On Extremal Limits and Duality Orbits of Stationary Black Holes

With reference to the effective three-dimensional description of stationary, single center solutions to (ungauged) symmetric supergravities, we complete a previous analysis on the definition of a general geometrical mechanism for connecting global symmetry orbits (duality orbits) of non-extremal solutions to those of extremal black holes. We focus our attention on a generic representative of these orbits, providing its explicit description in terms of D=4 fields. As a byproduct, using a new characterization of the angular momentum in terms of quantities intrinsic to the geometry of the D=3 effective model, we are able to prove on general grounds its invariance, as a function of the boundary data, under the D=4 global symmetry. In the extremal under-rotating limit it becomes moduli-independent. We also discuss the issue of the fifth parameter characterizing the four-dimensional seed solution, showing that it can be generated by a transformation in the global symmetry group which is manifest in the D=3 effective description.


Introduction
The seminal work by [1] has defined an effective D = 3 description of (asymptotically flat) stationary black holes in D = 4 supergravity theories [2], which unveiled a larger global symmetry (to be dubbed duality in the following) underlying these solutions. In fact this approach has provided a valuable tool for their classification [3,4,5,6,7,8,9,10,11] and consists in describing this kind of solutions as solutions to an effective D = 3 Euclidean sigma-model which is formally obtained by reducing the D = 4 theory along the time direction and dualizing the vector fields into scalars. The action of the global symmetry group (duality group) G of this Euclidean model has been extensively used in the literature as a solution-generating technique to construct non-extremal, rotating, electrically charged black hole solutions coupled to scalar fields [3] and, more recently, found application in the context of subtracted geometry [12,13,14,15].
Stationary, asymptotically flat, black holes can therefore be conveniently classified in orbits with respect to the action of G. We shall restrict ourselves here to the single-center case. In a recent paper [16] we defined a general geometrical mechanism for connecting the orbit corresponding to non-extremal solutions to those defining the extremal (i.e. zero-temperature) ones, and applied it, as a worked-out example, to the T 3 -model. Here we wish to complete this analysis by applying the same mechanism to explicit solutions to the STU model, thus proving it for the broad class of symmetric extended supergravities which share the STU model as a common universal truncation. These include all the extended (i.e. N ≥ 2) four-dimensional models whose scalar manifold is symmetric of the form G 4 /H 4 , and the isometry group G 4 ⊂ G, which defines the global symmetry (or D = 4-duality) of the four-dimensional theory, is 1 In the N = 2 case, the above condition in referred to the special Kähler manifold spanned by the scalar fields in the vector multiplets, since those in the hypermultiplets are not relevant to the black hole solutions under consideration. Moreover by specializing to the non-degenerate case (see the second of references [17]), we are excluding those models with G 4 = U(p, q) and vector field-strengths together with their magnetic duals transforming in the p + q + p + q, like the minimal coupling N = 2 models with G 4 = U (1, q) or the N = 3 supergravity with G 4 = U(3, q). 2 The explicit solutions used in the present paper were derived independently.
function of the values of the D = 4 scalar fields at radial infinity and of the electric-magnetic charges.
In Sect.s 5 and 6 we give the explicit form of the non-extremal rotating solutions to the STU model (with fully integrated vector fields) corresponding to the sets of charges p 0 , q i and q 0 , p i , i = 1, 2, 3, and study their limits to extremal static and under-rotating black holes. We conclude with a discussion of the 5 th invariant-parameter of a generic D = 4 single-center solution, with respect to the D = 4 global symmetry group, showing that it is not G-invariant and that it can be thus generated by means of a G-transformation not belonging to G 4 (an explicit calculation is given in Appendix B). Consequently the most general extremal, single-center solution to the D = 3 effective model, modulo G-transformations, is a 4-parameter one.

Stationary Single-center Solutions
We shall be working with a D = 4 extended (i.e. N > 1), ungauged supergravity, whose bosonic sector consists in n s scalar fields φ r (x), n v vector fields A Λ µ (x), Λ = 1, . . . , n v , and the graviton g µν (x), which are described by the following Lagrangian 3 : where e := |det(g µν )|. In symmetric supergravities, which we shall restrict to, the scalar fields φ s span a homogeneous, symmetric, Riemannian scalar manifold: where the isometry group G 4 is the symmetry group of the whole theory provided its non-linear action on the scalar fields is combined with a symplectic action, defining a representation R of G, on the vector field strengths F Λ = dA Λ and their magnetic duals G Λ . We shall be dealing with stationary, axisymmetric, asymptotically flat, single center solutions whose space-time metric, in a suitable system of coordinates, has the general form: where i, j = 1, 2, 3 label the spatial coordinates x i = (r, θ, ϕ) and U, ω ϕ , g ij are all functions of r, θ. The two Killing vectors are ξ = ∂ ∂t and ψ = ∂ ∂ϕ . As mentioned in the introduction, these solutions can be given an effective description in an Euclidean D = 3 model describing gravity coupled to n = 2 + n s + 2n v scalar fields φ I (r, θ) comprising, besides the D = 4 scalars φ s , the warp function U and 2n v + 1 scalars Z M = {Z Λ , Z Λ } and a originating from the time-like dimensional reduction of the D = 4 vectors and the dualization of the Kaluza-Klein vector ω ϕ into a scalar. The precise relation between the scalars a, Z M and the four-dimensional fields is [16]: where * 3 is the Hodge operation in the D = 3 Euclidean space, M (4) the symmetric, symplectic matrix characterizing the symplectic structure over M (4) scal (see Appendix A for an explicit construction). The effective D = 3 Lagrangian describes a sigma-model coupled to gravity and reads: where e (3) ≡ det(g ij ) and C is the symplectic-invariant, antisymmetric matrix. The scalar fields span a homogeneous, symmetric, pseudo-Riemannian manifold of the form scal as a submanifold. The isometry group G is a semisimple, non-compact Lie group which defines the global symmetry of the model, while H * is a non-compact real form of the maximal compact subgroup of G.
Stationary axisymmetric solutions are described by n functions φ I (r, θ), solutions to the sigma model equations, and characterized by a unique "initial point" φ 0 ≡ (φ I 0 ) at radial infinity and an "initial velocity" Q, at radial infinity, in the tangent space T φ 0 [M scal ], which is the Noether charge matrix of the solution. Since the action of G/H on φ 0 is transitive, we can always fix φ 0 to coincide with the origin O (defined by the vanishing values of all the scalars) and then classify the orbits of the solutions under the action of G (i.e. in maximal sets of solutions connected through the action of G) in terms of the orbits of the velocity vector Q ∈ T O (M scal ) under the action of H * . The Noether charge matrix Q is computed as: 10) J = J i dx i being the Noether current. The explicit form of J is given by the standard theory of sigma models on coset manifolds: where M(φ I ) = L(φ I )ηL(φ I ) † is an H * -invariant symmetric matrix built out of the representative L(φ I ) of G/H at the point φ I and η is a suitable H * -invariant matrix in the chosen representation of G (see Appendix A of [16] for the definition of the adopted conventions). 4 The 4 The coset geometry is defined by the involutive automorphism σ on the algebra g of G which leaves the algebra H * generating H * invariant. All the formulas related to the group G and its generators are referred to a matrix representation of G (we shall in particular use the fundamental one). The involution σ in the chosen representation has the general action: σ(M ) = −ηM † η, η being an H * -invariant metric (η = η † , η 2 = 1), and induces the (pseudo)-Cartan decomposition of g of the form: where σ(K * ) = −K * , and the following relations hold (2.13) scalar fields φ I define a local solvable parametrization of the coset, and the coset representative is chosen to be where T A = {H 0 , T • , T s , T M } are the solvable generators defined in Appendix A of [16]. 5 Since the generators T M transform under the adjoint action of G 4 ⊂ G in the symplectic duality representation R of the electric-magnetic charges, we shall use for them the following notation: The Noether matrix Q encodes all the conserved physical quantities associated with the solution, except the angular momentum M ϕ . In other words it contains no information about the rotation of the solution. In [18] we defined a new matrix Q ψ which describes the global rotation of the solution: ) being a representation-dependent constant. Both Q and Q ψ are matrices in the Lie algebra g of G. More specifically they belong to the space K * complement in g to the algebra H * of H * and isomorphic to T O (M scal ).
Being G the global symmetry group of the effective model, a generic element g of it maps a solution φ I (r, θ) into an other solution φ ′ I (r, θ) according to the matrix equation: (2.18) From their definitions (2.10), (2.16), and from (2.18), it follows that Q and Q ψ transform under the adjoint action of G as: Eq.s (2.17), and the last one in particular, allow to compute the angular momentum of the transformed solution without having to explicitly derive the latter from (2.18) and to compute the corresponding Komar integral on it. This is one of the main advantages of working with Q ψ . The presence of a non-vanishing Q ψ is a characteristic of the G-orbits of rotating solutions and therefore one cannot generate rotation on a static D = 4 solution using G ! 5 The structure of this solvable algebra is the following:

The Kerr Family
As proven in [1], the most general (non-extremal) stationary, axisymmetric single black hole solution to the model can be obtained from the Kerr solution through a G-transformation (more precisely through a Harrison transformation). The matrices Q and Q ψ for the Kerr solution, characterized by a mass m and an angular-momentum parameter α are diagonalizable and thus their G-orbits are uniquely characterized by their eigenvalues. In the pure Kerr solution, Q, Q ψ belong, modulo multiplication by α, to the same G-orbit. In fact we have: From (3.1) it follows that: .
Also the following matrix equations are satisfied: It is worth emphasizing that the equations (3.2), (3.3), , (3.4) together with the trace expression for m and α, are G-invariant and thus hold for any representative of the Kerr G-orbit. We can then define an extremality parameter c 2 in terms of the following G-invariant quantity [18]: .
In terms of c we can write the Hawking temperature of the black hole in the form: where S is the Bekenstein-Hawking entropy of the solution, expressed in the chosen units, in terms of the horizon area A, by the renown formula while ω H is defined as: The above expression allows to write the regularity bound for the Kerr solution in a G-invariant form which thus holds for any representative of the Kerr-orbit: .
(3.9) 7 If G is a real form of E C 8 , the fundamental and the adjoint representation coincide and the matrix equation becomes quintic in Q, [7]. 8 The constantc 2 in the case of the Kerr-Newmann-NUT black hole with electric and magnetic charges q, p and NUT charge n N UT , reads: Angular momentum and duality. Let us comment on the properties of the angular momentum M ϕ with respect to the four-dimensional duality symmetry G 4 . In our analysis, for the sake of simplicity, we have fixed the transitive action of G/H * on the solution by choosing the scalar fields at infinity to correspond to the origin O of the manifold. Let us relax this assumption in the present paragraph. All the formulas given in the previous section, including (2.17), clearly hold for generic "initial values" of the scalar fields.
In general, on a rotating black hole solution, the angular momentum would depend on the boundary values (φ s 0 ) of (φ s ) and on the electric-magnetic charges Γ M and be expressed in terms of Q ψ by the last of eq.s (2.17). Suppose now we transform the solution by means of an element g ∈ G 4 into another one with boundary values φ ′s 0 and charges Γ ′M Let us prove, by using the definition in (2.17), that M ϕ is not affected by the action of g. The matrix Q ′ ψ associated with the new solution is related to Q ψ by (2.19), so that the corresponding where we have used the property that G 4 commutes with the Ehlers group SL(2, R) E inside G, so that its elements commute with the sl(2, R) E generators {H 0 , T • , T † • }. We conclude that M ϕ , is a G 4 -invariant function of the scalar fields at radial infinity and the electric-magnetic charges. This is indeed what one would expect for the angular momentum of a solution: being a quantity related to its spatial rotation it should not be affected by a D = 4 duality transformation.
Clearly the above derivation would not hold for a generic global symmetry transformation in G. As we shall see below, in the under-rotating limit M ϕ is independent of φ s 0 and thus is expressed in terms of the G 4 -invariant of the electric-magnetic charges alone, namely the quartic invariant function I 4 (p, q). A similar thing happens for the horizon area (i.e. the entropy) by virtue of the attractor mechanism (see below). We conclude from this observation that there seems to be an "attractor mechanism" at work also for the angular momentum.
Finally let us notice that the simple proof (3.11) also applies to the ADM-mass and the NUT-charge, both given in (2.17). This is consistent with the duality invariance of M ADM proven in [27] (see eq. (29) therein) in a different and more sophisticated way.

Extremal Limits
The regularity bound c 2 ≥ 0 is saturated for the extremal solutions, which are thus characterized by a vanishing Hawking temperature (3.6). This bound can be saturated in essentially two ways: • Both sides of (3.9), though equal, stay different from zero. The extremality condition thus becomes a constraint on the two non-vanishing G-invariants. The resulting solution is called over-rotating extremal and retains, in this limit, the presence of an ergosphere. The two matrices Q and Q ψ are still diagonalizable; • Both sides of (3.9) vanish separately. The resulting solution can either be extremal underrotating [19,20,24,25,26] or extremal-static and has no ergosphere. In this limit [16] both Q and Q ψ become nilpotent, belonging to different G-orbits (or better H * orbits on We shall focus on the second limit, which has been considered in the literature in specific contexts: Heterotic theory [3,21]; Kaluza-Klein supergravity [19,20]. In [16] we defined a general geometric prescription for connecting the non-extremal Kerr-orbit to the extremal static or under-rotating ones, in a way which is frame-independent (i.e. does not depend on the particular string theory and compactification yielding the four-dimensional supergravity). This procedure makes use of singular Harrison transformations by means of which an Inönü-Wigner contraction on the matrices Q and Q ψ is effected, resulting in the nilpotent matrices Q (0) and Q In [16] we considered the maximal abelian subalgebra (MASA) of the space Span(J M ). This is a subspace whose generators J (N ) = {J ℓ } are defined by the normal form of the electric and magnetic charges, i.e. the minimal subset of charges into which the charges of the most general solution can be rotated by means of H c . Its dimension p is therefore just the rank of the coset H/H c . In the maximal supergravity, for example, p = rank SO * (16) U(8) = 4, the same being true for the half-maximal theory, p = rank SO(6,2)×SO(2,6+n) SO(2) 2 ×SO(6)×SO(6+n) = 4, and for the N = 2 symmetric models with rank-3 scalar, special Kähler manifold in D = 4 (for this class of theories, p = rank +1). The simplest representative of the latter class of models is the ST U one, which is a consistent truncation of all the others, besides being a truncation of the maximal and half-maximal theories. Therefore its space J (N ) is contained in the spaces of Harrison generators of all the above mentioned symmetric models. As a consequence of this, for the sake of simplicity, we can restrict ourselves to the simplest ST U model since the G-orbits of non-extremal and extremal regular solutions to the broad class of symmetric models mentioned above have a representative in the common ST U truncation. As for the restricted number of N = 2 symmetric models for which the rank of M (4) scal is less than 3 (p < 4), the following discussion has a straightforward generalization (the T 3 -model case with p = 2 was dealt with in detail in [16]). Depending on the symplectic frame, i.e. on the higher-dimensional origin of the four-dimensional theory, this normal form can consist of different kinds of charges. In all cases this normal form can be geometrically characterized as follows. If we express the Harrison generators in the form: where γ M are the 2n v roots of g such that γ M (H 0 ) = 1/2, the p generators J ℓ are defined by a maximal set {γ ℓ } of mutually orthogonal roots among the γ M : Symplectic frames and normal forms. Since the normal form of the electric and magnetic charges with respect to the group H c , for all the symmetric models mentioned above, is contained in the STU truncation, let us illustrate within the latter, the relevant symplectic frames. The STU model is a N = 2 supergravity coupled to three vector multiplets whose three complex scalars span a special Kähler manifold (2.2), where G 4 = SL(2, R) 3 and H 4 = SO(2) 3 . Upon time-like reduction to D = 3, the scalar manifold is enlarged to A for notations and technical  details about the STU model). If the STU model originates from Kaluza-Klein reduction from D = 5, the resulting symplectic frame corresponds to the following ordering of the roots γ M , M = 1, . . . , 8: If we embed the STU model in toroidally compactified Heterotic theory [3], one of the SL(2, R)s in G 4 has a non-perturbative (i.e. not block-diagonal) duality action in the R = (2, 2, 2), while the remaining two factors have a block diagonal symplectic representation. The corresponding symplectic frame is characterized by the following order of the roots γ M : 9 The two normal forms of the charge vector, being identified by the same sets of roots {γ ℓ } and {γ ℓ ′ }, now correspond to two electric and two magnetic charges: {p ′2 , p ′3 , q ′ 0 , q ′ 1 } and {p ′0 , p ′1 , q ′ 2 , q ′ 3 }. Finally one can consider the frame in which the generators of G 4 can be chosen to be represented symplectic matrices which are either block diagonal or completely block-off-diagonal (i.e. having entries only in the off-diagonal blocks). This is the frame originating from direct truncation of the N = 8 theory in which the SL(8, R) subgroup of E 7(7) has a block-diagonal embedding in Sp(56, R). It corresponds to the following order of the roots γ M : The two normal forms of the charge vector now correspond to either all electric or all magnetic charges: {p ′′ Λ } and {q ′′ Λ }. In all these cases, the MASAs of Span(J M ) are always defined by the same sets of generators 3,4,5 . We shall use in the following the first symplectic frame.
The procedure. Let us summarize the procedure defined in [18,16] where ℓ = 1, 6, 7, 8 and ℓ ′ = 2, 3, 4, 5. The matrices Q, Q ψ transform according to eq. (2.19): Next we perform, in the two cases, the rescaling: where σ ℓ , (σ ℓ ′ ) = ±1. We then send m to zero. This limit corresponds to an Inönü-Wigner contraction of Q ′ and Q ′ ψ which become nilpotent matrices Q (0) , Q ψ with a different degree of nilpotency, i.e. belonging to different H * -orbits: Q (0) has degree three while Q (0) ψ either vanishes or has degree two. This explains why, in the m → 0 limit, the ratio on the right hand side of eq. (3.9) goes to zero: the numerator Tr(Q 2 ψ ) vanishes faster than the denominator Tr(Q 2 ). The charge vector Γ M of the resulting solution, in the two cases, has 4 non-vanishing charges corresponding to the chosen normal form, i.e. {q 0 p i } or {p 0 q i }. Depending on the choice of the gradings σ ℓ (or σ ℓ ′ ) the charge vector Γ M can belong to any of the G 4 -orbits of regular solutions, characterized in terms of the G 4 -quartic invariant I 4 (p, q) of the representation R as follows [29] (see Appendix A for the explicit form of I 4 (p, q) in the STU model): BPS : I 4 (p, q) > 0 Z 3 -symmetry on the p i and the q i , non-BPS 1 : I 4 (p, q) > 0 no Z 3 -symmetry , non-BPS 2 : I 4 (p, q) < 0 .
For those choices of the gradings yielding I 4 > 0 we find both the BPS and a non-BPS solution and the resulting angular momentum is zero (extremal-static black hole, Q (0) ψ = 0). Only in the cases for which I 4 < 0 we find a rotating solution, which is the known under-rotating solution of [19,20,24,25,26]. Therefore we find, as a general result, that the extremal solutions obtained in this way have an angular momentum given by where I 4 = ε |I 4 | (the above equation was verified on the 5-parameter solution, see Appendix B). This formula makes the invariance of M ϕ under G 4 -transformations, proven for a generic solution at the end of the previous section, manifest, since both I 4 (p, q) and Ω = M (Kerr) ϕ /m 2 are G 4 -invariants, being the latter related to the original Kerr solution. Actually on our solutions we cannot see the dependence of the various quantities on the scalar fields φ I 0 , and in particular on the four-dimensional ones, at radial infinity, since these were fixed to zero. Having proven, however, in the previous section that M ϕ is a G 4 -invariant function of φ s 0 and Γ M , and having proven on our solutions that it is already an invariant function of the electric-magnetic charges alone, we conclude that, for the under-rotating solutions, M ϕ only depends on p Λ , q Λ .
Similarly one finds for the entropy, related to the horizon area and expressed in (3.7), the following form in the limit: 10 (4.10) The last expression, obtained by using (4.9), makes it manifest that S (extr) , as well as the whole near horizon geometry, is G 4 -invariant as M (extr) ϕ is. In the rotating extremal case (ε = −1) we further need to impose Ω < 1 in order for the solution to be well behaved.
We observe, however, that before the extremal limit m → 0 is effected, the expression of S is not G 4 -invariant. This can be explained by the fact that we generally made the G 4 "gauge" choice corresponding to fixing the 4D scalar fields at infinity at the origin of the moduli space, thus breaking the manifest G 4 invariance to H 4 . In the extremal under-rotating and static cases the attractor mechanism is at work [28], as a consequence of which the near horizon geometry becomes independent of the values of the scalar fields at radial infinity (which we have fixed to the origin) and only depends on the quantized charges p Λ , q Λ . In the non-extremal case, c 2 > 0, this is no longer the case and the near horizon geometry, as well as the entropy, depends on the scalar fields at infinity φ s 0 . We can then argue that S = S(p, q, φ s 0 ) is still invariant under G 4 , provided we transform both Γ M and φ s 0 simultaneously, just as it was proven at the end of last section to happen for the angular momentum. In other words, within our choice of scalar boundary conditions, S is expressed in terms of H 4 -invariants and, in the extremal limit, such expression should reduce to the only scalar-independent H 4 -invariant, namely to (4.10).
In the following sections, we work out the explicit solutions to the STU model, corresponding to the two normal forms, the complete description of which (including the integrated D = 4 vector fields), to our knowledge, were not present in the literature before [22] and which were derived by us independently. We then apply to them the general extremal limits discussed above, to derive extremal-static and under-rotating solutions. For a detailed algebraic description of the limit Q ′ , Q ′ ψ → Q (0) , Q ψ we refer the reader to Sect. 3 of [16].

Extremal Limits
Let us start redefining: where σ ℓ ′ = ±1, and introduce also the symbol ζ σ = ℓ ′ σ ℓ ′ . Next, send m to zero keeping the other parameters fixed. There are 16 different ways to rescale the four β ℓ ′ -parameters and the general results for the extremal limits of the ADM -mass, electric-magnetic charges Γ M , angular momentum M ϕ , entropy S and the quartic invariant I 4 are where we have used the short notation Γ M (extr) = (p 0 e , 0, 0, 0, 0, q 1 e , q 2 e , q 3 e ) for the extremal charges. The solutions can be classified as where, in the previous expressions, we have used the harmonic functions The extremal limits for the 4-D z scalar fields are while the limits for the 3-D scalar fields Z M (extr) read Introduce now the combination of the β-parameters: P c = c 1 c 6 c 7 c 8 , P s = s 1 s 6 s 7 s 8 .

Extremal Limits
Let us redefine: where σ ℓ = ±1 and ζ σ = ℓ σ ℓ . Then, send m to zero keeping the other parameters fixed. We find again 16 different ways to rescale the four β ℓ -parameters and the results for the extremal limits of the ADM -mass, electric-magnetic charges Γ M , angular momentum M ϕ , entropy S and the quartic invariant where we have used now the short notation Γ M (extr) = (0, p 1 e , p 2 e , p 3 e , q 0 e , 0, 0, 0) for the extremal charges. Also in this case the solutions can be classified as 5 independent parameters. These can be written in terms of five independent H 4 -invariants computed at radial infinity (depending on φ s 0 and p Λ , q Λ ). 12 This number 5 is nothing but the rank p of H * /H c , introduced in Section 4, plus one (in the T 3 -model p = 2 and the seed solution with respect to G 4 is a three-parameter one). In the D = 3 description, a larger symmetry group G is manifest. In particular on the charges we can act by means of the group H c = U(1) E × H 4 which contains, besides H 4 , an additional U(1) E -symmetry. Using it we can reduce the number of independent invariants characterizing the solution from p + 1 = 5 to p = 4, so that the seed solution with respect to G is a four-parameter solution characterized by electric and magnetic charges in one of the two normal forms of Γ M with respect to H c : {q 0 , p i } and {p 0 , q i }. In support of this argument we observe that the nilpotent H * -orbits of Q, corresponding to the extremal, regular, single-center solutions are unique and contain the 4-parameter solutions constructed here (see for instance [32]).
We have explicitly checked in the STU model that, acting on the 4-parameter BPS and non-BPS extremal solutions by means of a combination of U(1) E and Harrison transformations, the 5 th parameter can be generated. In the STU model the 5 H 4 -invariants can be constructed out of the central and matter charges (Z(φ s , p, q), Z I (φ s , p, q), I = 1, 2, 3), in terms of their moduli and overall phase and read (in the chosen symplectic frame): where the central and matter charges are defined as (see Appendix A): We have checked on the extremal solutions that the five invariants (7.1) are independent functions of q 0 , p i and α and therefore conclude that 5 th parameter can be generated by means of G. We refer the reader to Appendix B for an explicit calculation. As a general comment, let us observe that in order to find the 5-parameter solution we had to perform a set of non-commuting Ehlers and Harrison transformations on the 4-parameter solution, whose net effect is to modify topological properties of the D = 4 black hole. More precisely, we introduced the 5th parameter by a NUT-charge-generating U (1) E transformation in D = 3, and then converted it, by an appropriate Harrison transformation into a gauge charge non-commuting with the other gauge charges. In the D = 3 description, where all the bosonic degrees of freedom of the stationary black hole solution (corresponding to the metric, the gauge vectors and the D = 4 scalars) are collectively described by the scalar sigma-model G/H * , the above prescription is among the allowed symmetry transformations on the set of conserved charges. However, in the D = 4 description this transformation is highly non trivial: It generates the 5th parameter as a NUT charge, that is as a non-trivial topology of space-time, and then (in order to have an asymptotically flat black hole solution) trades it into a gauge charge thus adding to the non triviality of the gauge bundle. In our setting we have chosen 12 In other words, in these models, one can define a maximal set of five functionally independent functions I 1 , . . . , I 5 of φ s 0 and Γ M which are invariant under the action of G 4 on both the scalar fields at infinity and the electric-magnetic charges.
to fix the scalars at radial infinity to their origin, otherwise, for the extremal I 4 < 0 black hole, the same solution could have been converted, by the action of G 4 /H 4 , into one where the gauge bundle has commuting charges but the axions acquire a non trivial value at radial infinity [34,35,31,24]. Instead of referring to the D = 3 description, the other way adopted in [33,34,24], to find the D = 4 seed solution has been via Kaluza-Klein reduction from D = 5. Also in this case, the seed solution of D = 4 stationary, asymptotically flat black holes was found to correspond to a 5D NUT-charge configuration with angular momentum.
We conclude that in all its descriptions the seed, 5-parameter, solution should have an additional non-trivial topological feature with respect to the 4-parameter one. This distinction, at least for the I 4 < 0 extremal black hole in the static case, reflects itself in the different behavior of the harmonic functions H = (H M ) characterizing the solution: In the 4-parameter solution they obey the relation H T · C · ∂ r H = 0, while the 5-parameter seed solution satisfies H T · C · ∂ r H = 0. This shows that the transformation connecting the two cannot be a D = 4 global symmetry which would leave the symplectic product unaltered.
The study of the extremal limits, started in [16] and concluded here, was also a testing ground for the newly defined g-valued matrix Q ψ , which encodes the rotation property of the solution and which allows to directly compute the action of the symmetry group G on the angular momentum M ϕ . We have seen that, in spite of having the manifestly G 4 -invariant expression in (4.9), this quantity is far from being G-invariant. In the non-extremal case even the manifest G 4 -invariance of both M ϕ and of the entropy S, as functions of the electric-magnetic charges alone, is lost. We have argued at the end of Sect. 4 that, if we retain the dependence of these two quantities from the boundary values φ s 0 of the scalar fields at radial infinity (that we have fixed to zero in the present analysis), then as functions of both φ s 0 and Γ M , they could still be G 4 -invariant. This was proven for M ϕ at the end of Sect. 3 on general grounds. As pointed out earlier, in the class of models we have been considering here there are five independent G 4 -invariant functions (I n ) = (I 1 , . . . , I 5 ) of φ s 0 and Γ M , which reduce to those in (7.1) once we restrict to the STU truncation. We leave the determination of the explicit expression of M ϕ , M ADM and S in terms of I n in the Kerr-orbit to a future investigation. We just notice here that, once we solve this problem for the STU model, the same expressions in terms of I n hold for all the other symmetric models.

Acknowledgements
We wish to thank R. D'Auria for enlightening discussions. This work was partially supported by the Italian MIUR-PRIN contract 2009KHZKRX-007 Symmetries of the Universe and of the Fundamental Interactions.

A The STU model
The STU model is an N = 2 supergravity coupled to three vector multiplets (n s = 6, n v = 4) and with: We also consider the real parametrization {φ s } = {ǫ i , ϕ i }, related to the complex one by: The Kähler potential has the simple form: e −K = 8 e ϕ 1 +ϕ 2 +ϕ 3 . In the chosen symplectic frame (i.e. the special coordinate frame originating from Kaluza Klein reduction from D = 5), the special geometry of M (D=4) scal is characterized by a holomorphic prepotential F(z) = z 1 z 2 z 3 . The holomorphic Ω M (z) section of the symplectic bundle reads: while the covariantly holomorphic section is given by V M (z,z) = e K 2 Ω M (z). In terms of V M and of its covariant derivatives D i (D i V := ∂ i V + ∂ i K 2 V ) we write the central and matter charges (7.2) of a black hole solution with quantized charges Γ = (Γ M ) = (p Λ , q Λ ): Let us also give the explicit form of the quartic invariant for the STU model: I 4 (p, q) = −(p 0 ) 2 q 2 0 − 2 −2p 1 p 2 p 3 + p 0 q 3 p 3 + p 0 p 1 q 1 + p 0 p 2 q 2 q 0 − (p 1 ) 2 q 2 1 − p 2 q 2 − p 3 q 3 2 + + 2q 1 p 1 p 3 q 3 + q 2 p 1 p 2 − 2p 0 q 3 . (A.5) Upon timelike reduction to D = 3 the scalar manifold has the form G/H * with G = SO (4,4) and H * = SO(2, 2) 2 . We describe the generators of g = so(4, 4) in terms of Cartan H α and shift generators E ±α in the fundamental representation, with the usual normalization convention: [H α , E ±α ] = ±2 E ±α ; [E α , E −α ] = H α . (A.6) In our notation E −α = E † α = E T α . The positive roots of g split into: the root β 0 of the Ehlers subalgebra sl(2, R) E commuting with the algebra g 4 of G 4 inside g; the roots α i , (i = 1, 2, 3) of g 4 and eight roots γ M , m = 1, . . . , 8. The special coordinate parametrization of M (4) scal corresponds to a solvable parametrization of the manifold in which the real coordinates (φ s ) = (ǫ i , ϕ i ) are parameters of a solvable Lie algebra generated by (T s ) = (E α i , 1 2 H α i ). The coset representative L 4 is an element of the corresponding solvable group [37] defined by the following exponentialization prescription: We give, for the sake of completeness, the matrix form of φ s T s in the symplectic representation R: (A.10) The pseudo-Cartan involution σ defining the decomposition of g into H * and K * is defined by the matrix η = (−1) 2H 0 .