Black holes and general Freudenthal transformations

We study General Freudenthal Transformations (GFT) on black hole solutions in Einstein-Maxwell-Scalar (super)gravity theories with global symmetry of type $E_7$. GFT can be considered as a 2-parameter, $a, b\in {\mathbb R}$, generalisation of Freudenthal duality: $x\mapsto x_F= a x+b\tilde{x}$, where $x$ is the vector of the electromagnetic charges, an element of a Freudenthal triple system (FTS), carried by a large black hole and $ \tilde{x}$ is its Freudenthal dual. These transformations leave the Bekenstein-Hawking entropy invariant up to a scalar factor given by $a^2\pm b^2$. For any $x$ there exists a one parameter subset of GFT that leave the entropy invariant, $a^2\pm b^2=1$, defining the subgroup of Freudenthal rotations. The Freudenthal plane defined by span$_\mathbb{R}\{x, \tilde{x}\}$ is closed under GFT and is foliated by the orbits of the Freudenthal rotations. Having introduced the basic definitions and presented their properties in detail, we consider the relation of GFT to the global sysmmetries or U-dualites in the context of supergravity. We consider explicit examples in pure supergravity, axion-dilaton theories and $N=2,D=4$ supergravities obtained from $D=5$ by dimensional reductions associated to (non-degenerate) $ reduced$ FTS's descending from cubic Jordan Algebras.


Introduction
Recent observations consistent with expectations for the shadow of a Kerr black hole (BH) as predicted by general relativity. have been, for the first time, presented [1,2].
From the theoretical side, symmetries, duality transformations and the mathematical structures upon which they are realised, constitute important tools in the study of black hole solutions in general relativity and its supersymmetric extension, supergravity. For instance, the non-compact global symmetries of supergravity theories [3], or U-dualities in the context of M-theory [4,5], have played a particularly crucial role, starting with the work of [6]. For a large class of N ≥ 2 Poincaré supergravity theories with symmetric scalar manifolds 1 the U-duality groups are of "type E 7 ", a class of groups sharing the same algebraic structure as the second largest exceptional Lie group E 7 [8]. Groups of type E 7 are axiomatically characterised by Freudenthal triple systems (FTS) [8][9][10]. An FTS is a vector space F with, in particular, a symmetric four-linear form ∆(x, y, z, w) (see Sec. 2 for full details). The automorphism group Aut(F) of the FTS is the U-duality group G 4 of the associated 4d supergravity. The electromagnetic charges carried by the extremal black hole solutions in such theories correspond to elements x ∈ F and fall into linear representations of the associated U-duality groups. For such theories the leading-order Bekenstein-Hawking black hole entropy is given by S BH = π |∆(x)|, (1.1) where ∆(x) := ∆(x, x, x, x) is the unique U-duality invariant quartic polynomial of the BH charges.
In [11] it was shown that when the U-duality group is of type E 7 [8,12], these black hole solutions enjoy a nonlinear symmetry, named Freudenthal duality, acting on their associated charge vectors x. This holds for instance in all N > 2-extended, D = 4 supergravities, as well as in all N = 2 supergravities coupled to vector multiplets with symmetric scalar manifolds. However, supersymmetry is not a necessary ingredient (e.g. in the case of F(J C s 3 ) and F(J H s 3 ); cfr. Table 1). In [13] Freudenthal duality was then generalized to a symmetry not only of the Bekenstein-Hawking black hole entropy S BH , but also of the critical points of the black hole effective potential V BH : regardless of supersymmetry, such a formulation of Freudenthal duality actually holds for any Maxwell-Einstein system coupled to a non-linear sigma model of scalar fields, in four dimensions.
The role of Freudenthal duality in the structure of extremal black hole solutions was investigated in [14], in the framework of ungauged N = 8, D = 4 maximal supergravity; proving that the most general solution to the supersymmetric stabilization equations can be expressed in terms of the F-dual of a suitably defined real 56-dimensional vector, whose components are real harmonic functions in R 3 transverse space. Then, in [15] Freudenthal duality was also shown to be an onshell symmetry of the effective, one-dimensional action describing the dynamics of scalar fields in the background of a static, spherically symmetric and asymptotically flat black hole in N = 2, D = 4 supergravity.
In [16], it was then shown that the generalised, scalar-dependent Freudenthal duality introduced in [13] actually is a symmetry of the equations of motion of the full theory, and is not restricted to the extremal black hole solutions or their effective action. Remarkably, in [16] Freudenthal duality was also applied to world-sheet actions, such as the Nambu-Goto world-sheet action in any (t, s)-signature spacetime, then allowing for an F-dual formulation of Gaillard-Zumino duality [17][18][19] on the world-sheet.
It is also here worth remarking that, in recent years, groups of type E 7 , Freudenthal triple systems, and Freudenthal duality have also appeared in several indirectly related contexts, such as the relation to minimal coupling of vectors and scalars in cosmology and supergravity [20,21], Freudenthal gauge theory (in which the scalar fields are F-valued) [22], multi-centered BPS black holes [23], conformal isometries [24], Hitchin functionals and entanglement in quantum information theory [25][26][27][28][29] 2 .
Our focus here is on the notion of general Freudenthal transformations (GFT), introduced in [23]. In this work it was shown that F-duality can be generalised to an Abelian group of transformations x → x F = ax + bx. (1. 2) The GFT leave the quartic form invariant up to a scalar factor λ 2 = a 2 ± b 2 , The physical quantities of entropy or horizon area, ADM mass and, for multicenter solutions in some specific models, the inter-center distances scale up, respectively, as S BH → λ 2 S BH , M ADM → λ M ADM , r ab → λ r ab , (1.4) while the scalars on the horizon and at infinity are left invariant. The properties of General Freudenthal Transformations, in particular the invariance properties of the quartic FTS invariant (or the Bekenstein-Hawking entropy in physical terms), can be traced back to the existence and properties of Freudenthal planes in F. This notion first appears in the mathematical literature in [8]. Given an F-dual pair x andx we define the Freudenthal plane F x ⊂ F as the set of all elements y x = ax + bx, a, b ∈ R. (1.5) In the present work we introduce in detail these constructions and develop their applications to black holes in supergravity, as summarised here. An extended treatment of GFT is presented in Sec. 2-Sec. 5. In the following sections these mathematical tools will be applied to the physics of black holes solutions in supergravity. First, in Sec. 6 we will study the entropy properties of N = 2, D = 4 pure supergravity from the point of view of the FTS formalism. This provides an example of a degenerate FTS, where the quartic invariant is a positive definite perfect square. As a consequence the Freudenthal plane in this case coincides with the entire FTS and the GFT are transitive on the space of charges. The Freudenthal rotations correspond precisely to familiar electromagnetic duality. To go beyond electromagnetic duality we consider in Sec. 7 the axiondilaton model, an N = 2, d = 4 supergravity minimally coupled to one vector multiplet, which can be considered a consistent truncation of N = 4 supergravity. Again, this model is degenerate and cannot be uplifted to D = 5. This is reflected in the non-reduced character of the FTS; it is not built from an underlying cubic Jordan algebra. In Sec. 8 we proceed to the analysis of N = 2, D = 4 supergravities admitting a D = 5 origin. The mathematical structure of these models is that of a reduced FTS F ∼ = F(J 3 ) derived from a cubic Jordan Algebra, J 3 . In first place we study the T 3 model, or in Freudenthal terminology F(R).
In Subsec. 8. 4 we study the question of orbit stratification of the ∆ > 0 locus of F(R) and its preservation by GFT. In Sec. 9 we show, in different examples, how the action of GFT, and, in particular, Freudenthal duality can be realised by U-duality, or Aut(F), transformations that are "gauged" in the sense that they depend on the element of F to which they are applied. Finally in Sec. 10 we present some further physical discussion, summary and conclusions. We study the properties of asymptotically small interacting black holes. In the different appendices we present a summary of formulae used throughout the work and further technical details.

Freudenthal triple systems: definitions and properties
In 1954 Freudenthal [9,10] constructed the exceptional Lie group E 7 (of dimension 133) as the automorphism group of a structure based on the smallest, non-trivial E 7 irrepr. 56, in turn related to the exceptional Jordan algebra J O 3 of 3 × 3 Hermitian octonionic matrices (also referred to as the Albert algebra) [34]. Freudenthal's aforementioned construction is often referred to as a Freudenthal triple system (FTS) for reasons that shall become clear shortly.
At the end of 60's, Meyberg [35] and Brown [8] elaborated the axioms on which the, completely symmetric, ternary structure underlying an FTS is based; in fact, the E 7 irrepr. 56 is just an example of a class of modules, characterising certain Lie groups as of groups "of type E 7 ". The role of the FTS's in D = 4 Maxwell-Einstein (super)gravity theories was discovered later [36][37][38] to be related to the representation of the electric-magnetic (dyonic) charges of black hole solutions.
A FTS is axiomatically defined [8] as a finite dimensional vector space F over a field F 4 (not of characteristic 2 or 3), such that: 1. F possesses a non-degenerate antisymmetric bilinear form {x, y}. 4 In the following treatment, we will consider F = R (classical/(super)gravity level). The (quantum/Dirac-Schwinger-Zwanzinger-quantized) case (and further extensions thereof) will be investigated elsewhere. The complex case F = C is relevant for quantum qubit entanglement applications.
2. F possesses a completely symmetric four-linear form ∆(x, y, z, w) which is not identically zero. This quartic linear form induces a ternary product T (x, y, z) defined on F by {T (x, y, z), w} = 2∆(x, y, z, w).
3. For the ternary product T (x, y, z) it is required that 3{T (x, x, y), T (y, y, y)} = 2{x, y}∆(x, y, y, y). (2.1) In our case of interest, the semi-classical supergravity limit, the physical vector of charges x is to be regarded as continuous and the associated FTS is taken to be over R or C.
The automorphism group of an FTS is defined as the set of invertible F-linear transformations preserving the quartic and quadratic forms: as is easily verified [39]. The first of the conditions can be restated as {φ x, x } = 0. The F-linear map ϒ x : F → F defined by is in aut(F). This is a direct consequence of axiom III (Eq. 2.6). In fact note that Eq. 2.1 can be reexpressed as Note that, in particular, The linear map ϒ x was introduced in this Aut(F)-covariant form in [40]. For ∆(x) = 0 we may also define the normalized map Various D = 4 supergravities are listed in Table 1: the semi-simple cases F (R ⊕ Γ 1,n−1 ) and F (R ⊕ Γ 5,n−1 ) correspond to N = 2 resp. 4 Maxwell-Einstein supergravity, while F J A 3 ≡ F A correspond to the so-called N = 2 "magic" Maxwell-Einstein supergravities 8 [36]. Moreover, pertains to maximal N = 8 supergravity, and the simplest reduced FTS is F (R), related to the so-called T 3 model of N = 2, D = 4 supergravity (treated in Subsec. 8.1).
Finally, an FTS is said to be degenerate if its quartic form is proportional, in the global sense, to the square of a quadratic polynomial. Note that FTS on "degenerate" groups of type E 7 (as defined in [21], and Refs. therein) are not reduced and hence cannot be written as F(J 3 ); they correspond to theories which cannot be uplifted to D = 5 dimensions consistently reflecting the lack of an underlying rank-3 Jordan algebra J 3 .
Note that, although the map x →x is not a U-duality, the map ϒ x (or ϒ), for x fixed, is indeed: It follows from (c), that for ∆(x) > 0 the Freudenthal map x →x can be considered as an "xdependent" U-duality.
The Tand F-planes: Definition and general properties For a general element in F, respectively an element x ∈ F (4) , we define the associated Tand F-planes, respectively denoted T x , F x , as the R-linear spans of x, x or x,x. In each case: Naturally the F-plane F x is only defined as long as ∆(x) = 0 (maximal rank elements) meanwhile T x is defined for any x, although it degenerates to a T-line for rank x < 3 elements. If they both exist, T x and F x are the same space. It is advantageous to study the properties of the T-planes, and when needed, to specialise to F-planes. We will follow this strategy in what follows.

Linearity of T-transformations on the T-plane
We first show the linearity of the T-dual on the T-plane: T-planes are closed under T-transformations. For any linear combination, one has, because of the multi-linearity of T , (a, b constants, ∆ x = ∆(x)), Where we have used the properties ( [8], lemma 11.(abcf)): We can see that Eq. 3.4 is equivalent to, or simply summarizes, the relations Eq. 3.5-Eq. 3.7. Using Eq. 3.4 we can compute the map ∆ for any element on the T-plane. After a short explicit computation we have (using 2∆(x) = {x , x}) The sign of ∆ on the T-plane is constant. In any T-plane, there is always an element y ∈ T x such Similarly, we have the following expressions describing the behaviour of the map ϒ on the T-plane: Further mathematical properties of the T-planes are shown in Appendix A and Appendix B.
On the full FTS, for maximal rank elements, one can define an (Aut(F)-invariant) "metric" by the (non quadratic) expression and a "pseudo-norm" by If we fix x, and restrict ourselves to the F x plane we can use the expressions in Subsec. B.4 (see also further properties in Ref. [51]) and connect ||u|| with I 2 (u): andŜ is a linear map given in Appendix B. In particular We arrive to the same conclusions as for the T-plane. For ε = 1, the pseudo-norm (·, ·) (or I 2 (u)) is positive definite and the norm-preserving group is SO(2); thus, the F-plane F x undergoes a "spherical foliation". On the other hand, for ε = −1, the norm is positive semi-definite and the norm-preserving group is SO(1, 1); thus, the F-plane F x undergoes an "hyperboloid-like foliation". While the norm is timelike or null, the vector u can be timelike, spacelike and null according to η(x), the sign of I 2 (x).
As for the T-operation, the F-duality x →x, change the character of the vector. The vectors x,x are "I 2 -orthogonal", by Eq. B.29, (x,x) = I 2 (x,x) = 0.x is timelike (resp. spacelike) if x is spacelike (resp. timelike): x : lightlike ←→x : lightlike,x = ±x, (3.24) x : timelike(spacelike) ←→x : spacelike(timelike). (3.25) It is noted that, although the metric I 2 (x, y) is defined only inside a concrete F-plane, the character null, time or spacelike of a vector is an intrinsic property, as any given element belongs to one and only one F-plane, "its" plane, from the disjointness of the F-planes (see Subsec. B.2). 4 The orthogonal space F ⊥ x and the orthogonal plane F y⊥x In general {x,x} = −2 |∆ (x)| = 0. The bilinear form {·, ·} is non-degenerate on F x by construction, since x is neccesarily of maximal rank (∆(x) = 0). Consequently, for a given x ∈ F, the FTS F may be decomposed as where F x is the 2-dimensional F-plane and F ⊥ x = F/F x is its (dim F F − 2)-dimensional orthogonal complement w.r.t. the bilinear form {·, ·}: Hence, for a given x ∈ F, any element y ∈ F enjoys the decomposition where y x ∈ F x and y ⊥x ∈ F ⊥ x . Note that y x , y ⊥x (also denoted y , y ⊥ if there is no risk of confusion) are uniquely determined by x. The coordinates of y x = ax + bx are uniquely determined by the expressions, Or, in compact notation (with respect a fixed element x), The following properties hold (for the parallel component with respect a fixed x): ε{x, y} 2 + {x, y} 2 2 (4.9) (ay + bỹ) = ay + b(ỹ) (4.10) with η = sgn(ε{x, y} 2 + {x, y} 2 ). In particular, note the distributivity of (second expression). Note, however that Obviously, a similar construction can be performed for the T-plane (see [8], from pg. 89 on, where such a space is used to build a Jordan Algebra for reduced FTSs). This decomposition into "parallel" and "orthogonal" spaces and the further decomposition of the orthogonal space in orthogonal planes (to be defined in the next section) will appear important in what follows.
The F y⊥x plane Let us take an arbitrary reference vector x (of maximal rank for simplicity), a perpendicular vector y ∈ F ⊥ x , we define the space, F y⊥x := span{y, ϒ x (y)}. That is We will show that the "planes" F x and F y⊥x are {, }-orthogonal: For any y ∈ T ⊥ x , we can show that also ϒ x (y) ∈ T ⊥ x . We have indeed (using Equation (11c) in [8] in the first line and axiom 3 in the second line) We show next that, in the same case, successive powers of ϒ x acting on y belong to the orthogonal plane. In fact, ϒ n x (y) is proportional to y or ϒ x (y). We have for example (as for any In the second line we have used the Lemma 1(11e) in [8]. In general for any n, we have, by using induction (for any y ∈ F ⊥ x ), for n ≥ 1, ϒ 2n+1 x (y) = (−1) n ∆(x) n ϒ x (y). Let us remark that F y⊥x is not closed in general under T-transformations, it is not a sub-FTS with the operations inherited from the parent FTS. The plane F y⊥x is however closed under the ϒ x map. For any element belonging to it (u ∈ F y⊥x , u = ay + bϒ x (y)), = −b∆(x)y + aϒ x (y), (4.22) where in the second line we have used the equality expressed by Eq. 4.16 (see also [8]). According to this ϒ x (u) ∈ F ⊥ x .

Behaviour of ∆ on the F y⊥x plane
We are interested in the behaviour of ∆ on the F y⊥x plane. For any u ∈ F y⊥x , u = ay + bϒ x (y), we have, by combining Eq. B.10 with Eq. B.11 Or, in normalized terms   which relates the behaviour of the quartic invariant ∆ on the T x and F y⊥x planes. The behaviour of ∆ on the F x or F y⊥x planes is similar but with some important differences. In the F y⊥x case it depends on the signs of both ∆(x) and ∆(y). The overall sign of all the elements of the F y⊥x plane is the same as ∆(y) excluding the null elements such that (4.29) For example, any element of the form . We observe that the null elements of F x and any F y⊥x are aligned, they are given by the same Eq. 4.29 which it is independent of y.

Freudenthal rotations: The ϒ map and its exponential
The closure of F x under F-duality implies the existence of a one-parameter family of U-duality transformations stabilising F x as it will be shown in this section.
Recall, for any fixed x, ϒ x is in aut(F). In particular, the normalised version, ϒ x , given in Eq. 2.7, evaluated in x itself maps x into its F-dual, Note, we also have the relation (ε = sgn ∆(x)) then It is obvious that ϒ x (as well as ϒ ax+bx ) is a F x → F x map. Furthermore, the set of maps {ϒ ax+bx } a,b∈R for a fixed x forms an two-parametric automorphism subalgebra.
We are interested here in the action of the ϒ x map and the computation of its exponential. For this purpose, it is convenient to distinguish the action of any ϒ x on its particular associated F x plane and on the respective orthogonal complement F ⊥

The exponential map on the F x -plane
The action of the exponential of the (normalized) map ϒ x reads as follows. For any rank-4, fixed, where x ∈ F and exp is defined by the usual infinite series. The proof of Eq. 5.6 -Eq. 5.9 is based in the following properties: ϒ 2n+1 x (x) = εϒ 2n+1 x (x), (5.13) which are obtained by induction starting on with ϒ Hence, the set of transformations exp θ ϒ x belong to the automorphism subgroup Aut(F x ) ⊆ Aut(F) preserving the F x plane.
To summarise, as a consequence of Eq. 5.6 and Eq. 5.7, for any rank-4 x ∈ F, there exists a monoparametric subgroup σ (θ ) ∈ Aut(F x ) which is made of "rotations" in F x and whose generator is ϒ x : Let us study the details of the automorphism subgroup {σ (θ )} x depending on the sign of (2). The Freudenthal rotation with θ = π/2 is the U-duality transformation relating x to its F-dual. For N = 8 black holes with G 4 = E 7(7) the existence of a U-duality connecting x andx was guaranteed since all x with the same ∆(x) > 0 belong to the same E 7(7) orbit. For N < 8 not all x with the same ∆(x) > 0 necessarily lie in the same U-duality orbit; the orbits are split by further U-duality invariant conditions. See [39,52] and the references therein. Nonetheless, for ε = 1 the Freudenthal rotation given by Eq. 5.6 with θ = π/2 implies that x andx are in the same U-duality orbits for all FTS.
On the other hand, for ∆(x) < 0, (ε = −1) the subgroup σ (θ ) is SO(1, 1) which has three different kinds of orbits: the origin (a group fixed point), the four rays {(±t, ±t),t > 0}, and the hyperbolas such as a 2 − b 2 = 1. The Freudenthal rotation cannot relate x to its F-dual (by inspection of Eq. 5.7, the orbits of the exponential of the ϒ are hyperbolic). Therefore x,x lie in different branches. However, for any FTS, all x with the same ∆(x) < 0 lie in the same U-duality orbit [39,52]. Hence, there exists a U-duality transformation, which is determined by x, connecting x andx for ∆(x) < 0. But, this U-duality transformation is not represented by any member of σ (θ ). In fact, as we shall see there is a one-parameter family of U-dualities which connects x andx but does not preserve the F x -plane. We will return to this question in the next sections.
In summary, putting together the previous comments, we arrive to the conclusion that a) For all supergravities with any N, with E 7 -type duality groups, large Black Holes with charges x andx are in the same U-duality orbits, irrespective of the sign of ∆(x).
b) For ∆(x) > 0 the orbit of σ (θ ), which relates the F-dual BHs, is contained in the F x -plane. For ∆(x) < 0, the orbit of of the one-parameter subgroup, introduced later, connecting x and x leaves the F x -plane. It is might be "natural" to conjecture that this orbit only intersects the F x -plane only at x andx. We will come back to this point later on.
Note, a similar treatment can be performed for the case of small BHs, ∆(x) = 0. In this case the ϒ map and its exponential, the duality orbits, which corresponds to the null rays.
6 Pure N = 2, D = 4 supergravity and degenerate FTS The simplest example of a FTS (which is, being two-dimensional, a Freudenthal plane with ∆ (x) > 0) in supergravity is provided by the one associated to "pure" N = 2, D = 4 supergravity, whose purely bosonic sector is the simplest (scalarless) instance of Maxwell-Einstein gravity. In such a theory, the asymptotically flat, spherically symmetric, dyonic extremal Reissner-Nordström (RN) black hole (BH) solution has Bekenstein-Hawking entropy where p and q are the magnetic resp. electric fluxes associated to the unique Abelian vector field (which, in the N = 2 supersymmetric interpretation, is the so-called graviphoton). In this case, the associated FTS F N =2 "pure" has dim= 2 (i.e., it has N = 1, within the previous treatment); it is immediate to realize that this cannot be a reduced FTS, because 10 defining then the associated quartic invariant ∆(x) is defined by for any choice of p and q. This system can be considered a BPS (∆(x) > 0) prototype. Let us start by doing some, simple, explicit computations. For this purpose let us choose (without any loss of generality) a vector given by which corresponds to a purely magnetic extremal RN BH. For this configuration, The Freudenthal dual x of x can be computed by using and Eq. 6.3, to read A purely electric extremal RN BH is nothing else as the Freudenthal dual of purely magnetic extremal RN BH. The whole FTS F N =2 "pure" coincides with the Freudenthal plane F x associated to x : The transverse space is obviously empty A general Freudenthal transformation (GFT) depending on the real parameters a, b is given by or, in this case 14) The corresponding extremal RN BH is supersymmetric and 1 2 -BPS (in absence of scalar fields, supersymmety implies extremality). For a 2 + b 2 = 1, the general Freudenthal transformation leaves invariant the entropy of the black hole. In this context a GFT is nothing else as an instance of EM duality. Automorphism algebra and group element as ϒ and exp(θ ϒ) can be explicitly and easily computed.
"pure" provides the simplest case of degenerate FTS , in which 2∆ is the square of a quadratic polynomial I 2 : and thus it is always positive. In fact, "pure" N = 2, D = 4 supergravity is the n V = 0 limit of the sequence of N = 2, D = 4 supergravity "minimally coupled" to n V vector multiplets 11 [54] (see also [55,56]), in which the related FTS is degenerate ∀n V ∈ N ∪ {0}; the corresponding scalar manifold is CP n V .
In the formalism discussed in Sec. 3, in F N =2 "pure" = F x it holds that (I, J = 1, 2) The Euclidean nature of the metric structure defined on F N =2 "pure" = F x corresponds to a spherical foliation of F x for ∆ > 0. Degenerate FTS's never satisfy the reducibility condition [8], namely they are globally nonreduced; they have been treated e.g. in [57], and their application in supergravity has been discussed in [21] (see also and [56]). Other (infinite) examples of degenerate FTS's are provided by the ones related to the n-parameterized sequence of N = 3, D = 4 supergravity coupled to n matter (vector) multiplets [21,55,58]. On the other hand, N = 4 [59]and N = 5 12 , D = 4 "pure" supergravities have FTS's which do not satisfy the degeneracy condition (Eq. 6.15) in all symplectic frames, but rather (Eq. 6.15) is satisfied at least in the so-called "scalar-dressed" symplectic frame [55].
This FTS cannot be associated to any Jordan Algebra. Consistently, "pure" N = 2, D = 4 supergravity does not admit an uplift to D = 5, or conversely it cannot be obtained by dimensionally reducing any D = 5 theory down to D = 4. In general, degenerate FTS's are not built starting from rank-3 Jordan algebras, and therefore the corresponding Maxwell-Einstein (super)gravity models do not admit an uplift to D = 5; rather, degenerate FTS's are based on Hermitian (Jordan) triple systems (cfr. e.g. [37,57], and Refs. therein).
As discussed in Sec. 10 of [21], at least for the degenerate FTS's relevant to D = 4 supergravities with symmetric scalar manifold (i.e., N = 2 "minimally coupled" and N = 3 theories 13 ), Freudenthal duality is nothing but an anti-involutive U-duality mapping. This can be realized immediately in the aforementioned case of N = 2, D = 4 "pure" supergravity; let us consider (a = b = 1) The Freudenthal dual y of y can be computed (by recalling (Eq. 8.12) an using (Eq. 6.3)) to read where Ω 0 is nothing but the canonical symplectic 2 × 2 metric Ω 2×2 : Thus, Freudenthal duality in F N =2 "pure" is given by the application of the symplectic metric Ω ≡ Ω 0 , and it is thus an anti-involutive U-duality transformation. The relation (Eq. 6.18) defines a Z 4 symmetry in the 2-dim. FTS F N =2 "pure" = F x , spanned by x (Eq. 6.6) and its Freudenthal dual x (Eq. 6.8) : in fact, the iteration of Freudenthal duality yields This provides the realization of the Z 4 in the FTS F N =2 "pure" = F x , as a consequence of the antiinvolutivity of Freudenthal duality itself. The same symmetry will be also explicitly observed, for example, for the Freudenthal plane defined by the D0 − D6 brane charge configuration in reduced FTS's, to be studied in latter sections.
7 The axion-dilaton N = 2, D = 4 supergravity Let us consider now N = 2, D = 4 supergravity "minimally coupled" to one vector multiplet, in the so-called axion-dilaton (denoted by the subscript "ad") symplectic frame. Ultimately, this is nothing but the n V = 1 element of the sequence of CP n V "minimally coupled" models [54] , but in a particular symplectic frame, which can be obtained as a consistent truncation of "pure" N = 4 supergravity, in which only two of the six graviphoton survive (in this frame, the holomorphic prepotential reads F(X) = −iX 0 X 1 ; cfr. e.g. the discussion in [63], and Refs. therein).
The purely bosonic sector of such an N = 2 theory may be regarded as the simplest instance of Maxwell-Einstein gravity coupled to one complex scalar field. In the axion-dilaton symplectic frame, in the particular charge configuration obtained by setting to zero two charges out of four and thus having only two non-vanishing charges 14 , namely one magnetic and one electric charge p resp. q, the asymptotically flat, spherically symmetric, dyonic extremal BH solution has Bekenstein-Hawking entropy S ad π = |pq| , (7.1) and it is non-supersymmetric 15 (non-BPS). The expression (Eq. 7.1) is very reminiscent of the Bekenstein-Hawking entropy of a BH in a reduced FTS in the D0 − D6 charge configuration (to be treated later on, we refer to (Eq. 8.4)-(Eq. 8.5), I 4 ≡ ∆(x)): However, the N = 2 axion-dilaton supergravity model, as the "pure" N = 4, D = 4 supergravity from which it derives, cannot be uplifted to D = 5 (as instead all models related to reduced FTS's can), consistently with its "minimally coupled" nature : in fact, the charges P and Q do not have the interpretation of the magnetic resp. electric charge of the KK vector in the D = 5 → 4 dimensional reduction.
This truncated system can be described by a two dimensional FTS characterized by a quartic form (x ≡ (p, q) T ) One can recast this expression by defining in the following form S ad Let us start by choosing, without any loss of generality, a charge configuration given by The corresponding entropy is given by 14 In this case, the effective FTS F N =2 ad given by the truncation has dimension 2. 15 Indeed, in presence of scalar fields (in this context stabilized at the event horizon of the BH by virtue of the attractor mechanism), extremality does not imply BPS nature, and extremal non-BPS solutions may exist.
By virtue of Eq. 6.7, one can compute the Freudenthal dual x of x to read (ε ≡ sgn P 2 − Q 2 Thus, one can define a GFT transformations and the 2-dim. Freudenthal plane F x associated to x inside the whole 4-dim. FTS F N =2 ad : The corresponding extremal BH is non-supersymmetric (non-BPS). In particular for the Eq. 7.6 configuration The entropy is invariant for Therefore, notwithstanding the fact that N = 2, D = 4 axion-dilaton supergravity is nothing but the CP 1 "minimally coupled" model in a particular (non-Fubini-Study) symplectic frame and thus with (Eq. 6.15) holding true, in the peculiar (P, Q) charge configuration (Eq. 7.4), the corresponding F x ⊂ F N =2 ad can be considered as a "degenerate" limit of the ∆ < 0 prototype of Freudenthal plane for reduced FTS's. It is instructive to consider the explicit action of the Freudenthal duality in the Freudenthal plane F x ⊂ F N =2 ad . Let us start and consider (a = b = 1; we disregard the coordinates in The Freudenthal dual y of y can be computed (by recalling (Eq. 6.7)) to read 16 y = ε(Q, P) T = εÔy, (7.14) withÔ 16 By virtue of the discussion made at the end of th previous Subsection (also cfr. Sec. 10 of [21]), O (Eq. 7.15) can be completed to a 4 × 4 (consistently anti-involutive; cfr. discussion further below) transformation of the U-duality group U(1, 1).
Note thatÔ (Eq. 7.15) is involutive:Ô 2 = Id, (7.16) but since the Freudenthal duality on F x exchanges P and Q and thus flips ε(= sgn P 2 − Q 2 ), it follows that the correct iteration of the Freudenthal duality on F x ⊂ F N =2 ad is provided by the application of εÔ and then necessarily of −εÔ, thus corresponding to −Ô 2 = −Id acting on x, and thus correctly yielding As at the end of previous Subsection for "pure" N = 2, D = 4 supergravity, in this case due to the relations (Eq. 7.14)-(Eq. 7.15), we can define a Z 4 symmetry in the 2-dim. Freudenthal plane F x ⊂ F N =2 ad , spanned by x (Eq. 7.7) and its Freudenthal dual x (Eq. 7.8) : e.g., starting from ε = 1, the iteration of Freudenthal duality yields This provides the realization of the Z 4 in the Freudenthal plane F x ⊂ F N =2 ad , as a consequence of the anti-involutivity of Freudenthal duality itself.
We will now proceed to present an analysis of the (non-degenerate) reduced FTS's, of the properties of Freudenthal duality defined in them, and of the corresponding Freudenthal planes. Unless otherwise noted, we will essentially confine ourselves at least to (non-degenerate) reduced FTS's F = F(J 3 ), for which a 4D/5D special coordinates' symplectic frame can be defined.
A generic element x of the reduced FTS F splits as where the second renaming pertains to the identification of x with a dyonic charge configuration in D = 4 (super)gravity, where p's and q's are magnetic and electric charges, respectively; within the standard convention in supergravity, p 0 , p i , q 0 and q i will usually be called D6, D4, D2, D0 (brane) charges, respectively. The the symplectic product of two generic elements x and y in F reads where Ω is a simplectic matrix. We use a basis such that the 2N × 2N symplectic metric is realized as follows: where 0 and 1 denote the N × N zero and identity matrices, respectively. At least within (non-degenerate) reduced FTS's, the quartic polynomial invariant can be written 17 as follows 18 (cfr. e.g. [38,65,66] The symmetric quantities d i jk , d i jk follow the so-called adjoint identity of the Jordan algebra J 3 underlying the reduced FTS F (cfr. e.g. [40,66,67] and Refs. therein), which reads where the capital Latin indices span the entire FTS F, and K MNPQ = K (MNPQ) is the rank-4 completely symmetric tensor characterizing F [8,21,61]. Note that, from its very definition (Eq. 8.10), T (x, y, z) is completely symmetric in all its arguments [8].
Then, by using Ω to raise the symplectic indices, one can compute Let us now recall the definition (Eq. 3.1) of Freudenthal duality ∼ for groups of type E 7 [11,13], and make the symplectic indices explicit 19 : where we recall that ε ≡ sgn(I 4 ). By direct computation, one gets In Appendix D we present some explicit expressions for the triple product and other maps.

The T 3 Supergravity model and F(R)
The so-called T 3 model of N = 2, D = 4 supergravity is the smallest model in which the plane F y⊥x can be defined; such a model is comprised within all models based on (non-degenerate) reduced FTS's (cfr. e.g. (Table 1)). In this model, it holds that (i = 1, and p 1 ≡ T ) In the usual normalization of d-tensors used in supergravity literature 20 , it holds that (cfr. e.g. [66]) In this case we have (N = 2, i = 1, dimF = 4)): By direct computation, one gets which allows to compute the dual components by Eq. 8.12.
Since the T 3 model pertains to the unique reduced FTS for which N = 2 (cfr. Table 1), for this model dim F = 2N = 4 and the plane F y⊥x coincides with the whole space {, }-orthogonal to the Freudenthal plane F x : , gets decomposed as follows : where SO(1, 1) KK is related to the radius of the S 1 in the dimensional reduction from minimal (N = 2) D = 5 "pure" supergravity down to D = 4 (giving rise to the T 3 model).
Let us start first with a particular configuration with ∆(x) < 0. Specifying (Eq. 8.54) and (Eq. 6.7)-(Eq. 8.61) for the T 3 model, one has (ε = sgn(x 0 x 0 )) 20 Which, however, is not the one used e.g. in [61]. with Then, for a generic GFT transformation on x and therefore F x lies completely in the rank-4 ∆ < 0 orbit of Aut (F (J 3 = R)).
Anagolously, specifying (Eq. 8.65) and (Eq. 8.66) for the T 3 model, one obtains and, according to Eq. 8.20, where the strict inequality holds, because we assume y to be of maximal(= 4) rank in F T 3 . Note that, while x and x lie in the ∆ < 0 orbit of Aut(F(R)), y belongs to the other rank-4 orbit 21 .
Starting from the decomposition (Eq. 8.26), the Freudenthal plane F x related to x (Eq. 8.54) and the {, }-orthogonal plane F y⊥x = F ⊥ x can respectively be identified as follows: Nicely, within the interpretation of SO(1, 1) KK as a non-compact analogue of D = 4 helicity of a would-be spin-3 2 (Rarita-Schwinger) particle, the Freudenthal plane F x pertains to the two massless helicity modes.
Let us recall that, while F x (Eq. 8.33) is a quadratic sub-FTS of F T 3 (as discussed in Sec. is not a sub-FTS of F (as discussed in Sec. 5). This can be explicitly checked by relying on the treatment of Sec. 8.2; in fact, for the T 3 model, F y⊥x = F ⊥ x (8.34) is not closed under T . Out of the four cases 1-4 listed at the end of Sec. 8.2, only the last one (4) is to be considered : in this case, the condition of closure of F y⊥x = F ⊥ x (Eq. 8.34) under T is that y 1 is rank< 3 in J 3 = R and y 1 is rank< 3 in J 3 = R, namely Thus, the condition of closure of F y⊥x = F ⊥ x (Eq. 8.34) under T implies, in the case of the T 3 model, an absurdum, namely that the rank-4 element y ∈ F ⊥ x be the null element of the FTS F T 3 ≡ F(R). Therefore, F y⊥x = F ⊥ x (Eq. 8.34) is not closed under T . In other words, as also pointed out above, in order for y = 0, y 1 , 0, y 1 T ∈ F ⊥ x to be rank-4 (as assumed throughout), it must have both components non-vanishing; from (Eq. 8.32) one can 21 As pointed out above, there is a unique ∆ > 0 orbit in the T 3 model. observe that in the T 3 model y belongs to the rank-4, ∆ > 0, Aut(F T 3 ) = SL(2, R) orbit, unless y 1 = 0 and/or y 1 = 0, in which case it has rank< 4. Therefore an element y of the form is rank-4 (and necessarily in the unique ∆ > 0 orbit) iff y 1 = 0 and y 1 = 0. Furthermore, we are interested in the behaviour of the quartic invariant ∆ on the D4 ⊕ D2 F y⊥x plane. General results are presented in Appendix B, in particular in Eq. 4.23 and Eq. 4.24 which can be used here. According to these results 22 The sign of ∆ (ϒ x (y)) depends only on the sign of ∆(y) implying that ϒ x (y) belongs to the same rank-4 (∆ > 0) Aut(F(R))-orbit as y. Explicitly in this case For a generic element r one gets (Eq. 4.23 and Eq. 4.24) Thus, following the general behaviour explained in appendix B, r ∈ F y⊥x is not of the same (maximal = 4) rank orbit as y (and ϒ x (y)) only when The conditions for r ∈ F y⊥x ⊂ F ⊥ x to lie in the rank-3, rank-2 or even rank-1 orbits might be easily studied using expressions Eq. 4.23 and Eq. 4.24).
Let us study now the family of configurations with D4 − D0 charges. This family includes configurations with both ∆(x) > 0 and ∆(x) < 0 possibilities. Let us take 23 , then one obtains Thus the sign of Delta(x) equals the sign of x 0 x 1 : 22 We can explicitly write the ϒ map (see Eq. 2.6) 23 This can be seen as an special case of (Eq. 8.83) and (Eq. 8.85)-(Eq. 8.87) for the T 3 model.
For a positive sign, sgn(x 0 x 1 ) > 0, the dual is a D6 − D2 configuration, it reads 44) x and x belong to the same (rank-4, ∆ > 0) orbit of Aut((F = J 3 )), which is unique in this model (cfr. [68], and Refs. therein). For a generic element (8.45) implying that F x lies completely in the unique rank-4 ∆ > 0 orbit of Aut(F(R)). Then, let us pick a rank-4 element y ∈ F ⊥ x , that means which is {, }-orthogonal to x and x , one can show that the most general element of this kind is given by the charge configuration: whose quartic invariant is given by thus the signs of ∆(x) and ∆(y) are opposite In the case of (x 0 x 1 ) = 0 then F ⊥ x ∼ D6 ⊕ D4 (for x 0 = 0) and F ⊥ x ∼ D2 ⊕ D4 (for x 1 = 0) Moreover, according to Eq. 4.23 and Eq. 4.24 ∆ (ϒ x (y)) = ∆(x) 2 ∆(y). (8.50) The sign of ∆ (ϒ x (y)) depends only on the sign of ∆(y). Both of them are negative in our current case. For a generic element r r = ay + bϒ x (y) ∈ F y⊥x = F ⊥ x (a, b ∈ R), one gets (Eq. 4.23 and Eq. 4.24) implying that ϒ x (y) and for the case any ϒ x (r) lies in the same (maximal rank) Aut(F(R))-orbit as y. 24 24 Explicitly, from (Eq. 8.100), one obtains Let us consider a particular configuration with only D0 − D6 charges with an arbitrary number of them. We start by identifying x with the rank-4, strictly regular element of the FTS F given by the D0 − D6 brane charge configuration for any element of this configuration we have 25 One can compute the Freudenthal dual x. Using the expressions which allows to compute the dual components using Eq. 8.12. We arrive to (ε = sgn(p 0 q 0 )) Thus, depending on the sign of p 0 q 0 26 Note that and thus x belongs to the same (unique) rank-4 ∆ < 0 orbit of Aut(F(J 3 )) as x. Namely, when p 0 q 0 > 0, the action of Freudenthal duality on x D0D6 amounts to flipping p 0 only, whereas when p 0 q 0 < 0, the action of Freudenthal duality on x D0D6 amounts to flipping q 0 only.
Associated to a GFT transformation on x, one defines the Freudental plane F x ⊂ F (dimF x = 2), spanned by x and x , whose generic element is 25 This characterizes F as a reduced [8] FTS. 26 The result (Eq. 8.61) defines a Z 4 symmetry in the 2-dim. Freudenthal plane F x ⊂ F, spanned by x D0D6 (Eq. 8.54) and its Freudenthal dual x D0D6 (8.61) (or, equivalently, in the Darboux canonical basis, by the magnetic and electric charges p 0 and q 0 of the 5D → 4D KK Abelian vector -see below -) : e.g., starting from p 0 and q 0 both positive (denoted by "(+, +)"), the iteration of Freudenthal duality yields This provides a simple realization of the Z 4 symmetry characterizing every Freudenthal plane, as a consequence of the anti-involutivity of Freudenthal duality itself.
Within the choice above, F x is coordinatized by the charges of D0 and D6 branes, respectively being the electric and magnetic charges x 0 and x 0 of the KK Abelian vector in the reduction D = 5 → 4. In other words,within the position (which does not imply any loss of generality for reduced FTS's), the Freudental plane F x is spanned (in a canonical Darboux symplectic frame -see below -) by the electric and magnetic charges x 0 and x 0 of the D = 5 → 4 Kaluza-Klein Abelian vector (which is the D = 4 graviphoton in the N = 2 supersymmetric interpretation).
Note that F x lies completely in the (unique) rank-4 ∆ < 0 orbit of Aut(F(J 3 )), because (Eq. 3.9) This implies that s belongs to the same maximal (= 4) rank, ∆ < 0 Aut(F(J 3 ))-orbit as x and x, unless a 2 = b 2 . This observation actually yields interesting consequences for multi-centered black hole physics, as briefly discussed in Sec. 10.
One can choose a rank-4 element y ∈ F which is {, }-orthogonal to the generic D0 − D6 element x defined before and its dual and x. A possible, particular, choice is provided by a D2 − D4 brane charge configuration: One can compute the components of ϒ x (y) M (Eq. 2.6) as given by thus, ϒ x (y) is still given by a rank-4 D2 − D4 brane charge configuration, and it holds that Consequently, one can define the 2-dim. plane F y⊥x ⊂ F ⊥ x , spanned by y and ϒ x (y), whose generic element is r = ay + bϒ x (y) ∈ F y⊥x (a, b ∈ R, in our classical/supergravity treatment . In particular, note that ϒ x (y) belongs to the same Aut(F(J 3 ))-orbit as y, because (consistent with the general Eq. 4.23 and Eq. 4.24), it holds that ∆ (ϒ x (y)) = x 0 x 0 4 ∆(y) = (∆(x)) 2 ∆(y) ≷ 0. (8.73) whose sign depends only on the sign of ∆(y).
It is worth remarking that ϒ x (y) automatically satisfies (Eq. 8.72) for every pair y i and y i , with i = 1, ..., N − 1. In fact, regardless of d i jk and d i jk , when only a pair y i and y i for a fixed i is non-vanishing (among all y i 's and y i 's), then y is non-trivially of rank-4 in F, because generally ∆(y) = 0, since at least the term − y i y i 2 is present (cfr. (Eq. 8.4)-(Eq. 8.5)). Therefore, one can which implies r ∈ F y⊥x ⊂ F ⊥ x to be not of the same (maximal = 4) rank as y (and ϒ x (y)) only when (recall (Eq. 8.66)) The conditions for r ∈ F y⊥x ⊂ F ⊥ x to lie in the rank-3, rank-2 or even rank-1 orbits may be easily inferred.

Closure of the
The plane F y⊥x is not generally closed under the triple map T (or, equivalently, under Freudenthal duality ∼), see Sec. 3.
Within the framework under consideration, namely within the 4D/5D special coordinates' symplectic frame of reduced FTS's and within the choice given by Eq. 8.65 of the rank-4 element x ∈ F (with ∆(x) < 0) and of the rank-4 element y ∈ F ⊥ x = F/F x (with ∆(y) ≷ 0), we study now more in detail the condition of closure of the plane F y⊥x under T .
In order to determine the condition of closure of F y⊥x under T , we have to explicitly compute T (r) ≡ T (r, r, r) for a generic element r = ay + bϒ x (y) ∈ F y⊥x , and for any D2 − D4 configuration y. This is given by, ( see Appendix D) (c ≡ x 0 x 0 b, T (y) = T (y, y, y)).
T (r) M : (a − c) 3 d i jk y i y j y k = 0; (a + c) 3 d i jk y i y j y k = 0. There are various cases, as follows: 1. y i is rank-3 in J 3 and y i is rank-3 in J 3 , namely d i jk y i y j y k = 0, d i jk y i y j y k = 0. (8.79) In this case, no solutions exist to the system (Eq. 8.78), and F y⊥x is not closed under T .
2. y i is rank< 3 in J 3 and y i is rank-3 in J 3 , namely d i jk y i y j y k = 0, d i jk y i y j y k = 0. (8.80) In this case, T (r) 0 = 0 is automatically satisfied, while T (r) 0 = 0 has solution a = x 0 x 0 b. However, for a fixed x, this solution is a line in F 2 = R 2 spanned by (a, b), and thus is codimension-1 in F y⊥x . Therefore, only the 3. y i is rank-3 in J 3 and y i is rank< 3 in J 3 , namely d i jk y i y j y k = 0, d i jk y i y j y k = 0. (8.81) In this case, T (r) 0 = 0 is automatically satisfied, while T (r) 0 = 0 has solution a = −x 0 x 0 b. However, for a fixed x, this solution is a line in F 2 = R 2 spanned by (a, b), and thus is codimension-1 in F y⊥x . Therefore, only the 4. y i is rank< 3 in J 3 and y i is rank< 3 in J 3 , namely d i jk y i y j y k = 0 = d i jk y i y j y k . (8.82) In this case, the system (Eq. 8.78) is automatically satisfied ∀a, b ∈ R, and F y⊥x is therefore closed 27 under T . Note that the condition (Eq. 8.82) is not inconsistent with the assumption of y (8.65) to be a rank-4 element of F. In fact, if both y i and y i are rank-2 elements in J 3 resp. J 3 , then ∆(y) (Eq. 8.66) is still generally non-vanishing, with the second term vanishing iff d i jk d ilm y j y k y l y m = 0 (in this latter case, when non-vanishing, ∆(y) < 0, and y -and ϒ x (y) as well -would lie in the same ∆ < 0 Aut(F(J 3 ))-orbit as x and x). On the other hand, if y i and/or y i are rank-1 elements in J 3 resp. J 3 , still y can be a rank-4 element of F(J 3 ), because ∆(y) = − y i y i 2 0 in this case, and thus (when the disequality strictly holds), y -and ϒ x (y) as well -would lie, as above, in the same ∆ < 0 Aut(F(J 3 ))-orbit as x and x.

The general D0 − D4 sector
The Freudenthal plane F x We start by identifying x with the rank-4 element of the FTS F given by the 28 D0 − D4 brane charge configuration : 27 It could have also characterized as a 2-dimensional sub-FTS of F. 28 We might have as well started with a D2 − D6 configuration, and perform an equivalent treatment (obtaining, as evident from the treatment given below and from the anti-involutivity of Freudenthal duality, a D0 − D4 configuration as Freudenthal-dual of the starting D2 − D6 configuration). and we further impose that x D0D4 belongs to (one of) the ∆ > 0 Aut(F)-orbit(s) (see Subsec. 8.4).
∆(x D0D4 ) = 2 3 x 0 d i jk x i x j x k > 0. (8.84) From the definition (Eq. 8.12) (note that ε = 1 in this case), one can compute that the Freudenthaldual x D0D4 of the D0 − D4 configuration (Eq. 8.83) is a D2 − D6 configuration, namely 29 The dual is with (Eq. 8.12) d jkl x j x k x l ; (8.86) By exploiting the adjoint identity (see [40,66,67] and Refs. therein) of the Jordan algebra J 3 underlying the reduced FTS F Eq. 8.9 one can also check that ∆ is invariant under Freudenthal duality : and thus that x D0D4 would lies in the same Aut(F(J 3 ))-orbit as x D0D4 . Thus, one can define the Freudental plane F x (dimF x = 2), spanned by x (Eq. 8.83) and x (8.85), whose generic element is x F = ax + b x ∈ F x , a, b ∈ R. By using (Eq. 8.9) again, one can also compute that implying that F x lies completely in the rank-4 ∆ > 0 orbit 30 of Aut(F(J 3 )).
The orthogonal space F ⊥ x , and the plane F y⊥x ⊂ F ⊥ x Then, one can pick another rank-4 element y ∈ F which is {, }-orthogonal to x (Eq. 8.83) and x (8.85); the most general element of this kind is given by the charge configuration: x 0 x j y j d klm y k y l y m + d i jk d ilm y j y k y l y m ≷ 0. In (Eq. 8.92), the case of vanishing ∆ has been excluded because y is chosen to be of rank-4 in F. One can compute the components of ϒ x (y) M (Eq. 2.6) as given by (ϒ x (y)) 0 = −d i jk x i x j y k ; (8.93) (ϒ x (y)) i = −d klm d mi j x k x l y j + 2x j y j x i ; (8.94) which for this case can be written as By exploiting (8.9), one can then check that ϒ x (y) (Eq. 8.100) automatically satisfies x 0 x j y j d klm y k y l y m + d i jk d ilm y j y k y l y m ≷ 0.
implying that ϒ x (y) lies in the same Aut(F(J 3 ))-orbit as y.
The same holds for a generic element r = ay + bϒ x (y) ∈ F y⊥x (a, b ∈ R), which belongs to the same Aut(F)-orbit as y : indeed it can be checked that (consistently with Eq. 4.23 and Eq. 4.24) x 0 x j y j d klm y k y l y m + d i jk d ilm y j y k y l y m ≷ 0.

The canonical Darboux symplectic frame
We recall that in the 4D/5D special coordinates' symplectic frame a generic element Q of the reduced FTS F(J 3 ) splits as given by (Eq. 8.1), while the 2N × 2N symplectic metric is given by (8.3). By a simple re-ordering of rows and columns (amounting to a relabelling of indices , one can switch to a canonical Darboux symplectic frame 31 (in which the 4D/5D covariance is still manifest), in which x (Eq. 8.1) splits as follows: and in which the symplectic metric (Eq. 8.3) acquires the following form where ε is the 2 × 2 symplectic metric of the defining irrepr. 2 of Sp(2) ≈ SL(2) defined by (Eq. 6.19). At a glance, in a physical (Maxwell-Einstein) framework (Eq. 8.104) suggests that the choice of the (canonical) Darboux symplectic frame defined by (8.104) (or, equivalently, by (Eq. 8.105 and (Eq. 6.19)), amounts to making manifest the splitting of the electric-magnetic fluxes of the Abelian 2-form field strengths, grouped, within the symplectic vector Q (Eq. 8.104), into the KK vector's fluxes (magnetic p 0 and electric q 0 ), and into the fluxes (magnetic p i and electric q i , i = 1, ..., N − 1) of each of the N − 1 Abelian vectors with a D = 5 origin. When specifying such a generic (supersymmetry-independent) interpretation for minimal D = 5 supergravity dimensionally reduced down to N = 2, D = 4 supergravity, p 0 and q 0 are the magnetic resp. electric charges of the D = 4 graviphoton (the Abelian vector in the N = 2 gravity multiplet), whereas each of the N − 1 pairs p i , q i denote the magnetic resp. electric charges of the Abelian vector belonging to each of the N − 1 vector supermultiplets coupled to the gravity one (these all have a D = 5 origin, thereby comprising the D = 5 graviphoton, as well).
Thus, the (2N − 2)-dim. space F ⊥ x , {, }-orthogonal to the 2-dim. Freudenthal plane F x , gets decomposed into N − 1 2-dim. subspaces, all mutually orthogonal with respect to the symplectic product {, } defined by (8.105) : each of them corresponds to the electric-magnetic flux degrees of freedom of a vector supermultiplet in the corresponding N = 2, D = 4 supergravity, or, more generally, to the electric-magnetic fluxes of a D = 4 Abelian vector fields with a five-dimensional origin.

F-duality preserves the ∆ > 0 Aut(F)-orbits: The STU model
At least for (non-degenerate) reduced FTS's, Aut(F) has a transitive action on the ∆ < 0 locus of F, which thus corresponds to a unique one-parameter family of Aut(F)-orbits 32 ; consequently, Freudenthal duality trivially preserves the orbit structure for ∆ < 0.
The story is more complicated for the ∆ > 0 locus, which, again at least for (non-degenerate) reduced FTS's, generically (with the unique exception of the T 3 model) has two or more Aut(F)orbits. However, the existence of the Freudenthal rotations presented in ?? ensures that x andx always lie in the same orbit. Here we explicitly present the non-trivial orbit structure of the ∆ > 0 locus and its properties under Freudenthal duality for the STU model using only the (discrete) U-duality invariants characterising the orbits.
Even though the simplest reduced FTS exhibiting more than one ∆ > 0 Aut(F)-orbit is the one pertaining to the so-called ST 2 model of N = 2, D = 4 supergravity (namely, F(R ⊕ R) [73], we will explicitly treat the FTS F(R ⊕ R ⊕ R) related to the slightly larger STU model, because this can be considered as a genuine truncation of all (non-degenerate) reduced FTS's (with the the exception of the T 3 and ST 2 models, which are however particular "degenerations" of the STU model itself).
As determined in [73] (see also the treatment in Sec. F.1 of [68]), in the STU model there are two orbits with ∆ > 0, one supersymmetric and one non-supersymmetric (the one with vanishing central charge at the horizon : Z H = 0), and their coset expressions are isomorphic (even if they are SL(2, R) ×3 -disjoint orbits) : Following the treatment of [66], one can consider a D0 − D4 representative (also in the FTS representation [11]) of the orbits O ∆>0,BPS and O ∆>0,non−BPS,Z H =0 :

Linear realisations of general Freudenthal transformations
As consenquence of its definition, Freudenthal duality ∼ can only be consistently defined in the locus ∆ = 0 of the FTS F itself. In general, the group Aut(F) has a non-transitive action over such a locus, which undergoes a, (at least) twofold stratification, into a (always unique) ∆ < 0 orbit and into a ∆ > 0 sub-locus, which may in turn further stratify into Aut(F)-orbits.
While F-duality is a non-linear operation, as discussed in [11] its action can be realised by finite "local/gauged" U-duality transformations U : F → Aut(F), namely, as we will understand throughout the following treatment, that depend on the element of F they are applied onto.
More generally, Freudenthal duality, and so GFT, can be mimicked by finite transformations of at least three different kinds, the first two of which are not contained in the U-duality group: 9.1 Anti-symplectic realisation: ∆ < 0 At least within (non-degenerate) reduced FTS's, the quartic polynomial invariant ∆ of Aut (F (J 3 )) ≈ Con f (J 3 ) can be written as in Eq. 8.5 (see Sec. 8 for this and other expressions) (cfr. e.g. [38,65,66]; (i = 1, ..., N − 1, dimF = 2N)). Let us consider a D0 − D6 configuration, studied in Subsec. 8.2. As we can seen there (we refer to Eq. 8.54,Eq. 8.55,Eq. 8.60 and Eq. 8.61), the action of the Freudenthal duality ∼ on x D0D6 (Eq. 8.54) can be represented by a (maximal-rank) 2N × 2N matrix where A, B ∈ GL(N − 1, R). The action of a general Freudenthal transformations is then The transformation O ∈ GL(2N, R) is inherently not unique. Also, apart from the "∓" branching in (9.1), the whole realization of Freudenthal duality does not depend on p 0 nor on q 0 , and so it can be (loosely) considered an "ungauged" transformation in F. In particular, O is anti-symplectic, namely where 1 is the identity matrix in N − 1 dimensions. Note, however, that O is never symplectic (i.e., it always holds that O T ΩO = Ω ⇔ O / ∈ Sp (2N, R)). Let us consider a particularly simple anti-symplectic case, for which A = −1 ⇒ B = 1 : At least in all reduced FTS's F's based on simple and semi-simple rank-3 Jordan algebras (see Sec. 2 and Table 1 therein), it can be proved (cfr. App. D of [74]) that O (Eq. 9.5) realizes an outer automorphism of Aut (F(J 3 )), namely that where R F (g) denotes the 2N × 2N matrix representation of the element g of Aut (F(J 3 )) acting on F itself, and Aut, Inn and Out respectively denote the automorphism group, and its inner resp. outer components. Note that O is an involution: where here 1 denotes the the identity matrix in 2N dimensions. However, from (Eq. 6.7)-(Eq. 8. 61) it follows that the correct iteration of the Freudenthal duality on the D0 − D6 configuration x (Eq. 8.54) is provided by the application of ∓O and then necessarily of ±O , thus yielding to −O 2 = −1 acting on x, and correctly implying The anti-simplecticity of O (Eq. 9.5) implies that it does not preserve the symplectic structure of F (as neither Freudenthal duality does, as well [11]). This is consistent with the fact that O realizes an outer automorphism of the electric-magnetic U-duality group Aut (F(J 3 )), which in turn is generally realized in a symplectic way [17,75,76]: As observed in [74], at least for all automorphism groups of reduced FTS's over simple or semisimple rank-3 Jordan algebras it holds that (see e.g. [77]) Thus, all non-trivial elements of Out (Aut (F(J 3 ))) are implemented by anti-symplectic transformations. In [78] (also cfr. [79]), it was discussed that the global symmetry of the resulting Maxwell-Einstein (super)gravity contains the factor Z 2 , which can be offset by a spatial parity P transformation. In particular, from Eq. (2.118) of [78], it follows that the global symmetry group G of the resulting Maxwell-Einstein (super)gravity theory is given by where G 0 is the identity-connected, proper electric-magnetic (U-)duality, Aut (F(J 3 ))-part of G, whereas p corresponds to an element of G implemented by an anti-symplectic transformation. Interestingly, the above results relate a realization (not the unique one, though! -see Secs. Subsec. 9.2 and 9.3) of the Freudenthal duality on the well-defined representative D0 − D6 of the unique, non-BPS (non-supersymmetric) rank-4 ∆ < 0 orbit in F, to spatial parity transformations in the corresponding theory; in fact, anti-symplectic transformations, such as O (Eq. 9.5) are symmetries of the theory, provided that they are combined with spatial (3D) parity P. 9.2 Gauged symplectic realisation: ∆ < 0 It is also possible to find a (non-unique) symplectic transformation realizing the Freudenthal duality transformation (Eq. 6.7) on the D0 − D6 representative of the ∆ < 0 Aut (F)-orbit of F. However, this will necessarily be "gauged" in F, namely it will depend on the element of F it acts upon (i.e., in this case, on the D0 − D6 element (Eq. 8.54)).
9.3 Gauged Aut (F) realisation: ∆ < 0 In the ∆ < 0 locus of F (on which the action of Aut (F) is always transitive, thus defining a unique orbit O ∆<0 of F), we deal with the issue of mimicking the action of Freudenthal duality by an Aut (F) transformation (which will generally be local in F), and consider the following (commutative) diagram: Here, x can denotes a convenient "canonical" representative that can be defined in a uniform manner for all relevant FTS as in [39]. The corresponding Aut(F) transformations taking x andx to x can andx can , respectively, are denoted U and U . Similarly, M x and M x can are the gauged Aut (F) transformations that send x and x can tox andx can , respectively. Since the square commutes we free to pick a convenient canonical representative. Generally, at least for (non-degenerate) reduced FTS's, the homogeneous space O ∆<0 can be written as where we recall that Aut (F (J 3 )) Con f (J 3 ), Con f (J 3 ) and Str 0 (J 3 ) respectively denote the conformal and reduced structure groups of the cubic Jordan algebra J 3 .
For simplicity's sake, let us assume Str 0 (J 3 ) = Id (namely, there is no continuous nor discrete stabiliser for O ∆<0 , which thus is a group manifold: O ∆<0 ∼ = Aut (F)). Actually, this only holds for the T 3 model of N = 2, D = 4 supergravity, associated to the simplest example of (non-degenerate) reduced FTS [68,73]. Let us also choose a convinient representative of O ∆<0 . An obvious choice is given by the D0 − D6 configuration (Eq. 8.54), x can = x D0D6 , which makes the Str 0 (J 3 ) stabiliser of O ∆<0 manifest for all reduced FTS.
The assumption Str 0 (J 3 ) = Id implies U = U. From (Eq. 9.15) we have where the last step of the second line follows from the fact that Aut (F) and Freudenthal duality commute [11]. Thus, applying Freudenthal duality to (9.17) one obtains Consequently, any reasoning involving the diagram (Eq. 9.15) is independent of the choice of x can ; indeed, the Aut (F (J 3 )) transformation connecting any two elements of O ∆<0 = Aut (F (J 3 )) (say x can and x can ) will be unique, since Aut (F (J 3 )) is free on the orbit O ∆<0 by assumption.
With the generalisation to arbitrary reduced FTS in mind, it is useful to reexpress M x D0D6 (z), as defined by (Eq. 9.21), through the elementary Aut(F) transformations defined in [8,82] for generic reduced FTS ; where C, D ∈ J 3 and τ ∈ Str(J 3 ) s.t. N(τA) = λ N(A). In this form it is straightforward to generalise to arbitrary cubic Jordan algebra as follows. Consider the F-dual pair given by For base point E and c, d ∈ R we find is given by Hence, on setting d = − sgn(αβ )(β /α) 1/3 = −|β /α| 1/3 and c = sgn(αβ )(α/β ) 1/3 = |β /α| 1/3 we obtain as required. Using Eq. 9.15 this gives an explicit realisation of a U-duality relating any F-dual pair x,x with ∆(x) < 0. Explicitly, setting x can = w using U and then applying Note, since any x in a ∆(x) < 0 orbit has stabiliser Str 0 (J 3 ) the U-duality transformations are generically non-unique,x = SxU −1 SwM w S w US x x (9.32) for arbitrary S y ∈ Stab(y) ⊂ Aut(F). It is also straightforward to define aŵ ∈ F such that exp π 2 ϒŵ w =w. (9.33) whereŵ is determined by w. Recall, for x = (α, β , A, B), y = (γ, δ ,C, D), the Freudenthal product is defined by, and A ∨ B ∈ StrJ is defined by The action of Φ : F → F is given by The set of all such homomorphisms yields the automorphism Lie algebra, where the Lie bracket is given by generates an sl(2, R) subalgebra, we find which on setting d = −|β /α| 1/3 gives the F-dual transformation Eq. 9.30. Using 4x ∧ x(y) = 3T (x, x, y) + {x, y}x (9.43) and forŵ = 2(0, 0, As for the ∆(x) < 0 case treated above, when ∆(x) > 0 the action of Freudenthal duality on x can generally be realised by linear gauged Sp(2N, R) (recalling that dim R F = 2N) or Aut(F) transformations. For simplicity's sake, we confine ourselves here to the study of the gauged Aut(F) transformations 33 .
Concerning the ∆ > 0 locus of F, it is generally stratified in two or more orbits under the nontransitive action of Aut(F). In the following treatment, we will consider the particularly simple case of the T 3 model, in which such a stratification does not take place, and thus the ∆ > 0 locus of F(R) corresponds to a unique Aut(F) = SL(2, R) supersymmetric (1/2-BPS) orbit [68,73] : The non-trivial, discrete, stabiliser of the O ∆>0 orbits for the T 3 model is up to conjugation given by the Z 3 ⊂ SO(2) ⊂ SL(2, R), generated by [68] M ≡ 1 2 .
As computed below, the Freudenthal dual of the D0 − D4 representatives of the orbits O ∆>0 are given by D2 − D6 elements 33 At least in (non-degenerate) reduced FTS's, a similar treatment along the non-BPS Z H = 0 orbit (∆ > 0) can be given. Within (non-degenerate) reduced FTS's, the smallest model exhibiting a non-BPS ∆ > 0 orbit is the ST 2 model [73] In order to determine the gauged Aut (F (J 3 )) transformation M x D0D4 mimicking, along O ∆>0 , Freudenthal duality acting on x can = x D0D4 , i.e. such that x D0D4 = M x D0D4 x D0D6 (cfr. (Eq. 9.15)), we will again use the "T 3 degeneration" of the quantum information symplectic frame of the STU model, namely we search for a 2 × 2 matrix M such that where x D0D4 in the FTS parametrization (following the conventions of [68]) reads Long but straightforward algebra yields the twofold solution (y = q 0 /p > 0): (9.51) M x D0D4 (y) (defined by (Eq. 9.49)) can be realized in terms of the Aut(F(R)) transformations Eq. 9.26 as follows: It is interesting to note that for y = 1 ⇔ q 0 = p, the matrix M ± (y = 1) (Eq. 9.51) (and thus, through (Eq. 9.49), M x D0D4 ) does not belong to SL(2, Z): However, remarkably, only for y = 1 another solution to (Eq. 9.49) can be found (the subscript "add" stands for additional; recall (Eq. 9.23)): with ε defined by (Eq. 6.19). Due to the existence of the additional solution (Eq. 9.54), the integral (projective) case is obtained for y = 1 (by further setting p ∈ Z) (cfr. the treatment in the quantized charge regime, presented in [11]). Recall, since we are assuming ∆(x) > 0, we can use the Freudenthal rotation in Aut(F), Specialising to a generic 34 element in the D0D4 system , (9.56) 34 Generically, this is actually a larger class of charge configurations, but in the special cases of the T 3 , ST 2 , STU models it is precisely the D0D4 subsector. For N = 8, imposing A is diagonal restricts to the D0D4 subsector. and using x ∧ x, (9.57) we have Further restricting to the T 3 model and setting α = −q 0 , A = p we find, as given in Eq. 9.52.

Non-trivial orbit stabilizers
Here, generalizing the reasoning at the end of Subsec. 9.3, we want to reconsider the diagram (Eq. 9.15), and generalize the treatment to the case in which (regardless of the sign of ∆(x)) the Aut (F)-orbit to which x ∈ F belongs is endowed with a non-trivial stabilizer H , such that the corresponding homogeneous (generally non-symmetric) manifold can be written as Let us deal with the issue of mimicking the action of Freudenthal duality by an Aut (F) finite transformation (which will generally be "gauged" in F), and consider again the (commutative) diagram (Eq. 9.15). In general 35 (U,U ∈ Aut(F) H ), it holds that x can = Ux; x can = U x = U x, (9.63) where in the last step of the second line we used the commutativity of Freudenthal duality and Aut (F) . Thus, (Eq. 9.63) implies On the other hand: where again in the last step we used the commutativity of Freudenthal duality and Aut (F). Thus, (9.65) implies x can = U U −1 x can ⇔ U U −1 = Z x can ∈ Stab(x can ). (9.66) Let us observe that Stab(x) = Stab( x), (9.67) since for Z x ∈ Stab(x) and Z x ∈ Stab( x) we have and vice versa, implying (Eq. 9.67). By virtue of results (Eq. 9.64), (Eq. 9.66) and (Eq. 9.67), one can write Moreover, it is here worth commenting that, in presence of non-trivial H , any reasoning involving the diagram (Eq. 9.15) is actually dependent from the actual choice of x can ; indeed, the Aut (F (J 3 )) transformation connecting any two elements of O (say x can and x can ) will not be unique, for the reasons highlighted above.

Summary, concluding and further remarks
The purpose of this work has been to present, extend and clarify old and new results about General Freudenthal transformations (GFTs), of whose, Freudenthal duality is a discrete element, filling in when necessary details not yet appearing in the mathematical or physical literature. This work has presented an extended treatment of General Freudenthal transformations (GFT) (Sec. 2-Sec. 5), and, as a particular case of them, Freudenthal duality, applied to the physics of black holes solutions of supergravity theories. As it is shown here, there is a rich interplay between mathematical and physical properties.
In Sec. 6 we have studied the entropy properties of N = 2, D = 4 pure supergravity from the point of view of FTS formalism, a non reduced FTS with overall positive quartic invariant ∆. We have seen how the General Freudenthal Transformations, whose the Freudental duality is a special case, are nothing else as an instance of EM duality. Freudenthal duality is an anti-involutive U-duality transformation.
In Sec. 7 we have studied the axion-dilaton model, a N = 2, d = 4 supergravity minimally coupled to one vector multiplet, which can be considered a consistent truncation of N = 4 supergravity. The mathematical structure is in this case a two dimensional FTS with negative quartic invariant. The corresponding extremal BH is non-BPS. A SU(1, 1) subgroup of Freudenthal General transformations leave invariant the entropy. Freudenthal duality is in this case a involutive U-duality transformation. This model, as the pure supergravity studied before, cannot be uplifted to D = 5, which corresponds directly to the fact that the associated FTS is not reduced.
In Sec. 8 we proceeded to the analysis of N = 2, D = 4 supergravities of D = 5 origin. The mathematical structure of these models is that of a FTS derived from a cubic Jordan Algebra. In first place we study a T 3 model, or in Freudenthal terminology a F(J 3 = R) structure. This model is represented by the unique reduced FTS with dim F = 4 as vector space.The automorphism group is a four dimensional representation of SL(2, R) which can be decomposed into KK SO(1, 1) representations. The full space is split into two two-dimensional planes simplectically orthogonal between them which can be identified respectively with D6 General Freudenthal transformations, and in particular Freudenthal duality ∼, will preserve the orbit structure for ∆ < 0. This trivial since, at least for (non-degenerate) reduced FTS's, where, Aut(F) has a transitive action on the ∆ < 0 locus of F, which thus corresponds to a unique oneparameter family of Aut(F)-orbits. The situation is more complicated for the locus of ∆ > 0 configurations, for which, in general (with known exceptions) these are stratified in two or more automorphism orbits. In Subsec. 8.4 we study this question and conclude that the orbit stratification of the ∆ > 0 locus of F(R ⊕ R ⊕ R) is preserved under Freudenthal duality ∼, and by extension by General Freudenthal transformations. This is specially relevant as the STU model is always a genuine truncation of all models described by (non-degenerate) reduced FTS's.
In Sec. 9 we show, in different examples, how the action of General Freudenthal transformations, and, in particular, Freudenthal duality can be mimicked/undone by finite U-duality, or Aut(F), transformations which are gauged in that they depend on the element of F they are applied onto. We restrict to two situations. First for configurations with ∆ < 0 locus of F the action of Aut (F) is always transitive, thus defining a unique orbit O ∆<0 of F. The ∆ > 0 locus of F is generally stratified in two or more orbits under the non-transitive action of Aut(F). However it is possible to find particular cases where this stratification does not take place. One of these cases, a T 3 model where the unique orbit of the SL(2, R) automorphism group is (1/2)−BPS supersymmetric. This example is studied in Subsec. 9.4.
The entropy of a a linear a configuration which can be written as a linear combination is which is, not linearly, related to the entropies of individual configurations x and y plus different quantities which can be written in terms of FTS operations However for linear combinations of the form given by a General Freudenthal transformation, x F = ax + bx, the entropy of the composite object is simply related to that one of x Thus there is a family of configurations for which the entropy is the same to the entropy of x, those with a 2 + εb 2 = 1, with ε = sgn ∆(x). We can use these results to show how it is possible to construct, in different ways, asymptotically "small" (zero entropy) interacting black holes from an initial non trivial configuration.
In a first way, note that, when ∆(x) > 0 (ε = 1), the elements defined by (see also Eq. B.36) x F± = x ±x are rank-4 element of F x ⊂ F (see Eq. B.37 and Table 2). However, when ∆(x) < 0 (ε = −1) it follows that x F± are null elements 36 with vanishing Bekenstein-Hawking entropy This suggests the existence of a class of "two-centered black hole solutions" where each centre is "large" non-BPS (∆(x) = ∆(±x) < 0), they are interacting since {x, x} = 0, yet asymptotically their Bekenstein-Hawking entropy vanishes, so the total system (before crossing a line of marginal stability) belongs to a small nilpotent 37 orbit. The physical or geometric significance of such configurations remains unclear.
In a second way, other small Black Hole solutions can constructed by application of the properties of ϒ x ∈ aut(F) maps. The behaviour of S BH (or ∆) on the F or F y⊥x planes is similar (see Subsec. 4.1). The null elements of F x and any F y⊥x are "aligned". The locus of null entropy, is given by the same Eq. 4.29 which it is independent of y. upon an arbitrary charge vector with x . From Eq. 4.30, for example, any element of the form is null, ∆(z ± ) = 0. Then, this describes another class of two centered black hole configurations, which they are interacting since {y, ϒ x y} = 0, yet asymptotically their Bekenstein-Hawking entropy vanishes S BH y ± ϒ x (y) = 0. (10.9) 36 The elements x F± are actually rank-1 element of F x ⊂ F. In this case (Eq. B.37, Table 2) In particular it does not have a well definedF-dual. 37 In the D = 3 language; cfr. e.g. the nilpotent orbits of so(4, 4) acting on its adjoint irrep. 28 in [85].
Further work about these small BHs will be presented elsewhere.
It is a challenging problem to extend these results to systems with quantized charges, in this case the requirement that the set of charge vectors x f = ax + bx belongs to the charge lattice is extremely restrictive. Let us remind that for the case of Freudenthal duality, demanding that x,x are integers restrict us to a open subset of black holes where the entropy is necessarily an integer multiple of π. The complete characterisation of discrete U-duality invariants, which may or may not also be F-and GFT invariant, remains an open question and, hence, so does the F-dual invariance of higher order corrections to the entropy.
From the structure of this expression it is obvious that the requirement the entropy being a perfect square is automatically kept under a GFT. Finally, we note that the charges of five dimensional stringy black holes may be described in the context of (cubic) Jordan algebras. The cubic norm of one of these algebras determines the entropy to lowest order. The Jordan dual (introduced in [42]) is related to the Freudenthal dual of the corresponding 4D model. The generalization of this 4D/5D correspondence from a General Freudenthal transformation to, to be defined, "General Jordan Transformations" (GJT) , of which the Jordan dual is a particular case, will be treated elsewhere [83].

A Freudenthal triples: Assorted properties.
A summary of some FTS definitions, notation and properties used through this work. See Ref. [8] for additional ones and proofs.
For any two vectors u 1 , u 2 ∈ F y⊥x we have the following properties (u i = a i y + b i ϒ x (y) = α i+ y + + α i− y − , i = 1, 2) (2α i∓ = b i ± ia i / ∆(x)), The first two expressions are obtained by direct computation. To get Eq. B.10, we note that where we have used ( Eq. B.4, Eq. B.5). Finally Eq. B.11 is a particular case of Eq. B.10. Eq. B.9 can also be written as (for ∆(x) = 0), As consequence, for ε = 1, the mapΥ preserves the bilinear antisymmetric form {, } in each F y⊥x plane. This implies necessarily thatΥ x is a symplectic transformation on the plane.

B.2 Maximal rank T-Planes are disjoint
Let us have two non-degenerate (generated by maximal rank elements) planes T x 0 ,T x 1 generated by distinct elements x 0 , x 1 (x 0 = x 1 ). We will show that the two planes are, or the same, or disjoint. Suppose we can find a common element y ∈ T x 0 ∩ T x 1 , this implies that a) the signs of ∆(x 0 ), ∆(x 1 ) are the same and b) also y ≡ T (y) ∈ T x 0 ∩ T x 1 . Then, we can find coefficients The coefficients α i , β i are given in terms of the a i , b i by Eq. 3.4. Inserting the values of these coefficients we can see that these equations are invertible as long as If ∆(y) = 0 the equations are invertible: we can write any pair (x 0 , x 0 ) or (x 1 , x 1 ) as linear combinations of y, y . The existence of such linear combinations shows that T y = T x 0 = T x 1 . If the common element is not of maximal rank then the situation is different. If ∆(y) = 0, the equations are not invertible but then necessarily both ∆(x 0 ) < 0, ∆(x 1 ) < 0. y is a "null" or "light-cone" element, it can be seen it is then proportional to one of the two vectors ± | ∆(x 0 ) | x 0 + x 0 and one of the two We can use these last expressions to arrive to: "Some" elements of T x 0 (a one dimensional subset of them) can be written as linear combination of those of T x 1 .
In the case of degenerate planes (generated by non-maximal rank elements), one can show in the same way that the T-"lines" T x 0 , T x 1 , corresponding to elements x 0 , x 1 with ∆(x 0 ) = ∆(x 1 ) = 0 are either identical (if x 0 = x 1 = 0 or x 0 ∝ x 1 ) or disjoint (if x 0 ∝ x 1 ).

B.3 Rank on the T/F-plane
Let it be a generic vector x 0 of a given rank and its associated (possible degenerate) T-plane T x 0 ≡ T 0 ( we write ∆ 0 = ∆(x 0 )). We are interested in studying what it can be said about the rank of any element x ∈ T 0 on the plane. Let us consider the different cases.
Let us assume then a nonzero rank and consider the following possibilities: A) If ∆ 0 > 0 then, by Eq. 3.9, ∆(x) > 0, and Rank(x) = Rank(x 0 ) = 4, ∀x ∈ T 0 . B) If ∆ 0 < 0 then, by Eq. 3.9, we have two possibilities: ∆(x) < 0 and then Rank(x) = 4 or If ∆(x) = 0 then x is a null vector, it belongs to the light-cone generated by x ± = ± | ∆(x 0 ) | x 0 + x 0 , then, by Eq. 3.4, x = T (x) = 0. For the sake of concreteness, let us take x = x + . We can find at least one element, its associated element x − , such that similarly would happen if we start by assuming x = x − . We arrive to the conclusion that Rank(x) = 2 in this case.
C) If ∆ 0 = 0 (Rank(x 0 ) = 1, 2, 3) then, by Eq. 3.9, ∆(x) = 0. We are confronted with two selfexcluding possibilities: The results are summarized in Table 2. Similar results are obtained for the F-plane. Table 2: Relation of rank(x), the rank of any element x in the T-plane T 0 generated by x 0 .

B.4
The Tor F-plane as a (quadratic) sub-FTS system: Euclidean and hyperbolic planes Let us fix a maximal rank element x 0 (∆(x 0 ) = 0) and its T-plane T x 0 ≡ T 0 . As a consequence of Eq. 3.9, the restriction of the quartic map ∆ to T 0 can be written in terms of a symmetric bilinear form I 2 as follows where ε 0 = sgn ∆(x 0 ) (we call ε 0 the "signature of T 0 ", it does not depend on the plane maximal rank element chosen as generator). Choosing a basis on the plane and coordinates with respect it (x = x I e I , (I = 1, 2), e 1 ≡ x 0 , e 2 ≡ x 0 ), the bilinear form is given by Then the quadratic form The full quadrilinear map restricted to the plane is then of the form (for generic vectors x, y, z, w ∈ T 0 ) ∆ T 0 (x, y, z, w) = ε 0 12 (I 2 (x, y)I 2 (z, w) + I 2 (x, z)I 2 (y, w) + I 2 (x, w)I 2 (y, z)) .

(B.22)
To arrive to this expression we have used the multi-linearity of ∆ and the properties ∆(x, x, x, x ) = ∆(x, x , x , x ) = 0 (see properties in Appendix A). In the case of ∆ 0 = 0 the expressions reduce trivially to I 2 ≡ 0. We can quickly convince us that the T 0 plane is itself a two dimensional FTS of quadratic (or degenerate) type (see [51]) with a characteristic "signature ",ε 0 , and whose symmetric quadrilinear and antisymmetric bilinear forms are those inherited from the original FTS.
The explicit form of the matrix S is The trilinear T map, the F-dual maps restricted to T 0 are given by, in component form [51] T (x, y, z) = where it has been used the linear map defined byŜ(x) ≡ S I J x J e I and where η(x) ≡ sgn(I 2 (x)) = sgn (x 1 ) 2 + ε 0 (x 2 ) 2 | ∆ 0 | .
If ε 0 = 1 then η(x) = 1, ∀x ∈ T 0 . In summary any non-degenerate T (or F) plane can be considered as a two-dimensional quadratic FTS system, the three FTS axioms, see Sec. 2, are trivially satisfied by the T 0 -restricted maps ∆ T , {} T , T T (∆ F , {} F , T F ).

Euclidean, hyperbolic T-planes
The signature of the I 2 bilinear form coincides with the signature of the T 0 plane, the sign of ∆(x 0 ) with x 0 any maximal rank element in the plane. Thus I 2 defines an Euclidean or a Minkowskian R 1,1 (hyperbolic or split-complex) structure on the T-plane according to it.
Let us focus on the second case. Endowed by the metric I 2 , T 0 becomes a Minkowski plane. The set of all transformations of the hyperbolic plane which preserve the I 2 form is the group O(1, 1). This group consists of the hyperbolic subgroup SO + (1, 1), combined with four discrete reflections given by x → ±x, x → ±x .
Using standard notation, we say that a non-zero vector x ∈ T 0 is spacelike if I 2 (x) > 0, lightlike, or null, if I 2 (x) = 0 and timelike if I 2 (x) < 0.
In terms of these coordinates the quadrilinear map is given by For ∆(x) < 0 (ε = −1 in the previous expression) the basis thus defined is a null basis (∆(e ± ) = 0), α ± are "null" or "light-cone" coordinates, and then Similar coordinates will be defined on a T-plane.

B.7 The general exponential map
Let us consider now the action on the orthogonal complement of a given element x ∈ F. For any y ∈ T ⊥ x ( that means {x, y} = 0, {x , y} = 0, see Eq. 4.2), we have also seen that ϒ x (y) ∈ T ⊥ x . In conclusion, for any y ∈ T ⊥ x the action of successive applications of ϒ x is restricted to lie on F y⊥x , ϒ n x (y). The orbit of any y ∈ T ⊥ x under σ x (θ ) lies completely on F y⊥x . This can be seen from Eq. 4.18 and Eq. 4.19. These equations allow to explicitly compute the exponential map by summing an exponential series, for any rank-4 x, A similar series is obtained for (exp θ ϒ x )(ϒ x (y)). Summing the series, the exponential of the "normalized" map in the orthognal plane is fully determined by the expressions (ϒ x ≡ ϒ x /( | ∆(x) |,ε = sgn ∆(x)) The geometrical character of the orbits of the exponential of the ϒ x map in the F ⊥ x plane solely depends on ε, the sign of ∆(x), and not, for example, on the signature of y. They are closed (circles or elipses) or hyperbolic, respectively for ε = 1 or −1.

It can be explictly checked that
{e θ ϒ x (y), e θ ϒ x (ϒ x (y))} = {y, ϒ x (y)}. (B.43) We have also, according to Eq. 4.27 ∆(Υ n x (y)) = ∆(y). (B.44) Let us compute now, for a fixed element x, the exponential map exp θ ϒ x on a generic FTS element z, not necessarily on the orthogonal complement T ⊥ x . For that purpose, first we decompose the element on its F x parallel and orthogonal components z = z + z ⊥ .
Without loss of generality we can assume that z = x (if it is not so, we simply realign the F-plane by choosing z as the defining element of the plane: F x ≡ F z ). Then z = x + z ⊥ . The action of any power of ϒ x on z is, by linearity, with ϒ n x (x) ∈ F x and ϒ n x (z ⊥ ) ∈ F ⊥ x . As a consequence, the exponential of the ϒ x (orῩ x ) is of the form ( for z = x + z ⊥ ) : where any of summands is computed independently using the corresponding relations ( Eq. 5.6 and Eq. 5.7 for F x , or, Eq. B.41 and Eq. B.42 for F ⊥ x ). Putting together these relations, one arrives to In the previous sections we have seen how it is natural to define structures on the FTS space as F-planes and F x , their orthogonal complement F ⊥ x . Within any F ⊥ x it results also natural to define planes F y⊥x closed under the action of the ϒ x map. This decomposition of the F ⊥ x space can be performed in a systematic way providing a natural canonical form for any FTS, similar to the Darboux canonical form of any symplectic space.
The orthogonal space F ⊥ x can be furtherly decomposed in 2-dimensional subpaces orthogonal with respect to the antisymmetric bilinear form {, }.
Given a fixed initial element x 0 of maximal rank, let us first define for convenience the shorthand notationẋ ≡ ϒ x 0 (x), for the fixed element x 0 . In particularẋ 0 = 3x 0 = 3T (x 0 ). We will construct on continuation a series of mutually orthogonal vectors iterating the procedure used before (Eq. 4.6) in a sort of modified Gram-Schmidt procedure. Let us initially assume a number of pairs, formed by some vectors and their transforms, (x 0 ,ẋ 0 ), (x 1 ,ẋ 1 ), ...(x n−1 ,ẋ n−1 ), which are already mutually orthogonal, that means (for i, j = 0, n − 1) Where c i are nonzero constants.We now extend this set of pairs by iteration. We show that it is possible to find a pair (x n ,ẋ n ) orthogonal to the previous ones. Let us take an arbitrary vector z and decompose in parallel and orthogonal parts with respect all these vectors, z = z + z ⊥ . The parallel part is easily computed, it is the sum of the parallel parts to each of the individual pairs. It is given by (Eq. 4.6) Obviously, the z defined in this way is on the subspace generated by x i ,ẋ i , (i = 0, n−1). The vector z ⊥ = z − z is orthogonal by construction to all the subspace, It is also straightforward to show that (z ⊥ )˙≡ ϒ x 0 (z ⊥ ) = 3T (x 0 , x 0 , z ⊥ ) is also orthogonal to the full set : In the last line C 0 = 3,C i = 1, (i = 1, n − 1). So (z ⊥ ,(z ⊥ )) is the pair we were looking for, we redefine x n ≡ z ⊥ , (C.9) x n ≡ 3T (x 0 , x 0 , z ⊥ ) =(z ⊥ ). (C.10) The process is iterated as long as we exhaust the dimensionality of the vector space (n = N) or we cannot find vectors with non trivial pairs, (x i ,ẋ i ) = 0. In this way we reduce the sympletic form to a canonical Darboux form. In the basis formed by the vectors (x 0 ,ẋ 0 , x 1 ,ẋ 1 , ...) the the symplectic form is expressed by the matrix

D The Reduced F(J 3 ) case: explicit expressions
We present here some explicit formulas used in Sec. 8. By exploiting the results in App. D of [86], one can compute the components of T (x, y, z) M (Eq. 8.10) in the 4D/5D special coordinates' symplectic frame, characterizing every reduced FTS. Using obvious notation (see Sec. 8), these components read: Then T (r) M = Ω MN T (r) N = −T (r) 0 , −T (r) i , T (r) 0 , T (r) i T , (D. 28) where T (r) 0 , T (r) i , T (r) 0 and T (r) i are given by (8.77). Final expressions are given in the text, see Sec. 8.