Duality invariance in Fayet-Iliopoulos gauged supergravity

We propose a geometric method to study the residual symmetries in $N=2$, $d=4$ $\text{U}(1)$ Fayet-Iliopoulos (FI) gauged supergravity. It essentially involves the stabilization of the symplectic vector of gauge couplings (FI parameters) under the action of the U-duality symmetry of the ungauged theory. In particular we are interested in those transformations that act non-trivially on the solutions and produce scalar hair and dyonic black holes from a given seed. We illustrate the procedure for finding this group in general and then show how it works in some specific models. For the prepotential $F=-iX^0X^1$, we use our method to add one more parameter to the rotating Chow-Comp\`ere solution, representing scalar hair.


Introduction
Duality transformations have played, and continue to play, an important role in fundamental developments in string theory, supergravity, quantum field theory as well as in the physics of black holes. Perhaps the most relevant example for this is the fact that the five known string theories are actually all related by a web of dualities, and correspond just to perturbative expansions of a single underlying theory about a distinct point in the moduli space of quantum vacua, cf. e.g. [1] for a review. This web contains in particular weak/strong coupling dualities, of which the celebrated AdS/CFT correspondence [2] is another famous example.
Duality transformations have been instrumental also in the construction of black hole solutions in string theory. Typically one reduces a higher-dimensional theory (in presence of Killing directions) to lower dimensions, in particular to d = 3, where all vector fields can be dualized to become scalars. One gets then three-dimensional gravity coupled to a nonlinear sigma model, and employs the global symmetries of the latter to obtain new black holes from a given seed. This technique was used by Cvetič and Youm [3] to construct the most general rotating five-dimensional black hole solution to toroidally compactified heterotic string theory, specified by 27 charges, two rotational parameters and the ADM mass. In a similar way, Chow and Compère [4] obtained the most general asymptotically flat, stationary, rotating, nonextremal, dyonic black hole of four-dimensional N = 2 supergravity coupled to 3 vector multiplets (the socalled stu model). It generates through U-dualities the most general asymptotically flat, stationary black hole of N = 8 supergravity.
Note that this typical structure of getting, after a Kaluza-Klein reduction, threedimensional gravity coupled to a nonlinear sigma model, is also crucial to prove full integrability in some particular cases, cf. e.g. [5,6].
When (part of the) global symmetries of some given supergravity theory are gauged, as it typically happens in AdS supergravity, the sigma model target space isometries are generically broken by the presence of a scalar potential, so that the powerful solution-generating techniques described above seem to break down. An instructive example is the timelike dimensional reduction of four-dimensional Einstein-Maxwell gravity down to three dimensions, which gives Euclidean gravity coupled to an SU(2, 1)/S(U(1, 1) × U(1)) sigma model [7,8]. Adding a cosmological constant to the Einstein-Maxwell theory leads to a scalar potential in three dimensions, that breaks three of the eight SU(2, 1) generators, corresponding to the generalized Ehlers and the two Harrison transformations. This leaves merely a semidirect product of a one-dimensional Heisenberg group and a translation group R 2 as residual symmetry [9]. Although in this concrete example the surviving symmetries cannot be used to generate new solutions from known ones, they may nevertheless be useful in more general settings.
The aim of this paper is thus to provide a systematical and thorough investigation of the residual symmetries in N = 2, d = 4 U(1) Fayet-Iliopoulos (FI) gauged supergravity, elaborating on [10], where a particular stu model was considered. To this end, we shall use a geometric method, whose underlying idea is the following: The on-shell global symmetry group of the ungauged theory is called U-duality, and consists of the isometries of the special Kähler non-linear sigma model that act linearly also on the field strengths via the symplectic embedding [8]. For purely electric gaugings, the scalar potential generically spoils this invariance, but allowing also for dyonic gaugings one can recover the whole U-duality invariance, at the price of changing the vector of gauge couplings and so the physical theory. We will call this group U fi , that stands for fake internal symmetry group, which acts on a solution by mapping it to other solutions of other theories. Given U fi , we fix a generic choice of the coupling constants G. The true internal symmetry group U i of the gauged supergravity theory is then S G , the stabilizer of G under the action of U fi 1 . The remainder of this paper is organized as follows: In the next section, we briefly review the theory we are interested in, namely N = 2, d = 4 U(1) FI-gauged supergravity, and explain more in detail the general idea outlined above. In section 3 we explicitely determine the residual symmetry group for four different prepotentials that are frequently used, but we stress that our method is general, and can be applied to arbitrary prepotentials and extended to N = 4 and N = 8 gauged supergravity theories as well. After that, in section 4, it is shown how to apply the residual symmetries to generate new black hole solutions from a given seed in each of the four cases. In section 5 we comment on a possible extension of our work to include also gauged hypermultiplets. Section 6 contains our conclusions and some final remarks. Some supplementary material is deferred to two appendices. The bosonic sector of N = 2, d = 4 supergravity coupled to n V vector multiplets consists of the vierbein e a µ , n V + 1 vector fields A Λ µ with Λ = 0, . . . n V (the graviphoton plus n V other fields from the vector multiplets), and n V complex scalar fields z i (i = 1, . . . , n V ). The latter parametrize an n V -dimensional special Kähler manifold, i.e., a Kähler-Hodge manifold, with Kähler metric g i (z,z), which is the base of a symplectic bundle with the covariantly holomorphic sections 2 where K is the Kähler potential. V obeys the constraint Alternatively one can introduce the explicitly holomorphic sections of a different symplectic bundle, In appropriate symplectic frames it is possible to choose a homogeneous function F (X) of second degree, called prepotential, such that F Λ = ∂ Λ F . In terms of the sections v the constraint (2.2) becomes The couplings of the vector fields to the scalars are determined by the (n V +1)×(n V +1) period matrix N , defined by the relations If the theory is defined in a frame in which a prepotential exists, N can be obtained from we have the important relation between the symplectic sections and their derivatives, with The bosonic Lagrangian reads In the case of dyonic U(1) FI-gauging, the scalar potential has the form [12] where L = G, V , and G = (g Λ , g Λ ) t denotes the symplectic vector of gauge couplings (FI parameters).

Fake internal symmetries, stabilization and solutions
The kinetic part of (2.10) corresponds to the action of the ungauged theory, whose on-shell global symmetry group is called U-duality, consisting of the isometries of the non-linear sigma model that act linearly also on the field strengths via the symplectic embedding [8]. For purely electric gaugings, the scalar potential generically spoils this invariance, but, as is clear from (2.11), for dyonic gauging one recovers the whole Uduality invariance, at the price of changing the vector of gauge couplings and so the physical theory. We will call this group U fi , that stands for fake internal symmetry group 4 . The action of U fi on a solution is the mapping to other solutions of other theories, in the same way in which some elements of the symplectic group map solutions of theories with different prepotential into each other [12], cf. e.g. (B.2), (B.3). Given U fi , we fix a choice of the coupling constants G and, at least at the beginning, we suppose that they are generic. We want to underline that for abelian dyonic gaugings, the Maxwell equations remain homogeneous and so the action (2.10) doesn't have topological terms [13].
The true internal symmetry group U i of the gauged supergravity theory is S G , the stabilizer of G under the action of U fi , up to possible U(1) factors. This is obvious from the definition of the stabilizer, which means that we impose to stay in the same theory, and this restricts of course the group of internal symmetries. By acting with S ∈ S G on a given seed solution (V, G, F µν ) 5 of the equations of motion, we can generate another configuration via the map (V, G, F µν ) → (Ṽ,G,F µν ) := (SV, SG, SF µν ) = (SV, G, SF µν ) . (2.13) The transformed fields solve the field equations by construction 6 . In general, the scalars transform nonlinearly under the corresponding isometry, the field strengths are rotated and the metric is functionally invariant. 4 When the special Kähler manifold is symmetric we define the Lie algebra u fi of U fi through the equations (A.3). The corresponding definition for nonsymmetric special Kähler manifolds requires more care. 5 Actually we should write (V, G, F µν , g µν ), but since S G does not act on the metric, we shall suppress the dependence on g µν . 6 As is clear from the formalism introduced in [12], the application of S ∈ S G on a static solution of the BPS flow preserves the same amount of supersymmetry as the original configuration. In the rotating case, the same is true if one considers electric gaugings only [14].
Technically, in order to determine S G , it is simpler to work with the corresponding algebra s G = {a ∈ u fi | aG = 0} . (2.14) There are some cases in which U i strictly contains S G , and this depends on some particular symmetric structures of the model under consideration. Typically, this happens because the symmetry of the model allows to act with some symplectic matrices in a more general way than (2.13), leaving nevertheless the theory invariant.

Stabilization and symmetries for some prepotentials
Now we want to apply these techniques to some specific prepotentials. Each of them exhibits different peculiar features related to the geometry of the underlying special Kähler manifold, namely to the symplectic embedding of the isometry group of the non-linear sigma model (cf. app. B).

Prepotential
This prepotential encodes a particular special Kähler structure on the symmetric manifold SU(1, 1)/U(1). The symplectic section is V = (X 0 , X 1 , −iX 1 , −iX 0 ) t , and we fix the couplings in a completely electric frame, G = (0, 0, g 0 , g 1 ) t . The solution to (A.3) defines the algebra u fi , to be the U-duality su(1, 1) plus a u(1), generated by t 2 , which acts trivially on the z i , as we will see shortly. From the stability equation (2.14) one finds that s G is generated by On the other hand, the U(1) generated by t 2 is given by and it transforms the section V according to The projective special Kähler coordinates are thus insensible to its action. The matrix M defined in (2.7) transforms as One can thus act with T α on F µν only, leaving the equations of motion still invariant. T α is an example for a 'field rotation matrix' that is commonly used to generate non-BPS solutions, a technique first introduced in [15,16] and subsequently applied to gauged supergravity in [17,18]. In conclusion, the internal symmetry group of this model is with the two U(1) factors identified respectively with S and T α .

Prepotential
, describes a special Kähler structure on the symmetric manifolds SU(1, n V )/(U(1) × SU(n V )). The symplectic section reads Due to the linearity of V in the coordinates X Λ , one can easily construct the oneparameter subgroup , under which the section V transforms as we can add a new parameter to all the solutions of this model by acting with L α on F µν only. The stability equation is slightly more involved. Notice that the case with only one vector multiplet is symplectically equivalent to F = −iX 0 X 1 , and thus the results for n V = 1 can be obtained from the previous subsection by an appropriate symplectic rotation, cf. app. B. Let us discuss the general case of n V = n vector multiplets. Eq. (A.3) defining the algebra u fi is equivalent to These equations define an embedding of U(1, n) into Sp(2n + 2, R). To see this, let so ηB is symmetric. This suggests an embedding for any real α = 0. This is indeed an injective Lie algebra morphism, and its image consists of the elements of sp(2n + 2, R) which solve (A.3) with F Λ = i α η ΛΣ X Σ . In particular, (3.9) selects ι 2 . A basis for u(1, n) is given by the matrices where A a are a basis for the space of (n + 1) × (n + 1) real matrices A such that ηA is antisymmetric, and B k generate the space of (n + 1) × (n + 1) real matrices B such that ηB is symmetric, with B 0 = I, the identity matrix. The embedding extends obviously to the group level via the exponential map, and, in particular, notice that Let us now consider the symmetry group S G . If we set (3.14) with g = (g 1 , . . . , g n ), then we see that the invariance of G is defined by the equations which define a maximal compact subgroup 7 U(n) of U(1, n). To see this, let us first put 8ĝ and define Λ g ∈ SO(1, n) by Thus, A (or ηB t ) has g in the cokernel if and only if Λ g AΛ −1 g (or Λ g ηB t Λ −1 g ) has (ĝ, 0) in the cokernel. From this we immediately get that s G is generated by the elements of u(1, n) of the form where z ∈ u(1, n) has vanishing first row and first column. Thus, z g ∈ U(n). This provides also a way to realize an explicit construction of the group elements of S G . One can choose e.g. a generalized Gell-Mann basis [19] for su(n), add the identity matrix I n and then embed the basis into u(1, n) by adding a first row and column of zeros. If we call {z I } n 2 −1 I=0 such a basis for the compact subalgebra u(n) of su(1, n), then is a basis for s G 0 , where G 0 ≡ (0, 0,ĝ, 0). Then we can explicitly construct the group elements by means of the Euler construction of S G 0 9 , as in [19,21]. Finally we have For practical purposes we can take Λ g defined by whose inverse is obtained by the replacement g → − g.
Let us focus on the first nontrivial case SU(1, 2)/(U(1) × SU (2)). We fix the couplings in a completely electric frame, G = (0, 0, 0, g 0 , g 1 , g 2 ) t . A basis for u(2) (relative to the vector G 0 = (0, 0,ĝ, 0)) is which, by means of ι 2 , defines the basis of s G 0 Note that from which we immediately get the expression for a generic element of S G 0 , we get for a generic element of S G In order to have even more manageable expressions for the matrices, it may be convenient to change to the basis R µ defined by

Prepotential
This prepotential describes a special Kähler structure on the symmetric manifold (SU(1, 1)/U(1)) 3 , the well-known stu model. This is symplectically equivalent to the model with F = −2i(X 0 X 1 X 2 X 3 ) 1/2 , for which supersymmetric black holes with purely electric gaugings are known analytically [22]. After a symplectic transformation to F = −X 1 X 2 X 3 /X 0 , the electric gaugings considered in [22] become G = (0, g 1 , g 2 , g 3 , g 0 , 0, 0, 0) t , so we shall concentrate on this case in what follows. The symplectic section reads Let us now look at the solutions of (A.3). To this end, we define Since the lhs is a homogeneous polynomial of degree 6 in (X 0 , X 1 , X 2 , X 3 ), the coefficients of each monomial must be zero. The simplest way to get the general solutions is then to look at the powers of X 0 . The possible powers of X 0 in p S ≡ X X XSX X X, p R ≡ F F F RF F F and p Q ≡ X X XQ t F F F are (6, 5, 4), (2, 1, 0) and (4, 3, 2) respectively. Since S and R are symmetric, p S and p R can vanish only if S and R are zero. Thus, we are left with the following three possibilities: 1. R = 0 and p Q cancels p S . The only common power for X 0 is 4, so we have to take matrices which generate only this power and equal degrees for the remaining variables. A quick inspection gives the solutions 10 (3.30) 2. S = 0 and p Q cancels p R . The only common power for X 0 is 2, so we have to take matrices generating only this and equal degrees for the remaining variables. The solution is 3. R = S = 0 and Q satisfies p Q = 0. This implies that Q must be diagonal and that the space of such solutions is 3-dimensional. The simplest way to fix a basis of this space is to choose In this way the nine matrices S, T and U generate the group U fi = (SL(2, R)) 3 . In order to determine the symmetry algebra s G we have to consider the equation (using the same notation as in the previous subsection) whose general solution is given by for arbitrary x, z ∈ R. A convenient basis is which defines a two-dimensional abelian algebra. Notice that so that the algebra is compact (and thus defines the group U(1) × U(1)) if and only if g 0 g 1 g 2 g 3 < 0. One can easily verify that, unfortunately, none of these continuous symmetries survives for the truncation to the t 3 model [23,24] with prepotential F = −(X 1 ) 3 /X 0 . It is worth noting that a particular situation arises for g 1 = g 2 = g 3 = −g 0 ≡ g. As was shown in [10], there is an enhancement of the internal symmetry group in this case. This happens because the scalar potential V can be written in terms of fundamental objects that define the nonlinear sigma model of the non-homogeneous projective coordinates z i = x i + iy i [8,10], namely (3.36) In fact, the transformation property of M i , implies the invariance of the potential only if T T t = 1. Going back to the symplectic formalism we see that this condition is equivalent to require for the symmetry group to be orthogonal, which, in terms of the elements of u fi amounts to consider just the subspace of antisymmetric matrices. Thus, the symmetry algebra is generated by while the subalgebra leaving G fixed is generated by W 2 − W 1 and W 3 − W 2 . The full symmetry group is therefore an extension U i = U(1) 3 of S G = U(1) 2 .

Prepotential
The base manifold for this prepotential is neither symmetric nor homogeneous and it has been studied in [25]. The symplectic section is given by V = (X Λ , F Λ ) t , with (3.39) The solution to (A.3) is obtained by proceeding exactly like in the previous subsection. After introducing the vectors we reduce the equations to a polynomial identity, and looking at the coefficients we get a five-dimensional space of solutions generated by the symplectic matrices

(3.41)
A direct comparison with the results of [25] shows that this algebra strictly contains the U-duality algebra. This is due to the fact that the group of symmetries of the scalar potential is larger than the symmetry group of the whole Lagrangian. Indeed the generator D 2 does not leave the metric invariant. Thus, the U-duality group is generated by the algebra Notice that the S i are nilpotent of order 4 for i = 1 and order 2 for i = 2, 3. They are indeed eigenmatrices for the adjoint action of D 1 , all with eigenvalue −2. The stability equation (2.14) has a nontrivial solution only if A = −g 1 g 2 /(g 3 ) 2 . With this choice for A one gets a one-dimensional algebra s G generated by It is nilpotent of order 4 so that U i = S G is a unipotent group of order 4. It is worthwhile to note that for g 1 = g 2 = g 3 one gets A = −1, which is the physically most interesting case, since the corresponding prepotential arises in the context of type IIA string theory compactifed on Calabi-Yau manifolds [26].

Scalar hair and dyonic solutions
We shall now use the results of the previous section in order to generate new supergravity solutions from a given seed. The transformations in U i add new parameters to a given solution and leave not only the equations of motion invariant, but also some potential first-order flow equations (if these are satisfied by the seed). The transformed field configuration preserves thus the same amount of supersymmetry as the one from which we started.
As was stressed in [10], the latter statement is not true in the stu model for the additional U(1) that arises for equal couplings, whose action generically leads to a non-BPS solution. The same story holds also in the quadratic models for T α and L α , due to the properties (3.5) and (3.8) [18].
In what follows we will consider several relevant examples for some well-studied prepotentials, but there is no obstacle to extending this method to other solutions and prepotentials as well. We underline that in the static case, owing to the existence of the black hole potential V BH [27,28], one can directly rotate the charges Q instead of the field strengths F µν .

Prepotential
For this prepotential, we have U i = U(1) 2 , whose action on the static and magnetic BPS seed solution of [22] is Using the results of section 3.1 and the constraints on the seed parameters (cf. [22]), one getsQ = (p 0 cos α, p 1 cos α, −p 1 sin α, −p 0 sin α) t , The parameter β does not modify the supersymmetry of the solution; for α = 0 the new configuration satisfies again the BPS flow equations of [12,22]. For α = 0 one gets a solution that still obeys a first-order flow, but this time a non-BPS one [18], driven by the fake superpotential where U (r) and ψ(r) are functions appearing in the metric and L was defined in section 2.1. The first-order equations following from (4.3) imply the equations of motion provided the Dirac-type charge quantization condition holds [18]. From (4.2) we see that for α = 0 one generates a dyonic solution from a purely magnetic one, while β adds scalar hair to the seed. Note that this result was first obtained in [10]. As another example for the action of U i we consider the Chow-Compère solution [29], that solves the equations of motion following from the Lagrangian (2.12) of [29], which is obtained from (2.10) by setting and redefining 11 (4.8) 11 We assume g 0 /g 1 > 0.
The dyonic rotating black hole solution of [29] is given by where R(r) = r 2 − 2mr + a 2 + g 2 r 1 r 2 (r 1 r 2 + a 2 ) , W (r, u) = r 1 r 2 + u 1 u 2 , r 1,2 = r + ∆r 1,2 , u 1,2 = u + ∆u 1,2 , and ∆r 1,2 , ∆u 1,2 are constants defined by Below we shall also use the linear combinations The complex scalar field has the very simple form while the gauge fields and their duals read where the three-dimensional electromagnetic scalars are The solution is thus specified by the 7 parameters m, n, a, γ 1,2 and δ 1,2 that are related to the mass, NUT charge, angular momentum, two electric and two magnetic charges. Notice that a similar class of rotating black holes containing one parameter less was constructed in [30].
Let us now consider the action of S defined in (3.2). For the transformed scalar we getz (4.18) Note that the quantities Σ ∆r and Σ ∆u defined in (4.12) remain invariant under (4.18), while ∆ ∆r and ∆ ∆u transform as ∆ ∆r ∆ ∆u = cos 2β − sin 2β sin 2β cos 2β The transformed gauge fields can be easily inferred from (4.20) In conclusion, S adds one more parameter β to the solution of [29]. Under the action of T α (cf. (3.3)) the scalar z does not change. It turns out that the new gauge fields can again be written in the form (4.14), but with the three-dimensional electromagnetic scalars replaced by where we used the definitionŝ g = g 2 0 − g 2 1 − g 2 2 , f (g 1 , g 2 , z 1 , z 2 ) = g 0 g 1 + g 2 0 z 1 + g 1 g 2 z 2 − g 2 2 z 1 , h(g 1 , g 2 , z 1 , z 2 ) = iĝ g 2 1 + g 2 2 (2g 0 g 1 g 2 z 1 + g 2 1 (g 2 − g 0 z 2 ) + g 2 2 (g 2 + g 0 z 2 )) , k(g 1 , g 2 , z 1 , z 2 ) = g 0 g 1 (g 2 1 + g 0 g 1 z 1 + g 2 2 + g 0 g 2 z 2 ) . (4.22) The explicit expressions forQ are not particularly enlightening, so we don't report them here. One may apply the above transformations to the static and magnetic BPS seed given by eqns. (3.100) and (3.101) of [22] to generate dyonic and axionic solutions. Note that the form of (3.27) splits the dependence of the group coordinates from the couplings. Defining the section V g = (X X X g , F F F g ) t ≡Λ g V, the action of S G becomes V g = S 0 (x 0 , x)V g that more explicitly reads This split is independent of the parametrization of the group and so one can also use that of [19,21].

Extension to hypermultiplets
In this section we briefly comment on a possible generalization of our work to include also hypermultiplets. In this case the situation is more involved, since the coupling constants are replaced by the moment maps P x . However, when only abelian isometries of the quaternionic hyperscalar target space are gauged, the scalar potential can be cast into the form [31] where we defined Here, h uv denotes the metric on the quaternionic manifold, and D u is the covariant derivative acting on the hyperscalars. The most general symmetry transformation of the nonlinear sigma model is a linear combination of the isometries of the quaternionic and the special Kähler manifold. Let us define the formal operator where k u is a Killing vector of the quaternionic manifold, U an element of the Uduality algebra, k i the corresponding holomorphic special Kähler Killing vector, and A µ is the symplectic vector of the gauge potentials [31]. Then it is clear from (5.1) that a sufficient condition for δV = 0 is δL = 0 13 , that holds if and only where we added a hat to the quaternionic quantities that define the gaugings. Moreover the invariance of the kinetic term of the hyperscalars [11] leads to where L denotes the Lie-derivative. After choosing a specific model, these equations can in principle be solved for the parameters that define the linear combination of Killing vectors (5.2). In practice, (5.3) and (5.4) represent a highly constrained and very model-dependent system, and it is a priori not guaranteed that a nontrivial solution exists in general. In the FI limit, (5.3) boils down to the stabilization equation for the coupling constants G and (5.4) is trivially satisfied, as it must be. An interesting class of these models are the N = 2 truncations of M-theory described in [32,33]. In this case the solution of (5.3) and (5.4) could simplify the study of the attractor equations [31], necessary to work along the lines of [34], namely to compare the gravity side with the recent field theory results of [35][36][37].

Conclusions
In this paper we presented a geometric method to determine the residual symmetries in N = 2, d = 4 U(1) Fayet-Iliopoulos gauged supergravity. It involves the stabilization of the symplectic vector of gauge couplings, i.e., the FI parameters, under the action of the U-duality symmetry of the ungauged theory. We then applied this to obtain the surviving symmetry group for a number of prepotentials frequently used in the string theory literature, and showed how this group can be used to produce hairy and dyonic black holes from a given seed solution. Moreover, we pointed out how our method may be extended to a more general setting including also gauged hypermultiplets.
It would be very interesting to combine our results with dimensional reduction or oxidation as a solution-generating technique much like in the ungauged case discussed in the introduction. For instance one might think of starting from five-dimensional N = 2 gauged supergravity coupled to vector multiplets and then reduce to d = 4 along a Killing direction to get one of the models discussed here. One can then apply the residual symmetry group of the four-dimensional theory and subsequently lift back to d = 5 to generate new solutions. Notice that, for a timelike dimensional reduction, the scalar manifold of the resulting Euclidean four-dimensional theory is para-Kähler rather than Kähler [38], so that our results can not be applied straightforwardly, but require some modifications. Another direction for future work could be to reduce gauged supergravity theories to three dimensions and study in general the surviving symmetry preserved by the scalar potential. Work along these directions is in progress [39].

B Symplectic embedding
The choice of the symplectic embedding of the non-linear sigma model isometry group is necessary to completely specify the special Kähler structure over a manifold [11,20,23,40,41]. In what follows we shall summarize some properties used in the bulk of our paper.

(B.1)
This embedding is not unique since one can always act by conjugation with a symplectic matrix to construct a symplectically equivalent embedding. There are choices for the section V such that the isometry group sits in the symplectic group in a simple way, but the existence of a prepotential in that frame is in general not guaranteed. On the other hand, many symplectically equivalent embeddings are encoded by different prepotentials. Two physically interesting examples are [43,44] A physically less important transformation, which is nevertheless useful for practical purposes, is for instance One can also construct inequivalent embeddings over the same manifold, the simplest example being SU(1, 1)/U(1) [23]. Notice finally that symplectic equivalence does not mean physical equivalence. Even if it is possible to construct maps between the solutions of symplectically equivalent models, in general the solutions are physically different. The first of (B.1) is obtained by restricting the action of ι α ≡ C −1 α to the subgroup with C = 0. One can also explicitly verify that in this frame the prepotential exists and is given by F = − i 2α X Λ η ΛΣ X Σ .