Dualities Near the Horizon

In 4-dimensional supergravity theories, covariant under symplectic electric-magnetic duality rotations, a significant role is played by the symplectic matrix M({\phi}), related to the coupling of scalars {\phi} to vector field-strengths. In particular, this matrix enters the twisted self-duality condition for 2-form field strengths in the symplectic formulation of generalized Maxwell equations in the presence of scalar fields. In this investigation, we compute several properties of this matrix in relation to the attractor mechanism of extremal (asymptotically flat) black holes. At the attractor points with no flat directions (as in the N = 2 BPS case), this matrix enjoys a universal form in terms of the dyonic charge vector Q and the invariants of the corresponding symplectic representation RQ of the duality group G, whenever the scalar manifold is a symmetric space with G simple and non-degenerate of type E7. At attractors with flat directions, M still depends on flat directions, but not MQ, defining the so-called Freudenthal dual of Q itself. This allows for a universal expression of the symplectic vector field strengths in terms of Q, in the near-horizon Bertotti-Robinson black hole geometry.


Introduction
One of the most appealing properties of extended (ungauged) four-dimensional supergravities (i.e. locally supersymmetric models with no less than 8 supercharges) is their on-shell global symmetry which is conjectured to encode the known string/M-theory dualities [1]. The corresponding global symmetry group G, to be also dubbed U-duality, is the isometry group of the scalar manifold (i.e., global symmetry of the scalar field sigma-model), whose (non-linear) action on the scalar fields is combined with a linear symplectic action on the n electric field strengths F Λ µν , Λ = 0, . . . , n − 1, and their magnetic duals G Λ| µν [2] (electric-magnetic duality action of G). The latter action being defined by an embedding of G in the symplectic group Sp(2n, R), so that F Λ µν , together with G Λ µν , transform under electric-magnetic duality in a symplectic representation R Q of G. This embedding, which determines the couplings of the vector fields to all the other fields in the action, is built-in the definition of a flat symplectic bundle over the scalar manifold, which is a common mathematical feature of N 2-extended supergravities [3,4,5].
Solutions to these theories naturally arrange themselves in orbits with respect to the action of G, and important physical properties are captured by G-invariant quantities. A notable example are the extremal, static, asymptotically-flat black holes in D = 4, which have deserved considerable attention in the literature during the last 20 years or so, since they provide a valuable tool for studying string/M-theory dualities. These solutions are naturally coupled to scalar fields as a consequence of the non-minimal couplings of these to the vector fields in the supergravity action. Near the horizon, however, they exhibit an attractor mechanism [6,7]: the near-horizon geometry, which is described by an AdS 2 × S 2 Bertotti-Robinson space-time [8], is independent of the values of the scalar fields at radial infinity, and it only depends on the quantized magnetic and electric charges p Λ , q Λ . In particular the horizon area A H , which is related to the entropy S of the solution through the Bekenstein-Hawking formula [9], is expressed in terms of the quartic invariant I 4 (p, q) of the representation R Q of G, only depending on p Λ , q Λ (we set 8πG N = c = = 1): This is a consequence of the fact that the horizon represents an asymptotically stable equilibrium point for the radial evolution of those scalar fields which are effectively coupled to the solution and thus affect its geometry. In other words, such scalars flow from radial infinity to the horizon toward values which only depend on the quantized charges (fixed values). The horizon fixed point is defined by extremizing an effective potential V BH (ϕ; p, q) (ϕ generically denoting the scalar fields) [7]: where Q = (p Λ , q Λ ) is the vector quantized charges in the representation R Q of G. The value of this potential at the horizon defines its area, being equal to |I 4 (p, q)|. The scalar fields which are not fixed at the horizon are those which are not effectively coupled to the black hole charges, and they are flat directions of V BH . They will be denoted by ϕ f lat . In the above formula, M(ϕ) is a 2n × 2n symmetric, symplectic, negative-definite matrix-valued function of the scalar fields. In all extended supergravities it is defined by the flat symplectic bundle over the scalar manifold. In fact, it encodes all the information about the non-minimal couplings of the scalar to the vector fields in the action through the kinetic term of the latter and the generalized theta-term. Moreover it allows to define the so called Freudenthal duality [10], a recently studied on-shell symmetry [11,12,13] which we shall be dealing with in the following. An interesting question to be posed is what happens to the geometric structures associated with the scalar manifold, e.g. pertaining to its symplectic bundle, near the horizon. In the present investigation, we focus on the matrix M(ϕ), because of its relevance to the geometry of the supergravity model.
At the horizon M(ϕ) depends on Q, through the fixed scalars, and on the flat directions: As we shall prove in what follows, the dependence on the flat directions drops out already when we contract M H once with the charge vector. This implies the independence of the vector field-strengths at the horizon from ϕ f lat . On general grounds, using the properties of M(ϕ), one can show that if we act on the solution by means of an element g of G, which maps ϕ into ϕ ′ and Q into Q ′ , the matrix M(ϕ) at the horizon transforms as follows: 1 where, with an abuse of notation, we have denoted by g also the symplectic 2n × 2n matrix representing the corresponding G-element on contravariant vectors of R Q . In absence of flat directions, the above equation suggests that M H (Q) should be described in terms a symmetric, symplectic, negative-definite matrix defined on the G-orbit of Q, and thus constructed out of Q and of G-invariant tensors. Restricting our analysis to the case of simple groups G, with the exception of the STU model, for charge vectors Q with I 4 (Q) > 0 we could construct such a matrix M(Q) using a simple Ansatz, which involves only Q and G-invariant tensors, and imposing the following properties of M H : ∂Q , (1.5) where I 4 =: ǫ |I 4 |, and C is the symplectic invariant 2n × 2n antisymmetric matrix. Note that the second of (1.5) [11] implies (1. 6) however, it can be checked that this yields the same condition (namely, (A.1) further below) on the real coefficients A, B and C of the Ansatz (4.9)-(4.10). Starting from the same Ansatz we actually find two solutions to the above equations, denoted by M + (Q) and M − (Q). For charges with I 4 (Q) > 0 and no flat directions, we give arguments in favor of the identification of one of these matrices (M + ) with M H (Q). The other solution (M − ), on the other hand, is never negative definite and has the general form: This Hessian has been considered in the literature, see [14,15], though in different contexts. As far as regular BPS solutions in N = 2 supergravities are concerned, the two matrices M ± enjoy an interesting interpretation as the value at the horizon of two characteristic symplectic, symmetric matrices of the theories: the matrix M which is constructed out of the real and imaginary parts of the period matrix N ΛΣ (ϕ) (defining the generalized theta-term and the kinetic term for the vector fields, respectively), and a matrix M (F ) , constructed just as M, but in terms of the real and imaginary parts of a different complex matrix, namely the Hessian F ΛΣ of the holomorphic prepotential of the special Kähler manifold. In terms of the covariantly holomorphic section V = (V M ) of the special Kähler manifold describing the vector multiplet scalars z i , and of its covariant derivatives U i = D i V = (U M i ) (we use the notations of [16]), the two matrices have the following expressions: The former was given in [5] and [17], and it is the real part of the identity (1.16) of [11]. On the other hand, the latter expression follows from (1.13) of [12]; furthermore, Q T M (F ) (z,z)Q agrees with Eq. (57) of [18]. In N 2-extended supergravity, for charge orbits characterized by I 4 (Q) < 0, the two matrices M ± , though still satisfying the second of (1.5), are anti-symplectic, namely for them the following property holds: (1.10) The matrix M + , in particular, for all regular charge-orbits, as opposed to M − , has the notable property of being an automorphism of the U-duality algebra g, that is g, in the representation R Q , is invariant under the adjoint action of M + (if I 4 < 0, being M + anti-symplectic, it is an outer automorphism). On the other hand M − is still, in all regular orbits, identified with the Hessian (1.7). Moreover both M ± are invariant, up to a sign, under Freudenthal duality at the horizon. For a generic regular charge-orbit we will find the following relation between M H and the automorphism M + : where A is an involutive automorphism of G in the stabilizer of Q, depending in general on Q and ϕ f lat . For I 4 < 0, both M + and A are anti-symplectic outer-automorphisms of G, while for I 4 > 0, A ∈ G and, in absence of flat directions, it is the identity matrix. Besides the interpretation in terms of M at the horizon, which holds only for M + in specific orbits, the solution M − is the symplectic metric on the G-orbit of Q [15] and thus it has a mathematical relevance per se.
We then consider M H for solutions with flat directions, and prove a general factorization property: where the two factor-matrices are elements of G and commute. The former M H 1 (Q) is negative definite while the latter M 0 (ϕ f lat ) is in the stability group of Q. As anticipated above, from this it follows that the dependence on the flat directions drops out once we contract M H with a charge vector, so that the vector field strengths at the horizon are ϕ f lat -independent.
The relation between the decompositions (1.12) and (1.11) is that M H 1 (Q) can be written as the product M + (Q) A 0 (Q), A 0 being an involutive automorphism in the stabilizer of Q, so that A(Q, ϕ f lat ) in (1.11) is given by the product A 0 (Q) M 0 (ϕ f lat ).
The plan of the paper is the following. In Sect. 2, we recall some basic facts about extremal black hole solutions in extended supergravities, as well as their properties under the global symmetry of the models. This includes a review of the Freudenthal duality, and sets the stage for the discussion of our results. In Sect. 3, we recall the main properties of the independent lowest-order invariant tensors, namely C M N (symplectic metric) and K M N P Q (K-tensor), in the symplectic black hole charge representation R Q of the U-duality groups of symmetric four-dimensional Maxwell-Einstein (super)gravity theories (to which we restrict our present investigation). In Sec. 4, which focuses on the cases without flat directions, we construct, out of a general ansatz involving suitable contractions of the K-tensor and of the symplectic metric C M N with a number of charge vectors Q, a symmetric matrix M satisfying conditions (1.5). As anticipated above, restricting our analysis to simple U-duality groups, treated in Subsec. 4.1, we actually find, for I 4 (Q) > 0 two solutions M + and M − . The former is identified with M H , while the properties of the latter are studied at the end of the same Section. The definition of the matrices M ± is the generalized to the I 4 < 0 orbit, in Sec. 4.2; here general properties of M ± , in any regular charge-orbit I 4 = 0, are discussed. In Sec. 4.3 we consider N = 2 theories, where we show that M − , in the BPS-orbit, is identified with the matrix M (F ) . The peculiar cases of the non-BPS attractors in the T 3 -model as well as in N = 2 minimally coupled Maxwell-Einstein theory and in N = 3 supergravity are considered in Subsecs. 4.4 and 4.5, respectively. Then, in Sec. 5, flat directions [25] are taken into account, for both cases I 4 < 0 (Subsec. 5.1) and I 4 > 0 (Subsec. 5.2).
A summary of results and general properties of M + and M − , as well as their relation to the matrix M H , is finally given in Sec. 6.
Appendices A and B contain details of the derivation of some results of Sec. 4, while Appendix C containing a discussion of anti-symplectic outer-automorphisms of the U-duality algebra, concludes the paper.

Symmetry Properties of Extremal Black Holes in Extended Supergravities
One of the basic ingredients of the symplectic formulation of electric-magnetic duality in N 2extended supergravity theories in four dimensions, whose bosonic sector reads (in absence of gauging) where n V + 1 denotes the number of Abelian vector fields 2 , and I denotes the (n V + 1)dimensional identity matrix. I ΛΣ is the kinetic vector matrix, and R ΛΣ enters the topological theta term in (2.1); they are usually regarded as the imaginary resp. real part of a complex kinetic matrix N ΛΣ , such that (2. The symplectic structure of the generalized special geometry [4,5] of scalar fields yields that M can be equivalently rewritten as where L is an element of the Sp (2n V + 2, R)-valued symplectic bundle of generalized special geometry.
M enters the symplectic-covariant form of the Maxwell equations (twisted self-duality condition) [19] where (Λ = 0, 1, ..., n V ; in N = 2 theories, the naught index is reserved for the graviphoton) (2.5) 2 The notation "n V + 1" is actually relevant for N = 2 supergravity, in which one can distinguish between the graviphoton and the n V = n − 1 matter multiplets' vector fields. 3 Throughout this paper we use for the symplectic invariant matrix the following form: C = 0 n I n −I n 0 n .
is the symplectic vector of the 2-form Abelian field strengths and of their duals, and the fourdimensional Hodge dual is denoted, as usual, by * . Correspondingly, the equations of motion and Bianchi identities take the form dH = 0. (2.6) In the background of a static, spherically symmetric, asymptotically flat, dyonic extremal black hole (τ := −1/r) one can introduce the symplectic vector Q of asymptotic magnetic and electric fluxes of H as follows: Thus, the twisted self-duality condition (2.4) and the symplecticity of M imply that where F denotes the "non-critical", scalar-dependent generalization of the so-called Freudenthal duality [10], defined as a scalar-dependent involution on the symplectic vector Q [11]: It should be stressed that the anti-involutivity (2.12) of F is a direct consequence of the symplecticity of M itself. Thus, in every generalized special geometry [5], one can define a scalardependent almost-complex structure as follows [13]: such that F (Q) := −S (ϕ) (Q) . For U-duality 4 groups G of type E 7 [21], S (ϕ) ∈ Aut(F) ≡ G, where F denotes the corresponding Freudenthal triple system [13]; in these theories, S may be regarded as the projection onto the adjoint in the symmetric tensor product of the symplectic representation R Q of G, carried by F itself.
By virtue of the definition (2.11), the twisted self-duality condition (2.4) can be recast as [13] H = −F ( * H) = − * F (H) , (2.16) which is nothing but the "unfluxed" version of (2.10). (2.16) expresses the compatibility of the two different almost-complex (anti-involutive) structures defined by the Hodge * -duality (in D = 4 spacetime, * 2 = −Id) and by the operator F (in the symplectic vector G-space R Q spanned by H (2.5) or by its asymptotic flux Q (2.8); recall (2.12)). An equivalent restatement of such a compatibility is given by (2.10), as well. Furthermore, M is relevant to define the Abelian 2-form field strengths H in the background (2.7) (cfr. e.g. [22,23,24]) consistently with (2.16). Note that the dependence of H on scalars is completely encoded in M (ϕ), or, equivalently, in the "non-critical" Freudenthal duality F (2.11). M also defines the (positive definite) effective black hole potential (1.2), such that F (2.11) can equivalently be defined as The potential V BH (1.2) governs the radial evolution of the scalar fields ϕ(τ ) as well as of the warp factor U (τ ): (2.23) At the event horizon of the extremal black hole (τ → −∞), the attractor mechanism [6,7] yields that, regardless of the initial (asymptotic) conditions: up to flat directions [25]. It can be shown [11] that where S BH denotes the Bekenstein-Hawking entropy [9] of the extremal black hole (2.7) as Note that (2.26) implies M H to be homogeneous of degree zero in the charges. For U-duality groups of type E 7 [21],Q can also be written as [10,11] where K M N P Q is the so-called invariant K-tensor (see below, and cfr. Sec. 3), which allows for the definition of a quartic homogeneous polynomial I 4 in the charges Q: Therefore, at the event horizon of the extremal black hole, the symplectic field strengths vector where U H is the leading order contribution in τ of the near-horizon limit of U(τ ). (2.34) implies which is nothing but the evaluation of (2.19)-(2.21), or equivalently of the twisted self-duality condition (2.16), at the horizon. It should also be recalled that M occurs in the expression of the metric of the enlarged scalar manifold, obtained by dimensional reduction of the D = 4 Maxwell-Einstein-scalar Lagrangian density (2.1) down to D = 3 (and by subsequent dualization of the D = 3 vector fields to scalars Z) [19]: When (2.1) is regarded as the bosonic sector of N = 2, D = 4 (ungauged) Maxwell-Einstein supergravity coupled to n V Abelian vector multiplets, such a dimensional reduction is named c-map [26], and the ds 2 D=3 (2.36) is quaternionic-Kähler, which can also be considered as the metric of the quaternionic scalars of the N = 2 hypermultiplets.
In the present investigation, we spell out some additional basic properties of the matrix M. Along the radial evolution of the scalar flow, M has a complicate dependence on the scalars; for instance, in generic d-geometries, the expression of the real symmetric matrices I ΛΣ and R ΛΣ is given e.g. in Sec. 2 of [27].
In absence of flat directions, when all the scalars ϕ's are stabilized to a (purely) Q-dependent value ϕ H (Q) (2.24) at the (unique) event horizon of the extremal "large" black hole, the nonlinear action of an element g of the U-duality group G on the scalars induces a linear transformation on M (cfr. (1.4)): The attractor mechanism [6,7] is responsible for the stabilization of ϕ (through (2.24)), and thus of M (ϕ), in terms of Q, which transforms in the representation R Q of G. This allows for a group-theoretical characterization of the horizon expression M H (2.27) of M. Indeed, a U-duality transformation g ∈ G acting on Q yields to a symplectic transformation on M H itself: Certainly an Sp(2n V + 2, R)-covariant, symmetric matrix M(Q), only built out of Q and of G-invariant tensors in products of the representation R Q , satisfies the above transformation property. These invariant tensors include the symplectic metric C M N , which allows for the definition of the invariant product of two symplectic vectors B 1 and B 2 : 40) and the rank-4 completely symmetric invariant K-tensor K M N P Q (cfr. Sec. 3). In the next sections we address the problem of expressing M H in terms of a matrix M(Q) of this kind, restricting ourselves to D = 4 Maxwell-Einstein (super)gravity theories whose scalar manifold is a symmetric space G/H (which correspond to characterizing G as a group of type E 7 [21]). We find a simple identification for I 4 > 0 orbits in absence of flat directions. As we will elucidate in the subsequent treatment, things change in presence of flat directions (short-hand denoted by ϕ f lat ), namely, of scalar degrees of freedom which are not stabilized at the horizon of the extremal black hole, and thus for which the attractor mechanism does not hold, at least at Einsteinian level [25]. In this case, M depends on flat directions (unlike the ADM mass and V BH itself), also at the horizon (i.e., setting the scalar fields at their attractor value ϕ H (Q) (2.24)): On the other hand, we will prove that M H Q does not depend on ϕ f lat : (2.42) and (2.43) also yield a crucial difference between the "non-critical" Freudenthal duality F (2.11) [11] and its "critical", horizon limit F H (2.25) [10]: while the former depends on flat directions, the latter does not: (2.46)-(2.47) can be regarded as an attractor mechanism for the Freudenthal duality : F H (Q) =:Q does not depend on flat directions ϕ f lat , and thus it is purely Q-dependent.

The K-Tensor
Let us consider a D = 4 U-duality group G of real dimension d, with generators t α in the adjoint representation (α = 1, ..., d). The Gaillard-Zumino [2] symplectic maximal embedding defining the Cartan-Killing metric k αβ of G as N is a singlet of G, as expected by analyzing the product of the representations. In fact, at least all electric-magnetic duality groups consistent with (not necessarily symmetric nor homogeneous) generalized special geometry [5] share the property that where Adj denote the adjoint representation of G (e.g. 133 in E 7 ). A key property of t α M N defined by (3.2) is the even symmetry of the symplectic indices: At least for groups G "of type E 7 " [21], it is possible to construct the aforementioned rank-4 completely symmetric invariant tensor, dubbed K-tensor [28]: which can be generally defined as follows: Needless to say, the prototype of groups "of type E 7 " is E 7 itself (pertaining to N = 8 and N = 2 supergravity, in its real forms E 7(7) and E 7(−25) , respectively), with R Q = 56. By following the treatment of [28], one can prove that where the real constants ξ and τ have been introduced; the latter can be determined by imposing the skew-tracelessness condition C N P K M N P Q = 0, yielding [28] τ = d n(2n + 1) , (3.9) whereas, by consistency with the definitions used in literature (cfr. [29], taking into account the different normalization conventions), ξ is fixed as Thus, the following general expression for the K-tensor is obtained: The formula (3.11) will be relevant to many subsequent computations (most of them reported in Appendix A). By contracting the K-tensor with four charge vectors Q's, one obtains the quartic G-invariant homogeneous polynomial I 4 [30] (2.30) in R Q , which can therefore be rewritten as ∂Q . (4.1) For each regular orbit (I 4 > 0, I 4 < 0), we find two distinct solutions M ± with different properties. As mentioned in the Introduction, in the present investigation we will consider only simple U-duality groups G; we leave the treatment of semi-simple cases elsewhere. The I 4 < 0 attractor in the N = 2 T 3 -model, though still having no flat-directions, deserves a separate treatment which will be given in Subsec. 4.4. Indeed, in this case, the identification M H = M + cannot work since M + is antisymplectic.
Simple U-duality groups G "of type E 7 " [21] will be considered in Subsec.s 4.1, 4.2, 4.3 and 4.4. Here we first construct the solutions M ± for I 4 > 0, discuss their geometric properties and the relation of one of them to M H . Then we move to the definition of M ± in the I 4 < 0 case, generalizing some of their properties to all regular orbits.
The particular case of minimal coupling of Abelian vector multiplets to N = 2 supergravity, in which K M N P Q is reducible (corresponding to degenerate groups "of type E 7 " [33]), will be considered in Subsec. 4.5.
We use for M the following general Ansatz (A, B, C ∈ R): where: By recalling (2.28) [10,11], it holds that such that (4.4) can be rewritten as By exploiting the identity 5 the Ansatz (4.4) (or, equivalently (4.7)) can be further simplified as It should be remarked that a term proportional to Q (MQN ) cannot occur in (4.7) (or, equivalently, in (4.10)), because it is not consistent with (4.3) [11].
A consistent solution to (4.2)-(4.3) within the Ansatz (4.4) can be found only for ǫ = +1 ⇔ I 4 > 0, and it reads The splitting into "±" branches generally corresponds to two independent expressions, namely M + and M − , in terms of suitable contractions of the K-tensor itself and of the symplectic metric C M N with charge vectors Q's; note that M − lacks the term proportional to Q M Q N , because C − = 0. This "±" degeneracy can be removed when considering the relation to the negative-definite matrix M H . Indeed M − (Q) always has (at least) a positive eigenvalue and thus can never be identified with M H . This result is illustrated in App. B by a direct computation in the ST U model (and its rank-2 (ST 2 ) and rank-1 (T 3 ) "degenerations" determine the corresponding symmetric models), and thus holds at least in all rank-3 symmetric models of which the ST U one is a universal sector. This check allows one to conclude that only the "+" branch is consistent with the properties required for the matrix M (at the horizon).
Using (4.9)-(4.10), direct computation in some cases (recall I 4 > 0) suggests the following identification Using general properties of M + , to be discussed below, we shall give, at the end of Sect. 6, an alternative argument in favor of the above identification.
Let us comment on the properties of the above matrices under Freudenthal duality F (2.11), and in particular under its "critical"/horizon version F H ( 2.25). Using the property as it can be verified by exploiting the properties of groups "of type E 7 " [21], it is straightforward to show that Furthermore, the result (4.11), as discussed in App. A, is constrained by the consistency condition relating the dimension d of the adjoint irrep. Adj and the dimension 2n = 2n V + 2 of the black hole charge irrep. R Q of G. As observed in [28], (4.16) actually characterizes at least all the pairs (G, R Q ) related to simple rank-3 Euclidean Jordan algebras [35] (such pairs are example of simple, non-degenerate groups "of type E 7 " [33]).
The cases related to D = 4 Maxwell-Einstein gravity theories with local supersymmetry are reported in Table 1; within this class, the so-called ST U model [34] is an exception: the corresponding rank-3 Jordan algebra is semi-simple (R ⊕ R ⊕ R), but however it still satisfies (4.16).
Interpretation of M − . Interestingly, also can be given a meaning within the stratification of R Q into G-orbits. Indeed, M −,I 4 >0|M N (4.21) can be regarded as the metric of the non-compact pseudo-Riemannian rigid special Kähler manifold [15] [35]. Note that the ST U model [34], based on R ⊕ R ⊕ R, is reducible, but triality symmetric. All cases pertain to models with 8 supersymmetries, with exception of M 1,2 (O) and J Os 3 , related to 20 and 32 supersymmetries, respectively. The D = 5 uplift of the T 3 model based on R is the pure N = 2, D = 5 supergravity. J H 3 is related to both 8 and 24 supersymmetries, because the corresponding supergravity theories share the very same bosonic sector [35,38]. All data d and n satisfy the relations (4.16)-(4.18).
with real dimension 2n V +2; O I 4 >0 denotes the corresponding "large" G-orbit defined by the Ginvariant constraint I 4 > 0 on the charge representation R Q of G; the R + factor in (4.22) simply corresponds to the non-vanishing (strictly positive) values of I 4 itself. The signature along the R + -direction is negative, while the metric on O I 4 >0 is the opposite of the Cartan-Killing metric on the coset G/G 0 , G 0 being the stabilizer of Q, namely its positive and negative eigenvalues correspond to the non-compact and compact generators in the coset space, respectively.
In N = 2 (symmetric) theories, two G-orbits are defined by the constraint I 4 > 0 : the ( 1 2 -)BPS orbit, and the non-BPS Z H = 0 orbit [39]. Let us consider for instance the N = 2 exceptional magic theory [35] (G = E 7(−25) , R Q = 56), for which one can define the two pseudo-Riemannian 56-dimensional rigid special Kähler manifolds:  , R Q = 56) there is only one G-orbit defined by the constraint I 4 > 0, namely the 1 8 -BPS "large" orbit, which thus allows to define the pseudo-Riemannian 56-dimensional rigid special Kähler manifold [15]:  This can be proved by using where I 4 = ǫ |I 4 |, which generalizes (4.13) for any sign of I 4 . Note that properties (4.14) and (4.27) can be summarized, for any sign of I 4 , as follows: Interestingly, the two manifolds share the same signature. As opposite to M − , the adjoint action of M + defines, just as in the I 4 > 0 case, an automorphism of the U-duality group G: whereR Q denotes the 2n × 2n matrix representation of G in R Q . Since for I 4 > 0 M + thus belongs to the inner -automorphisms Inn(G) ⊂ Aut(G), whereas for I 4 < 0 the anti-symplecticity of M + implies that it belongs to the outer -automorphisms Aut(G)/Inn(G) 6 , see Appendix C, because the matrix realizationR Q of the elements of G in the representation R Q is symplectic: The same generally does not hold for the adjoint action of M − : We can define the matrix S + := CM + , which is still in Aut(G), since M + is. Moreover . We can then use (4.29) and write: from which we can easily derive the following property: 37) or, equivalently: Finally it can be easily shown from their definition in both I 4 > 0 and I 4 < 0 cases, that

Interpretation of M ± in N = 2 Theories
In the vector multiplet sector of an N = 2 supergravity, we can define two symmetric, symplectic matrices: one is the matrix M constructed out of the real and imaginary parts of N ΛΣ , as in (2.2), the other is a matrix M (F ) defined by having the same matrix form as in (2.2), but in terms of the real and imaginary parts of the complex n × n matrix F (X) being the holomorphic prepotential, homogeneous function of degree 2 of X Λ (z) (we use the notations of [16]). We can write then: where M[R, I] is the function of the matrices R, I defined in (2.2). As anticipated in the introduction, can write the matrix M(z,z) in the manifestly symplectic-covariant form [5,17] Note that the r.h.s. is the sum of two symmetric matrices: which satisfy the condition A 1 CA 2 = 0, which follow from the general properties: V T CU i = V T CU i = 0. Therefore, if M = A 1 + A 2 is symmetric and symplectic, also A 1 − A 2 is. The latter is just the matrix M (F ) : The relation between the two matrices being then 7 : which is consistent with the relation between the lower diagonal blocks of the two matrices given e.g. in [18]: where I n−1 denotes the (n − 1) × (n − 1) identity matrix.
Let us now evaluate relation (4.46) at the horizon of a regular BPS black hole (thus, with I 4 > 0) and show that it yields the relation between M ± , proving thus that, if M + coincides with the matrix M H , M − coincides with M (F ) at the horizon. To this end, we use the relations [5]: which hold at the horizon of the solution. Using the property that, at the horizon, |Z| 2 horizon = √ I 4 , we end up with which is the same relation holding between M + and M − . Indeed, from (6.6) and (6.4), it follows that 54) which for I 4 > 0 reduces to the same relation (4.53).

T 3 Model with I 4 < 0
Among the symmetric models, this is the unique case in which the non-BPS I 4 < 0 attractors do not exhibit any flat direction [39,25], and thus it deserves a separate treatment.
In this case, however, we can write where M + (Q) is the anti-symplectic matrix given by (4.25), and A 0 (Q) is defined as:  where Stab Q [GL(4, R)] denotes the stabilizer of Q in GL(4, R).
Since, as anticipated in Sec. 4.2 (see footnote 6) and as discussed in Sect. 6, the adjoint action of M + defines an outer -automorphism of G, and since M H is an element of G, it follows that also the adjoint action of A 0 (Q) defines an outer-automorphism of G (Out(G) := Aut(G)/Inn(G); cfr. App. C): 4.5 N = 2 minimally coupled and N = 3 As mentioned above, the cases of 1/2-BPS attractors in N = 2 minimally coupled supergravity, non-BPS attractors in T 2 model, 1/3-BPS attractors in N = 3 "pure" supergravity, and non-BPS attractors in the same theory coupled to 1 vector multiplet, deserve a separate treatment.
Besides not exhibiting any flat direction, they indeed share the peculiar property that the Ktensor is non-primitive : namely, it can be expressed in terms of a rank-2 symmetric invariant tensor S M N of the corresponding U-duality group.
The aforementioned cases correspond to particular cases (r = 1 and r = 3, respectively) of the class of pseudo-unitary duality group U(r, n) with symplectic black hole charge representation given by the (complex) fundamental representation r + n [33,40].
After [33], the following definition and relations holds 8 : Indeed, U(r, n) has an Hermitian invariant rank-2 tensor in its (complex) fundamental representation r + n, whose real and imaginary part defines S M N (symmetric) and the symplectic metric C M N (antisymmetric) (cfr. e.g. (2.33) of [33]). The quadratic invariant I 2 occurs in the Bekenstein-Hawking formula (2.31) for extremal black holes in the corresponding (super)gravity theories: As discussed in Sec. 10 of [33], in these cases the horizon/"critical" limit F H of the "noncritical" Freudenthal duality F (2.11) is nothing but a particular anti-involutive symplectic transformation of U(r, n).
By exploiting (4.60)-(4.64), the result (4.12) simplifies as follows: where ̺ := ζ/ |ζ|; indeed, let us remark that the result (4.66) holds for any sign of I 2 , and also that the sign of the real normalization constant ζ occurring in the definition (4.60) is not fixed (as discussed in [33], only ζ 2 is fixed).

M H with Flat Directions
In presence of flat directions ϕ f lat [25] (namely, in all -symmetric -cases not treated in the analysis of Sec. 4), the matrix M H is generally not G-covariant, i.e. it cannot be computed only in terms of purely Q-dependent covariant quantities. Thus, the procedure outlined in Sec. 4 cannot be applied. However, in the following treatment we will determine the most general expression of M H also in presence of flat directions. These generically occur in the case I 4 < 0 (genuine non-BPS attractors), to be dealt with in Subsec. 5.1, and in the case I 4 > 0 for non-BP S solutions and BP S ones in theories with N > 2, to be discussed in Subsec. 5.2. At least for symmetric d-geometries (or when U-orbit structure is present), by choosing a representative Q = (p 0 , q 0 ) belonging to the genuine non-BPS orbit (corresponding to I 4 (Q) < 0 in symmetric models), one obtains that

5.1
where a := aE −1 . Thus, up to suitable redefinitions of the axions (a → a := aE −1 ), L −1 Q is independent of flat directions (spanning the scalar manifold of the theory uplifted to D = 5), also off-shell [24]. Therefore, by recalling (2.3) and observing that G −1 (E) = G(E −1 ), one obtains the following off-shell decomposition: with E denoting the D = 5 kinetic vector matrix. Note that off-shell it generally holds that At the I 4 < 0 critical points (i.e., on-shell at genuinely non-BPS critical points of V BH ), in the (p 0 , q 0 ) charge configuration the axions can be set to zero without any loss of generality (a H = 0, a H = 0) [27], and therefore is due to the fact that M 1 depends on EE T only through a's. Thus, in contrast with (5.7), on-shell it generally holds that While in symmetric models the result above can be extended by duality all along the genuine non-BPS U-orbit (I 4 < 0), it should be stressed that for non-symmetric d-geometries, the result (5.8)-(5.9) strictly holds for the (p 0 , q 0 ) charge configuration. Indeed, for any d-geometry (see e.g. [27,37] In symmetric models, all these genuine non-BPS solutions (or suitable truncations thereof) can be uplifted to non-BPS I 4 < 0 solutions in N = 8, D = 4 supergravity, and a prototypical solution is given e.g. by the Kaluza-Klein black hole, as discussed in [41]. By using (5.11) and recalling the block-diagonal form of the generator D of the SO(1, 1) Kaluza-Klein dilatation (I here denoting the n V × n V identity): the expression of M H 1 can be recast as follows: By recalling (1.2), the relations above entail the dependence of the black hole effective potential V BH , as well as of related quantities, on the various kinds of D = 4 scalar fields: where (5.3) was used; this shows that, up to suitable redefinitions of the axions (a → a := aE −1 ), V BH is independent of flat directions, also off-shell [24] (and the same holds for the ADM mass, which can be computed as the asymptotic (radial) limit of the first order superpotential W [42]).
On the other hand, as anticipated in (2.46), the scalar-dependent, "non-critical" Freudenthal duality (2.11) does depend on flat directions off-shell : but, as anticipated in (2.47), it does not when evaluated on-shell : In order to determine the most general expression for M H I 4 <0 , one needs to generalize the expression M H 1 (5.9) to a generic representative of the I 4 < 0 U-orbit, in which all components of the charge vector Q are non-vanishing. This can be achieved by applying the most general set of axionic translations to M H 1 (5.9); such translations are a "universal" sector of the D = 4 electric-magnetic symmetry, common to all d-geometries (recently studied in [37]): where T I and T I denote the generators of the contravariant (namely, axionic) and covariant translations (with α I and α I denoting the corresponding parameters; T I T = T I ). After the treatment e.g. of [43,37], the explicit form of the finite "universal" translations g (5.17) can be computed to read where (5.20) In order to determine the expression of M H 1 when all charges are non-vanishing (but still constrained by I 4 < 0), the system:  must be solved, constrained by 10 − p 0 q 0 2 = I 4 (P 0 , P I , Q 0 , Q 0 ) < 0; (5.22) The solution of the system (5.21)-(5.23) is provided by the parameters α I , α I , as well as the charges p 0 and q 0 , expressed in terms of the charges P 0 , P I , Q 0 , Q I , and it can be computed to read 27) where no sum on repeated indices is assumed in the numerator of the r.h.s. of (5.26). It is amusing to observe that, in the "wrong" assumption I 4 > 0, (5.26) α I becomes complex and matches the attractor value of the D = 4 scalar fields at BPS critical points (cfr. the discussion in Subsec. 5.2). Therefore, by plugging (5.24)-(5.27) back into (5.18), one obtains the general expression M H 1,gen supported by a "generic" representative of the non-BPS I 4 < 0 U-orbit with all charges non-vanishing: As mentioned, M H 1,gen P 0 , P I , Q 0 , Q I is not covariant with respect to the D = 4 U-duality group G 4 , but its covariance is broken down to the non-compact stabilizer of the non-BPS U-orbit, which, in the symmetric N = 2 d-geometries, is nothing but the D = 5 U-duality group G 5 : Thus, by recalling (5.8) and (5.6), the general expression of M H I 4 <0 supported by a "generic" representative of the non-BPS I 4 < 0 U-orbit with all charges non-vanishing is given by: with M 0 and g respectively defined by (5.6) and (5.17).

I 4 > 0
At non-BPS I 4 > 0 and BPS critical points of V BH in symmetric d-geometries (in which flat directions generally occur), the situation is even simpler (in this section we only consider BPS attractors in N > 2 theories in which they exhibit flat directions). Indeed, all such solutions (or suitable truncations thereof) can be uplifted to ("large") 1 8 -BPS I 4 > 0 solutions in N = 8, D = 4 supergravity, and the corresponding prototypical solution is given by the Reissner-Nördstrom (RN) black hole, discussed e.g. in [41], which is consistent with an attractor solution in which all scalar fields (but not necessarily the flat directions) vanish.
Let us recall that the (non-BPS and BPS) I 4 > 0 U-orbits are generically described by a non-symmetric coset of the form In this case, the general expression of M H I 4 >0 (supported by a "generic" representative of the I 4 > 0 non-BPS or BPS U-orbit with all charges non-vanishing) is formally obtained by the same procedure described in Subsec. 5.1, with the important difference that in this case the KK SO(1, 1) is replaced by a compact U(1) symmetry, and different non-compact real forms do occur with respect to the previous treatment. The "translations" are now U(1)-charged, and indeed the corresponding parameters are complex (α I =α I ): whereα I andα I solves the system constrained by p 2 + q 2 2 = I 4 (P Λ , Q Λ ) > 0, (5.38) in which it should be remarked that d IJK and d IJK are the invariant tensors of G 0 , and not of G 5 (as instead they were in (5.23)). The solutionα I =α I P Λ , Q Λ of the system (5.37)-(5.38) is formally given by (5.26) with α I →α I , which indeed yields complexα I 's for I 4 > 0, as observed below (5.27) : in other words, amusingly,α I P Λ , Q Λ as well asα I P Λ , Q Λ =α I P Λ , Q Λ ) are nothing but the attracted values of the scalar fields (and of their complex conjugates) at

Summary of Results and General Properties
We have constructed two symmetric real matrices M ± (Q) satisfying the conditions (1.5): where I 4 =: ǫ |I 4 |. These matrices also satisfy relations (4.39) : The matrix which is never negative definite, enjoys an interpretation as symplectic metric of the corresponding G-orbit of Q (see above as well as the final part of Sec. 4.1). Moreover it does not belong to Aut(G). On the other hand, the matrix Let us illustrate some properties of A; as it follows from from Eq. (6.3), A(Q, ϕ f lat ) is in the stabilizer of Q in GL(2n, R). Moreover, since M + ∈ Aut(G) and M H ∈ G ⊂ Aut(G), and both are invariant under H 0 (denoting the stabilizer of ϕ f lat ), also A is, and thus we can write (4.59): An important property of A is the following: which follows from (6.9), but can be alternatively be proven using Eq.s (6.7), (2.13), (6.1), and (4.29): From this, it also follows that A is involutive: Note that a property analogous to (6.11) holds for M − : as it can be shown along the same lines as in (6.11) and using property (4.38). If I 4 < 0, M + (Q) is anti-symplectic, and thus (6.7) yields that A is anti-symplectic as well. Therefore, as M + (Q), it defines an outer -automorphism of G (see Appendix C for a discussion on anti-symplectic outer-automorphisms of the U-duality algebra), and one can write (Q ∈ O I 4 <0 (5.29); H 0 = H 5 ): M + (Q) ∈ Out(G); (6.14) where M 0 (ϕ f lat ) ∈ G 0 /H 0 was defined in Sec. 5.1, and A 0 (Q) is a purely charge-dependent antisymplectic outer-automorphism of G. By recalling the on-shell decomposition (5.8) discussed in Sec. 5.1: Then, one can generalize to a generic Q ∈ O I 4 <0 (5.29) by following the procedure outlined in Sec. 5.1.
Note that, at least in those cases 11 in which Out(G) ⊂ Z 2 , (6.19) which seems to be common for groups "of type E 7 " (including for instance E 7(7) itself) [21], all non-trivial outer-automorphisms are implemented by an anti-symplectic transformation. If I 4 > 0, M + (Q) (cfr. (6.1)) is symplectic, and thus (6.7) yields that A is symplectic as well. Therefore, as M + (Q), it defines an inner -automorphism of G, and one can write (with Q belonging to regular G-orbits with I 4 > 0; H 0 = mcs (G) /U(1) in the BPS case, while H 0 = H 0 in the non-BPS case (5.31)): (6.20) where M 0 (ϕ f lat ) was defined in Sec. 5.2, and here ϕ f lat denotes the flat directions at I 4 > 0 (generally different from the flat directions at I 4 < 0, considered above; cfr. the procedure outlined in Sec. 5.2), and A 0 (Q) is a purely charge-dependent symplectic inner-automorphism of G.
In absence of flat directions ϕ f lat (such as for N = 2 regular BPS orbit), namely in those cases considered in Sec. 4, G 0 = H 0 , M 0 (ϕ f lat ) = I and property (6.9) implies which is consistent with the identification M H = M + made in Sect. 4 (cfr. (4.12)). Let us conclude with a few comments. A special role in our discussion has been played by outer-automorphisms of the U-duality algebra which are implemented by anti-symplectic transformations. These should correspond, modulo U-dualities, to a discrete symmetry of ungauged extended supergravities, see Appendix C, which deserves a separate discussion [45].
Finally it would be interesting to extend our analysis to "small orbits" of R Q , for which I 4 = 0. A Computing the Coefficients A, B and C We will here report the derivation of result (4.11), which can actually be obtained in (at least) two equivalent ways.
A.1 With the Invariant Tensor S αβ M Q ...
We start from the condition (4.3), which can be easily recast as On the other hand, the implementation of the symplectic condition (4.2) requires some further manipulations. By exploiting (4.8), one can rewrite (4.2) as where the result (obtained by explicit computation) was used. The skew-trace of (A.2) yields to (recall n = n V + 1) where the result has been taken into account.
Since the l.h.s. of Eq. (A.2) is skew-symmetric, the only way to obtain from (A.2) a further constraint (not proportional to the skew-trace condition (A.4)) on the real coefficients A, B and C is to single out the terms not proportional to the symplectic metric C M Q itself. Group theoretical arguments (cfr. e.g. App. C of [28]) lead to the following decomposition: and the result has been used. Using the irreducible decomposition (where A is a constant to be determined), one can prove that the three terms in the r.h.s. of (A.6) are not independent. In fact, the following relation holds: thus implying (A.6) to reduce to Therefore, the finite symplecticity condition (A.2) for M H can be rewritten as follows: It is clear that t α|(A 1 A 2 S αβ M )(Q t β|A 3 A 4 ) contains t α|A 1 A 2 S αβ M Q t β|A 3 A 4 which, due to (A.7), is orthogonal to (and thus independent of) the symplectic metric C M Q . Thus, the related coefficient has to be set to zero. This argument leads to the following independent conditions: In these relations, the real constant A introduced in the decomposition (A.9) has been set to The result (A.15) can be achieved by noticing that, using (A.9), the following equation holds: Thus, in order to study its definiteness, it suffices to analyze the signs of its diagonal elements.
In the ST U truncation under consideration, it can be explicitly computed that the first diagonal element is strictly positive (I 4 = q 0 p 1 p 2 p 3 > 0): thus implying that M −|M N is not negative definite.
On the other hand, it can be calculated that M + (Q), given by (4.9)-(4.10) and (4.11) in the branch "+", is diagonal, with all strictly negative elements, and thus trivially negative definite. C Outer (Anti-symplectic) Automorphisms of g In symmetric extended D = 4 supergravities, the U-duality algebra g admit an automorphism implemented, in the representation R Q , by an anti-symplectic transformation. Consider the symplectic frame in which the subgroup H SO(n) ⊂ Sp(2n, R) has a block-diagonal representation. Such frame is obtained through a Cayley transformation of the complex basis in which the whole H is block-diagonal. In this frame the conjugation by the anti-symplectic matrix: O = I n 0 n 0 n −I n , (C.1) defines an automorphism: For instance, in the maximal theory, such transformation switches the sign of the generators in the 35 c (parametrized by the pseudo-scalars) and 35 s (compact generators in su(8) ⊖ so (8)), leaving the other generators unaltered [44]. Since all G transformations in R Q are implemented by symplectic matrices, O is not in G and defines a non-trivial outer automorphism of g: 1314 O ∈ Aut(G) Inn(G) = Out(G) .

(C.3)
We can give an alternative representation to O, for those supergravities admitting a D = 5 uplift, in the symplectic frame originating from the D = 5 → D = 4 reduction. In this frame the generators t α of g have a characteristic matrix form given in [37], defined by branching the D = 4 duality algebra with respect to O(1, 1) × G 5 , G 5 being the global symmetry group of the D = 5 parent theory. The algebra g decomposes accordingly:  13 Strictly speaking, to show that O is an outer-automorphism, one should prove that no other element of G can induce the same transformation on g. This is immediate if R Q is irreducible since any other real matrix inducing the same transformation, must be proportional to O, and thus non-symplectic. Inspection of supergravities in which R Q is reducible, however, leads to the same conclusion: No element of G can induce the same automorphism as O.
14 The simplest example of a real Lie group admitting a symplectic representation, in which an outer automorphism is implemented by an anti-symplectic transformation, is SL(2, R). The fundamental representation 2 is symplectic and the anti-symplectic matrix σ 3 = diag(+1, −1) implements an outer-automorphism.