Non-BPS Attractors in 5d and 6d Extended Supergravity

We connect the attractor equations of a certain class of N=2, d=5 supergravities with their (1,0), d=6 counterparts, by relating the moduli space of non-BPS d=5 black hole/black string attractors to the moduli space of extremal dyonic black string d=6 non-BPS attractors. For d = 5 real special symmetric spaces and for N = 4,6,8 theories, we explicitly compute the flat directions of the black object potential corresponding to vanishing eigenvalues of its Hessian matrix. In the case N = 4, we study the relation to the (2,0), d=6 theory. We finally describe the embedding of the N=2, d=5 magic models in N=8, d=5 supergravity as well as the interconnection among the corresponding charge orbits.


Introduction
Recently the study of the attractor equations for extremal black holes (BHs) [1]- [5] in four dimensions received special attention, especially in relation with new results on non-BPS, non-supersymmetric solutions [6]- [48].
Not much is known about non-BPS attractors in five dimensions, although general results for symmetric special geometries in BHs (and black strings) backgrounds were derived in [49]. More recently, it has been shown [36] that real special symmetric spaces have, in the non-BPS case, a moduli space of vacua, as it was the case for their d = 4 special Kähler descendants [21]. In four dimensions, massless Hessian modes for generic cubic geometries were shown to occur for the non-BPS case with non-vanishing central charge in [10,34]. Some additional insight on the correspondence among (the supersymmetry preserving features of) extremal BH attractors in four and five dimensions have been gained in [40], by relating the d = 4 and 5 BH potentials and the corresponding attractor equations. In particular, it was shown that the moduli space of non-BPS attractors in d = 5 real special symmetric geometries must be in the intersection of the moduli spaces of non-BPS Z = 0 and non-BPS Z = 0 attractors in the corresponding d = 4 special Kähler homogeneous geometries.
Aim of the present investigation is to perform concrete computations of the massless modes of the non-BPS d = 5 Hessian matrix, and further relate the d = 5 BH (or black string) potential to the d = 6 dyonic extremal black string potential and its BPS and non-BPS critical points, following the approach of [49] and [50]. This analysis reveals a noteworthy feature of the relation between d = 5 and d = 6. Namely, the moduli space of d = 6 non-BPS (with vanishing central charge 1 ) dyonic string attractors is a submanifold of the moduli space of d = 5 non-BPS attractors of symmetric real special geometries. The only exception is provided by the cubic reducible sequence of real special geometries, for which the non-BPS d = 6 and d = 5 moduli spaces actually coincide. It is worth pointing out that moduli spaces also exist, for particular non-BPS-supporting charge configurations, for all real special geometries with a d = 6 uplift [51]. This is the case for the homogeneous non-symmetric real special geometries studied in [52]. For N = 2, d = 5 magic supergravities, with the exception of the octonionic case, the non-BPS moduli spaces can also be obtained as suitable truncations of the moduli space of BPS sttractors of N = 8, d = 5 supergravity. In all cases, the Hessian matrix is semi-positive definite.
It is worth pointing out that in this work we consider only extremal black p-extended objects which are asymptotically flat, spherically symmetric and with an horizon geometry AdS p+2 × S d−p−2 [53]. Thus, we do not deal with, for instance, black rings and rotating BHs in d = 5, which however also exhibit an attractor behaviour (see e.g. [54]).
The paper is organized as follows.
In Sect. 2 we recall some relevant facts about N = 2, d = 6 self-dual black string attractors and the properties of the black string effective potential in terms of the moduli space spanned by the tensor multiplets' scalars. In Sect. 3 we discuss the d = 5 effective potential in a six-dimensional language for the d = 5 models admitting a d = 6 uplift (including all homogeneous real special geometries classified in [52]), in the absence (Sub-sect. 3.1) or presence (Subsect. 3.2) of d = 6 vector multiplets. In Subsubsect. 3.2.1 we perform an analysis of the attractors in d = 5, N = 2 magic supergravities, and comment on the moduli spaces of attractor solutions for such theories. Thence, in Subsects. 4.1, 4.2 and 4.3 we recall a similar analysis of the attractors respectively in d = 5, N = 8, 6 and 4 supegravities [49,55,56]. The analysis holds for all N = 2 symmetric spaces, as well as for homogeneous spaces by considering particular charge configurations. In Sect. 5 we comment on the conditions to be satisfied in order to obtain an anomaly-free (1, 0), d = 6 supergravity by uplifting N = 2, d = 5 theories. Sect. 6 is devoted to final remarks and conclusions.
The Appendix discusses some group embeddings, relevant in order to elucidate the relation between the N = 8, d = 5 BPS unique orbit and the non-BPS orbits of the N = 2, d = 5 theories obtained as consistent truncations of N = 8 supergravity. Such N = 2 theories include the magic supergravities based on the Jordan algebras J H 3 , J C 3 , J R 3 with n H = 0, 1, 2 hypermultiplets, respectively.
2 (1, 0), d = 6 attractors for extremal dyonic strings In d = 6, (1, 0) and (2, 0) chiral supergravities 2 there are no BPS BH states, because the central extension of the corresponding d = 6 superalgebras does not contain scalar central charges [58]. However, there are BPS (dyonic) string configurations, as allowed from the superalgebra, and extremal black string BPS attractors exist [50,49]. Such attractors preserve 4 supersymmetries, so they are the d = 6 analogue of d = 5 and d = 4 1 2 -BPS extremal BH attractors. Interestingly enough, extremal black string non-BPS attractors also exist in such d = 6 theories [49], as it is the case for (extremal BH attractors) in d = 5 and d = 4. The next sections are partially devoted to such an issue.
Let us start by recalling the general structure of the minimal supergravity in d = 6, the chiral (1, 0) theory. The field content of the minimal theory is: • Gravitational multiplet: (V a µ , ψ Aµ , B + µν ) ; ; (µ = 0, 1, · · · , 5; A = 1, 2) ; (2.1) • Tensor multiplets: The scalar fields in the tensor multiplets sit in the coset space [59] They may be parametrized in terms of q + 2 fields X Λ , (Λ = 0, 1, · · · , q + 1), contstrained by the relation where η ΛΣ = diag[1, −1, · · · , −1]. The kinetic matrix for the tensors is: whose inverse matrix is: As for any d = 6 theory, the field strengths of the antisymmetric tensors H Λ = dB Λ have definite self-duality properties: As a consequence, there is no distinction between the associated electric and magnetic charges • Vector multiplets: The kinetic matrix for the vector field strengths is given in terms of a given constant matrix C Λ αβ by [60]: (2.10) • Hypermultiplets: The hypermultiplets do not play any role in the attractor mechanism, and will not be discussed further here.
Since the vector multiplets do not contain scalar fields, the only contribution to the black string effective potential comes from the tensor multiplets, and reads [50]: (2.12) or equivalently, in terms of the dressed central and matter charges Z = (X Λ e Λ ) and Z i = P iΛ e Λ (where P ΛΣ , P ΛΣ X Σ = 0 is the projector orthogonal to the central charge): The criticality conditions for the effective black string potential (2.13) reads and therefore two different extrema are allowed, the BPS one for Z i = 0 ∀i, and a non-BPS one for Z = 0, both yielding the following critical value of V (6) : 3 N = 2, d = 5 attractors with a six dimensional interpretation In the absence of gauging, the minimal five dimensional theory generally admits the following field content (omitting hypermultiplets): • Gravitational multiplet: (V a µ , ψ Aµ , A µ ) ; (µ = 0, 1, · · · , 4; A = 1, 2) ; (3.1) • Vector multiplets: (a = 1, · · · , n) .
The scalar fields do not necessarily belong to a coset manifold, but their σ-model is described by real-special geometry. In particular, the scalar manifold is described by the locus where L I (φ), I = 0, 1, · · · , n are function of the scalars and V is the cubic polynomial: written in terms of an appropriate totally symmetric, constant matrix d IJK . Note that in order to have a d = 6 uplift the real special geometry must have a certain structure, as discussed in [62]. Namely This is always the case for the homogeneous spaces discussed in [52], where C Λαβ is written in terms of the γ-matrices of SO (q + 1) Clifford algebras. The kinetic matrix for the vector field-strengths has the general form: The BH effective potential in five dimensions is given by where q I = S 3 ∂L ∂F I are the electric charges and a IJ the inverse of (3.6).

No d = 6 vector multiplets
We are interested in finding the relation of the six dimensional attractor behavior to the five dimensional one. Let us first consider the simplest case of a six dimensional supergravity theory only coupled to q + 1 tensor multiplets (no vector multiplets). In this case, n = q + 1 and the scalar content is given by the six dimensional scalars X Λ plus the Kaluza-Klein (KK) dilaton z. The five dimensional scalar fields are related by the constraint (3.3), where the surface expression (3.4) takes here the simple form 3 : The constraint (3.3) then becomes: The components of the kinetic matrix are in this case: More precisely, setting:X we have: The matrix (3.1.3) is easily inverted giving: Then, in this case the BH effective potential takes the form: where (e z , e Λ ) ≡ q I denote the electric charges and, to obtain the last expression, we made use of (3.1.7). The physical interpretation of the charges e z and e Λ is the following: e z is the Kaluza-Klein charge and e Λ are the charges of dyonic strings wrapped around S 1 . The extrema of V (5) are found for: which is the stabilization equation for the KK dilaton, solved by: and for: ∂V (5) ∂X Λ = 0 ⇒ ∂V (6) ∂X Λ = 0 (3. 1.12) which shows that in this case the attractor solutions of the five dimensional theory are precisely the same of the parent six dimensional theory. The BH entropy is now given by [49]: The solution of Eqs. (3.1.12) depends on whether the d = 6 attractor is BPS or not. As previously mentioned, the d = 6 BPS attractors correspond to Z i = 0 ∀i, whereas the non-BPS ones are given by Z = 0 (and Z i = 0 for at least some i) [50,49]. Thus, all q + 1 d = 6 BPS moduli are fixed, while there are q non-BPS flat directions, spanning the d = 6 non-BPS moduli space SO(1,q) SO(q) [49]. The supersymmetry-preserving features (BPS or non-BPS) of the d = 6 attractors solutions depend on the sign of e Λ e Λ : it is BPS for e Λ e Λ > 0 and non-BPS for e Λ e Λ < 0. In this latter case, also the d = 5 solution is non-BPS, because in a given frame [51] e z e Λ e Λ = e z e + e − (with e ± ≡ e 1 ± e 2 ), and if e + e − < 0 the three charges cannot have the same sign [40]. On the other hand, if e Λ e Λ > 0 one can have both BPS and non-BPS d = 5 solutions [40].
Thus, we can conclude that for the "generic sequence" of d = 5 symmetric real special spaces the non-BPS moduli space, predicted in [36], does indeed coincide with the above mentioned d = 6 (tensor multiplets') non-BPS moduli space, found in [49].

Inclusion of d = 6 vector multiplets
Let us now generalize the discussion to the case where s extra vector multiplets: corresponding to the dimensional reduction of six dimensional ones, are present [62]. The reduction may be done preserving the SO(1, q + 1) symmetry when the number s of d = 6 vector multiplets coincides with the dimension of the spinor representation of SO(1, q+1): This implies that the kinetic matrix of the d = 6 vector fields is positive definite and no phase transitions, as discussed in [63,62], occur in this class of models. The extra scalars contribute to the general relations (3.6) and (3.7) via a modification of the cubic form V into [52]: The total number of five dimensional scalars is then q + 2 + s. Of particular interest are the four magic models which are associated with the simple Jordan algebras having an irreducible norm form (displayed in Table 4 of [36]). In these cases q = 1, 2, 4, 8 and s = 2q. Also the " generic sequence" L(0, P ) can be viewed as a particular case of Eq. (3.2.3) with q = 0 and s = P . The d = 6 origin of the second term in Eq. (3.2.3) is the kinetic term of the d = 6 vector fields, which reads [60,62] Thus, in the presence of d = 6 BH charges Q α , it originates an effective d = 6 BH potential of the form V Such a potential has run-away extrema at d = 6 [57]. This can be seen for instance in the case n T = 1 ⇔ q = 0, where Eq. (3.2.5) reduces to (α = 1, ..., P , X 0 = coshφ, (in the last step we used the fact that in the n T = 1 case we may set C 0 αβ = C 1 αβ = δ αβ without loss of generality). Consequently We then conclude that, besides BPS BH attractors, also non-BPS extremal BH attractors are excluded in (1, 0) supergravity in six dimensions. However, we can have a 0dimensional black object by an intersection of a d = 6 BH with a d = 6 black string. Its reduction to d = 5 gives a BH which carries both the string charge and the BH charge, with cubic invariant of the form [64] The inclusion of extra multiplets corresponding to d = 6 vector multiplets induces a significative complication in the model. In particular, the moduli space of the non-BPS attractors drastically changes with respect to the case described in section 3.1. As we shall prove below, in the magic models the number of moduli becomes equal to s = 2q instead of q as it was in the absence of these extra multiplets.
Before entering into the detail of the magic models, let us argue the existence, at least for the homogeneous spaces L(q, P ) (and, for q = 4m, L(q, P, P ′ )) [52], of particular non-BPS critical points where the same results of section 3.1 may still be directly applied. Indeed, it turns out that for the four magic models the non-BPS attractor moduli spaces of dimension 2q always contain as a subspace precisely the coset SO(1,q) SO(q) (that is the moduli space of d = 6 non-BPS attractors for q+1 strings, as discussed above). Such submanifold of the moduli space may be obtained by considering the particular critical point where Y α = 0. This critical point may always be reached because, as (3.2.3) and (3.6) show, the Y coordinates always appear quadratically in the effective potential (3.7). Then, for Y α = 0 the effective potential reduces to the one previously considered (Eq. (3.1.9)), whose non-BPS attractor solution is known to have q flat directions belonging to the coset SO(1,q) SO(q) . This is in fact only half the total number of flat directions for these solutions. It may be understood because the non compact stabilizer of the non-BPS orbit (that is for example F 4(−20) ⊃ SO (1,8) for q = 8 [65,66]), mixes the X with Y variables, so that the restriction {Y α } = 0 implies the reduction of the orbit to its subgroup SO(1, q). The same considerations may be directly extended, for charge configurations where the spinorial charges are set to zero, to the series of homogeneous non-symmetric spaces L(q, P ) (and, for q = 4m, L(q, P, P ′ )) [52], which always admit a non-BPS attractor point where all the spinorial moduli are zero. As before, this condition selects the submanifold SO(1,q) SO(q) of the non-BPS attractor moduli space, with the only difference that in this case the number q is not directly related to the number of spinorial moduli.

N = 2 magic models
For N = 2 supergravity, one can apply the general relations of real special geometry [61,49], so that the efffective potential takes a simpler form. Indeed, for N = 2 supergravity the vector kinetic matrix a IJ is related to the metric g xy of the scalar manifold via In terms of these quantities the central charge is and we can write the potential as where ∂ x Z = q I h I ,x = P a x Z a are the matter charges. The index x = 1, · · · , n V is a world index labelling the scalar fields while a is the corresponding rigid index. P a x denotes the scalar vielbein. The matter charges obey the differential relations: To make explicit computations of the attractor points of the potential and of the corresponding Hessian matrix, let us use the property that both T abc and g ab , written in rigid indices, are invariants of the group SO(q + 1), where q = 1, 2, 4, 8 for the magic models, corresponding to the symmetric spaces L(q, 1). The H-representation R of the scalar fields branch with respect to SO(q + 1) in the following way where R s is the real Clifford module of SO(q + 1) of dimensions dim(R s ) = 2, 4, 8, 16 corresponding to the four values of q. The index a split into the indices 1, m, α, where m = 1, . . . , q + 1 and α = 1, . . . , dim(R s ). Let us write the general form for T abc and g ab : where Γ n are the (symmetric, real) SO(q + 1) gamma matrices in the R s representation. The coefficients of T abc are determined in terms of the coefficients of g ab by the following relation: The potential V can be written in the following useful form: where the following short-hand notation is used: Z a ≡ g ab Z b . Let us now compute the extrema of V . Using eqs. (3.2.1.6) we find It is straightforward to see that the above expression has two zeroes corresponding to the two attractors: • BPS attractor: Z n = Z α = Z 1 = 0 and the potential at the extremum reads V 0 = Z 2 ; • non-BPS attractor: Z n = Z α = 0, Z = 1 4 3 α Z 1 and the potential at the extremum reads V 0 = 9 Z 2 .

Purely five dimensional analysis of attractors in N -extended theories
For any extended supergravity in five dimensions the BH potential enjoys the general expression in terms of the dressed charges [55,56]: where Z AB (A, B = 1, · · · N) are the antisymmetric, Sp(N)-traceless graviphoton central charges, X the trace part while Z I (I = 1, · · · , n) denote the matter charges (which only appear for N ≤ 4 theories). For all the models with a scalar sector spanning a symmetric space, the dressed charges obey some known differential relations in moduli space which allow to explicitly find the attractor condition as an extremum for the scalar potential in moduli space: We are going to study in the following the BPS and non-BPS attractors for the various cases. The scalar manifold is the coset G/H = E 6(6) Sp(8) .
(4.1.1) and the BH potential takes the form: The differential relations among the 27 central charges Z AB (satisfying Z AB Ω AB = 0), are: where the vielbein P ABCD = P ABCD,i dφ i satisfies the conditions The extremum condition is then To explicitly find the solution, it is convenient to put the central-charge matrix in normal form: and to truncate the theory to the "charged" submanifold spanned by the vielbein components that couple to the dressed charge in normal form, that is: P 1 ≡ P 1234 = P 5678 , P 2 ≡ P 1256 = P 3478 while P 3456 = P 1278 = −P 1 − P 2 .  This is a BPS attractor, supported by the unique BPS orbit [49] E 6(6) F 4 (4) , and the maximum amount of supersymmetry preserved by the solution at the horizon is 1/4 (N = 8 → N = 2).
As mentioned above, the two vielbein-components P 1 and P 2 span the submanifold of the moduli space which couples to the proper values of the central charge. This automatically projects out, in the N = 2 reduced theory, the 28 scalar degrees of freedom corresponding to the hypermultiplets.
2. e 1 = e 2 = 0, with Bekenstein-Hawking entropy V extr = 1 3 X 2 . It is a non-BPS attractor of the N = 6 theory, and leaves all the Sp(6) symmetry of the theory unbroken.
Since the bosonic sector of this theory coincides with the one of an N = 2 theory based on the same coset space [49], these are also the attractor solutions of the corresponding N = 2 model. In the N = 2 version, however, the interpretation of the attractor solutions as BPS and non-BPS are interchanged.
To study the stability of the solutions, let us consider the Hessian matrix and evaluate it on the two extrema. In the first case (BPS N = 6, non-BPS N = 2) the solution breaks the symmetry to Sp(4) × Sp (2), (A → (α, a), α = 1, 2, a = 1, · · · , 4), since at the horizon we find, for the central charge in normal form: (4.2.12) Corresponding to the group decomposition of the degrees of freedom: 14 → (5, 1) + (1, 1) + (4, 2) P AB → (P ab ; P ; P aα ) , (4.2.13) the scalar vielbein decomposes as where P ab is the SO(1,5) SO(5) vielbein, satisfying P ab Ω ab = 0. On the solution, the Hessian matrix (4.2.10) is then: (4.2.15) As expected, the directions corresponding to the scalars in the (4, 2) of Sp(4,2) Sp(4)×Sp(2) are flat. When the theory is interpreted as an N = 6 one, this is the BPS solution whose states, regarded as N = 2 BPS multiplets, have flat directions corresponding to the hyperscalar sector. On the other hand, in the N = 2 interpretation this is instead the non-BPS solution, and now the flat directions correspond to degrees of freedom in the vector multiplets' moduli space.
The second solution (non-BPS N = 6, BPS N = 2) leaves all the Sp(6) symmetry unbroken since the horizon value of the central charge matrix in normal form is now: Now the vielbein degrees of freedom do not decompose at all 14 → 14 P AB → P AB (4.2.17) and correspondingly all the scalar degrees of freedom become massive. Let us end this section by writing the quantities used here in the N = 2 formalism adopted in subsubsection 3.2.1. In this case the rigid index a labelling the tangent space directions are replaced by the antisymmetric traceless couple [AB] (recall that we use the convention that any summation over an antisymetrized couple always requires a factor 1/2) : where antisymmetrization in the couples (A 1 , A 2 ), (B 1 , B 2 ), (C 1 , C 2 ) is understood. As far as the central charges are concerned, we have the following correspondence: The scalar manifold is the coset The bare electric charges are a SO(5, n)-singlet e 0 and a SO(5, n)-vector e Λ (the weight with respect to SO(1, 1) is +2 for e 0 and −1 for e Λ ). The BH potential reads: (4.3.3) and the differential relations among the 5 central charges Z AB (satisfying Z AB Ω AB = 0), the singlet X and the n matter charges Z I are [55]: (4.3.5) The central charge matrix may be put in normal form: so that the BH potential takes the form 3.9) and the differential relations among the dressed charges become (dσ and P I ≡ P I12 = −P I34 are the components of the scalar vielbein coupling to the charges in normal form):  In both cases the Bekenstein-Hawking entropy turns out to satisfy [55] S (5) BH = (V | extr ) 3/4 = |e 0 e Λ e Λ |. (4.3.14) To study the stability of the solutions, let us consider the Hessian matrix and evaluate it on the two extrema.
On the BPS attractor solution, the R-symmetry Sp(4) is broken to Sp(2) × Sp(2) (A → (αα)) and the dressed charges in normal form become showing that the 4n scalars parametrized by P Iαα , which correspond to N = 2 hypermultiplets, have massless Hessian modes.
On the other hand, the non-BPS solution breaks the symmetry SO(n) to SO(n − 1) (I → 1, k; k = 1, · · · n − 1) so that the vielbein P IAB decomposes into (P 1AB , P kAB ). The Hessian matrix on the solution is: Note in particular that the 5(n − 1) scalars corresponding to the vielbein P kAB , spanning the submanifold SO(5,n−1) SO(5)×SO(n−1) , are flat directions. For the N = 4 theory it is easy to find a six dimensional uplift in terms of the IIB, (2, 0) chiral d = 6 theory coupled to n tensor multiplets [50] (at least for the anomalyfree case n = 21) on similar lines as performed in section 3. Indeed, similarly to the dimensional reduction of the N = 2 theory coupled to tensor multiplets only, in the dimensional reduction of the IIB theory from six to five dimensions the scalar content is incremented only by the KK-dilaton, which provides a O(1, 1) factor commuting with the SO(5,n) SO(5)×SO(n) coset. Moreover, the vector content in the gravitational multiplet is also incremented by one graviphoton (whose integral corresponds to the singlet charge X). Since the KK-dilaton is stabilized on the attractor solutions, then the five dimensional attractors are in one to one correspondence with the six dimensional ones: on the BPS attractor there are 4n flat directions (corresponding to the quaternionic manifold SO(4,n) SO(4)×SO(n) ), while on the non-BPS solution there are 5(n − 1) flat directions (spanning the coset SO(5) × SO(n − 1)). 5 Anomaly free (1, 0), d = 6 supergravity with neutral matter In this section we comment on the constraints that an N = 2, d = 5 supergravity should satisfy in order to be uplifted to an an anomaly-free N = (1, 0), d = 6 theory. It is well known that in a (1, 0) supergravity with neutral matter the absence of the gravitational anomaly demands a relation among the triple n T , n V , n H of possible matter multiplets (tensor, vector and hyper multiplets, respectively), namely [68,69] n H − n V + 29n T = 273. (5.1) Moreover, the consistency of the gauge invariance of tensor and (Abelian) vector multiplets requires that the gauge vector current is conserved, i.e. [62,51,70,71,72] Such a condition holds true for all symmetric real special manifolds [52], with the exception of the sequence L (−1, P ), P > 0 (whose corresponding Kähler and quaternionic sequences are not symmetric [73]). Disregarding such a sequence, among all homogeneous real special spaces (see e.g. the Table 2   Notice that for the J O 3 -based supergravity n H = 28, so its corresponding quaternionic manifold could be identified with the exceptional quaternionic Kähler coset [74] (which is the quaternionic reduction -or equivalently the hypermultiplets' scalar manifold -of the d = 4 J O 3 -based supergravity [74,52]). On the other hand, for the " generic sequence" there are two possible uplifts to d = 6, depending whether one starts with L (q, 0) or L (0, P ). Indeed, starting from L (q, 0) the condition (5.1) implies n H = 244 − 29q, (5.5) which demands 0 q 8, whereas starting from L (0, P ) the same anomaly-free condition yields n H = 244 + P, (5.6) which always admits a solution.
The (1, 0), d = 6 theory obtained by uplifting the real special symmetric sequence L (q, 0) has n V = 0 and n T = q + 1, and thus 1 n T 9. On the other hand, the anomaly-free (1, 0), d = 6 uplift of the real special symmetric sequence L (0, P ) has n T = 1 and n V arbitrary, thus it may be obtained from the standard compactification of heterotic superstrings on K 3 manifolds (see e.g. [75]).
The model L (−1, 0) admits an anomaly-free uplift to d = 6, having n V = n T = 0 and n H = 273.
All other homogeneous non-symmetric real special spaces do not fulfill the condition (5.2)-(5.3) in presence of only neutral matter, so they seemingly have a d = 6 uplift to (1, 0) supergravity which is not anomaly-free, unless they are embedded in a model where a non-trivial gauge group is present, with charged matter [76,77].

Conclusion
There are three theories with eight supercharges which admit black hole/black string attractors, namely N = 2 supergravity in d = 4, 5, 6 dimensions. For symmetric special geometries, the entropy is respectively given by the quartic, cubic and quadratic invariant of the corresponding U-duality group in the three diverse dimensions. In this paper we extend previous work [40] on the investigation of the BPS and non-BPS attractor equations of such theories, by relating them as well as the corresponding moduli spaces of (non-BPS) critical points.
Furthermore, we related the moduli space of the N = 8, d = 5 BPS unique orbit to the moduli space of N = 2, d = 5 non-BPS orbit for all magic supergravities, as well as for the " generic sequence" of real special symmetric spaces. This latter is directly related to the d = 6 tensor multiplets' non-BPS moduli space, which describes a neutral dyonic superstring in d = 6.
We also considered N = 4, d = 5 supergravity, and related its 1 4 -BPS and non-BPS attractors to the ones of (2, 0), d = 6 theory. Also in this case the moduli space of non-BPS attractors is spanned by the d = 6 non-BPS flat directions, studied in [50].