On the Moduli Space of non-BPS Attractors for N=2 Symmetric Manifolds

We study the ``flat'' directions of non-BPS extremal black hole attractors for N=2, d=4 supergravities whose vector multiplets' scalar manifold is endowed with homogeneous symmetric special Kahler geometry. The non-BPS attractors with non-vanishing central charge have a moduli space described by real special geometry (and thus related to the d=5 parent theory), whereas the moduli spaces of non-BPS attractors with vanishing central charge are certain Kahler homogeneous symmetric manifolds. The moduli spaces of the non-BPS attractors of the corresponding N=2, d=5 theories are also indicated, and shown to be rank-1 homogeneous symmetric manifolds.


Introduction
The issue of the Attractor Mechanism in extremal black holes [1,2,3,4,5] has recently received much attention, and a number of interesting advances has been performed [6]- [35]. Among the others, we cite here the OSV conjecture [36] (see also [32] and Refs. therein), relating black hole (BH) entropy to topological partition functions, and the entropy function formalism [6,8], which allows one to include the higher derivative (gravitational and electromagnetic) corrections to Maxwell-Einstein action (this is crucial specially for the so-called "small" BHs, with vanishing classical entropy). An important step has been the realization that the Attractor Mechanism allows for extremal non-BPS BH scalar configurations of different nature [5,17,21] (see also [18]).
The present investigation concerns the latter issue, and in particular the study of the "flat" directions of the Hessian matrix of the black hole potential V BH at its critical points [37,38,39,40,10,23,34]. Beside considering the case of N = 8, d = 4, 5 supergravity, we will deal with N = 2, d = 4, 5 Maxwell-Einstein supergravity theories (which in the following treatment we will simply call "supergravities") whose vector multiplets' scalar manifold is homogeneous symmetric. Indeed, for such theories a rather general analysis can be performed, determining the moduli space of the various species of non-BPS critical points of V BH , mainly by using group theoretical methods (see e.g. [41,42,43]). In fact such moduli spaces are closely related to the nature (of the stabilizer) of the "orbits" [47,44,33] of the background dyonic BH charge vector Q ≡ m Λ , e Λ (1.1) which supports the considered attractor, where m Λ and e Λ respectively stand for the magnetic and electric BH charges, and Λ = 0, 1, ..., n V , with n V being the complex dimension of the special Kähler scalar manifold. In the case of the stu model [50,51,23], our results are in agreement with the ones obtained in [21,33,34].
The paper is organized as follows.
In Sect. 2 we review the BPS and non-BPS critical points of V BH,N =2 for extremal BHs on homogeneous symmetric scalar manifolds, and the corresponding orbits of the supporting BH charges [21,33]. The resulting properties are summarized, in particular the existence of "flat" directions for the non-BPS case, related to the rank of the Hessian matrix of V BH,N =2 at the corresponding critical points of V BH,N =2 . Thence, in Sect. 3 we deal with the N = 8 theory, and derive the moduli spaces for non-singular 1 1 8 -BPS and non-BPS critical points of V BH,N =8 . In Sect. 4 we do the same for the N = 2 supergravities considered in Sect. 2, by taking into account that in general non-BPS critical points of V BH,N =2 can occur in two different species, depending on the vanishing of the N = 2 central charge Z. Thus, in Sect. 5 we consider the case d = 5, in particular the N = 8 theory (having only an 1 8 nonsingular class of attractors) and the N = 2 homogeneous symmetric supergravities (having an unique non-BPS class af attractors). Finally, some outlooks are given in Sect. 6.
2 N = 2, d = 4 Homogeneous Symmetric Supergravities: Attractors and Critical Hessian The symmetric special Kähler manifolds GV H0⊗U (1) of N = 2, d = 4 supergravities have been classified in the literature [45,46]. With the exception of the family whose prepotential is quadratic , all such theories can be obtained by dimensional reduction of the N = 2, d = 5 supergravities that were constructed in [52,53,54] (they will be treated in Sect. 5). The supergravities with symmetric manifolds that originate from 5 dimensions all have cubic prepotentials determined by the norm form of the Jordan algebra of degree 3 that defines them [52,53,54].
The vector multiplets' scalar manifolds of homogeneous symmetric N = 2, d = 4 supergravities are given in Table 1.
The irreducible sequence in the second row of Table 1 has quadratic prepotentials (and thus C ijk = 0). On the other hand, the reducible sequence in the third row, usually referred to as the generic Jordan family, has a 5-dim. origin, and it is related tot the sequence R ⊕ Γ n of reducible Euclidean Jordan Table 1: N = 2, d = 4 homogeneous symmetric special Kähler manifolds algebras of degree 3. Here R denotes the 1-dim. Jordan algebra and Γ n denotes the Jordan algebra of degree 2 associated with a quadratic form of Lorentzian signature (see 2 e.g. Table 4 of [21], and Refs. therein). Beside the generic Jordan family, there exist four other supergravities defined by simple Jordan algebras of degree 3. They are called magic, since their symmetry groups are the groups of the famous Magic Square of Freudenthal, Rozenfeld and Tits associated with some remarkable geometries [60,61]. J O 3 , J H 3 , J C 3 and J R 3 denote the four simple Jordan algebras of degree 3 with irreducible norm forms, namely by the Jordan algebras of Hermitian 3×3 matrices over the four division algebras, i.e. respectively over A = O (octonions), A = H (quaternions), A = C (complex numbers) and A = R (real numbers) [52,53,54,55,56,57,58,59]. By defining A ≡ dim R A (= 8, 4, 2, 1 for A = O, H, C, R, respectively), Tabel 1 yields that the complex dimension of the scalar manifolds of the magic N = 2, d = 4 supergravities is 3 (A + 1). Beside the analysis performed in [21], Jordan algebras have been recently studied (and related to extremal BHs) also in [62] and [63].
As found in [47], the 1 2 -BPS supporting charge orbit is GV H0 . By denoting by H and H two noncompact forms of H 0 , in [21] it was found that the non-BPS Z = 0 and non-BPS Z = 0 supporting BH charge orbits respectively are the cosets GV ; see [21] for further details).
The data of all the N = 2, d = 4 homogeneous symmetric supergravities are given in Tables 3 and 8 of [21].
In the following treatment we will denote by r the rank of the 2n V × 2n V Hessian matrix H of V BH . Since in N = 2, d = 4 supergravity the 1 2 -BPS critical points of V BH are stable, and H 1 2 −BP S has no massless modes [5], it holds that the rank is maximal: r 1 2 −BP S = 2n V . On the other hand, from the analysis performed in [21] for homogeneous symmetric N = 2, d = 4 supergravities, it follows that r non−BP S is model-dependent, and it also depends on the vanishing of the N = 2 central charge Z.
As it will be evident from the reasoning performed below, the massless modes of the Hessian of V BH,N =8 at its non-singular 1 8 -BPS and non-BPS critical points actually are "flat" directions of V BH,N =8 at the corresponding critical points. Such "flat" directions span the following real homogeneous symmetric sub-manifolds of USp (8) share the same structure: they are the coset of the (non-compact) stabilizer of the corresponding supporting BH charge orbit and of its m.c.s.. As yielded by the analysis performed in Sect. 4, this is also the structure of the moduli spaces of the two classes of non-BPS attractors of the homogeneous symmetric N = 2, d = 4 supergravities. Remarkably, USp (8) is the real manifold on which N = 8, d = 5 supergravity is based. Such a relation with the d = 5 parent theory is exhibited also by non-BPS Z = 0 moduli spaces of the homogeneous symmetric N = 2, d = 4 supergravities; see Sect. 4.
In order to understand that the "flat" directions of the Hessian of 2) 3 Following the notation of [46], by d-geometry we mean a special Kähler geometry based on an holomorphic prepotential function of the cubic form F (X) = d ABC at its critical points actually span a moduli space, it is useful to recall that the N = 8 central charge matrix Z AB (φ, Q) can be rewritten as [48] Z where φ denote the 70 real scalar fields parameterizing the coset G8 H8 = E 7 (7) SU (8) , Q is the N = 8 charge vector, and L Λ AB (φ) ∈ G 8 is the field-dependent coset representative, i.e. a local section of the principal bundle G 8 over G8 H8 with structure group H 8 . Thus, it follows that which shows that V BH,N =8 is not G 8 -invariant, because its coefficients (given by the components of Q) do not in general remain the same. Now, if we take g ≡ g Q ∈ H Q , where H Q is the stabilizer of one of the orbits G8 HQ spanned by the charge vector Q, then Q gQ = Q, and thus: (3.5) Let us now split the fields φ into φ Q ∈ HQ hQ (where h Q ≡ m.c.s. (H Q )) and into the remaining φ Q , paremeterizing the complement of HQ hQ in G8 HQ . By defining Since H Q is a non-compact group, this implies V BH,N =8 to be independent at its critical points on the fields φ Q parameterizing the coset HQ hQ . In other words, the (covariant) derivatives of V BH,N =8 , when evaluated at its critical points and with all indices spanning the coset HQ hQ , vanish at all orders. It is easy to realize that such a reasoning can be performed for all supergravities with N 1 based on homogeneous (not necessarily symmetric) manifolds 4 GN HN , also in presence of matter multiplets (and thus of matter charges). Indeed, such arguments also apply to a generic, not necessarily supersymmetric, Maxwell-Einstein system with an homogeneous (not necessarily symmetric) scalar manifold.
By choosing Q belonging to an orbit of the representation R V of G N which supports critical points of V BH,N , the previous reasoning yields the interesting result that, up to "flat" directions (at all orders in covariant differentiation of V BH,N ), all critical points of V BH,N in all N 0 Maxwell-Einstein (super)gravities with an homogeneous (not necessarily symmetric) scalar manifold (also in presence of matter multiplets) are stable, and thus they are attractors in a generalized sense.

N = 2, d = Symmetric Supergravities: Attractors and their Moduli Spaces
By using the arguments of the previous Section, we now determine the moduli spaces of non-BPS critical points of V BH,N =2 (with Z = 0 and Z = 0) for all N = 2, d = 4 homogeneous symmetric supergravities. As previously noticed, N = 2 1 2 -BPS critical points are stable, and at such points all the scalars are stabilized by the classical Attractor Mechanism, because H 1 2 −BP S has no massless modes at all [5]; thus, there is no 1 2 -BPS moduli space for all N = 2, d = 4 supergravities (as far as the metric of the scalar manifold is non-singular and positive-definite). This is qualitatively different from the previously considered case of N = 8 1 8 -BPS critical points.    Tables  3 and 8 of [21]).
In general, such a difference can be explained by noticing that for N = 2 the 1 N = 1 2 -BPS configurations are the maximally supersymmetric ones, i.e. they preserve the maximum number of supersymmetries out of the ones related to the asymptotically flat BH background. (see [21] for further details on notation). They are respectively given by Table 2 and 3.
Remarkably, the moduli spaces of non-BPS Z = 0 critical points are nothing but the N = 2, d = 5 homogeneous symmetric real special manifolds, i.e. the scalar manifolds of the d = 5 parents of the considered theories. Their real dimension dim R (rank r) is the complex dimension dim C (rank r) of the N = 2, d = 4 symmetric special Kähler manifolds listed in Table 1 Table 2 are consistent with the non-BPS Z = 0 "n V + 1 / n V − 1" mass degeneracy splitting found by Tripathy and Trivedi in [10] (and confirmed in [21,34,33]) for a generic special Kähler d-geometry of complex dimension n V .
Concerning the moduli spaces of non-BPS Z = 0 critical points, they are homogeneous symmetric (not special) Kähler manifolds. In the models st 2 and stu (n = 1 and n = 2 elements of the generic Jordan family) there are no non-BPS Z = 0 "flat" directions at all (see Appendix II of [21]). By recalling that A ≡ dim R A, Tabel Table 3: Moduli spaces of non-BPS Z = 0 critical points of V BH,N =2 in N = 2, d = 4 homogeneous symmetric supergravities. They are (non-special) homogeneous symmetric Kähler manifolds and Townsend [52,53]. As mentioned in the Introduction, all this is consistent with the results about the stu model [50,51,23] obtained in [21,33,34]: for such a model (n = 2 element of the generic Jordan family) there are 2 non-BPS Z = 0 "flat" directions (spanning the manifold (SO(1, 1)) 2 , as yielded by Table 2) and no non-BPS Z = 0 "flat" directions.  Considering now the case N = 2, the manifolds of the homogeneous symmetric N = 2, d = 5 supergravities are given by Table 2. As shown in [44], the 1 2 -BPS critical points are stable already at the Hessian level, as in the d = 4 case. There is an unique class of non-singular non-BPS critical points; by slightly modifying the notation introduced in [44], we denote by H 5 and K 5 the (non-compact) stabilizer of the corresponding non-BPS charge orbits and its m.c.s., respectively. It then follows that the moduli space of the unique class of non-singular non-BPS critical points of V BH,N =2 in homogeneous symmetric N = 2, d = 5 supergravities is given by the homogeneous symmetric manifold The explicit form of e H5 e K5 and its data for all homogeneous symmetric N = 2, d = 5 supergravities is given in Table 4. Such a Table yields that the the moduli spaces of non-singular non-BPS critical points of V BH,N =2 in magic N = 2, d = 5 supergravities have real dimension 2A. Their stabilizer contains the group spin (1 + A). Here we just point out that, unlike the case d = 4 [10,34], an explicit calculation of the "flat" directions of non-BPS critical points of V BH,N =2 in d = 5, despite some recent works on Attractor Mechanism and entropy function formalism in d = 5 supergravities (see e.g. [66], [67] and [68], and Refs. therein), is missing at the present time.

Conclusion
In the present investigation we have extended the analysis performed in [21] and [33] about the spectrum of non-BPS critical points of V BH,N =2 , their degeneracy and stability. For the case of d-geometries [10,34], and in particular for homogeneous symmetric special Kähler geometries [21,33], the Hessian matrix of V BH,N =2 at its non-BPS critical points generally has some strictly positive eigenvalues and some vanishing eigenvalues, corresponding to "flat" directions. For the non-BPS Z = 0 case, our analysis generalizes the findings of [34].
One should not be surprised by our result, because the existence of "flat" directions in the Hessian of V BH was pointed out also at BPS critical points (preserving 4 supersymmetries) in the framework of N > 2, d = 4 extended supergravities [39,33], the "flat" directions being associated to hypermultiplets' scalar degrees of freedom in the supersymmetry reduction N > 2 −→ N = 2 of the considered theory [39,40,38,33] (see [37] for an introduction to the Attractor Mechanism in N 2-extended supergravities).
We have shown that the geometrical structure of the non-BPS moduli spaces depends on the vanishing of the N = 2 central charge Z. As previously mentioned, for Z = 0 our results are in agreement with the ones of [10] and [34].
It is easy to realize that our results extend also to the case of homogeneous non-symmetric special Kähler geometries. Clearly, in such a framework the classification of the charge orbits supporting nonsingular critical points might be different from the symmetric case. Actually, as mentioned above, our results also hold for a generic, not necessarily supersymmetric, Maxwell-Einstein system with an homogeneous (not necessarily symmetric) scalar manifold.
For generic, non-homogeneous special Kähler d-geometries, the U -duality group has no longer a transitive action on the representation space of the BH charges, and the analysis is more complicated, and it might yield different results about stability. However, the non-BPS moduli spaces are still present at least in some particular cases, e.g. in the model called Kaluza-Klein BH (in M -theory language) [49] or D0 − D6 system (in Type IIA Calabi-Yau compactifications in the language of superstring theory) [10,34], in which the only non-vanishing charges are p 0 and q 0 . In this case, the moduli space is the corresponding real special manifold.
The existence of moduli spaces clarifies the issue of classical stability of non-BPS critical points of V BH,N =2 , at least for the analyzed case of homogeneous symmetric vector multiplets' scalar manifolds. All such non-BPS critical points are stable, with a certain number of "flat" directions, which however do not enter into the classical Bekenstein-Hawking [65] BH entropy S BH , whose U -invariant expression in the considered framework in d = 4 reads [39] S BH (Q) = π |I 2 (Q)| for quadratic models; S BH (Q) = π |I 4 (Q)| for cubic models, (6.1) I 2 (Q) and I 4 (Q) being the unique invariant (quadratic and quartic in the BH charges, respectively) of the representation R V of the U -duality group in which the charge vector sits.
It is conceivable that most of the "flat" directions will be removed by quantum effects, i.e. by higherderivative corrections to the classical BH potential V BH . However, this might not be the case for N = 8 BHs.
We conclude by saying that for the cases considered in the present investigation the existence of "flat" directions is closely related to the Lorentzian signature of the BH charge orbits supporting non-BPS critical points of V BH,N =2 , i.e. to the fact that the corresponding stabilizer is a non-compact group. The same phenomenon already happened for N > 2 also at non-singular BPS critical points [38,39,40].