d-Geometries Revisited

We analyze some properties of the four dimensional supergravity theories which originate from five dimensions upon reduction. They generalize to N>2 extended supersymmetries the d-geometries with cubic prepotentials, familiar from N=2 special K\"ahler geometry. We emphasize the role of a suitable parametrization of the scalar fields and the corresponding triangular symplectic basis. We also consider applications to the first order flow equations for non-BPS extremal black holes.

were first studied in [5], where they were shown to lead to supergravity couplings with flat potentials characterized by the completely symmetric rank-3 tensor d ijk . They are particularly relevant in connection with the large volume limit of Calabi-Yau compactifications of type IIA superstrings where the d-tensors are related to intersection forms of the Calabi-Yau manifold. Formally, the d-tensor appears in the expression for the curvature of any special Kähler manifold [6] R ikl = −g i g kl − g il g k + C ikp Clpg pp (1.2) since in "special coordinates" the covariantly holomorphic quantity C ijk is given by C ijk = e K(z,z) d ijk , with K(z,z) denoting the Kähler potential.
Notice that a generic d-geometry of complex dimension n V is not necessarily a coset space, but nevertheless it admits n V + 1 real isometries, corresponding to Peccei-Quinn shifts of the n V axions, and to an overall rescaling of the prepotential [3].
This paper aims to study d-geometries in a framework broader than N = 2, considering the r-map for N ≥ 2 extended supergravities along the lines of previous work on this 4D/5D JHEP02(2013)059 relation in the context of black hole supergravity solutions and their attractors [7][8][9]. Due to the structure of 5D spinors, these generalized d-geometries encompass all extended supergravities with a number of supercharges multiple of 8, and thus an even number of supersymmetries N = 2, 4, 6, 8. d ijk is an invariant tensor of the underlying classical duality group G 5 of the D = 5 action [10], corresponding to the continuous version of the non-perturbative string symmetries G 5 (Z) of [11]. The dimensional reduction yields interesting relations between the scalar manifolds and the isometries of the 5D and 4D theories: G 5 is embedded into the D = 4 electric-magnetic duality group G 4 , whose isometries are included in Sp(2n V + 2, R) (for generic N > 1, one has Sp(2n, R) for a theory with n vector potentials; for N = 2, n = n V + 1). More precisely, one always has the chain of embeddings Our main point is that the five-dimensional origin of all generalized d-geometries naturally selects a particular branching of the D = 4 scalars, given by the axions a I , the Kaluza-Klein scalar φ and the 5D scalars λ x : Φ = a I , φ, λ x .
( 1.4) When N > 2 these latter transform in a suitable representation of H 5 , the maximal compact subgroup of G 5 , which depends on N : for instance, in N = 8 there are 42 of them, sitting in the rank-4 antisymmetric skew-traceless representation 42 of USp (8), and there are 27 axions. Remarkably, only in N = 2 the number of axions exactly matches the number of scalars plus 1, so that the two sets can be combined to give complex scalars. For this case we will use a small index i rather than I, to emphasize its complex nature. We will illustrate that the a I and φ give rise to a universal sector which is present in any N = 2, 4, 6, 8 -extended supergravity in D = 4 endowed with generalized d-geometry for the vector multiplet sigma model.
In the study and classification of BPS and non-BPS extremal black hole supergravity solutions, the relation between 4D and 5D for cubic holomorphic prepotentials F (X) (1.1) was used in [7] to relate the two N = 2 effective black hole potentials and to derive the 4D attractors and Bekenstein-Hawking classical entropies from the 5D ones. The key idea was to reformulate the 4D effective black hole potential in terms of 5D real special geometry data, implementing the natural splitting (1.4) of the 4D scalar fields.
Some extra features arise in symmetric special geometries, where the d-symbols satisfy the relation [1] and one can define cubic , G 5 -invariant, and quartic, G 4 -invariant polynomials of electric (q 0 , q i ) and magnetic charges (p 0 , p i ) by [12]: (1.7)

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The simplest example of rank-3 symmetric d-geometry is provided in N = 2 by the stu model [13,14], with 3 complex scalar fields spanning the coset (SU(1, 1)/U(1)) 3 , which serves as the ubiquitous toy model in the context of black holes arising from superstring and M -theory. The generalization of N = 2 special geometry is achieved in terms of a generalized symplectic formalism, established in [15], which enlarges the rich geometric structure of special Kähler manifolds [3] to the other extended supergravities. In fact, an important difference between N = 2 and N > 2 extended theories is that for N > 2 the scalar sigma model is always given by a symmetric space G/H. The formalism of [15] hinges on the definition of generalized sections (f , h) of a flat symplectic bundle [16], which relates to N > 2 the flat bundle underlying special Kähler geometry [17]. Even in N = 2 the sections are fundamental, since they allow to describe also theories where the holomorphic prepotential F (X Λ ) does not exist [18,19]. More precisely, the sections V A = (f Λ A , h ΛA ), with Λ = 0, . . . , n V and A = 0, a, are square complex matrices defined in N = 2 supergravity by with (L Λ , M Λ ) = e K/2 (X Λ , F Λ ), D a denoting the flat covariant derivative in the scalar manifold: where N ΛΣ (z) is the 4D complex vector kinetic matrix. The sections encode a generic element L of the flat Sp(2n V + 2, R)-bundle over the D = 4 scalar manifold as [15] A 10) or the inverse transformation with the symplectic property L T ΩL = Ω = 0−1 1 0 yielding the conditions This paper studies in detail the properties of a certain parametrization (2.2), (2.13) of four-dimensional generalized d-geometries, which reflects their five-dimensional origin, yielding a lower-triangular structure (2.13) for the matrix L characterizing the flat symplectic bundle sigma model which generalizes the one of N = 2 special Kähler d-geometry to any for any N = 2, 4, 6, 8. This parametrization exploits nilpotent (of degree 4) translations [17,20,21] parametrized by axion scalars a I , and it acts on the same space where the d-tensor is defined. The sigma model is parametrized by additional block diagonal elements in the matrix L, one of them being a dilatation in terms of the KK radius φ, and JHEP02(2013)059 by a symmetric matrix, which depends on the 5D data and is related to the kinetic term of the 5D vector fields.
It should be stressed that the proposed basis turns out to be different from the standard parametrization of N = 2 d-geometry (1.8), although it leads to the same 4D vector kinetic matrix. We will emphasize that the two symplectic frames are in fact related by a unitary transformation M that was introduced in [9], which only depends on the 5D data. The unitary transformation M , that rotates the usual N = 2 complex basis of special geometry into the basis where f is real and L is lower triangular, allows to make a precise connection with the N = 2 stu model, viewed as a sub sector of the full N = 8 theory [15,22,23]. In the t 3 model, this unitary transformation is numerical (cfr. appendix B), because the relevant 5D uplifted theory is the pure N = 2, D = 5 supergravity.
Symmetric d-geometries can be related to Euclidean Jordan algebras of rank 3 [1,24], which were classified in [25]; in this case, the nilpotent axionic translations fit into a Jordan algebra irreducible representation. The reduction to D = 4 yields a Freudenthal triple system (see e.g. [12]).
Our results have interesting applications to non-BPS extremal black holes, that we illustrate by making a precise and non trivial comparison between the methods of [22] and [26] in the computation of the fake superpotential [27] for non-BPS solutions and (p 0 , q 0 ) charge configuration in the stu-truncation of N = 8 supergravity.
Beyond their interest in relation to supergravity structure and solutions, one may hope that these general properties of N ≥ 2 d-geometries and the corresponding triangular symplectic frame (with degree-4 nilpotent axionic translations) could play a role in understanding the symmetry structure of supergravity counterterms, in order to clarify the issue of ultraviolet finiteness of N = 8 and other extended supergravity theories in D = 4 space-time dimensions [28].
The paper starts in section 2 with the universal decomposition for the D = 4 symplectic element L in the proposed basis 1.4, where axion are singled out. Then, the relation between L and the matrix M entering the black hole effective potential is elucidated in section 3. Other geometrical identities in a 5-dimensionally covariant formalism are presented in section 4. The simpler case of N = 4, D = 4 pure supergravity (with no matter coupling) is discussed in section 5. For d-geometries based on symmetric spaces G/H, the computation of the Vielbein and of the H-connection is carried out in section 6, in particular focusing on N = 8 supergravity. Next, in section 7 the N = 2 axion basis is related to the reformulation of special Kähler geometry as flatness condition of a symplectic connection [17].
A detailed treatment of N = 2 d-geometries is then given in section 8, where we elaborate on the results of [9] on the unitary matrix M rotating the axion basis to the usual special coordinates one. Geometrical identities for M and the related matrix M are derived in section 9.
An application of the axion basis to the first order formalism for extremal black holes is considered in section 10. After a preliminary analysis for the stu model in sections 10.1.1 and 10.1.2 , explicit computations for the t 3 limit in the p 0 , q 0 (D0 − D6) charge configuration are performed in sections 10.1.3, and the known fake non-BPS superpotential is JHEP02(2013)059 retrieved in section 10.2. In table 1 we list the allowed Rank-3 Euclidean Jordan algebras J 3 and corresponding symmetric generalized d-geometries, characterized by a parameter q related to the number of vector and scalar fields for each N = 2, 4, 6, 8. Some appendices conclude the paper. In appendix A useful results on exponential matrices are collected, while appendix B contains some explicit computations in the t 3 model, displaying the matrix M . The purely imaginary nature of the Vielbein of the stu model and its consistent embedding into the N = 8 theory are discussed in appendix C. Finally, appendix D deals with the duality-invariant polynomial and the first order fake superpotential in the D0 − D6 configuration of the stu model with i 3 = 0.
2 Universal decomposition for the D = 4 symplectic element in the axion basis We are interested in general features of all D = 4 Maxwell-Einstein (super)gravity theories admitting an uplift to D = 5. The classification of the tensors d IJK associated to homogeneous Riemannian d-spaces was performed in [3]. For symmetric geometries, d IJK can be characterized as the cubic norm of an associated rank-3 Jordan algebra 1 [1,25]. In this case, the general properties are given in terms of a parameter q reported in table 1.
The number of D = 5 vectors is n V = 3q + 3, while the number of D = 4 2-form field strengths and their duals is 6q + 8. Only in N = 2 theories, the number of 5D real scalars is 3q + 2, while the number of 4D complex scalars is 3q + 3 (one for each 4D Abelian vector multiplet). Quite generally, the relation between the number of vector and scalar fields in theories derived from five dimensions is such that # 4D scalars = # 5D scalars + # 5D vectors + 1 # 4D vectors = # 5D vectors + 1 = n V + 1 , where the n V axions arise from the total number of 5D vectors. We will show that in these generalized d-geometries, the representation of the D = 4 axions a I is nilpotent of degree four and that, together with the Kaluza-Klein SO(1, 1) radius parametrized by the real scalar φ, it provides a universal sector of the scalar manifold of the D = 4 theory, regardless of its specific geometry. This reflects the property of special Kähler d-geometries [3], of always having as minimal isometry of the scalar manifold the n V axionic Peccei-Quinn translations and the SO(1, 1) overall rescaling.
To prove the above statement, we split the symplectic element L according to the decomposition of the D = 4 scalars (1.4), and we demonstrate that 2 In order to identify the various factors in (2.2), one must consider the definition (1.11) and complement it with the results of [9], where the 4D/5D connection was used for N = 8 1 With the exception of the non-Jordan symmetric sequence [29] of N = 2, D = 5 vector multiplets' scalar manifolds SO(1,n V ) SO(n V ) . 2 In the following we will switch the axion index from i into I, whenever our analysis holds for generic N 2 d-geometries.

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to determine the 28 × 28 symplectic sections (f Λ A , h ΛA ) in a five-dimensionally covariant symplectic frame, where the indices split as Λ = (0, I) and A = (0, a). They take the form: and where is the coset representative of the 5D scalar manifold G 5 /H 5 . Notice that in this basis the section f is real and it takes a lower triangular form, and that the 5D scalars enter the sections only through E(λ). By generalizing this 5D/4D approach to the class of theories under consideration and interpreting the indices Λ, A on the appropriate representations, we determine the generic expression for each factor in (2.2).
The axionic generators also appeared in [30] in the context of gauging of flat groups in 4D supergravity, and they are given by the 2(n V + 1) × 2(n V + 1) block-matrix It is easily checked that T (a) is nilpotent of order four: which, by definition (2.7), yields (2.10) As we will discuss in section 8, this is in agreement with the N = 2 interpretation of [21]. The 1-dimensional Abelian SO(1, 1) factor in (2.2) is given by

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whereas the (2n V + 2) × (2n V + 2) matrix G is By matrix multiplication of (2.10)-(2.12) according to (2.2), one finds that the symplectic matrix L (1.11) acquires the triangular form: We see that, in this particular basis, B = Im f = 0, since the f section is purely real: (2.14) On the other hand, one has along with the normalization Notice that the C sub-block is the only one depending on d IJK . Conversely, one can say that the formula (2.13) for the symplectic representative yields an explicit expressions for the symplectic sections f and h which match eqs. (2.3) and (2.4).
The symplectic sections (2.3) and (2.4) are given in the particular symplectic frame defined by the partial decomposition of L (2.13) in a solvable basis, which is covariant with respect to H 5 = USp(8), the local symmetry of the D = 5 uplifted theory. Furthermore, E (λ) is the coset representative of the rank-6 symmetric D = 5 scalar manifold

Relation between M and L
We now consider a further consequence of the symplectic structure of generalized special geometry [15], holding for every D = 4 Maxwell-Einstein supergravity even beyond dgeometries. It can be useful in the present context and in view of applications to black holes. The black hole effective potential for dyonic charges Q = (p Λ , q Λ ) is given by [33] where the central charges Z A =< Q, V A > are defined by the symplectic product in terms of the symplectic invariant metric The matrix M is given by where N = hf −1 is the D = 4 kinetic vector matrix.

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JHEP02(2013)059 whose symmetric and antisymmetric parts are given by (3.4) and Ω respectively. C is related to the symplectic sections (f , h) by: and therefore its action on the vector V A is given by expressing a twisted self-duality [34], recently used in [35].
Using the above relations, since both M and L are given in terms of the sections (f , h), one can see that they can be related by [36,37] (3.10) where the last step in (3.11) follows from the symplecticity of L itself. Notice that, since also M is symplectic, (3.10) implies that M = −LL T , withL ≡ ΩL. To prove (3.10)-(3.11), one just notices that L (1.11) can be rewritten as (with * here denoting complex conjugation) which, by (3.9) implies By sandwiching (3.10) with the dyonic charge vector Q, one also obtains where the real central charge vector Z satisfies which were derived in [9] for N = 8, but that we can here interpret as valid for all generalized d-geometries. The components with flat indices are obtained by so that the complex central charge vector with flat indices is and the effective black hole potential is written as [9] V (3.20)

5D-covariant identities
In the 5D covariant formalism introduced in [9], it was found that the kinetic vector matrix N ΛΣ in N = 8, D = 4 supergravity can be decomposed as: In virtue of the discussion of section 2, these formulae hold for any d-geometry. Note that ImN depends on the axions a I but not on d IJK , whereas ReN only depends on axions, and only through d IJK . It is immediate to realize that this is a consequence of the solvable decomposition (2.2) of L, as well as of the relation (3.10) between M and L. Indeed, using (3.5), the matrix A (2.10) can be rewritten as Then, since DG is a diagonal matrix, (3.10) implies

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Using (2.11), (2.12) and (4.2), one can check that (4.5) As mentioned, this explains the dependence of ImN on axions alone and not on the dtensor, and that of ReN on axions only through d IJK .
5 A related case: N = 4, D = 4 pure supergravity Although pure 4D N = 4 supergravity cannot be obtained from five dimensions by Kaluza-Klein reduction, which would always give rise to the coupling to matter multiplets, we mention it here because of the recent related work of [38] and as a simple instance of the splitting of scalar fields associated with (2.2). The vector kinetic matrix N ΛΣ in this case reads [39] (Λ, Σ = 1, . . . , 6) where the axio-dilatonic complex scalar field S of the gravity multiplet, spanning the rank-1 symmetric coset G/H = SL(2, R)/SO (2), is defined as A solvable basis can be defined also for this theory as in (5.1), and it is given by the axio-dilatonic symplectic frame , where the relevant matrices read such that the coset representative L of SL(2, R)/SO(2) satisfies In this case the axionic generator is nilpotent of order two rather than of order four, as for generic d-geometries: The different degree of nilpotency is due to the fact that this theory does not admit a 5D uplift and thus it is not a d-geometry in absence of matter coupling.

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6 Vielbein and H-connection in the axion basis When the d-geometry is not only an homogeneous but a symmetric cosets G/H , the Vielbein P µ and H-connection ω µ in a solvable decomposition can be simply computed from the (g ⊖ h)-valued Maurer-Cartan 1-form L −1 dL by standard methods where subscripts "s" and "a" denote the symmetric and antisymmetric part, respectively. The simplest example is provided by the axio-dilatonic coset G/H = SL(2, R)/SO(2) treated above, whose coset representative is given by (5.5), with Maurer-Cartan 1-form leading to the Vielbein P µ and U(1)-connection ω µ respectively given by In particular, one sees that the U(1) connection ω µ contains only the da differential. The kinetic term for the nonlinear σ-model SL(2, R)/SO(2) therefore reads [39] Tr We now consider in particular N = 8 supergravity, where the Cartan decomposition for the D = 4 scalar manifold (2.16) reads According to (2.18)-(2.18), the following usp(8)-covariant branchings take place: The coset Vielbein P µ is given by the non-compact generators while the compact ones give the SU(8)-connection ω µ

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The Maurer-Cartan 1-form gets generally decomposed as From the definitions (2.10), (2.11) and (2.12), one can compute This implies that the Maurer-Cartan 1-form L −1 ∂L does not depend on the axions a I explicitly, but only on their differential da I . According to (6.1) and (6.2), the Vielbein P µ and SU(8)-connection ω µ for the coset (2.16) are the symmetric and anti-symmetric part of (6.12), respectively. In particular, the component 27 k of P µ and the component 27 h of ω µ respectively read: (6.17)

Flat connections and axion basis
As shown in [17] and further investigated in [21], the defining identities of N = 2 special Kähler geometry can be viewed as the flatness condition of a non-holomorphic connection

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A I and can be encoded into a first-order matrix equation [21] ( where U is a non-holomorphic matrix (V, D i V, DīV , V ) with V = (X Λ , F Λ ). One can further choose a gauge where A i becomes holomorphic such that (7.1) can be recast as follows: with now an holomorphic solution matrix V containing V in the first row. In turn, the holomorphic flat connection A i can be decomposed as where Γ i is the diagonal part (which vanishes in special coordinates), and C i generates an Abelian subalgebra of sp(2n + 2, R) that is nilpotent of order four: The case of special Kähler d-geometry in the axion basis basis is analysed in appendix C of [21]. In particular, by recalling (2.8), one can compute the axionic generators of the solvable parametrization of the D = 4 scalar manifold treated above as Up to relabelling of rows and columns, (7.6) matches the expression of C i (for n = 27) given by (3.6) of [21]. For N = 2 special Kähler d-geometries (namely, for those special geometries admitting an uplift to D = 5) in the axion basis, this highlights the relation between the solvable parametrization of the D = 4 scalar manifold discussed in section 2 and the nilpotent connection of the reformulationà la Strominger in the holomorphic gauge (7.2). 8 N = 2 special Kähler d-geometry, symplectic sections and the unitary matrix M In this section we are going to make contact with N = 2 special Kähler d-geometries [3] in the symplectic frame defined by the cubic prepotential (1.1). We recall for convenience some results of [7] and we build on them. It has already been remarked that N = 2 special Kähler d-geometry differs from the higher N -extended theories in that the n V 5D axions a i exactly combine with the 5D scalars λ i = λ i (λ x , φ) in order to give complex 4D scalar fields JHEP02(2013)059 Moreover, in N = 2 the central charge can be readily computed from the cubic prepotential F (X) of eq. (1.1) by the usual formula (3.2) For N = 2 cubic geometry one finds [7] where 4) with the (real) Kähler potential and its (purely imaginary) derivatives given by Notice that i is a curved index of the 5D U-duality group G 5 , and Λ = (0, i). The connection with the universal basis is given by introducing n V 5D scalars asλ i = e −2φ λ i so that they satisfy d ijkλ iλjλk = 1. The n V complex 4D scalar components are then (a i , φ,λ i ) . The special Kähler metric is given by One can assemble Z and DīZ into a symplectic central charge vector Z α with a curved lower index Then, from Z α in (8.2) and (8.3) one can read off the components of V α , which are

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While it can be checked that we should better consider the normalized symplectic sections with flat tangent indices They are the components of Z A = (Z, D a Z), and they can be obtained by flattening the curved indices i by the G 5 -Vielbein e a i , 3 so that the orthonormalized symplectic sections f Λ A and h ΛA are given by It was emphasized in [9] that the symplectic sections f and h of (generalized) special geometry are defined only up to the action of a unitary matrix M , which preserves the form of the kinetic vector matrix N = hf −1 and the conditions (1.12) derived from symplectic invariance of L. Actually, the matrix M found in [9] to connect N = 2 with N = 8 is exactly the necessary one to rotate the usual basis of special geometry into the axion basis of any d-geometry. It can be written as where ∂K = 2iλ i g i ; (8.18) By further rescaling the D = 4 dilatons as the matrix M (8.16) can be recast as follows:

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Using (1.11), one can see that the action (8.15) of M induces the following transformation of the coset representative L: where the real symmetric and unitary matrix Indeed, since L is symplectic, one has checked that also Y is symplectic, but given (8.26), this leads to [Y, Ω] = 0 .

Unitarity relations for M and induced relations onM
The residual freedom in the definition of the symplectic section was found in [9] to imply that the symplectic vector Z A = Z, D a Z T of N = 2 special geometry, with a flat index

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This is obvious from the fact that the N = 2 sections in (8.10) are not lower triangular, as required in the axion basis in (2.3) where the the symplectic section f is real. Notice that the E 6(6) basis is related to the usual de Wit and Nicolai symplectic frame by a symplectic transformation [8] . However, under a change of symplectic basis, that is a duality transformation, the kinetic matrix transforms as N where a IJ is the kinetic vector matrix of N = 8, D = 5 supergravity. In the E 6(6) -frame of 4D N = 8 supergravity, the symplectic section with curved indices f read [9] where, in the symmetric gauge [8], Λ = 0, I and α = 0, I, where here I is a curved index spanning the 27 of E 6(6) . From (8.10), (9.5) and (9.3), one can compute the matrix [9] which does not depend on the axion fields. Moreover, using (8.14), (9.3) and (8.15), the relation betweenM and M is given by The unitarity of M entails the following identities forM , namely: In this section we show an interesting application of the axion basis to non-BPS extremal black holes. The unitary transformation M that rotates the usual N = 2 basis of special geometry Z A into the E 6(6) basis Z A allows to make a precise connection with the N = 2 stu model, where the three complex scalar fields z i = {s , t , u} span the rank-3 coset space viewed as a sub sector of the full N = 8 theory [15,22,23]. The aim is to illustrate the computation of the fake superpotential for non-BPS solutions and (p 0 , q 0 ) charge configuration in the stu-truncation of N = 8 supergravity. This example was discussed from two different viewpoints: in [22] the fake superpotential was computed for generic charges in terms of duality invariants of the underlying special geometry, while in [26] Bossard, Michel and Pioline (BMP) provided a procedure based on nilpotent orbits which lead to the fake superpotential as solution of a sixth order polynomial. The virtue of the axion basis is that, while showing the equivalence of the derivation of [26] and [22], we can read out the fake superpotential from the N = 8 central charge in the skew symmetric form. Here we start from the formula for the central charge derived in [9] using 4D/5D special geometry relations, and we look for a suitable SU (8) transformation that brings it to the form given by eq. (2.68) of [26] In particular, we study the effect of such a rotation with respect to the decomposition 28 → 1 C + 27 C , which is common to the central charge normal frame of both [9] and [26]. We identify this transformation in the t 3 -truncation where it depends only on one angle χ, purely given in terms of duality invariant quantities. When this rotation is used to match the central charge in [9] and that of [26], we consistently retrieve the non-BPS fake superpotential for the N = 2 t 3 model, within the (p 0 , q 0 ) charge configuration in presence of non zero axions. This is a non-trivial consistency check for the 4D/5D formalism based on the matrices M and M [9] detailed in previous sections.
The key point of this analysis is that the 28 components of the N = 8 central charge matrix Z AB can be traded for the symplectic vectors Z A (with flat lower index) or Z α (with a curved one) reflecting the splitting 28 = 1 C + 27 C of the axion basis. Since Z AB can always be brought to the skew-diagonal form one has to relate the eigenvalues z 1 , z 2 , z 3 , z 4 with the complex components of Z α = (Z 0 , Z I ) [9], with I = 1, 2, 3, 0 + iZ 0 (m) ) , In order to find the skew eigenvalues z 1 , z 2 , z 3 , z 4 in (10.3), one needs the inverse metric, which in this case is factorized as as well as the purely imaginary Vielbein (see appendix C) 9) and the Kähler connection (10.10) Using ( (10.14) By recalling the definition λ i V −1/3 = λ i e −2φ ≡λ i (cfr. section 8), and defining one computes The 4D/5D covariant splitting is thus manifest in the following form of the central charge matrix 4 [9] This result, compared with formulae (3.2) of [9], explains the definition in which Ω = ǫ ⊗ id 4 , given in eq. (4.7) of the same reference; notice that the overall phase i is uninfluential.

Residual U(1) 3 symmetry of the skew-diagonal Z AB
The form of the central charge, as derived in the previous section, reflects the more general structure of the 28 → 1 C + 27 C decomposition of SU(8) ⊃ USp (8) representation. The central charge matrix for the p 0 , q 0 configuration in N = 8 Supergravity has been given in [26], in the same symplectic frame. The reason why this is a suitable frame to study the non-BPS orbit is related to the choice of orbit representative. The moduli space of the non-BPS p 0 , q 0 solution is indeed the moduli space of the 5 dimensional theory, namely E 6(6) /USp (8) . By solving a nonstandard diagonalization problem, the authors of [26] identify the fake-superpotential in the singlet of the axion-base decomposition of the central charge matrix. However, the form of Z AB is unique up to SU(8) transformations,

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and the choice of symplectic frame is not covariant with respect to the action of SU (8), since the singlet is not left invariant by R-symmetry rotations.
Starting from the form of the central charge in (10.23), we look for the transformation that rotates Z AB in such a way that the transformed matrix can be identified with the one of [26]. The goal is to determine the SU(8) rotation in terms of the scalar fields, and then read from the transformed singlet the explicit form of the fake superpotential.
Because of the residual USp (8)   In the non-BPS p 0 , q 0 charge configuration (corresponding to D0 − D6 in Type II language), the dressed charges of the N = 8 theory read (3.17) Thus, the N = 8 skew-diagonal Z AB (10.3) in the p 0 , q 0 charge configuration can then be written as where α i ≡ a i /λ i is the axion/dilaton ratio, with λ i = e 2φλi , andλ 1λ2λ3 = 1. When a i = 0, one recovers the KK solution studied in [9].
To proceed further, it is convenient to define the following quantities: We can write Thus, by recalling (10.26), Z AB can be decomposed as This parametrization of the central charge matrix will allow us to perform the necessary rotation to identify the fake superpotential.

U(1) 3
The matrix Z AB (10.32) has a residual U(1) 3 ⊂ SU(8)/USp(8) symmetry. More precisely, U(1) 3 can be considered as the Cartan subalgebra of the symmetric, rank-3 compact manifold SU(8)/USp(8) (dim R = 27); indeed, U(1) 3 -transformations do not generate offdiagonal elements, and they leave the skew-diagonal form of Z AB invariant. We choose to parametrize such a U(1) 3 matrix as a 4 × 4 matrix acting on the diagonal part of Z AB , namely (χ i ∈ R) (10.33) Note that, consistently, the sum of the four diagonal phases vanishes. Therefore, by the exponential mapping, one obtains 34) which, analogously to Z AB (10.32), enjoys the following decomposition: where all matrices are reciprocally commuting.

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Under U(1) 3 (10.34), Z AB (10.32) transforms as Without loss of generality, one can therefore just redefine the χ i 's by a factor of 2, and consider the transformation Each single U i actually reads and induces the following transformation on Z AB (10.32): Consequently, U (10.35) has a well defined action on the coefficients of the matrices (10.31); for example, by acting with only U 1 gives rise to the following transformations of Y 0 and Y i 's: such that the U 1 -transformed central charge matrix (10.32) can be rewritten as The complete action of U (10.35) on (10.32) reads where the ζ I 's are defined as

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Within the same p 0 , q 0 axionful charge configuration, it is interesting to compare the U(1) 3 -transformed Z AB (10.42)-(10.44) with the "non-standard" skew-diagonalized Z (BM P ) AB obtained by Bossard, Michel and Pioline (BMP) in [26] Z (BM P ) AB which can equivalently be recast in the following form: by introducing the quantities: For simplicity's sake, we will here confine ourselves to solve such a system within the "t 3degeneration" of the formalism under consideration, which amounts to choosing three equal phases χ i 's, corresponding to the diagonal U(1) diag inside U(1) 3 .

t 3 model
As mentioned, at the level of U -transformation, the "degeneration" procedure from stu to t 3 model amounts to identifying

Duality invariants
One can also relate the parameters entering the solution (10.73) to the duality invariants I 4 , i 1 , i 2 and i 3 defined e.g. in [40]. Using the relations (3.6)-(3.10) of [22], one finds where i 2 = b + 3i 1 , and the "±" choice has to be consistent with the positivity of e 6φ . We notice that α is a duality invariant quantity by itself, as well as the combinations q 0 e −3φ and p 0 e 3φ (recall √ −I 4 = p 0 q 0 ). Thus, the expression (10.73) is explicitly duality invariant.

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Substituting the expression of τ ≡ tan χ as in (10.73), one finds that which yields the following explicit expression: Notice that the overall minus in (10.88) is totally irrelevant, since it can be eliminated with a U(1) diag -rotation through the matrix −ǫ ⊗ id 4 .
Equation (10.88), up to a factor of 1/2, coincides with the formula of the non-BPS fake superpotential for the (p 0 , q 0 ) configuration in the t 3 model computed in [22]. The difference of a factor 1/2 is simply due to the different normalization used for the normal form central charge in our notation (which coincides, for example, with the one in eq. (3.13) of [15]) with respect to the one used in [26], as one can read from eq. (2.11) therein. This implies that the correct identification would be Imµ 0 = 1 2 Imζ 0 . Consequently, the correctly normalized fake superpotential becomes finally This computation is a non-trivial consistency check for the formalism based on the axionindependent matrices M and M introduced in sections 8 and 9, as well as for the results on the phase χ obtained above.

(D.1)
It is worth remarking that that these four invariants collapse to a single one, in the axionless case (α i ≡ a i /λ i = 0). The black hole potential for this system is given in terms of the invariants by and it admits the fake superpotential [22,26,41] W = 1 2 √ i 1 + i s 2 + i t 2 + i u as well as the analogous ones concerning derivatives with respect to the scalars t and u, and by recalling that (recall (C.4)) C stu 2 = g ss g tt g uū ,