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Charge Orbits of Extremal Black Holes in Five Dimensional Supergravity

We derive the U-duality charge orbits, as well as the related moduli spaces, of"large"and"small"extremal black holes in non-maximal ungauged Maxwell-Einstein supergravities with symmetric scalar manifolds in d=5 space-time dimensions. The stabilizer groups of the various classes of orbits are obtained by determining and solving suitable U-invariant sets of constraints, both in"bare"and"dressed"charges bases, with various methods. After a general treatment of attractors in real special geometry (also considering non-symmetric cases), the N=2"magic"theories, as well as the N=2 Jordan symmetric sequence, are analyzed in detail. Finally, the half-maximal (N=4) matter-coupled supergravity is also studied in this context.


Introduction
Five-dimensional supergravity theories with non-maximal supersymmetry (2 N < 8), emerging from Calabi-Yau compactifications of M -theory, admit extremal black p-brane solutions in their spectrum [1]. In particular, ungauged theories admit extremal black holes (p = 0) and black strings (p = 1) which are asymptotically flat, and reciprocally related through U -duality 1 . These objects have been intensely studied along the years, due to the wide range of classical and quantum aspects they exhibit.
For asymptotically flat, spherically symmetric and stationary solutions, the Attractor Mechanism [3,4,5,6] proved to be a crucial phenomenon, determining, in a universal fashion, the stabilization of scalar fields in the near-horizon geometry in terms of the fluxes of the two-form field strengths of the Abelian vector fields coupled to the system. Moreover, the Attractor Mechanism turned out to be important also to unravel dynamical properties such as split attractor flows [7] and wall crossing [8], and to gain insights in the microstate counting analysis (see e.g. [9], and Refs. therein), also in relation to string topological partition functions [10] (see also [11] for a recent account and list of Refs.). In d = 5 space-time dimensions, progress has been achieved also with the discovery of new attractor solutions (see e.g. [12]), as well as with the formulation of a first-order formalism governing the evolution dynamics of non-supersymmetric scalar flows [13].
For supergravity theories with scalar manifolds which are symmetric cosets, the extremal solutions of the ungauged theory can be classified through the orbits of the relevant representation space of the U -duality group, in which the corresponding supporting charges sit. The relation between U -invariant BPS conditions and charge orbits in d = 5 supergravities has been the subject of various studies along the years [14,15,16,17,18,19,20].
The present paper extends to d = 5 space-time dimensions the 4-dimensional investigation of [21].
We derive the U -duality charge orbits, as well as the related moduli spaces, of "large" and "small" extremal black holes and black strings in ungauged Maxwell-Einstein supergravities with symmetric scalar manifolds. The stabilizer groups of the various classes of orbits are obtained by determining and solving suitable U -invariant sets of constraints, both in "bare" and "dressed" charges bases, as well by exploitingİnönü-Wigner contractions and SO (1, 1)-gradings.
It is here worth pointing out that in this paper we will not deal with maximal N = 8, d = 5 supergravity, because a complete analysis of extremal black hole attractors and their "large" and "small" charge orbits is already present in literature, see e.g. [17,14,15,16,18,22,23,20,24,25,26]. We will just mention such a theory shortly below Eq. (4.3).
The plan of the paper is as follows.
We start and give a résumé of real special geometry in Sect. 2, setting up notation and presenting all formulae needed for the subsequent treatment of charge orbits and attractors.
In Sect. 3 extremal black hole (black string) attractors are studied in full generality within real special geometry. Starting from the treatment of [19], various refinements and generalizations are performed, in particular addressing the issue of generic, non-symmetric vector multiplets' scalar manifolds. In Subsect. 3.1 we analyze the various classes of critical points of the effective potential V , also within the so-called "new attractor" approach (see Subsubsect. 3.1.4). Then, in Subsect. 3.2 we compute the higher order covariant derivatives of the previously introduced rank-3 invariant tensor T xyz , which will play a key role in the subsequent developments and results, exposed in Subsects. 3.3 and 3.4, respectively dealing with generic and homogeneous symmetric real special manifolds. A general analysis of the Hessian matrix of V , crucial in order to establish the stability of considered attractor points, is then performed in Subsect. 3.5. In Sect. 4 all "small" charge orbits of symmetric "magic" real special geometries are explicitly determined and classified, by exploiting the properties of the functional I 3 introduced in Subsubsect. 3.3.3. Note that "small" charge orbits support non-attractor solutions, which have vanishing Bekenstein-Hawking [27] entropy in the Einsteinian approximation. Nevertheless, they can be treated by exploiting their symmetry properties under U -duality.
Sect. 5 analyzes the "duality" relating the N = 2 "magic" theory coupled to 14 Abelian vector multiplets and the N = 6 "pure" supergravity, both based on the rank-3 Euclidean Jordan algebra J H 3 and thus sharing the very same bosonic sector. Then, Sect. 6 is devoted to the analysis of the "large" (Subsect. 6.1) and "small" (Subsect. 6.2) charge orbits of N = 2 Jordan symmetric sequence. Similarly, Sect. 7 provides a detailed treatment of the "large" (Subsect. 7.1) and "small" (Subsect. 7.2) charge orbits of the half-maximal (N = 4) matter coupled supergravity. The analysis of both Sects. 6 and 7 is made in the "bare" charges basis, and various subtleties, related to the reducible nature of the d = 5 U -duality group and disconnectedness of orbits in these two theories, are elucidated. Some Appendices conclude the paper, containing various details concerning the determination of the "small" orbits in symmetric "magic" real special geometries.
The resolution of U -invariant defining (differential) constraints, both in "bare" and "dressed" charges bases, is performed in App. A.
Finally, we point out that all results on charge orbits can actually be obtained in various other ways, including the analysis of cubic norm forms of the relevant Jordan systems in d = 5; this will be investigated elsewhere.
In the present Section, we recall some basic facts about RSG, setting up notation and presenting all formulae needed for the subsequent treatment of charge orbits and attractors. Apart from a slight changes in notation, we will adopt the conventions of [19], which are slightly different from the ones used in [34] (see the observations in [34] itself).
We start by specifying the kind and range of indices being used. i = 0, 1, ...n V is the index in the "ambient space" (in which M 5 is defined through a cubic constraint; see Eq. (2.5) below). "0" is the index pertaining to the ("bare") d = 5 graviphoton, and n V stands for the number of Abelian vector multiplets coupled to the supergravity multiplet. On the other hand, x = 1, ..., n V , and a = 1, ..., n V respectively denote "curved" and (local) "flat" coordinates in M 5 .
The metric a ij in the "ambient space" (named g ij in [34]) can be defined as follows: is the volume of M 5 itself, and d ijk = d (ijk) is a rank-3 completely symmetric invariant tensor (see further below). In turn, the λ i 's are some real functions (with suitable features of smoothness and regularity) of the set of scalars φ x of the theory, coordinatizing M 5 : They do satisfy the inequality (2.2). As elucidated e.g. in [34], the λ i 's are nothing but the (opposite of the) imaginary ("dilatonic") part of the complex scalar fields of the special Kähler geometry (SKG) based on a cubic holomorphic prepotential (usually named d-SKG; see e.g. [32,36]), endowing the Abelian vector multiplets' scalar manifold of N = 2 Maxwell-Einstein supergravity in 4 space-time dimensions. In this respect, the "ambient space" in 5 dimensions is nothing but the "dilatonic sector" of the d-SKG in 4 dimensions.
It is now convenient to introduce rescaled variables as follows: Thus, the metric of M 5 is the pull-back of a ij on the hypersurface V (λ) ≡ 1 (2.5) in the "ambient space", namely: where (the semicolon denotes Riemann-covariant differentiation throughout) Notice that the constraint (2.4) implies ∂V λ ∂φ x = 3d ijk λ i ,x λ j λ k − √ 6d ijk λ i x λ j λ k = 0. (2.8) Let us now introduce T xyz , a rank-3 completely symmetric invariant tensor, related to d ijk through the definition whose inversion reads In other words, T xyz is the φ-dependent "dressing" (through λ i ,x (φ)'s) of the constant (φ-independent) tensor d ijk . It is here worth anticipating that Eqs. (2.9) and (2.10) play the key role to relate the formalism based on "bare" charges with the formalism based on the "dressed" charges (see further below).
T xyz enters the so-called "RSG constraints", relating in M 5 the Riemann tensor R xyzu to the metric tensor g xy , as follows: It is worth noticing a direct consequence of such "RSG constraints": the sectional curvature (see e.g. [37] and [38]) of matter charges in RSG globally vanishes: This is trivially due to the symmetry properties of the Riemann tensor R xyzw (which are the ones for a generic Riemann geometry), and it is a feature discriminating RSG from SKG (in which R (Z) generally does not vanish; see e.g. [39,40]). As a consequence of the constraints (2.12) (within the metric postulate), the definition of M 5 to be an homogeneous symmetric manifold R xyzu;t = 0 (2.14) yields where the index-raising through the contravariant metric • a ij has been explicited.

Attractors in RSG
The present Section is dedicated to the study of attractors in RSG. This has been firstly treated in [19] (and then reconsidered in [20], in connection to d = 6).
Starting from the treatment of [19], we will generalize and elaborate further various results obtained therein.
It is worth recalling that no asymptotically-flat dyonic solutions of Einstein Eqs. exist in d = 5. Thus, the d = 5 asymptotically flat black holes (BHs) can only carry electric charges q i . Their magnetic duals are the d = 5 asymptotically flat black strings, which can only carry magnetic charges p i .
We will perform all our treatment within the electric charge configuration. Due to the mentioned BH/black string duality, this does not imply any loss of generality. Furthermore, we will study attractors within the Ansätze of asymptotical (Minkowski) flatness, staticity, spherical symmetry and extremality of the BH space-time metric (if no scalars are coupled, this is nothing but the so-called Tangherlini extremal d = 5 BH). The near-horizon geometry of extremal electric BHs and extremal magnetic black strings respectively is AdS 2 × S 3 and AdS 3 × S 2 .

Classes of Critical Points of V
From the general theory of Attractor Mechanism [3,4,5,6], the stabilization of scalar fields in proximity of the (unique) event horizon of a static, spherically symmetric and asymptotically flat extremal BH in N = 2, d = 5 Maxwell-Einstein supergravity is described by the critical points of the positive-definite effective potential function where the N = 2, d = 5 central charge function Z and its Riemann-covariant derivatives ("matter charges") have been defined as follows: The definitions (3.2) and (3.3) can be inverted, obtaining the fundamental identities of RSG (in electric formulation) [19]: The identities (3.4) relate the basis of "bare" (φ-independent) electric charges q i to the basis of "dressed" (central and matter) charges {Z, Z x }, which do depend on the scalars φ x , as yielded by definitions (3.2) and (3.3). By recalling definitions (3.2) and (3.3), one obtains that Therefore, by using Eq. (3.5) the criticality conditions (alias Attractor Eqs.) for the effective potential V can be easily computed to be [19]: A priori, the classes of critical points of V which are non-degenerate (i.e. with V | Vx=0 = 0) are three: This class is defined by the sufficient (but not necessary) criticality constraint

Non-BPS
This class is defined by the constraints which are critical provided the following algebraic constraint among Z and Z x 's hold: At least in symmetric RSG, this implies [19] V = 9Z 2 . (3.11)

Remark
It is here worth recalling the Bekenstein-Hawking entropy-area formula [27], implemented for critical points of V : The Attractor Mechanism [3,4,5,6] is known to hold only for the so-called "large" BHs, which, through Eq. (3.12), have a non-vanishing Bekenstein-Hawking entropy.
Therefore, attractors in strict sense are given by non-degenerate critical points of V . On the other hand, degenerate critical points of V , namely critical points such that V | ∂V =0 = 0 are trivial. Indeed, by virtue of the positive definiteness of V (inherited from the strictly positive definiteness of • a ij throughout all its domain of definition), it holds that (3.13) which is the trivial limit of the theory with all (electric) charges switched off. The same reasoning can be repeated in the magnetic case. Thus, only "large" BHs do exhibit a (classical) Attractor Mechanism, implemented through non-trivial (alias non-degenerate) critical points of the effective potential itself [6].

"New Attractor" Approach
Through the so-called "new attractor" approach [43], an equivalent form of the n V real criticality conditions (i.e. of the so-called Attractor Eqs.) for the various classes of critical points of V can be obtained by plugging the criticality conditions themselves into the n V + 1 real RSG identities 2 (3.4). By so doing, one respectively obtains: • BPS Attractor Eqs.: (3.14) While Eqs. (3.7) are n V real differential ones, the n V +1 real Eqs. (3.14) are purely algebraic.

Derivatives of T xyz
Now, in order to proceed further, it is convenient to compute the Riemann-covariant derivative of the invariant tensor T xyz , namely T xyz;w , a quantity which will be relevant in the subsequent treatment. By using the definition (2.9), one obtains Consequently, the condition (2.16) for the real special manifold M 5 to be a symmetric coset can be equivalently recast as follows (see e.g. page 14 of [19], and Eq. (3.2.1.9) of [20]): T r(yz T xw)s g rs = 1 2 g (yz g xw) . (3.17) One can then proceed further, and compute T xyz;w;q . Starting from Eq. (3.16) one obtains (within the metric postulate) T xyz;w;q = T (xyz;w);q = −2 √ 6T r(yz|;q T |xw)s g rs = −2 √ 6T r(yz;q T xw)s g rs = T (xyz;w;q) .
(3.18) Through Eq. (3.16), this result can be further elaborated to give: One can now introduce the following rank-5 completely symmetric tensor E xyzwq , which is the "RSG analogue" of the so-called E-tensor 3 of SKG: satisfying by definition the relation

Generic RSG
Let us now consider the value of the effective potential V at the various classes of its critical points. By recalling its very definition (3.1), Eqs. (3.7) and (3.10) yield the following results: Through Eq. (3.12), this yields to 3.3.2 Non-BPS and the "Dressed" Charges' Sum Rule Consequently at non-BPS Z = 0 critical points of V it generally holds that: where the real quantity has been introduced. This latter is the "RSG analogue" of the complex quantity ∆ introduced in SKG [44] (see also [45,39,21,40]). As ∆ enters the "dressed" charges' sum rule at non-BPS (Z = 0) critical points of V BH in SKG (see e.g. Eqs. (282)-(284) of [44]), so ∆ enters the "dressed" charges' sum rule (3.28) at non-BPS critical points of V in RSG, which further simplifies to (3.11) at least in symmetric RSG (having ∆ = 0 globally). Notice that, through Eq. (3.27) and definition (3.29), the (assumed) strictly positive definiteness of g xy (throughout all M 5 , and in particular at the considered class of critical points of V itself) yields Through Eq. (3.12), Eq. (3.28) yields Within a generic RSG, let us now consider the function In general, I 3 is a diffeomorphism-and symplectic-invariant function of the scalars φ x in M 5 , or equivalently a functional of the "dressed" charges {Z, Z x } in M 5 . Its derivative reads (recalling Eq. (3.16)) From the definition (3.29), it thus follows that The computation of I 3 and I 3,x (respectively given by Eqs. (3.32) and (3.33)) at the various classes of critical points of V (specified by Eqs. (3.7)-(3.10)) respectively yield to the following results.

BPS
Thus, by recalling Eqs. (3.23) and (3.24), it follows that Non-BPS Eq. (3.27) and definition (3.29) yield On the other hand, by recalling Eqs. (3.16) and (3.10), the term T xyz Z x Z y Z z can be further elaborated at non-BPS Z = 0 critical points of V as follows: Thus, definition (3.32) yields the following expression of I 3 at non-BPS Z = 0 critical points of V : Thus, by recalling Eqs. (3.28) and (3.31), it follows that

Symmetric RSG and "Large" Charge Orbits
Let us now consider the case in which 4 is a symmetric coset. (At least) in this case, d ijk is the unique G 5 -invariant rank-3 completely symmetric tensor, whereas T xyz is the unique H 5 -invariant rank-3 completely symmetric tensor.
"Magic" symmetric M 5 's are reported in Table 1 (see e.g. [32], and Refs. therein; see also [46] for a brief review and list of Refs.).
Besides these four isolated cases, there are two infinite sequences of other symmetric real special manifolds, namely the so-called Jordan symmetric sequence 44) and the non-Jordan symmetric sequence [41] M nJ,5,n ≡ SO (1, n) SO (n) , n = n V ∈ N, Table 1: Homogeneous symmetric real special vector multiplets' scalar manifolds M 5 of N = 2, d = 5 "magic" supergravity. M 5 's also are: 1) the non-BPS Z = 0 moduli spaces of N = 2, d = 4 special Kähler symmetric vector multiplets' scalar manifolds [23]; and 2) the "large" 1 2 -BPS charge orbits O BP S,large 's of N = 2, d = 5 Maxwell-Einstein supergravity itself [19]. The "large"non-BPS Z = 0 charge orbits O nBP S,large =M * 5 (see e.g. Table 5 of [48] and Refs. therein) and the related non-BPS Z = 0 moduli spaces M nBP S,large are reported, as well. The rank r of the orbit is defined as the minimal number of charges defining a representative solution. As observed in [23], for "magic" supergravities n V = dim R M 5 = 3q+2, whereas dim R M nBP S,large = 2q, and Spin (1 + q) ⊂ h 5 . See text for more details n V being the number of Abelian vector supermultiplets coupled to the N = 2, d = 5 supergravity one.
The sequence (3.45) is the only (sequence of) symmetric RSG which is not related to Jordan algebras of degree three. It is usually denoted by L (−1, n − 1) in the classification of homogeneous Riemannian d-spaces (see e.g. [32], and Refs. therein). It will not be further considered here, because it does not correspond to symmetric spaces in four dimensions. G 5 and H 5 can respectively be interpreted as the reduced structure group Str 0 and the automorphism group Aut of the corresponding Euclidean Jordan algebra of degree three (see e.g. [47] for a recent review, and Refs. therein): Furthermore, (at least 5 ) in symmetric RSG, due to Eqs. (2.16) and (3.33), it holds that In other words, I 3 is independent on all scalars φ x . Furthermore: where I 3 is the unique cubic invariant of the relevant electric (ir)repr. R Q of d = 5 U -duality G 5 , defined by (7.2). As mentioned above, d ijk is G 5 -invariant in all RSG, whereas d ijk is G 5 -invariant at least in symmetric RSG. In this framework, by virtue of the relations (7.27) or (7.31), the Bekenstein-Hawking entropyarea formula (3.12) can be completed as follows (recall Eq. (3.48)): Furthermore, in RSG based on symmetric cosets G 5 H 5 the representation space of the irrepr. of G 5 in which the (electric or magnetic) charges sit admit a stratification in disjoint orbits [15,19]. Such orbits are homogeneous, in some case symmetric, manifolds.
The charge orbits supporting non-degenerate (in the sense specified above; see the end of Subsect. 3.1) critical points of V are called "large" orbits, because they correspond to the previously introduced class of "large" BHs with non-vanishing Bekenstein-Hawking entropy-area (see Eq. (3.12)). On the other hand, charge orbits corresponding to "small" BHs (having vanishing Bekenstein-Hawking entropy-area) are correspondingly dubbed "small" orbits.
In the treatment of symmetric RSG performed in present Subsection, only "large" orbits, firstly found in [19], are considered.
In Sect. 4, through the properties of the function I 3 defined by Eq. (3.32), the stabilizers of all "small" charge orbits of symmetric RSG will be derived, by suitably solving G 5 -invariant (sets of) defining differential constraints, as well as by performing suitable group theoretical procedures.
We can now specialize the results obtained in Subsect. 3.3 and in Subsubsect. 3.3.3 to "magic" symmetric RSG. The detailed treatment of N = 2 Jordan symmetric sequence (3.44) will be given in Sect. 6. Actually, the "large" charge orbits of (3.44) have been already considered in [19] (see also [23] for the study of corresponding moduli spaces), but in Sect. 6 the treatment is further refined.

Eqs. (3.35) and (3.48) yield to
and thus: Such a "large" BH is supported by (electric) charges belonging to the "large" charge orbit (homogeneous symmetric manifold) [19] O BP S,large = The compactness of H 5 yields the absence of a moduli space related to 1 2 -BPS "large" attractor solutions, a fact that can be seen also from the expression of the Hessian matrix of V evaluated along the BPS criticality constraints (3.7) (see Eq. (3.72) below).
It is worth remarking that M 5 's also are the non-BPS Z = 0 moduli spaces of N = 2, d = 4 special Kähler symmetric vector multiplets' scalar manifolds [23].
Notice that in general As observed in [23], for "magic" supergravities (based on Euclidean Jordan algebras of degree three J A 3 over the division algebras A) it holds: Indeed, from its very definition, in this framework it globally holds that ∆ = 0, (3.56) and thus (recall Eq. (3.11)): Through Eq. (3.49), it thus follows that Such a "large" BH is supported by (electric) charges belonging to the "large" charge orbit (homogeneous symmetric manifold) where H 5 is the unique non-compact, real form of H 5 = mcs (G 5 ) which admits a maximal symmetric embedding into G 5 : The homogeneous symmetric pseudo-Riemannian manifold M * 5 is the " * -version" of M 5 , obtained through timelike d = 6 → 5 reduction from the corresponding anomaly-free uplifted N = (1, 0), d = 6 chiral theory (see e.g. As observed in [23], for "magic" supergravities it holds (see e.g. also Table 8 of [46], and Refs. therein): dim R M nBP S,large = 2q; where Spin (1 + q) is the spin group in 1 + q dimensions. Notice that 2q is the number of d = 6 (scalarless) vector multiplets needed for an anomaly-free uplift of the considered N = 2, d = 5 "magic" Maxwell-Einstein supergravity to the corresponding N = (1, 0) chiral quarter-minimal "magic" supergravity in d = 6 (see e.g. Sect. 5 of [20], and Refs. therein). Thus, by recalling (3.54), the number of "non-flat" scalar degrees of freedom along O nBP S,large is (3.64) The "large" non-BPS Z = 0 charge orbits O nBP S,large = M * 5 , and the related non-BPS Z = 0 moduli spaces M nBP S,large for "magic" models are reported in Table 1. Furthermore, it should be recalled that the Jordan symmetric sequence (3.44) is related to the reducible rank-3 Euclidean Jordan algebra R⊕Γ 1,n , where Γ 1,n is the rank-2 Jordan algebra with a quadratic form of Lorentzian signature (1, n), i.e. the Clifford algebra of O (n, 1) [49].

Hessian Matrix of V
From its very definition (3.1), the first derivative of V reads (recall Eq. (3.6)) By further differentiating, the global expression of the real Hessian n V ×n V matrix of V in a generic RSG can be computed as follows: Then, further elaboration of Eq. (3.66) is possible for Z = 0. Indeed, in such a case Eq. (3.67) implies that (recall Eq. (3.16)) Notice that the symmetry properties I 3;x;y = I 3;(x;y) and T xzw;y Z z Z w = T (xzw;y) Z z Z w are not manifest respectively from Eqs. (3.67) and (3.68), due to the presence of E xyzwr , T xyz;w , and I 3;x;y itself. By plugging Eq. (3.68) back into Eq. (3.66), the following result is achieved: This is the global expression of the real Hessian n V × n V matrix of V (at least) in symmetric RSG, and indeed it matches the result given by Eq. (5-1) of [19] (see also [20]). Thus, Eqs. (3.66) and (3.70) yield to the following result:

Evaluation at Critical Points of V
We will now proceed to evaluate the Hessian matrix of V given by Eq. BPS The necessary and sufficient BPS criticality constraints (3.7) plugged into Eq. (3.66) yield Eq. (3.72) holds for a generic RSG, and it matches the result given by Eq. (5-2) of [19]. For a strictly positive definite g xy (as it is usually assumed), it implies that the Hessian matrix of V at its BPS critical points has all strictly positive eigenvalues. As mentioned above, the lack of Hessian massless modes at 1 2 -BPS critical points of V determines the absence of a moduli space in BPS attractor solutions, which thus have all scalar fields φ x stabilized at the (unique) event horizon of the considered (electric) d = 5 extremal BH.
Non-BPS It is here worth noticing that Eq. (3.10) yields to By recalling the "dressed" charges' sum rule given by Eq. (3.27) and definition (3.29), Eq. (3.73) implies 32 3 On the other hand, by using Eq. (3.16), one can compute also that By dividing by Z x Z x = 0, one then obtains the "dressed" charges' sum rule given by Eq. (3.27). However, one can also interpret Eq. (3.75) as a quadratic Eq. in the unknown Z x Z x , obtaining the result When I 3;x = 0 (i.e. -at least -for symmetric RSG) Eq. (3.76) consistently yields [19]: (3.77)

"Small" Charge Orbits and Moduli Spaces in Symmetric "Magic" RSG
In the treatment of symmetric RSG performed in Subsect. 3.4, only "large" charge orbits, supporting solutions to the corresponding Attractor Eqs. (and firstly found in [19]; see also [20]), have been considered.
In the present Section, by exploiting the properties of the functional I 3 introduced in Subsubsect. 3.3.3, all "small" charge orbits of "magic" symmetric RSG will be explicitly determined through the resolution of G 5 -invariant defining (differential) constraints both in "bare" and "dressed" charges bases, as well as through group theoretical techniques. By definition, I 3 (= I 3 in symmetric RSG, as discussed in Subsect. 3.4, see Eq. (3.48)) vanishes for all "small" charge orbits. Consequently, such orbits do not support solutions to the Attractor Eqs. (alias criticality conditions of the effective potential V ; see Eqs. (3.7)-(3.10), or Eqs. (3.14)-(3.15) in the so-called "new attractor" approach). In other words, the (classical) Attractor Mechanism does not hold for "small" charge orbits, which indeed do support BH states which are intrinsically quantum, in the sense that the effective description through Einstein supergravity fail for them.
Besides the condition of vanishing I 3 , further conditions, formulated in terms of derivatives of I 3 in some charge basis, may be needed to fully characterize the class of "small" orbits under consideration. It is here worth pointing out that the (sets of) G 5 -invariant constraints which define "small" charge orbits in homogeneous symmetric real special manifolds G 5 H 5 are characterizing equations for charges (in both "bare" and "dressed" bases), but they actually are identities in all scalar fields φ x , and thus they hold globally in G 5 H 5 . This is to be contrasted with "large" charge orbits, which are defined through the Attractor Eqs. themselves, which are at the same time characterizing equations for charges (in both "bare" and "dressed" bases) and stabilization equations for the scalars φ x at the event horizon of the extremal BH.
As it is well known [23], at non-BPS Z = 0 critical points of V some scalars are actually unstabilized at the (unique) event horizon of the corresponding "large" extremal BH solutions. Such unstabilized φ x 's span the moduli space M nBP S,large (given by Eq. (3.62)), associated with an hidden compact symmetry of the non-BPS Z = 0 Attractor Eqs. themselves, which can be traced back to the non-compactness of the stabilizer of the non-BPS Z = 0 "large" charge orbit O nBP S,large (see Eq. (3.59), to be contrasted with Eq. (3.52)).
The "small" charge orbits are homogeneous manifolds of the form: where denotes semi-direct group product throughout, and T is the non-semi-simple part of the stabilizer of O small , which in all symmetric RSG (with some extra features characterizing the symmetric Jordan sequence, see Sect. 6) can be identified with an Abelian translational subgroup of G 5 itself.
One can associate a moduli space also to "small" charge orbits, by observing that the noncompactness of S max T yields the existence of a corresponding moduli space defined as 6 Note that, differently from "large" orbits, for "small" orbits there exists a moduli space M small = T also when S max is compact. As found in [50,51] for "large" charge orbits of N = 2, d = 4 stu model, and recently proved in a model-independent way in [52], the moduli spaces of charge orbits are defined all along the scalar flows, and thus they can be interpreted as moduli spaces of unstabilized scalars at the event horizon (if any) of the extremal BH, as well as moduli spaces of the ADM mass of the extremal BH at spatial infinity. In the "small" case, the interpretation at the event horizon breaks down, simply because such an horizon does not exist at all (at least in Einsteinian supergravity approximation). In general, the number of "non-flat" scalar degrees of freedom supported by a ("large" or "small") charge orbit O with associated moduli space M is defined as follows: As an example, let us briefly consider the maximal N = 8, d = 5 supergravity, whose "large" and "small" charge orbits have been classified in [15]. The scalar manifold of the theory is 1. The unique "large" charge orbit is 1 8 -BPS: Thus, the number of "non-flat" directions along O 1 8 −BP S reads (4.7) Since the charge orbit is "large", 1 8 −BP S also expresses the actual number of scalar degrees of freedom which are stabilized in terms of the electric (magnetic) charges in the near-horizon geometry of the extremal black hole (black string) under consideration.
with corresponding moduli space Thus, the number of "non-flat" directions along O 1 4 −BP S reads (4.10) with corresponding moduli space where M (2,2),d=6 is the scalar manifold of maximal (non-chiral) supergravity in d = 6. Thus, the number of "non-flat" directions along O 1 2 −BP S reads (4.13) As we will point out more than once in the treatment below, result (4.13) expresses the pretty general fact that the unique "non-flat" direction along maximally supersymmetric (namely,  Table 2: Q q and A q for the various N = 2, d = 5 "magic" supergravities (based on J A 3 , In the treatment of Subsect. 4.1, the G 5 -invariant constraints defining all classes of "small" charge orbits in all symmetric RSG will be derived. Then they will be solved both in "bare" and "dressed" charge bases in Subsect. A. Furthermore, in App. B the origin of "small" charge orbits (and in particular of T ) will be elucidated through group theoretical procedures (namely, Inönü-Wigner contractions [57,58] and SO (1, 1)-three grading).
While the treatment of Subsect. 4.1 holds for all symmetric RSG, the treatments given in Apps. A and B strictly fit only the isolated cases of symmetric RSG provided by the so-called "magic" symmetric RSG's [28,29,30,31]. The main results of Apps. A and B are reported in Tables 3 and 4 (the symmetric Jordan sequence (3.44) is considered in Sect. 6). In the "magic" octonionic case J O 3 (q = 8), the results of [15] are matched. Below we summarize the main results of Apps. A and B.
• The "small" lightlike BPS charge orbit (dim R = 3q + 2) thus with T lightlike,BP S = R (spin(q+1),spin(Qq)) . (4.16) Q q and A q , a further factor group in S max , are given by Table 2. Furthermore, we define with Spin (q + 1) and Spin (Q q ) respectively denoting the spinor irreprs. in q + 1 and Q q dimensions. It is worth remarking that A q is independent on the space-time dimension (d = 3, 4, 5, 6) in which the quarter-minimal symmetric "magic" (Maxwell-Einstein) supergravity (classified by q = 8, 4, 2, 1) is considered. It also holds that where the groups G cent and G paint are usually introduced in the treatment of supergravity billiards and timelike reductions (for recent treatment and set of related Refs., see e.g. [48]; see also Table 5 therein, also for subtleties concerning the case q = 8 in d = 5, 6). The moduli space corresponding to (4.14) is purely translational: Thus, by recalling (3.54), the number of scalar degrees of freedom on which the ADM mass • The "small" lightlike non-BPS charge orbit (dim R = 3q + 2) The related moduli space reads (dim R = 3q) where M nJ,5,q is the q-th element of the generic non-Jordan symmetric sequence (3.45). Thus, by recalling (3.54), the number of scalar degrees of freedom on which the ADM mass depends where is the U -duality group of the corresponding (1, 0), d = 6 chiral supergravity theory. Thus: The related moduli space reads (dim R = 3q + 1) Thus, by recalling (3.54), the number of scalar degrees of freedom on which the ADM mass depends along O critical,BP S is The unique scalar degree of freedom on which the ADM mass depends can be interpreted as the Kaluza-Klein radius in the d = 5 → d = 4 reduction. Furthermore, it is worth observing that: is the manifold of tensor multiplets' scalars in the corresponding (1, 0), d = 6 theory (see e.g. Sect. 5 of [20] for a recent treatment).
It should also be noticed that O nBP S,large (given by Eq. (3.59)) and O critical,BP S (given by Eq. (4.29)) share the same compact symmetry, or equivalently that M nBP S,large (given by Eq. (3.62)) and M critical,BP S (given by Eq. (4.32)) share the same stabilizer group (apart from an A q commuting factor), but they do not coincide. This is due to the fact that H 5 and G 6 × A q share the same mcs, namely In the case A = R (q = 1), the following further results holds (see also Tables 3 and 4): (4.36) Notice that J R 3 is the unique case, among J A 3 in d = 5, in which M nBP S,large and M critical,BP S not only share the same stabilizer, but they actually do coincide (up to R 2 ). Moreover, M nBP S,large,J R 3 also coincides with M lightlike,nBP S,J C 3 (up to R (2,2) ), because the respective charge orbits O nBP S,large,J R 3 and O lightlike,nBP S,J C 3 share the same semi-simple, namely non-translational, part of the stabilizer (apart from a commuting A 2 = SO (2) factor), i.e. SO (2, 1).
The Jordan symmetric infinite sequence [28,29,30,31,32,35,36] given by Eq. (3.44) needs some extra care (also at the level of "large" charge orbits), because of the factorization of G 5 . The "large" and "small" charge orbits for such a sequence will be treated in Sect. 6. This treatment refines and complete the ones given e.g. in [19,23,20]).

G 5 -invariant Defining Constraints
As mentioned above, "small" charge orbits in all symmetric RSG are all characterized by the constraint (recall Eq. (3.48)): Table 3: "Small" lightlike charge orbits O lightlike,BP S and O lightlike,nBP S (with associated moduli spaces) in symmetric "magic" RSG Table 4: "Small" critical charge orbit O critical,BP S (with associated moduli space M critical,BP S ) in symmetric "magic" RSG where I 3 = I 3 is the unique cubic scalar invariant of the relevant electric representation R q of the d = 5 U -duality group G 5 (in which the electric charges q i 's sit). By recalling definitions (3.32) and (7.2), the "smallness" condition (4.37) can be recast as follows: in the "dressed" and "bare" charges basis, respectively. It is here worth noticing that Eq. (4.38) can be recast as a cubic algebraic equation: with polynomial discriminant Thus, for D > 0 one gets one real and two complex conjugate (unacceptable) roots, whereas for D < 0 all roots are real and unequal. In the particular case all roots are real, and at least two equal. Let us proceed further, by differentiating the functional I 3 with respect to the "dressed" charges as well as function I 3 with respect to the "bare" charges {q i }. One respectively obtains: (4.44) where it should be recalled once again that here we are considering symmetric real special manifolds (4.46) . (4.47) By further differentiating, one then obtains: . (4.49) Starting from the fourth order of differentiation, all derivatives vanish. This is no surprise, because I 3 is a functional polynomial homogeneous of degree three in "dressed" charges Z, as well as (equivalently) I 3 is a polynomial homogeneous of degree three in "bare" charges q i 's.
At this point, it is possible to classify the various "small" charge orbits through G 5 -invariant conditions involving I 3 and its non-vanishing functional derivatives with respect to Z, or equivalently through G 5 -invariant conditions involving I 3 and its non-vanishing derivatives with respect to q i 's.

"Small" Lightlike Orbits
The "small" lightlike charge orbits are defined by the constraints (recall Eqs. (4.38) and (4.39)): 50) or equivalently: (4.51) The sets of constraints (4.50) and (4.51) are both G 5 -invariant, but their manifest invariance is different. Indeed, the "dressed" charge basis Z is covariant with respect to H 5 , and thus the set of constraints (4.50) exhibits a manifest H 5 -invariance. Instead, the "bare" charge basis {q i } is G 5 -covariant, and thus the set of constraints (4.50) is manifestly G 5 -invariant. In the "dressed" charge basis, it is immediate to realize that two classes of "small" lightlike charge orbits exist, namely: • "small" lightlike charge orbit for which the constraints (4.50) are solved with Z = 0: and / or T x yz Z y Z z = 0 (at least for some x). (4.52) Notice that the constraint Z x Z x = 0 is automatically satisfied, because: 1) g xy is assumed to be strictly positive definite, and 2) Z x = 0 at least for some x (otherwise, since Z = 0, one would obtain the trivial limit in which all charges vanish).

"Small" Critical Orbit
The "small" critical charge orbit is defined by the constraints (recall Eqs. (4.38) and (4.39)): 54) or equivalently: is actually manifestly G 5 -invariant. Once again, in the "dressed" charges basis it is immediate to realize that only one class of "small" critical charge orbits exists, namely: • "small" critical charge orbit for which the constraints (4.54) are solved with Z = 0: Notice that, for the same reason the constraint ∂ I 3

∂Z Z=0
= 0 is automatically satisfied for the "small" lightlike charge orbit whose a representative in the "dressed" charges basis is given by Eq. (4.52), a "small" critical charge orbit with representative having Z = 0 cannot exist. Indeed, such an orbit should have Z = 0 and Z x Z x = 0. Due to the assumed strictly positive definiteness of g xy , this would be possible only in the trivial limit of the theory in which all charges do vanish. This can be formally stated as follows: The rank-3 Euclidean Jordan algebra J H 3 (q = 4) is related to two different theories, namely an N = 2 theory coupled to 14 Abelian vector multiplets and the N = 6 "pure" theory. These two theories share the same bosonic sector [15,19,59], but their fermionic sectors, exploiting the supersymmetric completion of the bosonic one, are different.
Thus, it also follows that the supersymmetry-preserving features of the "large" and "small" charge orbits of the relevant irrepr. 15 of G 5 = SU * (6) are different. The N -dependent supersymetry properties of the various orbits are given in Table 5 (notice they are consistent with the results of [53]). In the "large" (attractor) cases, these match the results of [20].

N = 2, d = 5 Jordan Symmetric Sequence
The Jordan symmetric sequence of N = 2, d = 5 supergravity coupled to n V = n + 1 vector multiplets reads (dim R = n + 1, rank= 2, n ∈ N ∪ {0}) This sequence is associated to the rank-3 Euclidean reducible Jordan algebra R ⊕ Γ 1,n . In the following treatment, we will determine the "large" and "small" orbits of the irrepr. (1, 1 + n) of the U -duality group SO (1, 1) × SO (1, n). For brevity's sake, we will do this only through an analysis in the "bare" charges' basis. Without any loss in generality, one can choose to treat only d = 5 extremal (electric) BHs. Indeed, due to the symmetricity of the reducible coset (6.1), the treatment of d = 5 extremal (magnetic) black strings is essentially analogous.
Two disconnected geometric structures emerge in the treatment, namely: Table 5: N -dependent supersymmetry-preserving features of "large" and "small" charge orbits of the irrepr. 15 of the d = 5 U -duality group SU * (6), related to J H 3 . This corresponds to two "twin" theories, sharing the same bosonic sector: an N = 2 Maxwell-Einstein theory and the N = 6 "pure" theory. The subscript "H" stands for "(evaluated at the) horizon" • Timelike two-sheet hyperboloid T n , with the two disconnected sheets T ± n respectively related to q 0 ≷ 0: • Forward/backward light-cone Λ n of (n + 1)-dimensional Minkowski space with metric η IJ defined by (6.5), with two (forward Λ + n and backward Λ − n ) cone branches, respectively related to q 0 ≷ 0: 0 here denoting the origin of Λ n itself.
Due to such structures, as well as to the lower (N = 2) supersymmetry, the case study of "large" and "small" charge orbits in N = 2, d = 5 Jordan symmetric sequence exhibits some subtleties absent in the N = 4, d = 5 theory analyzed in Sect. 7.
In the "bare" charges' basis, the electric cubic invariant of the (1, 1 + n) of SO (1, 1) × SO (1, n) reads as follows (I = 0, i, where i = 1, ..., n, throughout; "0" pertains to the d = 5 graviphoton field, which through the dimensional reduction d = 5 → d = 4 becomes the Maxwell vector field of the axio-dilatonic vector multiplet): where q H is the electric charge of the dilatonic vector multiplet: it is an SO (1, n)-singlet, with SO (1, 1)-weight +2. On the other hand, the SO (1, n)-vector q I has SO (1, 1)-weight −1, such that I 3,el defined by (6.4) is SO (1, 1) × SO (1, n)-invariant. Notice that the action of the U -duality group does not mix q H and q I , and this originates more charge orbits with respect to the irreducible cases. Moreover, η IJ = η IJ is the Lorentzian metric of SO (1, n): In N = 2, d = 5 Jordan symmetric sequence, as well as in N = 4, d = 5 theory, the reducibility of the associated rank-3 Jordan algebra gives rise to many subtleties and differences with respect to the theories associated to irreducible Euclidean rank-3 Jordan algebras. In the N = 2 case under consideration, the major difference consists in a higher number of "large" and "small" orbits with respect to the "magic" supergravities.
and thus the hyperboloid (6.2) and light-cone (6.3) structures gets respectively factorized as follows ("+", "−" and "0" respectively denote strictly positive, strictly negative and vanishing values): (6.10) This is invariant under triality permutation symmetry of q H , q + and q − , and it is consistent with the analysis of [34].
• non-BPS (3-charge) orbit, with Z = 0 at the horizon, defined as follows: q 0 > 0. The supersymmetry properties of O BP S,large and O nBP S,large,I can be understood by noticing that the flip of the sign of q H amounts, in the dressed charges' basis, to the exchange Z ←→ ∂ s Z, where s is the real dilaton scalar field, parametrizing SO (1, 1) of (6.1).
It is worth pointing out both the N = 2 orbits O BP S,large and O nBP S,large,I (respectively given by (6.7) and (6.14)) uplift to the same N = 4 orbit O 1 4 −BP S,large,N =4,d=5 given by Eq. (7.4). As mentioned, this is due to the fact that in N = 4, d = 5 q H > 0 ←→ q H < 0 amounts to exchanging the two gravitinos in the gravity multiplet, i.e. the two (opposite) skew-eigenvalues of the skew-traceless central charge matrixZ AB (A, B = 1, ..., 4).

"Small" Orbits
Let us now consider the "small" orbits, and compute the criticality and double-criticality conditions on I 3,el defined by (6.4): (6.23) is shorthand for the vector of electric charges. As expected from the fact that I 3,el is homogeneous of degree three, (6.24) implies that the unique doubly-critical orbit is the trivial one with all charges vanishing, because The "small" orbits of the (1, 1 + n) of the U -duality group SO (1, 1) × SO (1, n) list as follows: 1. BPS lightlike (I 3,el = 0, q 0 < 0.  Analogously to what holds for symmetric "magic" RSG (noted below Eq. (4.33)), the unique scalar degree of freedom on which the ADM mass depends can be interpreted as the Kaluza-Klein radius in the d = 5 → d = 4 reduction. Furthermore, as in the corresponding N = 4, d = 5 "small" orbit (given by Eq. (7.34)), the sign of q H does not matter here. Orbit (6.55) is originated by the d = 6 → d = 5 reduction of (1, 0) theory with all charges switched off. Indeed, q H is the electric charge of the Kaluza-Klein vector in the reduction d = 6 → d = 5.
In particular, for n = 0, this orbit reads Thus, the stratification structure of the (1, 1 + n)-repr. space of the d = 5 U -duality group SO (1, 1) × SO (1, n) can be given through the following two chains of relations, proceeding (left to right) from 1-charge orbits to 2-charge and then 3-charge orbits: Thus, summarizing, N = 2, d = 5 Jordan symmetric sequence admits six "small" charge orbits describing the flux configurations supporting static, spherically symmetric, asymptotically flat "small" BHs: four 1 2 -BPS and two non-BPS. Furthermore, the "large" orbits are three, namely one 1 2 -BPS and two non-BPS (with Z = 0 at the horizon).

N = 4, d = 5 Supergravity
The scalar manifold of N = 4, d = 5 supergravity coupled to n V = n ∈ N ∪ {0} matter (vector) multiplets reads (dim R = 1 + 5n, rank= 1 + min (5, n)) This theory is associated to the rank-3 Euclidean reducible Jordan algebra R⊕Γ 5,n . In the following treatment, we will determine the "large" and "small" orbits of the irrepr. (1, 5 + n) of the U -duality group SO (1, 1) × SO (5, n). For brevity's sake, we will do this only through an analysis in the "bare" charges' basis. Without any loss in generality, one can choose to treat only d = 5 extremal (electric) BHs. Indeed, due to the symmetricity of the reducible coset (7.1), the treatment of d = 5 extremal (magnetic) black strings is essentially analogous.

"Large" Orbits
• 1 4 -BPS (3-charge) orbit, defined by a timelike q I vector, with q H of any sign: q H ∈ R 0 , q I q J η IJ > 0. In N > 2-extended supergravity theories, in general 1 N -BPS attractors have a related moduli space [23]. It corresponds to the hypermultiplets' scalar manifold in the supersymmetry reduction N > 2 −→ N = 2 of the theory under consideration. In this case, it is amusing to observe that M 1 4 −BP S,large given by (7.5) is the c-map of the vector multiplets' scalar manifold of the N = 2, d = 4 Jordan symmetric sequence: .

"Small" Orbits
The conditions on I 3,el defined by (7.2) are formally the same as the ones holding in N = 2, d = 5 Jordan symmetric sequence, and given by Eqs. (6.23) and (6.24). Thus, analogously to the case of N = 2, d = 5 Jordan symmetric sequence, and as expected from the fact that I 3,el is homogeneous of degree three, (6.24) implies that the unique doubly-critical orbit is the trivial one with all charges vanishing (namely, 0-charge orbit; recall Eq. (6.26)). The "small" orbits of the (1, 5 + n) of the U -duality group SO (1, 1) × SO (5, n) list as follows: 1. Lightlike (I 3,el = 0,   Thus, the number of "non-flat" scalar degrees of freedom along O nBP S,small is independent on n 1: Thus, the number of "non-flat" scalar degrees of freedom along O 1 2 −BP S,small,II is independent on n 1:  Thus, the number of "non-flat" scalar degrees of freedom along O 1 2 −BP S,small,II is independent on n 1: given by Eq. (7.30). Indeed, the n = 0 limit of (7.29) is given by (7.33) itself. Furthermore, analogously to what holds for symmetric "magic" RSG (noted below Eq. (4.33)) and for N = 2, d = 5 Jordan symmetric sequence treated in Sect. 6, the unique scalar degree of freedom on which the ADM mass depends can be interpreted as the Kaluza-Klein radius in the d = 5 → d = 4 reduction. Orbit (7.34) is originated by the d = 6 → d = 5 reduction of (2, 0) theory with all charges switched off. Indeed, q H is the electric charge of the Kaluza-Klein vector in the reduction d = 6 → d = 5. Notice that in the pure theory (i.e. n = 0) M 1 2 −BP S,small,III vanishes, and thus: Thus, the stratification structure of the (1, 5 + n)-repr. space of the d = 5 U -duality group SO (1, 1) × SO (5, n) can be given through the two chains of relations, proceeding (left to right) from 1-charge orbits to 2-charge and then 3-charge orbits: For pure N = 4, d = 5 supergravity, such a stratification structure simplifies as follows: Thus, summarizing, N = 4, d = 5 supergravity theory admits five "small" charge orbits describing the flux configurations supporting static, spherically symmetric, asymptotically flat "small" BHs: one 1 4 -BPS, three 1 2 -BPS and one non-BPS. The "large" orbits are two, namely one 1 4 -BPS and one non-BPS (withZ AB = 0 at the horizon).
The relations among the charge orbits of N = 4, d = 5 supergravity and the charge orbits of N = 2, d = 5 Jordan symmetric sequence can be determined through the supersymmetry reduction d = 5 : yielding to the results summarized in Table 6. Finally, it is worth summarizing the results obtained about the number of "non-flat" scalar degrees of freedom, within the symmetric RSG studied in previous Sections. For the "magic" supergravities, it holds whereas for N = 4 supergravity and N = 2 Jordan symmetric sequence the results are reported in Table 6. As pointed out above, in the symmetric RSG's under consideration the unique scalar degree of freedom on which the ADM mass depends along the 1-charge 1 2 -BPS (maximally symmetric) charge orbits can be interpreted as the Kaluza-Klein radius in the d = 5 → d = 4 reduction.   Table 6: "Large" (rank= 3) and "small " (rank= 1 and 2) charge orbits of the repr.
(1, 5 + n) and (1, 1 + n) of the d = 5 U -duality groups SO (1, 1) × SO (5, n) and SO (1, 1) × SO (1, n) of N = 4 supergravity (based on R⊕Γ 5,n ) and N = 2 Jordan symmetric sequence (based on R ⊕ Γ 1,n ), respectively. The rank r of the orbit is defined as the minimal number of charges defining a representative solution. " * " denotes the fact the orbits are related through a flip of the sign of q H . The disconnected timelike hyperboloid T n and lightcone Λ n structures are defined by (6.2) and (6.3), respectively. , defined in (4.3), denotes the number of "non-flat" scalar degrees of freedom supported by the charge orbit

A Resolution of G 5 -invariant Constraints
In this Appendix, we explicitly solve the G 5 -invariant defining constraints of "small" charge orbits in "magic" symmetric RSG, both in the "bare" (Sub-App. A.1) and "dressed" (Sub-App. A.2) charges bases.

A.1 "Bare" Charges Basis
Let us start by noticing that for each of the four "magic" symmetric RSG's a unique maximal symmetric embedding into G 5 exists containing a factor SO (1, 1). It reads (recall Eq. (4.30)) [54] where the group A q has been defined in Table 2. Notice that, in the cases q = 4 and 2, G 6 ×SO (1, 1) is not embedded maximally (also considering non-symmetric embeddings [55]) into G 5 itself. When removing A q in the cases q = 4 and 2 (and thus losing the maximality), the embedding (A.1) has a nice interpretation in terms of truncation of the "magic" supergravity to theories belonging to the Jordan symmetric sequence (3.44) [19]: where it should be recalled that (q = 8, 4, 2, 1; see e.g. [47]) In order to solve the "small" lightlike G 5 -invariant defining constraints (4.51) in "bare" charges in a way consistent with an orbit representative having Z = 0, let us further embed the mcs of the group in the right-hand side of Eq. (A.1), thus obtaining Thus, under the "branching" (A.4) the irrepr. R Q of G 5 in which the electric charges q i 's sit decomposes as follows: This in turn entails the "branching" In the first line of (A.5) subscripts denote the weight with respect to SO (1, 1), whereas in the second line they just discriminate between the two singlets of SO (q + 1) × A q . Also recall that, as given in Table 2, A q and Q q are absent for q = 8 and q = 1. Therefore, with respect to SO (q + 1) × A q , one obtains: • two singlets (note that (1, 1) I is a singlet of SO (q + 1, 1) × A q , as well); • one vector (q + 1, 1); • a (double-)spinor (Spin (q + 1) , Spin (Q q )).
The representation decomposition (A.5) yields that d ijk , the rank-3 completely symmetric G 5invariant tensor (namely, the unique singlet in the tensor product (R Q ) 3 ) decomposes in a such way that (1, 1) II and (q + 1, 1) have the same couplings inside (R Q ) 3 .
Details concerning the various "magic" symmetric RSG's are given further below. The position which solves (with maximal -compact -symmetry SO (q + 1) × A q ) the "small" lightlike G 5 -invariant defining constraints (4.51) in "bare" charges (and in a way consistent with an orbit representative having Z = 0) reads as follows: Since SO (q + 1)×A q is the unique group maximally (and symmetrically) embedded into G 6 ×A q × SO (1, 1) which has SO (q + 1) × A q as (in this case improper) mcs, it follows that SO (q + 1) × A q is also the maximal semi-simple symmetry of O lightlike,BP S , which is thus given by Eq. (4.14).
The origin of the non-semi-simple Abelian (namely, translational) factor R (spin(q+1),spin(Qq)) in the stabilizer of O lightlike,BP S will be explained through the procedure of suitableİnönü-Wigner contraction performed in Sub-App. B.1.

A.1.2 O critical,BP S
Eq. (A.4) and subsequent ones are also relevant for the resolution of the "small" critical G 5invariant defining constraints (4.55) in "bare" charges in a way consistent with an orbit representative having Z = 0 (which is the unique possible case; see treatment above). In this case, the position which solves (with maximal -non-compact -symmetry G 6 × A q ) the constraints (4.55) in "bare" charges reads as follows: At least for the relevant values q = 8, 4, 2, 1 it holds that spin (q + 2) = spin (q + 1) (recall definition (4.17)). Therefore, since q (1,1) II = 0; q (q+1,1) = 0; ⇔ q (q+2,1) = 0, (A. 9) it follows that the position (A.8) exhibits maximal -non-compact -symmetry G 6 × A q , which then is the maximal semi-simple symmetry of O critical,BP S , which is thus given by Eq. (4.29). The origin of R (spin(q+2),spin(Qq)) in the stabilizer of O critical,BP S will be explained through the procedure of suitable SO (1, 1)-(three-)grading performed in Sub-App. B.2.
The representation decomposition (A.11) yields that d ijk decomposes in such a way that the manifest "triality" exhibited by the "branching" of R Q is removed, and the two (double-)spinors are put on a different footing with respect to the vector. As a consequence: • (1, 1) II , (1, 1) III and (q, 1); • Spin (q) , Spin (Q q ) and Spin (q) , Spin (Q q ) separately have the same couplings inside (R Q ) 3 . The position which solves (with maximal -compact -symmetry SO (q) × A q ) the "small" lightlike G 5 -invariant defining constraints (4.51) in "bare" charges (and in a way consistent with an orbit representative having Z = 0) reads as follows:    q (q,1) = 0; q (Spin (q),Spin(Qq)) = 0; q (Spin (q),Spin(Qq)) = 0, (A.14) with the three singlets q (1,1) I , q (1,1) II and q (1,1) III constrained by Notice that in (A.14) the charges related to the vector and to the two (double-)spinors are on equal footing, thus exhibiting a "triality symmetry", as already mentioned above. Notice that SO (q, 1)×A q is the unique group which is maximally (if one consider also the factor SO (1, 1)) and symmetrically embedded into G 6 × A q × SO (1, 1), and also which has SO (q) × A q as mcs. Therefore, it follows that SO (q, 1) × A q is also the maximal semi-simple symmetry of O lightlike,nBP S , which is thus given by Eq. (4.24).
As mentioned above, the origin of R (spin(q+1),spin(Qq)) in the stabilizer of O lightlike,nBP S will be explained through the procedure of suitableİnönü-Wigner contraction performed in Sub-App. B.1.

A.1.4 Details
We now explicit some details of the treatment of symmetric "magic" RSG.
where the the normalizations and conventions of Table 58 of [55] have been adopted.

A.2 "Dressed" Charges Basis
Concerning the resolution of the G 5 -invariant (sets of) constraints in the basis of "dressed" charges, one should notice that for each of the four "magic" symmetric RSG's a unique non-compact, real form H 5 of the compact group H 5 ≡ mcs (G 5 ) exists with maximal symmetric embedding into G 5 (see e.g. [54]; also recall Subsect. 3.4 and Table 1): In order to solve the "small" lightlike G 5 -invariant defining constraints (4.51) in "dressed" charges in a way consistent with an orbit representative with Z = 0, let us further embed thus obtaining where the brackets denote the auxiliary nature of the embedding. Thus, under the "branching" (A.33) R Q decomposes as follows: where R is an irrepr. of throughout. Therefore, with respect to SO (q + 1) × A q , one obtains: • two singlets; • one vector (q + 1, 1); • one (double-)spinor (Spin (q + 1) , Spin (Q q )).
As mentioned in the resolution in the basis of "bare" (electric) charges q i 's, the origin of R (spin(q+1),spin(Qq)) in the stabilizer of O lightlike,BP S will be explained through the procedure of suitableİnönü-Wigner contraction performed in Sub-App. B.1.

A.2.2 O lightlike,nBP S
In order to solve the "small" lightlike G 5 -invariant defining constraints (4.51) in "dressed" charges in a way consistent with an orbit representative having Z = 0, the embedding (A.31) has to be further elaborated as follows: is the unique non-compact form of h 5 (defined by (A.32)) to be embedded maximally and symmetrically into H 5 (see e.g. [54]).
The position which solves (with maximal -compact -symmetry SO (q) × A q ) the "small" lightlike G 5 -invariant defining constraints (4.50) in "dressed" charges (and in a way consistent with an orbit representative having Z = 0) reads as follows: Besides SO (q + 1)×A q , the only other group which is maximally (and symmetrically) embedded into H 5 and which has SO (q) × A q as (m) cs, is SO (q, 1) × A q . Therefore, SO (q, 1) × A q is also the maximal semi-simple symmetry of O lightlike,BP S , which is thus given by Eq. (4.24).
The origin of R (spin(q+1),spin(Qq)) in the stabilizer of O lightlike,BP S will be explained through the procedure of suitableİnönü-Wigner contraction performed in Sub-App. B.1.
The results obtained in Sub-Apps. B.1 and B.2 (and reported in Tables 3 and 4) allows one to conclude that all "small" charge orbits of symmetric "magic" RSG (classified by q = 8, 4, 2, 1) share the same non-semi-simple, Abelian (namely, translational) part of the stabilizer. Therefore: