All partial breakings in ${\cal N}=2$ supergravity with a single hypermultiplet

We consider partial supersymmetry breaking in ${\cal N}=2$ supergravity coupled to a single vector and a single hypermultiplet. This breaking pattern is in principle possible if the quaternion-K\"ahler space of the hypermultiplet admits (at least) one pair of commuting isometries. For this class of manifolds, explicit metrics exist and we analyse a generic electro-magnetic (dyonic) gauging of the isometries. An example of partial breaking in Minkowski spacetime has been found long ago by Ferrara, Girardello and Porrati, using the gauging of two translation isometries on $SO(4,1)/SO(4)$. We demonstrate that no other example of partial breaking of ${\cal N}=2$ supergravity in Minkowski spacetime exists. We also examine partial-breaking vacua in anti-de Sitter spacetime that are much less constrained and exist generically even for electric gaugings. On $SO(4,1)/SO(4)$, we construct the partially-broken solution and its global limit which is the Antoniadis-Partouche-Taylor model.


Introduction
Field theories invariant under global or local N = 2 supersymmetry allow very large classes of vector, hyper or tensor multiplet interactions characterized by specific sigma-model geometries. The existence of realizations in which zero or one supersymmetry remains unbroken at the ground state of the theory is then a relatively vast and complicated subject which cannot be addressed in theories with more supersymmetries in which the class of allowed matter and gauge couplings is fatally restrictive.
Consider for instance the simplest global N = 2 Maxwell theory, defined by an arbitrary prepotential F(z). Since its scalar fields cannot break the SU(2) R symmetry, a spontaneous breaking to N = 1 is clearly impossible. 1 Antoniadis, Partouche and Taylor [1] (APT) have however invented many years ago a realization with partial breaking in which the SU(2) R symmetry is violated by electric and magnetic Fayet-Iliopoulos (FI) terms inducing a nonlinear deformation of the second supersymmetry variation of one gaugino, defining it as the (single) goldstino. The ingredients of the model are then a non-canonical holomorphic prepotential and the FI constants. More recently [2], a similar mechanism has been shown to exist for a single hypermultiplet on a specific class of hyper-Kähler manifolds with a (translational) isometry, using its off-shell single-tensor dual formulation [3].
Local N = 2 supersymmetry is more involved in several aspects. Firstly, the supergravity multiplet includes the graviphoton and electric-magnetic duality in the local super-Maxwell theory is extended and powerful [4]. 2 Partial breaking requires the generation of a massive gravitino N = 1 multiplet, with two spin one fields in a 6 B + 6 F (bosonic plus fermionic degrees of freedom) on-shell content. Fully spontaneous partial breaking requires then at least one physical Maxwell multiplet (for the second massive spin one state) and one hypermultiplet for the SU(2) R breaking. The minimal case of one hypermultiplet on the SO(4, 1)/SO(4) quaternion-Kähler manifold coupled to a single Maxwell multiplet has been studied in detail. It was shown that a partial breaking of N = 2 supersymmetry can be realised for a generic prepotential, so that the APT model is obtained in an appropriate rigid globally supersymmetric limit [5]. A necessary ingredient 3 is the gauging of N = 2 supergravity along magnetic directions of vector fields, or alternatively a standard electric gauging in a non-prepotential field basis [8] 4 .
A more general analysis was also performed [11,12] in a class of quaternionic manifolds of dimension 4(n + 1) that are obtained by the so-called C-map from a special Kähler manifold of dimension 2n, corresponding to the effective supergravity of the perturbative type II superstring compactified on a Calabi-Yau threefold [13]. The special Kähler manifold is associated to the scalars of vector multiplets of the mirror theory, while the extra scalar components are the 2n Ramond-Ramond fields and the universal hypermultiplet of the string dilaton parametrising for n = 0 an SU(2, 1)/SU(2) × U(1) space broken to a quaternionic manifold with four isometries upon inclusion of the perturbative (one-loop) corrections [14][15][16]. For n = 0, it was shown that partial breaking can always be realised in either Minkowski or anti-de Sitter (AdS) vacuum by an appropriate choice of the embedding tensor that defines the directions of the gauging [17,18], which should have again some non-vanishing magnetic component. Finally, for the case of the single universal hypermultiplet (n = 0), no Minkowski N = 1 vacuum was found.
In this work, we perform a general analysis of the N = 2 partial breaking in supergravity theories containing a single hypermultiplet with two commuting isometries, gauged by the graviphoton and an additional vector multiplet. We work in a prepotential frame and use the embedding-tensor formalism [17], for dyonic gaugings of the graviphoton and of the vector multiplet along two commuting isometries of the hypermultiplet manifold. Our goal is to provide a generic treatment for N = 1 Minkowski vacua for arbitrary quaternion-Kähler manifolds, special-Kähler metrics and dyonic gaugings. In addition, we would like to obtain the APT model [1] as an off-shell gravity-decoupling limit. A general quaternionic manifold of dimension four with two commuting isometries can be parametrised by the Calderbank-Pedersen (CP) metric [19], where we find a no-go result for N = 1 Minkowski vacua for a general special Kähler manifold of the vector multiplet, which seems to be in contradiction with the results obtained for the hyperbolic space SO(4, 1)/SO (4). We prove that this contradiction is only apparent because the latter space cannot be written in a CP form, with its torus symmetry identified within the three-dimensional abelian sub-algebra of SO(4, 1), as a single exception.
The outline of this paper is as follows. In Sec. 2, we present a brief review of the mattercoupled N = 2 supergravity. We first present the ungauged case exhibiting the electromagnetic duality transformations in the symplectic formalism ( § 2.1). In passing, we show that a non-prepotential frame can exclusively arise from a magnetic duality transformation of the theory defined by the superconformal prepotential F = −iX 0 X 1 . We then summarize the gauging of isometries for the hypermultiplet manifold using the embedding-tensor formalism ( § 2.2); in particular, we exhibit the relation of the scalar potential to the fermion shifts that provide a convenient way to look for partial supersymmetry breaking N = 1 vacua. In Sec. 3, we make a systematic analysis in the case of one hypermultiplet with two isometries. We present the CP metric ( § 3.1) and compute the fermion shifts upon gauging its isometries ( § 3.2) proving a no-go theorem for partial breaking in Minkowski space ( § 3.3).
We also show that partial breaking in AdS is generically possible and we give an explicit example using a standard electric gauging of two shift isometries in the case of the universal dilaton hypermultiplet in type II superstrings compactified on a Calabi-Yau threefold ( § 3.4). We then identify an obstruction for bringing the hyperbolic space in CP coordinates that allows partial breaking in Minkowski space ( § 3.5). In Sec. 4, we return to the general analysis of the SO(4, 1)/SO(4) which is actually the only quaternionic manifold that does not admit a CP metric when the two commuting isometries are shifts in the Poincaré coordinates ( § 4.1). We construct explicitly the partial breaking Minkowski vacuum and study its off-shell gravity-decoupling limit ( § 4.2), as well as non-supersymmetric Minkowski vacua ( § 4.3). Section 5 contains some concluding remarks. Finally, we include four appendices.
Appendix A contains useful formulae for the gauging of quaternionic manifolds with isome-tries, App. B elaborates the hyperbolic space in CP coordinates, App. C discusses coordinate transformations used to derive the CP metric and App. D proves a result on N = 0 vacua of the SO(4, 1)/SO(4) model stated in § 4.3.

The kinetic terms
The N = 2 target space M describing the scalar-field kinetic terms of a single hypermultiplet and n V vector multiplets is factorized, and both metrics only depend of the scalar fields of their respective multiplets, a property which by supersymmetry extends to all kinetic terms. The hypermultiplet scalar dynamics is encoded in the four-dimensional quaternion-Kähler metric M H with coordinates q u = (q 1 , q 2 , q 3 , A generic quaternion-Kähler manifold for n H hypermultiplets is Einstein with holonomy 6 and dimension 4n H . For n H = 1, since Sp(2) × SU(2) ∼ SO(4), a particular characterization is needed: the quaternion-Kähler metric is Einstein with an (anti-) selfdual Weyl curvature tensor. As we will see in the next sections, four-dimensional metrics with these properties have been studied quite extensively when they admit one or several continuous isometries, which is the case of interest here.
The N = 2 Maxwell sector is conveniently constructed in the superconformal formulation: it is then defined in terms of a prepotential F(X I ) of n V + 1 complex scalar fields, with Weyl weight one. The index I = 0, . . . , n V includes a compensating multiplet. Its component fields include the propagating graviphoton, while its two gauginos and complex scalar are used to gauge-fix superconformal symmetries and solve field equations of auxiliary fields in the Weyl multiplet. 7 It is a common but unnecessary choice to set I = 0 as the compensator direction. Superconformal invariance requires that F(X I ) has Weyl weight two: and f (z a ) is an arbitrary function of the zero-weight scalar fields z a = X a /X 0 in the n V physical Maxwell multiplets: the Poincaré theory is formulated in terms of the scalars z a . 5 We use hypermultiplet scalars and metric with dimension mass 0 . 6 Or G × SU(2), G ⊂ Sp(2n H ). 7 These appear linearly in the lagrangian and then impose constraints.
There is however a subtlety: electric-magnetic duality acts in the Maxwell sector as Sp(2(n V + 1), R) linear transformations of the vector of sections Choosing the section vector V (as a function of a given set of scalar fields) defines a symplectic frame: electric-magnetic duality would imply that the (ungauged, abelian) theory can be equivalently formulated in each symplectic frame obtained by the action of Sp(2(n V + 1), R) on V. 8 In a prepotential symplectic frame, there exists F(X) such that the sections are where f a = ∂ ∂z a f (z). In a prepotential frame, the symplectic-invariant product i(X I F I − F I X I ) and Y will appear in the Kähler potential of the Poincaré fields z a . Note that there is an ambiguity: this quantity vanishes if with real coefficients α I J and two prepotentials differing by F describe the same theory.
One may wonder if all frames in the symplectic orbit of a prepotential frame admit a prepotential, or if there exists orbits which relate prepotential and non-prepotential frames. The question of the existence of a prepotential frame has been discussed in general in Ref. [20] 9 , but the simple case n V = 1, which is of interest here is very simple to solve explicitly. Consider a symplectic transformation relating sections V and V, assuming that V defines a prepotential frame with prepotential F(X 0 , X 1 ) and Poincaré scalar z = X 0 /X 1 . Assuming that we identify the compensators in both frames, X 0 = X 0 , Sp(2, 2, R) duality reduces to Sl(2, R) transformations which are electric-magnetic if m 2 = 0 or m 3 = 0. We wish to find a Poincaré scalar z = X 1 / X 0 and a prepotential F( X I ) = −i( X 0 ) 2 g( z), which identify sections V as a prepotential 8 The symplectic orbit of V. 9 Summarized in [21], § 21.2.2, page 474. See also ref. [22]. frame: The three equations relating f and z with g and z are generated by the Legendre transforma- and it is singular only if f (z) (or g( z)) is linear. Hence, the symplectic frame with sections V is a prepotential frame with Poincaré field z and prepotential F = −i( X 0 ) 2 g X 1 X 0 with a single exception, for which (with m 1 = m 4 = 0) and i(X I F I − F I X I ) = −αX 0 X 0 (z + z) (2.14) leading to Kähler potential K = − ln(z + z). This simple discussion agrees with the general argument given in Refs. [20] and [21]. 10 The conclusion is that in the n V = 1 case, all symplectic orbits connect exclusively prepotential frames, with the single exception of the orbit of F(X) = −iX 0 X 1 which includes non-prepotential frames. The first example of partial N = 2 breaking in supergravity [8] was found using precisely the sections (2.13). An electric gauging of two translation isometries of the hypermultiplet manifold SO(4, 1)/SO(4) ∼ Sp(2, 2)/SU(2) × SU(2) in this non-prepotential frame leads to a two-coupling theory with zero potential and N = 0 for generic values of the couplings, N = 1 when a linear relation is verified by the couplings, and N = 2 for zero couplings.
Since the prepotential (2.12) is in the symplectic orbit of the non-prepotential frame (2.13) 10 But disagrees with statements in Ref. [5] for instance. and since all other orbits include prepotential frames only, we are always allowed to work in a prepotential frame with sections (2.5) and to gauge isometries in this frame: since gauging fixes the electric-magnetic duality symmetry, the theory will then depend on the prepotential, the gauge couplings and the choice of hypermultiplet manifold.
The kinetic terms of the helicity 0, ± 1 2 fields 11 in Poincaré Maxwell multiplets have a Kähler metric with Kähler potential (2.15) For instance, for scalar fields (in a prepotential frame), the superconformal lagrangian includes e −1 L kin. = −g µν N I J D µ X I D ν X J , (2.16) with and with a covariant derivative D µ X I = (∂ µ − iA µ )X I involving the gauge field of the superconformal U(1) R symmetry. Eliminating this auxiliary vector field delivers 12 using the homogeneity of the prepotential. The Poincaré theory can then be obtained in field coordinates X I = X 0 (1, z a ), X 0 = κ −1 y(z, z) and sections V = yU = y(Z I (z), F I (z)) once the dilatation and U(1) R gauge-fixing conditions have been applied. In terms of z a then, which leads to expression (2.15). The same metric appears in the kinetic terms of the Poincaré gauginos λ ia . However the kinetic terms of gauge fields include further contributions due to the graviphoton: 11 Propagating or auxiliary. 12 Omitting fermions.
with n V + 1-dimensional metric Notice that ImN I J is negative on physical fields.
In the following, we will explicitly consider the case n V = 1 only.

Fermion shifts, scalar potential, supersymmetry breaking
In N = 2 supergravity, the scalar potential appears when isometries of the theory are gauged. With the graviphoton and the gauge field of a vector multiplet (n V = 1), we can gauge two commuting isometries, as required if partial supersymmetry breaking is envisaged [11]. This of course implies that two commuting isometries should exist and this defines a class of scalar manifolds for a single hypermultiplet for which explicit metrics are available. The problem of partial breaking can then be analytically studied in general.
The scalar potential in supergravity theories has a particular structure. The supersymmetry variation of all fermions ψ A I is of the form where the fermion shift M A Ij is a function of scalar fields (A runs over all supermultiplets, I over all fermions in multiplet A, j over all supersymmetries; in N = 2 theories fermions are always in SU(2) doublets and I = i). If the supermultiplet admits an off-shell realization, as the N = 2 Maxwell or single-tensor multiplets, the fermion shifts are in general auxiliary scalar fields. For instance, in a Maxwell N = 2 multiplet with gauginos λ i , and Y ij is the SU(2) triplet of real (electric) auxiliary fields. For the gravitinos, The scalar potential is then symbolically [23] V = e ∑ coeff. × fermion shifts † × metric × fermion shifts, where the sum is over all fermions and the coefficients are negative for gravitinos and positive for spin-1 2 fields and depend on the normalization chosen for the fermion fields. Hence, fermion shifts define the ground state of the theory and a nonzero value of a spin- 1 2 fermion shift at the ground state indicates the presence of a goldstino, or several goldstinos, and then indicates spontaneous supersymmetry breaking. Analyzing the structure of the fermion shifts is fundamental when studying the breaking phases of a supersymmetric theory.
In order to obtain the fermion shifts, we need to specify the gauging applied in the theory.
The gauge generators (associated with gauge field I) and the gauge variations can be defined by electric-magnetic symplectic vectors Θ I a ξ a : the embedding tensor Θ I a specifies a linear combination of the (commuting) isometries ξ a = ξ u a ∂ u of the quaternion-Kähler metric h uv ; it defines the coupling constants of the gauged theory. The index I defines Θ I a as a fixed symplectic vector associated with each isometry, but we will rather use Θ I a = Ω I J g Ja , (2.27) and the coupling constants are the numbers g Ia .
Consistency of the gauging is guaranteed by the locality constraint on the embedding The hypermultiplet scalar fields are coordinates q u on a (four-dimensional) quaternion-Kähler space with metric h uv . For each isometry vector δ a q u = ξ u a , one can derive an SU(2) triplet of prepotentials (or moment maps) solving the differential equation in terms of the triplet of complex structures J x . 13 As usual, to describe the hypermultiplet fermions (hyperinos), we need a vielbein f iA u , which for N = 2 is defined by (i and A are respectively SU(2) and Sp(2n H ) = Sp(2) doublet indices and hyperinos carry index A). Then, for given quaternion-Kähler metric h uv , complex structures J x , isometries ξ u a and prepotentials we obtain the following expressions for the fermion shifts: 14 Gravitinos: Hyperinos: and of their conjugates (P aij = P ij a * ): Gravitinos: Gauginos: Hyperinos: (2.33) The embedding tensor always appears in the combination Θ a I V I = κ −1 e K/2 Θ a I U I . The notation ∇ α stands for Kähler-covariant derivatives. Since Kähler transformations act as Supersymmetry imposes the identity and the gauging and fermion shifts lead then to the following N = 2 scalar potential [21,24,25]: 14 These shifts hold for fermions with dimension mass 1 2 . Similarly, the scalars and the metrics g αβ and h uv are dimensionless. We use Weyl spinors, in terms of the Kähler potential K. For later use, we find useful to express the scalar potential (2.37) in terms of the anti-selfdual covariant derivatives k − auv defined in App. A: Supersymmetry breaking is then easily discussed. Firstly, at the ground state defined by the scalar potential, nonzero shifts of the spin-1 2 fermions indicate the presence of zero, one or two goldstinos, for a spontaneous breaking into N = 2, 1 or 0 unbroken supersymmetry(ies). Secondly, if one or two supersymmetries remain unbroken, the value of the gravitino shift S ij indicates the spacetime geometry of the ground state (AdS or Minkowski). Partial breaking N = 2 → N = 1 implies that there should be one (and only one) spinor ǫ 1i for which three conditions must be fulfilled: and the scalar curvature of AdS spacetime is given by R = 4Λ, Λ = −3|µ| 2 . Furthermore, the second supersymmetry with spinor parameter ǫ 2 should verify either In the next sections, we analyze these conditions on a special Kähler geometry with arbitrary prepotential F(X I ) and a generic quaternion-Kähler geometry for a single hypermultiplet.

The hypermultiplet with isometries and partial breaking
For one hypermultiplet, the four-dimensional quaternion-Kähler geometry is defined as an Einstein space with constant Ricci curvature proportional to κ 2 [26] 15 and (anti-) selfdual Weyl curvature. With one or two isometries, metrics for generic quaternion-Kähler spaces have been thoroughly discussed. We will use two canonical forms: the Przanowski-Tod (PT) [27,28] and the already quoted Calderbank-Pedersen (CP) [19]. Both are defined in terms of a solution of a differential equation, nonlinear (Toda) for the PT metric for spaces with one isometry, linear in the CP metric with two commuting isometries. Since we are interested in the latter case, we first consider the hypermultiplet metric in CP coordinates. 15 For metric g uv = κ −2 h uv as defined in Eq. (2.2).

The CP metric
According to Calderbank and Pedersen [19], a four-dimensional quaternion-Kähler metric with two commuting isometries can be written in a set of coordinates ρ > 0, η, ψ and ϕ for every solution F(ρ, η) of the linear equation with isometries acting as shifts of ψ and ϕ. The line element ds 2 = h uv dq u dq v is The metric determinant is and positivity requires 4ρ The CP metric describes a conformally antiselfdual Einstein space 16 with scalar curvature normalized to R = −12. It is endowed with a triplet of SU(2) selfdual 2-forms J x (complex structures) which are covariantly constant with an SU(2) connection ω x [19]: On the metric (3.2), Calderbank and Pedersen [19] write: 17 "Any selfdual Einstein metric of nonzero scalar curvature with two linearly independent commuting Killing fields arises locally in this way (i.e., in a neighbourhood of any point, it is of the form (3.2) up to a constant multiple)." We will see that a slight inaccuracy in this statement allows for an exception which is of fundamental importance in our subject. 16 An Einstein metric with anti-selfdual Weyl curvature. 17 It is the point (ii) of their main theorem 1.1.
The two Killing vectors of the CP metric are by construction ξ 1 = ∂ ϕ and ξ 2 = ∂ ψ . Using Eq. (2.29), the triplets of Killing prepotentials (or moment maps) are The standard vierbein one-forms of the metric (3.2) are (3.7) We will need the corresponding symplectic vielbeins f iA u obtained from relations and we have checked that the f iA u 's satisfy Eq. (2.30). We will use the + sign below.

Fermion shifts
Consider a generic dyonic gauging of the two isometries, described by: where the embedding tensor Θ I a is compatible with the locality condition (2.28) and g 0,1,2,3 are the gauge couplings. We use a prepotential frame and formulas (2.5) apply. The corresponding Kähler potential is given in Eq. (2.15) for a single z: In order to evaluate the fermion shifts given in (2.32), we use the results of § 3.1. Defining we find for the gravitino shift, and for the shifts of spin-1 2 fermions. 18

Partial breaking in flat space
The first condition for partial breaking is certainly that the ground state does not lead to two goldstinos.
with zero eigenvector 19 16) and this ǫ i is also eigenvector of S ij with The eigenvalue should vanish for a Minkowski ground state: turning condition (3.15) into and then to The determinant of N i A turns out to be proportional to using the Minkowski conditions (3.18) and (3.15). The conditions for the positivity of the CP metric, ρ, F, F 2 ρ + F 2 η > 0 and condition (3.18) imply that N i A does not have a zero eigenvalue. Hence, the partial breaking of N = 2 supersymmetry in Minkowski spacetime is excluded whenever the hypermultiplet can be described in the CP field coordinates and metric. According to Ref. [19], this would be always the case.
There is an apparent contradiction between this conclusion and the known existence [8] of a partial breaking on the SO(4, 1)/SO(4) hypermultiplet in Minkowski spacetime. We will see shortly that for this quaternion-Kähler space, and only for this space, there exists a pair of isometries for which the coordinates used by Calderbank and Pedersen [19] do not exist. This is the earlier quoted exception, leading to a statement of uniqueness for partial breaking with a single multiplet and two gauged isometries.

Partial breaking in AdS
The obstruction found for partial breaking into Minkowski spacetime does not exist for AdS ground states. Partial breaking in this case requires at the first place Eq. (3.15). With this condition, the gaugino and gravitino shifts read at the ground state. For the unbroken supersymmetry parameter ǫ (the zero eigenvector of and the cosmological constant is The Minkowski condition (3.18) which cancels Λ does not apply and the second condition for partial breaking is that ǫ is also a zero eigenvector of the hyperino shift matrix (3.14): Solutions to Eqs. (3.15) and (3.25) would lead to stable AdS ground states. 20 An example. We can realize the above conditions for N = 1 AdS vacua in a specific example. We consider for this a CP metric with F η = 0, i.e.
This metric has extended isometry Heisenberg ⋉ U(1) and it describes the scalar manifold of the universal hypermultiplet in type II strings, including the one-loop perturbative corrections, as obtained in Ref. [14]. The case σ = 0 is the tree-level SU(2, 1)/SU(2) × U(1). From the expressions of A 2 and A 3 in Eq. (3.14), one obtains:

27)
and thus the condition (3.25) implies: where we dropped the symbols of expectation values. On the other hand, condition (3.15) yields − g 2 + g 3 Im f zz = ±g 0 ρ Im K z , Re c = −g 3 Re f zz ± g 0 ρ Re K z . (3.30) It follows that there are four equations that can be solved for the four expectation values of ρ, η and z. Indeed, using the expression of c in Eq. (3.12), one obtains: which can be used in the remaining two equations (right part of (3.29) and (3.31)) for deter- 20 Their stability is carefully discussed in the appendix B of Ref. [11]. An early example was given in ref. [10].
mining z. For instance, for g 3 = 0, on finds the solution: where the last two equations determine z, using the expression (3.11) for K. Note that partial N = 2 supersymmetry breaking in AdS can be realised without introducing magnetic FI coupling terms (g 3 = 0).

Hyperbolic space and Calderbank-Pedersen coordinates
We now come back to the issue met at the end of § 3.3. To resolve the case, we will show that the hyperbolic space cannot be described by a CP metric with shift isometries on (ϕ, ψ) generated by pairs of elements in the three-dimensional abelian subalgebra of SO(4, 1). 21 34) and the dependence on the quaternion-Kähler space is in the function F(ρ, η) only.
Consider now the simplest quaternion-Kähler space, This hyperbolic space admits coordinates in which the line element is This is a conformally-flat space with R uv = −3 h uv . The corresponding symplectic vierbeins two-forms J x (complex structures) is covariantly constant up to the SU (2) connection ω x Their Killing prepotential triplets follow from Eq. (2.29): (3.41) In these coordinates and for these isometries, we find the following expression for the scalars appearing in Eq. (2.40): The comparison with the generic values (3.34) obtained for the CP metric indicates that for these isometries, CP coordinates cannot be found. 22 The origin of this obstruction is located in the derivation of the CP metric given in Ref. [19]. This metric is a consequence of the Joyce description for anti-selfdual conformal metrics with a U(1) × U(1) symmetry [29], the Jones-Tod correspondence for four dimensional anti-selfdual spaces with at least one isometry [30], and the use of Przanowski-Tod (PT) theorem to determine which metrics, among the Joyce metrics, are Einstein spaces [27,28,31]. In short, to identify the Einstein representatives among the conformal structures with anti-selfdual Weyl tensor, one employs the PT form where the metric is generated by a 22 A similar conclusion, technically more involved though, derives from comparing the inner products of the two isometries, i.e. ξ u a ξ v b h uv .
function Ψ(X, Y, Z) solving the continual Toda equation [27,28,31] 23 The PT line element is then (3.44) The quaternion-Kähler metric has one isometry acting on the fourth coordinate ψ and generated by ∂ ψ . The simplest solution is of course for which This is the metric (3.36) with b 0 = Z, b 1 = e C/2 X, b 2 = e C/2 Y, b 3 = ψ + g and the isometry of the PT metric shifts b 3 .
In order to make contact with the CP metric, we assume the existence, in the PT description, of a second isometry generated by ∂ ϕ . The case of primary interest for us is a shift isometry acting on b 1 or b 2 . We choose a translation isometry of the Y coordinate. Assume then that ϕ = Y and that Ψ does not depend on Y. Finding the CP coordinates is possible using a transformation due to Ward [32]: 24 X, Z; e Ψ(X,Z) =⇒ (ρ, η; V(ρ, η)) , and Killing vectors are used for constructing the N = 2 supergravity lagrangian. In Ref. [8] they first worked in the non-prepotential frame described in Sec. 2, Eqs. (2.13) and (2.14). Then they reworked the example in a generic frame with arbitrary prepotential function f (z) [5]. Our objective in this section is to complete the description of the model by showing explicitly that the N = 2 supergravity theory (at finite κ then) admits a stable ground state with partial breaking which continuously deforms to the APT model in the gravity-decoupling limit κ → 0. This can be seen as deriving off-shell the APT lagrangian as the κ → 0 limit of the SO(4, 1)/SO(4) supergravity lagrangian. 25 Using the embedding tensor (3.10), the prepotential frame (2.5) leading to Kähler potential (2.15) and coordinates b u with metric (3.36) for the hypermultiplet, the supergravity potential reads: with c defined as (4.4) and c z = −ig 2 + g 3 f zz . Since hypermultiplet scalars only appear in the prefactor b −2 0 , the ground state of the potential, in order to escape the runaway of b 0 , requires Minkowski geometry, V = 0. In Ref. [8], the authors consider the particular case g 1 = g 2 = 0, f (z) ∼ z, K = − ln(z + z) and then V ≡ 0.

Notice that the scalar potential (4.3) vanishes if
Since f zz is imaginary, K zz = K z K z and V = 0.

N = 1 Minkowski vacua
The fermion shifts (2.32) induced by this gauging read for gravitinos and hyperinos. They verify the relation 26 For gauginos, where we have used The conditions for partial breaking are then easily stated. To have a common zero eigenvector for W ij and N i A , we need a dyonic (electric and magnetic) gauging with g 3 = 0 = g 0 and (4.10) The first condition c = ±g 0 leads to a zero eigenvector ǫ of N i A while the second condition c z = 0 ensures that the same ǫ is a zero eigenvector of W ij z . This second condition for partial breaking also implies S ij ǫ j = 0 and V = 0 and then partial breaking can only exist in Minkowski spacetime. The conditions (4.10) define an N = 1 supersymmetric stable ground state. In Ref. [8] where g 1 = g 2 = f zz = 0, these conditions reduce to g 3 = ±g 0 = 0.
Solving the conditions for partial breaking commonly impose, for a given choice of f (z), particular values or relations on the coupling constants. For instance, a linear f = z, as used in Ref. [8], has partial breaking only if g 0 = ±g 3 = 0, g 1 = g 2 = 0. The conditions may be impossible to solve: f = z 2 forces all g i to be zero (but this example is irrelevant since the Kähler metric K zz ≡ 0). For a generic prepotential f (z), one usually finds that two couplings are determined in terms of the other two.
The spectrum of the partially broken theory includes N = 1 supergravity (2 B + 2 F onshell states), a massive N = 1 gravitino multiplet (gravitino, the two spin-one fields, one fermion, 6 B + 6 F ), a massless chiral multiplet (2 B + 2 F ) and a chiral multiplet with the scalar z and mass proportional to the free parameter f zzz , precisely as in the APT model, see below. The four hypermultiplet scalars are massless (two are Goldstone bosons) and the mass matrix reduces to z only with where Y is defined through the Kähler potential (3.11): Hence, the mass of the scalar z (and of its fermion partner) is given by At the N = 1 ground state, the value of the hypermultiplet scalar b 0 is an arbitrary parameter. From the gravitino shift matrix 27 or from the expression of the scalar potential however, the mass of the massive gravitino scales as 14) and the theory has two order parameters, b 0 and f zzz for the massive gravitino and chiral (with z) multiplets respectively.
In order to discuss the gravity-decoupling limit κ → 0 of the supergravity theory and make contact with the APT model, we first redefine the hypermultiplet scalars ( b 0 = 0): 28 whereμ is a mass scale (and κμ ∼μ M P is dimensionless). The hypermultiplet kinetic terms 27 Which in a Minkowski ground state is proportional to the mass matrix, S ij ∼ κ −2 m ij 3 2 . 28 This is a simplistic use of the procedure described in Refs. [15,16,34,35]. are then (4.16) and in the limit κ → 0, the kinetic metric is the trivial hyper-Kähler h uv =μ 2 δ uv . 29 For the vector multiplet kinetic term, we need as κ → 0 where F (x) is the dimension-two prepotential of the rigid N = 2 theory and x is a dimension-one scalar. In other words, we need and the Kähler potential of the rigid theory will be K(x, x) = −ixF x + ixF x . Following Ref. [5], this is obtained from the formal κ expansion, (4.19) and the definition F (x) =μ 2 F( x µ ), (4.20) with F x =μ F z and F xx = F zz . The arbitrary complex number λ will get a precise value later on.
With the rescaling (4.15) of the hypermultiplet scalars, a corresponding rescaling of the Killing vectors, or equivalently a (first) rescaling of the coupling constants, is needed: leading to the scalar potential where c and c z are expressed in terms of g i (instead of g i ), c = −i( g 1 + g 2 z) + g 3 f z and c z = −i g 2 + g 3 f zz . Before expanding in powers of κ, we perform a second redefinition of the gauge couplings, g 0 = κμ g 0 , g 1 = κμ g 1 , g 2 = (κμ) 2 g 2 , g 3 = g 3 . (4.23) 29 We could as well define dimension-one fields withμ b u → b u .
The leading terms in the quantities c, c z and K z appearing in the potential (4.22) are then (4.24) and, to leading order in κ, the potential reads where (4.26) The scalar potential of a globally supersymmetric theory is not expected to have an irrelevant additive constant and we cancel ζ M 2 − C by choosing (4.27) which also implies This is the leading term in the first condition (4.10) for partial breaking (related to the gravitino and hyperino shift matrices). The shift matrix for canonically normalized (mass 3 2 dimension) gauginos Λ i becomes where K z , c z and c are given in Eqs. (4.24) and (4.28). 30 Up to here, the analysis has been off-shell only. We now expect that the second condition (4.10) for partial breaking, which indicates that only one gaugino is a goldstino, follows from the minimum of the potential, which is at 31 This vacuum equation is also the leading order term of c z = 0, the second condition for partial breaking (4.10). And for c z = 0, the gaugino shift matrix (4.29) has one zero eigenvalue. At the ground state, the vector multiplet metric is (4. 31) and the deformation parameter of the supersymmetry variation in the goldstino direction is then δΛ goldstino = M 2 + · · · = g 3μ 2 + · · · . (4.32) The coupling constant g 3μ 2 is the magnetic FI term at the origin of the partial breaking. The N = 2 multiplet splits in a massless N = 1 Maxwell, including the goldstino, and a chiral N = 1 multiplet with mass as expected from Eq. (4.13).
In conclusion, we have shown that this N = 2 supergravity theory possesses for all values of κ an N = 1 ground state which coincides in the limit κ → 0 with the APT lagrangian and its N = 1 vacuum.

N = 0 Minkowski vacua
The scalar potential (4.3) can also be written Non-supersymmetric vacua will then follow by solving (4.35) supplemented by stability conditions and the existence of two goldstinos. Since h is real, Eqs. (4.35) give three conditions for the two real components of z. Hence, for a given f (z) one expects at least one non-trivial condition on the gauge coupling constants: once z is fixed by ∂ z h = 0, the number V must vanish to avoid runaway in b 0 . Since c zz = g 3 f zzz , with the definition (3.11) of Y, it is immediate that f zzz = 0 solves ∂ z h = 0. We have already observed that c z = 0 leads to h = 0. Hence, f zzz = c z = 0 is a solution of conditions (4.35). Since c z = 0 also implies that the gaugino shift matrix has a zero eigenvector, the determinant of N i A should be nonzero in an N = 0 ground state: This happens in the FGP model [8] with f (z) = z, g 1 = g 2 = 0 and then c = g 3 : all z are stable ground states since V ≡ 0, the generic ground state has N = 0 and partial breaking occurs when g 0 = ±g 3 . Hence we may think that partial-breaking solutions are surrounded (in the parameter space of the solutions of a model) by N = 0 solutions. However, assuming leads to the scalar potential

Outlook
In summary, our motivation was to classify spontaneous (partial) supersymmetry breaking in the minimal case of N = 2 supergravity, containing one hypermultiplet and one vector multiplet. The former could describe the universal dilaton of type II superstrings compactified on a Calabi-Yau threefold, while the latter should gauge together with the N = 2 graviphoton two commuting isometries of the hypermultiplet quaternion-Kähler manifold, which is necessary in order to obtain a massive N = 1 spin-3/2 multiplet. The analysis can be done in a general way, since a four-dimensional quaternionic manifold with a two-torus isometry can be put in the Calderbank-Pedersen metric form [19].
To our surprise, using this approach we found a no-go theorem on the existence of N = 1 32 For a proof, see App. D.
Minkowski vacua, which would also hold for any number of abelian vector multiplets. This result seems in conflict with the well-known example of the hyperbolic space SO(4, 1)/SO(4) [5]. However, we proved that the hyperbolic space cannot be written in a Calderbank-Pedersen form, where its torus symmetry lies within the three-dimensional abelian subalgebra of SO(4, 1). We furthermore showed that it is easy to obtain N = 1 vacua of partially broken supersymmetry in AdS space.
Finally, we revisited the hyperbolic space for gaugings within the three-dimensional Abelian subalgebra of SO(4, 1), while for the scalars of the vector sector we considered a generic holomorphic prepotential. We worked out the details for a generic gauging leading to a supergravity theory with potential (4.3) and possessing N = 1 or N = 0 Minkowski vacua for all values of κ. For the N = 1 vacua, we also worked out their off-shell gravity-decoupling limit, and obtained the APT lagrangian [1].
Some open questions remain to be answered, which are outside of our present scope.
Regarding the gauged isometries, on the one hand, one may consider gauging isometries of the special-Kähler manifold of vector multiplets, or (part of) the SU (2)

A Four-dimensional quaternionic manifolds with isometries
Consider a four-dimensional quaternionic space, described by an Einstein metric with antiselfdual Weyl curvature normalized with R = −12. This space is endowed with a triplet of SU(2) self-dual complex structures J x uv , which are covariantly constant modulo an SU(2) one-form connection ω x The complex structures J x uv are normalized to satisfy: Assume that the quaternionic space has some isometries generated by ξ a = ξ u a ∂ u . As a consequence of the Bianchi identity for the Riemann tensor, condition (A.1) leads to in terms of the (anti)-selfdual covariant derivatives These (anti)-selfdual covariant derivatives obey the following identities valid for any four-dimensional metric. 33

B The hyperbolic space and its Calderbank-Pedersen coordinates B.1 The hyperbolic space in global and Poincaré coordinates
The SO(4, 1) isometry algebra of the hyperbolic space H 4 obtained as SO(4, 1)/SO(4) includes six compact SO(4) generators X uv and four noncompact SO(4, 1)/SO(4) generators Y u = X u5 , with η 55 = −1 = −η uu . It has a three-dimensional abelian subalgebra related to noncompactness. In the standard notation or the SO(4, 1) algebra, 34 The commuting r i = X i4 + Y i form a vector of SO(3) ⊂ SO(4). 33 They follow from SO(4) group theory. 34 The same would hold for X i4 − X i5 .
We can describe H 4 in global coordinates. In this set of coordinates the SO(4) acts linearly and the line element takes the form Its ten isometry generators are: Generators of SO (4) : Generators of SO(4, 1)/SO (4) : where ∂ u = ∂ ∂x u . The three-dimensional abelian subalgebra is generated by In these coordinates, the SO(4) generators act as simple linear variations but the action of the commuting r i is more involved. The curvature is directly related to the SO(4, 1)-invariant quantity 1 − x v x v , and these coordinates are then convenient for describing the (flat or rigid) gravity-decoupling limit.
An alternative coordinate system is the Poincaré patch . The metric takes the form (3.36) In these coordinates, the generators of the three-dimensional abelian subalgebra act as translations of b i : The two sets of coordinates are related by

B.2 Calderbank-Pedersen coordinates
The isometry algebra SO(4, 1) admits a variety of pairs of commuting generators and for each pair, according to Ref. [19], there should exist CP coordinates ρ, η, ϕ, ψ. Examples of (inequivalent) pairs are: i. A pair of isometries in the three-dimensional abelian subalgebra, for instance r 1 and r 2 .
ii. The Cartan subalgebra of SO(4), chosen as the compact generators X 12 and X 34 , or a compact and a non compact SO(4, 1) generator, like X 12 and Y 4 = X 45 .
iii. A compact generator of SO(4) and one of the r i 's, for instance X 23 and r 1 .
We next identify ξ 1,2 with the Killing vectors of the CP metric or with linear combinations of them, and use the identity (3.34) for recognizing the change of coordinates. There are actually several possibilities and we focus on two cases, following Ref. [19].

C Ward transformation
Assume that we have a solution V(ρ, η) of the equation where indices denote derivatives with respect to η or ρ. A further derivative with respect to ρ leads to and F(ρ, η) generates via Eq. (3.2) a quaternion-Kähler metric in Calderbank-Pedersen coordinates. Coordinates (ρ, η) can be traded for (X, Z) by a double Legendre transformation: This implies firstly As usual, ∂x i ∂x j = δ i j for each set of coordinates delivers the relations between the second derivatives of V and K. This procedure has been elaborated by Ward in Ref. [32] and used in the derivation of the CP metric [19]. It allows in particular to find CP coordinates for a quaternion-Kähler metric with two isometries expressed in PT coordinates, for a given Toda solution Ψ.
The case where Ψ is a constant is clearly excluded.

D A proof
In § 4.3 on N = 0 vacua of the SO(4, 1)/SO(4) model, we claim that all solutions of ∂ z h = h = 0 with f zzz = 0 are N = 1 vacua. 35 We give here a proof of this statement. Recall that, for a given prepotential function f (z), and assuming f zzz = 0, one finds the factorization: This quantity should vanish for a solution of ∂ z h = h = 0. The solutions are either c z = 0 or g 3 Y + (z − z)(c − c) = 0 and in both cases ∂ z h = 0 requires c = ±g 0 .
• If c z = 0 and c = ±g 0 the vacuum has N = 1 supersymmetry: the two conditions for partial breaking (4.10) are fulfilled.
• If c z = 0, the vacuum state would be at g 3 Y + (z − z)(c − c) = 0 and c = ±g 0 = c. Then Y = e −K = 0, which is excluded.
Hence, Minkowski N = 0 vacua with f zzz = 0 do not exist.