56) or equivalently, using the vielbein V AA A A , where (4.59) Thus, the non-BPS, Z * = 0 solutions are related to the BPS ones by a permutation of the four SL(2) factors. 24 This is in accordance with the fact that these solutions lift to 1/8-BPS solutions in maximal supergravity [48]. Using the explicit form of the solutions given in Appendix C, one may check that the Noether charge is nilpotent of degree 3 in the vector representation, and corresponds to the nilpotent orbits [(+ − +) 2 (−) 2 ] II , [(− + −) 2 (+) 2 ] I,II in cases (a,b,c,d), respectively.
In particular, for axion-free solutions with Z * < 0 and Z i, * > 0, both (B.17) and (2.88) reduce to the formula proposed in [12,18]  if g (i) = 0 for i odd; in this case n is even, and g C = g (2i) yields a n + 1-grading). Correspondingly, g C decomposes into a sum of irreducible representations of SL(2) with spin j ≤ n/2. e is an element of g (2) , and therefore nilpotent of degree 2n + 1. Its complex adjoint orbit O C (e) is P C \G C , where P C is the stabilizer of e, a non-reductive subgroup of the parabolic subgroup i≥0 g (i) ⊂ g C which contains in particular g (n−1) ⊕ g (n) . The complex dimension of O C (e) is equal to dim g − dim g (0) − dim g (1) . Complex orbits can be labelled by their Dynkin indices, which are the coefficients of h on the standard generators of the Cartan subalgebra of g. These can take values amongst 0, 1, 2 only. The subalgebra g (0) ⊂ g R commuting with h is the direct sum of abelian factors associated to the nodes with a non-zero label, and of the semi-simple Lie algebra whose Dynkin diagram coincides with the set of zero nodes. The set of nilpotent orbits admits a partial ordering, with e ≤ e whenever e lies in the closure of the orbit through e. It is convenient to display them in a Hasse-type diagram, with arrows pointing from e to e when e ≤ e .
We are interested in nilpotent orbits of g R , where g R is the Lie algebra of a noncompact real form G R of G C , with maximal compact subgroup K R . The maximal compact Lie algebra k R of g R is obtained as the (−1)-eigenspace of the Cartan involution † (not to be confused with the adjoint), so that elements of k R are of the form e−e † where e ∈ g R . The Kostant-Sekiguchi correspondence relates G R -orbits in g R to K C -orbits in g C k C . More precisely, the Kostant-Sekiguchi homeomorphism where N is the variety of nilpotent elements inside g C . Its inverse can be obtained by use of the complex conjugation * , under the convention that the generators of g R are real, The representative of a real orbit O R (e) can be chosen so that its standard triplet (e, f , h) is a Cayley triplet, i.e. verifies In practice, f = e † is obtained from e via the Cartan involution, after having normalized e properly. The Kostant-Sekiguchi homeomorphism maps this sl 2 subalgebra of g R to the following sl 2 subalgebra of g C , such that the nilpotent element 1 2 (e + f + ih) lies in g C k C , while the semi-simple element i(e − f ) lies in k C .
It turns out that the G R -orbit of e is uniquely characterized by the K C -conjugacy class of i(e−e † ). Moreover, one can always choose a representative e c of the orbit O R (e) such that i(e c − e † c ) is in a fixed Cartan subalgebra of k C . The weighed Dynkin diagram associated to O R (e) is then the Dynkin diagram of K decorated by the eigenvalues of the simple roots of k with respect to the element i(e c − e † c ). This diagram labels the real nilpotent orbit uniquely if k is semi-simple, otherwise it has to be supplemented with extra labels associated to the Abelian factors. In contrast to G, the labels of the weighed Dynkin diagram of K are not bounded by 2, although they are still positive.
For applications to black holes, we are however more interested in the orbit of an element e lying in N ∩ (g R k * R ) under K * R , where K * R is a non-compact real form of K C defined by the Cartan involution ‡ . When the intersection O R (e) ∩ (g R k * R ) is nonempty, there exists a "starred" Cayley triplet (e c * , f c * = e ‡ c * , h c * ) such that e c * , f c * ∈ g R k * R and h c * lies in a fixed Cartan subalgebra of k * R . The orbit is characterized by the eigenvalues of the simple roots with respect to h c * , which furnish a weighted Dynkin diagram for K * . The semi-simple generator h c * defines a graded decomposition of the Lie algebra k * .
A given nilpotent G C -orbit in general corresponds to several G R -orbits labelled by K C -weighted Dynkin diagrams d i . Similarly, the K C -orbit associated to d i corresponds to several K * R -orbits, labeled by the same 25 K C -weighted Dynkin diagrams d j . Thus, each K * R -orbit is labelled by a pair (d i , d j ). It turns out, although we do not know a proof of this fact, that extremal black holes correspond to choosing d i = d j . This condition on P complements the nilpotency condition on Q mentioned in Section 1.4.

Minimal orbit, dimension 58
We start with the minimal orbit of E 8 (8) , with weighed Dynkin diagram 0 0000001 . It corresponds to the graded decomposition of the adjoint representation, familiar from the dimensional reduction from 4 to 3 dimensions. The 3875 representation is also five-graded with respect to this decomposition, therefore the elements in the grade 2 component are nilpotent of degree 3 in both the adjoint representation and the 3875. To see that such elements can be chosen inside e 8(8) so * (16), we decompose (A.6) further into This exhibits the embedding so * (16) ⊂ e 8(8) , associated to the weighted Dynkin diagram 0 · 1000000 of the real E 8 (8) orbit The weighted Dynkin diagram of the real orbit labels the compact element of the Cartan subalgebra of so(16) that appears in the Cayley triplet associated to the nilpotent element e, namely e − e † ∈ 1 (2) − 1 (−2) . As it turns out, it also labels a non-compact element of the Cartan subalgebra of so * (16) which is h = [e, e ‡ ] ∈ gl (0) 1 for a particular choice of normal triplet.

Nilpotent orbit of dimension 112
The next nilpotent orbit, with weighted Dynkin diagram 0 0000010 , is slightly more intricate. Its graded decomposition is the seven-graded decomposition in SL(2, R) × E 6(6) irreducible representations To exhibit a singlet in the grade 2 component, one should decompose further into F 4(4) irreducible representations, It follows from the seven-graded decomposition that such a nilpotent element vanishes to the fourth power in the adjoint representation. In order to identify the existence of a corresponding Spin * (16) orbit inside e 8(8) so * (16), one should further decompose 27 The embedding of so * (16) corresponds to the decomposition 27 We added indices to the 14 2 and the 14 3 representations of U Sp (6) to differentiate the real 2-form representation from the pseudo-real 3-form representation.
The resulting orbit has as isotropy subgroup of Spin * (16) and correspond to black holes for which some of the central charges are larger than the mass.

Nilpotent orbits of dimension 114
There are two 114-dimensional orbits which lie in the same complex orbit of E 8 associated to the weighted Dynkin diagram 0 0000002 . They are associated to the five-graded decomposition (A.6), however the weight 2 indicates that the five grading must be considered as an even nine grading, or equivalently that the corresponding nilpotent elements lie inside the grade 1 component with respect to the five-grading. Because the 3875 representation decomposes as well into a five-graded decomposition, the representative of these orbit is nilpotent of degree five in both the adjoint and the 3875 representation. It turns out that the two real orbits are associated to the decomposition of E 7(7) under the two real forms E 6(2) and E 6(6) of E 6 , respectively. In the first case, The nilpotent element can be chosen in the real component of C (1) , therefore its nilpotency degree in the adjoint representation is 5. In order to understand the relevant embedding of Spin * (16) one should further decompose and so * (16) into as suggested by the weighted Dynkin diagram 0 · 0000020 . This way one determines the isotropy subgroups inside Spin * (16) and E 8(8) respectively (2) .
It is also interesting to consider the isotropy subgroup inside the E 7(7) subgroup of E 8(8) which intersects with Spin * (16) on an SU (8) subgroup. From the embedding one finds that this isotropy subgroup is E 6(2) . This implies that ♦(W − 1 2 Z) > 0 on the corresponding Spin * (16) orbit.
The other nilpotent orbit of dimension 114 is associated to the decomposition where the internal grading is associated to the three-graded decomposition of e 7(7) into irreducible representations of e 6 (6) . It follows that the nilpotency degree of an element of 1 (1,3) in the adjoint representation is 3. In order to get a nilpotent element of degree 5 in the adjoint representation, one could consider an element in 1 (1,−3) + 1 (1,3) . In order to ensure that this element lies in e 8(8) so * (16), we have to further decompose e 6(6) ∼ = sp(4)⊕42 and consider the embedding of so * (16) associated to its decomposition These decompositions enable us to compute the corresponding isotropy subgroups inside Spin * (16) and E 8(8) , The isotropy subgroup inside the E 7(7) subgroup of E 8(8) which intersects with Spin * (16) on an SU (8) subgroup is E 6(6) , as follows from the decomposition This implies that ♦(W − 1 2 Z) < 0 on the corresponding Spin * (16) orbit.
Using similar techniques for different decompositions of so * (16), one gets additional Spin * (16) orbits with the following isotropy subgroups all of them corresponding to black holes with naked singularities. are manifestly triality invariant. The nilpotent orbits O R (e) under the real group 29 G = SO 0 (p, q, R) can be obtained by attaching ± signs to each of the box of the Young tableau, alternating along each 29 The subscript 0 denotes the component connected to the identity. Recall that O(p, q, R) has four connected components when pq > 0. The connected component of the identity SO 0 (p, q, R) is homotopy equivalent to SO(p, R) × SO(q, R). row, such that the total number of (+, −) signs is (p, q) and moreover, such that every row of even length starts with +. A given signed Young tableau corresponds to 4 different orbits labelled (I, I), (I, II), (II, I), (II, II) when all rows have even length, 2 different orbits labelled I, II when all rows with odd length have an even number of + signs, 2 different orbits labelled I, II when all rows with odd length have an even number of − signs, or else a single orbit. All orbits with the same signed Young tableau are related by O(p, q, R). The closure ordering involves not only the shape of the Young tableaux but also the sign and latin letter assignments, and is discussed in detail in [56]. For G = SO 0 (4, 4), the nilpotent orbits of real dimension 18 and lower are summarized in Figure 6.
Let us briefly discuss the fate of these orbits under the two-step restriction

B. Solving the non-BPS diagonalization problem
The fake superpotential for extremal non-BPS black holes is given by (2.78), where is proportional to the component of the central charge in the parametrization (2.68). In this section, we explain how to compute W from the SU (8) invariants ρ 0 , ρ 1 , ρ 2 , ρ 3 and ϕ. The latter can be computed from the central charge Z ij as explained below (2.11).

B.1 General case
To compute , let us define four real positive variables x n as In order to determine the variables x n , we equate (2.11) and (2.68) (keeping in mind that the rotations R i j andR i j are distinct for the two decompositions), obtaining ρ n 2 = W 2 x n − e 2iα 2 = W 2 x n − cos(2α) 2 + sin 2 (2α) .
which gives the algebraic equation At the point ϕ = π/4, it is straightforward if tedious to check that the polynomial (B.14) at ε = 0 admits the six roots .