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The Hesse potential, the c-map and black hole solutions

We present a new formulation of the local c-map, which makes use of the real formulation of special Kahler geometry and the associated Hesse potential. As an application we use the temporal version of the c-map to derive the black hole attractor equations from geometric properties of the scalar manifold, and we construct various stationary solutions for four-dimensional vector multiplets by lifting instanton solutions of the time-reduced theory.


Introduction
The special Kähler geometry of N = 2 vector multiplets [1] plays a central role in the study of the non-perturbative properties of gauge theories [2,3], string compactifications [4,5,6], and of black holes, in particular the attractor mechanism [7], black hole entropy [8,9,10,11] and the OSV conjecture [12,13,14]. Its distinguished feature is the existence of a single holomorphic function, the prepotential, which encodes all vector multiplet couplings. The power of holomorphicity is a key property, which sets N = 2 theories apart from N = 1 theories where the Kähler potential is not related to an underlying holomorphic function. While at first glance our knowledge of special Kähler geometry appears to be comprehensive, there are still aspects which deserve further study.

Projective special Kähler geometry in real coordinates
It is known that effective supergravity actions are subject to non-holomorphic corrections [15], which enter into the relation between the supergravity effective action and string amplitudes. This has consequences for black hole entropy and the OSV conjecture [16,17,13,14,18]. In this context it became clear that it is sometimes preferable to formulate special Kähler geometry in terms of special real instead of special holomorphic coordinates [19,14]. This real formulation has been used to develop a manifestly duality covariant approach to the OSV conjecture [14,20,21].
While the real formulation of the affine special Kähler geometry of rigid vector multiplets is straightforward, the real formulation of the projective special Kähler geometry of local vector multiplets leaves room for improvements. For affine special Kähler manifolds N the special real coordinates are Darboux coordinates, and the special Kähler metric is Hessian [19]. The Hesse potential is obtained by applying a Legendre transformation to the imaginary part of the prepotential [22]. Electric-magnetic duality, which is a central feature of N = 2 vector multiplets, acts by symplectic transformations. While the prepotential is not a symplectic function, the Hesse potential is, and the special real coordinates form a symplectic vector. In [25] a real formulation of projective special Kähler geometry was worked out, and it was shown that only part of the symplectic covariance of the underlying affine manifold could be kept manifest. However, in applications such as black hole solutions and the study of non-holomorphic corrections one would like to have the full symplectic covariance manifest.
In this paper we obtain a real formulation of projective special Kähler geometry which is symplectically covariant. We make use of the superconformal formalism which employs the gauge equivalence between a theory of n + 1 superconformal vector multiplets with scalar manifold N and a theory of n vector multiplets coupled to Poincaré supergravity, with scalar manifold N = N/ * = M//U (1), see for example [26] for a review. The main idea is to keep the U (1) gauge invariance of the superconformal formulation intact, which amounts to working on N or on the associated Sasakian S, which is a U (1) principal bundle overN , instead of working onN itself. We derive explicit expressions for the scalar and vector kinetic terms as real symmetric tensor fields on N . These tensor fields can be expressed in terms of the Hesse potential and are related to one another and to the metric of the associated superconformal theory by adding differentials dual to the vector fields generating the * -action.

The c-map
The special geometries of five-dimensional vector multiplets [27], four-dimensional vector multiplets [1] and of hypermultiplets [28] are related to one another by dimensional reduction. The corresponding maps between the scalar manifolds are called the r-map and the c-map respectively [29,30,31,32]. Both maps have rigid and local versions, depending on whether rigid or local supersymmetry is considered. Moreover, by reducing over time rather than space one obtains 'temporal' versions of the r-and c-map [51,45,33,34,24], which can be used for generating stationary solitonic solutions by lifting Euclidean, instantonic solutions [33,34,35,36,24,47,37,37], and to study the radial quantization of BPS solutions [33,35]. The local c-map is also an important tool for investigating the non-perturbative dynamics of hypermultiplets [40,41,42], which shows interesting phenomena such as wall crossing [43,44].
The geometry underlying the rigid r-map and rigid c-map [29,45] is well understood: for both maps the scalar manifold of the higher-dimensional theory is simply replaced by its tangent bundle (or, equivalently, its cotangent bundle) and the special structures on both manifolds are related in a canonical way. The metric induced on the (co-)tangent bundle is a version of the so-called Sasaki metric, where the special connection rather than the Levi-Civita connection is used to define the vertical distribution. To be specific, the rigid r-map between the scalar manifolds M, N ≃ T M of five-and four-dimensional rigid vector multiplets takes the following form in terms of adapted coordinates σ i , b i on T M : The geometry of the local r-map and c-map is more complicated because the supergravity multiplet contributes additional degrees of freedom to the scalar manifold. For the local r-map the metric on the vector multiplet manifoldN can nevertheless be brought to the above Sasaki form [47,46,26]. The reason is that the Kaluza-Klein scalar combines with the five-dimensional scalars precisely in such a way that the scalar manifoldM of five-dimensional vector multiplets coupled to supergravity is extended to the scalar manifold M of the associated superconformal theory, but with the superconformal Hesse potential replaced by its logarithm. The scalar manifoldN of the four-dimensional vector multiplet theory is then obtained by applying the rigid r-map to M .
The local c-map [29,30] has an even more complicated structure. It relates projective special Kähler manifoldsN of dimension 2n to quaternion-Kähler manifoldsQ of dimension 4n + 4. In three dimensions abelian gauge fields, including the Kaluza-Klein vector can be dualized into scalars, which become part of the scalar manifoldQ. Using special holomorphic coordinates onN , the metric onQ was obtained in [30]. While completely explicit, this expression is rather complicated, and not covariant with respect to the symplectic transformations of the underlying vector multiplet theory.
In this paper we reformulate the local c-map and obtain an explicit expression for the metric in terms of the Hesse potential of the associated vector multiplet theory which is symplectically covariant and only differs from the Sasaki form by simple universal terms. This is done using the ideas introduced above: (i) we show that the Kaluza-Klein scalar can be identified with the radial direction of the * -bundle N overN . Thus as in the case of the local r-map there is a natural way to combine the four-dimensional scalars with the Kaluza-Klein scalar. (ii) To preserve symplectic covariance we avoid U (1) gauge fixing, which amounts to working on a principal U (1) bundleQ over the quaternion-Kähler manifoldQ. In complete analogy to the vector multiplet case, the metric ofQ is lifted horizontally to a symmetric (degenerate) tensor field on the total space ofQ. (iii) We use our real formulation of projective Kähler geometry to express everything in terms of real coordinates and the Hesse potential.
Our construction is different from other 'covariant' c-maps, which use the hyper-Kähler cone and twistor space associated to every quaternion-Kähler manifold [38,39,34]. In particular, the U (1)-bundleQ and the systematic use of horizontal lifts and of special real coordinates are specific to our approach.
One advantage of our formulation is that we obtain an explicit and relatively simple expression for the quaternion-Kähler metric itself. In contrast, other constructions provide expressions for the hyper-Kähler potential of the hyper-Kähler cone, or for the Kähler potential of the twistor space, in terms of either the holomorphic prepotential [39] or the Hesse potential [34]. This leaves the still complicated step of lifting data fromQ to the hyper-Kähler cone or twistor space, or projecting data down from there toQ. Being able to work directly onQ has immediate advantages for constructing solutions, as we will explain below.

Solitons and Instantons
Dimensional reduction is a standard tool for generating solutions with (at least) one Killing vector field [50]. In particular, dimensional reduction over time allows to lift Euclidean, instantonic solutions to stationary, solitonic solutions.
Therefore we include the case of time-like reduction when working out the cmap. For temporal reduction the resulting manifold has split signature and is expected on general grounds to be para-quaternion Kähler [51,45].
Our main motivation in studying solutions is to develop a formalism which does not depend on supersymmetry (Killing spinor equations), and applies to general c-map spaces, without the assumption that the scalar manifold is symmetric or homogeneous. This continues work done previously in [24,47,48,49] for five-dimensional vector multiplets. For symmetric spaces group theoretical methods have led to a detailed understanding of extremal BPS and non-BPS solutions [36,37]. For general c-map spaces such methods are not applicable and need to be replaced by other methods. Solving the reduced, three-dimensional equations of motion is equivalent to finding a harmonic map from the threedimensional base space (i.e. the reduced space-time) into the scalar target space. Particular solutions to this problem are given by harmonic maps onto totally geodesic submanifolds [50,24]. The simplest choice for the base manifold is to take it to be flat, which for non-rotating black hole solutions corresponds to imposing extremality. In this case the scalar submanifold must be totally isotropic, so that the classification of BPS and non-BPS non-rotating solutions corresponds to the classification of totally geodesic, totally isotropic submanifolds.
In this paper we only consider three-dimensional base spaces which are Ricciflat, and, hence, flat. We do not impose spherical symmetry, unless when considering specific examples. One advantage of our approach is that, for flat base spaces, spherical symmetry is not needed to solve the field equation, i.e. it is as easy to obtain multi-centered solutions as single-centered solutions. This is different in the approach of [34], where only single centered BPS black holes were constructed, while multi-centered solutions were left as an open problem.
For this type of problem it is advantageous that we do not need to lift solutions to the twistor space or to the hyper-Kähler cone.
The structure of our expression for the (para-)quaternion-Kähler metric immediately suggests that in order to restrict fields to a totally isotropic submanifold we should make an ansatz of the form ∂ µ q a = ±∂ µq a , where the two sets of scalars correspond to the positive and negative directions of the scalar metric. By lifting to four dimensions we recognize that this is equivalent to the BPS condition imposed by the vanishing of the gaugino variation, and we can also verify that in this case the ADM mass is equal to the central charge. can be generalized to a larger class of prepotentials. The reason is that the ability to solve the attractor equations only depends on certain homogeneity properties of the prepotential. A similar observation allowed the construction of new solutions in five dimensions [47].
Extremal non-BPS solutions are associated to totally geodesic, totally isotropic submanifolds different from the universal ones described above. Since we want to include target spaces which are not symmetric, we cannot use the group theoretical methods of [36,37] to find non-BPS solutions. Another method is to replace the central charge by a 'fake superportential' by applying a charge rotation matrix [63,64]. Within our approach we can modify the ansatz by allowing a constant field rotation matrix, ∂ µ q a = R a b ∂ µq b , as was done for the local r-map in [47]. For non-rotating solutions we show that this generalized ansatz works, but only if a compatibility condition between the field rotation matrix and the metric is satisfied. At first glance this makes it hard to say anything about the existence of non-BPS extremal solutions for general, nonsymmetric target spaces, without considering specific models. However, for the class of prepotentials already mentioned above, which includes the very special prepotential as a subclass, we can demonstrate the existence of a non-trivial field rotation matrix for axion-free solutions. In contrast, for rotating solutions the presence of a non-trivial field rotation matrix always requires to generalize the ansatz by admitting a curved three-dimensional base space.
2 Review of rigid vector multiplets

Rigid vector multiplets and the rigid c-map
To set the scene, we will review rigid N = 2 vector multiplets and the rigid c-map [51,45]. We will use the conventions of [45], except for a relative minus sign in the relation between the scalar metric N IJ and ImF IJ . 1 Vector multiplets (A Î µ , λ I i , X I ) contain vector fields, a doublet of fermions, and complex scalars. Here and in the followingμ,ν, . . . = 0, 1, 2, 3 are Lorentz indices, i = 1, 2 is the SU (2) R -index, and I labels the vector multiplets. The 1 In our present convention the kinetic terms for scalar and vector fields are positive definite if ImF IJ is positive definite. Note that if we use the superconformal approach to construct a supergravity theory, N IJ must be chosen indefinite, with the negative directions corresponding to conformal compensators. relevant terms in the bosonic Lagrangian are are the (anti-)selfdual projections of the field The Hodge-dualization of field strength is given bỹ F Î µν = 1 2 ǫμνρσF I|ρσ . All couplings in the Lagrangian can be expressed in terms of the holomorphic prepotential F (X I ). 2 Denoting the derivatives of the prepotential as

the scalar metric is
This is an affine special Kähler metric, because the Kähler potential K(X,X) for the metric can be expressed in terms of the holomorphic prepotential, The additional, 'special' structure of the scalar geometry is a consequence of the electric-magnetic duality transformations, which leave the field equations (but not the action) invariant. Electric-magnetic duality acts by symplectic transformations, see [56] for a concise summary. The quantities where the dual gauge fields are defined by Here N IJ is the inverse of N IJ , and µ, ν, . . . = 0, 1, 2 for space-like and µ, ν, . . . = 1, 2, 3 for time-like reductions.
The map induced by dimensional reduction between the respective scalar manifolds M and N is called the rigid c-map. For space-like reductions N is hyper-Kähler [29], as required for rigid hypermultiplets [57]. For time-like reductions one obtains a para-hyper-Kähler manifold, as required for Euclidean hypermultiplets [45]. In both cases the manifold N can be interpreted as the cotangent bundle of M , N = T * M , equipped with a natural metric, which one might call the '∇-Sasaki' metric. 3 This becomes manifest if one uses special real coordinates instead of special holomorphic coordinates on M , see (10) below.
Since special real coordinates will play an important role in the following, we will review them in some detail.

The real formulation of affine special Kähler geometry
The intrinsic definition of affine special Kähler geometry [19] states that a Kähler manifold is affine special Kähler 4 if it is equipped with a flat, torsion-free, symplectic connection ∇, such that the complex structure I satisfies d ∇ I = 0. The affine coordinates (x I , y I ) of this flat connection are Darboux coordinates, and are called special real coordinates in the following. They are related to the special holomorphic coordinates X I by: x I = Re(X I ) , y I = Re(F I ) .
Conversely, the special holomorphic coordinates X I and the quantities F I , which complete them into a complex symplectic vector, can be decomposed as We remark that we could also take the imaginary parts u I , v I as real coordinates and x I , y I to be functions of u I , v I . More generally we could take the real parts of e iα (X I , F I ) as Darboux coordinates. Affine special Kähler manifolds always admit not just one special connection, but an S 1 -family which is generated by [19] ∇ (α) = e αI • ∇ • e −αI .
Neither physics, nor geometry depends on the choice of the special connection from this family, but each connection in the family has its own system of special real coordinates. The 'dual' special real coordinates u I , v I are flat Darboux coordinates with respect to the special connection ∇ (π/2) . By computing the Jacobian of the coordinate transformation (X,X) ↔ (x, y), and using F IJ = F JI , one obtains the following relations: Affine special Kähler manifolds are Hessian manifolds, and the Hesse potential is proportional to the Legendre transform of the imaginary part of the holomorphic prepotential [22]. This transformation replaces u I = ImX I by y I = ReF I as independent variables: H(x, y) = 2ImF (X(x, y)) − 2y I u I (x, y) .
Taking derivatives of 2 Im(F ) with respect to (x, y) we find Using these results, we find that the derivatives of the Hesse potential are proportional to the dual real coordinates: Taking second derivatives we find This allows us to express the Hessian metric H ab in terms of the second derivatives of the prepotential: We will also need the relation between the differentials of the special holomorphic and the special real coordinates: Next, we compute the derivatives of the Hesse potential with respect to the special holomorphic coordinates. This is not needed for the real formulation of affine special Kähler geometry, but will be important later for the real formulation of projective affine special Kähler geometry.
and by a similar calculation Taking second derivatives we find Using equations (6) and (7) it is straightforward to verify verify that which shows that N IJ and H ab represent the same metric in terms of special holomorphic and special real coordinates, respectively. It is easy to show that the inverse of the Hessian metric is given by Moreover, it is useful to note that where Ω ab := 0 I −I 0 is the matrix representing the fundamental form (Kähler form) in special real coordinates. 5 With these results it is straightforward to express the reduced Lagrangian (4) in terms of special real coordinates. Defining (q a ) = (s I , 2p I ), we find [45,58] It is now manifest that the metric on N is the canonical positive definite (for ǫ = −1) and split signature (for ǫ = 1) metric on the cotangent bundle of M , respectively. Using special real coordinates has further advantages. All objects appearing in the above Lagrangian transform linearly under symplectic transformations: q a ,q a are contravariant and covariant vectors, respectively, while H ab and H ab are symmetric tensors [46]. In contrast, F IJ = 1 2 (R IJ +iN IJ ) transforms fractionally linearly under symplectic transformations.

Vector multiplets coupled to 4d supergravity
The coupling of vector multiplets to supergravity can be constructed using the superconformal calculus, which exploits the gauge equivalence between a locally superconformal theory of n + 1 vector multiplets and n vector multiplets coupled to Poincaré supergravity. 6 This is reviewed, for example, in [59,60]. We will use elements of this approach, and focus on the bosonic fields and the underlying scalar geometry. The first step in the construction is to write down a theory of n + 1 rigidly superconformal vector multiplets. Compared to the previous section, this amounts to the additional constraint that the prepotential is homogeneous of degree two. The resulting scalar manifold is a conical affine special Kähler manifold [19,24], which is an affine special Kähler manifold with a holomorphic homothetic action of * = Ê >0 · U (1): where Aμ is the U (1) connection. In principle we should also include the connection bμ of local dilatations, but it is known that the terms containing this connection cancel within the Lagrangian. Alternatively, one can impose the gauge condition bμ = 0, known as the K-gauge. The gravitational term is not canonical, since the Ricci scalar is multiplied by the dependent field which acts as a compensator for local dilatations.
The gauge couplings are given by the real and imaginary parts of the complex matrix This differs from the gauge couplings F IJ = 1 2 (R IJ + iN IJ ) of the rigid theory by terms which arise from integrating out an auxiliary field (the tensor field of the Weyl multiplet). Note that N IJ is manifestly U (1) invariant, so that by imposing the D-gauge we obtain tensor fields on S andN . This is in fact a Kähler quotient, becauseN inherits a Kähler metric from N .
ManifoldsN , which are obtained by this construction from conical affine special Kähler manifolds, are called projective special Kähler manifolds.
It is well known from work on black hole solutions that it is often advantageous to use the gauge equivalence, and to work on the larger space N rather than on the physical scalar manifoldN . One particular advantage is that this keeps symplectic covariance manifest. Fixing a U (1) gauge corresponds to selecting a hypersurface of the Sasakian S, which can be done by choosing any condition which is transversal to the U (1) action (for example ImX 0 = 0). However, choosing a symplectically invariant condition corresponds to selecting, at each point, the direction orthogonal to the U (1) action. But this is the contact distribution of the Sasakian and therefore not integrable. For this reason a hypersurface corresponding to a U (1) gauge cannot be selected in a symplectically invariant way. 7 In the following we will keep the local U (1) gauge invariance intact, and for reasons that will become clear later we also postpone imposing the D-gauge.
The above Lagrangian contains the U (1) gauge field, which makes its local U (1) invariance manifest. However, the U (1) connection is a non-dynamical, auxiliary field, and we now eliminate it by its equation of motion Now the gauged sigma model is replaced by the ungauged sigma model We have used that the prepotential is homogeneous of degree 2 and therefore X(∂μN )X = 0. The Lagrangian still contains terms proportional to ∂μK because we have not yet imposed the D-gauge. Observe that the tensor field g IJ is degenerate on the large space N , because This is not a problem, because the directions along which g IJ is degenerate correspond to the unphysical degrees of freedom normal toN ⊂ N . Geometrically, these are the vertical directions of the * -bundle N overN , i.e. the radial direction of the Riemannian cone and the orbits of the U (1) isometry. While g IJ is not a metric on N , we obtain a non-degenerate metric by projecting itN .
In other words, g IJ is the horizontal lift of the projective special Kähler metric gN to N , and, if we impose the D-gauge, to S.
The well known formula for the Kähler potential of the projective special Kähler manifoldN can be obtained by using coordinates X 0 , z i on N , where z i = X i /X 0 are special coordinates onN . Rewriting K given in (14) as a function of X 0 , z i , one finds that the dependence on X 0 can be removed by a Kähler transformation. This shows explicitly that the tensor g IJ is degenerate on the two vertical directions. Defining F (z) = (X 0 ) −2 F (X I ), we obtain the Kähler potential of the projective special Kähler metric ofN : To obtain a theory with positive definite kinetic terms for the physical scalars, the projection of g IJ ontoN must be positive definite, while positive definite kinetic terms for the vector fields require that I IJ is negative definite, see (11). It is known that both conditions are satisfied if the metric N IJ of the conical affine special Kähler manifold N has complex Lorentz signature (− − + · · · +) [61,46]. The negative directions, which are the directions normal toN ⊂ N , correspond to conformal compensators. We remark that −I IJ can be interpreted as a positive definite metric on N , and that the relation between the indefinite metric N IJ and the definite metric −I IJ has a natural geometric interpretation, which is analogous to the relation between the Griffith and Weil intermediate Jacobians for Calabi-Yau threefolds [46].

The real formulation of projective special Kähler geometry
In section 2 we have reviewed the real formulation of affine special Kähler geometry. It is not straightforward to obtain a real formulation of projective special Kähler geometry which preserves symplectic covariance. The reason is that the physical scalars of the super-Poincaré theory correspond to special coordinates is not, and only part of the symplectic covariance can be kept manifest [25].
In this section we show how a manifestly symplectic real formulation can be obtained by preserving the U (1) gauge invariance. This amounts to expressing the degenerate tensor g IJ and the vector kinetic matrix N IJ in terms of special real coordinates on N and in terms of the Hesse potential H. In doing so we will get a clearer understanding of the geometrical meaning of these tensor fields.
Since the theory associated with N is now superconformal, we have additional relations in addition to those derived in section 2. The prepotential and the Hesse potential are now homogeneous of degree two in special holomorphic and special real coordinates, respectively. This implies Also note that The affine special Kähler manifold is now a complex cone, at least locally.
This means that there is a homothetic and holomorphic action of * , which is given by the the homothetic Killing vector field ξ and the U (1) Killing vector field Iξ, where I is the complex structure. The explicit expressions with respect to special holomorphic and special real coordinates are: In special real coordinates the complex structure itself is given by I a c = 1 2 Ω ab H bc in terms of the Kähler form Ω ab and the metric H ab .
We remark that the q a are special real coordinates with respect to a fixed, but arbitrary special connection. For conical affine special Kähler manifolds the U (1) gauge transformations preserve the metric, the symplectic and the complex structure, but they rotate the special connections, and the associated special real coordinates, among themselves.
Our first task is to rewrite the tensor where in terms of special real coordinates. Using (8) and (9) we find Using (18), we find By the results of section 2, the first term gives To evaluate the second term, we observe that where we used (5) together with homogeneity. Using further results from section 2, this implies To proceed, we substitute (7) into the second term on the right hand side of (19), with the result We now observe that Using this, the second term becomes Combining the two terms, we find that where ab is the horizontal lift of the projective special Kähler metric, expressed in special real coordinates.
Before we proceed to express N IJ in real coordinates, let us analyze what the above calculation tells us about the underlying geometry. Solving (20) for the affine special Kähler metric H ab , we obtain: ab , which by projection gives the projective special Kähler metric, and two negative definite terms 8 which correspond to the directions generated by ξ = q a ∂ a and Iξ = 1 2 H a Ω ab ∂ b . As we will see, all relevant tensor fields on N are related to the metric H ab by adding terms proportional to the squares of the one-forms H a and Ω ac q c . These one forms are obtained by contracting the homothety ξ with the metric and the Kähler form, respectively (equivalently by contracting ξ and Iξ with the metric). It is an advantage of the real formalism that the directions generated by ξ and Iξ can be described in such a simple way.
We now introduce one further tensor field on N , which will play an important role for the c-map. As we have seen before, the Kähler potential (14) of the supergravity theory is obtained by taking the logarithm of the Kähler potential (3) of the corresponding superconformal theory. The logarithm effectively encodes the superconformal quotient. This motivates us introduce the tensor obtained by taking the second derivatives of the logarithm of the Hesse potential H of the rigid theory. Specifically, we setH = − 1 2 log(−2H) andH ab = ∂ 2 a,bH . Then Since we know that the right hand side is positive definite in the horizontal directions and degenerate in the vertical directions generated by ξ and Iξ, it follows immediately thatH ab is a non-degenerate Hessian metric which is negative definite along the U (1) direction generated by Iξ and positive definite in all other directions. The homogeneity properties of the Hesse potential (15) also imply that the matrixH satisfies the identity q a q bH ab = 1 .
This will not be used in this paper, but may be useful for produce 4d nonextremal black hole solutions as a similar identity was needed in the 5d case [49].
We now turn to the vector kinetic matrix N IJ . It is known how to express this complex matrix, which transforms fractionally linearly under symplectic transformations in terms of a real matrixĤ ab , which transforms as a symmetric tensor of rank 2. In the conventions of [46], the relation iŝ It is known that the tensor −Ĥ ab is positive definite, given that H ab has complex Lorentz structure, and therefore it can be interpreted as a positive definite metric on N . In [46] it was shown that in terms of complex geometry the indefinite and definite metric are related by a transformation that exchanges Griffith and Weyl flags. We would now like to relateĤ ab to the other tensor fields in terms of real coordinates.
Below we will prove that the tensorsH ab , H ab andĤ ab are related by: Given that H ab has complex Lorentz signature, it is manifest that −Ĥ ab is positive definite. In contrast to the indefinite metrics H ab andH ab , the definite metricĤ ab is not Hessian. However it is uniquely determined by the Hesse potential H. It is the above identity which will be critical in matching up moduli fields with the electric/magnetic potentials in order to produce black hole solutions.
It remains to prove (22), which requires some effort. To start we need the explicit relations between the real and imaginary parts of F IJ = 1 2 (R IJ + iN IJ ) and N IJ = R IJ + iI IJ : where (XN X) = N MN X M X N , etc. It is straightforward to verify that the inverse of I IJ is where we used 2H = (XNX) and the decomposition X I = x I + iu I , F I = y I + iv I . Next, one can verify Finally, one can also verify that Putting everything together we have Expressing this in terms of the special real coordinates q a usingĤ ab , H ab and Ω ab this becomesĤ which proves (22).
In summary, we have found the real tensor fields H For later use we now derive the expression for the graviphoton in terms of real coordinates. The graviphoton is the vector field which in the Poincaré supergravity theory belongs to the supergravity multiplet and therefore is invariant under symplectic transformations. Its field strength is given by where the dual field strength are Adding the self-dual part and expressing everything in real variables, we obtain These terms are not independent, we can either use the real coordinates (x I , y I ) together with the field strength F Î µν , G I|μν , or the dual real coordinates (u I , v I ) together with the Hodge-dual field strengthF Î µν ,G I|μν . Using the definitions of these quantities, one can verify that The local c-map and the Hesse potential We now turn to the dimensional reduction of four-dimensional vector multiplets coupled to supergravity. We perform the reduction using the complex formulation of the four-dimensional scalars, and use the gauge equivalence to describe them in terms of the scalars X I taking values in N . The reductions over space and time are performed in parallel. After dualizing the three-dimensional vector fields we systematically express all quantities in terms of special real coordinates.
Our overall strategy is to obtain an expression which comes as close to the 'metric on the cotangent bundle form' (10) of the rigid c-map. Therefore we express the couplings in terms of the logarithm of the Hesse potential. We will see that all terms that cannot be brought to this form are universal, in the sense that their couplings only contain constant matrices and the Kaluzu-Klein scalar.

Dimensional reduction
Our starting point is the Lagrangian representing the bosonic part of fourdimensional N = 2 supergravity coupled to n vector multiplets, where g IJ = ∂ I ∂J K. We have eliminated the U (1) gauge field by its equation We perform the reduction of the Lagrangian over a time-like and space-like dimension simultaneously, differentiating between the two cases by In order to reduce directly into the Einstein frame we decompose the metric as which implicitly defines (e φ , V µ , g µν ) in terms of the four-dimensional metriĉ gμν. The reduced Lagrangian is given by where the terms descending from the four-dimensional metric appear in the first line, the four-dimensional scalars in the second line, and the gauge fields in the third and fourth line. We have denoted the field strength of the Kaluza Klein-vector by V µν , and the scalar fields ζ I = A I 0 (A I 3 ) are the components of the four dimensional vectors along the reduced timelike (spacelike) direction.
The Lagrangian at present still contains the bare Kaluza Klein-vector V µ , which prevents the associated abelian gauge symmetry from being manifest. Therefore we make the field redefinition The Lagrangian now takes the manifestly gauge invariant form, where we have dropped the primes and gathered together like terms.

Conformal rescaling
In order to obtain a canonical Einstein-Hilbert term we perform the conformal rescaling g µν = e 2Kg µν .
The various terms in the Lagrangian have the following transformation rules in three dimensions: The Lagrangian itself becomes One can see that by redefining the KK-scalar φ ′ = φ − K, the field K decouples from all other fields besides gravity. We will now set this field to be constant, and drop the primes. 9 This amounts to imposing the D-gauge. We could of course have done this at an earlier stage, but we found it instructive to demonstrate how the radial degree of freedom K of the cone N decouples.

Dualization of vector fields
Since we are working in three dimensions, and the vector fields in the Lagrangian only appear via their field strengths, it is possible to dualise the vector fields into scalar fields (A I , V ) ∼ (ζ I ,φ). This is achieved by adding the Lagrange The variation of L 3 + L Lm gives the algebraic equations of motion (note that Substituting the above expressions back intoL 3 = L 3 + L Lm we are left with the dual Lagrangiañ

A field redefinition
We would now like to bring the Lagrangian into a form that resembles (10) more closely. From [46] we know that by setting (q a ) = 1 2 (ζ I ,ζ I ) and using the real tensorĤ ab , the terms in the third line of (24) are proportional to 1 HĤ ab ∂ µq a ∂ µqb . 10 Using (22) we can express this in terms of the Hessian metricH ab up to model independent terms. To proceed, we need to re-organize the remaining scalars X I , φ,φ into 2n + 2 real scalars q a which transform as a symplectic vector and balance the 2n + 2 real scalarsq a . The counting of degree of freedom works out, because the n + 1 complex scalars are subject to two conditions, and therefore correspond to 2n independent real scalar fields.
Moreover, by going to special real coordinates on N , we can relate them to a symplectic vector. But what about φ andφ?
We proceed by making use of an observation that was made in the context of the local r-map, which relates the scalar manifolds of five-dimensional and fourdimensional vector multiplets [47]. There it is possible to absorb the Kaluza Klein-scalar into the manifold parametrized by the higher-dimensional (in the case of the r-map, the five-dimensional) scalars. This amounts to lifting the constraint imposed by the D-gauge. The Kaluza Klein-scalar is identified with the radial direction of the cone N over S, which is promoted from a gauge degree of freedom to a dynamical degree of freedom. This idea can be implemented in the four-dimensional setting by defining a new set of complex scalars Y I by The Kaluza Klein-scalar is now a dependent field, determined by the expression Since φ transforms by a shift under the global isometry group, we find that the new scalar fields must transform by a scale factor under these isometries 10 Actually, in [46] the dual coordinatesqa and the inverse metricĤ ab were used, but this is simply a different parametrization.
The Lagrangian can now be written as where φ is a dependent field.
The Lagrangian is still invariant under local U (1) transformations of the fields (Y,Ȳ ), and the equations of motion transform by an overall phase factor. This is shown using that the tensor g IJ has two null directions By differentiation we obtain the identities Under phase transformations the derivatives of the metric transform by a phase and the Kaluza Klein-scalar is invariant It follows that the Lagrangian is U (1) invariant, the (Y,Ȳ ) equations of motion transform by an overall phase, and all other equations of motion are invariant.
When comparing (24) to (26) both Lagrangians look identical, except that X I have been replaced by Y I . It is instructive to check that substituting X I = e −φ/2 Y I into (24) gives indeed (26). Due to the peculiar properties of the degenerate tensor g IJ , no derivative terms involving φ are generated by the substitution. All factors e φ/2 cancel, because g IJ is homogeneous of degree −2, and because I IJ and R IJ are homogeneous of degree 0. Of course the essential difference between (24) and (26) is that φ is now a dependent field.
One might wonder whether the dualized Kaluza Klein-vectorφ could be treated in a similar way as the Kaluza Klein scalar φ. This is not so, because the symmetries carried by the reduced gravitational degrees of freedom φ,φ do not match with the symmetries of the affine special Kähler manifold N . The fields φ,φ parametrize the coset with isometry group SL(2, Ê). A two-dimensional solvable subgroup generated by shifts in φ andφ extends to a symmetry of the full Lagrangian once theq a are included. In contrast, the manifold N has a homothetic action of * . Upon taking the logarithm of the Hesse potential, the dilatation becomes an isometry (rather than a homothety) of the 'metric' g IJ . 11 Above we observed that the we proceed differently, by keepingφ as an independent field, and, consequently, keeping the local U (1) gauge invariance. We will see later that when we construct solutions, the local U (1) gauge symmetry is gauge fixed while preserving symplectic covariance and the isometries of scalar metric. We will also see that for our solutions it will always be possible to expressφ in terms of the other fields.
We can interpret our treatment of the scalar fields geometrically as follows.
If we freeze the scalarsq a descending from the four-dimensional gauge fields, then the scalar manifold parametrized by the physical four-dimensional scalars z i , and by φ andφ is Using the gauge equivalence, we can describeN in terms of the fields X I using the Kähler quotient: Now we absorb φ into N . This 'un-does' the D-gauge, and re-institutes the radial degree of freedom of the cone N over S, while leaving the U (1) isometry intact.
The coset SL(2Ê)/SO (2) is broken up and the remaining one-dimensional piece is parametrized by the scalarφ, with a metric depending on φ. The scalar manifold can be represented as a deformed product Here N/U (1) is the quotient of N with respect to its U (1) isometry rather than the Kähler quotient. The advantage of this way of organising the fields becomes apparent once we use real special coordinates on N .

The real parametrization
The kinetic term of the complex scalar fields takes precisely the same form as considered previously in section 4 We make the real decomposition and use the previous results to write this term in the Lagrangian as where q a = (x I , y I ) T . Note that our previous calculations are still applicable after the replacement X I → Y I , due to homogeneity.
The Kaluza Klein-scalar is given in terms of the real variables by which is homogeneous of degree two in q a = (x I , y I ) T . The kinetic term for the Kaluza Klein-scalar can then be written as and this term cancels against the second term in (27). When rewriting the terms descending from the four-dimensional gauge fields using the variableŝ q a = ( 1 2 ζ I , 1 2ζ I ) T , they take the form We can now put together all terms and write the Lagrangian in terms of real fields asL This formula is one of our main results, and provides a new formulation of the We prefer not to fix the U (1) gauge and to work on the U (1) bundle, because, as we explained in section 3, a U (1) gauge fixing would spoil the manifest symplectic covariance. In the following section we will show that instantonic solutions can be constructed and be lifted to solitonic solutions, such as black holes, while preserving symplectic covariance. We will then revisit the issue of U (1) gauge fixing.

Stationary Solutions
We now turn to finding stationary solutions of the four-dimensional Lagrangian.
Four-dimensional stationary BPS solutions for general vector multiplet couplings have been constructed some time ago by imposing invariance under part of the supersymmetry transformations [53,54,62,11]. We expect to recover these solutions and to obtain further non-BPS solutions. To this end we reduce over a time-like dimension, therefore making the choice ǫ = 1 in the formula for the reduced Lagrangian (29). We will find that in flat backgrounds we can give solutions to generic models in terms of harmonic functions.
Before embarking into the details, let us explain the overall strategy. Since the three-dimensional Lagrangian is a combination of perfect squares, we will try to reduce the field equations to Bogomol'nyi equations which follow from imposing that the squares vanish individually. We will focus on solutions where the three-dimensional metric is Ricci-flat, and, hence, flat. This restricts the fields to take values in totally isotropic submanifold, and therefore we will call the corresponding ansatz the isotropic ansatz. After lifting to four dimensions we will obtain four-dimensional extremal static black hole solutions as well as over-extremal (singular) rotating solutions. The structure of the Bogomol'nyi equations can be read off from the Lagrangian (29). One of the Bogomol'nyi equations results from imposing that the first line of (29) vanishes, which gives a relation of the form ∂ µ q a = ±∂ µq a between the q a and theq a , which is identical to the relation found for fivedimensional black holes [47]. If the scalar metric satisfies a certain compatibility condition, one can instead impose the more general condition where R a b is a constant 'field rotation matrix'. Such solutions are non-BPS, and will be discussed in a separate section. Once either of these condition is imposed, the terms in the second line combine into one term, which, however, has a similar structure as a term within the square in the third line. The most general ansatz only requires that the second and third line vanish in combination, while a more restricted ansatz requires that the second and third line vanish independently. The restricted ansatz corresponds to static solutions, because imposing that the third line vanishes is equivalent to the vanishing of the field strength of the Kaluza-Klein vector. Without this restriction, we obtain stationary rotating solutions. We will refer to solutions obtained from our isotropic ansatz as isotropic solutions. Note that they will in general neither be BPS (since we admit a field rotation matrix), nor extremal (since rotating solutions are over-extremal).
As in the five-dimensional case [47], we will be able to demonstrate that the equations of motion can be reduced to decoupled harmonic equations by choosing suitable 'dual' coordinates. Therefore the solution will be given in terms of a set of harmonic functions. We will also see that this way we naturally obtain the generalized stabilization equations of four-dimensional black holes, in their algebraic and manifestly symplectic form.

Equations of motion
We will now derive all the field equations of the Lagrangian (29) and show explicitly how they are solved by imposing Bogomol'nyi equations.
Firstly, we perform the variation of the equation (29) with respect to the field q a to obtain the equation of motion Next, the equation of motion for theq a fields The equation of the fieldφ, which descends from the Kaluza Klein-vector, is given by This equation is nothing but the Bianchi identity for V µν , the field strength of the Kaluza Klein-vector, which allow us to write the field strength in terms of a gauge potential V µν = ∂ µ V ν − ∂ ν V µ . Finally, from the variation of the metric we find the Einstein equations

Dual coordinates
The Hessian matrixH ab allows us to define a natural set of dual coordinates By the chain rule we find the expression for the derivative of the dual coordinates The existence of these dual coordinates is critical for obtaining solutions to generic models in terms of harmonic functions. Note that the definition of the dual coordinates is completely analogous to the five-dimensional case [47].

The isotropic ansatz
A flat three-dimensional geometry requires that the energy-momentum tensor must vanish identically. To achieve this we must impose an appropriate ansatz for the fields, which consists of two distinct parts. The first part of our ansatz is to identify the vectors ∂ µ q a and ∂ µq a up to an overall sign Upon imposing this ansatz the vacuum Einstein equations reduce to The second part of our ansatz is now clear: we must make the identification where the choice of sign is important. One can interpret this as fixingφ in terms of other fields which are independent φ = 2(q a ∓q a )Ω ab q b .
Note that our first ansatz means that q q ∓q a is a constant in spacetime. By construction the ansätze (34) and (35)  Next, we need to consider the effect this ansatz has on the other equations of motion. Firstly, from theφ equation of motion we find the condition Turning our attention to the q a equation of motion, we see that the second term will drop out, the third line will simplify, and due to (36) the derivative in the second line will only act on q c . We are left with The fourth term then cancels with the derivative acting on the Hesse potential in the third term Since Ω ab is antisymmetric the second term cancels with the third term, and writing the first term in terms of the dual coordinates q a we are finally left with ∆q a = 0 .
This is the Laplace equation for the dual coordinates q a , with respect to the flat Euclidean three-dimensional metric. Solutions are given by harmonic functions. Now let us consider theq a equation of motion. From (36) we see that the derivative in the second term will only act on q c , and the second and third term will simplify to give The second and third term cancel due to antisymmetry of Ω ab , and we again get the Laplace equation on the dual coordinates (37).
Let us finally check that the solutions to the q andq equations of motions are consistent with theφ equation of motion (36). Using the identity q a Ω ac = − 1 4 H a Ω ab H bc , we can write the LHS of (36) in terms of dual coordinates as It is clear that for solutions satisfying the Laplace equation the RHS will vanish.
We conclude that upon imposing our ansätze all equations of motion reduce to the Laplace equation on the dual coordinates (37).
When rewriting the isotropic ansatz (34) in terms of four-dimensional quantities one recovers a well known relation which for four-dimensional BPS solutions follows from supersymmetry. First, it is useful to note that the threedimensional scalarsq a = 1 2 (ζ I ,ζ I ) are related to four-dimensional field strength by While the first relation holds by definition, the second requires one to combine and manipulate various of the relations in this section. The above relations show that the scalar fields ζ I ,ζ I can be interpreted as electro-static potentials for the field strength and Hodge-dual field strength. Combining this with (q a ) = 1 2 Y I +Ȳ I , F I (Y ) +F I (Ȳ ) = 1 2 e φ/2 X I +X I , F I (X) +F I (X) , the isotropic ansatz (34) becomes Thus the isotropic ansatz implies that the real part of the symplectic vector (X I , F I ) is proportional to the gauge potentials. For supersymmetric solutions this follows from the gaugino variation, see for example [11], while here we obtain it as the Bogomol'nyi equation associated to the first line of (29).

Remarks on the local U (1) symmetry
The ansatz (34) where H I , H I are harmonic functions. Rewriting this in terms of the original four-dimensional fields X I , we obtain Using the D-gauge −i(X IF I − F IX I ) = 1, we can verify that We remark that there is an alternative description which allows to keep the U (1) invariance manifest and effectively decouples the U (1) gauge degree of freedom. As in [11], one can modify the definition of Y I as where h is a phase factor which transforms with the same charge under U (1) as X I . Note that in [11] a different convention is used, which corresponds in our notation to taking Y I = e −φ/2h X I , instead of the above relation. This only alters how e φ depends on the independent coordinates Y I , but has no baring on our discussion. When comparing to [11], note that the Kaluza-Klein scalar e φ is related to the functions f, g used there by e φ = e −2f = e 2g .
The effect of the modified definition (43) is that Y I is now a U (1) invariant field. Due to the degeneracy of g IJ and the homogeneity properties of the functions involved, this modification does not change the calculations presented above. In particular, no derivative term for the field h is generated. When rewriting (40), by replacing the U (1) invariant variables Y I by the original variables X I , which are subject to U (1) transformations, we obtain: as in [11], except for a different normalization of the Y I . 12 Using again the D-gauge −i(X IF I − F IX I ) = 1, this implies which determines the compensating phase h for our solution.

Remarks on attractor behaviour and gradient flow equations
The equations (40) Recently the BPS equations for four-dimensional N = 2 gauge theories were reformulated, using the Hesse potential, in symplectically covariant form, for general non-spherical solutions [66].
Our formalism by-passes the gradient flow equations and we directly obtain solutions in terms of harmonic functions. While we leave a comprehensive discussion of the relation between our approach and gradient flow equations for future work, we would like to expand a little on the discussion given in [47], where we observed that the field equation can be recast in first order form. One way of re-writing the second order equations of motion into first order form is to rewrite the Lagrangian as a (possibly alternating) sum of squares. This can be done systematically within our formalism, as follows. Upon inspection of the Lagrangian (29) we see that the second and third line are already written as the 12 And, of course, in the present paper we do not consider higher derivative terms.
sum of square terms. We then only need to consider the first line, which we can write as where H a are harmonic functions. In the spherically symmetric case one can dimensionally reduce the Lagrangian to one dimension, where derivatives of harmonic functions are just constants, which can be identified with the conserved charges carried by the solution. One then obtains gradient flow equations, which are driven by the central charge in the supersymmetric case and by a fake superpotential in general. We refer to [47] for a discussion of the spherically symmetric case and proceed without imposing spherical symmetry.
The first part ∂ µ q a = ±∂ µq a of the isotropic ansatz can be seen as imposing that the squares displayed above vanish. The second part of the isotropic ansatz matches the remaining squares, which appear with a relative sign difference, and, When we instead eliminate the harmonic functions, we recover the isotropic ansatz. We can also make contact with relations recently found in [66] by with q a =H a . Then the left-hand side is related to the gradient of the Hesse potential, while the right-hand side is This can be related to the expression for the graviphoton in terms of real coordinates by Hodge-dualizing the field strength Thus we obtain a relation between the gradient of the Hesse potential and the magnetic components of the graviphoton, or, equivalently, the electric components of the Hodge-dual of the graviphoton This relation appears to be the local analogue of an equation for the gradient of the Hesse potential recently found in [66] for BPS dyons in rigid N = 2 theories.
As the unique symplectically invariant contraction between scalars and gauge fields, the graviphoton plays the role of the central charge vector field used in [66].

Rotating solutions
We now have an ansatz for finding stationary isotropic solutions (flat 3d metric) to completely generic models in terms of the dual coordinates. However, in order to write down these solutions explicitly in terms of the four-dimensional fields one must disentangle them from the dual coordinates. This is equivalent to solving the generalised stabilisation equations, and is not always possible in closed form. In this section we will discuss solutions which lift to rotating over-extremal solutions in four-dimensions, with the STU model as an explicit example. These solutions are characterised by axial symmetry and the requirement that they are asymptotic to Minkowski space at infinity.
The results of the previous section show that upon imposing our isotropic ansatz (34) and (35), the equations of motion reduce to ∆q a = 0, and solutions are given in terms of the dual coordinates by harmonic functions We wish to disentangle the four-dimensional metric from this solution, and show that it corresponds to a rotating solution. We can do this by retracing our steps in the dimensional reduction procedure to find The first equation is trivial; the second is model dependent and we will look into it in more detail later. For now let us focus on the third equation, or more accurately set of equations. These are entirely independent of the details of the model, i.e. choice of prepotential. Following the method for producing rotating isotropic solutions used in [53,54], we impose that solutions are axially symmetry about the coordinate ϕ in an oblate spheroidal coordinate system, defined by x = r 2 + α 2 sin θ cos ϕ , y = r 2 + α 2 sin θ sin ϕ , z = r cos θ .
In this coordinate system the third set of equations in (45) become Since solutions should be asymptotically flat, we must require that ∂ [µ V ν] −→ 0 as r −→ ∞. We will come back to this shortly. Single-centred harmonic functions in oblate spheroidal coordinates can be written as where R = r 2 + α 2 cos 2 θ. It is understood that (h I , h I , m I , m I , p I , q I ) are all independent integration constants. While h I , h I determine the values of the scalars at infinity and p I , q I are the magnetic and electric charges, m I , m I are the dipole momenta [54]. In [54] a restricted class of harmonic functions was considered, which corresponds to switching off half of the integration constants appearing in the expressions above. This restricted class of solutions was taken in order to satisfy the condition that the field strength of the U (1) connection vanishes. In our formalism it is clear that we do not need impose this condition to produce solutions.
Integrating the equations (46) and (47) we find an explicit model independent expression for the only non-zero component of the KK-vector where C is an arbitrary constant. We observe that all three independent symplectic constructions of the vectors (h I , h I ), (p I , q I ) and (m I , m I ) of integration constants appear in this expression. The term in the second line is the angular momentum of the black hole, while n = 1 2 (h I p I − h I q I ) is the NUT charge, as can be seen by comparison with [67,68]. The term in the third line does not carry a particular name, but is known to occur in rotating solutions [67]. For static solutions all these terms are absent, which beside m I = m I = 0 imposes the constraint h I p I − h I q I = 0 on the integration constants. Note that upon imposing this condition the KK-vector reduces to Since this is proportional to α it will vanish in the static limit. In the general case V ϕ does note vanish for r → ∞ unless we impose h I p I − h I p I = 0 (and C = 0). However, since the field strength ∂ [µ V ν] goes to zero, such a term could be eliminated by a coordinate transformation. In addition to requiring the KKvector to vanish asymptotically, we also need to ensure the KK-scalar behaves appropriately, i.e. e φ −→ 1 as r −→ ∞. This will place one more restriction on the integration constants (h I , h I ). Since the KK-scalar is a model dependent field we will need to look at specific examples if we wish to write this constraint explicitly.
The formula for the ADM mass for axially symmetric solutions is given by Expanding in descending orders of r we have Computing the ADM mass one finds a particularly simple dependence on the Hesse potential We would now like to investigate the relation between the mass and central charge. For solutions with vanishing NUT charge one has r 2 q a Ω ab ∂ r q b → 0 asymptotically, which implies that r 2 q a Ω ab ∂ r q b → 0 asymptotically. We can then write the mass as This confirms that these solutions are BPS.
Before we enter into a discussion of specific models, we need to make a few comments about this class of rotating solutions. It contains the rotating supersymmetric solutions of [54], which are not black holes but have naked singularities. As is well known, for rotating four-dimensional solutions the extremality bound is higher than the supersymmetric mass bound, so that rotating supersymmetric solutions are necessarily singular. Besides the ring singularity at r = 0, a non-vanishing NUT charge can introduce further singularities [68].
We also remark that time-independence might imply further constraints on the allowed charges [62,66]. Due to such constraints and the presence of naked singularities, the physical relevance of these rotating solutions is not immediately clear, in contrast to the static solutions to be considered later. For us they are interesting for technical reasons, because they show how rotating solutions can be obtained within the framework of dimensional reduction over time.
To obtain physically relevant rotating solutions without naked singularity our method needs to be extended to solutions which take values along non-isotropic submanifolds. This is similar to the problem of deforming static extremal into non-extremal black holes, and both problems will be addressed in future work.
We conclude this section by giving the explicit solution for the STU model.

The ST U model
For the STU model we can find solutions explicitly in closed form. The model is characterised by the prepotential The name ST U -models derives from the conventional notation S, T, U = Y i Y 0 , i = 1, 2, 3 for the physical scalars. The corresponding Hesse potential is given in terms of the imaginary parts of Y I , F I by where d ABC = |ǫ ABC |. A detailed derivation of this expression is given in appendix A.1.
Rotating isotropic solutions to this model correspond to taking 1 H u I = H I and 1 H v I = H I . Using the expression e φ = −2H we can write the KK-scalar for the STU model explicitly in terms of harmonic functions In order that the solution is asymptotically Minkowski space we must impose a constraint on the integration constants At first glance it also appears that the KK-vector (48) will not vanish asymptotically, as is required for Minkowski space. However, since the field strength of the KK-vector vanishes asymptotically we can make a change of coordinates so that spacetime is Minkowski.
For completeness, let us remark on the remaining four-dimensional fields for this solution. The original complex scalar fields are given by where Y I are given in terms of u I , v I through with These expressions have been adapted from similar expressions derived in [78].
To obtain static solutions we will impose precisely the same isotropic ansatz as for stationary solutions, but in order to link to previous work we will reverse the order in which we apply the two parts of the ansatz. We first impose only the second part of the isoptropic ansatz (35), which in this case is simply It is then clear that the equations of motion simplify considerably. Only the first line of each equation is relevant, and we are have left with The equation of motion corresponding to the KK-vector is clearly solved automatically. The effective action for these equations is given by the first line of The equations of motion (56) (57) and (58) take precisely the same form as when one reduces five-dimensional vector-multiplets over a timelike dimension in static, purely electric backgrounds. Both isotropic and non-isotropic solutions have been found in this case, and can be shown to lift to electrically charged extremal black holes [47] and non-extremal black holes respectively [49]. In order to obtain non-isotropic solutions one must modify (34), the part of our ansatz that relates ∂ µ q and ∂ µq , by a universal 'non-extremality' factor. In this case the three-dimensional spacetime metric is no longer flat but conformally flat. The machinery for producing these non-isotropic solutions takes a slightly different form than in the isotropic case, and for that reason we will not consider these solutions in this paper. We remark that is possible to use the techniques established in [49] to produce non-isotropic solutions which lift to non-extremal black holes in four-dimensions, which we have found for particular models, but we leave a detailed discussion of this topic to future work.
In order to produce isotropic solutions to these equations of motion in flat three-dimensional backgrounds we must again impose the ansatz It is clear by inspection that in this case all equations of motion reduce to the Laplace equation for the dual coordinates ∆q a = 0 .
In this case condition (55) places one constraint on the integration constants of q a .
The formula for the ADM mass for is given by Since e φ → 1 at spatial infinity we can write this as Using the fact that the NUT charge vanishes q a Ω ab ∂ µ q b = 0, which implies that q a Ω ab ∂ µ q b = 0, we can write this as These extremal black hole solutions therefore satisfy the BPS bound.

Examples of extremal black hole solutions
We will now consider explicit solutions to the equations of motion in static backgrounds. We impose the ansätze (55) and (59)  We will first consider a class of extremal black hole solutions of the STU model that are obtained by taking the static limit of the rotating solutions discussed in the previous section. We will then present axion-free solutions to a wider class of models which have prepotentials of the form This class of models includes those that have a 'very special' form, where f (Y 1 , . . . , Y n ) is a homogeneous cubic polynomial. Such models can which be obtained by the dimensional reduction of five-dimensional theories. While axionfree solutions for very special prepotentials are well known [52], our derivation shows that to obtain solutions it is enough to assume that f is homogeneous, and so we can obtain axion-free solutions for a larger class of prepotentials.
We end by giving explicit solutions to models where f = ST U + aU 3 . This is a deformation of the ST U -model which is still of the very special form, but the target space is no longer symmetric. The model with a = 1 3 corresponds to a particular Calabi-Yau compactification and its heterotic dual [80,79].

The ST U model
We first consider the static limit of the rotating solutions found in the previous section. This will give us extremal black hole solutions to the STU model.
Taking the static limit amounts to setting α → 0 and imposing the constraint where the harmonic functions are given by The asymptotic integration constants h I , h I satisfy the two constraints Like in the case for rotating solutions, using the expression (28) we can write u I , v I explicitly in terms of harmonic functions by The original four-dimensional scalar fields are given by where Y I are given in terms of harmonic functions through (53) and (61). Again, the gauge fields are given by the expressions (38), (39).
The above extremal black hole solutions of the STU model are spherically symmetric as they were obtained by taking the static limit of axially-symmetric rotating solutions, but this need not be the case in general. If we do not impose any symmetry constraints on spacetime then we will obtain the same expressions for the four-dimensional fields, but with the harmonic functions which are completely general. Multi-centered black hole solutions with centers are at x α correspond to the choice

Models of the form
A class of models for which we can find explicit extremal black hole solutions are those where the prepotential takes the form where f is real when evaluated on real fields. Since F is a homogeneous function of degree 2 it follows that f is a homogeneous function of degree 3. If f is in is of the 'very special' form which derives from five-dimensional supergravity by reduction.
We will consider a restricted set of solutions that are characterised by the requirement that Y A are purely imaginary and Y 0 is purely real, which implies that the four-dimensional scalars Z A = Y A /Y 0 are purely imaginary. For very special prepotentials, where the real part of Z A corresponds to a fivedimensional gauge potential, this means that such solutions are 'axion-free.' Therefore we , we impose this condition in the following, and 'axion-free solutions' are characterized by the consistent reality condition on the fields which impose that Y 0 and F A are purely real while Y A and F 0 are purely imaginary.
In terms of real variables this corresponds to imposing that which defines a particular submanifold of the scalar manifold. In terms of dual With the above assumptions we can write Y I , F I in terms of the dual real coordinates as where and f A = ∂f ∂Y A . Using (16) and (28) we obtain expressions for the KK-scalar and Hesse potential (evaluated on axion-free configurations) The real parts of Y I , F I can be read off from (64) as This amounts to solving the generalised stabilisation equations, and is the reason why we can find solutions explicitly in closed form.
Solutions to these models are given in terms of harmonic functions by The harmonic functions are given by The asymptotic integration constants h I , h I must satisfy only one constraint We can write v 0 , u A explicitly in terms of harmonic functions by The original four-dimensional scalar fields are given by which can be written in terms of harmonic functions using (64) and (65). The gauge fields are given by the expressions (38), (39).

The ST U + aU 3 model
We now turn to a specific one-parameter family of models of the form F = , which are characterised by the prepotential This is a deformation of the ST U -model where the target space is no longer symmetric. Specialising to solutions with x 1 = x 2 = x 3 = 0 and y 0 = 0 we have where Solutions are given in terms of harmonic functions by where the harmonic functions are again defined to be with H A = H 0 = 0. The asymptotic integration constants h I , h I must satisfy the constraint We can write v 0 , u A explicitly in terms of harmonic functions by The original four-dimensional scalar fields can be determined through the expressions which one can write explicitly in terms of harmonic functions using (66) and (67). The gauge fields are given by the expressions (38),(39).

Field rotations and non-BPS solutions
Four-dimensional extremal non-BPS have been studied in the past [69,70,71], and more recently there has been increased interest in this topic, starting from [72,73,74]. As in the five-dimensional case [47], the ansatz ∂ µ q a = ±∂ µq a with a universal sign does not necessarily exhaust all solutions. To obtain further solutions we can adapt the observation that new solutions can be generated by flipping signs of charges [70], or, more generally, by 'rotating charges' [63,64]. BPS solutions correspond to particular combinations of signs, while other choices lead to non-BPS solutions.
As we have seen above the ansatz ∂ µ q a = ±∂ µq a leads to BPS solutions.
For static solutions we can use the same generalization of the ansatz as in fivedimensions [47] and introduce a constant field rotation matrix This is the analogue of 'rotating charges' in our framework. By inspection of the field equations, we find that this ansatz only works if the following compatibility condition between the scalar metric and the field rotation matrix holds If this condition is satisfied, then the solution for the dual scalar fields q a ,q a is again given by harmonic functions, but now the harmonic functions for q a are related to those forq a through the constant matrix R b a , which is the transposed of the inverse of R a b : Equivalently, the relations (38) and (39) between four-dimensional scalars and gauge fields are modified by the presence of this matrix. Decomposing the field rotation matrix R T,−1 into blocks the expressions for the gauge fields become In particular, the electric and magnetic charges appear rotated relative to the solutions of the scalar fields. Note that in general not only the charges but also the asymptotic behaviour of solutions changes [75]. This is necessary in order to avoid introducing naked singularities. 13 The presence of a non-trivial field rotation matrix also modifies the ADM mass (60): This makes it manifest that such solutions are not BPS. Note that we saw above that the R = ±Id leads precisely to the relation between four-dimensional scalars and gauge fields which is implied by the BPS condition.
Since a field rotation matrix only provides a solution if the compatibility condition (69) is satisfied, it is in general not clear that non-BPS solutions can be obtained by this ansatz. For symmetric spaces non-BPS solutions can be obtained in a systematic way using group-theoretical methods [36,37]. For non-symmetric target spaces these methods do not apply, and therefore it is interesting to ask under which conditions one can guarantee the existence of a non-trivial field rotation matrix which satisfies (69).
Geometrically, this is equivalent to the problem of identifying totally geodesic, totally isotropic submanifolds of the scalar target space. We have seen that for c-map spaces there is a universal solution, given by the ansatz ∂ µ q a = ±∂ µq a , which corresponds to BPS solutions. Finding non-BPS solutions amounts to 13 We thank the authors of [75] for bringing this to our attention.
finding further such submanifolds, which correspond to the non-BPS branches that one can identify in symmetric target spaces by group theoretical methods.
In the following section we establish that a non-trivial field rotation matrix exists for non-axionic solutions of models with a prepotential of the form where f is real when evaluated on real fields, i.e. for the class of examples considered above. Before we turn to the details, we remark that we do not only need to impose a condition on the model (i.e. on the form of the prepotential), but also on the field configurations, by restricting to axion-free solutions. This corresponds to restricting to lower-dimensional submanifolds of the scalar manifold. If no such restriction is imposed, the compatibility condition (69) implies that the field rotation matrix acts by an isometry. Requiring the existence of such an isometry imposes a condition on the prepotential. By restricting to field configurations which are axion free, the compatibility condition (69) becomes less restrictive and we can establish the existence of a field rotation matrix under much milder assumptions on the form of the prepotential.
But the resulting totally geodesic, totally isotropic submanifold corresponding to the axion-free non-BPS solution is of lower dimension than the submanifold corresponding to BPS solutions, which has maximal dimension. It would be interesting to clarify whether this is a generic feature of non-BPS solutions in models with non-symmetric target spaces.
Finally, we mention that in the rotating case one cannot simply adapt the isotropic ansatz in the same way when a field rotation matrix is available, as this no longer produces a solution to the equations of motion. In order to produce non-BPS rotating solutions one needs to relax the condition that threedimensional metric is flat. We will not consider such solutions in this paper, and leave the investigation of such solutions to future work.

Non-BPS solutions to
For models with prepotentials of the form , where f is real when evaluated on real fields, there always exists a non-trivial field rotation matrix for solutions satisfying the conditions (62). For the remainder of this section we will focus on the specific case where n = 3, but the solutions can be extended to arbitrary n ≥ 1 without loss of generality.
To see why a field rotation matrix always exists for this class of models we must analyze the matrixH ab in some detail. Firstly, one observes that the conditions (62) imply that the matrixH ab decomposes intõ where a * represents a possible non-zero entry. To see why this is the case, consider, for example, the matrix elementH 10 . Let us denote by ♯ the restriction of solutions to (62). We can writeH 10 as In the second line we used that the variable x 0 does not enter into the axion-free condition ♯, which amounts to setting other variables to constant (zero) values.
Therefore we can take the derivative with respect to x 0 after imposing the axion free condition ♯. In the third line we used the fact that ∂H ∂x 1 = −v1 H . This is valid irrespective of the condition ♯ by definition of the dual coordinates. The same argument is true for any matrix element containing one index in {0, 5, 6, 7} and one index in {1, 2, 3, 4}.
When expressed in terms of u I , v I , the axion free ansatz (62) implies that v 1 = v 2 = v 3 = u 0 = 0. Consequently, the corresponding harmonic functions vanish H 1 = H 2 = H 3 = H 0 = 0, and the central 4 × 4 block appearing in (72) completely decouples from the equations of motion, and is of no relevance to the remaining discussion.
Actually, the matrixH ab decomposes even further. Using the formula for the Hesse potential (81) for this class of solution, which is derived in appendix A.2 one observes thatH ab takes the more restrictive form where the entries in the bottom-right block depend only on y 1 , y 2 , y 3 .
For such solutions, these modes always admit a non-trivial field rotation matrix of the form One can therefore find non-BPS solutions to these models generically.

Non-BPS solutions to ST U + aU 3 model
Since this model falls into the category of it admits the nontrivial field rotation matrix given by (74), and we can obtain non-BPS solutions.
The non-BPS solutions are given explicitly by where the harmonic functions are again given by with H A = H 0 = 0. The asymptotic integration constants h I , h I satisfy the constraint 4h 0 (h 1 h 2 h 3 + a(h 1 ) 3 ) = 1 .
We can write v 0 , u A explicitly in terms of harmonic functions by The original four-dimensional scalar fields can be determined through the expressions which one can write explicitly in terms of harmonic functions using (66) and (75). The expressions for the gauge fields remain unchanged, and are given by (38), (39).

Conclusions and Outlook
In this paper we have shown how four-dimensional N = 2 vector multiplets coupled to supergravity can be described in terms of a real formulation of special Kähler geometry using the gauge equivalence with conformal supergravity.
Key technical points, which allowed us to preserve symplectic covariance, were to avoid U (1) gauge fixing, and the use of the degenerate metric obtained by integrating out the auxiliary U (1) gauge field. Geometrically this corresponds to working on the Sasakian S or the conical affine special Kähler manifold N , and to use a horizontal lift for the metric. We expect that this formulation will be useful for studying non-holomorphic corrections. While the canonical version ∂ µ q a = ±∂ µq a of the ansatz always works and leads to BPS solutions, non-BPS solutions can be obtained if a non-trivial field rotation matrix exists, which must satisfy a compatibility condition with the metric. For non-symmetric target spaces the existence of such a matrix is nontrivial, but we were able to show that it exists for axion-free solutions for a class of prepotentials, which contains the very special ones as a subclass. An interesting future direction is to develop the understanding of non-BPS solutions for non-symmetric target spaces. Since symmetric spaces are contained in our formalism as special cases, one promising strategy is to translate the grouptheoretical characterisations of BPS and non-BPS solutions into geometrical properties of totally geodesic, totally isotropic submanifolds and then to investigate whether these conditions have natural generalizations for non-symmetric spaces.
Another direction is the generalization to non-extremal static black holes, which for the five-dimensional case was discussed in [49], and, more recently, in [76]. Deforming extremal into non-extremal solutions corresponds to deforming isotropic into non-isotropic submanifolds. It is currently not clear to us to which extent this can be done in a universal way. However, specific examples suggest that our method can be generalized, and we plan to report on this in a future publication. Non-extremal four-dimensional black holes in N = 2 supergravity have been recently discussed in [75] from a different though related point of view. For N = 4 supergravity the full class of stationary point-like solutions is known [77].
We have also shown how rotating solutions can be obtained, and recovered the known rotating supersymmetric solutions. In this case the use of field rotation matrices to produce non-BPS solutions requires to generalize the ansatz and to admit a curved three-dimensional base space. Moreover, these solutions have naked singularities, and making non-singular will also require to go beyond the isotropic ansatz considered in the second part of this paper.

A Hesse potentials
A.1 Hesse potential for STU model In this section we derive the Hesse potential for the STU model. Due to the relation between the Hesse potential and BPS black hole entropy [14], this is equivalent to solving the attractor equations. However, the relation between Hesse potential and prepotential is 'off-shell', and does not require to impose a particular background solution. Therefore we find it instructive to present the derivation in a form where this is manifest. Technically we closely follow [52], but instead of charges and horizon values of fields we use fields without imposing supersymmetry or any of the field equations.
The STU model in special complex coordinates is characterised by the holomorphic prepotential Introducing the inhomogeneous coordinates Z A = Y A /Y 0 one can write the Kähler potential for the STU model as e −K = −i Y IF I − F IȲ I = 8Y 0Ȳ 0 Im(Z 1 )Im(Z 2 )Im(Z 3 ) .
Our strategy will be to write the individual fields Y 0 , Z 1 , Z 2 , Z 3 in terms of x I = Re(Y I ) and y I = Re(F I ).
Firstly, by direct calculations one can show that Combining these three expressions one gets the quadratic equation for Z 1 : where y.x = y I x I . Solving this we find an expression for Z 1 purely in terms of x I , y I : where W = −(y.x) 2 + 4y 1 x 1 y 2 x 2 + 4y 1 x 1 y 3 x 3 + 4y 2 x 2 y 3 x 3 + 4x 0 y 1 y 2 y 3 − 4y 0 x 1 x 2 x 3 .
We now restrict ourselves to physically relevant configurations, where the RHS is strictly positive. Since H = − 1 2 e −K we can write the Hesse potential explicitly in terms of x I , y I as H(x, y) = −2 − (y.x) 2 + 4y 1 x 1 y 2 x 2 + 4y 1 x 1 y 3 x 3 + 4y 2 x 2 y 3 x 3 + 4x 0 y 1 y 2 y 3 − 4y 0 x 1 x 2 x 3 1/2 . (77) One can use a similar procedure to determine the Hesse potential in terms of the imaginary parts of Y I , F I , which we denote by u I = Im(Y I ) and v I = Im(F I ). What one obtains is precisely the same expression: The reason why we obtain the same result is that the Hesse potential is independent of the phase of Y I , i.e. it is invariant under U (1) transformations Y I → e iα Y I . The imaginary parts of Y I , F I are simply the real parts of e −iπ/2 Y I , e −iπ/2 F I , which describe the same Hesse potential.
A.2 Hesse potential for models of form F = f (Y 1 ,...,Y n ) Y 0 We now extend the previous discussion to models with a prepotential of the form Since F is a homogeneous function of degree 2 it follows that f is homogeneous function of degree 3. In this case it is not possible to obtain an expression for the Hesse potential in closed form. However, one can still show that the Hessian metricH ab takes the from (73) when restricting to axion-free field configurations (62). The part of the Hessian metric relevant for this subspace can consistently be obtained by setting half of the variables of the Hesse potential to zero, and for this truncated Hesse potential we can obtain an explicit expression.
Recall the definition of x I , y I and u I , v I : We will now impose the conditions (62), which restrict us to the particular class of solutions for which Y A are purely imaginary, Y 0 is purely real and F 0 is purely imaginary. In this case the fields x I , y I can be given explicitly in terms x 0 x 1 . . .
One must choose the negative sign in the expression for λ in order to ensure that the Hesse potential is strictly negative. The Kähler potential can be written as and since e −K = −2H we have the following explicit expression for the Hesse potential in terms of u I , v I : We would now like to find an equivalent expression for the Hesse potential in terms of x I , y I . Here we cannot use the same trick of making a U (1) rotation as in the STU model, since imposing the conditions (62) implicitly brakes the U (1) invariance of the system. Geometrically the condition selects a lower dimensional hypersurface which no longer has this U (1) isometry.
Finding an explicit expression for the Hesse potential in terms of x I , y I would involve inverting the relations (79), which in general cannot be calculated in closed form. However, we will now show that the Hesse potential can be consistently restricted to the subspace of axion-free solutions, where it separates into two distinct factors: H(x, y) = √ x 0 h(y 1 , . . . , y n ) , where h is some homogeneous function of degree 3/2. This property is crucial in demonstrating the existence of non-BPS solutions to such models.
Firstly, on the subspace of axion-free solutions, half of the variables x I , y I are zero. We denote the restricted Hesse potential by H(x, y) = H(x 0 , y 1 , . . . , y n ) .
Next, observe that for axion-free field configurations Taking partial derivatives with respect to x 0 we find ∂H ∂x 0 = 4v 0 + 4x 0 ∂v 0 ∂x 0 .
Note that it does not make a difference whether we impose the axion-free condition before or after taking derivatives with respect to x 0 , because the axion-free condition does not involve this variable. But we know from (5) that and, hence, for some specific, but as yet undetermined, function h. The restriction of the Hesse potential to axion-free configurations is therefore given by (81). This allows us to determine components ofH ab which we need to go from (72) to (73).