Two-Centered Magical Charge Orbits

We determine the two-centered generic charge orbits of magical N = 2 and maximal N = 8 supergravity theories in four dimensions. These orbits are classified by seven U-duality invariant polynomials, which group together into four invariants under the horizontal symmetry group SL(2,R). These latter are expected to disentangle different physical properties of the two-centered black-hole system. The invariant with the lowest degree in charges is the symplectic product (Q1,Q2), known to control the mutual non-locality of the two centers.


Introduction
Multi-centered black-hole solutions of supergravity theories in d = 4 space-time dimensions have recently received much attention, especially in connection to the classification of non-perturbative string BPS states and their brane interpretation [1]. A generalisation of the attractor Mechanism [2,3] (for a review, see e.g. [4]) has been shown to occur, as firstly pointed out by Denef [5], called split attractor flow for BPS N = 2 black holes [5,6,7,1].
Attempts to generally classify the two-centered solutions of supergravity theories with symmetric scalar manifolds and electric-magnetic duality (U -duality 1 ) symmetry given by classical Lie groups have been considered [10,11,12]. In particular, within the framework of the minimal coupling [13] of vector multiplets to N = 2 supergravity, it was shown in [11] that different physical properties, such as marginal stability and split attractor flow solutions, can be classified by duality-invariant constraints, which in this case involve two dyonic black-hole charge vectors, and not only one.
This leads one to consider the mathematical issue of the classification of orbits of two (or more) dyonic charge vectors in the context of multi-centered black-hole physics. For the theories treated in [11,12], the charge vector lies in the fundamental representation of U (1, n) (minimally coupled N = 2 supergravity [13]) and in the spinor-vector representation of SL (2, R) × SO (q, n), corresponding to reducible cubic N = 2 sequence [14,15] for q = 2, and to matter-coupled N = 4 supergravity for q = 6.
In [11], the two-centered U -invariant polynomials of the minimally coupled theory were constructed, and shown to be four (dimension of the adjoint of the two-centered horizontal symmetry U (2)). The same was done for the aforementioned cubic sequence in [12], where the number of U -invariants were computed to be seven for n 2, six for n = 1 and five for the irreducible t 3 model.
It is the aim of the present investigation to generalise these results to four-dimensional supergravity theories with symmetric irreducible scalar manifolds, in particular to the N = 8 maximal theory and to the N = 2 magical models.
We find that when the stabilizer of a two-centered charge orbit is non-compact, the corresponding orbit is not unique. As we will consider in Section 4, this feature is also exhibited by the classification of the orbits of two non-lightlike vectors in a pseudo-Euclidean space E p,q of dimension p + q and signature (p, q). A prominent role is played by an emergent horizontal symmetry SL h (2, R), whose invariants classify all possible two-vector orbits.
In this respect, the aforementioned t 3 model, whose U -duality group is SL (2, R), provides a simple yet interesting example, because it may be obtained both as rank-1 truncation of the reducible symmetric models and as first, non-generic element of the sequence of irreducible N = 2 symmetric models, which contains the four rank-3 magical supergravity theories mentioned above. The twocentered configurations and the generic (BPS) orbit O = SL (2, R) of t 3 model were studied in Sec. 7 of [12], in which it was pointed out that, as it occurs also for the one-centered case [16], no stabilizer for the two-centered orbit exists 2 . The five components of the spin s = 2 horizontal tensor I abcd (defined in (3.12) below, and explicitly given by (3.15)-(3.19)) form a complete basis of duality-invariant polynomials [12]; as a consequence, the counting (2.2) for p = 2-centered black hole solutions in the t 3 model simply reads 5 + 3 − 0 = 4 × 2, because I p=2 = 5 and dim R (G p ) = 0. Moreover, there exist only two independent [SL h (2, R) × SL (2, R)]-invariant polynomials, which can be taken to be the symplectic product W (of order two in charges, defined in (3.9) below) and I 6 (of order six in charges, defined in (3.24) below); an alternative choice of basis for the SL (2, R)-invariant polynomials is thus e.g. given by three components of I abcd out of the five (3.15)- (3.19), and the two horizontal invariants W and I 6 .
The plan of the paper is as follows. In Section 2 we give a group theoretical method (based on progressive branchings of symmetry groups, considered as complex groups) to find the multi-centered charge orbits of a theory with a symmetric scalar manifold; we then apply it to all irreducible symmetric cases. The analysis of this section will not depend on the real form of the stabilizer of the orbit, and the results will then hold both for BPS and all the non-BPS orbits of the given model. In Section 3 we propose a complete basis for U -duality polynomials in the presence of two dyonic black-hole charge vectors in irreducible symmetric models, and we also consider the role of the horizontal symmetry in this framework. Section 4 extends the analysis of Section 2 to different non-compact real forms of the stabilizer of one-centered charge orbits related to Jordan algebras over the octonions, namely to N = 8 theory (whose 1 8 -BPS one-centered stabilizer is E 6(2) ) and for exceptional magical N = 2 theory (whose BPS and non-BPS I 4 > 0 one-centered stabilizers are the compact E 6(−78) and the non-compact E 6(−14) , respectively).
Possible extensions of the present investigation may also cover composite configurations with "small" constituents, as well as a detailed study of the multi-centered charge orbits in N = 5, 6extended supergravity theories.

Little Group of p Charge Vectors in Irreducible Symmetric Models
We consider a p-center black hole solution in a Maxwell-Einstein supergravity theory in d = 4 spacetime dimensions.
The p dyonic black-hole charge vectors can be arranged as where Q M a sits in the irreducible representation (p, Sympl (G 4 )) of the group SL h (p, R) × G 4 . p is the fundamental representation (spanned by the index a = 1, . . . , p) of the horizontal symmetry group [12] SL h (p, R) (see Section 3), while Sympl (G 4 ) is the symplectic irreducible representation of the black-hole charges, spanned by the index M = 1, . . . , f of the U -duality group G 4 , where f ≡dim R (Sympl (G 4 )).
Suppose there are I p independent G 4 -invariant polynomials constructed out of Q a , and let G p denote the little group of the system of charges, defined as the largest subgroup of G 4 such that G p Q a = Q a ∀a. Then, the following relation 3 holds [12]: Some preliminary general observations are in order: • The group theoretical analysis of the present Section does not depend on the real form of G 4 and G p . We will then generally consider the complex groups. From a physical point of view, the BPS and non-BPS cases in various supergravity theories correspond to different choices of non-compact real forms of G p (and of G 4 , as well). However, for BPS orbits in N = 2 symmetric models, and in particular for magical models, the stabilizer is always the compact form of the relevant group (see Table 1).
• We shall generally assume Q 1 to be in a representation corresponding to a "large" black hole 4 , namely such that the quartic invariant I 4 Q 4 1 = 0.
• We shall consider "generic" orbits, in which all I p invariants are independent.
3 A necessary but not sufficient condition for Eq. (2.2) to hold is p < f , such that the p dyonic charge vectors can all be taken to be linearly independent. 4 Multi-center configurations with "small" constituents [7,17,18] can be treated as well, and they will be considered elsewhere.
• There are two relevant cases, corresponding to different behaviors in the counting of invariants: a) The largest subgroup commuting with In the case b), all the singlets in the decomposition of G 4 → G p correspond to p-center G 4invariant polynomials of Sympl (G 4 ). On the other hand, in the case a) the number of singlets corresponds to the number of p-center G 4 -invariant polynomials, plus one if some of them are charged with respect to U (1), because one of the singlets can still be acted on by the corresponding U (1)-grading.
• The general method for working out G p and thus I p , having solved the problem for p − 1 centers, is to consider the p th charge vector Q p as transforming in a (reducible) representation of the little group G p−1 of the former p − 1 charges, and solve the corresponding one-charge-vector problem.
In the next Subsections we will consider the cases p = 1 and p = 2 in all irreducible symmetric cases pertaining to supergravity theories in d = 4 dimensions (with the exception of the rank-1 t 3 model, treated in [12]). In the case p = 1, we will retrieve the well known result I p=1 = 1, whereas in the p = 2 case we will obtain I p=2 = 7 in all cases under consideration. denotes the "conformal" group of J A 3 (see e.g. [19], and Refs. therein). By introducing A = R, C, H, O, it is worth remarking that the stabilizer group G p=2 J A 3 and the automorphism group Aut (t (A)) of the normed triality t (A) in dimension dim R A = 1, 2, 4, 8 (given e.g. in Eq. (5) of [20]) share the same Lie algebra. In other words, g p=2 J A 3 ∼ tri (A), where tri (A) denotes the Lie algebra of Aut (t (A)) itself (see e.g. Eq. (21) of [20]).

J
Let us start considering the exceptional case, based on the Euclidean degree-3 Jordan algebra J O 3 on the octonions O. Since, as mentioned earlier, we actually work with complex groups, this case pertains also to maximal N = 8 supergravity, based on the Euclidean degree-3 Jordan algebra J Os 3 on the split octonions O s .
• Let us first solve the one-center problem (p = 1). G 1 is a real form of E 6 ; the 56 branches with respect to E 6 as follows (subscripts denote the U (1)-charges throughout): and correspondingly the charge vector Q 1 (defined as p Λ , q Λ throughout) decomposes as follows: Note that the branching (2.3) contains two E 6 -singlets, and According to the previous discussion, one of the singlets can be freely acted on by the U (1). Thus, by acting with G 4 /G 1 = E 7 /E 6 , the 1-center charge vector Q 1 can be reduced as follows: One is then left with only one independent singlet charge I (1) related to the 1-center quartic invariant I 4 Q 4 1 ; therefore, I 1 = 1, as expected. This analysis is consistent with the general formula (2.2), which in this case reads: (2.6) • Let us now proceed to deal with the two charge-vector problem (p = 2). The second charge vector is denoted as Q 2 ≡ (m Λ , e Λ ) throughout. Having solved the problem for p = 1, we can decompose Q 2 with respect to G 1 = E 6 using (2.3), obtaining the decomposition and then determine the corresponding little group inside E 6 . The little group of the irreducible representation 27 of E 6 is F 4 , under which Note in particular that F 4 is a maximal (symmetric) subgroup of E 6 , so that all singlets correspond to extra E 7 -invariant polynomials, and that m 26 can be set to zero through the action of G 1 /F 4 = E 6 /F 4 , thus yielding the result: (2.10) • The 26 of F 4 has little group SO (8), which does not commute with a U (1) in F 4 . Under this non-maximal embedding, the 26 branches as and correspondingly e 26 → (I (6) , I (7) , e 8v , e 8s , e 8c ) . (2.12) Therefore, by acting with F 4 /G 2 = F 4 /SO(8), Q 2 can then be put in the form −→ (I (2) , I (4) , 0 26 , I (3) , I (5) , I (6) , I (7) , 0 8v , 0 8s , 0 8c ). (2.13) In conclusion, we found that the little group of a 2-centered black-hole solution is G 2 = SO(8), and the corresponding 2-centered charge orbits correspond to different real forms of the quotient of complex groups . (2.14) The E 7 -invariant polynomials for a 2-centered configuration are seven: I 2 = 7; indeed, the general formula (2.2) gives: This model is based on the Euclidean degree-3 Jordan algebra J H 3 on the quaternions H, and it is "dual" to N = 6 "pure" theory, because these theories share the same bosonic sector [14,23,21,22,24].
( 2.27) In conclusion, we found that the little group of a 2-centered black-hole solution is G 2 = [SU (2)] 3 , and the corresponding 2-centered charge orbit reads (in complexified form) The SO (12)-invariant polynomials for a 2-centered configuration are seven: I 2 = 7; indeed, the general formula (2.2) gives: Let us now consider the model based on the Euclidean degree-3 Jordan algebra J C 3 on C. Since, as mentioned earlier, we actually deal with groups on the complex field, this case pertains also to "pure" N = 5 supergravity, which is based on M 1,2 (O), the Jordan triple system (not upliftable to d = 5) generated by 2 × 1 matrices over O [14].
• Let us first solve the problem for p = 1. G 1 is a real form of SU (3), the relevant (maximal symmetric) embedding is and the 14 ′ accordingly branches as corresponding to the charge decomposition Once again, the analysis here is analogous to the cases treated above. The branching (2.44) contains two SU (3)-singlets, but, by virtue of (2.43), one of the singlets can be freely acted on by the U (1). By acting with G 4 /G 1 = U Sp (6) /SU (3), Q 1 can be reduced to  where 0 3 collectively denotes such three singlets set to zero.
In conclusion, we found that the little group of a 2-centered black-hole solution is the identity itself: G 2 = I, and the corresponding 2-centered charge orbit reads (in compact form) The U Sp (6)-invariant polynomials for a 2-centered configuration are seven: I 2 = 7; indeed, the general formula (2.2) yields: (6)) − dim R (I) = 7 + 21 − 0 = 28 = 2 · 14. (2.54) 3 Invariant Structures and the role of the Horizontal Symmetry SL h (2, R) We now propose a candidate for a complete basis of G 4 -invariant polynomials for the p = 2 case, highlighting the role of the horizontal symmetry group [12] in the classification of multi-center invariant structures.
Our treatment applies at least to the irreducible cubic geometries of symmetric scalar manifolds of d = 4 supergravity theories [15]  are invariant structures, symmetric in the symplectic indices (for the notation, see [26]). In particular, one can construct the so-called K-tensor 6 [27] where τ is a G 4 -dependent constant defined as with d ≡dim R Adj (G 4 ) and f ≡dim R (Sympl (G 4 )). From its definition (3.2), the K-tensor is a completely symmetric rank-4 G 4 -invariant tensor of Sympl (G 4 ). 5 As mentioned above, the irreducible rank-1 cubic case (the so-called N = 2, d = 4 t 3 model, associated to the trivial degree-1 Jordan algebra R) has been treated in [12]. 6 With respect to the treatment given in [27], we fix the overall normalization constant of the K-tensor to the value , as needed for consistency reasons.
In the presence of a single-centered black-hole background (p = 1), associated to a dyonic black-hole charge vector Q M in Sympl (G 4 ), the unique independent G 4 -invariant polynomial reads [27] On the other hand, in the presence of a multi-centered black-hole solution (p 2), the horizontal symmetry SL h (p, R) [12] plays a crucial role in organizing the various G 4 -covariant and G 4 -invariant structures.
In the following treatment we will consider the 2-centered case (p = 2), the index a = 1, 2 spanning the fundamental representation (spin s = 1/2) 2 of the horizontal symmetry SL h (2, R).
By using the symplectic representation (3.1) of the generators of G 4 , one can introduce the tensor (homogeneous quadratic in charges) where 3 is the rank-2 symmetric (spin s = 1) representation of SL h (2, R). In irreducible models, T α|ab is the analogue of the so-called T-tensor, introduced in [12] for reducible theories. Under the centers' exchange 1 ↔ 2, T α|11 ↔ T α|22 , while T α|12 is invariant. Interestingly, one can prove that the quantity is not independent from lower order invariants. Indeed, at least in the aforementioned irreducible cases, it holds that Thus, from (3.5) and (3.6), it follows that is the symplectic product of the charge vectors Q 1 and Q 2 , which is a singlet (1, 1) of SL h (2, R) × G 4 (manifestly antisymmetric under 1 ↔ 2). An important difference between the reducible models (studied in [12]) and the irreducible treated in the present investigation is that, while the former generally have a non-vanishing horizontal invariant polynomial X , the latter have it vanishing identically. Indeed, the analogue of X (defined by Eq. (4.13) of [12]) for irreducible models can be defined as where result (3.8) was used in the last step. The t 3 model mentioned in the Introduction is a nongeneric irreducible model (studied in Sec. 7 of [12]); in this case, the vanishing of X is given by Eq. (7.16) of [12]. By using the K-tensor (3.2), one can also define the tensor (homogeneous cubic in charges) By further contracting with a 2-centered charge vector, one can introduce the tensor (homogeneous quartic in charges) (3.14) Note that only the completely symmetric part Q By virtue of the various relations considered above, these three approaches give equivalent results, which we now specify for the sake of clarity: (3.17) The subscripts in the G 4 -invariant polynomials I +2 , I +1 , I 0 , I −1 and I −2 defined by (3.15)-(3.19) denote the polarization with respect to the horizontal symmetry SL h (2, R), inherited from the components of I abcd (3.12); indeed, the five G 4 -invariant polynomials (3.15)- (3.19) sit in the rank-4 symmetric representation (spin s = 2) 5 of SL h (2, R) itself [12].
In order to proceed further, it is worth mentioning the decomposition [27] t N α|M t β|N Q = −t α|M P t β|N Q C P N = 1 2n denotes an invariant primitive tensor of G 4 . From (3.20), the following identity for the K-tensor can be derived [27] (recall Footnote 6): A G 4 -invariant polynomial homogeneous sextic in charges can then be defined as follows: Note that I 6 is manifestly antisymmetric under 1 ↔ 2. The first line of (3.24) is manifestly [SL h (2, R) × G 4 ]invariant, the second and third lines provide explicit expressions, and in the fourth line the "master" identity (3.22) was exploited. If the symplectic product W = 0 (defined in (3.9)), the two charge vectors Q M 1 and Q M 2 are mutually non-local. The concept of mutual non-locality is very important in the treatment of marginal stability in multi-center black holes (see e.g. [5,6,7,1,28,17,18]).
The above treatment suggests that a candidate for a complete basis of G 4 -invariant polynomials in the irreducible cases under consideration is given by the seven polynomials: where [12] Tr I 2 = I +2 I −2 + 3I 2 0 − 4I +1 I −1 ; (3.27) Indeed, the spin s = 2 representation 5 of SL h (2, R), whose components are the G 4 -invariant polynomials I +2 , I +1 , I 0 , I −1 and I −2 (defined by (3.15)-(3.19)), can be re-arranged as a 3 × 3 symmetric traceless matrix I [12]. (3.27) and (3.28) (respectively homogeneous of order eight and twelve in charges) are the only independent SL h (2, R)-singlets which can be built out of such a 3 × 3 symmetric matrix I, due to its tracelessness [12]. Note that Tr I 2 and Tr I 3 are both invariant under 1 ↔ 2.
It is worth pointing out that the analysis of Secs. 2 and 3 can be easily generalised to p 3 centers. The two-centered representation of spin s = J/2 of SL h (2, R) is then replaced by the completely symmetric rank-J tensor representation R J of SL h (p, R) (J = 1, 2, 3, 4 are the values relevant for the above analysis). On the other hand, W and I 6 generally sit in the R 2 , 1 representation of SL h (p, R) × G 4 , where R 2 is the rank-2 antisymmetric representation of SL h (p, R) (which, in the case p = 2, becomes a singlet). However, due to the tree structure of the split flow in multi-center supergravity solutions [5,6,7,1], to consider only the case p = 2 does not imply any loss in generality (as far as marginal stability issues are concerned). For N = 2 BPS two-centered extremal black holes, the stabiliser of the supporting charge orbit is always compact, so the orbit is unique (see Table 1 for magical models). This is no longer the case when the stabiliser is non-compact, as it holds for N = 2 two-centered solutions with two non-BPS centers characterised by I 4 Q 4 1 > 0 and I 4 Q 4 2 > 0, and for N 3 two-centered solutions with two 1 N -BPS centers. These are interesting cases, in which a split attractor flow through a wall of marginal stability has been shown to occur [31,30]. We will consider here the 1 8 -BPS two-centered orbits in the maximal N = 8 theory (based on J Os 3 ) and the non-BPS two-centered orbits (of the aforementioned type) in the exceptional N = 2 magic model, based on J O 3 . These two cases can be obtained by repeating the analysis of Section 2.1 and choosing suitable non-compact real forms of G 4 and G 2 .
The 1-centered charge orbits respectively read [32,16]:  Table 1), the Lie algebra so (8) of the stabilizer of O N =2,J O 3 ,nBPS,p=2,I (4.7) is nothing but the Lie algebra tri (O) of the automorphism group Aut (t (O)) of the normed triality over the octonionic division algebra O (see e.g. Eq. (21) of [20]). It is here worth observing that the Lie algebra so (4, 4) of the stabilizer of O N =8, 1 8 -BPS,p=2,I (4.4) enjoys an analogous interpretation as the Lie algebra tri (O s ) of the automorphism group Aut (t (O s )) of the normed triality over the split form O s of the octonions. On the other hand, a similar interpretation seems not to hold for the stabilizer of O N =8, 1 8 -BPS,p=2,II (4.5) as well as for the stabilizer of O N =2,J O 3 ,nBPS,p=2,II (4.8). We expect the N = 8 orbits (4.4) and (4.5), as well as the N = 2 orbits (4.7) and (4.8), to be defined by different constraints on the four SL h (2, R) × G 4 invariant polynomials given by Eq. (3.26); we leave this interesting issue for further future investigation.
Here, we confine ourselves to present parallel results on pseudo-orthogonal groups, which may shed some light on the whole framework. Let us consider two vectors x and y in a pseudo-Euclidean (p + q)-dimensional space E p,q with signature (p, q) and p > 1, q > 1. The norm of a vector is defined as, say 9) and the scalar product as x · y ≡ x 1 y 1 + ... + x p y p − x p+1 y p+1 − ... − x p+q y p+q . It is intuitively clear that the two-vector orbits do depend on the nature of the vectors themselves. Let us start and consider two timelike vectors (x 2 > 0 and y 2 > 0), whose one-center orbits are separately given by O p=1,timelike . It is straightforward to show that the two-center orbits supporting this configuration are SO (p, q) SO (p − 2, q) if x 2 y 2 > (x · y) 2 ; (4.13) (4.14) If both vectors are spacelike (x 2 < 0 and y 2 < 0), the two-center orbits read SO (p, q) SO (p, q − 2) if x 2 y 2 > (x · y) 2 ; (4.15) SO (p, q) SO (p − 1, q − 1) if x 2 y 2 < (x · y) 2 . (4.16)