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All null supersymmetric backgrounds of N=2, D=4 gauged supergravity coupled to abelian vector multiplets

The lightlike supersymmetric solutions of N=2, D=4 gauged supergravity coupled to an arbitrary number of abelian vector multiplets are classified using spinorial geometry techniques. The solutions fall into two classes, depending on whether the Killing spinor is constant or not. In both cases, we give explicit examples of supersymmetric backgrounds. Among these BPS solutions, which preserve one quarter of the supersymmetry, there are gravitational waves propagating on domain walls or on bubbles of nothing that asymptote to AdS_4. Furthermore, we obtain the additional constraints obeyed by half-supersymmetric vacua. These are divided into four categories, that include bubbles of nothing which are asymptotically AdS_4, pp-waves on domain walls, AdS_3 x R, and spacetimes conformal to AdS_3 times an interval.


Introduction
Supersymmetric solutions to supergravity theories have played, and continue to play, an important role in string-and M-theory developments. This makes it desirable to obtain a complete classification of BPS solutions to various supergravities in diverse dimensions. Progress in this direction has been made in the last years using the mathematical concept of G-structures [1]. The basic strategy is to assume the existence of at least one Killing spinor ǫ obeying D µ ǫ = 0, and to construct differential forms as bilinears from this spinor. These forms, which define a preferred G-structure, obey several algebraic and differential equations that can be used to deduce the metric and the other bosonic supergravity fields. Using this framework, a number of complete classifications [2][3][4] and many partial results (see e.g. [5][6][7][8][9][10][11][12][13][14][15][16][17] for an incomplete list) have been obtained. By complete we mean that the most general solutions for all possible fractions of supersymmetry have been obtained, while for partial classifications this is only available for some fractions. Note that the complete classifications mentioned above involve theories with eight supercharges and holonomy H = SL(2,H) of the supercurvature R µν = D [µ D ν] , and allow for either half-or maximally supersymmetric solutions.
An approach which exploits the linearity of the Killing spinors has been proposed [19] under the name of spinorial geometry. Its basic ingredients are an explicit oscillator basis for the spinors in terms of forms and the use of the gauge symmetry to transform them to a preferred representative of their orbit. While the equivalent G-structure technique leads to nonlinear equations which might be difficult to interpret and to solve in some cases, the spinorial geometry approach permits to construct a linear system for the background fields from any (set of) Killing spinor(s) [20]. This method has proven fruitful in e.g. the challenging case of IIB supergravity [21][22][23]. In addition, it has been adjusted to impose 'near-maximal' supersymmetry and thus has been used to rule out certain large fractions of supersymmetry [24][25][26][27][28]. Finally, a complete classification for type I supergravity in ten dimensions has been obtained in [29], and all half-supersymmetric backgrounds of N = 2, D = 5 gauged supergravity coupled to abelian vector multiplets were determined in [30,31]. Spinorial geometry was also applied to de Sitter supergravity [32], where interesting mathematical structures like hyper-Kähler manifolds with torsion emerge.
In the present paper we shall finish the classification of supersymmetric solutions in four-dimensional N = 2 matter-coupled U(1)-gauged supergravity initiated in [33], generalizing thus the simpler cases of N = 1, considered recently in [34,35], and minimal N = 2, where a full classification is available both in the ungauged [36] and gauged theories [37]. A strong motivation for our work comes from the AdS 4 /CFT 3 correspondence, which has been attracting much attention in the last months, after the discovery of superconformal field theories describing coincident M2-branes [38,39]. In this context, supergravity vacua with less supersymmetry correspond on the CFT side to nonzero vacuum expectation values of certain operators, or to deformations of the CFT. Disposing of a systematic classification of supergravity vacua is thus particularly useful. Of special interest in this context are domain wall solutions interpolating between vacua preserving different amounts of supersymmetry, because they can describe a holographic RG flow.
The case where the Killing vector constructed from the Killing spinor is timelike was considered in [33], so we will now concentrate on the null class. Note that this is more than a mere extension of [33]: The timelike case typically contains black hole solutions, while the lightlike class includes gravitational waves and domain walls, whose importance in an AdS/CFT context was just explained.
The outline of this paper is as follows. In section 2, we briefly review N = 2 supergravity in four dimensions and its matter couplings, whereas in 3 the orbits of Killing spinors are discussed. In section 4 we determine the conditions coming from a single null Killing spinor and give explicit examples for supersymmetric backgrounds. Finally, in section 5, we impose a second Killing spinor and obtain the additional constraints obeyed by half-supersymmetric solutions. It is shown that half-BPS geometries are divided into four classes, that include bubbles of nothing which are asymptotically AdS 4 , pp-waves on domain walls, AdS 3 × R, and spacetimes conformal to AdS 3 times an interval. Appendices A and B contain our notation and conventions for spinors.

Matter-coupled N = 2, D = 4 gauged supergravity
In this section we shall give a short summary of the main ingredients of N = 2, D = 4 gauged supergravity coupled to vector-and hypermultiplets [40]. Throughout this paper, we will use the notations and conventions of [41], to which we refer for more details.
Apart from the vierbein e a µ and the chiral gravitinos ψ i µ , i = 1, 2, the field content includes n H hypermultiplets and n V vector multiplets enumerated by I = 0, . . . , n V . The latter contain the graviphoton and have fundamental vectors A I µ , with field strengths The fermions of the vector multiplets are denoted as λ αi and the complex scalars as z α where α = 1, . . . , n V . These scalars parametrize a special Kähler manifold, i. e. , an n V -dimensional Hodge-Kähler manifold that is the base of a symplectic bundle, with the covariantly holomorphic sections where K is the Kähler potential and D denotes the Kähler-covariant derivative 1 . V 1 For a generic field φ α that transforms under a Kähler transformation K(z,z) → K(z,z) + Λ(z) + Λ(z) as φ α → e −(pΛ+qΛ)/2 φ α , one has D α φ β = ∂ α φ β + Γ β αγ φ γ + p 2 (∂ α K)φ β . Dᾱ is defined in the same way. X I transforms as X I → e −(Λ−Λ)/2 X I and thus has Kähler weights (p, q) = (1, −1). obeys the symplectic constraint To solve this condition, one defines .
F is a homogeneous function of degree two, called the prepotential, whose existence is assumed to obtain the last expression. This is not restrictive because it can be shown that it is always possible to go in a gauge where the prepotential exists via a local symplectic transformation [41,42] 2 . The Kähler potential is then The matrix N IJ determining the coupling between the scalars z α and the vectors A I µ is defined by the relations the following differential constraints hold: Here, C αβγ is a completely symmetric tensor which determines also the curvature of the special Kähler manifold. We now come to the hypermultiplets. These contain scalars q X and spinors ζ A , where X = 1, . . . , 4n H and A = 1, . . . , 2n H . The 4n H hyperscalars parametrize a quaternionic Kähler manifold, with vielbein f iA X and inverse f X iA (i. e. the tangent space is labelled by indices (iA)). From these one can construct the three complex structures with the Pauli matrices σ j i (cf. appendix A). Furthermore, one defines SU(2) connections ω X by requiring the covariant constancy of the complex structures: where the Levi-Civita connection of the metric g XY is used. The curvature of this SU(2) connection is related to the complex structure by Depending on whether κ = 0 or κ = 0 the manifold is hyper-Kähler or quaternionic Kähler respectively. In what follows, we take κ = 1. The bosonic action of N = 2, D = 4 supergravity is where C I,JK are real coefficients, symmetric in the last two indices, with Z I Z J Z K C I,JK = 0, and the covariant derivatives acting on the scalars read Here k α I (z) and k X I (q) are Killing vectors of the special Kähler and quaternionic Kähler manifolds respectively. The potential V in (2.12) is the sum of three distinct contributions: with 15) and the triple moment maps P I (q). The latter have to satisfy the equivariance condition which is implied by the algebra of symmetries. The metric for the vectors is given by where F IJ = ∂ I ∂ J F , and F denotes the prepotential. Finally, the supersymmetry transformations of the fermions to bosons are where we defined In (2.19), A µ is the gauge field of the Kähler U(1), with the moment map function and The major part of this paper will deal with the case of vector multiplets only, i. e. , n H = 0. Then there are still two possible solutions of (2.16) for the moment maps P I , which are called SU(2) and U(1) Fayet-Iliopoulos (FI) terms respectively [41]. Here we are interested in the latter. In this case where e is an arbitrary vector in SU(2) space and ξ I are constants for the I corresponding to U(1) factors in the gauge group. If, moreover, we assume f IJ K = 0 (abelian gauge group), and k α I = 0 (no gauging of special Kähler isometries), then only the V 3 part survives in the scalar potential (2.14), and one can also choose C I,JK = 0. Note that this case corresponds to a gauging of a U(1) subgroup of the SU(2) R-symmetry, with gauge field ξ I A I µ .

Orbits of spinors under the gauge group
A Killing spinor 3 can be viewed as an SU(2) doublet (ǫ 1 , ǫ 2 ), where an upper index means that a spinor has positive chirality. ǫ i is related to the negative chirality spinor ǫ i by charge conjugation, ǫ C i = ǫ i , with Here C is the charge conjugation matrix defined in appendix B. As ǫ 1 has positive chirality, we can write ǫ 1 = c1 + de 12 for some complex functions c, d. Notice that c1 + de 12 is in the same orbit as 1 under Spin(3,1), which can be seen from e γΓ 13 e ψΓ 12 e δΓ 13 e hΓ 02 1 = e i(δ+γ) e h cos ψ 1 + e i(δ−γ) e h sin ψ e 12 .
Having fixed ǫ 1 = 1, also ǫ 1 is determined by ǫ 1 = ǫ 1C = e 1 . A negative chirality spinor independent of ǫ 1 is ǫ 2 , which can be written as a linear combination of odd forms, ǫ 2 = ae 1 + be 2 , where a and b are again complex valued functions. We can now act with the stability subgroup of ǫ 1 to bring ǫ 2 to a special form: (1 + µX + νY )(ae 1 + be 2 ) = be 2 + [a − 2b(µ + iν)]e 1 . 3 Our conventions for spinors and their description in terms of forms can be found in appendix B.
In the case b = 0 this spinor is invariant, so the representative is ǫ 1 = 1, ǫ 2 = ae 1 (so that ǫ 2 =ā1), with isotropy group R 2 . If b = 0, one can bring the spinor to the form be 2 (which implies ǫ 2 = −be 12 ), with isotropy group I. The representatives 4 together with the stability subgroups are summarized in table 1. Given a Killing spinor ǫ i , one can construct the bilinear with the Majorana inner product A defined in (B.4), and the sum over i is understood. For ǫ 2 = ae 1 , V A is lightlike, whereas for ǫ 2 = be 2 it is timelike, see table 1. The existence of a globally defined Killing spinor ǫ i , with isotropy group G ∈ Spin(3,1), gives rise to a G-structure. This means that we have an R 2 -structure in the null case and an identity structure in the timelike case. In U(1) gauged supergravity, the local Spin(3,1) invariance is actually enhanced to Spin(3,1) × U(1). For U(1) Fayet-Iliopoulos terms, the moment maps satisfy (2.23), where we can choose e x = δ x 3 without loss of generality. Then, under a gauge transformation the Killing spinor ǫ i transforms as which can be easily seen from the supercovariant derivative (cf. eq. (2.19)). Note that ǫ 1 and ǫ 2 have opposite charges under the U(1). In order to obtain the stability subgroup, one determines the Lorentz transformations that leave the spinors ǫ 1 and ǫ 2 invariant up to arbitrary phase factors e iψ and e −iψ respectively, which can then be gauged away using the additional U(1) symmetry. If ǫ 2 = 0, one gets in this way an isotropy group generated by X, Y and Γ 13 obeying i. e. G ∼ = U(1)⋉R 2 . For ǫ 2 = ae 1 with a = 0, the stability subgroup R 2 is not enhanced, whereas the I of the representative (ǫ 1 , ǫ 2 ) = (1, be 2 ) is promoted to U(1) generated by Finally, notice that in U(1) gauged supergravity one can choose the function a in ǫ 2 = ae 1 real and positive: Write a = R exp(2iδ), use e δΓ 13 1 = e iδ 1 , e δΓ 13 ae 1 = e −iδ ae 1 = e iδ Re 1 , and gauge away the phase factor exp(iδ) using the electromagnetic U(1). Table 1: The representatives (ǫ 1 , ǫ 2 ) of the orbits of Weyl spinors and their stability subgroups G under the gauge groups Spin(3,1) and Spin(3,1) × U(1) in the ungauged and U(1)gauged theories, respectively. The number of orbits is the same in both theories, the only difference lies in the stability subgroups and the fact that a is real in the gauged theory. In the last column we give the vectors constructed from the spinors.
Note that in the gauged theory the presence of G-invariant Killing spinors will in general not lead to a G-structure on the manifold but to stronger conditions. The structure group is in fact reduced to the intersection of G with Spin(3,1), and hence is equal to the stability subgroup in the ungauged theory.
The representatives, stability subgroups and vectors constructed from the Killing spinors are summarized in table 1 both for the ungauged and the U(1)-gauged cases.
4. Null representative (ǫ 1 , ǫ 2 ) = (1, ae 1 ) In this section we will analyze the conditions coming from a single null Killing spinor, and determine all supersymmetric solutions in this class. We shall first keep things general, i. e. , including hypermultiplets and a general gauging, and write down the linear system following from the Killing spinor equations. This system will then be solved for the case of U(1) Fayet-Iliopoulos terms and without hypers, while the solution in the general case will be left for a future publication. As was explained before, it is always possible to choose a real and positive, so we shall set a = e χ in what follows.

Conditions from the Killing spinor equations
From the vanishing of the hyperini variation one obtains whereas the gaugino variation yields It is straightforward to show that the equations (4.3)-(4.6) imply that Finally, from the gravitini we get with the gauged SU(2) connection From equations (4.13) and (4.14) one obtains and with As the (n V + 1) × (n V + 1) matrix (Z I , DᾱZ I ) is invertible [41], eqns.
Moreover, antiselfduality implies that so that all fluxes except F −I+• =: ψ I are fixed. Using (4.18), eqns. (4.11) and (4.12) become from which one can determine some components of the spin connection and the gauge potential A as follows: First of all, (2.15) permits to decompose ψ I in a graviphoton part ψ and matter vector part ψ α as where Then, the sum of the real parts of (4. 19) and (4.20) yields On the other hand, the difference of the real parts of (4. 19) and (4.20) gives where we defined Plugging (4.24) into (4.23) one gets From the sum of the imaginary parts of (4.19) and (4.20) we have Finally, the difference of the imaginary parts of (4.19) and (4.20) yields Summarizing, the components ω −• , ω +− and ω •• are fixed by the supersymmetry conditions, while the remaining components will be determined below by imposing vanishing torsion.
In order to obtain the spacetime geometry, we consider the spinor bilinears where the Majorana inner product is defined in (B.4). The nonvanishing components are This yields for the trace part Using the identities it is straightforward to shew that the linear system (4.11) -(4.14) implies the following constraints: These equations are easily shown to be equivalent to The next step is to impose zero torsion. The torsion two-form reads From the vanishing of T − one gets E − ∧ dE − = 0, so by Fröbenius' theorem there exist two functions H and u such that locally Let us introduce a coordinate v such that Since V is proportional to E + as a vector, and E + , E − = 0, u is independent of v, and thus can be used as a further coordinate. Taking into account that Up to now, our discussion is completely general, i. e. , it includes hypermultiplets and a general gauging. In the remainder of this paper, we shall specialize to the case without hypers and no gauging of special Kähler isometries (k α I = 0). The inclusion of hypermultiplets will be studied in a forthcoming publication. This leaves two possible solutions for the moment maps [41], namely SU(2) or U(1) Fayet-Iliopoulos (FI) terms. We shall consider here the latter, which satisfy (2.23), where e x = δ x 3 without loss of generality 6 . One has then and D µ z α = ∂ µ z α . Equ. (4.18) implies then for the fluxes while (4.8) leads to the flow equation for the scalars. Notice that the special U(1)⋉R 2 orbit with representative (ǫ 1 , ǫ 2 ) = (1, 0), that can be obtained in the limit χ → −∞, cannot occur in the FI case with nontrivial scalar 6 e x = δ x 3 can always be achieved by a global SU(2) rotation (which is a symmetry of the theory).
fields: Multiplying (4.20) with e χ and letting χ → −∞ yields (with A 1 2 = 0) gS 12 = 0, so that either g = 0 (ungauged case) or ξ I Z I = 0, which implies ξ I D α Z I = 0, so that the scalars are constant (cf. section 4.3.1). In the presence of hypermultiplets and general gauging however, this orbit might occur, so there would be one more representative to consider.
In order to proceed, it is convenient to distinguish two subcases, namely dχ = 0 and dχ = 0.

Constant Killing spinor, dχ = 0
If dχ = 0, equation (4.24) reduces to and thus Imψ = 0 and C 3 = 0, which implies χ = 0. Let us denote the remaining two coordinates by w,w (withw the complex conjugate of w) and define G ≡ E + u , so that the null tetrad reads To simplify E • , first perform a diffeomorphism This eliminates E •w . Notice that due to the Cauchy-Kovalevskaya theorem, it is always possible to solve (4.39) locally for w ′7 . Finally, the component E • u can be removed using the residual gauge freedom, given by the stability subgroup R 2 of the null spinor. To see this, consider an R 2 transformation with group element where X and Y are given in (3.3). Defining α = µ + iν, this can also be written as Given the ordering A, B = +, −, •,•, the Lorentz transformation matrix a AB corresponding to Λ ∈ R 2 ⊆ Spin(3, 1) reads (4.41) The transformed vierbein α E A = a A B E B is thus given by without loss of generality. Then the inverse tetrad reads In what follows we shall set E • w ≡ ρe iζ . Equ. (4.27) reduces to while (4.7) and (4.8) lead to and respectively. (4.45) implies that the scalars are independent of v and thus A v = 0. The vanishing of the torsion gives the missing components ω +• of the spin connection (that we do not list here), plus the additional constraints on the null tetrad.
All that remains to be done at this point is to impose the Bianchi identities and the Maxwell equations, which read respectively The fluxes can be obtained by setting χ = 0 in (4.37), One finds that the Bianchi identities imply ∂ v ψ I = 0 and Note that imposing dF I = 0 is actually not sufficient; we must also ensure that ξ I F I = ξ I dA I , because the linear combination ξ I A I is determined by the Killing spinor equations (cf. (4.44)). This leads to the additional condition In conclusion, the null tetrad (with the exception of E + u = G), gauge fields and scalars are determined by the coupled system (4.45)-(4.50) and (4.53)-(4.55). Finally, the wave profile G is fixed by the uu component of the Einstein equations, which are given in (C.1) of [33], where in our case Then, as was shown in [33], all other equations of motion are automatically satisfied.
Notice that the above system is Kähler-covariant, as it must be: Under a Kähler transformation the Killing spinors transform as In order for our representative (ǫ 1 , ǫ 2 ) = (1, e 1 ) to be invariant, this must be compensated by a Spin (3,1) Taking into account (4.57), (4.59), as well as it is easy to show that the system (4.45)-(4.50) and (4.53)-(4.55) is Kähler-covariant.
In what follows, we shall obtain explicit solutions in some special cases.

Constant scalars
If we assume ξ I D α Z I = 0 , (4.60) the flow equation (4.46) implies ∂ w z α = 0. Actually, since the scalar potential (4.60) forces the scalars to be constant, i. e., they do not depend onw and u either 8 . One has then A µ = 0, so that (4.50) and the complex conjugate of (4.48) give and thus By a diffeomorphism w ′ = F (u, w) combined with a local Lorentz transformation (4.40) to eliminate E • u one can thus set (after dropping the primes) E • = H −1/2 dw without loss of generality, so that ρ = H −1/2 , ζ = 0. From (4.47) one obtains and hence E + w = ∂ w m for some real function m that can always be set to zero by shifting v and G. Finally, (4.48) yields where we defined the constant and A(u) is an arbitrary real function. The metric has the form of a Lobachevski wave on AdS, Note that, by shifting w, G and v appropriately, one can always achieve A(u) = 0.
To obtain the gauge fields, observe that the Bianchi identities (4.53) imply that ∂ w (ψ I H −3/2 ) is real, From the Maxwell equations (4.54) one concludes that N IJ λ J must be real as well, and thus (Im N ) IJ λ J = 0 . (4.67) As Im N is invertible, this yields λ I = 0, so that for some function ρ I (u, w). Taking into account (4.60), equ. (4.21) gives Since ψ must be real, this implies that the linear combination ξ I ρ I can depend on u only. Then, one easily shows that (4.55) is automatically satisfied. Analogous to (4.21), we can decompose ρ I = D α X I ρ α + iX I ρ , (4.69) where (using ξ I ρ I = iξ IX I ρ) The fluxes are thus given by with ρ I satisfying (4.69) and (4.70). Note that this solution with constant scalars includes the one in minimal gauged supergravity found in [7].

Prepotential
We now consider a simple model determined by the prepotential that has n V = 1 (one vector multiplet), and thus just one complex scalar z. Choosing Z 0 = 1, Z 1 = z (cf. [41]), the symplectic vector v reads (4.73) The Kähler potential, metric and kinetic matrix for the vectors are given respectively by e −K = 2(z +z) , Note that positivity of the kinetic terms in the action requires Rez > 0. For the scalar potential one obtains which has an extremum at z =z = |ξ 0 /ξ 1 |. In what follows we assume ξ I > 0. The Kähler U(1) is In order to solve the system (4.45)-(4.50) and (4.53)-(4.55) we shall take z =z (this includes the extremum of the potential, and thus the AdS vacuum) and ζ = E + w = E +w = ψ I = 0. Then, A µ = 0, and the only nontrivial equations are (4.46), (4.48) and (4.50), which become (4.80) (4.79) and the complex conjugate of (4.80) can be combined to give and hence where g(u,w) denotes an arbitrary function. Because ρ √ H is real, g(u,w) can depend only on u. As was explained in section 4.3.1, one can set g(u) = 1 without loss of generality by a combination of a diffeomorphism w → w/g(u) and a local Lorentz transformation (4.40), so that ρ = H −1/2 . From (4.78) and (4.79) we get with f (u) an arbitrary function that we will take equal to one in the following. Since z is real, (4.78) yields ∂ x H = 0, where we introduced the real coordinates x, y by w = x + iy. Let us further assume that also ∂ u H = 0, so that H (and thus, by virtue of (4.83), also z) depends only on y. Then, the flow equation (4.78) together with ρ = H −1/2 and (4.83) implies where the integration constant was chosen such that the scalar goes to its critical value for y → 0. The metric becomes where G is determined by the uu component of the Einstein equations. (4.85) describes a gravitational wave propagating on a domain wall. For z → ξ 0 /ξ 1 (y → 0), the geometry becomes that of a wave on AdS 4 . Note that perhaps some of the assumptions made above (like reality of z) can be relaxed while maintaining integrability of the equations. Another possible generalization is the inclusion of nonvanishing gauge fields. This will be done in section 5.2.

Ungauged case
Finally let us check if we correctly reproduce the results of [4] in the ungauged case. If g = 0, the flow equation (4.46) implies ∂ w z α = 0, and thus z α = z α (w, u). Using (2.20), this gives so that (4.50) leads to This can be integrated to give with h(w, u) an arbitrary function that can be set to zero without loss of generality by a combination of a diffeomorphism w → w ′ (u, w) and a local Lorentz transformation (4.40). From (4.48) we have ∂ w H = ∂wH = 0, and hence H = H(u) so that we can set H = 1 by a redefinition of u.
In conclusion, the metric is given by The scalars are arbitrary functions ofw, u, and the gauge fields read with ψ I determined by Plugging this into T − = 0 we obtain d e 2χ − 1 E − = 0 , and therefore one can introduce a function u such that On the other hand, (4.27) gives where we defined A = A I ξ I .
We already introduced the coordinates u, v. Using (4.92) together with V = ∂ v = − √ 2(1 + e 2χ )E + , we get ∂ v , dχ = 0, and thus ∂χ/∂v = 0, so that χ is independent of v. Furthermore, (4.92) and (4.93) imply that du ∧ dχ = 0, therefore the function χ must depend nontrivially on the two remaining coordinates. This allows to choose χ as a further coordinate. Finally, the fourth coordinate will be called Ψ. Notice that due to V, E • = 0, A has no v-component, A v = 0. Now we employ the R 2 stability subgroup of the null spinor (cf. (4.42)) to set and hence Imψ = 0. Using also we can proceed to impose vanishing torsion. T • = 0 determines the following components of the spin connection: whereas T + = 0 gives together with the constraint that determines E + χ and E + Ψ . In (4.98) we introduced the indices m, n, . . . = χ, Ψ, and the convention ǫ χΨ = 1. The Kähler-covariant derivatives D u appearing in (4.98) are defined as (As we remarked in section 4.2, in order for the spinor representative to be invariant under a Kähler transformation, one must compensate with a Spin(3,1) transformation, which acts also on E • , E•).
Finally we have to ensure that the Maxwell equations and Bianchi identities hold. From ξ I F I = ξ I dA I we obtain Imposing the Bianchi identities, dF I = 0, one gets ∂ v ψ I = 0 and The Maxwell equations dRe N IJ F +J = 0 yield in addition Notice that (4.7) implies ∂ v z α = 0. Using this together with the fact that ∂ v is Killing, one easily shows that all components of the vierbein do not depend on v either. The flow equation (4.38) becomes In conclusion, the coupled system (4.98), (4.99), (4.100), (4.101), (4.102) and (4.103) determines the components of the null tetrad (except E + u = G), the functions ψ I and the scalar fields z α . The fluxes F I are then given by (4.37). Finally, the wave profile G is fixed by the uu component of the Einstein equations (cf. (C.1) of [33]), where in our case and we defined Then, as was shown in [33], all other equations of motion are automatically satisfied. Note that R uu in (4.104) can be rewritten in a manifestly real form, but then the expression becomes considerably longer.
In the next subsection we shall obtain an explicit solution to the above equations.

Explicit solutions for dχ = 0
If one sets ψ I = A u = A χ = E + χ = E + Ψ = 0 and z α = z α (χ), A Ψ = A Ψ (χ), the only nontrivial equations are (4.99) and (4.103), which reduce to Note that (4.106) can also be written in the form sinh 2χ∂ χ z α = g αβ ∂βW , (4.107) with the superpotential W = ln(ξ IZ I ) + K. In what follows, we solve (4.105) and (4.106) for the simple model with prepotential F = −iZ 0 Z 1 introduced in section 4.3.2. Assuming in addition that the single scalar z = Z 1 is real, (4.105) and (4.106) become with the solution where c,c are integration constants. Finally, for the metric and the nonvanishing components of the fluxes one obtains respectively Notice that z ± in (4.108) are related by the strong-weak coupling duality that sends z + to z − and vice versa. (4.112) is actually a residual Z 4 symmetry that remains of the full symplectic duality group Sp(4, R) after the gauging: In the notation of [41], it corresponds to Let us now briefly discuss the properties of the spacetime (4.110). Introducing the new radial coordinate ρ = √ coth χ, we have ρ → ∞ for χ → 0 + and ρ → 1 for χ → ∞. Asymptotically for ρ → ∞ one has z ± → ξ 0 /ξ 1 (the vacuum), and the metric approaches which represents a Lobachevski wave on AdS 4 . On the other hand, for ρ → 1, z + → 2c, z − → ξ 2 0 /(2cξ 2 1 ), A ± Ψ goes to a constant, and where we defined R = arcoshρ. From this it is evident that in order for the metric to be regular at ρ = 1, one must identify 9 As the spacetime ends at ρ = 1, (4.110) can be interpreted as a (wave on a) bubble of nothing [43,44] that asymptotes to (a wave on) AdS 4 . Notice that, in order to have a well-defined limit for the case of constant scalars (c = 0), one must choosec ∝ c −1/2 .

Half-supersymmetric backgrounds
Let us now investigate the additional conditions satisfied by half-supersymmetric vacua. Again, we will do this separately for dχ = 0 and dχ = 0. As the stability subgroup of the first Killing spinor was already used, the second one cannot be simplified anymore, and is thus of the general form where a, b, c, d are complex-valued functions.

Case dχ = 0
From δψ i + = 0 one obtains while δψ i − = 0 leads to (using also (5.1)-(5.4)) The integrability conditions of the system (5.1)-(5.8) imply that Making use of (5.9) and (5.10), the remaining gravitino variations δψ i and The vanishing of the gaugino variations yields the additional conditions as well as ∂ u z α = 0. This implies A u = 0, so that (5.11) simplifies to where in the last step we used the gauge condition (4.95). Thus, equ. (5.19) reduces to Because ψ = 2i (ImN ) IJ X I ψ J , we have moreover Since the (n V + 1) × (n V + 1) matrix (X I , DᾱX I ) is invertible, (5.21) and (5.22) imply while (5.17) and (5.18) yield Using (5.23) and (5.28), it is easy to show that (5.27) is equivalent to (4.100) that follows from ξ I F I = ξ I dA I . Moreover, the Bianchi identities (4.101) are automatically satisfied once (5.23) and (5.28) hold. Note also the similarity between (5.27) and (4.99). Equ. (4.98) becomes where Ξ(u, χ, Ψ) denotes an arbitrary function obeying ∂ Ψ Ξ = 0 11 . Furthermore, (5.25), (5.26) and (5.29) imply for E + with Λ(u, χ, Ψ) again a function that can be chosen at will. By shifting v one can set Λ = 0 without loss of generality. Finally, we may employ the residual gauge freedom related to the choice of the coordinate Ψ that consists in sending Ψ → f (u, χ, Ψ), where the only constraint on f is ∂ Ψ f = 0 12 . Choosing f = Ξ we have then and thus E + u = E + χ = E + Ψ = 0. This means that the most general half-supersymmetric background in this class is given by , (5.33) The bilinear V µ = A(ǫ i , Γ µ ǫ i ) associated to the second covariantly constant spinor has norm squared which is in general negative, so that the solution belongs also to the timelike class studied in [33]. For a 0 = 0 and χ > 0, the second Killing vector is given by V = ∂ u . Note that the uu component of the Einstein equations is identically satisfied for the half-supersymmetric backgrounds. This is not surprising, since they belong also to the timelike class, where the Killing spinor equations imply all the equations of motion [33].

Case dχ = 0
From the vanishing of the gravitino variation we obtain the system while the gaugino supersymmetry transformations yield that will be useful later. Equ. (5.44) allows to determine ψ I , where f (u, w) denotes an arbitrary function that can be set to one without loss of generality by a reasoning analogous to that following (4.63). Plugging (5.50) into (4.47) leads to E + w = ∂ w m, with m some real function. By shifting v and G appropriately, one can always achieve m = 0. (4.48) simplifies to and thus the scalars are independent of w +w as well. The remaining integrability conditions for the system (5.38)-(5.41) turn out to be 0 = X ·ξ and Making use of (5.58) in the complex conjugate of (4.55) (recall that ψ is real) yields hence ψ = ψ(w −w, u). Plugging the eqns. (5.45) as well as the contraction of (5.46) with ξ I into (5.55), one finds that the latter is identically satisfied. From (5.46) and (5.56) one obtains and the constraint where we used that follow from (4.55) and (5.52). Note that above we assumed that g αβ D α X I DβX J ξ I ξ J is nonvanishing. If this expression were zero, then ξ I D α X I = 0, since the Kähler metric g αβ is non-degenerate. As a consequence, the scalars are constant, as can be seen from (4.61). This case was considered in section 4.3.1 and will not be pursued further here. In addition to (5.59)and (5.62), the function G must obey the uu component of the Einstein equations, that becomes where l I (u) are arbitrary real functions of u obeying the constraint l I ξ I = 0 due to (5.58), and the factor 2 was chosen for later convenience. By virtue of (5.54), N IJ is likewise independent of w +w, and hence (4.54) implies with m I (u) again some real functions. Since (Im N ) IJ is invertible, (5.65) together with (5.66) give an expression for ψ I in terms of l I and m I , In what follows, we shall solve the above equations for the simple model with prepotential F = −iZ 0 Z 1 introduced in section 4.3.2. In this case one has together with k(u) = 0, i.e., The scalar field z and metric are given by (4.84) and (4.85) respectively, and the fluxes read Note that for y → 0, when the scalar goes to its critical value, we have H → 8ξ 0 ξ 1 g 2 y 2 , and thus (if we choose h(u) = 0) the wave profile becomes which means that the solution reduces to a subclass of the charged generalization of the Kaigorodov spacetime found in [45]. This concludes the explicit example of a half-supersymmetric background with dχ = 0, ψ = 0.
Next we have to consider the case when ψ = 0. Then, (4.44) and (4.55) give Contracting (5.46) with ξ I , taking into account that ψ = ψ I ξ I = 0 and assuming as before that g αβ D α X I DβX J ξ I ξ J = 0 (otherwise, as was explained above, the scalar fields would be constant), we get with λ a constant. Let us first assume that bX·ξ +dX·ξ = 0, so that X·ξX·ξ is constant due to (5.45). Then, the Killing spinor equations for b and d simplify to so that E + is constrained by Integrating the last two equations, one has with E(u) some imaginary function. This implies for some real function m. By shifting v and G appropriately, we can thus set E + w = E +w = 0. Then, (5.96) gives The latter relation determines E + u , 2ig 2X·ξX·ξ +Ê(u) exp 2ig 2X·ξX·ξ(w −w) , (5.102) withÊ(u) real and otherwise arbitrary. Note that E + u is independent of w +w. Eqns. (4.46), (5.42) and (5.43) boil down to ∂ u z α = 0 and While the lhs is independent of x, the prefactor of E + u on the rhs depends in general nontrivially on x. This is compatible only if E + u = 0, hence α(u) is constant due to (5.101). The function H appearing in the metric follows from (4.48), yielding where y = (w −w)/2i, and we can always choose h(u) = −2 √ 2 by redefining the coordinate u.
In conclusion, the metric is given by ds 2 = 2 exp 4g 2X·ξX·ξy dudv + dy 2 + dx 2 , (5.105) which is simply AdS 3 ×R. The dependence of the scalars on the R-coordinate x is governed by (5.103), that can be rewritten as where the constant C is defined by The solution to the Killing spinor equations reads and norm squared which is negative unless λ 1 = 0 or Re α = 0, so in general the solution again belongs also to the timelike class.
The final case to consider is ψ = 0, bX·ξ +dX·ξ = 0. Then, equ. (5.47) together with (5.90) implies that One easily shows that ∂ µ A ν − ∂ ν A µ = 0, and thus A µ = ∂ µ ς for some v-independent function ς. Using this and (4.48), we can integrate (4.50) to obtain where f (u, w) denotes an arbitrary function that, as was explained before, can be set to unity without loosing generality. Then one has ρ = H −1/2 and ζ = −ς, and (4.47) yields E + w = E +w = 0. The Killing spinor equations for b reduce to where κ is a constant that satisfiesλκ = λκ due to (5.88). The Killing spinor equations for a boil down to whereâ satisfies where ω 1 and ω 3 are real. Shifting the coordinate v one can set ω 3 = 0 without loss of generality. Plugging back this expression for E + u into (5.119) one gets Note that E + u must in addition satisfy the uu component of the Einstein equations, namely Solving these equations in general seems to be difficult. A simplification can be made by assuming a =c, which happens for ∂ ub = ∂ uâ = 0, λâ +λā = λκ. If we take in addition ω 1 = ω 2 = ω 3 = 0 (and thus E + u = 0), (5.119) is satisfied. Moreover, from (5.114) one gets ∂ u H = 0, and (5.42) yields ∂ u z α = 0. Note thatb can be set to zero by a constant shift of w, cf. (5.113). Using (5.116), the equations (4.46) and (5.43) simplify to which imply (w∂ w +w∂w)z α = 0 , (5.124) i.e., ∂ r z α = 0, where we introduced polar coordinates r, θ according to w = re iθ . The scalar fields depend thus on the angular coordinate θ only. This, in turn, gives A u = A r = 0 and ς = ς(θ). By means of the separation ansatz √ H = rh(θ), (4.48) becomes Solving (5.125) for X·ξe iς and plugging the result into (5.127), one finds that (5.127) holds identically. In conclusion, the unknown functions z α , h and ς are determined by the system of ordinary differential equations (5.125) and (5.126). In the following, we shall solve these equations for the SU(1, 1)/U(1) model with prepotential F = (Z 1 ) 3 /Z 0 .
The Kähler potential and metric are given respectively by so we must have Imz > 0. For the scalar potential one obtains Notice that this model permits to introduce a gauging without having a scalar potential, by choosing ξ 1 = 0, ξ 0 = 0. Solving (5.125) for e i(θ−ς) and plugging into (5.126) gives in general Positivity of Imz restricts θ to the range 0 < θ < π/2. Using (5.135) in the sum of (5.132) and its complex conjugate allows to determine also Rez. Eventually this leads to Antisymmetric tensors are often contracted with Γ-matrices as in Γ · F ≡ Γ ab F ab . Moreover, we defined X·ξ ≡ X I ξ I . i, j, . . . = 1, 2 are SU(2) indices, whose raising and lowering is done by complex conjugation. The Levi-Civita ǫ ij has the property where in principle ǫ ij is the complex conjugate of ǫ ij , but we can choose ǫ = iσ 2 , such that ǫ 12 = ǫ 12 = 1 . (A.6) The Pauli matrices σ xi j (x = 1, 2, 3) are given by They allow to switch from SU(2) indices to vector quantities using the convention At various places in the main text we use σ-matrices with only lower or upper indices, defined by σ ij ≡ σ k i ǫ kj , i σ ij = (i σ ij ) * . (A.9) Notice that both σ ij and σ ij are symmetric. Spinors carrying an index i are chiral, e.g. for the supersymmetry parameter one has Γ 5 ǫ i = ǫ i , Γ 5 ǫ i = −ǫ i , (A.10) and the same holds for the gravitino ψ i µ . Note however that for some spinors, the upper index denotes negative chirality rather than positive chirality, for instance the gauginos obey Γ 5 λ αi = −λ αi , as is also evident from the supersymmetry transformations. The charge conjugate of a spinor χ is χ C = Γ 0 C −1 χ * , (A. 12) with the charge conjugation matrix C. Majorana spinors are defined by χ = χ C , and chiral spinors obey χ C i = χ i .
One easily checks that this representation of the gamma matrices satisfies the Clifford algebra relations {Γ a , Γ b } = 2η ab . The parity matrix is defined by Γ 5 = iΓ 0 Γ 1 Γ 2 Γ 3 , and one finds that the even forms 1, e 12 have positive chirality, Γ 5 η = η, while the odd forms e 1 , e 2 have negative chirality, Γ 5 η = −η, so that ∆ c decomposes into two complex chiral Weyl representations ∆ + c = Λ even (U ⊗ C) and ∆ − c = Λ odd (U ⊗ C). Note that Spin(3,1) is isomorphic to SL(2, C), which acts with the fundamental representation on the positive chirality Weyl spinors. Let us define the auxiliary inner product In many applications it is convenient to use a basis in which the gamma matrices act like creation and annihilation operators, given by The