BPS Black Hole Entropy and Attractors in Very Special Geometry. Cubic Forms, Gradient Maps and their Inversion

We consider Bekenstein-Hawking entropy and attractors in extremal BPS black holes of $\mathcal{N}=2$, $D=4$ ungauged supergravity obtained as reduction of minimal, matter-coupled $D=5$ supergravity. They are generally expressed in terms of solutions to an inhomogeneous system of coupled quadratic equations, named BPS system, depending on the cubic prepotential as well as on the electric-magnetic fluxes in the extremal black hole background. Focussing on homogeneous non-symmetric scalar manifolds (whose classification is known in terms of $L(q,P,\dot{P})$ models), under certain assumptions on the Clifford matrices pertaining to the related cubic prepotential, we formulate and prove an invertibility condition for the gradient map of the corresponding cubic form (to have a birational inverse map which is an homogeneous polynomial of degree four), and therefore for the solutions to the BPS system to be explicitly determined, in turn providing novel, explicit expressions for the BPS black hole entropy and the related attractors as solution of the BPS attractor equations. After a general treatment, we present a number of explicit examples with $\dot{P}=0$, such as $L(q,P)$, $1\leqslant q\leqslant 3$ and $P\geqslant 1$,or $L(q,1)$, $4\leqslant q\leqslant 9$, and one model with $\dot{P}=1$, namely $L(4,1,1)$. We also briefly comment on Kleinian signatures and split algebras. In particular, we provide, for the first time, the explicit form of the BPS black hole entropy and of the related BPS attractors for the infinite class of $L(1,P)$ $P\geqslant 2$ non-symmetric models of $\mathcal{N}=2$, $D=4$ supergravity.


Introduction
In the last decades, the theoretical and phenomenological implications of the physics of black holes [GR] had a profound and fertile impact on many branches of science, from astrophysics, cosmology, particle physics, to mathematical physics [Mo], quantum information theory [D], and, recently, number theory [BDM]. Remarkably, the singularity theorems proved by Penrose and Hawking [HP] imply that the black holes are an unavoidable consequence of Einstein's theory of General Relativity, as well as of its modern generalizations such as supergravity [BH, Sen, Cve], superstrings and M-theory [BHst]. Classically, the gravitational force inside the event horizon of a black hole is so strong that nothing, not even light, can escape. However, in the 70s Hawking showed that quantum effects cause black holes to thermally radiate, and eventually evaporate [Haw].
While the frontiers of physics are progressing also the 21st century, it should not be forgotten that the physics of the 20th century is conceptually founded on two theories which are mutually incompatible. On one side, Quantum Mechanics governs the microscopic world of the basic constituents of matter, such as molecules, atoms, nuclei and beyond. On the other side, General Relativity describes gravity and the macroscopic, largescale structures, ranging from planetary orbits to the Universe in its entirety. As the energy increases, Quantum Mechanics and General Relativity inevitably meet, giving rise to startling, even paradoxical, consequences.
A tantalizing aspect of the physics of a black hole is that its thermodynamical features seem to encode fundamental insights of a not yet formulated theory of Quantum Gravity, which should necessarily arise from the reconciliation of the two aforementioned apparently contradictory physical theories. In this framework, a crucial relevance owes to the Bekenstein-Hawking entropy-area formula [HB]: where 1 k B is the Boltzmann constant, ℓ 2 P = G /c 3 is the squared Planck length, whereas A H denotes the area of the event horizon of the black hole itself. This formula relates a thermodynamical quantity (the entropy S) to a geometric quantity (the area A H ), and after much theoretical work it still puzzles the scientific community. In fact, a crucial issue that Quantum Gravity must necessarily address concerns the origin of S at a fundamental level. At classical level, (1.1) yields that black hole entropy is determined by the area of the event horizon, which is a macroscopic and geometric quantity; however, black hole entropy must also enjoy a microscopic, statistical derivation, accounting for fundamental microscopic degrees of freedom.
Since superstring theory and M-theory are the most serious candidates for a theory of Quantum Gravity, they are expected to provide a microscopic, statistical explanation of the entropy-area law (1.1) [Micro]. Black holes are typical non-perturbative objects, since they describe a physical regime in which the gravitational field is very strong; thus, only a non-perturbative approach can successfully deal with them. Progress in this direction came after 1995 [W], through the recognition of the role of string dualities, which allow one to relate the strong coupling regime of one superstring model to the weak coupling regime of another. Remarkably, there is evidence that the string dualities are all encoded into the global symmetry group (the U -duality, also named electric-magnetic, group) of the low energy supergravity effective action [HT].
Black holes, and in particular their extremal configurations [Extr], are embedded in a natural way in supergravity theories, which, being invariant under local super-Poincaré transformations, include General Relativity, providing a consistent description of the graviton coupled to other fields in a supersymmetric framework. Extremal black holes have become objects of crucial relevance in the context of superstrings after 1995 [Micro, BHst, BH, Ort] : the classical solutions of supergravity that preserve a fraction of the original supersymmetries can be interpreted as non-perturbative states, necessary to complete the perturbative string spectrum and make it invariant under the many conjectured duality symmetries [Str, Sch, HT, GMV, VSch]. In such a framework, extremal black holes, as well as their parent p-branes in higher dimensions, are conceived as additional particle-like states that compose the spectrum of a fundamental quantum theory. Similar to monopoles in gauge theories, these non-perturbative quantum states originate from regular solutions of the classical field equations, i.e. the very same Einstein equations on which General Relativity relies. The crucial new ingredient, in this respect, is Supersymmetry, which requires a precise balance between vector fields and scalar fields in the bosonic spectrum of the theory. As such, the general framework we are going to deal with is provided by the so-called Einstein-Maxwell-scalar theories, whose global mathematical thorough treatment has been recently given in [LS] (see also [AMS]).
Supergravity theories provide a low-energy effective theory description of superstring and M-theory, holding at the lowest order in the string loop expansion, when the space-time curvature is much smaller than the typical string scale (string tension). Consequently, the supergravity description of extremal black holes can be trusted only when the radius of the event horizon is much larger than the string scale, corresponding to the regime of large charges. We will not be dealing with further corrections, introduced by string theory, which give rise to higher derivative terms in the low energy effective action, such that the black hole entropy is expected to be corrected by subleading terms in the limit of small curvature :it is well known that these corrections determine a deviation from the area law for the entropy [Wa, CdWM].
The cosmic censorship conjecture [CCC] is naturally realized by conceiving extremal black holes as solitonic solutions of N -extended locally supersymmetric theory of Einstein gravity : in fact, denoting with N the number of spinor supercharges in 3 + 1 space-time dimensions, when N 2 the so-called BPS (Bogomol'nyi-Prasad- where M and Q respectively are the mass and the magnetic (or electric) charge of the black hole, is just a consequence of the supersymmetry algebra, implying that no naked singularities can occur. When the black hole solution is embedded into a N -extended supergravity theory, the model is characterized by a certain N -dependent number of scalar fields, collectively denoted by φ. In this framework, the charge Q is to be replaced by the maximum eigenvalue of the N × N central charge matrix appearing in the r.h.s. of the supersymmetry algebra (depending on the expectation value φ H (p, q) of the scalar fields on the event horizon, where p's and q's respectively denote magnetic and electric charges of the black hole) : In the present paper, we will be dealing only with extremal black holes, in which the BPS bound (1.2)-(1.3) is saturated. Extremal black holes enjoy the following peculiar and crucial feature: despite the fact that the dynamics depends on scalar fields, the event horizon of the black hole loses all information about the scalars, and this holds regardless of the supersymmetry-preserving features of the solution. This phenomenon is described by the so-called attractor mechanism [AM, BFMb]: independently of their boundary conditions at spatial infinity, scalar fields flow to a fixed point given by a certain ratio of electric and magnetic charges, when approaching the event horizon. In this framework, the scalar fields are moduli, i.e. they are continuous parameters which can be freely specified at infinity, raising the dangerous possibility that the black hole entropy might depend on their values. Indeed, such a dependence presumably would lead to a violation of the second law of thermodynamics, since it would allow one to quasi-statically decrease the entropy by varying the moduli. Instead, the black hole entropy turns out to depend only on the values acquired by the scalar fields at the event horizon, which in turn only depend on the conserved charges (p and q) associated to gauge invariance of the black hole solution itself : in this sense, the entropy of extremal black holes is a topological quantity, because it is fixed in terms of the quantized electric and magnetic charges, while it does not depend on moduli.
For extremal black holes in Maxwell-Einstein supergravity theories with N = 2 local supersymmetry in 3 + 1 space-time dimensions [ADFM], the saturation of the BPS bound (1.3), M (p, q) = |Z(φ H (p, q) , p, q)|, (1.4) yields, for the black hole entropy, S (p, q) = A H (p, q) 4 = π|Z(φ H (p, q) , p, q)| 2 , (1.5) where Z is the N = 2 central charge function [CDF]. The attractor values of the scalar fields at the event horizon, here collectively denoted by φ H (p, q), arise as solutions to the so-called BPS attractor equations : where D φ denotes the Kähler-covariant differential operator acting on the scalar manifold (target space of moduli fields). The entropy S generally enjoys a U -duality-invariant expression (homogeneous of degree two) in terms of electric and magnetic charges, only depending on the nature of the U -duality groups and on the appropriate representations of electric and magnetic charges [ADF2]. Through the years, the attractor mechanism has been discovered to have a broader application [Ort2,FGK,nBPS,BFM,BFGM] beyond the BPS cases, being a peculiarity of all extremal black-holes, BPS or not. Even for these more general cases, because of the topological nature of the extremality condition, the entropy formula turns out to be still given by a U -duality invariant expression built out of electric and magnetic charges.
The present paper is devoted to the determination of the explicit expression of two purely charge-dependent quantities, characterizing the physics of BPS extremal black holes : the Bekenstein-Hawking entropy S (p, q) and the attractor values, collectively denoted by φ H (p, q), acquired by the scalar fields when approaching the (unique) event horizon (regardless of the boundary conditions of their evolution dynamics). We will consider ungauged Maxwell-Einstein supergravity theories with N = 2 extended local supersymmetry in 3 + 1 spacetime dimensions, in the case in which the special Kähler geometry of the vector multiplets' scalar manifold is determined by a cubic holomorphic prepotential (very special geometry). In fact, in such a framework only the models in which the scalar manifold is a symmetric coset have been thoroughly investigated : exploiting the relation to the theory of cubic (simple and semisimple) Jordan algebras and related Freudenthal triple systems, which hold in all cases but the so-called Luciani models (with quadratic prepotential), the explicit expressions of S (p, q) and of φ H (p, q) have been explicitly computed for extremal, both BPS and non-BPS, black holes (cfr. e.g. [FGimK] and refs. therein).
On the other hand, very little is known in the case in which the (vector multiplets') scalar manifold is not symmetric. In very special geometry, the BPS entropy and the BPS attractor values of the scalar fields have been computed by Shmakova [Shm], up to the solution of an inhomogeneous system of quadratic algebraic equations, named BPS system. A noteworthy, countably infinite class of cubic non-symmetric models is provided by homogeneous non-symmetric models, which have been classified in [dWVP1] (in a mathematical context, see also the subsequent classification in [Cor]). These models are quite interesting from a physical point of view, because some of them naturally occur in the four dimensional effective supergravity description of brane dynamics, when their brane and bulk degrees of freedom get unified. To the best of our knowledge, only [DFT] dealt with such a class of models, but did not investigate the explicit determination of S (p, q) and φ H (p, q). In the present work, we will rely on the existing classification of homogeneous non-symmetric special manifolds, and we will formulate a (sufficient but not necessary) condition for the BPS system to be explicitly solved. Thus, within the validity of such a condition (which we will prove to actually hold for an infinite, countable number of models), we will explicitly determine the expression of the Bekenstein-Hawking (semi)classical black hole entropy S (p, q) as well as of the purely charge-dependent attractor values φ H (p, q) acquired by the scalar fields at the event horizon of asymptotically flat, spherically symmetric, static, dyonic, extremal BPS black holes.
The paper is organized as follows. In Secs. 2-3 we introduce the Bekenstein-Hawking BPS black hole entropy and the BPS attractor values of scalar fields in N = 2 very special geometry, relating their explicit expressions to the solution of the corresponding BPS system, or equivalently to the inversion of the gradient map of the cubic form defining the corresponding cubic holomorphic prepotential. Then, in Sec. 4 we specialize the treatment to homogeneous very special geometry, briefly recalling some basic facts on symmetric and non-symmetric spaces in Secs. 4.1 resp. 4.2. In Sec. 4.3 we review the classification of homogeneous special d-spaces [dWVP1], and in Sec. 4.3.1 we briefly consider the class L(q, 1). Sec. 5, which is in turn split into eight Subsecs., is then devoted to the introduction and review of another important ingredient of our treatment : Euclidean Clifford algebras. Next, Secs. 6 and 7 contain the main results of the present paper : after enouncing an invertibility condition for the gradient map of the cubic form in Sec. 6.2 (then proved in Sec. 7), the BPS system is explicitly solved in Secs. 6.4-6.5, and the explicit expressions of the BPS entropy and of the BPS attractor values of scalar fields at the horizon are computed in Sec. 6.6 Then, Sec. 6.7 introduces the so-called complete models, whose known examples coincides with the symmetric d-spaces, recalled in Sec. 6.5.1. Secs. 8, 9 and 10 present a threefold wealth of models in which the invertibility condition of Sec. 6.2 holds true, and in some cases, such as the models L(1, 2) and L(1, 3) in Secs. 9.5 resp. 10.6 (then generalized 2 as L(1, P ) with P 2 in Sec. 10.7), the corresponding BPS system is explicitly solved in full detail, by thus providing the full fledged expressions of the BPS entropy and attractors. The non-uniqueness of the matrices Ω K 's occurring in the invertibility condition of Sec. 6.2 is discussed in Sec. 11,and in Sec. 11.3 the difference between descendant models and submodels is highlighted. Moreover, Sec. 12 discusses the unique model of the present paper which has a non-vanishingṖ : namely, the model L(4, 1, 1). A brief discussion of cubic models determined by Kleinian (rather than Lorentzian) quadratic polynomials (thus giving rise to a non-special geometry) is provided in Sec. 13. Finally, Sec. 14 deals with the model L(9, 1), as an example of model in which the invertibility criterion of Sec. 6.2 does not seem to be applicable, and thus other approaches are needed to prove or disprove the invertibility of the BPS system. Finally, some outlook and hints for further developments are provided in the concluding Sec. 15.

BPS Black hole entropy and attractors in very special geometry
A large class of four-dimensional Maxwell-Einstein gravity theories with local N = 2 supersymmetry can be obtained by an S 1 -compactification of five-dimensional minimal supergravity. In such a case, the Kähler-Hodge geometry of the vector multiplets' scalar manifolds in D = 4 is named very special [SKG, Stro, ABCDF, Fre], and it is determined by an holomorphic prepotential of the type 3 where d ijk is a completely symmetric real tensor, and the X Λ 's, Λ = 0, i, are the contravariant symplectic sections of the Kähler-Hodge target space of scalar fields. The symplectic frame in which F (X) (2.1) is specified is the one of the so-called "4D/5D special coordinates" (see for instance [ABCDF, CFM]) : the manifest symmetry is the electric-magnetic (U -) duality 4 group of the parent theory in D = 5 (which leaves d ijk invariant). From the treatment given in [Shm], in the general case in which all electric and magnetic charges associated to the black hole solution are non-vanishing, the Bekenstein-Hawking entropy of static, spherically symmetric, BPS extremal dyonic black holes in ungauged N = 2, D = 4 Maxwell-Einstein supergravity whose vector multiplets' scalar manifold displays a very special Kähler geometry, reads 5 where p 0 , p i , q 0 and q i are the magnetic resp. electric black hole charges, and Note that, as it must be, S (2.2) is homogeneous of degree 2 in the black hole charges. Furthermore, q 0 enters the expression (2.2) only through the quantity p · q (2.4). The x i 's appearing in (2.2) are the solutions x i (∆ j ; d klm ) of the system of algebraic quadratic equations In Sec. 10.7.2 we will also briefly present a geometric point of view on the factorization of the inverse map of the gradient map, whose detailed investigation goes beyond the scope of this paper.
3 Einstein summation convention on repeated indices is understood throughout. Lowercase Latin indices run 1, ..., N throughout; N denotes the number of vector multiplets. The index 0 pertains to the (D = 4) graviphotonic sector. 4 In this paper, U -duality is referred to as the "continuous" symmetries of [CJ]. Their discrete versions are the U -duality non-perturbative string theory symmetries introduced by Hull and Townsend [HT].
5 This formula fixes a typo in Eq. (12) of [Shm]. For further, recent insight on the BPS entropy in very special geometry, see [AMS]. which we will henceforth name the BPS system 6 pertaining to the model of N = 2, D = 4 ungauged supergravity (coupled to vector multiplets 7 ) under consideration. Since the ∆ i 's are homogeneous of degree two in the black hole charges p i , p 0 and q i , (2.6) implies that the x i 's are homogeneous of degree one in the same variables: Furthermore, the x i 's contribute to the black hole entropy only through the square of the quantity 8 The number of equations in the system (2.6) is equal to the number N of complex scalar fields, which in the large volume limit of Calabi-Yau compactifications of type II superstrings correspond to the Calabi-Yau moduli fields. The l.h.s. of (2.6) is given by quadratic forms with coefficients d (i)jk , while the r.h.s. is arbitrary and depends on the values of electric and magnetic charges of the extremal BPS black hole. In other words, it defines N quadratic hypersurfaces (each of dimension N − 1) in an N -dimensional space, and the intersection of these hypersurfaces is the set of solutions of the system (2.6). Therefore, the BPS system may also not admit any analytical (or in closed form) real solution at all. Note that the condition is a consistency condition for the BPS entropy (2.2) to be well defined. We will see below what is the general (set of) BPS condition(s) in N = 2 ungauged supergravity with cubic prepotential (cfr. (2.14) below). By switching the notation of scalar fields (at the horizon) from φ H to z i H , and denoting with i the imaginary unit of C, the explicit solutions to the BPS Attractor Equations read [Shm] z i H p 0 , p k , q 0 , q k = 3 2 where S is given by (2.2), and the subscript "H" denotes the evaluation at the (unique) event horizon of the extremal BPS black hole. Finally, it is here worth recalling that in very special geometry, by construction, the quantity d ijk Im z i H Im z j H Im z k H must have a definite sign, say negative, and this imposes a further constraint on the sign of x i 's : By exploiting (2.6), condition (2.11) can be rewritten as and thus (2.10) gets simplified to 9 (2.14) 3 BPS systems and the gradient map ∇ V Given a rank-3 completely symmetric tensor d ijk = d (ijk) in N dimensions (i, j, k = 1, ..., N ), we recall the cubic form defined in (2.8) : From the Euler formula, since V (3.1) is a homogeneous polynomial of degree 3 in the x's, it follows that More subtly, since the x i 's are themselves homogeneous of degree 1 in the black hole charges p i , p 0 and q i , it holds that where we used (2.7).
The BPS system (2.6) can then be defined as the non-homogeneous system of N quadratic equations in N unknowns x i 's (i = 1, ..., N ) is a real quantity depending, for a given d ijk , on some real given (background) constants, namely the magnetic and electric charges p i , p 0 and q i of the extremal black hole. By (3.5), (3.4) can be rewritten as Thus, by introducing the gradient operators ∇ x := {∂ x i } i=1,...,N and ∇ p := ∂ p i i=1,...,N , the BPS system (2.6) (or, equivalently (3.4)) can be cast as follows 10 : (3.9) 10 We recall that the symplectic bundle of special geometry is flat [Stro, ADF, CFGM].
All quantities involved in these formulae are real, and, for N , d ijk and ∆ i (p, q) given in input, real solutions x i = x i (∆ j ; d klm ) to the system (3.9) are searched in closed form, in such a way that, when plugged into (2.2), one can obtain a closed form expression for the Bekenstein-Hawking entropy (2.2) as well as for the attractor values of scalar fields (2.13) of extremal BPS black holes also in homogeneous non-symmetric very special geometry. Of course, as already mentioned, the system (3.9) might have no analytical real solution in the general case : it describes N quadratic hypersurfaces (each of dimension N − 1) in R N , and the intersection of these hypersurfaces is a solution of the system. An important quantity is the Hessian matrix of the cubic form V (x) (3.1) : which can be regarded as the Jacobian matrix of the quadratic map represented by the gradient map Since each coefficient H ij (x) of the symmetric matrix H is a linear homogeneous polynomial in the x's, the determinant of the N × N matrix H(x) is a homogeneous polynomial of degree N in the x's, as evident from (3.10). Even if we will not exploit it in the subsequent treatment, the condition of non-vanishing det H (x) (i.e. H(x), and thus ∇ V , of maximal rank) can be used to establish whether the gradient map ∇ V can be inverted; in fact, Dini's Theorem ensures that if det H (x) = 0 then locally the map ∇ V is a diffeomorphism, and therefore x is an isolated point in the fiber ( 4 Homogeneous very special geometry

Symmetric d-manifolds
The tensors d ijk as in (2.1) giving rise to homogeneous very special Kähler spaces (which, for this reason, have been named d-manifolds) have been classified in [dWVP1] (see also [dWVVP,dWVP3]). A noteworthy subclass is represented by the symmetric d-manifolds [dWVVP], whose d ijk 's have been reconsidered also e.g. in [FGimK,BMR2,DHW]. Symmetric d-manifolds are characterized by a purely numerical (constant) contravariant tensor d ijk such that the so-called "adjoint identity" of cubic Jordan algebras holds [GST, CVP]: a solution to the BPS system (2.6) is given by (see Sec. 3.3.1 of [Hal]) Indeed, by exploiting the adjoint identity (4.1), (4.3) yields to Furthermore, by using (4.1), one computes that Thus, the consistency condition (4.2) can be recast as follows : Moreover, the condition (2.12) can be rewritten as implying that only the branch "+" of (4.3) is consistent; thus selecting it (for d lmn ∆ l ∆ m ∆ n > 0), (4.8) one obtains that the explicit solutions (2.10) to the BPS Attractor Equations read By substituting (4.3) into (2.2), one obtains the BPS entropy S in terms of the quartic invariant polynomial I 4 (cfr. e.g. [CFM] and Refs. therein), which can be proved to (finitely) generate the ring of invariant polynomials of the non-transitive action of the 4D U-duality group over the black hole charges' representation space [Kac]. By defining and recalling (2.4) and (2.5), the expression (2.2) the BPS entropy can be remarkably simplified into the following formula : and therefore (4.9) further simplifies to It should be remarked that the condition I 4 > 0 may be weaker than the actual BPS condition; in fact, it satisfied by both BPS and non-BPS attractors (these latter with vanishing central charge). Indeed, in the symmetric d-spaces the strictly BPS conditions (2.14) can be specified as the following system of inequalities : (4.14)

Non-symmetric d-manifolds
In non-symmetric d-manifolds, the tensor d ijk still exists 12 , but it generally depends on the rescaled imaginary parts (denoted below byλ i ) of the scalar fields z i ; indeed, within the conventions of [CFM], the "dual" d ijk cubic tensor is generally defined as d ijk : = a il a jm a kn d lmn ; (4.15) (4.16) where the scalar fields read This can also be obtained by the generalization of the adjoint identity (4.1) in non-symmetric very special geometry 13 , which reads [BMR2]: where the so-called "E-tensor" is defined as [BMR1] E p ijmn = a pk E p|ijmn ; (4.22) (4.24) 4.3 Classification of homogeneous d-manifolds : L(q, P,Ṗ ) In the present investigation, we will focus on the solution of the BPS system (2.6) in the case in which the d-manifolds (coupled to N = 2, D = 4 supergravity) are homogeneous (and thus the U -duality group has a non-linear but transitive action on the target space of scalar fields) but 14 non-symmetric 15 . To this aim, we now recall some basic facts on the classification homogeneous d-spaces.
In [dWVP1,dWVP3,dWVVP], homogeneous very special Kähler spaces arising as non-compact Riemannian scalar manifolds of vector multiplets in N = 2, D = 4 supergravity have been classified 16 . They are denoted as "L-spaces" (or "L-models"), specified by three sets of integer parameters : L(q, P,Ṗ ). In such a classification, the index i = 1, ..., N is partitioned as 17 {i} = s, {I} , {α} ; where D q+1 is a certain function of q valued in N (see e.g. where, for I = 0, 1, ..., q + 1, the square matrices Γ I , of size D q+1 · P +Ṗ and components (Γ I ) αβ are defined as follows : Γ 0 := I Dq+1·(P +Ṗ ) , and {Γ I } I =0 are Γ-matrices that provide a real representation of the Euclidean Clifford algebra Cl (q + 1, 0); see Sec. 5. All the other, unwritten, components of d ijk vanish. We should also 14 Notice that all (known and classified) homogeneous non-symmetric manifolds are of d-type, even if, as far as we know, there is no proof that homogeneous non-symmetricity implies d-type. 15 Within the black hole effective potential formalism, a discussion of the various classes of attractors and related black hole entropies in homogeneous scalar manifolds of N = 2, D = 4 supergravity has been given in [DFT]. Therein, in (2.23)-(2.25) the explicit expression of the E-tensor (which is non-vanishing for homogeneous non-symmetric cases) has been computed. 16 In [Cor] this classification has been rephrased in terms of normal J-algebras; for a recent survey, see also [AMS]. 17 In the present paper, attention should be paid to the three different uses of '·' : it indicates a scalar product involving naught and i-indices (as in (2.4)), or an algebraic multiplication (as in the third row of (4.25)), or a scalar product involving only i-indices (split into s, and x-and y-indices), as in (6.30)-(6.33). We hope that such different meanings are easily inferred from the context. recall that non-vanishing (and independent) values ofṖ are possible only when q = 4m, with m ∈ N ∪ {0}. Indeed, for q = 0 mod 4 the representations Γ I and −Γ I are not equivalent, and a reducible representation is and thus characterized by the multiplicity of each of these representations, namely P andṖ ; of course, an overall sign change of all the gamma matrices can always be re-absorbed, and this is the reason why L(4m, P,Ṗ ) = L(4m,Ṗ , P ). If the representation consists of copies of only one version of the irreducible representations, then we denote it by L(4m, P ).
In correspondence with the splitting (4.25) of the index set, we introduce new variables s, x I and y α , and define ξ := T s, x I , y α . (4.29) Then, (3.1) and (3.5) can be written as (4.33) The BPS system (2.6) (or, equivalently, (3.9)) is an inhomogeneous system of N quadratic equations in N unknowns (s, x I , y α ), and it acquires the following form : 2sη IJ x J + Q I (y) = 3∆ I ; 2x I (Γ I ) αβ y β = 3∆ α .
Within homogeneous very special geometry, we should also mention the so-called T 3 model (corresponding to pure minimal supergravity in D = 5), which is absent in the discussion of Sec. 5 (and Table 2) of [dWVVP], and then mentioned in [dWVP3] (see Sec. 9 therein).

Basics on Euclidean Clifford algebras
We recall the definition of a Clifford algebra and the basic results on (matrix) representations of such algebras. We then discuss Γ-matrices and Clifford sets of Γ-matrices, which define such representations. We recall the Pauli matrices and introduce quadratic forms defined by symmetric Γ-matrices that are tensor products of Pauli matrices. These quadratic forms are building blocks in the definition of the cubic forms in the L(q, P,Ṗ )-models.

Euclidean Clifford algebras
The Euclidean Clifford algebra Cl(n, 0) is the quotient of the tensor algebra on V = R n by the relations q n (v) = v ⊗ v for all v ∈ V , where q n (v) is the Euclidean quadratic form (4.36) : We identify V with the corresponding subspace in Cl(n, 0) and we write xy for the product of x and y ∈ Cl(n, 0). If e 1 , . . . , e n is an orthonormal basis of V and we write v = x 1 e 1 +. . .+x n e n , then in Cl(n, 0) we have q n (v) = v 2 , so: x 1 2 + . . . + (x n ) 2 = (x 1 e 1 + . . . + x n e n ) 2 , (5.2) and expanding the right hand side we see that in Cl(n, 0) we have (here I, J = 1, ..., n) e 2 I = 1, e I e J + e J e I = 0 (I = J) . (5. 3) The Euclidean Clifford algebra Cl(n, 0) is a vector space of dimension 2 n with basis the e I1 e I2 . . . e I k with I 1 < I 2 < . . . < I k and k = 1, 2, . . . , n.
19 Notice that one can take "diagonal blocks" of Γ-matrices to produce new ones.
Notice also that changing the sign of all Γ I , I = 1, . . . , n, one obtains again a representation φ − : Cl(n, 0) → M m (R) (with φ − (e I ) = −Γ I ) but now, since n is odd, we have φ − (c) = −φ(c). So, as is well-known, the representations φ and φ − are not equivalent. More generally, changing the sign of an odd number of the Γ I defines a representation which is not equivalent to φ since it changes the sign of φ(c).

Quadratic forms and Γ-matrices
Recall the definition of the four 2 × 2 (Pauli) matrices: γ 00 := I 2 = 1 0 0 1 , γ 10 := σ 1 = 0 1 1 0 , (5.21) The notation is chosen such that (we recall that i, j, k, l = 0, 1) where the indices are summed modulo 2. Notice that Recall that the tensor product of the square matrices M = (M ij ) and N = (N kl ), of size m × m and n × n respectively, is the matrix of size nm × nm given by the (block) matrix By a Γ-matrix (of size 2 g × 2 g , and characteristic [ i1...ig j1...jg ]), in this paper we intend the following tensor product : Such a Γ-matrix is symmetric iff the sum of the products i k j k is zero modulo 2 : The quadratic form (with characteristic [ i1...ig j1...jg ]) in y 1 , . . . , y 2 g associated to the Γ-matrix (5.25) is defined as If g a=1 i a j a = 0 modulo 2, the parity of the characteristic is said to be even, and then (5.26) shows that the Γ-matrix is symmetric. The (anti)symmetric Γ matrices of size 2 g are a basis of the vector space of (skew)symmetric matrices of size 2 g .

Clifford sets of Γ-matrices
One easily verifies: are even and the sum of any two distinct characteristics is odd.

Quadratic identities between quadratic forms
Whereas the quadratic forms Q[ i j ] (with even characteristics of length g) are a basis of the vector space of all quadratic forms in m = 2 g variables, their squares, homogeneous polynomials of degree four, are not linear independent. The most basic example, known as Jacobi's identity, is: which is easily verified since, by definition, and thus we indeed have the identity For g = 2, 3, 4 there are similar identities between 4, 6, 10 respectively such quadrics, see (8.61), (8.37), (8.5) respectively. A remarkable fact, and crucial for the invertibility results in this paper, is that in all these four cases the the Gamma-matrices of the quadrics are I m and the remaining Γ I form a (maximal) Clifford set. Actually, these identities are classical identities between theta functions. The case g = 1 was known to Jacobi, and the other cases were already known to Max Noether, see [Noe,p.332,p.334]. Unfortunately, for g > 4 it seems that the squares of the quadrics defined by I m and a maximal Clifford set are no longer linearly dependent.

Heisenberg groups
The Γ-matrices generate a finite (non-Abelian) subgroup of GL(2 g , C), which is called a Heisenberg group (cfr. e.g. [CG]) or a Clifford group. The subgroup has order 2 · 2 2g and each element is of the form ±Γ where Γ is one of the 2 2g Γ-matrices. See [CG, App. A] for the quadratic forms Q m = Q[ ǫ ǫ ′ ]. The quadratic relations among the Q m 's that we discussed can also be found in A.3 of [CG], where they are discussed in the context of Hopf maps.

BPS entropy and attractors in L (q, P ) models
In the present paper we focus on BPS systems related to models L(q, P,Ṗ ) of homogeneous very special geometry [dWVP1]: we will thus only consider the case of a Lorentzian quadratic form in the x i and a related Clifford set of Γ-matrices. In particular (see Sec. 4.3), we consider V (cfr. (4.35)) to have Lorentzian (mostly plus) signature and dim V = q + 2, (6.1) such that (cfr. (4.27)) For later convenience, in order to highlight the Lorentzian (mostly plus) signature, we also shift the labeling of the q + 2 I-indices from 1, ..., q + 2 to 0, 1, ..., q + 1; moreover, the matrix Γ 0 will be nothing but the identity matrix of size m = D q+1 · P +Ṗ .
Physically, the L(q, P,Ṗ ) models of homogeneous very special geometry determine the scalar manifolds (i.e., the target spaces of scalar fields) in ungauged N = 2 Maxwell-Einstein supergravity theory coupled to vector multiplets in three, four or five Lorentzian space-time dimensions (the corresponding spaces are quaternionic, Kähler and real, respectively).
Since the closed form expression of the Bekenstein-Hawking entropy (2.2) as well as of the attractor values of scalar fields (2.13) of extremal BPS black holes are already known for symmetric spaces (see e.g. [FGimK]), in the present paper we will focus on homogeneous non-symmetric spaces.
In the present section, we will start from a Clifford set, and we will define a certain cubic form; then, we will show that under certain conditions its gradient map is invertible; consequently, the corresponding BPS system (2.6) can be explicitly solved by (6.53), thus allowing for a novel, closed form expression of the Bekenstein-Hawking entropy (2.2) as well as of the attractor values of scalar fields (2.13) of extremal BPS black holes, respectively given by (6.66) and (6.73)-(6.75) below.
In particular, the models L(1, 2) and L(1, 3) will be analyzed in full detail in Secs. 9.5 and 10.6, respectively, and their analysis will be explicitly generalized to the class of models L(1, P ) with P 2 in Sec. 10.7. In this respect, such Secs. extend the treatment given 20 in Sec. 4 of [Shm], by providing explicit expressions for the BPS black hole entropy as well as for the BPS attractors in such an infinite class of non-symmetric (homogeneous) models of N = 2, D = 4 supergravity with cubic prepotential.

The invertibility condition
Given the Clifford set {Γ 1 , . . . , Γ q+1 }, in order to invert the gradient map of V, we need the existence of further symmetric m × m matrices Ω 1 , . . . , Ω r , which anti-commute with the matrices in the Clifford set: Moreover, if we denote the associated quadratic forms defined by these Ω K by then the following Lorentzian quadratic identity should hold: If auxiliary matrices Ω K with all these properties exist, then the gradient map (with I = 0, ..., q + 1, and α = 1, ..., m) is invertible, with (birational) inverse given by polynomials of degree 2 if r = 0 and of degree 4 if r > 0. An explicit expression of the birational inverse map of the gradient map ∇ V will be given in Sec. 6.4. This results in a closed form expression of the solution to the BPS system (2.6), given in Sec. 6.5. In turn, in Sec. 6.6 this will allow for a closed form expression of Bekenstein-Hawking entropy (2.2) as well as of the attractor values of scalar fields (2.13) of extremal BPS black holes in the homogeneous non-symmetric very special geometry characterizing the corresponding model of ungauged N = 2 Maxwell-Einstein supergravity theory coupled to vector multiplets in four space-time dimensions.

Remarks
Notice that only in the case that we do have a Lorentzian identity −Q 0 (y) 2 + q+1 I=1 Q I (y) 2 = 0, there is no need for the extra Ω K 's (and one can then take r = 0).
As it will be seen in Sec. 6.7, this is a quite special case, corresponding to some symmetric very special spaces, and to the very rich geometry related to simple cubic Jordan algebras [PR1,PR2]. In the present investigation, since we want to solve the BPS system (2.6) and study the expression of the BPS black hole entropy (2.2) in cases not treated in literature, we will be interested in some classes of homogeneous non-symmetric spaces which all have r > 0.
It seems rather restrictive (and mysterious) to request the existence of the (Lorentzian) quadratic identity (6.8), but it is crucial for us in order to show the existence of a (birational) inverse map of the gradient map of the corresponding cubic form V, and thus to provide a closed form expression of the solution to the associated BPS system (2.6).
It is here worth remarking that we will not impose any condition on products involving only Ω-matrices; in particular, for P 3, we will consider symmetric m × m matrices Ω j 's that are not invertible (i.e. whose rank is less than m), so they cannot be Γ-matrices (see Sec. 10).
We conclude with some remarks on the inverse of the gradient map ∇ V . This gradient map is given by the partial derivatives of V which are homogeneous polynomials of degree two in ξ := (s, x, y). Therefore ∇V(ξ) = ∇V(−ξ) and this implies that the inverse image of the image of any non-zero point contains at least two points which differ by a sign.
The inverse map ∇V −1 will be given by homogeneous polynomials of degree four in general, and thus the composition ∇V −1 • ∇V has coordinate functions that are homogeneous of degree eight. Therefore it cannot be the identity map (since the identity map has coordinate functions that are homogeneous of degree one), but one has: (cfr. (6.13)), with f (ξ) homogeneous of degree 2 · 2 + 3 = 7. In particular, for the points ξ with f (ξ) = 0 the inverse of the gradient map does not provide useful information. All this should not be surprising, in fact in the simple cubic Jordan algebra models the gradient map is given by

Explicit inversion of ∇ V
When r > 0, the inverse of the gradient map ∇ V is given as a composition of two maps, namely : 13) The map α, which has q + 3 + m + r components that are homogeneous polynomials of degree 2 in the variables z 1 , . . . , z q+3+m , is given by where the r quadratic forms R K (z)'s depend only on the last m variables z q+3+1 , . . . , z q+3+m : The composition α • ∇ V will be explicitly computed in Sec. 7.2 below, and it is given by Next, the map µ, which has q + 3 + m components that are homogeneous polynomials of degree 2 in the variables t, u 0 , . . . , u q+1 , v 1 , . . . , v m , w 1 . . . , w r , is given by where the definition of the last m components involves all the q + 2 symmetric Γ-matrices {Γ I } as well as the r auxiliary symmetric matrices {Ω K }, required in the invertibility condition enounced in Sec. 6.2; all these matrices have size m × m.
Recalling the relabelling (4.29), we will verify in Sec. 7 that the composition of maps µ • α • ∇ V is given by (cfr. (6.5)) As we will see below, in complete models (see Sec. 6.7) r = 0 by definition, and one can omit the map α (because it becomes proportional to the identity map), and then (6.19) reduces to The identity (6.19) implies that the composed map can be regarded as the inverse map of the gradient map ∇ V . In order to determine the general form of the map µ • α (6.21), the map µ (6.17) must be evaluated on the image of the map α (6.14) : The replacement of the variables t, u 0 , . . . , u q+1 , v 1 , . . . , v m , w 1 . . . , w r with the corresponding components (homogeneous polynomials in the variables z 1 , . . . , z q+3+m ) in the image of α (6.14) reads as follows (I = 0, 1, ..., q + 1, α = 1, ..., m, K = 1, ..., r): one can easily compute : where the last line of the r.h.s. contains the product of the m × m matrices Γ I and Ω K with the m × 1 vector z (6.24). From the definitions (6.3) and (6.7), one computes (6.26) and therefore (6.25) can be further elaborated as follows : where (by recalling (6.7) and (6.15)) R K (ẑ) = Tẑ Ω Kẑ . (6.28) Each of the q + 3 + m components of this composed map is given by an homogeneous polynomial of degree 4 in the q + 3 + m variables z 1 , . . . , z q+3+m . As it is evident, the explicit form of such polynomials depends on q + 2 (symmetric) Γ-matrices Γ I (such that Γ 0 = I m ) as well as on the r symmetric auxiliary matrices Ω K and the corresponding quadratic forms R K defined in (6.7) and in (6.15). Note that both the Γ I 's and the Ω K 's, as well as the corresponding quadratic forms Q I 's (6.4) and R K 's (6.7), occur in the invertibility condition enounced in Sec. 6.2 (cfr. Eqs. (6.6) and (6.8), respectively), which is assumed to hold throughout the treatment of this Section, as well as of the subsequent Secs. (6.5)-(6.7) and in the whole Sec. 7.

Solution of the BPS system
From (6.18)-(6.19), by replacing ∇ V with 3∂ p ∆ = 3 T (∆ s , ∆ I , ∆ α ), one obtains and recalling the Euler formula (3.2), one can replace ∇ V (ξ) with 3∂ p ∆ and use (6.29) in order to obtain Then, from (6.29) and (6.35), it follows that is the general solution of the BPS system (2.6). However, we must also recall the condition (2.12), which in this case reads which thus implies that only the branch "+" of (6.37) is consistent. Summarising, at least in those homogeneous d-spaces [dWVP1,dWVVP] in which the invertibility condition enounced in Sec. 6.2 is satisfied, there exists a quartic homogeneous polynomial map (i = 1, ..., q + 3 + m, where namely (cfr. the index splitting 21 (4.29) as well as the result (6.27)) : where η II (no sum on the repeated index I) denotes the non-vanishing (diagonal) components of the (q + 2)dimensional (mostly plus) Lorentzian metric η IJ introduced in (4.28).
Consequently, (6.46) yields that if the solution to the system (2.6) reads, in vector notation, or, more explicitly (recall (4.29)) where ∆ i ≡ ∂ p i ∆ has been defined in (2.3) (see also (3.5) and (3.7)), and all sums have been made explicit in the numerators; the quantity (∂ p ∆) · (µ • α) (∂ p ∆) in the square root in the denominator is given by (6.46).
In order to prove that (6.53) is a solution to the BPS system (2.6), we compute Thus, by recalling the Euler formula which is the BPS system (2.6) itself.
Thus, in all models explicitly treated below, after the checking that the invertibility condition enounced in Sec. 6.2 holds true, the crucial data to be known are the symmetric m × m Γ-matrices and the symmetric, auxiliary m × m matrices Ω K such that (6.6) and (6.8) both hold true.

The symmetric case
It should be remarked that those homogeneous d-spaces [dWVP1,dWVVP] in which the invertibility condition enounced in Sec. 6.2 is satisfied include the noteworthy class of homogeneous symmetric d-spaces (cfr. Sec. 4.1), in which it holds that where d ijk is defined in (4.15)-(4.19), and in the case of symmetric d-spaces it is a constant (numerical) tensor (i.e., it does not depend on any scalar fields' degree of freedom). (6.61) implies that Indeed, by plugging (6.62) into (6.53), one obtains thus matching (4.8) (recalling (4.29)).

BPS black hole entropy and attractors
Let us recall that ξ enters the expression (2.2) of the black hole entropy only through the quantity (2.8), and (6.57) holds true. Remarkably, by virtue of (2.8), the solutions of the BPS system (2.6) enter the expression of the black hole entropy only through the square of the quantity (6.57) : Then, by recalling (2.2), the Bekenstein-Hawking entropy of static, spherically symmetric, BPS extremal dyonic black holes in the model under consideration of N = 2, D = 4 Maxwell-Einstein supergravity has the following expression : In both formulae (6.65) and (6.66) the scalar product (∂ p ∆) · (µ • α) (∂ p ∆) is given by (6.46). Note that, as it must be, S (6.66) is an homogeneous positive function of degree 2 in the black hole charges. The consistency conditions for (6.65) and the BPS black hole entropy (6.66) to hold formally read as follows : where again the scalar product (∂ p ∆) · (µ • α) (∂ p ∆) is given by (6.46). By exploiting the results (6.37) and (6.65) and defining the expression of BPS attractor points (2.13) is given, in vector notation, by (6.72) or, more explicitly : where all sums are explicitly indicated in the numerator of (6.75), and (∂ p ∆)·(µ • α) (∂ p ∆) and S are respectively given by (6.46) and (6.66).
6.7 Complete models: r = 0 The classification, completed in [dWVP1] (see also [dWVVP]), shows that any L(q, 1, 0) ≡ L(q, 1), q −1 model of homogeneous very special geometry is defined by Γ 0 = I m and a Clifford set Γ-matrices, of size m × m, namely by {Γ 1 , . . . , Γ q+1 }, where m = D q+1 (given e.g. in Table 1 of [dWVVP] and in (4.26)). For fixed q, such matrices are unique (up to a choice of basis in R m ). We will choose in Sec. 8.1 a Clifford set {Γ I } I=1,...,9 of symmetric Γ-matrices. The associated quadratic forms {Q I } I =0 's have the additional pleasant property that putting the last m/2 coordinate y α 's equal to zero, some quadrics vanish identically, whereas the non-vanishing ones are the quadrics associated to an L(q ′ , 1) model with q ′ < 8.
A complete model is defined to be a model in which r = 0 in the invertibility condition enounced in Sec. 6.2. Then, (6.8) implies that the associated quadratic forms satisfy a Lorentzian quadratic relation where (4.37) has been recalled. Defining the associated cubic form V as in (6.5), the main result then states that the gradient map ∇ V will be invertible, with the (birational) inverse map being polynomial of degree 2. Indeed, since for r = 0 all coordinate functions of the map α are multiples of z 1 and µ is homogeneous, so, as mentioned above, one can redefine α to be the identity map. As a consequence, µ, which is given by quadratic polynomials, is the birational inverse of ∇ V ; cfr. (6.20), which can be regarded as a consequence of the so-called "adjoint identity" (4.1) of cubic Jordan algebras.
The only 22 complete models known to us have q = 1, 2, 4, 8 and m = 2q, so N = 3q + 3 = 6, 9, 15, 27 respectively; such models have been discussed at the end of Sec. 4.3.1, and they will be further discussed below. In these models, which correspond to the "magic" class of symmetric d-manifolds, the cubic forms are the well known norm forms on simple cubic Jordan algebras J A 3 , for A = R, C, H, O [Rus]. These complete models correspond to the models L(q, 1) [dWVP1,dWVVP] with q = 1, 2, 4, 8 = dim R A for A = R, C, H, O respectively, provided that the (q + 2)-dimensional vector has a (mostly plus) Lorentzian signature (1 − , (q + 1) + ), which can always be arranged.
As we will see in Sec. 13, also quadratic forms in q + 2 = 4, 6, 10 dimensions of Kleinian signatures (2 + , 2 − ), (3 + , 3 − ) and (5 + , 5 − ) can be considered: they are associated to simple cubic Jordan algebras over split composition algebras J As 3 , for A = C s , H s , O s . Moreover, they correspond to non-supersymmetric Maxwell-Einstein theories (for C s and H s ), as well as to maximal supergravity (in the case of O s ); cfr. [MPRR, MR].

Verifying the inverse map
In this section we prove that, if the condition enounced in Sec. 6.2 is satisfied, then the formulas in Sec. 6.4 indeed provide the (birational) inverse map of the gradient map of the cubic form V under consideration.

Factorization of R K 's
First of all, we start and derive a useful property of the quadratic polynomials R K 's, defined by the extra symmetric matrices Ω K , with 1 K r (with r 0), of size m.
The last m = 2 g coordinate functions of ∇ V are denoted by (y = T (y 1 , . . . , y m )) We show that, upon substituting ∇ y V into R K (6.7), the following factorization holds : In fact, by definition, one has: Recall that Γ 0 = I m and that from (6.6) we have Γ I Ω K = −Ω K Γ I for I = 1, . . . , q + 1. Therefore, for I > 0 it holds that If I, J > 0 and I = J, we have Γ I Γ J = −Γ J Γ I since we have a Clifford set, and thus Furthermore, when I = J > 0 we have Γ I Γ J = Γ 2 I = I m , and thus Therefore, all terms with I = J in (7.3) cancel, and we are left with which proves (7.2).

The map α•∇ V
For ξ = (s, x, y) ∈ R q+3+m (cfr. (4.29)), we now verify that the image (t, u, v, w) = α(∇ V (ξ)) in R q+3+m+r is given by (6.16). The first q + 3 + m components of α(z) are the z 1 z a 's, with a = 1, . . . , q + 3 + m. Since the first component of ∇ V is ∂V/∂s = q(x), we see that the first q + 3 + m components of α(∇ V (ξ)) : The last r components of α (z) are the R K 's, evaluated on the last m variables. Hence, the last r components of α•∇ V (ξ) are the R K 's evaluated on m-vector ∇ y V. From (7.2) we see that these components are −4q(x)R K (y), K = 1, ..., r.
2. For the next q + 2 components of the map (µ • α•∇ V ) (s, x, y), one needs to prove that (cfr. (7.16)): (7.20) We claim that Indeed, it holds that If I = 0, then Γ 0 = I m and thus (7.22), combined with (5.12), leads to x J Q J (y) + 2q(x)Q 0 (y). (7.23) On the other hand, if I = 1, ..., q + 1 then Γ I and Γ J commute only for J = 0 or J = I, and they anti-commute otherwise; consequently, by moving Γ I to the right, it follows that x J Q J (y) . (7.24) Hence, the claim (7.21) is proven. Now, we show that the equalities in (7.20) follow from the claim (7.21). In fact, by recalling (6.5) for I = 0 one obtains Analogously, for I = 1, ..., q + 1 it holds that x J Q J (y) = 4x I V. (7.26) This concludes the verification for the q + 2 components under consideration.
3. Finally, for the last m components of (µ • α•∇ V ) (s, x, y) one must check that First of all, we substitute V 0 = −2sx 0 + Q 0 (y) and V I = 2sx I + Q I (y) for I = 1, . . . , q + 1, in the r.h.s. of (7.27), obtaining x I Q I (y), the formula (7.27) follows if we verify the following two identities: We recall that ∇ y V = 2( q+1 I=0 x I Γ I )y. The first identity (7.29) is easy to verify by substituting this and using (5.12); in fact, its r.h.s. can be elaborated as follows : (7.31) In order to prove the second identity (7.30), we observe that, since Γ 2 I = I m , the first term in its r.h.s. can be elaborated as The second term in the r.h.s. of (7.30) reads So, the second identity (7.30) follows from (7.32) and (7.33) if we can show that 34) or, comparing the (matrix) coefficients of the x I 's, equivalently, for all I = 0, 1, . . . , q + 1: To verify (7.35), we use the identity (6.8), which we write as with, as above, Q I (y) = T yΓ I y, R K (y) := T yΩ K y; note that F(y) is identically zero as a polynomial in y = (y 1 , . . . , y m ). Therefore all partial derivatives of F w.r.t. the y α are also identically zero (as cubics in y). Notice that 0 = ∇ y F (y) = 2 (−Q 0 (y)Γ 0 + Q 1 (y)Γ 1 + . . . + Q q+1 (y)Γ q+1 + R 1 (y)Ω 1 + . . . + R r (y)Ω r ) y .
The results in Sec. 7, holding when the condition enounced in Sec. 6.2 is satisfied, provide the proof of the invertibility of the gradient map ∇ V by giving its explicit birational inverse map, as discussed in Sec. 6.4. 8 Examples, I : L(q, 1), q = 1, ..., 8 Whenever P = 1, 2, the various L(q, P ) models for which we can prove invertibility, namely those with q = 1, .., 8 and P = 1 as well as those with q = 1, 2, 3 and P = 2, are conveniently presented below as suitable linear sections (also named descendants 23 ) of the complete model L(8, 1). Note that the invertibility of these descendants is a priori by no means guaranteed by the invertibility of L(8, 1) (which is basically the J O 3 and thus is well-known to be invertible, cfr. Sec. 8.1); however, the general treatment given above does allow us to verify invertibility of such descendants. Furthermore, the models L(q, P ) with q = 1, 2, 3 and P 3 will then be treated with a different method in Sec. 10.
In this case, the Bekenstein-Hawking entropy and the attractor values of scalar fields of BPS extremal black holes are explicitly known; see e.g. [FGimK]. Thus, we will not consider the solution of the BPS system in this model, but rather we will give some treatment useful to discuss some descendants from L(8, 1) itself (cfr. Sec. 8.2).
23 But not necessarily submodels; cfr. Sec. 11.3. 24 In fact, one could in principle write down an explicit 3 × 3 Hermitian matrix M with octonionic components such that V = −N (M ), where N is the cubic norm (generalizing the determinant) of M , but we refrain from doing so; see [Kru].
( 8.2) The cubic form of this model is Since the quadratic forms Q 0 , . . . , Q 9 satisfy the Lorentzian quadratic relation − Q 0 (y) 2 + Q 1 (y) 2 + . . . + Q 9 (y) 2 = 0, (8.5) the L(8, 1) model satisfies the invertibility condition of Sec. 6.2 with r = 0 (so no extra Ω K are needed) : in fact, as mentioned above, L(8, 1) is a complete model. Thus, the gradient map ∇ V L(8,1) of V L(8,1) is invertible and the inverse map µ is given by homogeneous polynomials of degree two. It holds that The gradient map can be identified with the adjoint map here M ♯ is the adjoint matrix of M ∈ J O 3 . This adjoint map # is (birationally) invertible, with inverse given by the map M ♯ → (M ♯ ) ♯ , which is thus essentially the map µ.
The model L(4, 1) corresponds to a symmetric space, and it is related to the simple cubic Jordan algebra over the quaternions, J H 3 . In this case, the Bekenstein-Hawking entropy and the attractor values of scalar fields of extremal BPS black holes are explicitly known; see e.g. [FGimK]. In the present treatment, we will highlight its relation to the complete 'parent' model L(8, 1).
The model L(2, 1) corresponds to a symmetric space, and it is related to the simple cubic Jordan algebra over the complex numbers, J C 3 . In this case, the Bekenstein-Hawking entropy and the attractor values of scalar fields of extremal BPS black holes are explicitly known; see e.g. [FGimK]. In the present treatment, we will highlight its relation to the complete 'parent' models L(8, 1) and L(4, 1).
The determinant of J C 3 Again, it is worth making more explicit the relation between the complete model L(2, 1) and the Euclidean simple cubic Jordan algebra J C 3 . If i denotes the imaginary unit of C, then to ξ = T (s, x 0 , x 1 , x 2 , x 3 , y 1 , y 2 , y 3 , y 4 ) (cfr. (4.29)) we define the 3 × 3 complex Hermitian matrix This model corresponds to a symmetric space [dWVP1], and it is related to the simple cubic Jordan algebra over the reals, J R 3 , namely the algebra of symmetric real 3 × 3 matrices. In this case, the Bekenstein-Hawking entropy and the attractor values of scalar fields of extremal black holes are explicitly known; see e.g. [FGimK]. In the present treatment, we will highlight its relation to the complete 'parent' models L(8, 1), L(4, 1) and L(2, 1).
(10.2) Given the matrices Ω K of size m × m of the L(q, 2) model (q = 1, 2, 3) discussed in Sec. 9, let as before R K be the quadratic form in 2m variables defined by Ω K and let Ω (kl) K be the symmetric matrix of size mP defined by the quadratic form R (k,l) K in mP variables, so that R (k,l)

Invertibility
The Γ-matrices, of size mP × mP , of the L(q, P ) (P 3) models are the q + 2 matrices Γ (P ) I := Γ I ⊗ I P , I = 0, . . . , q + 1, where the Γ I are the Γ-matrices of the L(q, 1) model. Now, we claim that if we consider the mP × mP matrices Ω (kl) K , 1 k < l P where the Ω K are the extra matrices in the L(q, 2) model, then the conditions (6.6) and (6.8) for the invertibility of the gradient map of these L(q, P ) models (P 3) are all satisfied.
The anti-commutativity condition follows rather trivially from the fact that in Sec. 9 we checked that the Γ (2) I , I = 1, . . . , q + 1, anti-commute with all the Ω K 's of the L(q, 2) models we have considered, and therefore also the Γ-matrices Γ (P ) I of the L(q, P ) models (with P 3) anti-commute with all the Ω (kl) K , so (6.6) is satisfied. On the other hand, in Sec. 10.3 below, we will check the existence of the Lorentzian identity (6.8), namely we will verify the following Lorentzian identity between quadratic forms in mP variables for all P 3: Thus, the invertibility condition of Sec. 6.2 is satisfied and the corresponding gradient map (and BPS system) can be inverted for any q = 1, 2, 3 and P 3.
Again, from the treatment of Sec. 6, the solution of the BPS system of L(q, P ) for q = 1, 2, 3 and P 3 is then given by (6.54)-(6.56), and (6.66) and (6.73)-(6.75) yield the corresponding expression of the BPS black hole entropy and of the BPS attractors, respectively. We refer to Sec. 9.5, 10.6 for examples of lower dimensional models L(q, P ) (namely, L(1, 2) and L(1, 3)) which we explicitly work out in detail (see also the generalization to L(1, P ) models with P 2 in Sec. 10.7)

Proof of the Lorentzian identity (10.4)
The definition (9.5) of Q (P ) I 's allows us to rewrite the first terms in the l.h.s. of (10.4) as follows: (10.5) where the remaining cross terms are defined as follows (no sum on repeated indices, I = 0, 1, ..., q + 1) : S I,k,l := 2Q I (y (k) )Q I (y (l) ) . (10.6) • We do the cases q = 1, 2 first. The Lorentzian quadratic relations (8.73) resp. (8.61) hold, thus for any k we find that Consequently, it remains to show that In the Lorentzian identity (9.7) for the L(q, 2) models, the vector y is in fact y (12) ; so, replacing it by y (kl) , one obtains the identities, for any 1 k < l P : Recalling the definition (9.5) of the quadratic forms Q (2) I , one gets Q (2) I (y (kl) ) 2 = Q I (y (k) ) + Q I (y (l) ) 2 = Q I (y (k) ) 2 + Q I (y (l) ) 2 + 2Q I (y (k) )Q I (y (l) ) .
can then be found following the procedure discussed in Sec. 6.4.
Consequently, from the treatment of Sec. 6, the full fledged expression of the solution (6.53) of the BPS system of L(1, P ) for arbitrary P 2 is given by (∆ 1 ≡ ∆ s , and recall the definition (2.3)): (10.88) . . . (10.92) Then, (6.66) and (6.73)-(6.75) respectively yield the corresponding full fledged expression of the BPS black hole entropy and of the BPS attractors : (10.95) with (∂ p ∆) · (µ • α) (∂ p ∆) given by (10.94), and p · q = p 0 q 0 + p 1 q 1 + .... + p 2P +4 q 2P +4 ; (10.104) Consistently, it should be remarked that for P = 2 and P = 3 the above formulae allow one to retrieve the explicit results obtained in Secs. 9.5 and 10.6, respectively. It is here worth remarking that the above expressions provide, for the first time to the best of our knowledge, the explicit form of the BPS black hole entropy and attractors in an infinite class of (homogeneous) nonsymmetric models of N = 2, D = 4 supergravity with cubic prepotential. 10.7.2 On the geometry of (∇ V L(1,P ) ) −1 , P 1 We will now describe briefly some properties of the factorization (10.107) in order to have a different, geometric perspective on its nature and on the arising of the quadratic forms R k 's (or R (k,l) K 's) appearing in the Lorentzian quadratic identities (6.8), (9.7), or (10.4). Since the map α : R 2P +4 → R 2P +4+( P 2 ) is defined by homogenous polynomials of degree two, the map α has image Z ⊂ R 2P +4+( P 2 ) defined by homogeneous polynomials. Moreover, the dimension of Z, as an algebraic variety, is 2P + 4, as it is easily seen by looking at the expression of α. From the parametrization α of Z, we can deduce that it is a cone over a Grassmann variety G(2, P + 2). Moreover, the inverse of α (as a birational map from R 2P +4 to Z) is the restriction to Z of the projection π(t, u, v, w) = T (t, u, v). Indeed, the map α : R 2P +4 → Z is such that, letting z = T (z 1 , . . . , z 2P +4 ), (π • α)(z) = z 1 z . (10.108) The map µ : R 2P +4+( P 2 ) → R 2P +4 restricted to Z induces a map µ : Z → R 2P +4 . The inverse (as a birational map) is the map φ : A similar description of the images Z's of the corresponding maps α's and the definition of the corresponding φ's can actually be provided for all the models L(q, P ) treated until now; however, a thorough treatment goes well beyond the scope of this paper, and we leave it for further future work.
This geometric approach has many interesting applications for the study of the geometry of the singular locus of the cubic hypersurface V L(q,P ) in the associated projective space, which in many cases is a notable algebraic variety (for example in the complete case, but not only). In some cases, the quadratic homogeneous polynomials defining the map φ are a basis of all homogenous quadratic polynomials vanishing on (some irreducible component of) the singular locus of the cubic. It may also happen that there are more quadratic equations vanishing on the singular locus than the partial derivatives of V L(q,P ) which define it, and this explains why correspondingly r > 0. This geometric point of view and its applications to Algebraic Geometry will be considered elsewhere.

Non-uniqueness of Ω's
The models L(1, P ) for P 4 can be handled with the general results obtained in Sec. 10.2, as outlined in the previous Section. However, those with P 8 can also be embedded into the complete L(8, 1) model, and this provides an alternative way to determine the inverse of their gradient maps.
In this respect, we should stress the fact that the extra Ω-matrices, required for non-complete systems by the invertibility condition enounced in Sec. 6.2, may not be unique; even the number r of such matrices is not determined uniquely by the model.
We illustrate this for P = 4 and 8, respectively in Secs. 11.1 and 11.2.

Descendant Submodel
In the treatment given above, we have discussed various cases in which a model L(q, P,Ṗ ) can be regarded as a model L(q ′ , P ′ ,Ṗ ′ ) with a larger number of variables (namely, q + P +Ṗ · D q+1 < q ′ + P ′ +Ṗ ′ · D q ′ +1 ) with some linear constraints (and possibly with some renamings of variables). L(q, P,Ṗ ) has thus been defined as a descendant of L(q ′ , P ′ ,Ṗ ′ ), denoted by L(q, P,Ṗ ) ⊂ L(q ′ , P ′ ,Ṗ ′ ), (11.15) and this relation has been instrumental in proving the invertibility of the corresponding gradient map by using the invertibility condition enounced in Sec. 6.2. Here, we want to point out that (11.15) does not necessarily imply that L(q, P,Ṗ ) is a submodel of L(q ′ , P ′ ,Ṗ ′ ) (while the converse is trivially true).
holds, as required by the aforementioned condition, which then implies that the gradient map of L(4, 1, 1) is invertible. We now present the quadrics of the L(8, 1) system, as given in (8.2), with the appropriate renamings suitable for the L(4, 1, 1) model:

Kleinian signatures and split algebras
As already mentioned at the end of Sec. 6.7, one can also find sets of symmetric Γ-matrices defining a Clifford algebra representation of a quadratic form in q + 2 = 4, 6, 10 dimensions with Kleinian "neutral" signatures (2 + , 2 − ), (3 + , 3 − ) and (5 + , 5 − ) for q = 2, 4 and 8 respectively : these Kleinian signatures correspond to simple cubic Euclidean Jordan algebras over split composition algebras J As 3 , for A = C s , H s and O s , respectively. These cases do not belong to the homogeneous special manifolds classified by L(q, P,Ṗ ) : in fact, they pertain to the so-called 'magic' non-supersymmetric Maxwell-Einstein theories, as well as to the maximal supergravity (in the case of split octonions O s ); cfr. [MPRR, MR] : they can be regarded as the 'Kleinian counterparts' of the magic N = 2 supergravity theories discussed in Sec. 5 (recall (4.38) therein) [Has, MPRR, MR]  Notice that, as the magic L(q, 1) models (q = 1, 2, 4, 8) are related to Euclidean Clifford algebras Cl(q + 1, 0) (cfr. Sec. 5), their 'Kleinian counterparts' models (existing for q = 2, 4, 8) are related to Clifford algebras Cl q 2 + 1, q 2 in q + 1 dimensions with (mostly minus) signature q 2 + 1, q 2 ; in fact, the reality properties of the spinors are the same for the (q + 1, 0) and q 2 + 1, q 2 signatures in q + 1 dimensions. The extremal black hole entropy in maximal supergravity is explicitly known (see e.g. [FGimK]), and by suitable truncations one obtains the same quantity in the J Hs 3 -and J Cs 3 -based theories. For completeness's sake, we consider here the Kleinian model based on the exceptional cubic Jordan algebra J Os 3 , which pertains to maximal supergravity; in this case, x I Q I (y) with q(x) = x 1 2 + . . . + x 5 2 − x 6 2 − . . . − x 10 2 .
However, in striking contrast to the complete models related to Γ-matrices of size m = 2 g with g = 1, 2, 3, 4, discussed in Sec. 6.7 and respectively treated in Secs. 8.2.7,8.2.6,8.2.4 and 8.1, there is no (Lorentzian) quadratic identity between the 11 quadratic forms Q 0 , . . . , Q 10 . Therefore, one cannot exploit the invertibility condition enounced in Sec. 6.2 in order to determine the invertibility of the gradient map ∇ V L(9,1) . Of course, such a condition provides a sufficient but not necessary condition for invertibility, so the lack of a suitable quadratic identity of quadrics does not necessarily imply the non-invertibility of the gradient map of the corresponding cubic form.
At any rate, other approaches to prove invertibility or non-invertibility of the gradient map ∇ V L(9,1) should be found, but they are beyond the scope of the present investigation.

Final remarks and outlook
We have considered the issue to obtain an explicit expression of the attractor values of scalar fields as well as of the Bekenstein-Hawking entropy, of static, asymptotically flat, dyonic, BPS extremal black holes in ungauged N = 2 Maxwell-Einstein supergravity theories in four space-time dimensions, coupled to non-linear sigma models of scalar fields endowed with very special geometry; this class of theories encompasses all four-dimensional N = 2 theories which can be obtained as an S 1 -compactification of five-dimensional minimal supergravity theories. After [Shm], this problem can be translated into the issue of solving certain algebraic inhomogeneous systems of degree two, named BPS systems.
Within the so-called 'very special' geometry (related to cubic holomorphic prepotentials), we have focused on homogeneous non-compact Riemannian spaces. For homogeneous symmetric spaces, which are related to (simple and semi-simple) cubic (Euclidean) Jordan algebras, the solution to the BPS system is explicitly known, as is the expression of the BPS entropy and attractors (cfr. [FGimK], and Refs. therein) : they can be formulated only in terms of a unique quartic invariant 34 polynomial in the black hole electric-magnetic charges. On the other hand, not much is known for the homogeneous non-symmetric spaces; in fact, to the best of our knowledge, only [DFT] and [ADFT] briefly treated, within a different formalism, the models L(1, 2) and L(2, 2). Therefore, in the present investigation we have focussed on homogeneous non-symmetric very special geometry, which has been classified, in terms of Euclidean Clifford algebras, in [dWVP1].
In Sec. 6.2 we have formulated a (sufficient, but not necessary) condition for the invertibility of the gradient map of the cubic form defining the homogeneous non-symmetric very special geometry (and thus for the resolution of the related BPS system) : this condition requires the existence of a suitable Lorentzian quadratic identity involving the quadratic forms defined by the symmetric Γ-matrices of the corresponding Euclidean real Clifford algebra, as well as some other quadratic forms defined by symmetric auxiliary matrices denoted by Ω K . Subsequently, we have thus provided in Sec. 6.4 an explicit expression for the (birational) inverse map of the gradient map of the models for which the invertibility condition holds; the inverse map is a homogeneous polynomial map of degree four. Then, in Sec. 6.5, we have presented, within the assumption that the aforementioned condition holds true, a procedure for the explicit solution of the related BPS system, determining in Sec. 6.6 an explicit formula for the BPS Bekenstein-Hawking entropy of extremal black holes, as well as for the attractor values of the scalar fields in such a background. It is also here worth remarking that the explicit solution of the BPS system is also relevant for the solution of the attractor equations in asymptotically AdS, dyonic, extremal 1 4 -BPS black holes of U (1) Fayet-Iliopoulos gauged Maxwell-Einstein N = 2 supergravity in four space-time dimensions [Hal, HG].
In particular, the models L(1, 2) and L(1, 3) have been worked out in full detail in Secs. 9.5 resp. 10.6, and in Sec. 10.7 their treatment has been generalized (in a P -dependent manner) to the infinite class of L(1, P ) P 2 non-symmetric models of N = 2, D = 4 supergravity. In this respect, we have extended the treatment given in Sec. 4 of [Shm], by providing, for the first time to the best of our knowledge, the explicit form of the BPS black hole entropy and of the BPS attractors in an infinite class of (homogeneous) non-symmetric models of N = 2 supergravity with cubic prepotential.
We leave the treatment of such classes to future work. It would also be interesting to investigate the invertibility of the gradient map, and thus the solution to the corresponding BPS system, of cubic forms associated to noteworthy classes of non-homogeneous spaces.
Also, we would like to recall that in Sec. 10.7.2, we have briefly considered a geometric perspective on the factorized nature of the inverse map of the gradient map of the cubic forms pertaining to the models L(1, P ) with P 1. We conjecture that this holds essentially true for any L(q, P ) model, thus providing an explanation to r > 0 in non-complete models; in future works, it will be interesting to discuss this geometric point of view in detail, as well as to study various subsequent applications to Algebraic Geometry.
Within this research venue, it would be interesting to investigate the geometric aspects of the examples of non-homogeneous very special geometry discussed by Shmakova in Sec. 4 of [Shm] (cfr. Refs. therein, as well), as well as of the non-homogeneous 2-moduli cubic models in which non-trivial involutory matrices determining multiple attractor solutions exist [MT, MMT]; we leave these tasks for further future work.