IIB backgrounds with five-form flux

We investigate all N=2 supersymmetric IIB supergravity backgrounds with non-vanishing five-form flux. The Killing spinors have stability subgroups $Spin(7)\ltimes\bR^8$, $SU(4)\ltimes\bR^8$ and $G_2$. In the $SU(4)\ltimes\bR^8$ case, two different types of geometry arise depending on whether the Killing spinors are generic or pure. In both cases, the backgrounds admit a null Killing vector field which leaves invariant the $SU(4)\ltimes \bR^8$ structure, and an almost complex structure in the directions transverse to the lightcone. In the generic case, the twist of the vector field is trivial but the almost complex structure is non-integrable, while in the pure case the twist is non-trivial but the almost complex structure is integrable and associated with a relatively balanced Hermitian structure. The $G_2$ backgrounds admit a time-like Killing vector field and two spacelike closed one-forms, and the seven directions transverse to these admit a co-symplectic $G_2$ structure. The $Spin(7)\ltimes\bR^8$ backgrounds are pp-waves propagating in an eight-dimensional manifold with holonomy $Spin(7)$. In addition we show that all the supersymmetric solutions of simple five-dimensional supergravity with a time-like Killing vector field, which include the $AdS_5$ black holes, lift to $SU(4)\ltimes\bR^8$ pure Killing spinor IIB backgrounds. We also show that the LLM solution is associated with a co-symplectic co-homogeneity one $G_2$ manifold which has principal orbit $S^3\times S^3$.


Introduction
Supersymmetric IIB backgrounds with "active" five-form fluxes and with vanishing oneand three-form field strengths have been extensively investigated in the context of string theory, branes and black holes. Examples of such backgrounds are the Freund-Rubin space AdS 5 × S 5 [1], the D3-brane [2] and the maximally supersymmetric plane wave [3] solutions which have been instrumental in the formulation and understanding of the AdS/CFT correspondence [4]. More recently, in the same context many new solutions of increasing complexity have been found preserving some supersymmetry. These include the bubbling solutions of [5] and the lift of AdS 5 black holes [6] to IIB supergravity [7,8].
Motivated by the above widespread applications, we present a systematic investigation of all supersymmetric IIB supergravity backgrounds with active five-form flux F . This is based on our solution [9,10] of the Killing spinor equations of IIB supergravity [11,1,12] for one Killing spinor, using the spinorial geometry technique of [13]. Although, the supersymmetric backgrounds with F flux are special cases of the N = 1 IIB backgrounds, there are some differences. Unlike generic IIB supersymmetric backgrounds, backgrounds with (only) F flux always preserve an even number of supersymmetries. So the backgrounds we shall investigate will preserve at least two supersymmetries, N ≥ 2. In addition, the vanishing of one-and three-form fluxes imposes additional conditions on the geometry. It turns out that the geometry of the supersymmetric backgrounds with F flux is rather restricted and the five-form field strength F takes a simple form.
In analogy with the generic N = 1 IIB backgrounds, the N = 2 supersymmetric IIB backgrounds with F flux can be separated into three classes distinguished by the stability subgroups of the Killing spinors in Spin(9, 1). These are Spin(7) ⋉ R 8 , SU(4) ⋉ R 8 and G 2 . We find that the Spin(7) ⋉ R 8 backgrounds are pp-waves with rotation and null F flux propagating on a holonomy Spin(7) manifold whose metric depends on a wave profile coordinate.
The geometry of the SU(4) ⋉ R 8 backgrounds is more subtle. These are further divided into two subclasses, the generic and the pure spinor backgrounds, which have distinct geometries. The Killing spinors of the former backgrounds do not obey additional conditions apart from those imposed by SU(4) ⋉ R 8 invariance up to a possible conjugation with a Spin(9, 1) transformation. The Killing spinors of the latter are pure SU(4) ⋉ R 8 -invariant spinors. One consequence of this is that the stability subgroups of the spinors in Spin(9, 1) are not sufficient to characterize uniquely the geometry of the supersymmetric backgrounds. The geometry rather depends on the embedding of the Killing spinor bundle into the spinor bundle of IIB supergravity up to a Spin(9, 1) rotation as has been explained in [16]. In both cases, the spacetime admits a null Killing vector field X with non-vanishing twist or rotation, and an almost complex structure with compatible (4,0)-form leading to an SU(4) structure in the directions transverse to the lightcone. In the generic case, the twist takes values in R 8 and so it is trivial, the almost complex structure is not integrable, the W 1 , W 4 and W 5 classes associated with the SU(4) structure are determined in terms of functions of the spacetime, and W 2 is related to W 3 . In the pure spinor case, the twist takes values in su(4) ⊕ s R 8 and so it is not trivial, the almost complex structure is integrable, i.e. W 1 = W 2 = 0, W 4 = W 5 is given in terms of the twist of X, and W 3 is not restricted by the Killing spinor equations. of such backgrounds. These include the AdS 5 black holes. In section five, we show that the Spin(7) ⋉ R 8 backgrounds are pp-waves. In section six, we solve the Killing spinor equations of G 2 backgrounds and relate their geometry to co-symplectic G 2 manifolds. In section seven, we describe how families of co-calibrated G 2 manifolds can be used to construct solutions of IIB supergravity with emphasis on those of co-homogeneity one. We find that the bubbling AdS solutions are such an example. In section eight, we give our conclusions. In appendix A, we present the linear systems associated with the N = 2 IIB supersymmetric backgrounds. In appendix B and C, we summarize some results on null and G 2 structures in ten-dimensions.

Geometry and supersymmetry
The Killing spinors of IIB supergravity are complex positive chirality Weyl spinors, S + C . In the absence of one-form P and three-form G field strengths, P = G = 0, the Killing spinor equations are linear over the complex numbers. This means that if ǫ is a Killing spinor, then iǫ is also Killing. Therefore IIB backgrounds with non-vanishing five-form flux preserve an even number of supersymmetries. This in particular implies that if one sets P = G = 0 in the N = 1 backgrounds of [9] and [10], one can obtain the conditions for the most general N = 2 backgrounds with five-form F fluxes. As a consequence, there are three classes of N = 2 supersymmetric backgrounds with F fluxes distinguished by the stability subgroups Spin(7) ⋉ R 8 , SU(4) ⋉ R 8 and G 2 of the Killing spinors in Spin (9,1).
The maximal number of SU(4) ⋉ R 8 -and G 2 -invariant spinors in S + C is four. If the structure group of spacetime is one of these groups, then it admits a rank four subbundle I of the spin bundle S + C spanned by the invariant spinors. Since we are investigating backgrounds that admit two SU(4) ⋉ R 8 -or G 2 -invariant Killing spinors, the Killing spinor bundle K is a subbundle of I. Choosing a basis of spinors in I, one can write the embedding of K in I. As we shall demonstrate confirming the analysis in [10], the conditions on the geometry of spacetime imposed by supersymmetry depend on the embedding of K in I up to a Spin(9, 1) automorphism of S + C . Before we proceed, we shall explain how the geometries of the different embeddings of K in I can be identified. Although the spinors in I are not Killing, nevertheless they are well-defined on the spacetime because of the reduction of its structure group. Consequently, one can use a basis in I to construct the form spinor bi-linears that describe the SU(4) ⋉ R 8 structure of the spacetime. As we shall see the restrictions on the geometry of the spacetime that arise from the Killing spinor equations can be written as conditions on the covariant derivatives of these bi-linears. Moreover, the Killing spinor equations imply that (some of) the components of the five-form flux are also written in terms of these bi-linears and their covariant derivatives.
Killing spinors with isotropy group K ⋉ R 8 , K = Spin(7), SU(4), are associated with a null Killing vector field X. In the complement of the zero locus of X, the cotangent bundle T * M of the spacetime M, dim M = 10, admits a real trivial rank one null subbundle I spanned by the associated one-form κ to X, and a subbundle P = {α ∈ Γ(T * M) | i X α = 0} of rank 9. Moreover, one has that where T ⋆ is of rank 8 and its dual T is identified as the bundle of the "transverse directions" to the lightcone. Observe that P is not canonically decomposed in T ⋆ and I. It is also possible to define the bundle of higher-degree "transverse" forms of the spacetime. We shall not explain this construction further here because it can be found in the appendices of [30]. A basis of SU(4) ⋉ R 8 -invariant Killing spinors is {1, e 1234 } and it can be shown, up to a local Spin(9, 1) transformation 2 , that the SU(4) ⋉R 8 -invariant Killing spinors (ǫ, iǫ) are given by where f, g 1 , g 2 are real spacetime functions which describe the embedding of K in I. Our spinor conventions can be found in [9,10]. If g 2 = 0, then the spinor is Spin(7) ⋉ R 8 invariant. We shall show that there are two classes of N = 2 backgrounds with SU(4) ⋉ R 8 -invariant Killing spinors. One such class are the generic backgrounds for which there is no restriction on the spacetime functions f, g 1 and g 2 , i.e. the embedding of K in I is generic. However, there is another class of supersymmetric backgrounds with different geometry for which g 1 = 0 and f = ±g 2 . The Killing spinors are pure, i.e. they are annihilated by a maximally isotropic subspace. Since the geometry of these two classes is different, the isotropy group of the spinors in Spin(9, 1) and the number of supersymmetries are not sufficient to characterize the supersymmetric backgrounds. The N = 2 backgrounds with Spin(7) ⋉ R 8 -invariant Killing spinors can be thought of as a special case of N = 2 backgrounds with SU(4) ⋉ R 8 -invariant Killing spinors. As we have mentioned, these arise by setting g 2 = 0 in (2.2). It is clear from the results of [10] that if the isotropy group of the Killing spinors is G 2 , the tangent bundle of the spacetime decomposes as where I 3 is the trivial vector bundle of rank three. Moreover, one direction in I 3 is spanned by a time-like Killing vector field. As in the previous case, T , which has rank 7, is the "transverse" bundle or the "transverse directions" of the spacetime. Unlike the previous cases, the decomposition of T * M is the orthogonal decomposition with respect to the spacetime metric. However, it is not always the case that there is a submanifold B in M such that the restriction of T on B is its tangent bundle. One can take the rank 4 bundle I of G 2 invariant spinors to be spanned by (1 + e 1234 , e 51 + e 5234 ). Moreover, up to a Spin(9, 1) gauge transformation, the G 2 -invariant Killing spinors of N = 2 backgrounds can be written as where f, g are real spacetime functions. In this case, all the embeddings of K in I give the same geometry.
3 Generic N = 2 SU (4) ⋉ R 8 backgrounds 3.1 Conditions on the geometry The linear system associated with the Killing spinor equations in this case has been presented and solved in appendix A.1. The solution has been found using the property that for these backgrounds the functions f, g 1 , g 2 which determine the Killing spinors are generic. Here, we shall investigate the consequences that the supersymmetry conditions have on the geometry. To analyze the geometry and fluxes, we introduce the pseudohermitian frame (e + , e − , e α , eᾱ), α = 1, 2, 3, 4, adapted to the description of spinors in terms of forms and write the metric and fluxes as where Φ is a anti-self dual, and Ψ is a self-dual, four-form in the eight directions transverse to the light-cone directions, and X is a three-form, i, j = 1, 2, 3, 4, 6, 7, 8, 9. The last term in the expression for F is required by the self-duality 3 of F , * F = F , and it is completely determined by the spacetime metric and X . Choosing a basis {1, e 1234 } in the space of SU(4) ⋉ R 8 -invariant spinors I , one can show that the spacetime admits the form spinor bi-linears where ω = −e 1 ∧ e 6 − e 2 ∧ e 7 − e 3 ∧ e 8 − e 4 ∧ e 9 = −iδ αβ e α ∧ eβ , are the fundamental hermitian and (4,0) SU(4) forms, respectively. To continue, it is convenient to carry out the description of the geometry in the gauge f 2 + g 2 1 + g 2 2 = 1. Moreover, it is convenient to separate the conditions that arise from supersymmetry into those that involve the light-cone directions and those that involve the transverse directions only. To describe the former, let (e + , e − , e i ) be the co-frame of (e + , e − , e i ), e A (e B ) = δ A B . Then the conditions in A.1 can be rewritten as where X = e + and H = log(1 −4f 2 g 2 2 ). Therefore X is Killing and leaves the SU(4) ⋉R 8 structure invariant. The last condition in (3.4) follows from the torsion free condition of the metric de − + Ω − = 0 and the conditions in A.1. It implies that the rotation of X is trivial, i.e. e − ∧ de − = 0.
The remaining geometric conditions along the transverse directions can be expressed as where W 1 , W 2 , W 3 and W 5 are the Gray-Hervella classes, see [17,18], which can be expressed in terms of fundamental SU(4) forms ω and χ as described in appendix B.2. Since W 1 and W 2 do not necessary vanish, one concludes that the almost complex structure that arises from the metric and ω in the transverse directions is not integrable.
There is an additional condition that arises from the Killing spinor equations which restricts the functions f, g 1 , g 2 that determine the Killing spinors. This is most easily expressed by adapting a coordinate u along X, X = ∂ ∂u , and introduce coordinates (u, v, y I ) on the spacetime M such that the metric is written as where all components depend on v, y I . The Killing spinor equations then imply that f, g 1 , g 2 depend only on the coordinates y, v and the ratio depends only on v. To summarize, the Killing spinor equations of generic SU(4) ⋉ R 8invariant spinors imply the geometric conditions (3.4) and (3.5), and (3.7). The conditions that arise from the transverse directions are not particularly illuminating and we have not found a way to simplify them. It is more straightforward to understand the conditions along the light-cone directions (3.4). In particular, since X has trivial twist there is a coordinate, which we again denote by v, such that the metric can be written as where all components depend on v, y I . This concludes the investigation of the geometry.

Fluxes
To find the conditions on the fluxes imposed by supersymmetry, we decompose the forms Φ, Ψ and X in (3.1) that determine the five-form field strength into SU(4) representations. Using the fact that Φ is anti-self-dual in the transverse directions, one can show that where α is a traceless (1,1)-form, α βγ = α β β = 0, s is a (2,0) and (0,2) symmetric tensor, s ij = s ji , s αβ = 0 and∧ denotes inner derivation 4 . In turn, one finds that Similarly using the self-duality of Ψ, one can write either andΨ 2,2 is a traceless (2,2)-form. Moreover, the three-form X can be written as andX 2,1 andX 1,2 are traceless (2,1)-and (1,2)-forms, respectively. The supersymmetry conditions imply restrictions on the various irreducible representations of SU(4) that appear in the above decompositions. In particular, it turns out that by inspecting the conditions in A.1 one obtains and so The remaining conditions 5 give 4 Let π be a k-form, then s∧π = 1 (k−1)! s j ii π ji2...i k e i1 ∧ e i2 ∧ · · · ∧ e i k . 5 We use (α · β) j1...
Observe that Ψ 2,2 is not restricted by the Killing spinor equations. One can substitute the above expressions into the formula for F . We shall not do this here because it does not lead to a simplification for the expression of F . However, as we shall show, F is simplified in some special cases.

Special cases
A large class of backgrounds consists of those for which the transverse metric is independent of v. Using the torsion free condition for the frame (e − , e + , e i ), one finds that provided that 6 de + ∈ su(4) ⊕ R 8 . Assuming also that Killing spinors are taken to be independent of v as well, we have that In such a case, the flux can be written as To construct solutions in this case, one must find almost Hermitian manifolds with an SU(4) structure which satisfy the conditions (3.5), and then write the metric as (3.8).
One must also impose the closure of F , dF = 0 and the E −− = 0 component of the Einstein equations.

Geometry and fluxes
In the solution of the Killing spinor equations with SU(4) ⋉ R 8 invariant Killing spinors, a special case arises whenever the Killing spinor is pure [9]. In particular, the Killing spinors (2.2) are pure if one sets We consider the case f + g 2 = 0, the investigation of the other case is similar. The solution of the linear system can be found in appendix A.2.1. Here we shall investigate the conditions on the geometry and write the fluxes in a closed form.
It is convenient to investigate the geometry in the gauge h = 1. First write the metric and fluxes as in (3.1) using the pseudo-hermitian frame (e − , e + , e α , eᾱ). Next consider the form bi-linears (3.2) and observe that the conditions on the geometry that involve light-cone directions can be written as i.e. X = e + is Killing vector field and preserves the SU(4) ⋉ R 8 structure. Unlike the generic SU(4) ⋉ R 8 case, e − ∧ de − = 0. The remaining geometric conditions along the transverse directions are which can be recast in terms of the SU(4) structures, see appendix B.2, as The vanishing of W 1 , W 2 can be interpreted as integrability of the almost complex structure along the transverse directions. In particular, one can show using the torsion free conditions of the frame, that (e − , e α ) span an integrable distribution of co-dimension 5.
As we have seen, this is unlike what happens in the generic SU(4) ⋉ R 8 backgrounds.
The conditions (4.2) and (4.4) constitute the full set of restrictions that supersymmetry imposes on the geometry of spacetime. Next let us turn to the flux F . Using (3.1) and the results of appendix A.2.1 F can be written, after some work, as It is clear that all the components of F are determined in terms of the geometry apart from those ofΨ 2,2 which is not restricted by the Killing spinor equations. In this case, Φ may not vanish.
One can adapt coordinates along the Killing vector field X, X = ∂/∂u, and write the metric as (3.6). However unlike the generic case no further simplification is possible because the twist of X may not be trivial, i.e. e − ∧ de − = 0. If one imposes e − ∧ de − = 0, one finds additional restrictions on the geometry that are not implied by the Killing spinor equations.

Special cases and examples 4.2.1 Special cases
As in the class of generic backgrounds, one can take the transverse metric to be independent of v and impose de + ∈ su(4) ⊕ R 8 to find (3.19). In such cases, F can be written as Further simplification of the metric and fluxes occurs when the rotation of X is trivial, i.e. e − ∧ de − = 0. This is equivalent to requiring that there is an one-form l such that de − = l ∧ e − or equivalently de − ∈ R 8 . Then the Frobenius theorem implies that there is a function H = H(y, v) such that e − = e H(y,v) dv for some coordinate v which is related to that denoted with the same symbol in (3.6) by a coordinate transformation. The metric takes the form A large class of known backgrounds have metric and fluxes given by (4.7) and (4.6), respectively. These include the D3-brane and intersecting D3-brane configurations as we shall see below.
Moreover, the geometric conditions are also simplified. In particular one finds that the geometric conditions in (4.4) become The spacetime in this case can be reconstructed from an eight-dimensional Hermitian manifold with a SU(4) structure for which the W 4 and W 5 classes satisfy the conditions above. Assuming that H depends only on y, H = H(y), and setting W 1 = W 2 = 0 in (B.12) which expresses the exterior derivatives of ω and χ in terms of the W classes, we find that the remaining geometric conditions can be rewritten as Observe that the rescaled forms ω 0 = e 1 3 H ω and χ 0 = e H χ for dH = 0 are not canonically normalized, so such Hermitian manifolds do not contain a Calabi-Yau in their conformal class. The expressions for the metric (4.7) and the flux (4.6), and the conditions (4.9) are the full content of the Killing spinor equations for the case that the transverse metric and H are independent of v.
Of course additional conditions are imposed on the backgrounds from the Bianchi identity of F , dF = 0 and the vanishing of the E −− component of the Einstein equations, In what follows, we consider examples which admit a null Killing vector with a trivial and a non-trivial twist.

D3-branes and intersecting branes
The D3-brane [2] and its intersections [20] are examples of backgrounds with pure SU(4) ⋉ R 8 -invariant Killing spinors. The metric of the former is where h is a (multi-centred) harmonic function of R 6 . (h should be distinguished from the function h that multiplies the Killing spinor which we have set equal to 1.) The background preserves 16 supersymmetries. Clearly, this metric is of the form (4.7). To rewrite the metric as (3.6), set and change coordinates as to find Observe that the pure spinor 1 satisfies the projection condition that arises in the D3brane Killing spinor equations. It is then easy to find that where ω(R 2 ) and ω(R 6 ) are the constant Kähler forms on R 2 and R 6 , respectively. Substituting all these into the flux and takingΨ 2,2 = 0, one can easily show that This is precisely the five-form flux of the D3-brane. Moreover observe that the space transverse to the lightcone directions is Hermitian, the closure conditions (4.9) are satisfied and the rotation of X is trivial e − ∧ de − = 0. The latter also follows from a direct inspection of the D3-brane metric. One consequence of these conditions is that H 3 × S 5 , which is the near-horizon geometry of the D3-brane restricted on the transverse directions to the lightcone, admits a relatively balanced SU(4) Hermitian structure. One can also show that this class includes the delocalized D3-brane intersections. We shall demonstrate this for the configuration of two intersecting D3-branes at a string [20]. The computation is straightforward for the remaining cases. The metric is [21] where h 1 , h 2 are (multi-centred) harmonic functions of the transverse space R 4 . Changing coordinates as the metric can be written in the standard light-cone form with The hermitian form can be chosen as Substituting this into the the expression for the flux, one finds that Similarly observe that the transverse space is Hermitian, the closure conditions (4.9) are satisfied and the rotation of X is trivial e − ∧ de − = 0. One can easily extend the above results to include D3-brane configurations with a null rotation by taking n = 0 and superposed with a pp-wave V = 0. One can also alloŵ Ψ 2,2 = 0 which may lead to resolved D3-brane configurations.

Uplifted Five-Dimensional Solutions
Solutions of minimal gauged five-dimensional supergravity which admit a timelike Killing vector, ∂ ∂t , associated with a Killing spinor have spacetime geometry [14] where F is a function and Ψ = Ψ m dx m is a 1-form, ds 2 N = h mn dx m dx n is a metric on a Kähler 4-manifold N, and (t, x m ), m = 1, 2, 3, 4, are spacetime coordinates. The components of the metric depend only on x n . In addition, the one-form gauge potential is where ℓ is constant and R N = dP is the Ricci form of the Kähler manifold N. The function F is determined in terms of the Ricci scalar R N of N via where G + , G − are self and anti-self-dual 2-forms on N, the Ricci form R N is constrained by 25) where J N is the Kähler form of N.
The uplifted metric is given by [15] ds 2 10 = ds 2 5 + ℓ 2 dα 2 + cos 2 αdβ 2 + sin 2 α cos 2 α(dξ 1 − sin 2 βdξ 2 − cos 2 βdξ 3 ) 2 + cos 2 α sin 2 β cos 2 β(dξ 2 − dξ 3 ) 2 It is convenient to define and rewrite the metric as where is the Kähler-Einstein metric on CP 2 which has constant holomorphic sectional curvature. Q is the potential for the Ricci form of the metric (4.28), where 29) and the Kähler form of CP 2 is J CP 2 = 1 6 dQ. The Ricci scalar of (4.28) is R CP 2 = 24. Note that in the uplifted solution, the Killing vector ∂ ∂t is null. In order to write the uplifted solution as a pure spinor SU(4) ⋉ R 8 background, we set ds 2 = 2e − e + + ds 2 8 , where The calculation we present below can also be carried out if CP 2 is replaced with any other four-dimensional Kähler-Einstein manifold E such that R E = 24. However, we shall continue the analysis using CP 2 .
The complex structure along the transverse directions to the lightcone is identified as the direct sum of the complex structure on CP 2 together with the complex structure of the Kähler base manifold N of the 5-dimensional solution. It is clear that the null Killing vector field ∂/∂t of the ten-dimensional solution has non-trivial twist.
It will be convenient to split the SU(4) indices α into α = (a, µ) for a, b = 1, 2 and µ, ν = 3, 4 and choose a Hermitian frame along the transverse directions as for real constants k 1 , k 2 to be fixed, whereê a is a Hermitian frame of the Kähler base manifold N, We also setê In addition, we have ds 2 CP 2 = 2δ µνê µêν and J CP 2 = −iδ µνê µ ∧êν. It is also convenient to write ω = ω N + ω CP 2 , where Using the expression for the spin connection which we give in A.2.2, it is straightforward to show that the constraints on the geometry imposed by supersymmetry are satisfied provided that Next, note that the formula for the uplifted five-form is [15] where µ 1 = sin α, µ 2 = cos α sin β, µ 3 = cos α cos β and with e 0 = F (dt + Ψ). In addition, * (5) denotes the 5-dimensional Hodge dual taken with respect to dvol 5 and * denotes the standard Hodge star operation of M whose volume form can be rewritten as dvol M = 1 4 e + ∧ e − ∧ ω N ∧ ω N ∧ ω CP 2 ∧ ω CP 2 . Note the change in normalization for the 5-form when comparing with the Killing spinor equation in [7]. In order to simplify this expression, note that We then find the uplifted five-form is given by In comparing this expression with the 5-form obtained from the pure spinor classification, it is straightforward to see that Also note that Hence, the formula for the uplifted five-form matches the expression for the five-form given in (4.5). It is remarkable that in establishing the equality of the uplifted five-form with that in (4.5), we have seen that all components of the latter give a non-trivial contribution. It remains to consider the Bianchi identity; note that one can rewrite F as where ⋆ N denotes the Hodge dual taken on the Kähler base with metric h and volume form − 1 2 J N ∧ J N . Hence So the Bianchi identity implies that As expected, this constraint is equivalent to that implied by the five-dimensional gauge field equations obtained in [14]. Observe that as F is inversely proportional to the Ricci scalar R N , this constraint is a highly non-linear constraint on the geometry of the Kähler base space N. The AdS 5 black holes found in [6] belong to the class of five-dimensional backgrounds which we have uplifted to ten dimensions. In fact, it was shown in [7] that the uplifted black hole solutions preserve exactly 1/16 of the supersymmetry in IIB supergravity. Hence it follows that the Killing spinors of the uplifted black hole solution are (1, i1) of the pure spinor SU(4) ⋉ R 8 backgrounds. This can also be seen by an inspection of the conditions given in [7].
5 Spin(7) ⋉ R 8 backgrounds As in previous cases, it is convenient to carry out the analysis without loss of generality in the gauge f 2 + g 2 = 1. A direct inspection of the geometric conditions in appendix A.3 reveals that the holonomy of the Levi-Civita connection, ∇, of the spacetime is contained in Spin(7) ⋉ R 8 , This is equivalent to requiring that the forms are ∇-parallel, where φ is the fundamental Spin (7) self-dual four-form given by φ = Re χ − 1 2 ω ∧ ω. In particular, the null vector field X is Killing and twist free. Adapting coordinates along the parallel vector field, X = e + = ∂/∂u, one finds that the spacetime metric can be written as where the components depend on the coordinates v, y I . It remains to specify the fluxes. One can show using the results of appendix A.3 that To derive this, we first remark that F = e − ∧ Ψ, where Ψ is a self-dual four-form in the eight-transverse directions to the light-cone. Then we use the decomposition of the four-forms in R 8 under Spin (7) representations as Λ 4 ( i.e. the functions f, g depend only on v. It is clear from the above that the spacetime is a pp-wave with rotation propagating on an eight-dimensional manifold which has holonomy Spin (7). The metric of the transverse space depends on the wave-profile coordinate v. Imposing the Bianchi identity for F one finds that F is closed iff dΨ 27 = 0 up to forms of the type e − ∧ µ. The only equation that remains to be imposed to find solutions is the vanishing of the E −− component of the Einstein equations, E −− = 0. This equation can be easily recovered from that in [16] by setting the one-form and three-form field strengths to zero. 6 G 2 backgrounds

Geometry
We have presented the linear system for G 2 backgrounds and its solution in appendix A.4. In particular, we have given the solution in SU(3) ⊂ G 2 representations. Here, we shall investigate the consequences that the supersymmetry conditions have on the geometry and fluxes of the theory. For this, we introduce a frame 7 (e + , e − , e 1 , e i ), i = 2, 3, 4, 6, 7, 8, 9, where (e + , e − , e 1 ) span the trivial subbundle of T M and (e i ) the transverse directions of the spacetime. The metric and fluxes written in this frame are where Φ is a anti-self dual, and Ψ is a self-dual, four-form in the eight directions transverse to the light-cone directions, and X is a three-form. The expression for the flux is similar to that which we have given for F in is the fundamental G 2 three-form, and e ± = (1/ √ 2)(±e 0 + e 5 ). It is clear that T * M = I 3 ⊕ T where I 3 is a trivial bundle of rank 3 and T are the remaining "transverse" directions of the co-tangent bundle of the spacetime.
The geometric conditions of appendix A.4 can be expressed in G 2 representations as where the one-form associated with V is κ = f 2 e 0 , and X 1 , X 2 , X 3 and X 4 have been defined in appendix B.3. The ⋆ Hodge duality operation is with respect to the "transverse" volume form dvol = e 2 ∧ e 3 ∧ e 4 ∧ e 6 ∧ · · · ∧ e 9 . Clearly V is a time-like Killing vector field as may have been expected from the results of [9,10]. The class X 3 , which is associated with the traceless symmetric representation of G 2 , is not restricted by the supersymmetry conditions. To show that X 2 = 0, one has to compute Π 14 (⋆d ⋆ ϕ) , (6.6) and demonstrate that it vanishes using the conditions, whered denotes the exterior derivative restricted to the transverse directions, and Π 14 is the projection to the 14dimensional representation in the decomposition of Λ 2 (R 8 ) = Λ 7 ⊕ Λ 14 in G 2 irreducible representations, Λ 14 = g 2 . In particular, Moreover, one can immediately see using (C.3) that Thus the transverse directions are co-symplectic or co-calibrated. Moreover, it turns out that using the conditions (6.4), one can show that L V ϕ = 0. It is sometimes more convenient to re-express the conditions on the geometry as Observe that the above constraints imply that f , e + , e − , e 1 and ϕ are all invariant under the action of V . Therefore V preserves the G 2 structure of spacetime. One can introduce some special coordinates on the spacetime. Solving the closure conditions, we write e 1 = f −2 dx 2 and e 5 = f −2 dx 1 , and adapt a coordinate t along the Killing vector field V , V = ∂/∂t, where the functions x 1 and x 2 can be thought of as spacetime coordinates. The metric of the spacetime M can be written as where ds 2 7 is the metric along the "transverse" directions, e i = e i S dx S + e i I dy I , m = V S dx S + m I dy I , and y I are the remaining coordinates of the spacetime.
The spacetime is foliated by eight-dimensional Lorentzian manifolds N given by x S = const. In turn N is a real line bundle over a seven-dimensional base space B. A special case arises whenever m I = 0. In such a case M is foliated by seven-dimensional manifolds B given by t, x S = const. Moreover the conditions (6.9) imply that B is a co-symplectic G 2 manifold. This is perhaps the most significant geometric property of this class of IIB supersymmetric backgrounds. As we shall see the D3-brane and the LLM solutions are of this type.

Fluxes
To find the conditions that supersymmetry imposes on the flux F in (6.1), it is convenient to write Φ = e 1 ∧ α − ⋆α , Ψ = e 1 ∧ β + ⋆β , X = e 1 ∧ γ + δ , (6.11) where α, β, δ are three-forms and γ is a two-form. Then using the decomposition Λ 3 (R 7 ) = Λ 3 1 ⊕ Λ 3 7 ⊕ Λ 3 27 in G 2 irreducible representations, α, β, δ can be written as respectively, where s 1 , s 2 and s 3 are symmetric traceless 2-tensors associated with the 27-dimensional representation of G 2 . In particular, one has and similarly for the rest. Using Λ 2 (R 7 ) = Λ 7 ⊕ g 2 , one has γ = u∧ϕ +γ , (6.14) whereγ ∈ g 2 and The IIB Killing spinor equations imply that all the five-form flux is determined in terms of the geometry. In particular a calculation reveals that where s Γ is defined in the appendix B.3. Similarly, we get Substituting these expressions back into the flux and after some computation, one finds that (6.19) and κ = f 2 e 0 . This is a remarkably simple expression for the flux. To summarize the solution of the Killing spinor equations is given by (6.10) and (6.19) subject to the conditions (6.9).
7 Co-calibrated G 2 manifolds and supergraviy backgrounds 7.1 Co-calibrated G 2 manifolds Co-calibrated or co-symplectic G 2 manifolds are seven-dimensional manifolds with a G 2 structure which have the property that i.e. the dual of the fundamental G 2 form is closed. If one in addition requires that X 3 = 0, then from the results of appendix B.3 it follows thatdϕ = l ⋆ϕ, l = 0. These are the weak holonomy or nearly parallel G 2 manifolds. So the co-symplectic G 2 manifolds include the nearly parallel G 2 manifolds, which in turn include the tri-Sasakian ones. Nearly parallel G 2 manifolds have appeared before in the context of eleven-dimensional supergravity compactifications, see e.g. [34,35].  (3) is given as diag(e ikθ 1 , e iℓθ 2 , e −i(θ 1 +θ 2 ) ) and SO (3) is maximal in SO (5). These are the only strictly nearly parallel G 2 compact homogeneous manifolds [24]. However, there are additional homogeneous tri-Sasakian and Einstein-Sasakian manifolds, see e.g. [37,38,24]. More examples of nearly parallel G 2 manifolds can be constructed by squashing the tri-Sasakian ones along the orbits of so(3) [40,24]. Using the tri-Sasakian examples of [39], one can construct infinite families of such manifolds which are neither homogeneous nor tri-Sasakian.
There are co-symplectic G 2 manifolds which are not nearly parallel. This can be easily seen from the results of [19]. Unlike the nearly parallel case, there is no classification of compact homogeneous co-symplectic G 2 manifolds. It is expected that the class of compact homogeneous co-symplectic G 2 manifolds is large 8 .
The co-homogeneity one co-symplectic G 2 manifolds are more tractable. A classification has been given in [19]. It has been found that the principal orbits, which are six-dimensional homogeneous manifolds, are one of the following spaces up to discrete identifications. The metric and fundamental form of co-homogeneity one manifolds can be written as ds 2 7 = h 2 (y)dy 2 + ds 2 6 (µ(y)) , ϕ = ϕ(h(y), µ(y), θ(y)) , (7.3) where µ are the homogeneous moduli of the G 2 structure which have been promoted to functions of the inhomogeneous coordinate y. We shall explain the θ dependence later. Using a y coordinate transformation, one can set h = 1 as was done in [19]. However, we shall not do so because in the associated supergravity backgrounds h may depend on another two coordinates. Setting e 6 = hdy, it is known that the stability subgroup of a vector in the seven-dimensional representation of G 2 is SU(3). There is a unique fundamental SU(3) two-form and the space of fundamental SU(3) 3-forms is two-dimensional. Using this, the G 2 invariant three form can be written as ϕ = Re(e iθχ ) + e 6 ∧ω , (7.4) whereω is the Hermitian form andχ is the (3,0)-form. The isotropy groups of the homogeneous spaces may leave more forms invariant, and so there may be many ways to construct a G 2 invariant 3-form ϕ. As we shall demonstrate below, the bubbling AdS solutions are examples of backgrounds based on co-symplectic co-homogeneity one G 2 manifolds whose principal orbit is S 3 × S 3 . In fact one uses a very special family of such manifolds with isometry SO(4) × SO(4). Solutions based on various families with principal orbits given in (7.2) will be presented elsewhere [41].

D3 brane
The D3-brane solution can be viewed as a special case of solutions with G 2 -invariant Killing spinors. For this write the metric as where (t, z) are the worldvolume and (x, y, w, q) are the transverse coordinates, respectively and h is the usual harmonic function h = h(w, x, y, q). The time-like Killing vector field is κ = ∂/∂t and so It is straightforward to verify thatd ⋆ ϕ = 0. Moreover Y r = X 1 = 0. The remaining conditions are also satisfied. A consequence of this is that the seven-dimensional "transverse" space B admits a G 2 structure for which the only non-vanishing class is X 3 . Asymptotically, B is flat space and at the origin is H 4 × S 3 .

Bubbling AdS Solutions
The solution found by Lin, Lunin and Maldacena in [5] is an example of a G 2 IIB supergravity background and so it is associated with a co-symplectic G 2 geometry. The co-symplectic geometry is of co-homogeneity one and has principal orbit S 3 × S 3 . To show this, we write the metric in the form given in (6.10) by introducing a frame e 0 = h −1 (dt + V S dx S ) , e 5 = hdx 1 , e 1 = hdx 2 , where σ a ,σ a , a = 1, 2, 3, are left-invariant 1-forms on SU(2) × SU(2) given by σ 1 = − sin ψdθ + cos ψ sin θdφ , σ 2 = cos ψdθ + sin ψ sin θdφ , σ 3 = dψ + cos θdφ , andσ a are defined in exactly the same way, but with θ, φ, ψ replaced withθ,φ,ψ throughout, and the rest of the components of the metric depend on y, x S . Moreover, the functions h and G are related by and Writing z = 1 2 tanh G, z is further constrained by The five-form is given by where F (1) , F (2) are two 2-forms. In turn these are given by and with where ⋆ 3 denotes Hodge duality on R 3 with metric dy 2 + (dx 1 ) 2 + (dx 2 ) 2 with positive orientation given by dy ∧ dx 1 ∧ dx 2 .
In this basis the transverse metric is ds 2 7 = (e 2 ) + (e 3 ) 2 + (e 4 ) 2 + (e 6 ) 2 + (e 7 ) 2 + (e 8 ) 2 + (e 9 ) 2 , (7.17) equipped with the fundamental G 2 form ϕ = Re e iH (e 2 + ie 7 ) ∧ (e 3 + ie 8 ) ∧ (e 4 + ie 9 ) − e 6 ∧ (e 2 ∧ e 7 + e 3 ∧ e 8 + e 4 ∧ e 9 ) , (7.18) where Observe that the expression for the fundamental form is as expected in the context of co-homogeneity one G 2 structures (7.4). The frame we have chosen is not adapted in the G 2 structure because the components of ϕ depend on H. However, we can adapt a frame by rotating the frame as 20) and similarly for the other two pairs while leaving e 6 as it is. Observe that the metric remains invariant under this rotation. However, in what follows in this section we shall continue to work with the frame we have originally introduced. It is straightforward to verify that ϕ satisfies This establishes that the LLM solution is associated with a co-symplectic G 2 structure. Moreover, it is straightforward to see by setting y, x S = const that the principal orbit is S 3 × S 3 . Observe that this principal orbit is one of the possibilities that appear in the classification of co-homogeneity one co-symplectic manifolds in [19] and are listed in (7.2). In fact, it is associated with the most symmetric family of such co-symplectic structures which has SO(4) × SO(4) symmetry. There are families of the same orbit that possess fewer isometries. Although ϕ is co-closed on the transverse space,dϕ = 0, and we find that The function f appearing in (6.10) is given by It remains to find the forms α, β, δ which appear in the decomposition of F , (6.11). These are Moreover, γ vanishes. It is then straightforward to verify, using (7.10), that all of the remaining geometric constraints are satisfied, and the components of F are determined in terms of the spin connection as set out in the previous section. There are other solutions which are extensions of the LLM solution and so they are associated with a co-symplectic G 2 structure. In particular observe that (6.10) is compatible with some recent results in [28] where bubbling solutions have been investigated that preserve 4, 8 and 16 supersymmetries.

Concluding remarks
IIB backgrounds with N ≥ 2 supersymmetry and active five-form flux have four distinct types of geometries. Killing spinors with isotropy group SU(4) ⋉ R 8 give rise to two types of geometry depending on whether the Killing spinors are generic or pure. In both cases, the spacetime admits a null Killing vector field and an almost complex structure on the transverse directions to the lightcone. For generic backgrounds, the twist of the Killing vector field is trivial but the almost complex structure non-integrable, while in the pure case the twist is non-trivial but the almost complex structure is integrable. In the latter case, if one assumes that the twist of the Killing vector field is trivial, then all the geometric conditions take a very simple form. In particular, the cubic power of a certain rescaled Hermitian form and a certain rescaled fundamental (4,0)-form must be closed. Examples of backgrounds that admit pure Killing spinors are the D3-brane and the lifts of supersymmetric backgrounds of gauged five-dimensional supergravity which include the AdS 5 black holes. The null Killing vector field of the D3-brane background has a trivial twist but that of the AdS 5 black hole does not. N ≥ 2 backgrounds with Killing spinors which have isotropy group Spin(7) ⋉ R 8 are pp-waves propagating on a manifold with holonomy contained in Spin (7).
The remaining type of geometry is associated with IIB backgrounds that admit G 2invariant Killing spinors. The tangent bundle of such spacetimes decomposes with respect to the spacetime metric into a trivial bundle of rank three and a bundle of rank seven which are the transverse directions of the spacetime. One direction in the trivial bundle is associated with a time-like Killing vector field and the other two directions are associated with closed one forms. The transverse geometry is that of a co-symplectic G 2 manifold. We give a description of how one can construct a supersymmetric IIB background from a family of co-symplectic G 2 manifolds . In addition, we show that the co-symplectic G 2 manifold associated with the bubbling AdS solution of [5] has principal orbit S 3 × S 3 . The transverse structures of the supersymmetric backgrounds are summarized in table 1.

Stab
Transverse structure Conditions Spin(7) ⋉ R 8 Holonomy Spin(7)dφ = 0 G 2 co-symplectic G 2d ⋆ ϕ = 0 Table 1: The columns contain the stability subgroups of the Killing spinors in Spin(9, 1), the type of transverse geometry, and the conditions on the fundamental forms, respectively.
The holonomy of the supercovariant connection of IIB supergravity with only F flux is contained in SL (16, C). This can be easily seen from the computation of the super-covariant curvature for this case in [23]. This is a subgroup of SL(32, R) which is the holonomy group of the IIB supercovariant connection [25]. Arguments based on the reduction of holonomy in the presence of parallel spinors similar to those of [26,27] suggest that there may exist supersymmetric IIB backgrounds with F flux with any even number of supersymmetries. We have identified the geometries of N = 2 backgrounds and the N = 32, maximally supersymmetric, backgrounds have been classified in [23]. It is also tractable to investigate the existence of N = 30 backgrounds by applying the technique developed in [29] to this case. If backgrounds exist with strictly N = 30 supersymmetry, they are expected to be homogeneous spaces [31]. We hope to report the outcome of this investigation in the future.
Next some of the conditions of D − ǫ = 0, their complex conjugate and dual give Some of the D − ǫ = 0 and some of the remaining D α ǫ = Dᾱǫ = 0 conditions give In addition, one finds that

A.2.2 Spin connection
The non-vanishing components of the spin connection along the transverse directions of the spacetime associated with the uplift of the five-dimensional supersymmetric solution can be expressed as ,bc . (A.14) Here hatted and unhatted frame indices are with respect to the bases e a andê a , and Ωâ ,bĉ ,Ωâ ,bc ,Ωâ ,bc are the components of the spin connection of N with respect to the basisê a . As N is Kähler with Kähler form J N , we haveΩâ ,bĉ = 0 andΩâ ,bc = 0. Similarly, we have for the directions along CP 2 .

B.1 Null vectors and SO(n) ⋉ R n structures
The stability subgroup in the special Lorentz group SO(n + 1, 1) of a nowhere vanishing null vector X is SO(n) ⋉ R n . Therefore, geometrically the structure of a Lorentzian manifold that admits a non-vanishing null vector reduces to SO(n) ⋉ R n . Topologically, the structure reduces further to the maximal compact subgroup SO(n). Let X be a non-vanishing null vector field on the spacetime. It is always possible to introduce a frame e + , e − , e i such that where e A is the co-frame, e A (e B ) = δ A B . It is convenient to use the Lorentzian metric to construct the associated null one-form κ = e − to X. Next consider the covariant derivative of κ, ∇κ with respect to the Levi-Civita connection of the Lorentzian metric. One way to determine the SO(n) ⋉ R n structures is to decompose ∇κ under the irreducible representations of either the geometric structure group SO(n) ⋉ R n or the topological structure group SO(n). Since the representation of SO(n) ⋉ R n on the space of oneforms is reducible but indecomposable, a more refined characterization of the geometry can be achieved by decomposing ∇κ under the topological structure group SO(n). In particular, one finds that Clearly, if κ is parallel, then all the classes vanish. If X is Killing, then Y 1 = Y 2 = Y 3 + Y 4 = Y 5 = Z 2 = Z 3 = 0, and similarly if X is self-parallel, Y 1 = Y 5 = 0. In all, there are 2 8 possible structures.

(B.4)
It is worth pointing out that the stability subgroup of e − and ω in the special Lorentz group SO(2n + 1, 1) is not U(n) ⋉ C n and so the introduction of the null three-form σ is necessary.
To determine the different U(n) ⋉ C n structures on the spacetime, we decompose ∇κ and ∇σ under representations of U(n), the maximal compact subgroup of U(n) ⋉ C n , which is the topological structure group of the spacetime 9 . It turns out that all independent structures can be found by considering the covariant derivative of κ, ∇ A κ B , and the component, ∇ A ω ij = ∇ A σ −ij − ∇ A κ − ω ij , of ∇σ. In particular the SO(2n) ⋉ R 2n classes which we have previously investigated are further decomposed as and W 1 , W 2 , W 3 , W 4 which are associated with ∇ i ω jk . The latter are related to those of Gray-Hervella for almost Hermitian manifolds, see [17]. A Cauchy-Riemann structure is a null U(n) ⋉ C n structure. A Cauchy-Riemann structure determines an integrable distribution spanned by e − , eᾱ. Using the torsion free conditions, we observe that this requires that Ω [α,β]+ = Ω +,+α = Ω [α,β]γ = Ω [+,β]γ = 0 . (B.7) In turn these give Ω α,βγ = Ω α,β+ = Ω +,+α = Ω +,αβ = 0 . (B.8) Observe that in this setting κ is not self-parallel. This in addition will require that Ω +,+− = 0. Therefore a Cauchy-Riemann structure is specified by the vanishing of the classes If κ is self-parallel, then in addition Y 1 = 0. The null SU(n) ⋉ C n structures can be investigated in a similar way. The associated nowhere vanishing forms in the basis introduced for the U(n) ⋉ C n case are κ = e − , σ = e − ∧ ω , ρ = e − ∧ χ , (B.10) where χ is the SU(n)-invariant holomorphic (n,0)-form. It turns out that the decomposition of ∇κ is as in the U(n) ⋉ C n case above. In addition, one can also define the classes V 1 and V 2 . There are two new additional classes The remaining classes are determined by ∇ i ω jk and ∇ i χ j 1 ...jn . These can be expressed in terms of the W 1 , W 2 , W 3 , W 4 and W 5 classes of the SU(n) structures, see e.g. [18]. The normalization which we use for these classes is that of appendix C in [32]. In particular for the SU(4) case which is relevant to the results of this paper, we have that Observe that any Hermitian eight-dimensional manifold with an SU(4) structure satisfies these conditions. As we have seen supersymmetry imposes in addition that W 4 = W 5 . We refer to these as relatively balanced Hermitian SU(4) structures because the difference of the two Lee forms vanishes. One consequence of (B.13) is that if B admits such relatively balanced Hermitian structure, thendW 4 =dW 5 , is a trace-less (1,1)-form.