Small Black Hole Constituents and Horizontal Symmetry

By exploiting the role of the horizontal symmetry SL(2,R), we extend the analysis and classification of two-centered extremal black hole charge configurations to the case of"small"single-centered constituents. These latter are seen to decrease the number of independent horizontal-invariant polynomials from four to one, depending on the rank of the charge orbit supporting each of the two centers. Within U-duality groups of type E7, both reducible and irreducible symmetric supergravity models in four space-time dimensions are considered, thus encompassing N = 2 and N = 8 theories.


Introduction
Multi-centered black holes (BH) are a fascinating subject, dealing with several aspects of quantum gravity, when this theory is defined through supergravity, and its high-energy, fundamental completions, namely superstrings and M -theory. The discovery of the split attractor flow and of walls of marginal stability [1,2,3] and the corresponding issue of microstate BH counting [4]- [6], have been some remarkable achievements in this field, also characterized by some puzzling and yet not fully understood features, such as anti-marginal stability and entropy enigmas (see also [7]- [17]; for studies on N > 2, see [18]- [20]). Earlier studies on composite (super)gravity solutions and marginal stability, were done in [21], while [22] provides a recent review on wall-crossing formulae.
This paper continues the investigation of geometric aspects of BH physics, by exploiting the (classical) duality symmetries of the underlying supergravities [23], which are the continuum limit of the U-duality [24] governing the non-perturbative string dynamics, in the context of two-centered BH solutions. The ultimate aim is to show that different aspects of multi-centered BH dynamics are encoded into different values of (and constraints among) certain multi-centered duality polynomial invariants. These duality invariants characterize some multi-centered charge orbits, which generalize the electric-magnetic charge orbits encoding all the main features of single-centered BH solutions, such as the Bekenstein-Hawking entropy [25], the ADM mass [26] and the BPS (supersymmetry-preserving) properties [27,28,29,30].
The single-centered orbits [27] are known to provide a stratification of the space of the irrep. R of the d = 4 classical U -duality group G 4 of the f -dimensional electric-magnetic charge vector Q ≡ p Λ , q Λ (Λ = 1, ..., f /2). Analogously, the p-centered orbits are expected to stratify the space R 1 × · · · × R p , for p BH constituents, with electro-magnetic fluxes given by Interestingly, it has been recently uncovered [31,32] that a p ( 2)-centered BH solution in d = 4 space-time dimensions enjoys an "horizontal" symmetry SL h (p, R) among the centers of the BH constituents. As a consequence of this symmetry, further invariant polynomials in the charges beside the usual U -duality invariants acquire an important role, and they provide a tool to achieve a finer classification of the allowed two-centered configurations 1 .
As commented in [32], the analysis can be performed for a generic number p of centers, but, due to the structure of the split flow in multi-center (super)gravity solutions [1,2,3,4], the case p = 2 is already fully illustrative, at least regarding marginal stability. It is then natural to explore the charge orbits for the horizontal doublet (Q 1 , Q 2 ), and to classify the allowed quantum numbers for the extremal BH two-centered compounds by suitable invariant constraints among horizontal invariantpolynomials.
For p = 2, the simplest example of such "horizontal invariants" is provided by the symplectic product of two charge vectors [1,32] W ≡ Q 1 , where a, b = 1, 2, C M N is the symplectic metric and ǫ is the usual SL (2, R) ∼ Sp (2, R) rank-2 invariant tensor. By considering (Q 1 , Q 2 ) as a doublet Q a (spin s = 1/2, fundamental irrep. 2) of SL h (2, R) [31], it is evident from (1.2) that W is manifestly invariant under both the U -duality and the horizontal symmetry. It is known that W enters the description of many physical properties of the two-centered BH compound states, such as the equilibrium distance between the two centers, the intrinsic overall (orbital) angular momentum, and the marginal stability condition [1,3]. Indeed, a crucial feature of two-centered BH physics is that their physical properties turn out to depend not only on Q 1 + Q 2 , as for the BPS ADM mass [34], but also on other combination of charges, such as the symplectic product (1.2). Another important instance is the entropy at the (split) horizon, which is the sum of the entropies of the two single-centered BH constituents [3]: where I 4 is the unique quartic invariant polynomial of the irrep. R of the U -duality group G 4 , and K M N P Q is the so-called K-tensor [35]. As a natural two-centered generalization of the quartic invariant I 4 (Q), for p = 2 one considers the contraction of the K-tensor with four a priori different charge vectors, which gives rise to the symmetric I abcd tensor [31,32], sitting in the spin s = 2 irrep. 5 of SL h (2, R): If G 4 is a simple group, one can define the symmetric object in the horizontal indices [32] T where t α M N (α = 1, ..., d ≡dim R (Adj (G 4 ))) is the symplectic representation of the generators of the Lie algebra g 4 of G 4 . Using these tensors, one has where τ is a model-dependent parameter [35]: . (1.9) A similar structure arises also when G 4 is a semisimple group factorized as SL(2, R) × SO(m, n), characterizing the reducible symmetric models listed in Table 1. Indeed, within the so-called Calabi-Vesentini basis [36,37] symplectic frame, one can define the T-tensor [31] T where indices are raised with the the pseudo-Euclidean SO(m, n) metric η ΛΣ . This is a symmetric object in the horizontal indices, whose components are the triplet of T-tensors of the reducible models (called T 1 , T 2 , T 12 in [31]). They constitute a two-centered generalization of the product T ΛΣ = p Λ q Σ − q Λ p Σ , which appears in the single-centered fourth order invariant written as [38,39,40] where "Tr η " denotes the η-trace, namely the trace in which the indices are contracted with η. For two centers, in reducible models it generalizes to The study of two-center extremal BH charge orbits associated to a generic horizontal doublet Q 1 , Q 2 has been initiated in [31,32]. It was found that in d = 4 supergravity theories with symmetric scalar manifolds and for generic charge vectors for each of the two centers, the dimension of a complete basis of U -duality invariant-polynomials is seven, and it includes both the horizontal singlet W and the quintet I abcd . The minimum number of invariant polynomials has been shown to decrease to four if only polynomials invariant under both U -duality and horizontal symmetry are taken into account. The dimension of this G 4 × SL h (2, R) invariant basis further reduces in some specific cases, for instance, for some rank-2 and rank-1 symmetric scalar manifolds pertaining to the so-called st 2 and t 3 , N = 2, d = 4 models, and for theories whose BH charge irrep. admits a quadratic invariant polynomial |I 2 | = |I 4 |, namely, N = 2 minimally coupled [50] and for N = 3 supergravity theories [41], which will not be dealt with in this paper. Interestingly, the "pure" N = 4, d = 4 supergravity, despite having a scalar manifold SL(2,R) U (1) of rank 1, has a complete basis formed by seven duality invariants [31].
The present investigation completes the previous analysis in that it determines the restrictions and constraints on such invariants when at least one of the two centers is occupied by a "small" black hole, having I 4 (Q) = 0. Small black holes, corresponding to horizonless solutions in Einstein two-derivatives (super)gravity, have zero entropy and they don't show an attractor behavior [27]. However, they are an interesting sector of the BH spectrum that has been recently examined in [42,29], in particular regarding the different supersymmetry features of the allowed orbits.
Large black hole charge orbits, with I 4 = 0 are described by a minimum of four charges, and in this sense the corresponding orbits are "rank four". More precisely, the rank of the orbit (or of the charge vector Q spanning it) is here the minimal number of charges which compose an orbit representative. Mathematically speaking, this defines the rank of Q as element of the associated Freudenthal triple system [43,44]. However, large orbits can become "lightlike" when they satisfy the condition I 4 = 0, and their rank reduces to three. If a further differential constraint ∂I 4 /∂Q = 0 is imposed, the rank further reduces to two (critical orbits), and it becomes one for doubly-critical orbits, having a suitable projection of ∂ 2 I 4 /(∂Q) 2 vanishing [45]. In Sec. 2 we shall revisit and add new results to the manifestly U -invariant constraints defining the rank, ranging from 4 to 1, of the single-centered charge vector Q, which gives rise to the stratification of the representation space of the U -duality group. When combining two centers, these constraints will clearly reflect in a number of combinations for the orbits of the compound system, which will be thoroughly investigated.
In Sec. 3 we shall deal with relations and properties of the invariant polynomials characterizing a two-centered (extremal) BH compound, which admit a natural interpretation and classification in terms of the horizontal symmetry group SL h (2, R).
Then, the results of Secs. 2 and 3 are used to perform a detailed analysis of all possible twocentered (extremal) BH charge configurations, by considering all possible combinations of the ranks of the charge vectors Q 1 and Q 2 pertaining to the two single-centered BH constituents.
This analysis is carried out for all supergravity theories (with symmetric scalar manifold) whose d = 4 U -duality group G 4 is a "group of type E 7 ", namely a group with a symplectic representation R admitting a completely symmetric rank-4 invariant structure q such that the invariant polynomial I 4 can be defined as 2 [47] It is also worth recalling that the "group of type E 7 " G 4 is a symmetry group of Jordan algebra-related structures, namely: where Aut (M (J 3 )) is the automorphism group of the vector space M (J 3 ) ≡ R⊕R⊕J 3 ⊕J 3 constructed over the Euclidean rank-3 Jordan algebra J 3 , whose conformal group is Conf (J 3 ) (see e.g. [48,49] for recent reviews and lists of Refs.). On the other hand, the G 4 's of minimally coupled N = 2 , and N = 3 [51] supergravities, omitted in this investigation, do not enjoy an interpretation in terms of (rank-3 Euclidean) Jordan algebras. The groups of type E 7 G 4 's (which are U -duality groups of supergravity theories in d = 4 spacetime dimensions with symmetric scalar manifolds) may be grouped into two classes, depending on whether G 4 is a semisimple Lie group, or it is a simple Lie group itself. The former case, analyzed in Sec. 4, corresponds to the reducible symmetric models, whose scalar manifolds 3 are grouped into two infinite sequences, reported in Table 1. The first (n = 1) element of the N = 2 sequence in Table 1, namely the so-called st 2 model, is non-generic, and it deserves a separate treatment, given in Sec. 4.2. The corresponding two-centered (extremal) BH charge orbits with both Q 1 and Q 2 "large" have been studied in [31]. Generally, the number of independent G 4 -invariant is seven, and a complete basis can be taken to be [31] W, X , I abcd , (1.15) where the quartic polynomial X is defined by Eq. (3.20). The latter case, analyzed in Sec. 5, corresponds to the irreducible symmetric models, namely to the so-called N = 2 t 3 model, to the N = 2 magical Maxwell-Einstein supergravity theories as well as to N = 5, 6, 8 "pure" supergravities, whose scalar manifolds are reported in Table 2. The t 3 model is non-generic and it deserves a separate treatment, given in Sec. 5.2.
The two-centered (extremal) BH "large" charge orbits of generic irreducible symmetric models and of t 3 model have been studied respectively in [32] and in [31]. Generically, the number of independent G 4 -invariant is seven, and a complete basis can be taken to be [32] W, I abcd , I 6 , (1. 16) where the sextic polynomial I 6 is defined by Eq. (3.15). The main difference between the sets (1.15) and (1.16) is that X = 0 (3.21) in irreducible cases, where also the constraint (4.8) of degree twelve Table 1: Reducible symmetric d = 4 supergravity models. "mcs" stands for maximal compact subgroup (with symmetric embedding). The rank of the scalar manifold, as well as the related reducible Euclidean rank-3 Jordan algebra J 3 are also given (for further elucidation, see e.g. [56] and Refs. therein). The subscript "v" stands for "vertical", and it has been introduced in order to distinguish the S-duality SL v (2, R) group from the horizontal symmetry group SL h (2, R) Table 2: Irreducible symmetric d = 4 supergravity models. N = 2 magical quaternionic Maxwell-Einstein supergravity and N = 6 "pure" supergravity are "twin", namely they share the same bosonic sector [57,58,59,60,61]. M 1,2 (O) is the Jordan triple system (not upliftable to d = 5) generated by 2 × 1 matrices over O [62]. Note that, with the exception of the reducible -but triality symmetricstu model [39,63], irreducible models are all ones for which the treatment of [35] holds (see e.g. Table  1 therein) in the charges does not hold. It will be emphasised in Sec. 3 that the existence of X in reducible models can be traced back to the semi-simple nature of the U -duality group G 4 , giving rise to two independent quartic polynomials I ′ and I ′′ with vanishing horizontal helicity, and related to I 0 by Eq. (3.9).
The outcome of this analysis is that the number of independent duality-and horizontal-invariant polynomials is decreased when one or both charge vectors are "small", since the vanishing of the quartic form (1.5) for one center or both gives rise to various polynomial relations among otherwise independent invariants. For example, as yielded by the analysis of Secs. 4 and 5, if at least one of the two charge vectors Q 1 and Q 2 is doubly-critical (i.e. rank 1, see Sec. 2; in the N = 8 language, this is 1 2 -BPS), then only one independent horizontal invariant, say the symplectic product W (1.2), exists. This is one of the few examples known in the literature [3].
Notice that the results concerning the st 2 and t 3 models are consistent with the " stu → st 2 → t 3 reduction" (see e.g. the discussion in [52,53,31], and the recent analysis in [33]).
Finally, in Sec. 6 we make some comments on some "small + small" two-centered charge configurations corresponding to I 4 (Q 1 + Q 2 ) < 0, i.e. to a BH compound that, regarded as a single-centered solution, is a "large" non-BPS (extremal) BH, making contact with recent literature, such as [11] and [54].

U -Invariant Constraints on Q
We start by giving a résumé (with original results) of the U -invariant constraints defining the charge orbits of a single-center (extremal) BH, namely of the G 4 -invariant conditions defining the rank of the dyonic charge vector Q ∈ R as an element of the corresponding Freudenthal triple system (FTS) (see [43,44], and Refs. therein); as mentioned above, G 4 is the d = 4 U -duality group "of type E 7 " [47], and R is its relevant BH charge irrep.. The symplectic indices M = 1, ..., f (f ≡dim R R (G 4 )) are raised and lowered with the symplectic metric C M N defined by (1.3). By recalling the definition (1.5) of the unique (quartic) G 4 -invariant polynomial constructed with Q ∈ R, the rank of a non-null Q as an element of F T S (G 4 ) range from four to one, and it is manifestly G 4 -invariantly characterized as follows: 1. rank (Q) = 4: "large" extremal BHs, with non-vanishing area of the event horizon: 2. rank (Q) = 3: "small" lightlike extremal BHs, with vanishing area of the event horizon: 3. rank (Q) = 2: "small" critical extremal BHs: = 0, at least for some M, N .
(2.5) 4. rank (Q) = 1: "small" doubly-critical extremal BHs [45,64]: (2.6) Let us consider the doubly-criticality condition (2.6) more in detail. At least for "groups of type E 7 " [47] which are U -duality groups G 4 of irreducible symmetric models in d = 4, it holds that where the subscripts "s" and "a" clearly stand for symmetric and skew-symmetric. The presence of the singlet (which is nothing but C M N defined in (1.3)) in the skew-symmetric part (2.8) characterizes the BH charge irrep. R to be symplectic. For example, for G 4 = E 7 (R = 56, Adj = 133) one gets (see e.g. [65]; the subscripts "s" and "a" respectively stand for symmetric and antisymmetric) For such groups, one can construct the projector operator on Adj (G 4 ): (2.12) where (recall (2.7)) (2.14) (2.15) where the model-dependent parameter τ is defined by (1.9). The explicit expression of P CD AB reads 4 (α = 1, ..., d): where the relation [35] (see also [55]) is the symplectic representation of the generators of the Lie algebra g 4 of G 4 . Notice that τ < 1 defined in (1.9) is just the ratio of the dimensions of the adjoint Adj and rank-2 symmetric (R × R) s (2.7) reps. of G 4 , or equivalently the ratio of upper and lower indices of t α M N 's themselves. It should also be noted that, with respect to the treatment given in [35], the result (2.18) has been supplemented with the relation ξ = − 1 3τ [32], obtained as a consistency condition within the computations yielding to (2.17).
The result (2.17) is a direct consequence of the fact that t α M N is the projector of (R × R) s onto Adj (recall (2.7)). More precisely, it holds that where the normalization (see e.g. Eq. (2.5) of [35]) has been used.

Two-Centered Relations
In order to study multi-centered charge configurations, it is worth considering some general relations for p( 2)-center (extremal) BHs, which are manifestly covariant under the horizontal symmetry SL h (p, R) introduced in [31]; we will here focus on the case p = 2. From [31,32] and the Introduction, we recall I abcd and respectively sitting in the spin s = 2 and s = 3/2 of SL h (2, R) (the horizontal indices a = 1, 2 are raised and lowered with ǫ ab , with ǫ 12 ≡ 1). For clarity's sake, we report the explicit expressions of the various components of I abcd , as well as their relations with the components of Q abc (3.1) (the subscripts "+2, +1, 0, −1, −2" denote the horizontal helicity of the various components [31,32]): Note that in reducible symmetric models, due to the semi-simple nature of G 4 , there are two independent components of I abcd with vanishing horizontal helicity, namely I ′ and I ′′ [31]: and related to I 0 by the relation (4.4) of [31]: Thus, one can consider the following symplectic product of spin 3/2 horizontal charge tensors:

10)
A priori, Q abc , Q def should project onto spin s = 3, 2, 1, 0 irreps. of SL h (2, R) itself; however, due to the complete symmetry of the K-tensor (and to the results of [47,35]), the projections on spin s = 3 and 1 do vanish: where the symplectic product W is defined by (1.2), and in (3.14) the definition of the sextic horizontal polynomial I 6 [32] (given by Eq. (3.24) of [32]) has been recalled: The complementary relation to (3.15), namely 1 4 Q 111 , Q 222 − 3 4 Q 122 , Q 112 consistently turns out to be proportional (through W) to the zero helicity component of I abcd ; indeed, by setting (a, b, e, f ) = (1, 1, 2, 2) in (3.12), one obtains: Furthermore, at least in reducible symmetric models (listed in Table 1), the sextic invariant is naturally defined within the T-tensor formalism as follows (see Sec. 3 of [31] for further detail): As denoted by the prime, I ′ 6 (3.17) does not coincide with the I 6 given by (3.14) (or equivalently by (3.15)); indeed, irreducible and reducible symmetric models, the following relation respectively holds: (3.18) This can be traced back to a crucial difference (pointed out in Sec. 3 of [32]) between reducible and irreducible symmetric models, concerning the horizontal invariant polynomial (of degree four in charges) X . In reducible models, X is defined by Eq. (4.13) of [31]: and it generally does not vanish. On the other hand, in irreducible models it is defined by Eq. (3.10) of [32], and it vanishes identically: Thus, it is here worth remarking that the relations (3.11)-(3.16) hold both in irreducible models (characterized by (3.21) and in reducible models (generally having non-vanishing X (3.20)). On the other hand, in presence of X = 0, I ′ 6 undergoes the renormalization I ′ 6 → I ′ 6 + 1 6 X W, and this explains Eq. (3.19) from Eq. (3.18).
Before analyzing and classifying the two-center extremal BH configurations and the corresponding defining constraints in terms of G 4 -and [SL h (2, R) × G 4 ]-invariant polynomials, in light of previous definitions and findings, we conclude this Section by pointing out some consequences of the rank of a charge vector, say Q 1 , on the set of G 4 -and [SL h (2, R) × G 4 ]-invariant polynomials of the two-centered configuration (Q 1 , Q 2 ), both in irreducible and reducible symmetric models: X = 0 (in reducible models).

Reducible Models
As given by Table 1, the reducible symmetric d = 4 supergravity models 5 have the following d = 4 U -duality group:  where the isomorphism (see Sec. 8 of [31]) has been used. The corresponding scalar manifolds thus belong to the sequence ST [m, n], of particular relevance for superstring compactifications (see e.g. the analysis in Sec. 3.1 and App. C of [68], and Refs. therein). We now give a complete analysis of all possible two-center charge configurations (Q 1 , Q 2 ) (with symplectic product (1.2) W = 0, i.e. mutually non-local), by providing for each configuration a "minimal" sets of independent G 4 -invariant and [SL h (2, R) × G 4 ]-invariant polynomials. The analysis will be carried out in the bare charges Q basis, by exploiting, for each of the two centers, the dualityinvariant definitions of rank of Q recalled in Sec. 2. The definitions and notation of [31] are used. The prototype of a generic (N = 2) reducible symmetric model is the stu model, studied in some detail in Sec. 2 of [31]. The non-generic case of the N = 2, d = 4 st 2 model is considered in Sec. 4.2.

Generic Reducible Models
1. (Q 1 , Q 2 ) = (rank 4, rank 4). This is the configuration supporting the generic 2-center charge orbits, studied in some detail in [31]. The number of independent G 4 -invariant is seven, a complete basis can be taken to be given by Eq. (1.15). On the other hand, the number of independent [SL h (2, R) × G 4 ]-invariant is four, and one can choose a complete basis to be [31] [SL h (2, R) × G 4 ] -inv (# = 4) : W, X , T r I 2 , T r I 3 , (4.3) where The choice (4.6)-(4.7) is characterized by the fact that the horizontal basis is a subset of the duality basis. Furthermore, the duality basis contains quantities related both to the singlecenter BH entropy π |I 4 (Q 1 + Q 2 )| and to the two-centered BH entropy (1.4). In general, I ′ 6 is related to the elements of the basis (4.3) by means of the polynomial constraint of degree twelve in charges given by Eq. (5.6) of [31], which we recall here (see also the analysis in [33]):  where T r I 2 is given by (4.11), and T r I 3 is further simplified to   This case actually splits in two subcases. Indeed, by plugging (3.23) into (4.8), this latter factorizes as W 2 − 4X + 6I 0 W 2 + 2X + 6I 0 = 0, (4.18) thus admitting two solutions:
10. (Q 1 , Q 2 ) = (rank 1, rank 1). Eq. (4.24) still holds true, and one obtains that the only relevant polynomial invariant is the symplectic product W. An example of this configuration is the D0 + D6 two-constituents solution of the stu model studied in [11] (see also [9]).

The st 2 Model
We now proceed to consider the non-generic case of the N = 2, d = 4 st 2 model, which has a rank-2 vector multiplets' scalar manifold, namely SL(2,R) , first (n = 1) element of the N = 2 sequence in Table 1. Thus, the U -duality group is This model may be obtained as a rank-2 truncation (also named " st 2 degeneration"; see e.g. [52,53,31]) of the prototype of generic reducible N = 2 symmetric models which, as observed above, is the stu model.
1. (Q 1 , Q 2 ) = (rank 4, rank 4). This is the configuration supporting the generic 2-center orbit, studied (in the BPS case) in some detail in Sec. 6 of [31]. The number of independent G 4invariant is six, a complete basis can be taken to be [31] [SL v (2, R) × SO (2, 1)] -inv (# = 6) : W, X , I ±2 , I ±1 . (4.31) On the other hand, the number of independent [SL h (2, R) × G 4 ]-invariant is three, and a complete basis can be taken to be [31] [SO v h (2, 2) × SO (2, 1)] -inv (# = 3) : W, X , T r I 2 . (4.32) Two polynomial constraints, of degree sixteen and eight in charges, hold in the st 2 model; they are given by Eqs. (6.10)-(6.12) of [31], which we recall here (see also the analysis in [33]): By means of (4.33) and (4.34), I ′ 6 and T r I 3 can be expressed in terms of the horizontal invariants of the basis (4.32). where T r I 2 is given by (4.11) and T r I 3 is further simplified to (4.15). For this configuration, there is actually a second subcase given by the subcase III (4.24) of the generic models, in which then W is the only relevant polynomial invariant. In the st 2 model, (4.24) is realized in the configuration (Q 1 , Q 2 ) = (rank 2, rank 2) for instance by setting the charges of the two vector multiplets to coincide; in the "d = 4/d = 5 special coordinates" symplectic frame, a particular realization of this is the "t 3 degeneration" (see e.g. [52,53,31]) in which p 1 = p 2 ≡ p and q 1 = q 2 ≡ q/2.  10. (Q 1 , Q 2 ) = (rank 1, rank 1). Eq. (4.24) does hold for Q 1 and for Q 2 , and Eq. (4.29) also holds, i.e. the only relevant polynomial invariant is W.

Irreducible Models
We now proceed to consider the irreducible symmetric models (see Table 2), by providing a complete analysis of all possible two-center charge configurations (Q 1 , Q 2 ) (with symplectic product (1.2) W = 0, i.e. mutually non-local), in the very same way as done for reducible symmetric models in Sec. 4. The definitions and notation of [31] and [32] are used. The non-generic case of the N = 2, d = 4 t 3 model is considered in Sec. 5.2. 4, rank 4). This is the configuration supporting the generic 2-center orbits, studied in some detail in [32]. As mentioned in Sec. 3 and derived in [32] (see Eq. (3.10) therein), the quartic horizontal-invariant polynomial X irred vanishes identically in the class of models under consideration. Furthermore, (an analogue with X irred = 0 of) the polynomial constraint (4.8) does not hold in these models. As discussed in [32], the number of independent G 4 -invariant is seven, a complete basis can be taken to be given by Eq. (1.16). On the other hand, the number of independent [SL h (2, R) × G 4 ]-invariant is four, and one can choose the following complete basis [32]:

Generic Irreducible Models
As for the reducible models analyzed in Sec. 4, different choices are of course possible. E.g., an equivalent duality-invariant complete basis respectively reads The choice (5.2) is characterized by the fact that the horizontal basis is a subset of the duality basis. Furthermore, in this case the duality basis contains quantities related both to the singlecenter BH entropy and to the two-centered BH entropy.

The t 3 Model
We now proceed to consider the non-generic case of the N = 2, d = 4 t 3 model, which has a rank-1 vector multiplets' scalar manifold, namely SL(2,R) U (1) (see the first line of Table 2. Thus, the U -duality group is As mentioned above, this model provides a simple yet interesting example, because it may be obtained both as the rank-1 truncation of the reducible N = 2 symmetric models, as well as the first (q = 0), non-generic element of the sequence of irreducible N = 2 symmetric models, which contains the four rank-3 magical supergravity theories [62].
1. (Q 1 , Q 2 ) = (rank 4, rank 4). This is the configuration supporting the generic 2-center orbit 6 , studied (in the BPS case) in some detail in Sec. 7 of [31] (see also the comment in the Introduction of [32]). The number of independent G 4 -invariant is five, a complete basis can be taken to be [31] SL v (2, R) -inv (# = 5) : W, I 6 , I ±2 , I +1 .  Three polynomial constraints, of degree sixteen, eight and four in charges, hold in the t 3 model; they are given by Eqs. (6.12), (7.18), (7.17) and (7.16) of [31] (see also App. A therein, and the analysis in [33]), which we recall here: Note that in (5.25) X ≡ X irred , and its identical vanishing consistently characterizes the t 3 model as a non-generic irreducible symmetric model. Due to (5.25), I ′ 6 and I 6 are related through Eq. 4. (Q 1 , Q 2 ) = (rank 2, rank 4). This charge configuration has been considered in [3]. The complete duality and horizontal bases can respectively be taken to be ⇔ I 6 = 1 12 W 3 ; (5.32) Consistent with the fact that the t 3 model can be obtained as a "rank-1 degeneration" (see e.g. [52,53,31]) of the reducible symmetric N = 2, d = 4 models, (5.32) matches the X = 0 limit of subcases I (4.19) and II (4.20) or, equivalently, it matches the subcase III (4.24).
5. (Q 1 , Q 2 ) = (rank 2, rank 3). The complete duality and horizontal bases can be taken to coincide, and W is the only relevant invariant polynomial in charges: 6. (Q 1 , Q 2 ) = (rank 2, rank 2). Again, the complete duality and horizontal bases can be taken to coincide and be given by (5.33).
Since the t 3 model lacks of an independent doubly-critical "small" charge orbit (namely, criticality implies doubly-criticality in this model; see e.g. [69] for a recent account within a d = 3 timelikereduced formalism), the cases given by points 7, 8, 9 and 10 are all missing for this model.

A Comment on Bound States with Negative Discriminant
The treatment given in Secs. 4 and 5 allows one to discuss in fair generality two-centered extremal BH compound states with a given value of I 4 (Q 1 + Q 2 ). Here, we would like to comment shortly on some two-centered charge configurations corresponding to a negative I 4 (Q 1 + Q 2 ), i.e. to a BH compound that, regarded as a single-centered solution, is a "large" non-BPS (extremal) BH.
By recalling the sum rule (cf. Eq. (4.7) of [31]) from the analysis of previous Secs. one can single out some two-charge configurations which necessarily imply Note that in this case I 4 (Q 1 + Q 2 ) becomes horizontal invariant, as well. For all symmetric reducible models (see Table 1), these configurations are the ones with i.e. points 9 and 10 of Sec. 5.1. In the non-generic irreducible t 3 model, treated in the Sec. 5.2, the unique configuration of the kind under consideration is given by rank (Q 1 , Q 2 ) = (2, 2) (point 6 above), because, as mentioned in Sec. 5.2, in such a model criticality implies doubly-criticality.
with a negative quartic duality invariant I 4 (Q 1 + Q 2 ). These configurations are (the semiclassical limit, with real, continuous charges of) some of the cases recently analyzed by Sen in 8 [54] in N = 4, d = 4 supergravity; furthermore, the case rank (Q 1 , Q 2 ) = (1, 1) encompasses the D0 + D6 configuration in the N = 2 stu model, which has been studied in [11]. Since both (6.2) and (6.9) are manifestly horizontal-invariant expression, the most general supporting two-centered configurations can be computed by acting on the relevant case of the heterotic configuration (6.5) with a generic [SO v h (2, 2) × SO (m, n)]-transformation (also, recall (4.2)).