Special Vinberg Cones and the Entropy of BPS Extremal Black Holes

We consider the static, spherically symmetric and asymptotically flat BPS extremal black holes in ungauged N = 2 D = 4 supergravity theories, in which the scalar manifold of the vector multiplets is homogeneous. By a result of Shmakova on the BPS attractor equations, the entropy of this kind of black holes can be expressed only in terms of their electric and magnetic charges, provided that the inverse of a certain quadratic map (uniquely determined by the prepotential of the theory) is given. This inverse was previously known just for the cases in which the scalar manifold of the theory is a homogeneous symmetric space. In this paper we use Vinberg's theory of homogeneous cones to determine an explicit expression for such an inverse, under the assumption that the scalar manifold is homogeneous, but not necessarily symmetric. As immediate consequence, we get a formula for the entropy of BPS black holes that holds in any model of N = 2 supergravity with homogeneous scalar manifold.


Introduction
The first purpose of this paper is to determine an explicit formula which gives the Bekenstein-Hawking entropy of a static, spherically symmetric and asymptotically flat BPS extremal black hole in terms of its electric and magnetic charges in ungauged N = 2 D = 4 supergravity theory, under the assumption that the scalar manifold of the vector multiplets is homogeneous. As a secondary goal, we want to offer a gentle introduction to Vinberg's theory of homogeneous cones associated with irreducible invariant cubic polynomials and to illustrate how this purely mathematical theory can be combined with some fundamental theoretical physics results, such as Bekenstein-Hawking entropy-area formula or Ferrara, Kallosh and Strominger's BPS algebraic attractor equations, to establish new non-trivial results on black holes.
The paper starts with a discussion of the invariant real cubic polynomials d(y) = d abc y a y b y c , (y a ) ∈ R n , which are associated to the holomorphic prepotentials F (X) = d abc X a X b X c X 0 , (X I ) = (X 0 , X a ) ∈ C n+1 , (1.1) that determine homogeneous scalar manifolds of the vector multiplets of ungauged N = 2 D = 4 supergravity. By known results on prepotentials and associated scalar manifolds ( [18,11,12,2]), any irreducible cubic polynomial of this kind corresponds to a rank 3 homogeneous convex cone V ⊂ R n of dimension n. This is in turn representable as the cone of positive Hermitian matrices in an appropriate space of 3 × 3 matrices, with vector and spinor valued entries, given by a special Vinberg T -algebra (see [40,3] and § 2.2 -2.3). Aiming to a presentation that might be accessible to any reader with no previous knowledge of Vinberg's theory, the first section starts with a selfcontained exposition of the main definitions and properties of special Vinberg T -algebras, of the corresponding cones of positive Hermitian matrices and of the invariant cubic polynomials which are important objects associated with these cones. We also introduce the notions of dual cones and associated dual invariant homogeneous rational functions of degree 3. These new objects are later used to determine the explicit general expressions for the inverses to the quadratic maps, that appear in the explicit entropy formula for BPS extremal black holes established in the third part.
The second and third parts of the paper provide a short review of special Kähler geometry and of the BPS extremal black holes in ungauged Maxwell-Einstein N = 2 supergravity with prepotentials of the form (1.1). The presentation is structured for readers who are not familiar with supergravity. Here is a short outline. Consider the class of metrics on the 4-dimensional space-time M of the form [34,31], ds 2 = −e 2U (r) dt 2 + e −2U (r) dr 2 + r 2 (dθ 2 + sin 2 θdϕ 2 ) . (1.2) They are solutions to the Euler-Lagrange equations of the bosonic sector of the aforementioned supergravity and describe static, spherically symmetric, asymptotically flat, dyonic extremal black holes with unique event horizon at r = 0 (as implied by extremality). The magnetic and electric charges of any such black hole are the fluxes of the electromagnetic fields F I = F I µν dx µ ∧ dx ν and their duals G J = ⋆ 1 2 δL δF J with respect to the Lagrangian L of the theory, Due to its rotational invariance and time independence, the dynamics of the scalar and electromagnetic fields of this kind of black holes can be described by means of a reduced 1-dimensional Lagrangian, characterised by the effective black hole potential [19] V BH (p I , q J , z, z) := |Z| 2 + g ab Z a Z b with Z a := ∂Z ∂z a + 1 2 ∂K ∂z a Z , (1.4) where (i) z = (z a ) is the map that represents the scalar fields and takes values in the scalar manifold S ⊂ C n ; (ii) g = (g ab ) is the Kähler metric of the scalar manifold S; (iii) K is the Kähler potential of g, (iv) Z = Z(p I , q J , z, z) is the N = 2 central charge, which is a function of the magnetic and electric charges (p I , q J ) and the scalar fields z = (z a ). We recall that, for any extremal black hole (1.2), the central charge function Z = Z(p I , q J , z, z) satisfies the identity where (X I ) = X 0 X a = X 0 1 z a ∈ C n+1 with z a = X a X 0 , and F J := ∂F ∂X J (X I ) .
For given charges p I , q J in an appropriate set, it is known that the values of the scalar fields z at r = 0 are stable critical points for V BH . By the famous Attractor Mechanism [24,39,22,23,19], such criticality condition determines a (locally invertible) relation between the horizon values of the scalar fields and the magnetic and electric charges of the black hole.
Let us now focus on the black holes (1.2) This is a system of equations that relates the set of the 2n+ 2 real numbers p I , q J and the set of the n + 1 complex numbers (Z o , z 1 o , . . . , z n o ). Such a system is always locally solvable in terms of the second set [38].
On the other hand, by the Bekenstein-Hawking entropy-area formula [7,28], the above property (a) of the BPS metrics (1.2) implies that the entropy of any such black hole is related with the central charge Z o at the horizon by [19] Hence any (local) inversion of the algebraic relation (1.6) (thus giving Z o as a function of p I and q J ) provides a formula for the black hole entropy S in terms of its magnetic and electric charges. The fourth part of our paper is devoted to the solution of such inversion problem.
More precisely, in our fourth section, we analyse in detail the correspondence (z a o , Z o ) → (p I , q J ) determined by (1.6), which we name BPS map. After a brief discussion about the invertibility of such a map in the (simpler) case with p 0 = 0, we tackle the situations with p 0 = 0. For them, we show that the BPS map is always a local diffeomorphism. This is obtained using Shmakova's formulas in [38] (which we newly derive in detail by means of a different line of arguments) that reduce the inversion problem for the BPS map to the (somehow simpler) inversion problem of the quadratic map From these formulas, the BPS map is proved to be a local diffeomorphism by checking that h d has a non-vanishing Jacobian and thus, due to the Inverse Function Theorem, it is locally invertible with smooth inverse. In the cases in which d(y) is an irreducible polynomial and is associated with homogeneous scalar manifolds, a globally defined inverse for h d : R n → R n * has been explicitly determined in the first part of this paper. Therefore, the results of our first part immediately yield to explicit formulas for global inverses to the BPS map, one per each of the two connected regions corresponding to the condition p 0 = 0. These two inverse maps give expressions for the horizon values Z o , z o = (z a o ) in terms of the electric and magnetic charges only. By (1.7), they also give an expression for the entropy of the black hole.
At the best of our knowledge, explicit expressions for the inverse map h −1 d and, consequently, for the entropy S of the above BPS black holes, was so far known only when d(y) = d abc y a y b y c and the corresponding prepotential determine a homogeneous symmetric scalar manifold S. Our results can be thus considered as a completion of such previous results, providing a solution to the above BPS black hole entropy problem for all cases in which S is homogeneous, regardless whether it is symmetric or non-symmetric.
The expression for the entropy S we obtain for the considered cases is S = π I 4 , where I 4 = I 4 (p 0 , p a , q 0 , q b ) is the homogeneous rational function of degree 4 Here d ′ (w) = d abc w a w b w c where d abc = δ ae δ bf δ cg d ef g and a(p 0 , p a ,q 0 , q a ), b(p 0 , p a ,q 0 , q a ) are homogeneous rational functions of degree 0 and 3, respectively (vanishing identically if the scalar manifold is a homogeneous symmetric Hermitian space), explicitly given in (5.34), and all of these terms are uniquely determined by the invariant cubic polynomial d(y) and the dual invariant degree 3 rational function d * = d + d associated with d(y), which we define in Definition 2.7 and we explicitly determine in Theorem 2.8. As we have already pointed out, formula (1.8) was previously known just in the cases in which (S = G/K, g) is a homogeneous symmetric Hermitian space. In all such cases it was observed by the second author in [32] that, up to a factor, the natural extension of the polynomial I 4 to C 2n+2 coincides with the unique generator of the ring of the relative invariants of the standard representa- is an appropriate embedding of G into Sp 2n+2 (R). This remarkable property is a consequence of the following three facts: (a) the equations of motion of the supergravity that we consider are invariant under the group G e.m. of the electric-magnetic dualities (also called generalised duality transformations or U-dualities in supergravity literature); (b) G e.m. is a subgroup of Sp 2n+2 (R) acting in a standard way on the space R 2n+2 of the magnetic and electric charges of the above BPS black holes and admitting a natural isomorphism has an open orbit. Note that (c) is precisely the property that implies that I 4 is the unique generator (up to a scaling) for the ring of the relative invariants.
Since (a) and (b) are true whenever (S = G/K, g) is a homogeneous (not necessarily symmetric) manifold, we immediately have that in all these cases I 4 is a relative invariant for the action of (R + × ρ(G)) C , G C ⊂ Sp 2n+2 (C), on C 2n+2 . But we claim that also (c) is true for any homogeneous scalar manifold. In fact, from the explicit expression of the BPS map and recalling the explicit form of the linear action of G on C n+1 (which projects onto the scalar manifold S under projectivisation), one can directly check that the isotropy subgroup of the representation ρ(G) = G e.m. on R 2n+2 has one dimension less than the isotropy H of the scalar manifold S = G/H. Thus the regular orbits ρ(G)·(p I , q J ) ⊂ R 2n+2 have dimension dim R ρ(G) · (p, q) = dim R S + 1 = 2n + 1, a property which implies (c). Combining these three simple observations, we may conclude that (up to a rescaling) I 4 is the unique generator of the relative invariants of (R + × ρ(G)) C for all cases in which S is homogeneous. Details on this and other aspects of the homogeneous rational function I 4 of degree 4 are left to a future work. Special T -algebras. Let (V, g V ) be a Euclidean vector space with associated Clifford algebra Cℓ(V ), constructed according to the Clifford relation v·w + w · v = −2g V (v, w). Let also S = S 0 + S 1 be a Z 2 -graded Cℓ(V ) module equipped with a Euclidean scalar product g S satisfying the following two conditions: In the terminology of [1,2], a scalar product satisfying these conditions is called symmetric admissible scalar product of type τ = −1.
We call the space (S, g S ) a metric Cℓ(V )-module.
In the following, to simplify notation, we will denote both the metric g V +g S of V +S and the corresponding induced metric on V ′ +S ′ =Hom(V + S, R) by ·, · . We will also denote by (·) ♭ : by (·) ♯ : V ′ + S ′ → V + S the inverse map of (·) ♭ and by ·, · V ′ +S ′ = g V ′ + g S ′ the scalar product on V ′ + S ′ , induced by (·) ♭ . A similar notation will be later used for any other Euclidean vector space.
Definition 2.1. The special T -algebra determined by (V, g V ) and (S = S 0 + S 1 , g S ) is the direct sum of vector spaces equipped with the product "·" and the Euclidean scalar product (·, ·) defined as follows. Consider the notation and define "·" as the product such that the only non-trivial multiplications are given by the bilinear maps · : A ij × A jℓ −→ A iℓ that are determined by the Clifford multiplication µ : V × S → S as follows: together with the rules This completely determines all of the remaining products.
Finally, we set (·, ·) to be the Euclidean scalar product on A, with respect to which all subspaces A ij are orthogonal one to the other, is the standard scalar product of R on each subspace A 11 = A 22 = A 33 = R, and is equal to ·, · V ′ +S ′ + ·, · on the subspace (S ′ 0 + V ′ + S ′ 1 ) + (V + S 1 + S 0 ).
In [2] it is proven that any special T -algebra, as defined above, satisfies the axioms of T -algebras of a rank 3 in the sense of Vinberg. This can be also directly checked, by observing that the Lie algebra equipped with the Euclidean scalar product (·, ·) N×N , is an N-algebra of rank 3 in the sense of [40,§III.7]. Indeed, by Vinberg's results, any T -algebra of rank 3 is uniquely determined by its nilpotent part (N = A 12 +A 23 +A 13 , (·, ·)), which is required just to be associative and with isometric product · : The subsets of A = A(V, S), defined by are closed under the multiplication · and such a product defines the structure of a simply connected solvable Lie group on each of them.
Definition 2.2. The group G is called Vinberg triangular group and G * is called dual (triangular) Vinberg group.
By the results in [40], G and G * are both simply connected and solvable and the map ı : G → G * , ı(A) := (A * ) −1 , is a Lie group isomorphism.
2.1.2. The standard matrix representation of a special T algebra. Let The product · of A defines the (non-associative) product between these matrices, determined by the standard matrix multiplication, x ij ·y jm .
Since this matrix product satisfies for any x, y ∈ A , the linear map is a linear representation, called the (standard) matrix representation of A.

2.1.3.
The space of Hermitian matrices H and the representations of the group G and G * in H. In terms of the matrix representation Mat = Mat(V, S), the involution (·) * of the T -algebra A is given by Using this map, we can express the scalar product (·, ·) of A in terms the matrix representation by the formula (x, y) = tr (X(x)·Y * (y)) (2.11) The space of Hermitian matrices in Mat is the subspace defined by It has a natural algebra structure determined by the Jordan multiplication The commutative algebra H is called the Hermitian Vinberg algebra associated with the metric Cℓ(V )-module (S, g S ).
In the matrix representation, the (upper) triangular group G and the dual (lower) triangular group G * are represented by the non-degenerate upper triangular matrices 14) and the lower triangular matrices The Lie algebra g (resp. g * ) of the group G (resp. G * ) consists of all upper triangular (resp., lower triangular) matrices in Mat ≃ A. The groups of upper and lower triangular matrices (2.14) and (2.15) are called standard realisations of G and G * , respectively.

Special Vinberg cones and their dual cones
The linear actions of G and G * on the space of Hermitian matrices H. As we mentioned above, the Lie algebras g = Lie(G) and g * = Lie(G * ) consist of arbitrary upper and lower triangular matrices in Mat ≃ A. As it can be directly checked using the axioms of T -algebras, the formula defines two linear representations T : g×H → H and T : g * ×H → H of these Lie algebras on the vector space H. Since the Lie groups G, G * are solvable and simply connected, these Lie algebras representations integrate to linear representations of the groups G and G * given by Due to the non-associativity of the algebra A, in general this action of the elements A = exp(B) in G or G * on the elements X ∈ H cannot be reduced to the standard expression A·X·A * .

2.2.2.
The linear representation of G on the dual space H ′ = Hom(H, R).
Given the vector space of Hermitian matrices H ⊂ Mat, we use the Euclidean metric X, Y := tr X·Y of H to identify the dual vector space The elements A of the group G act on H ′ by the dual transformations We denote by G ′ the group of these dual transformations. It is the exponential of the dual action of the elements B of the Lie algebra g = Lie(G) given by Indeed, for any Y ∈ H, Exponentiating both sides of (2.18), it follows that for any A = exp(B) ∈ G, (2.20) Hence, under the above identification The special Vinberg cones and their dual and adjoint cones. By Vinberg's results in [40] the following holds.
Theorem 2.4. The orbits V = G(I) and V * = G * (I) of the identity matrix I ∈ H are equal to They are both homogeneous convex cones, on which the groups G and, respectively, G * act simply transitively.
Definition 2.5. The convex cones V = G(I) and V * = G * (I) are called special Vinberg cone and its dual cone, respectively, associated with the metric Z 2graded Cℓ(V )-module (S, g S ).
The dual cone V * has the following important geometric role. Consider the adjoint cone of the cone V ⊂ H, that is the cone in H ′ ≃ H defined by It can be proved that V ′ = V * = G * (I). We finally recall the following

Group coordinates, adapted orthogonal coordinates and De Wit and Van Proeyen coordinates
Since both cones V = G(I) and V * = G * (I) are in natural bijection with G, we may consider the diffeomorphisms ξ G : V → G and ξ * G : V * → G given by We call them the group coordinates of V and V * .
The relations between the group coordinates and the corresponding elements in V and V * are as follows. If X ∈ V corresponds to the element , the entries of The inverses to the (2.22) are given by (see also [40,1]) while the inverses to the (2.23) are given by Note that (2.24) imply that V (resp. (2.25) imply that V * ) coincides with the convex set characterised by the following three inequalities which generalise Sylvester's criterion for positive definiteness.
We now observe that H is isomorphic (as a vector space) to the vector space Therefore each basis B for W naturally determines an associated system of coordinates for H ≃ W , which we call standard. If B has the form with 1 i standard basis of A ii = R and (e j ), (f 0|α , f 1|β ) orthonormal bases of (V, g V ) and (S = S 0 + S 1 , g S ), respectively, the corresponding coordinates on H are called adapted orthogonal coordinates on H. A basis B ′ for the space W ′ = Hom(W, R), which is dual to a basis B for W , determines coordinates for H ′ ≃ W ′ , which we call dual coordinates associated with B.
For a given system of adapted orthogonal coordinates the associated De Wit-Van Proeyen coordinates are the coordinates w = (w I ) determined by the linear transformation rules The corresponding dual transformation rules define the De Wit-Van Proeyen dual coordinates.

The invariant and dual invariant cubic polynomials
The Lie group G of upper triangular matrices is the direct product G = (R + I) × G 0 of the dilatation subgroup R + I and the unimodular subgroup Similarly G * is the direct product G * = (R + I) × G * 0 of the dilatation subgroup R + I and the unimodular subgroup G * 0 ⊂ G * = ı(G) Definition 2.7. A non-zero rational function p : H → R is called invariant cubic (resp. dual invariant cubic rational function) if it is a polynomial (resp. a homogeneous rational map) and satisfies Proof. The invariant cubic polynomials and the dual invariant cubic rational functions are unique up to a scaling because the subgroups G 0 and G * 0 have codimension one orbits. We now prove that, if d(I) = d * (I) = 1, then the formulas for d(X) and d * (X) are given by (2.33). By real analyticity, it suffices to prove this for X = A·A * ∈ G(I) and X * = A * ·A ∈ G * (I) for some A = are G 0 -invariant and G * 0 -invariant, respectively, and they are both equal to 1 on A = I. They are therefore the unique G 0 -and G * 0 -invariant functions d : V → R and d * : V * → R satisfying the normalising condition. The inverse maps (2.24) and (2.25) show that the expressions of these two functions in terms of the entries of the matrix X are given by (2.33).
In adapted orthogonal coordinates the invariant rational functions (2.33) are and where γ iαβ is the Γ-matrix representing the Clifford multiplication µ(e i , ·). Let us now denote by γ 3 = (γ 3ij ) and γ µ = (γ µij ) the square matrices, acting on the n S -dimensional space S = S 0 + S 1 (whose elements have components denoted by (s i ) = (s α 0 , s β 1 ) for short) given by By the transformation rules (2.30), the expression for d in de Wit -Van Proeyen coordinates w = (w I ) = (w 1 , w 2 , w 3 , w µ , w i ) becomes where µ runs between 3 and 3 + n and i, j run between 4 + n and 3 + n + n S ( 1 ). Note that -the γ µ are the Dirac matrices of the representation of the Clifford algebra A non-zero rational function p ′ : H ′ → R is called invariant dual cubic rational function if p ′ tA ′ (X) = t 3 p ′ (X) for any tA ∈ G = (R + I) × G 0 .
The isomorphism (·) ♭ : H → H ′ maps each invariant cubic rational function d * = d + d on H onto a uniquely associated invariant dual cubic rational function d ′ +d ′ on H ′ . Thus, Theorem 2.8 implies that up to a scaling, there is a unique invariant dual cubic rational function d ′ + d ′ : H ′ → R, which is determined as follows. Given an adapted system of coordinates x = (x a ) on R n = H, if we denote by g = (g oab ) the components of the Euclidean scalar product ·, · in the coordinate basis, by d abc the (symmetric in all indices) coefficients of the cubic polynomial d(x) = d abc x a x b x c and by g −1 . This means that, if the coordinates are orthogonal (thus with g o = (g oab ) = (δ ab )) and the components of d in these coordinates are denoted by d abc , then the components d abc of the dual polynomial d ′ in the associated dual adapted orthogonal coordinates are obtained by simply raising the indices with the 1 Mind the differences in indices conventions: on one hand in adapted orthogonal coordinates we denote the vector (resp. spinor) indices by Latin letters as i, j (resp. Greek letters like α, β); on the other hand, following a traditional choice of physics literature, in the de Wit -van Proeyen coordinates the vector (resp. spinor) indices are denoted by Greek letters of the sequence µ, ν, etc. (resp. Latin letters i, j and so on). matrix δ ab . In particular, if the coordinates are not just orthogonal, but also adapted (i.e. as in (2.29)), then the expression for d ′ +d ′ is On the other hand, since the De Wit -Van Proeyen coordinates (w I ) are obtained from the adapted orthogonal coordinates (y a , v j , s α 0 , s β 1 ) by means of the non-orthogonal transformation (2.30), the entries of the matrix (g oIJ ) and of its inverse in such new coordinates are Hence the expression for d ′ + d ′ in the De Wit -Van Proeyen coordinates is where µ runs between 4 and 3 + n and i, j run between 4 + n and 3 + n + n S .

Quadratic maps associated with invariant cubic polynomials and cubic rational functions, and their inverses
In what follows, we identify any cubic polynomial d : H → R and any dual cubic polynomial d ′ : H ′ → R with the associated symmetric tenors d ∈ S 3 H ′ and d ′ ∈ S 3 (H ′ ) ′ = S 3 H determined by polarisation.
(2.39) With no loss of generality from now on we assume k > 0 and k ′ > 0. Actually, given d = kd V , we constantly denote by d ′ + d ′ the canonically associated dual rational function, which is defined by As we announced in the introduction, the problem of determining the entropy of extremal BPS black holes in the supergravity theories of this paper can be reduced to the mathematical question of finding the inverses of certain quadratic maps determined by cubic polynomials (see [26]). In the cases in which the scalar manifold is a homogeneous space determined by an irreducible cubic polynomial, such a cubic polynomial is the invariant polynomial d associated with an appropriate special Vinberg cone. In these cases, the quadratic map that needs to be inverted is defined as follows.
Definition 2.9. Given a invariant cubic polynomial d = kd V and its canonically associated dual cubic rational function (2.40) The next lemma and theorem solve the mentioned inversion problem and leads to the solution to our BPS black hole entropy problem under the assumption that the scalar manifold is homogeneous, but not necessarily symmetric (see § 5.4).
and with h, h ′ + h ′ corresponding quadratic maps, as defined above. Let also denote h A(X) = A ′ h(X)) and Proof. (i) For the first identity it suffices to observe that The proof of the second identity is similar.
(iii) We first prove the first identity of (2.42) in case X ∈ V, i.e. under the assumption that X = tA(I) for some t ∈ R + and A ∈ G 0 . Then )=t 3 k(tA(I))=d(X)X.
A similar argument proves the second identity in case X ♭ ∈ V ′ . Since both sides of the two identities of (2.42) are rational functions, it follows that they hold for any X ∈ H for which both sides are well defined.

Projective-special Kähler manifolds and very special cones
The scalar manifolds (i.e. the target spaces of the maps representing the scalar field sector of the Maxwell-Einstein 4-dimensional supergravity theories considered in this paper) are Kähler manifolds of a particular kind, the socalled projective-special Kähler manifolds. For reader's convenience, we briefly review some properties of these manifolds and of their fundamental relations with the special Vinberg cones (for further information, see [11,12,13]; see also the extensive discussion in [30] and references therein).

Conical scalar manifolds, projective scalar manifolds and special Vinberg cones
A conical affine special Kähler (for short, conical scalar) manifold is a Kähler manifold (M, J, g M ) equipped with a flat torsion free connection ∇ and a homothetic vector field ξ such that: (iii) ∇ξ = Dξ = Id where D is the Levi-Civita connection of (M, g M ); (iv) the metric g M is positively defined on D := span(ξ, Jξ) and it is negatively defined on D ⊥ ; (v) the commuting holomorphic vector fields ξ, Jξ are complete and define a free holomorphic C * -action in which {x → e iϑ ·x = e ϑJξ (x)} is a one-parameter isometry group.
In this case the hypersurface with an induced Lorentzian metric g| T S of signature (1, 2n) and a sub-Riemannian structure ( Jξ ⊥ , −g S Jξ ⊥ ) of rank 2n, which are both preserved by the S 1 -action. By S 1 -invariance, the sub-Riemannian structure ( Jξ ⊥ , −g S Jξ ⊥ ) projects onto a Kähler structure (J, g S ) on the The Kähler manifolds (S, J, g S ) determined in this way are called projective special Kähler manifolds or, motivated by physics terminology, projective scalar manifolds.

Description of projective scalar manifolds in terms of conical holomorphic coordinates and prepotentials
Let ((M, J, g M ), ∇, ξ) be a conical scalar manifold (= conical affine special Kähler manifold) of dim R M = 2n + 2 and with associated projective scalar 2n-manifold (S, J, g S ). Around any point of M, there exists a neighbourhood U ⊂ M, on which there exist a distinguished system of holomorphic coordinates X = (X I ), I = 0, 1, · · · , n, called conical special coordinates, and a homogeneous of degree 2 holomorphic function F : U → C, called prepotential, which locally determines all data of M as follows.
(a) the complex structure J is given by the multiplication by i = √ −1 on the holomorphic vector fields ∂ ∂X I ; (b) g M is the metric given by and has the real function ϕ = − log(g M (ξ, ξ)| U ) as Kähler potential; (c) ξ is the vector field ξ = X I ∂ ∂X I + X I ∂ ∂X I . (d) using the notation x I := Re(X I ), y J := Im(X J ) and y J := Re(F J ), the connection ∇ is uniquely determined by the conditions that x I and y J are ∇-flat, i.e. ∇ ∂ ∂x I = ∇ ∂ ∂y J = 0. Note that, since y J = y J , in general the vector fields ∂ ∂X I = ∂ ∂x I − i ∂ ∂y I are not ∇-flat.
The C * -invariant functions z a = X a X 0 , 1 ≤ a ≤ n, are holomorphic coordinates on the corresponding open subset U ⊂ U/C * in the projective-special Kähler manifold (S = M/C * , J, g S ) and any C * -invariant map f : U → R corresponds to a unique function f U : U ⊂ S → R given by (in other words, in physics jargon, the Kähler gauge freedom is fixed such that X 0 = 1). In the open subset U ⊂ S, the metric g S has the Kähler potential X I m IJ (X,X)X J .

The supergravity r-map
Let W = R n+1 be an n + 1-dimensional vector space. A homogeneous cubic polynomial d : W → R is called hyperbolic if the Hessian ∂ 2 d x has signature (1, n) at some x ∈ W . A hypersurface T ⊂ W is called very special real if there exists a homogeneous cubic polynomial d such that T ⊂ T d = {d = 1} and the tensor field g T := −∂ 2 d| T T is a Riemannian metric on T. Note that if this occurs, then d is hyperbolic.
The supergravity r-map is the correspondence which associates with each very special real manifold (T, g T ) the projective scalar manifold (S T , J, g S T ) where: (i) S T is the open submanifold called Siegel domain, and (ii) J is the standard complex structure of C n and (iii) g S T is the Kähler metric, given in terms of the standard coordinates (z a = x a + iy a ) on C n by The triple (S T , J, g S T ) is the projectivisation of the conical scalar manifold with Kähler metric, homothetic vector field and connection ∇, which are uniquely determined by the coordinates (X I ) = (X 0 , X 1 = X 0 · z 1 , . . . , X n = X 0 · z n ) of C n+1 and the prepotential The projective scalar manifold (S T = M T /C * , J, g S T ) is the image of the special cubic (T, g T ) under the r-map. Note that, up to local equivalences, the r-map is essentially surjective, namely Theorem 3.1 ([12, Prop. 1.6 & 1.10]). Any 2n-dimensional projective scalar manifold (S, J, g S ) is locally isometrically biholomorphic to the image under the r-map (S T , J, g S T ) of a special cubic (T, g T ) in W = R n .
If the special cubic cone V = R + ·T of a special cubic (T, g T ) ⊂ R n is convex and homogeneous, the special cubic manifold (T, g T ) is called Vinberg cubic. supergravity theory As we mentioned in the introduction, our discussion is focused on the BPS black holes in theories of ungauged N = 2 supergravity on a 4-dimensional Lorentzian space-time M. For comprehensive overviews of the contents of such supergravity theories we refer to the foundational papers [16,17,14] and to the vast subsequent literature on the topic (see e.g. [4,30] and references therein). For our purposes we need to know just a few basic features, which we now recall.
For what concerns the physical contents of such supergravity theories, we are interested only in the gravitational field (i.e. a Lorentzian metric g on the space-time M) and in the bosonic sector of the vector multiplets, namely: • the scalar fields, which are mathematically represented by the coordinate components of a map z : M → S into an appropriate target complex nmanifold S, called scalar manifold; • the vector fields of n + 1 abelian gauge fields; up to gauge transformations these fields are given by the components A I µ , 0 ≤ I ≤ n, of the potential of a connection on a principal bundle π : P → M of the abelian group The scalar manifold S is always assumed to be a projective scalar manifold (S = S T , J, g S ) of complex dimension n which is image of a special cubic (T, g T ) under the r-map, as defined in §3.1. Such a scalar manifold S = S T is therefore a complex manifold of the form S = R n + iV where V = R + ·T ⊂ {d > 0} is a special cubic cone associated with T ⊂ {d = 1} ⊂ R n . By Theorem 3.2, if d is irreducible and V is convex and homogeneous, then V is a special Vinberg cone according to the definition in §2.2.
The field strengths of the abelian gauge fields (= the electromagnetic fields) of the theory are geometrically given by the n+ 1 components of the curvature 2-form where (y µ ) are coordinates for the space-time M.
In addition to the scalar and the abelian gauge fields, there is another important function which has to be considered: the central charge Z. It is a C-valued field Z : M → C on the space-time which is determined by the physical fields of the theory and satisfies a continuity equation.
The notion of the central charge was first introduced in [41,37] for generic solutions of supersymmetric theories and gives important information on the physical properties of the solutions of the field equations. For an extensive discussion, we refer the interested reader to the original papers. For our purposes, we need to recall just a couple of facts concerning its relation with the entropy of the black holes. We briefly review them in §4.3.

Static and spherically symmetric black holes and their electromagnetic charges
From now on, we assume to be working in a fixed supergravity theory on a 4-dimensional space time of the kind described in §4.1. Moreover, by "black hole" we constantly understand a static, spherically symmetric and asymptotically flat gravity field that solves the equations of the theory and has the singularity of an isolated black hole. In mathematical terms this means that the metric g and the bosonic fields associated with the black hole satisfy the following conditions.
(i) The Lorentzian manifold (M, g) admits a set of globally defined coordinates (t, r, ϑ, ϕ), where: (a) the vector field ∂ ∂t is time-like, (b) each level set M t=to = {t = t o } is space-like and diffeomorphic to R 3 , (c) (r, ϑ, ϕ) are spherical coordinates for each submanifold M t=to ≃ R 3 . (ii) The metric has the form g = −e 2U (r) dt 2 + e −2U (r) dr 2 + r 2 dθ 2 + sin 2 θdϕ 2 (4.1) such that g tends to the flat metric for r → ∞, and so that there is an event horizon at r = 0 (see for instance [5,21]). The geometry in the near-horizon limit r → 0 + is the Bertotti-Robinson geometry AdS 2 × S 2 [8,36]. The area of the S 2 which equals the area of the unique event horizon reads The scalar fields and the electromagnetic fields are t-independent and invariant under the standard SO 3 -action on each space-like submanifold M t=to ≃ R 3 .
By saying that "there is a black hole horizon at r = 0" we mean that the value r = 0 determines the boundary of an SO 3 -invariant and time independent region of the space-like manifold M t=to ≃ R 3 , from which no light ray might escape. where G is a tensor field which is uniquely determined by the electromagnetic field F by means of an appropriate generalised Hodge star operator. This operator depends in a non-trivial way on the scalar fields, but for our purposes we do not need to know its explicit expression. By (iii) and Gauss' Theorem, these integrals are independent on the choice of t o and r o and can be considered as the sources (= "charges") of the n + 1 electromagnetic fields F I := F I µν dy µ ∧ dy ν at large distances from the black hole. Indeed the 2-forms F I behave precisely as if they were classical electromagnetic fields, generated by corresponding magnetic monopoles and/or electric charges, all of them located at r = 0. The values p I , q I are called the magnetic charges and the electric charges of the black hole, respectively. If there are two charges p I , q I , which are both non trivial, the corresponding electromagnetic field F I behaves as if it were generated by a dyon and the black hole is called dyonic.

BPS black holes, their entropy and the "inverse relation" map
The first of the two properties of the central charge, which we need to recall, is that for each black hole satisfying (i), (ii), (iii), the function Z : M → C depends just on the coordinate r and, for r ≥ 0, the absolute value |Z(r)| is bounded from above by the mass of the black hole ( 2 ) m ≥ |Z(r)| . (4.6) The black holes for which the inequality (4.6) is "saturated" (i.e. such that On the other hand, by the Bekenstein-Hawking entropy-area formula ( [28,7]), the thermodynamical entropy S of a black hole is completely determined by the area A H of the (unique and time independent) event horizon located at r = 0. More precisely, in natural units This relation, together with Eq. (4.3) and with the fact that, for an extremal BPS black hole (see for instance [5,21]) implies that for such a black hole the following relation between the entropy and the central charge holds : The second important property of the central charge that we need to recall is a crucial phenomenon of the BPS black holes. Consider one such black hole, with magnetic and electric charges p I , q J and mass m = 0. We recall that the scalar fields associated with such a (static and spherically symmetric) black hole are given by a map, depending only on the radius r, z = z(r) : M → S T = R n + iV = R n + iR + · T into the projective scalar manifold S T = M/C * , determined from a special cubic cone of T ⊂ {d = 1} ⊂ R n through the r-map, which is the projectivisation of a corresponding conical scalar manifold M ⊂ C n+1 . The map z = z(r) is canonically associated with the map into the conical scalar manifold M z 1 (r), . . . , z n (r)) .
We also recall that the potential of the Kähler metric g S T of the scalar manifold S T is the function and that the prepotential F , which characterises the conical scalar manifold M ⊂ C n+1 , is the holomorphic function Let now r o ≥ 0 be the smallest radius such that m = |Z o | and denote by the values at r o of the central charge Z(r), of the scalar fields map z(r) and of the canonical associated lifted map X(r) = (1, z(r)), respectively. The following crucial properties hold: . (4.10) The equalities (4.10) completely determine the magnetic and electric charges in terms of the scalar fields and the central charge of the BPS black hole and they are consequences of the celebrated theory of the attractor mechanism [25,22]. We call them BPS relations.
We derive our main results from (4.9) and (4.10). In fact, in the next section, we will show that the problem of inverting the relations (4.10) boils down to determining the inverse map to the quadratic map y → h(y) = (h a (y) := d abc y b y c ) .
On the other hand, in Theorem 2.11 we determined the explicit expression of such inverse map in case d is an invariant cubic polynomial. Combining these results, we are able to obtain the explicit expression for the modulus |Z o | of the central charge of a BPS black hole -and thus, due to (4.9), for its entropy S -in terms of its electric and magnetic charges only, provided that the scalar manifold S = R n + iV is homogeneous. The formula we determine is valid for any choice of the homogeneous scalar manifold and extends the previously known expression, determined only for the cases in which S T is a symmetric manifold.

5.
Recovering the central charge and the scalar fields from the electric and magnetic charges of a BPS black hole

The BPS relations as a map
In this section, we discuss some aspects of the map, which determines the BPS relations between central charge and scalar fields on one side and the magnetic and electric charges on the other side. In order to make fully clear that here and in the next two subsections we address purely mathematical properties of this map -whose arguments and values are tuples of just (real or complex) numbers, not quantities with a prescribed physical meaning -we adopt the following notational conventions.
• An index running from 0 to n (resp. from 1 to n) is always denoted by a capital letter as I, J, K, etc. (resp. small letter as a, b, c, etc.).
• When no ambiguity may occur, n-tuples as (z a ), (p a ), etc., are denoted with no index, i.e. by z, p, etc.
• The standard pairing between a 1-form w = (w a ) ∈ R n′ := Hom(R n , R) and a vector y = (y a ) ∈ R n is denoted by w, y := w a y a .
• The standard complex coordinates of C n and the standard real coordinates of R 2n+2 are indicated by z = (z a ) and by (p 0 , p a , q 0 , q a ) = (p 0 , p, q 0 , q), respectively In what follows, we decompose the standard coordinate of C * as Z = te iϑ , with t = |Z| > 0 and ϑ ∈ R mod 2π, and we use the short-hand notation . In this way, using the above notational convention, the (5.2) take the form We now consider the purely mathematical questions of finding the domains U ⊂ C * ×S on which the BPS map f is locally invertible and, on such domains, determining the explicit expressions for the (local) inverses of this map. For this purpose, it is convenient to split the domain C * × S of the BPS map into the union of the following four disjoint subsets: -the (2n + 1)-dimensional hypersurfaces -the (2n + 2)-dimensional open subsets We discuss the behaviour of the restrictions f| C ± and f| A ± separately.

5.2.
The maps f| C ± take values into {p 0 = 0} and are globally invertible Due to (5.3), the map f sends both hypersurfaces C ± into the hypersurface {p 0 = 0} of R 2n+2 . Moreover we have the following is a diffeomorphism onto its image. More precisely, if (p 0 = 0, p, q 0 , p) is a point of f(C ± ), then there is a unique (t, z) ∈ C ± such that (p 0 = 0, p, q 0 , p) = f(t, z). This point is given by ( 4 ) where D a (p, q) := D ab (p)q b with D ab (p) inverse matrix of (d abc p c ). 4 Note that (5.8) gives the well known expression S = π 1 3 d(p) ( q, D + 12q 0 ) for the entropy of a BPS black hole with p 0 = 0 due to Shmakova [38]. We also remark that, from (5.7) and the fact that d(p) < 0, we also have that z = 1 6 D(p, q) + i p 2 S π 1 d(p) . This amends an error in [38,Formula (27)] Proof.
We give the proof only for f| C + , the other being similar. Given (p 0 = 0, p, q 0 , p) ∈ f(C + ), any pre-image (t, z = x + iy) ∈ C + satisfies Replacing (5.9) into (5.10) and (5.11), we get It follows that . Then α o belongs to the cone h(V d ) ⊂ h({d > 0}) and there are two neighbourhoods W 1 ⊂ R n and W * 2 ⊂ R n * of v o and α o , respectively, such that the restriction h| W 1 : W 1 → W * 2 is a diffeomorphism between such two neighbourhoods. Then U ′ is the open set U ′ := {(p I , q J ) : p 0 = 0 , 3h(p)−p 0 q ∈ W * 2 , sign(p 0 ) = sign(p 0 o ) } . (5.17) The corresponding neighbourhood U ⊂ C n+1 of (t o e iϑo , z o ) is given by the images of U ′ under the map (5.14) -(5.16).
In these new coordinates, the components (5.4) and (5.6) of f become (here, x := Re( z) and y := Im( z)) p = −ct y , q = 3ct Re(ie iϑ h( z)) . On the other hand, from (5.3) we may replace p 0 = −ct sin ϑ at all points. Since y = Im(e iϑ z) = sin ϑ x + cos ϑ y, we have that (5.19) implies 3c 2 t 2 p 0 h(y) = 3ct sin ϑ h sin ϑ x + cos ϑ sin ϑ y = = 3ct sin ϑh x + cos ϑ sin ϑ y = 3 p 0 h(p) − q . We now observe that, at each point y ∈ V d , the Jacobian of the map h is Jh| y = 2 (d abc y c ). Being d associated with a special cubic, this matrix is non-degenerate. Hence, by the Inverse Function Theorem, there exists a the sign being equal to +1 in case ϑ ∈ (0, π) (that is, in case p 0 > 0) and which matches [38,Formula (24)] ( 5 ).
The local invertibility property established in Theorem 5.2 has in practice the following meaning. Assume that p I , q J are the values of the magnetic and electric charges of a BPS black hole and that Z o = t o e iϑo := Z(r o ) and z o := z(r o ) are the corresponding values of the central charge and of the scalar fields map at the horizon r o = 0. Since the result does not guarantees that f| A ± is globally invertible, it does not exclude the possibility that there is some choice for the cubic polynomial d, which allows the existence of several different BPS black holes, all of them with the same electric and magnetic charges, but also each of them with distinct horizon values for the central charge or scalar fields. In other words, for an appropriate choice of d, it might be that there are distinct BPS black holes with charges and horizon values for the central charge and the scalar fields with . . constitute a discrete set of points in A + ∪ A − ⊂ C * × S T . Indeed, for each such pair there must be a neighbourhood on which the BPS map is one-to-one.
We finally stress the fact that the proof of Theorem 5.2 shows that there exists a global inverse each map map f : A ± → f(A ± ) ⊂ R 2n+2 if and only if the restriction to V of h admits a global inverse, namely Corollary 5.4. If the restriction h| V : V → h(V) ⊂ R n′ admits an inverse (h| V ) −1 : h(V) → V, then each of the two maps f| A ± : A ± → f(A ± ) ⊂ R 2n+2 is a diffeomorphism onto its image, with inverse given by (5.14) -(5.16).

The entropy of BPS black holes in case of homogeneous scalar manifolds
By Corollary 5.4, Theorem 3.2, Theorem 2.11, Lemma 2.10, if the scalar manifold S T = R n +iV, V = R + ·T ⊂ { d > 0}, is homogeneous and determined by an irreducible invariant cubic polynomial d = kd V : R n → R, then V is a special Vinberg cone and the corresponding quadratic map h| V : V ⊂ R n → V ′ ⊂ R n * is globally invertible with inverse given by with d ′ + d ′ = dual invariant cubic rational function and (h ′ + h ′ ) (y)(·) = 1 3 d ds (d ′ + d ′ )(y + s(·)) s=0 (5.33) As a consequence, the entropy S = π|Z| 2 of an extremal BPS black hole with magnetic charge p 0 = 0 is uniquely determined by the black hole charges 5 There is just a sign change that is due to our different assumptions. In fact, according to them, we have d(y) > 0 -and not d(y) < 0 -at the points z = x + iy of S.