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 E7. GFT can be considered as a 2-parameter, a, b ∈ ℝ, generalisation of Freudenthal duality: x→xF=ax+bx˜\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ x\to {x}_F= ax+b\tilde{x} $$\end{document}, where x is the vector of the electromagnetic charges, an element of a Freudenthal triple system (FTS), carried by a large black hole and x˜\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \tilde{x} $$\end{document} is its Freudenthal dual. These transformations leave the Bekenstein-Hawking entropy invariant up to a scalar factor given by a2 ± b2. For any x there exists a one parameter subset of GFT that leave the entropy invariant, a2 ± b2 = 1, defining the subgroup of Freudenthal rotations. The Freudenthal plane defined by spanℝ{x,x˜\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \tilde{x} $$\end{document}} 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 symmetries or U-dualities 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]. This result demonstrates yet again the effectiveness of general relativity, but also serves to emphasise the need to address the long-standing puzzles presented by its BH solutions. A classical stationary BH solution is characterised by its mass M , angular momentum J and charge Q alone. In particular, its horizon area is a simple function of these three quantities. Identifying the horizon area as an entropy (determined up to a numerical constant of proportionality), the classical mechanics of BHs obeys a set of laws are directly analogous to those of thermodynamics [3][4][5]. Hawking's prediction [6,7] that BHs quantum mechanically emit thermal radiation at the semiclassical level fixes the Bekenstein-Hawking area/entropy relation to be precisely (where the usual constants c = = G = 1), and suggests that the thermodynamic interpretation of BH mechanics is more than a mere analogy. However, it also presents an immediate question. A large BH carries a huge entropy, yet is classically characterised entirely by M, J and Q. Where, then, are the microscopic degrees of freedom underpinning the entropy?. Any complete theory of quantum gravity should address this challenge in some way or at least advance in this direction. String/M-theory provides an answer for a very special class of extremal dyonic BHs, where the calculations are made tractable by the presence of some preserved supersymmetries [8]. This result and its generalisations depend on a range of mathematical and theoretical insights. In particular, 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, which provides the low-energy effective field theory limit of string/M-theory. For instance, the non-compact global symmetries of supergravity theories [9], or U-dualities in the context of M-theory [10,11], have played a particularly crucial role, starting with the work of [12]. 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 [14]. Groups of type E 7 are axiomatically characterised by Freudenthal triple systems JHEP07(2019)070 (FTS) [14][15][16]. An FTS is a vector space F with, in particular, a symmetric four-linear form ∆(x, y, z, w) (see section 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 static 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 where ∆(x) := ∆(x, x, x, x) is the unique U-duality invariant quartic polynomial of the BH charges.
In [17] it was shown that when the U-duality group is of type E 7 [14,18], 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 Cs 3 ) and F(J Hs 3 ); cfr. table 1). In [19] Freudenthal duality was then generalised 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 [20], in the framework of ungauged N = 8, D = 4 maximal supergravity. In particular, the most general solution to the supersymmetric stabilisation equations where shown to be given by the F-dual of a suitably defined real 56-dimensional vector, whose components are real harmonic functions in R 3 transverse space. Then, in [21] Freudenthal duality was also shown to be an on-shell 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 [22] it was shown that the generalised, scalar-dependent Freudenthal duality introduced in [19] 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 [22] 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 [23][24][25] 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 [26,27], Freudenthal gauge theory (in which the scalar fields are F-valued) [28], multi-centered BPS black holes [29], conformal isometries [30], Hitchin functionals and entanglement in quantum information theory [31][32][33][34][35]. 2

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Our focus here is on the notion of general Freudenthal transformations (GFT), introduced in [29]. In this work it was shown that F-duality can be generalised to an Abelian group of transformations x → x F = ax + bx. (1. 3) The GFT leave the quartic form invariant up to a scalar factor λ 2 = a 2 ± b 2 , The entropy, ADM mass and, for multicenter solutions in some specific models, the intercentre distances scale as S BH → λ 2 S BH , M ADM → λM ADM , r ab → λr ab , (1.5) while the scalars on the horizon and at infinity are left invariant. The properties of GFT, in particular the 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 [14]. 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. (1.6) The F x -plane is closed under GFT. From (1.4) we see that the quartic form and, thus, the Bekenstein-Hawking entropy, is invariant under the special set of GFT with λ = ±1.
In particular, for any x, ∆(x) = 0 there exists a one-parameter subgroup of Aut(F) that preserves the F x -plane and the Bekenstein-Hawking entropy. These will be referred to as Freudenthal rotations. Although GFT are non-linear, there always exists a linearly acting "gauged" U-duality 3 transformation that sends x to x F . 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 sections 2-5. In the following sections these mathematical tools will be applied to the physics of black holes solutions in supergravity. First, in section 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 section 7 the axion-dilaton 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 section 8 we proceed to the analysis of N = 2, D = 4 supergravities admitting a D = 5 origin. The JHEP07(2019)070 mathematical structure of these models is that of a reduced FTS, which may be derived from a cubic Jordan Algebra, J 3 , so that F ∼ = F(J 3 ). In first place we study the T 3 model, or in Freudenthal terminology F(R).
In subsection 8. 4 we study the question of orbit stratification of the ∆ > 0 locus of F(R) and its preservation by GFT. In section 9 we show, in different examples, how the action of GFT, and, in particular, Freudenthal duality can be realised by U-duality transformations that are "gauged" in the sense that they depend on the element of F to which they are applied. Finally in section 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 [15,16] 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) [40]. 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 [41] and Brown [14] 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 [42][43][44] to be related to the representation of the electric-magnetic (dyonic) charges of black hole solutions.
A FTS is defined [14] 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}.
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).
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.

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The Rank 1 conditions appeared before in [15]. We define the sets of elements of a given rank F (k) ≡ {x ∈ F | Rank(x) = k}. The rank of a element can be related to the degree of supersymmetry preserved by the solution (see [48] and references therein).
Supergravity and the classification of FTS: an outline. An FTS is said to be reduced if it contains a strictly regular element: ∃ u ∈ F such that T (u) = 0 and u ∈ Range L u,u where L x,y : F → F; L x,y (z) ≡ T (x, y, z). It can be proved [14,49] that every simple reduced FTS F is isomorphic to an FTS F(J 3 ), where with J 3 denoting a rank-3 Jordan algebra. All algebraic structures in F(J 3 ) can be defined in terms of the basic Jordan algebra operations [14,49] (also cfr. [50] and refs. therein).
In a Maxwell-Einstein physical framework, the presence of an underlying Jordan algebra J 3 corresponds to the fact that the D = 4 Maxwell-Einstein (super)gravity theories can be obtained by dimensional reduction of a D = 5 theory, whose electric-magnetic (U-)duality 6 is nothing but the reduced structure group of J 3 itself. For F(J A 3 ), the automorphism group has a two element centre, and its quotient yields the simple groups listed e.g. in table 1 of [22], whereas for F(R ⊕ Γ m,n ) one obtains the semi-simple groups SL(2, R) × SO(m + 1, n + 1) [14,46,54]. In all cases, F fits into a symplectic representation of Aut(F), with dimensions listed e.g. in the rightmost column of table 1 of [22].
By confining ourselves to reduced FTS's F(J 3 ) related to simple or semi-simple rank-3 Jordan algebras J 3 , one can exploit the Jordan-Von Neumann-Wigner classification [40], and enumerate the possible FTS's, depending on their dimension dimF = 2N . 7 A summary of this classification is presented in table 1.
Finally, an FTS is said to be degenerate if its quartic form is identically proportional to the square of a quadratic polynomial. Note that FTS on "degenerate" groups of type E 7 (as 6 Here U-duality is referred to as the "continuous" symmetries of [51,52]. Their discrete versions are the U-duality non-perturbative string theory symmetries introduced by Hull and Townsend [53]. 7 Reduced FTS's have at least dimension 2N = 4, namely they contain at least N = 2 Abelian vectors in D = 4. Within the N = 2 interpretation, they are the 5D → 4D Kaluza-Klein (KK) vector (aka the D = 4 graviphoton) and the D = 5 graviphoton (which becomes a matter photon in D = 4). 8 The theories based on Lorentzian cubic Jordan algebras J A 2,1 and J As 2,1 correspond to certain classes of N = 2 supergravities with non-homogeneous vector multiplets' scalar manifolds (cfr. e.g. [55,56]).  defined in [27], 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 .

Freudenthal dualities and planes
F-duality. We have defined already the transformation x = T (x), valid for a vector of any rank (see eq. (2.3)). For rank-4 charge vectors x ∈ F (4) , the black hole charge Freudenthal duality is defined by ( ≡ (x) ≡ sgn ∆(x)) : The Freudenthal duality has the following elementary properties [48]: • It is an anti-involution:x = −x; • It is not a U-duality, since it is non-linear and generically {x,ỹ} = {x, y}. Also, in general, {x, y} + {x,ỹ} = 0.
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 "x-dependent" 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), while 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 ( [14], lemma 11.(abcf)): We can see that eq. (3.4) is equivalent to, or simply summarizes, the relations eqs. (3.5)-(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. Hence, in any F-plane, there is an element y ∈ F x such ∆(y) = 0 if and only if ∆(x) is negative.

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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 section A and section 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

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If we fix x, and restrict ourselves to the F x plane we can use the expressions in subsection B.4 (see also further properties in ref. [58]) and connect ||u|| with I 2 (u): where = sgn ∆(x), η(x) = sgn I 2 (x) andŜ is a linear map given in section B. In particular We arrive at the same conclusions as for the T-plane. For = 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 = −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 subsection 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 (4) , the FTS F may be decomposed as where F x is the 2-dimensional F-plane and F ⊥ x is its (dim F F − 2)-dimensional orthogonal complement w.r.t. the bilinear form {·, ·}:

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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): (ay + bỹ) = ay + b(ỹ) (4.10) with η = sgn( {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 [14], 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 be important in what follows.
The F y⊥x plane. Consider an arbitrary reference vector x (of maximal rank for simplicity) and a perpendicular vector y ∈ F ⊥ x . We define the space, F y⊥x := span{y, Υ x (y)}. That is (4.13)

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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 [14] 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 y ∈ F ⊥ x ) In the second line we have used the Lemma 1(11e) in [14]. 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). (4.19) 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)), Υ x (u) = 3T (x, x, ay + bΥ x (y)) + {x, ay + bΥ x (y)}x (4.20) = 3aT (x, x, y) + 3bT (x, x, Υ x (y)) (4.21) = −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 [14]). 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) ∆ (Υ x (y)) = ∆(x) 2 ∆(y).

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Or, in normalized terms TheΥ x map thus preserves both the bilinear and quartic invariants in each of the F y⊥x planes. Applying twice eq. (4.25) we arrive to ∆(Υ 2 x (y)) = ∆(y) (4.26) and in general ∆(Υ n x (y)) = ∆(y). 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 For example, any element of the form is null, ∆(z ± ) = 0 (for ∆(x) < 0, y ∈ F ⊥ x ). 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 Uduality 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), acting on x itself maps x into its F-dual, Note, we also have the relation ( = sgn ∆(x)) 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 ⊥ x . 9

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, where x ∈ F and exp is defined by the usual infinite series. The proof of eqs. (5.6)-(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 x (x) = 3x , Υ 2 x (x) = 9T (x, x, x ) = −3∆(x)x. By linearity exp θΥ x can be extended to the full F x plane.

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Hence, the set of transformations exp θΥ x form an automorphism subgroup Aut(F x ) ⊆ Aut(F) preserving the F x plane.
To summarise, as a consequence of eqs. (5.6) and (5.7), for any rank-4 x ∈ F, there exists a monoparametric subgroup σ x (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 x (θ) depending on the sign of ∆(x). For ∆(x) > 0, ( = 1) the subgroup σ x (θ) is SO (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 [45,59] and the references therein. Nonetheless, for = 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, ( = −1) the subgroup σ x (θ) is SO 0 (1, 1) which has three different kinds of orbits: the origin (a group fixed point), the four rays {(±t, ±t), t > 0}, and the hyperbolae a 2 −b 2 = ±r 2 . 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 [45,59]. 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 x (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 E 7 -type duality group of any N , large BH have charges x andx in the same U-duality orbit, irrespective of the sign of ∆(x). b) For ∆(x) > 0 the orbit of σ x (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 andx does not preserve the F x -plane. It would perhaps 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 group generated by Υ x has orbits corresponding to null rays.

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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, Introducing a basis 11 {e M } dim F M=1 , the Freudenthal dual x of x can be computed [17,19] by using where we recall that ≡ sgn ∆(x). Note, we have introduce here the dim

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For a generic FTS, we can choose a basis such that Ω is realized as follows: where 0 and 1 denote the N × N zero and identity matrices, respectively.
In the present case and from eq. (6.3) we find 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: and transverse space is obviously empty F ⊥ x = ∅. A general Freudenthal transformation (GFT) depending on the real parameters a, b is given by or, in this case The corresponding extremal RN BH is supersymmetric and 1 2 -BPS (in absence of scalar fields, supersymmetry 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(θΥ) can be explicitly and easily computed. Υ x (ax + bx) ∝ ax + bx. F N =2 "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 12 [61] (see also [62,63]), in which the related FTS is degenerate ∀n V ∈ N ∪ {0}; the corresponding scalar manifold is CP n V .

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In the formalism discussed in section 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 [14], namely they are globally non-reduced; they have been treated e.g. in [64], and their application in supergravity has been discussed in [27] (see also and [63]). Other (infinite) examples of degenerate FTS's are provided by the ones related to the n-parameterised sequence of N = 3, D = 4 supergravity coupled to n matter (vector) multiplets [27,62,65]. On the other hand, N = 4 [66] and N = 5, 13 D = 4 "pure" supergravities have FTS's which do not satisfy the degeneracy condition eq. (6.17) in all symplectic frames, but rather eq. (6.17) is satisfied at least in the so-called "scalar-dressed" symplectic frame [62].
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. [43,64], and refs. therein).
As discussed in section 10 of [27], 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 14 ), 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. (6.7) an using eq. (6.3)) to read y = (−q, p) T = Ω 0 y, (6.20) 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 13 A particularly interesting case is provided by N = 5, D = 4 supergravity [67], which is seemingly related to a non-reduced FTS which is non-degenerate, but also to a triple system denoted by M2,1(O) ∼ M1,2(O) [42,68,69] which deserves a particular study. 14 These cases pertain to simple, degenerate FTS's [27]. No examples of semi-simple or non-semi-simple degenerate FTS's relevant to (super)gravity (D = 4) models are known to us.

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eq. (6.20) 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. (6.11): in fact, the iteration of Freudenthal duality yields This provides the realisation of the Z 4 in the FTS F N =2 "pure" = F x , as a consequence of the anti-involutivity 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 [61] , 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 [70], 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 axiondilaton 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, 15 namely one magnetic and one electric charge p resp. q, the asymptotically flat, spherically symmetric, dyonic extremal BH solution has Bekenstein-Hawking entropy and it is non-supersymmetric 16 (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 eqs. (8.3)-(8.4), 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.

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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 Let us start by choosing, without any loss of generality, a charge configuration given by The corresponding entropy is given by By virtue of eq. (6.7), one can compute the Freudenthal dual 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 Within the formalism discussed in section 3, in F x ⊂ F N =2 ad , it holds that . The Kleinian nature of the metric structure defined on F x ⊂ F N =2 ad corresponds to an hyperbolic (i.e., hyperboloid-like) foliation of F x for ∆ < 0. 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. (6.17) holding true, in the peculiar JHEP07(2019)070 (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 The Freudenthal dual y of y can be computed (by recalling eq. (6.7)) to read 17 withÔ 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 (= 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 Ô and then necessarily of − Ô , 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 eqs. (7.14)-(7.15), we can define a Z 4 symmetry in the 2dim. 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 = 1, the iteration of Freudenthal duality yields This provides the realisation 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.
8 N = 2, D = 4 supergravities from D = 5: the reduced F case 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. 17 By virtue of the discussion made at the end of th previous subsection (also cfr. section 10 of [27]), 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).

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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. In the canonical basis the symplectic product of two generic elements x and y in F reads where Ω is a symplectic matrix.
At least within (non-degenerate) reduced FTS's, the quartic polynomial invariant can be written 18 as follows 19 (cfr. e.g. [44,72,73]; The symmetric quantities d ijk , d ijk follow the so-called adjoint identity of the Jordan algebra J 3 underlying the reduced FTS F (cfr. e.g. [46,73,74] and refs. therein), which reads where the capital Latin indices span the entire FTS F, and K M N P Q = K (M N P Q) is the rank-4 completely symmetric tensor characterizing F [14,27,68]. Note that, from its very definition eq. (8.9), T (x, y, z) is completely symmetric in all its arguments [14].

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Then, by using Ω to raise the symplectic indices, one can compute 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 (nondegenerate) reduced FTS's (cfr. e.g. (table 1)). In this model, it holds that (i = 1, and In the usual normalization of d-tensors used in supergravity literature, 20 it holds that (cfr. e.g. [73]) 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. (6.7). 20 Which, however, is not the one used e.g. in [68].

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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 (8.52) and (6.7)-(8.59) for the T 3 model, one has ( = sgn(x 0 x 0 )) 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)).
Analogously, specifying eq. (8.63) and eq. (8.64) for the T 3 model, one obtains and, according to eq. (8.18), ∆(y) = 1 3 y 1 y 1 2 > 0, (8.30) 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. (8.24), the Freudenthal plane F x related to x eq. (8.25) and the {, }-orthogonal plane F y⊥x = F ⊥ x can respectively be identified as follows: As pointed out above, there is a unique ∆ > 0 orbit in the T 3 model.

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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. (8.31) is a quadratic sub-FTS of F T 3 (as discussed in is not a sub-FTS of F (as discussed in section 5). This can be explicitly checked by relying on the treatment of subsection 8.2; in fact, for the T 3 model, Out of the four cases 1-4 listed at the end of subsection 8.2, only the last one (4) is to be considered: in this case, the condition of Thus, the condition of closure of F y⊥x = F ⊥ x eq. (8.32) 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 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. (8.30) one can 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 section 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 ∆ (Υ x (y)) = ∆(x) 2 ∆(y) = 1 3 x 0 x 0 4 y 1 y 1 2 > 0 (8.36) For a generic element r one gets eq. (4.23) and eq. (4.24) 22 We can explicitly write the Υ map (see eq. (2.6)) (Υx(y)) 0 = 0; (Υx(y)) 1 = 1 2 x 0 x0y 1 ; (Υx(y)) 0 = 0; (Υx(y)) 1 = − 1 2 x 0 x0y1 .

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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 For a positive sign, sgn(x 0 x 1 ) > 0, the dual is a D6 − D2 configuration, it reads x and x belong to the same (rank-4, ∆ > 0) orbit of Aut((F = J 3 )), which is unique in this model (cfr. [75], and refs. therein). For a generic element (8.43) 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

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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). (8.48) 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 , one gets eq. (4.23) and eq. (4.24) ∆(r) = a 2 + ∆(x)b 2 2 ∆(y). (8.49) 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 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. (6.7). We arrive to ( = sgn(p 0 q 0 )) Thus, depending on the sign 26 of p 0 q 0 . 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 Freudenthal plane F x ⊂ F (dimF x = 2), spanned by x and x , whose generic element is 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 Freudenthal 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 section 10. This provides a simple realisation of the Z4 symmetry characterizing every Freudenthal plane, as a consequence of the anti-involutivity of Freudenthal duality itself.

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before and its dual and x. A possible, particular, choice is provided by a D2 − D4 brane charge configuration: In eq. (8.64), the case of vanishing ∆ has been excluded because y is chosen to be of maximal(= 4) rank in F. By recalling (8.2), one can immediately check that 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. (8.71) whose sign depends only on the sign of ∆(y). It is worth remarking that Υ x (y) automatically satisfies eq. (8.70) for every pair y i and y i , with i = 1, . . . , N − 1. In fact, regardless of d ijk and d ijk , 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. Moreover, note that F y⊥x intersects at least three orbits of Aut(F). Indeed, it holds that, using eq. (4.23) and eq. (4.24), (c ≡ x 0 x 0 b) 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. (8.64)) 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.

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Closure of the D2 − D4 F y⊥x under T . The plane F y⊥x is not generally closed under the triple map T (or, equivalently, under Freudenthal duality ∼), see section 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. (8.63) 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 : 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 ijk y i y j y k = 0, d ijk y i y j y k = 0. (8.77) In this case, no solutions exist to the system eq. (8.76), 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 ijk y i y j y k = 0, d ijk y i y j y k = 0. (8.78) 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 x-dependent 1−dimensional 3. y i is rank-3 in J 3 and y i is rank< 3 in J 3 , namely d ijk y i y j y k = 0, d ijk y i y j y k = 0. (8.79) 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 x-dependent 1−dimensional locus a = −x 0 x 0 b in F y⊥x ⊂ F ⊥ x , and not F y⊥x itself, is closed under T .

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4. y i is rank< 3 in J 3 and y i is rank< 3 in J 3 , namely d ijk y i y j y k = 0 = d ijk y i y j y k . (8.80) In this case, the system eq. (8.76) is automatically satisfied ∀a, b ∈ R, and F y⊥x is therefore closed 27 under T . Note that the condition eq. (8.80) is not inconsistent with the assumption of y (8.63) 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. (8.64) is still generally non-vanishing, with the second term vanishing iff d ijk 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 inequality strictly holds), y -and Υ x (y) as wellwould 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: and we further impose that x D0D4 belongs to (one of) the ∆ > 0 Aut(F)-orbit(s) (see section 8.4).
From the definition eq. (6.7) (note that = 1 in this case), one can compute that the Freudenthal-dual x D0D4 of the D0−D4 configuration eq. (8.81) is a D2−D6 configuration, namely 29 The dual is with eq. (6.7) By exploiting the adjoint identity (see [46,73,74] and refs. therein) of the Jordan algebra J 3 underlying the reduced FTS F eq. (8.8) 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 . 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
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. (8.81) and x (8.83); the most general element of this kind is given by the charge configuration: One can also compute that In eq. (8.90), 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 which for this case can be written as  Moreover, (consistently with eq. (4.23) and eq. (4.24)) one can compute that x 0 x j y j d klm y k y l y m + d ijk 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)) 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 (6.9). 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: x = − p 0 , q 0 , p 1 , q 1 , . . . , p N −1 , q N −1 T , (8.102) and in which the symplectic metric eq. (6.9) 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. (6.21). At a glance, in a physical (Maxwell-Einstein) framework eq. (8.102) suggests that the choice of the (canonical) Darboux symplectic frame defined by (8.102) (or, equivalently, by eq. (8.103 and eq. (6.21)), amounts to making manifest the splitting of the electricmagnetic fluxes of the Abelian 2-form field strengths, grouped, within the symplectic vector Q eq. (8.102), 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 31 For some applications of the canonical Darboux frame to supergravity, see e.g. [77][78][79], and refs. therein. JHEP07(2019)070 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 (8.103): each of them corresponds to the electricmagnetic 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 ST U model
At least for (non-degenerate) reduced FTS's, Aut(F) has a transitive action on elements with a given ∆ < 0. Thus, the ∆ < 0 locus corresponds to a 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 (nondegenerate) 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 section 5.1 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 ST U model using only the (discrete) U-duality invariants characterising the orbits.
The ST U model, introduced independently in [81,82], provides an interesting subsector of string compactification to four dimensions. This model has a low energy limit which is described by N = 2 supergravity coupled to three vector multiplets interacting through the special Kähler manifold [SL(2, R)/ SO(2)] 3 . (In the version of [81], the discrete SL(2, Z) are replaced by a subgroup denoted Γ 0 (2)). The three complex scalars are denoted by the letters S, T and U , hence the name of the model [82,83]. The remarkable feature that distinguishes it from generic N = 2 supergravities coupled to vectors [84] is its S − T − U triality [82]. There are three different versions with two of the SL(2)s perturbative symmetries of the Lagrangian and the third a non-perturbative symmetry of the equations of motion. In a fourth version all three are non-perturbative [82,83]. All four are onshell equivalent. If there are in addition four hypermultiplets, the ST U model is selfmirror. Even though the simplest reduced FTS exhibiting more than one ∆ > 0 Aut(F)orbit is given by F(R ⊕ R), which corresponds to the ST 2 model of N = 2, D = 4 supergravity [80], we will explicitly treat F(R ⊕ R ⊕ R) corresponding to the slightly larger JHEP07(2019)070 ST U model, because it can be considered as a genuine truncation of all (non-degenerate) reduced FTS (with the exception of the T 3 and ST 2 models, which are however particular "degenerations" of the ST U model itself), thus covering all such cases.
As determined in [80] (see also the treatment in section F.1 of [75]), in the ST U 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 [73], one can consider a D0 − D4 representative (also in the FTS representation [17]) of the orbits O ∆>0,BP S and O ∆>0,non−BP S,Z H =0 :

Linear realisations of general Freudenthal transformations
As consequence 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 [17] 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: 1. Gauged anti-symplectic transformations, as we will discuss, within non-degenerate, reduced FTS's, for the ∆ < 0 orbit in subsection 9.1; 2. Gauged Sp(2N, R) transformations, where dim R F = 2N . We will discuss these, within the T 3 model (N = 2; cfr. (table 1)), for the ∆ < 0 orbit in subsection 9.2; 3. Gauged Aut(F) transformations, as we will discuss, within the T 3 model, for the ∆ < 0 and ∆ > 0 orbit in subsections 9.3 and 9.4, respectively.
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 realisation 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 section 2 and table 1 therein), it can be proved (cfr. appendix D of [85]) 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: 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 [17]). 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 [23,86,87]: ) . (9.10)

Anti-symplectic symmetries and parity transformations
As observed in [85], at least for all automorphism groups of reduced FTS's over simple or semi-simple rank-3 Jordan algebras it holds that (see e.g. [88]) Thus, all non-trivial elements of Out (Aut (F(J 3 ))) are implemented by anti-symplectic transformations. In [89] (also cfr. [90]), 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 [89], 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 realisation (not the unique one, though! -see subsections 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. (8.52)).

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 [45]. The corresponding Aut(F) transformations taking x andx to x can andx can , respectively, are denoted U and U . Similarly, M x and M xcan 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

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where we recall that Aut (F (J 3 )) Conf (J 3 ), Conf (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 [75,80]. Let us also choose a convenient representative of O ∆<0 . An obvious choice is given by the D0 − −D6 configuration eq. (8.52), 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 [17]. 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.

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where the Lie bracket is given by 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

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Concerning the ∆ > 0 locus of F, it is generally stratified in two or more orbits under the non-transitive 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 [75,80]: 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 [75] 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 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 ST U model, namely we search for a 2 × 2 matrix M such that where x D0D4 in the FTS parametrization (following the conventions of [75]) 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):

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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. (6.21). 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 [17]).
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 and using as given in eq. (9.52). 34 Generically, this is actually a larger class of charge configurations, but in the special cases of the T 3 , ST 2 , ST U models it is precisely the D0D4 subsector. For N = 8, imposing A is diagonal restricts to the D0D4 subsector.

Non-trivial orbit stabilizers
Here, generalizing the reasoning at the end of subsection 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 = U x; 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 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. 35 If U and U are more general finite transformations, such as "gauged" symplectic (not belonging to Aut(F)) or anti-symplectic transformations, things become more complicated, since, for instance, finite transformations of the pseudo-Riemannian non-compact non-symmetric coset Sp(2n, R)/ Aut(F) do not generally preserve the Aut(F)-orbit structure of F, and thus they generally do not commute with Freudenthal duality.

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10 Summary, concluding and further remarks The purpose of this work has been to present, extend and clarify old and new results concerning general Freudenthal transformations, which generalise Freudenthal duality, while filling in necessary details not yet appearing in the mathematics or physics literature.
We begin with a detailed and self-contained treatment of FTS, groups of type E 7 , F-duality and GFT in sections 2-5, laying the groundwork for the subsequent sections that apply the formalism to Einstein-Maxwell-Scalar (super)gravity theories.
In section 6 we study the entropy properties of N = 2, D = 4 pure supergravity from the point of view of FTS formalism, where is it given by a non-reduced FTS with positive-definite quartic invariant. In this case, general Freudenthal transformations are nothing other than the familiar U(1) electromagnetic duality and Freudenthal duality is an anti-involutive duality transformation.
In section 7 we considered the axion-dilaton model, an 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-semi-definite quartic invariant. The corresponding extremal BH is non-BPS. A SO 0 (1, 1) subgroup of the GFTs leave invariant the entropy. Freudenthal duality reduces in this case to a 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 section 8 we proceeded to the analysis of N = 2, D = 4 supergravities with a 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 the 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 SO(1, 1) representations, making the D = 5 origin manifest. The full space is split into two two-dimensional planes orthogonal (with respect to the symplectic bilinear form), which can be identified respectively with purely D6 − D0 or Freudenthal transformations forming a D6 − D4 plane, which is Υ-mapped to a D2 − D0 plane. Similar results are obtained in general theories beyond T 3 containing an arbitrary number of charges, they are studied in full detail in subsection 8.2 and subsection 8.3.
General Freudenthal transformations, and in particular Freudenthal duality ∼, will preserve the orbit structure for ∆ < 0. This is trivially true since, at least for (nondegenerate) reduced FTS, Aut(F) has a transitive action on elements with a given ∆ < 0. The situation is more complicated for the locus of ∆ > 0 configurations, for which, in general (with known exceptions) are stratified in two or more automorphism orbits. In subsection 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 result generalises to all cases with (nondegenerate) reduced FTS, as they can either be invectively embedded in, or truncated to, the ST U model.
In section 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 (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 within the ∆ < 0 locus of F, the action of Aut (F) is always transitive on elements of a given ∆, thus defining a unique one-parameter family of orbits {O} ∆(x)<0 of F. The ∆ > 0 locus of F is generally stratified in two or more orbits for any fixed ∆ > 0 under the non-transitive action of Aut(F). However it is possible to find particular cases where this stratification does not take place. For example, the T 3 model has a unique orbit for all ∆ > 0 under the SL(2, R) automorphism group. Hence, for the T 3 model all BH with ∆ > 0 are (1/2)−BPS. This example is studied in subsection 9.4.
The entropy of a linear superposition of configurations x, y is given by 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 + b 2 = ±1, with = sgn ∆(x). We can use these results to show that it is possible to construct asymptotically "small" (zero entropy) interacting black holes from an initial non-trivial configuration. First, when ∆(x) > 0 ( = 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 ( = −1) it

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follows that x F ± are null elements 36 ∆(x F ± ) ≡ ∆(x ±x) = 0. (10.6) 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. Alternatively, small BH solutions can constructed by the 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 subsection 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. For ∆(x) < 0, y ∈ F ⊥ x , any element of the form is null, ∆(z ± ) = 0. This describes another class of two-centred black hole configurations, which are interacting since {y, Υ x y} = 0, yet asymptotically their Bekenstein-Hawking entropy vanishes S BH y ± Υ x (y) = 0. (10.9) Further work on such small BHs will be presented elsewhere. The extension of these results to systems with quantized charges is challenging. 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 recall 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 preserved 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 defining the relevant algebra determines the BH entropy to lowest order. The Jordan dual (introduced in [48]) is related to the Freudenthal dual of the corresponding 4D model. The generalization of this 4D/5D correspondence from a general Freudenthal transformation to the corresponding putative "General Jordan Transformations" (GJT), of which the Jordan dual is a particular case, will be treated elsewhere [94].

Acknowledgments
The work of LB is supported by a Schrödinger Fellowship.

A Freudenthal triples: assorted properties
A summary of some FTS definitions, notation and properties used through this work. See ref. [14] for additional ones and proofs.
For any two vectors u 1 , u 2 ∈ F y⊥x we have the following properties The first two expressions are obtained by direct computation. To get eq. (B.10), we note that where we have used eqs. (B.4), (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),

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As consequence, for = 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 ).

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where 0 = sgn ∆(x 0 ) (we call 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 ) ∆ T0 (x, y, z, w) = 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 section 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 [58]) with a characteristic "signature ", 0 , and whose symmetric quadrilinear and antisymmetric bilinear forms are those inherited from the original FTS.
If 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 section 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 ( = −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 θΥ x )(Υ x (y)). Summing the series, the exponential of the "normalized" map in the orthogonal plane is fully determined by the expressions (Υ x ≡ Υ x /( | ∆(x) |, = sgn ∆(x)) exp θΥ x (y) = cos √ θ y + √ sin √ θ Υ x (y) , (B.41) exp θΥ x Υ x (y) = − √ sin √ θ y + cos √ θ Υ x (y) . (B.42)

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The geometrical character of the orbits of the exponential of the Υ x map in the F ⊥ x plane solely depends on , the sign of ∆(x), and not, for example, on the signature of y. They are closed (circles or ellipses) or hyperbolic, respectively for = 1 or −1.
It can be explicitly checked that {e θΥx (y), e θΥx (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 θΥ 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 ⊥ ): (exp θΥ x )(z) = (exp θΥ x )(x) + (exp θΥ x )(z ⊥ ) (B. 46) where any of summands is computed independently using the corresponding relations (eqs. (5.6) and (5.7) for F x , or, (B.41) and (B.42) for F ⊥ x ). Putting together these relations, one arrives to e θΥx (z) = cos √ θ 3 In particular, for any z ⊥ in the orthogonal space, the vector of the exponential map (strictly, we have to deal with a complexified FTS for = −1, see subsection B.1): The map exp(θΥ x ) over a generic element y in the FTS can be obtained from the previous formula performing a suitable rotation in the F x plane bringing x to a generic y , JHEP07(2019)070 C FTS Darboux canonical form: a foliation on F y⊥x planes 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 further 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 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) z = i=0,n−1 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:

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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 symplectic 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 section 8. By exploiting the results in appendix D of [97], one can compute the components of T (x, y, z) M eq. (8.9) in the 4D/5D special coordinates' symplectic frame, characterizing every reduced FTS. Using obvious notation (see section 8), these components read:
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