On Invariant Structures of Black Hole Charges

We study"minimal degree"complete bases of duality- and"horizontal"- invariant homogeneous polynomials in the flux representation of two-centered black hole solutions in two classes of D=4 Einstein supergravity models with symmetric vector multiplets' scalar manifolds. Both classes exhibit an SL(2,R)"horizontal"symmetry. The first class encompasses N=2 and N=4 matter-coupled theories, with semi-simple U-duality given by SL(2,R) x SO(m,n); the analysis is carried out in the so-called Calabi-Vesentini symplectic frame (exhibiting maximal manifest covariance) and until order six in the fluxes included. The second class, exhibiting a non-trivial"horizontal"stabilizer SO(2), includes N=2 minimally coupled and N=3 matter coupled theories, with U-duality given by the pseudo-unitary group U(r,s) (related to complex flux representations). Finally, we comment on the formulation of special Kaehler geometry in terms of"generalized"groups of type E7.


Introduction
We consider reducible symmetric supergravity models in D = 4 space-time dimensions 1 , which we will here dub Calabi-Vesentini (CV) models, for reasons which will be evident from treatment below. As also given by Table 1, these models are characterized by the following U -duality 2 and "horizontal" [5] symmetries: Considering an array of p charge vectors Q M a (a = 1, ..., p) pertaining to a p-centered solution, the U -duality group acts on the index M in a symplectic representation: where C M N is the symplectic-invariant metric (defined in (2.87) below). On the other hand, the "horizontal" symmetry acts on the index a as a linear transformation on the p vectors: The "horizontal" symmetry, which is not a symmetry of the Lagrangian formulation of the theory, proves to be useful in the classification of multi-charge orbits, which are relevant for the dynamics of multi-centered (black hole) solutions in supergravity [5,6,7]. For the two-centered case (p = 2) considered in the present investigation, the lowest-order duality-and "horizontal"-invariant polynomial is of order 2 in the charges, and it is nothing but the usual Schwinger symplectic product W of two dyonic charge vectors (see (2.16) below). As evident from (1.1), we anticipate that the case of N = 4 theory coupled to n V,N =4 = n − 2 0 matter (vector) multiplets can be recovered by shifting n → n + 4 in all formulae of the treatment below. The semi-simple nature of G 4 justifies the name "reducible", whereas "symmetric" is due to the fact that the corresponding scalar manifolds 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 [8], and Refs. therein).
Let us now reconsider the "T -tensor formalism" for CV models, introduced in Secs. 3 and 4 of [5], which will be further extended, until order 6 included, in Sec. 2. A key feature of CV models is the fact that the electro-magnetic splitting of the symplectic vector of the 2-form field strengths' fluxes (also named magnetic and electric charges) can be implemented with full manifest covariance with respect to G 4 (1.1). Namely, Q sits in the (2, m + n − 2) bi-fundamental irrep. of G 4 , and it is thus an electro-magnetic doublet 2 of the "vertical" SL v (2, R); the symplectic index M thus splits as follows (cfr. Eq. (3.7) of [5]) M = αΛ, α = 1, 2, Λ = 1, ..., m + n − 2.
From (1.1), the ("horizontal" × U -duality) group of a 2-centered solution in D = 4 CV models reads where we recall that "h" and "v" respectively stand for "horizontal" and "vertical". In the N = 2 case, the number of (Abelian) vector multiplets coupled to the gravity multiplet is n V,N =2 = n − 1.
We will throughout consider 2-centered 0-brane (BH) solutions, and thus the relevant representation of G p=2 × G 4 in which the corresponding 2-form field strengths fluxes Q Λ aα sit is which is thus amenable to a Gramian treatment, as considered in Sec. 8 of [5]. In the following treatment, the 2 of SL h (2, R) is spanned by the Latin lowercase indices a = 1, 2, the 2 of SL v (2, R) is spanned by the Greek lowercase indices α = 1, 2, and the vector n of SO (2, n − 2) is spanned by the uppercase Greek indices Λ = 0, 1, ..., n − 1 ("0" pertaining to the graviphotonic fluxes, as mentioned above). If no further decomposition with respect to proper subgroups is considered, maximal G 4covariance is manifest, and, as stated, the symplectic frame under consideration is usually dubbed CV [11] frame [12].
Aim of the present note is to give a complete treatment of 2-centered G 4 -("duality") and (G p=2 × G 4 )-(dubbed "horizontal") invariant homogeneous polynomial structures in the CV symplectic frame at order 2, 4 and 6 in the fluxes, thus clarifying, generalizing and completing the treatment given in [5], whose notation and formulae we will often refer to (reporting some of them, for ease of consultation). We will also briefly comment on the 2-centered "horizontal" symmetry of supergravity models with pseudo-unitary U -duality groups, refining the analysis of [21].
The plan of the note is as follows.
Then, in Sec. 3 we study duality-and "horizontal"-invariant two-centered polynomials in D = 4 symmetric supergravity models with U -duality group G 4 given by the pseudo-unitary group U (r, s).
The final Sec. 4 contains various remarks and observations, concerning the CV models (Sec. 4.1) and models with G 4 = U (r, s) (Sec. 4.2). Moreover, in Sec. 4.3, by suitably generalizing the notion of groups ''of type E 7 " [19,24,23], we comment on their relation to special Kähler geometry. We start and consider the rank-2 tensor product where "s" and "a" denote the symmetric and antisymmetric parts throughout. Ç a va sans dire, the obvious symmetry of Q Λ aα Q Σ bβ under the exchange aαΛ ↔ bβΣ restricts the analysis to the symmetric part of such a tensor product. Since in (pseudo-)orthogonal groups the symmetric rank-2 repr. can be further irreducibly split in traceless S 0 and trace 1 irreps. (with the naught denoting η-tracelessness in SO (2, n − 2) throughout; see Footnote 4 of [5]): (2.1) can be further elaborated as: where the following notation for SO (2, n − 2) irreps. has been introduced: The total real dimension of Q Λ aα Q Σ bβ is 2n (4n + 1). Thence, one can assign to each irreps. a tensor structure (with "#" denoting the corresponding dimension): (1, 1, S 0 + 1) : [ab] [αβ] . # = 1 2 n (n + 1) ; (each of them) Below, we will also use the further irreducibly split (1, 1, S 0 + 1) (as (2.2)), reading: Explicit expressions in terms of the flux vector Q Λ aα read 3 : (2.10) We will also make use of the following SO (2, n − 2)-matrix notations: (ab) ; (2.13) and analogous ones for the ǫ-traces. By further taking (half of) the η-trace of T (ΛΣ) , one obtains the symplectic product W of the two charge vectors Q Λ 1α and Q Λ 2α (cfr. e.g. (4.12) of [5], as well as (3.9) of [6]): which is evidently "horizontal" (i.e. (G p=2 × G 4 )-) invariant (actually, as yielded by the analysis of [5] and [6], the unique "horizontal"-invariant polynomial at order 2 in the fluxes). Clearly, (2.15) is a specification for CV models of the general formula (cfr. e.g. Eq. (3.9) of [6], and Sec. 3 therein for notation) where C M N is the symplectic-invariant metric defined in (2.87) below. The ǫ-traced tensors in (2.6) have been introduced in order to develop the subsequent treatment. Indeed, due to the very structure of (2.2), the irreducible splitting (2.2) is not relevant in order to classify and relate duality-and "horizontal"-invariant homogeneous polynomials in the BH fluxes (see the treatment of Secs. 2.2 and 2.3).

Order 4
Since there is no duality-nor "horizontal"-invariant polynomial structure at order 3, next we proceed to analyze the order 4 in the fluxes of the 2-centered BH solution in the framework under consideration. By exploiting the associativity of the (ir)reps.' tensor product, in each of the SL h (2, R)-and SL v (2, R)-sectors one gets whereas in the SO (2, n − 2) sector, recalling that S 0 × S 0 = 1 s + Adj a + S 0,s + ...; (2.18)

20)
3 In order to make contact with the notation of [5], we observe that T [ΛΣ] (ab) (2.9; also see (2.13)) and T (ΛΣ) (2.11; also see (2.14)) respectively correspond to T ≡ (T11 ≡ T1, T12, T22 ≡ T2) and 2T ΛΣ a , in turn given by Eqs. (3.3)-(3.6) resp. (4.2) of [5]. it holds Thus, the duality-or "horizontal"-invariant homogeneous polynomials at order 4 in the fluxes arise from the following tensor products: there are no other sources of duality-invariant polynomials involving tensor products of (3, 3, S 0 ). By using the SO(2, n−2)-matrix notation (2.12), the order-4 G 4 -invariant polynomial from (2.22) can reducibly be rewritten as the SL h (2, R)-bi-triplet It is convenient to introduce the following tensors (see also point 2 below): such that Eq. (2.23) can be rewritten as Then, J (ab)(cd) (2.24) can be SL h (2, R)-irreducibly decomposed as . (2.27) By making use of the cyclic property of Tr η and of the distributivity of the sum with respect to it, one obtains and therefore J (ab)(cd) can be rewritten as X is the order-4 "horizontal" invariant homogeneous polynomial defined by (4.13) of [5], which we report here, in the current notation (recall (2.12)-(2.14), as well as Footnote 3): In order to get an "horizontal" invariant polynomial homogeneous of order 4 in the fluxes, one has e.g. to ǫ-trace both sides of (2.31), obtaining (as a consequence of the ǫ-tracelessness of J (abcd) (2.30)): There are no other sources of duality-invariant polynomials involving tensor products of (3, 3, 1). The order-4 G 4 -invariant polynomial from (2.34) can reducibly be rewritten as the SL h (2, R)-bitriplet J (ab)(cd) defined by (2.25), which enjoys a decomposition analogous to the one of J (ab)(cd) . 3.

(2.54)
Observing that the definition (2.15) can be rewritten as definitions (2.6) imply that On the other hand, an explicit computation yields Therefore, by inserting (2.55)-(2.57) into (2.54), the following expression of the corresponding order-4 "horizontal" invariant polynomial is achieved: Note that, from (1.1) and observations below, the coefficient of W 2 in (2.58) is strictly positive in all CV models. However, without any loss of generality, one can instead consider the tensor product of the corresponding SL h (2, R) and SL v (2, R) ǫ-traces (once again, the unique independent manifestly SO v h (2, 2)-invariant combination), obtaining where Eq. (2.55) was used.

Summary
The above analysis completes, at order 4 in the fluxes, the treatment given in [5] and [7]. Besides W 2 and X , no other "horizontal" invariant homogeneous polynomials of order 4 in the BH fluxes Q Λ 1α and Q Λ 2α can be introduced. Concerning duality-invariant homogeneous polynomials of order 4, the Dixmier tensor I (abcd) [22], sitting in the spin s = 2 irrep. 5 of the "horizontal" symmetry SL h (2, R), generally defined by (2.43) and present in the analysis of [5,6], is (due to (2.48)) the unique algebraically independent duality-invariant tensor sitting in an irrep. of SL h (2, R) itself. Other duality-invariant tensors of mixed "horizontal" symmetry, such as I (ab)(cd) (2.37) and J (ab)(cd) (2.24) (related by (2.47)) can be introduced, but they do not sit in "horizontal" irreps..

Summary
The above analysis completes, at order 6 in the fluxes, the treatment given in [5] and [7]. Besides W 3 and WX , also the "horizontal" invariant I 6 (2.66) can be introduced. Concerning this, it is worth recalling here that two "horizontal" invariant order-6 homogeneous polynomials can be naturally introduced in CV models: • the I 6 defined by (2.66) above [5]; • the I ′ 6 given by (3.11) and (3.24) of [6], whose manifestly "horizontal"-invariant formulation in the CV symplectic frame [12] reads where, the symplectic-invariant C-structure reads consistent with the CV splitting (1.5).
The general 6 relation between I 6 and I ′ 6 is discussed in Sec. 3 of [7]; in CV models, such a relation is given by (also recall (2.44) and (2.45)) which, besides (2.76), provides a manifestly "horizontal"-invariant expression of I 6 in the CV symplectic frame.

Invariants of Pseudo-Unitary U -Duality
We now discuss the "horizontal" symmetry of D = 4 supergravity models with U -duality group G 4 given by the pseudo-unitary group U (r, s) for some r and s. Confining ourselves to theories with symmetric scalar manifolds, these supergravity theories are: • the N = 2 minimally coupled Maxwell-Einstein theory [25,26] (r = 1), with scalar manifold and vector 2-form field strengths sitting in the (complex) fundamental irrep. s + 1 of G 4 = U (1, s); • the N = 3 matter-coupled theory [27] (r = 3), with scalar manifold and vector 2-form field strengths sitting in the (complex) fundamental irrep. s + 3 of G 4 = U (3, s).
It is here worth recalling that N = 2 supergravity minimally coupled to 3 vector multiplets is "bosonic twin" to (i.e. shares the very same bosonic sector of) N = 3 supergravity coupled to 1 vector multiplet [26,28] (for a discussion of split flows and marginal stability in extended D = 4 supergravities, see e.g. [29]).
As observed in [21] (in which the split attractor flow and marginal stability features of theories (3.1) were investigated), the presence of an "extra" U (1) symmetry, acting only on the complex(ified) flux vector Q but not on the (complex) scalar fields (see e.g. (2.35)-(2.36) of [21]) is due to the fact that such theories are the only ones in which the "pure" theory limit (corresponding to the case in which only the graviton multiplet present) can be obtained by simply setting to zero the number s of matter (Abelian vector) multiplets. As such, the "extra" U (1) global factor 7 (which is not a global isometry of the scalar manifold) is nothing but the U (1) symmetry gauged by the complex scalars, which becomes global in their absence [30] (recall that the N = 2 and N = 3 graviton multiplets do not contain scalar fields). Moreover, in the N = 2 case, such a U (1) can also be interpreted as the symmetry of the graviphotonic electro-magnetic system.

SL (p, C) × U(r, s)
Before dealing with the actual "horizontal" symmetry of these theories, it is instructive to consider the group SL (p, C) × U (r, s), (3.3) and the orbits of complex vectors V A i (A = 1, ..., r + s, i = 1, ..., p) in the complex bi-fundamental irrep. (p, r + s) of SL (p, C) × U (r, s). This treatment can be considered the "complexified version" of the treatment given in the second part of Sec. 4 of [6], and also the results will be analogous.
There are only p 2 algebraically-independent U (r, s)-invariant homogeneous polynomials, all of order 2 in the fluxes, given by where "·" denotes the scalar product determined by the pseudo-Euclidean metric η AB of U (r, s). By respectively denoting with I p and G p the dimension of a complete basis of U (r, s)-invariant polynomials and the orbit of the irrep. r + s of U (r, s), the counting is consistent with the general counting rule [5,6]: because G p generally is a suitable non-compact form of the compact coset On the other hand, out of the p 2 order-2 U (r, s)-invariant polynomials U ij (3.4), one can construct a unique algebraically-independent [SL (p, C) × U (r, s)]-invariant homogeneous polynomial, of order 2p, defined as (3.8) The notation "det G" indicates the fact that the "horizontal"-invariant polynomial (3.8) is the determinant of the Hermitian-analogue of the Gramian matrix G introduced in (8.4)-(8.5) of [5] (see also the treatment of Sec. 8 therein). In the 2-centered case (p = 2), (3.8) reduces to is consistent with the general counting rule: because G p generally is the direct product of the Riemannian symmetric non-compact coset (SU (p) = mcs [SL (p, C)]) SL (p, C) SU (p) , dim R = p 2 − 1 (3.12) and of a suitable non-compact form of the compact coset (3.7): It is immediate to realize that SL (p, C) cannot be the "horizontal" symmetry of a p-centered BH solution of the models under consideration. Indeed, the total symmetry SL (p, C) × U (r + s) exhibits no invariants of order 2 in charges, as instead the symplectic product W (2.16) should generally be.

SL h (2, R) × U(r, s)
The number and the structure of p-centered (2 p r+s) algebraically independent duality-invariant homogeneous polynomials in the N = 2 minimally coupled theory have been already discussed in Sec. 4.2.1 of [21]. We now give a unified treatment (holding for both the theories (3.1) and (3.2)) of both p-centered (2 p r + s) "horizontal"-and duality-invariant polynomials.
As mentioned above, the 2-form field strengths and their dual (and thus the corresponding fluxes) sit in the complex fundamental irrep. r + s of G 4 = U (r, s). When considering a p-centered BH solution, the complex(ified) flux vector Q A i (A = 1, ..., r + s, i = 1, ..., p) sits in the bi-fundamental irrep. (p, r + s) of the "horizontal" × U -duality group : SL h (p, R) × U (r, s). (3.14) As noticed in [21] for the N = 2 minimally coupled case, there are only p 2 algebraically-independent U (r, s)-invariant homogeneous polynomials, all of order 2 in the fluxes, given by where "·" denotes the scalar product determined by the pseudo-Euclidean metric η AB of U (r, s), and Note that, with respect to (3.4), the "horizontal" indices i's here belong to the real fundamental irrep. p of SL h (p, R). By respectively denoting with I p and G p the dimension of a complete basis of duality invariant polynomials and the orbit of the irrep. r + s of G 4 , one obtains the very same counting given by Eqs. (3.5)-(3.7). Let us now consider the issue of "horizontal" invariants in the 2-centered (p = 2) case. In this case, there are p 2 = 4 order-2 U (r, s)-invariant polynomials v ij (3.15), namely [21]: where i, j = 1, 2, and I 2 (Q) is the unique algebraically-independent 1−centered U (r, s)-invariant polynomial (homogeneous of order-2 in the charges Q's; see e.g. [31,26], and Refs. therein): Out of (3.18), one can easily determine that the "minimal degree" basis of homogeneous [SL h (2, R) × U (r + s)]-invariant polynomials is composed by one invariant of order 2, namely W (2.16), and by the following invariant of order 4 (ǫ 12 ≡ 1): In order to show this, we start and compute By using the Schouten identities for SL h (2, R) δ e [a ǫ cd] = 0, (3.22) it is immediate to obtain is consistent with the general counting rule: because G 2 generally is the direct product of the Riemannian symmetric "horizontal" non-compact coset (SO (2) = mcs [SL (2, R)]) and of a suitable non-compact form of the compact coset (3.7): Thus, at least in the p = 2 case, an important feature of the models under consideration is that the "horizontal" sector coset (3.12) has a non-trivial stabilizer SO (2), differently e.g. from the CV (alias reducible symmetric) [5,7] and from the irreducible symmetric [6,16,7] models (also recall Footnote 4).
We leave the detailed investigation of the cases p 3 for future further study.
Consistent with analysis of [5] (and with the general results of [41]), a complete basis of homogeneous "horizontal" invariant polynomials for the CV models is given by (cfr. Eq. (8.2) of [5], as well as the treatment of Sec. 4 of [7]) W, X , I 6 , Tr I 2 0 , (4.1) where the order-8 "horizontal" invariant polynomial Tr I 2 0 is defined by Eq. (4.9) of [5]. It is worth remarking that, as yielded by the general analysis of [41], besides being "of minimal order" in the fluxes Q Λ 1α and Q Σ 2β of the two BH centers, the basis (4.1) is also freely generating the ring of "horizontal" invariant (homogeneous) polynomials: in other words, all other "horizontal" invariant polynomials are themselves polynomials in W, X , I 6 and Tr I 2 0 , with no syzygial constraints 8 . As for purely duality-invariant polynomials, we observe that their analysis at order-6 in the fluxes has not been performed in Sec. 2.3. This is due to the fact that, from the treatment given in Sec. 5 of [5], it is known that a(n in general non-freely generating) complete basis "of minimal order" in the Q's for the purely duality-invariant (homogeneous) polynomials is 9 (cfr. Eq. (1.15) of [7]) and thus one does not need to seek for order-6 (and/or higher) purely duality-invariant homogeneous polynomials. We would like also to point out that the non-generic example of CV model provided by the so-called st 2 model is treated in Sec. 6 of [5] and in Sec. 4.2 and App. B of [7]. Moreover, we recall that the so-called t 3 model (treated in Sec. 7 of [5] and in Sec. 5.2 and App. B of [7]) is, as it is well known, an isolated case in the classification of symmetric special Kähler geometries (see e.g. [42], and [43] for a list of Refs.); as such, it does not belong to the CV models (1.1). However, as shown in app. B of [5], it can be reformulated in terms of a "constrained" CV symplectic frame.

On Pseudo-Unitary U-Duality
On the other hand, the analysis carried out in Sec. 3 (in turn refining the treatment given in Sec. 4.2.1 of [21]) yields that the (symmetric) D = 4 supergravity models with U -duality group G 4 given 8 It should be pointed out that, due to the order-12 syzygial constraint given by Eq. (5.6) of [5] (holding in all CV models), the "horizontal"-invariant basis (cfr. Eq. (8.1) of [5], as well as the treatment of Sec. 4 of [7]) is not freely generating.
For irreducible symmetric models [6,16], due to the vanishing of X mentioned above, the complete basis "of minimal order" in the Q's for "horizontal"-invariant (homogeneous) polynomials is given by and it is freely generating [41]. Therefore, apart from the peculiar case of the so-called t 3 model (treated in Sec. 7 and App. B of [5], as well as in Sec. 5.2 of [7]), the X = 0 limit of the order-12 constraint given by Eq. (5.6) of [5] does not hold in irreducible symmetric models. 9 Of course, other choices are possible; see e.g. Sec. 4 of [7]. For irreducible symmetric models, due to the vanishing of X mentioned above, the (in general non-freely generating) complete basis "of minimal order" in the Q's for the purely duality-invariant (homogeneous) polynomials is given by Eq. (1.16) of [7] (also in this case, other choice are of course possible; see e.g. Sec. 5 of [7]). by the pseudo-unitary group U (r, s) have a much simpler case study concerning the duality-and "horizontal"-invariant homogeneous polynomials constructed out of the BH fluxes' irrep., at least for p = 2.
It is here worth commenting also about the N = 2, D = 4 "magic" Maxwell-Einstein supergravity based on the irreducible cubic Jordan algebra J C 3 [44,45] (see also e.g. [9,10] for a recent account) and on the N = 5, D = 4 "pure" theory [46]. Despite the fact that their U -duality groups are suitable noncompact forms of the (special) pseudo-unitary group SU (6) (namely, SU (3, 3) respectively SU (1, 5)), these theories do not belong to the class of models treated in Sec. 3. Indeed, they do not have an "extra" global symmetry U (1) under which their 2-form field strengths' fluxes are charged; this is also related to the fact that their magnetic and electric fluxes sit in a self-real irrep., namely the rank-3 completely antisymmetric 20 of SU (6) (and not in the complex fundamental irrep. 6 of the analogue would-be model of the type treated in Sec. 3).
It should also be noticed that (non-compact forms of) CP n spaces as moduli spaces of string compactifications have appeared in the literature, either as particular subspaces of complex structure deformations of certain Calabi-Yau manifold [47,48] or as moduli spaces of some asymmetric orbifolds of Type II superstrings [49]- [52], or of orientifolds [53].
Finally, we observe that the D = 4 supergravity models considered in Sec. 3 are not included in the analysis of [41]. In fact, only the "real (pseudo-orthogonal) analogues" of such models (in which the analogue of W ij vanishes; see the treatment given in the second part of Sec. 4 of [6]) can be found in Table II
Here we would like to discuss the characterization of SKG in terms of a suitable "generalization" of the groups of type E 7 [24] (for some preliminary discussion, see Sec. 4 of [23]).
As obtained in [19] (see Eq. (5.36) therein), the following real function, which we dub "entropy functional", can be defined on the vector multiplets' scalar manifold 10 M: Z is the central extension of N = 2, D = 4 local supersymmetry algebra, and Z i ≡ D i Z are the so-called "matter charges" (D i stands for the Kähler-covariant differential operator; see e.g. [35] and [8] for notation and further elucidation): with V M denoting the vector of covariantly-holomorphic symplectic sections of SKG, and V M i ≡ D i V M . Furthermore, C ijk is the rank-3, completely symmetric, covariantly holomorphic tensor of SKG (with Kähler weights (2, −2)) (see e.g. [36,37]): the last property being a consequence, through the covariant holomorphicity of C ijk and the SKG constraint on the Riemann tensor (see e.g. [36,37,38]) of the Bianchi identities satisfied by the Riemann tensor R ijkl . Furthermore, I 4 is an order-4 homogeneous polynomial in the fluxes Q; this allows for the definition of the Q-independent rank-4 completely symmetric tensor Ω M N P Q [23], whose general expression we explicitly compute here: where the SKG defining relation (see e.g. [36,37,38]) has been used in order to recast (4.8) in terms of V M , V M i and D i V M j only. Some further elaborations are possible; e.g., by using (4.7), I 4 (4.4) and Ω M N P Q (4.9) can respectively be rewritten as where the sectional curvature of matter charges (cfr. Eq. (5.3) of [39]; also note that (4.13) is different from the definition given by Eq. (3.1.1.2.11) of [40])) and the corresponding rank-4 completely symmetric tensor have been introduced. Note that R M N P Q can be regarded as the completely symmetric part of the "symplectic pull-back" (through the symplectic sections V M i ) of the Riemann tensor R ijkl of M. Thus, SKG can be associated to a generalization of the class of groups of type E 7 [24], based on I 4 and the corresponding (generally field-dependent, non-constant) Ω-structure: SKG :  Within SKG, through the constraint (4.7), this implies the covariant constancy of the C-tensor (4.6): D (i C j)kl = D (i C jkl) = 0, (4.17) which in turn yields the relation: C p(kl C ij)n g nn g pp C npm = 4 3 g (l|m C |ijk) ⇔ g nn R (i|m|j|n C n|kl) = − 2 3 g (i|m C |jkl) . (4.18) Equivalently, symmetric SK manifolds can be characterized by stating that their Ω M N P Q is is independent on the scalar fields themselves, and it matches the K-tensor K M N P Q defining the rank-4 invariant K-structure of the corresponding U -duality group of type E 7 [24] (see also e.g. [16], and Refs. therein). Consequently, the corresponding "entropy functional" I 4 (4.4) is independent on the scalar fields themselves, and it is thus a constant function in M, given by the 1-centered limit of the Dixmier tensor I abcd (2.43), which is nothing but the unique algebraically-independent 1-centered U -duality invariant polynomial I 4 : symmetric SKG  In turn, within symmetric SKG, the pseudo-unitary U -duality group U (1, s) (corresponding to N = 2 minimally coupled Maxwell-Einstein theory [25,26]) is "degenerate", in the sense that the corresponding I 4 actually is the square of the order-2 U (1, s)-invariant polynomial I 2 (3.19). Indeed, N = 2 minimally coupled supergravity is characterized by 11 C ijk = 0, which plugged into (4.4) (by taking (4.19) into account) yields: symmetric SKG where the normalization of [21] (see Eq. (2.15) therein) has been adopted. We conclude by recalling that, as noticed in [19] and in [23], the "entropic functional" I 4 (4.4) is related to the geodesic potential defined in the D = 4 → 3 dimensional reduction of the considered N = 2 theory. Under such a reduction, the D = 4 vector multiplets' SK manifold M (dim C = n V ) enlarges to a special quaternionic Kähler manifold M (dim H = n V + 1) given by c-map [42,54] of M itself : M = c (M). By specifying Eq. (4.4) in the "4D/5D special coordinates' " symplectic frame, I 4 matches the opposite of the function h defined by Eq. (4.42) of [55], within the analysis of special quaternionic Kähler geometry. This relation can be strengthened by observing that the tensor Ω M N P Q given by (4.8)-(4.9) is proportional to the Ω-tensor of quaternionic geometry, related to the quaternionic Riemann tensor by Eq. (15) of [56]; for further comments, see [23].