Symmetries, a systematic construction of invariant fields and AdS backgrounds

We give a systematic local description of invariant metrics and other invariant fields on a spacetime under the action of a (non-abelian) group. This includes the invariant fields in a neighbourhood of a principal and a special orbit. The construction is illustrated with examples. We also apply the formalism to give the R-symmetry invariant metrics of some AdS backgrounds and comment on applications to Kaluza-Klein theory.


Introduction
It is well known that most explicit solutions of Einstein equations are invariant under some group acting on the spacetime. These include black holes and cosmological solutions as well as large classes of solutions that have applications in string theory and AdS/CFT correspondence. In this article, we shall propose a systematic way to construct invariant metrics given the action of a (non-abelian) group on a spacetime. Similar results are obtained for other invariant fields.
The construction presented below applies generally to all problems that require a description of invariant fields under some group action. Nevertheless the initial motivation for this work has been the classification of AdS backgrounds in 10-and 11-dimensional supergravity theories which have applications in AdS/CFT correspondence, for a review see [1]. It is known that AdS n backgrounds admit a SO(n − 1, 2) × G group of symmetries that leave all fields invariant, where the R-symmetry group G acts on the internal space of the background. The groups G are compact and their action on the internal space of an AdS solution is non-linear. The main difficulty to give an expression for the invariant fields under some group action is that it requires a model of how a group acts on a manifold. This is resolved with the application of slice and principal orbit theorems which we shall describe below.
There is much progress in the classifications of supergravity AdS solutions. In particular the maximally supersymmetric AdS solutions of 10-and 11-dimensional supergravities have been classified in [2]. Moreover those that preserve strictly more than 16 supersymmetries have been classified in [3,4,5,6] using either global methods or the homogeneity theorem of [7] and the classification of all homogenous spaces up to dimension 9, see e.g. [8]. Some progress has also been made towards the classification of AdS solutions that preserve 16 supersymmetries. In particular, there is a partial classification 1 of such AdS 7 backgrounds [9,10,11]. Moreover, there are no smooth AdS 6 solutions with compact without boundary internal space in 11-dimensional and (massive) IIA supergravities 2 [12]. Furthermore if IIB AdS 6 backgrounds exist, the R-symmetry group, which has Lie algebra so(3), must have codimension 2 principal orbits as well a non-empty set of special orbits and both IIB scalars, axion and dilaton, must be non-constant functions on the internal space. However, there are several AdS 6 solutions with non-compact internal space and/or with singularities, see e.g. [13]- [18]. For the remaining AdS backgrounds systematic results are sparse although many explicit solutions are known with widespread applications, see e.g [19] for review and references within.
Further progress on the classification of AdS backgrounds, specially those that preserve less than half of supersymmetry, depends on understanding how the R-symmetry groups act on the internal spaces of such a solution. The Lie algebras of all R-symmetry groups of warped AdS backgrounds under some mild assumptions have been found in [20]. This together with the classification of all homogenous spaces up to dimension 9, see e.g [8], allow for the identification of all orbit types of the R-symmetry group in the internal space of the backgrounds up to a possible discrete identification. We shall use these data to give a systematic construction of invariant metrics on the internal space of AdS backgrounds. This paper is organized as follows. In section two, the slice and principal orbit theorems are described. In section three, our main result is given after a detailed description of the invariant geometry of G/H spaces which is required for the proof. In section four, some examples of group actions are given with orbits S 2 and S 3 which illustrate the use of slice theorem. Then some applications to AdS n backgrounds, for n = 6, 5, 4 are presented. In section 5, we give our conclusions and comment on applications to Kaluza-Klein theory.

Slice and principal orbit theorems
The properties of a group action on a manifold have been extensively investigated [21,22,23,24], see [25] for a recent review and also [26] for other applications. The following two main results will be used here.
1. The slice theorem or tubular equivariant neighbourhood theorem states under some compactness assumptions 3 that the neighbourhood of an orbit N of a group G acting smoothly on a manifold M is equivariantly diffeomorphic to an invariant vector bundle E over the orbit. E is identified with the normal bundle of N in M. This in particular means that the group action on a manifold can locally be modelled as a lift of the action of G on N to E.
2. The principal orbit theorem states that the union of all principal orbits 4 , i.e. those of maximal dimension, is a dense set of the manifold. In addition away from some special orbits, the manifold is a bundle with fibre the typical principal orbit N and base space B.
The slice theorems provides a model how a group G acts on a neighbourhood of an orbit N in M. As G acts transitively on N, N can be identified with a homogeneous space, N = G/H. The slice theorem then states that on a neighbourhood of a G/H orbit one can adapt as coordinates, those of E, and that on the fibre coordinates of E, G acts with a rotation. This gives a very concrete description of the vector fields generated by the group action in the neighbourhood of G/H. In turn this allows for the identification of the invariant metrics and other invariant fields in the vicinity of G/H in M.
To give a very brief sketch of the idea behind the slice theorem, suppose that N is a point p and M is equipped with a G-invariant metric ds 2 . First consider the exponential map exp : V ⊂ T p M → W ⊂ M, that is a diffeomorphism of the open sets V and W , p ∈ W , constructed using the geodesics of ds 2 parameterized with the affine length. As the metric ds 2 is invariant under the action α g of G, α g : W → W , α g will map geodesics of ds 2 to the geodesics of ds 2 . So for any two points q, z ∈ W with α g (q) = z, one has α g (exp v q ) = exp v z , where v q , v z ∈ V with exp v q = q and exp v z = z. As the distance between p and q, and p and z is the same as a consequence of the invariance of metric under G, and the geodesics are straight lines in the coordinates of V , v q and v z have the same length. As a result the induced action exp −1 •α g • exp on V ⊂ T p M is an orthogonal transformation. Thus the action of G on a neighbourhood V of the fixed point p is modelled with rotations.
One of the consequences of the principal orbit theorem is that the normal bundle of a principal orbit G/H is topologically trivial. This is because it can be identified with the pull back of the tangent bundle of the base space B restricted on the orbit G/H. Another consequence is that one should only consider as principal orbits G/H spaces for which G acts (almost) effectively on G/H. Otherwise, G will not act (almost) effectively on M. Though G may not act effectively on special orbits, e.g. G does not act effectively on fixed points.

A Frobenius approach
Before, we proceed with the use of the above two theorems to describe the invariant geometry of a manifold, let us consider an (almost) effective action of a group G on a M which generates the vector fields ξ r , r = 1, . . . , dim g, where g is the Lie algebra of G. The Lie bracket algebra of these vector fields closes as where f are the structure constants of g. The task is to write down the most general form of a metric on M which is invariant under the action of ξ. For this let us explore first the Frobenius theorem. In particular assume that there is an open subspace U ⊆ M such that ξ's span a subbundle L of T U of rank k. The Frobenius theorem states that ξs define a regular foliation. This means that there is a submanifold N of U, called the leaf of the foliation, such that T N = L| N . Moreover U admits an atlas with coordinates x M = (w I , z a ) and patching conditions w I α = w I αβ (w J β , z a β ), z a α = z a αβ (z b β ) such that Note that the components of the vector fields may depend on both the coordinates y of the leaf N and those x of the base space B of the bundle L.
Using these coordinates the most general metric on U can be written as where all components of the metric γ, Γ and g depend on both y and x coordinates. The metric retains its form under the patching conditions provided that Clearly g restricted on N is a metric on N and γ is a fibre metric on π * T B, where π is the projection π : U → B. Furthermore Γ is a non-linear connection [27] and gives a splitting j : π * T B → T U of the sequence In particular the horizontal vector fields which are identified with the sections of jπ * T B in the decomposition Imposing the invariance of the metric (3) under the action of the vector fields (2), one finds that The above conditions on the components of the metric can be simplified dramatically if the vectors fields ξ can be arranged to be independent of z a coordinates. In particular γ would have been independent of the w coordinates and so it would have been the pull back of a metric on B and the non-linear connection Γ would have been strictly invariant under the action of G instead of being invariant up to a gauge transformation as above. However in general, the isometry conditions (6) do not have an explicit solution unless some additional assumptions are made on the vector fields ξ. As a result, it is not apparent how to explicitly express an invariant metric under a group action on a manifold in this approach.
Before we proceed to resolve this puzzle using the slice and principal orbit theorems, notice that locally the metric (3) can be rewritten as whereg IJ +γ abΓ In this form, the metric is locally adapted again to a fibration but now with fibre B and base space N. Furthermore the new fibre twists over N with connectionΓ. It turns out that this is the approach taken by the slice theorem.

Lifting group actions
To explore further the consequences of the slice theorem, let us investigate the invariant vector bundles E over G/H. These are bundles for which left action a g of G on G/H can be lifted to the bundle space E. Before we investigate the lifting of group actions to vector bundles, let us first consider the lifting of group actions to principal bundles. A principal bundle P (K) on G/H with fibre K admits a lifting a ↑ g : P (K) → P (K) of the left action a g on G/H iff there is a group action a ↑ g of G on the bundle space P (K) such that π • a ↑ g = a g • π and a ↑ g (pk) = a ↑ g (p)k, where π : P (K) → G/H is the projection, p ∈ P (K) and k ∈ K. Denote with P (H) the "master" principal bundle which arises from the right action of the subgroup H on G, H → G → G/H. Clearly the bundle space of P (H) is G. One can demonstrate the following [28,29,30].
1. All principal bundles P (K) that admit a lifting a ↑ g of the left action a g of G on G/H are associated bundles 5 P (K) = G × α K of the master principal bundle P (H), where α : H → K is a group homomorphism given by a ↑ h (p 0 ) = p 0 α(h), h ∈ H and p 0 a fixed point in P (K) with π(p 0 ) = eH.
2. The lifting a ↑ g of a g to an associated bundle Typically there are several inequivalent lifts a ↑ g of a g on G/H to a principal bundle P (K) over G/H [28,30]. For every such lift a ↑ g , there is a group homomorphism α such that a ↑ g ′ (g, k) α = (g ′ g, k) α . In particular viewing E D as an associated bundle of P (H), where q : G × V → E D is the standard projection, q * is the push-forward map and R r are the right-invariant vector fields on G.
The identification of a neighbourhood U of a G/H orbit in a manifold M with an invariant bundle E D over G/H and the identification of the action of G on U with the , v) D of G on E D has several consequences. One is that the vector fields ξ generated by the action of G on U can be explicitly written down in terms of the right invariant vector fields on G/H and those generated on the fibres of E D by the representation α • D of H. The simplicity of the group action of G in these coordinates allows for the systematic construction of the invariant metrics on U ⊂ M utilizing the invariant geometry on G/H and E D .

Invariant geometry of homogeneous spaces
Having identified E D as the neighbourhood of an orbit of G in M, it remains to construct the most general invariant metric and other invariant fields on E D . As in many applications of interest, like in AdS/CFT, the Lie algebra of isometries is known rather than the group, we shall focus on the construction of the invariant fields under the action of a Lie algebra instead of a group. The construction presented below can be adapted to include invariance of the fields under groups too but we shall not explore this further here, see e.g. [29]. To proceed take G/H to be reductive 6 so that the commutators of g = h ⊕ m are where h is the Lie algebra of H.
Denote the generators of h with h α , α = 1, 2, ..., dim h and a basis in m as m A , A = 1, ..., dim g − dim h. In this basis, the brackets of the Lie algebra g take the following form If f AB C = 0, that is [m, m] ⊂ h, the space is symmetric. Let θ be the Maurer-Cartan form on G. Thus θ(X A ) = A for all vector left-invariant fields X A generated by the right action of A ∈ g on G. One can define a canonical connection and a frame on G/H by decomposing θ along h and m on G and then pulling back the resulting expression on U ⊂ G/H with the local section s : U ⊂ G/H → G. In particular, one has where ℓ A is a local left-invariant frame and Ω α the canonical left-invariant connection. Note that Ω = s * θ| h , where θ| h is the canonical principal bundle connection on H → G → G/H, see e.g. [29]. The curvature and torsion of the canonical connection are respectively, where the equalities follow after taking the exterior derivative of (12) and using (11). If G/H is symmetric space, then the torsion vanishes. A left-invariant p-form ω on G/H can be written as where the components ω A 1 ...Ap are constant and satisfy The latter condition is required for invariance under the right action of H on G. All left-invariant forms are parallel with respect to the canonical connection. It remains to describe the metrics of G/H which are left-invariant. These are written as 6 Some of our constructions can be extended to the non-reductive case but we shall not elaborate here.
where the components g AB are constant and satisfy For symmetric spaces, the canonical connection coincides with the Levi-Civita connection of invariant metrics.
To present the invariants metrics on E D , one also needs to describe the invariant connections on P (K). For this let as denote with ρ the Lie algebra homomorphism ρ : h → k induced from the Lie group homomorphism α : H → K which characterizes the principal bundle P (K) = G × α K. Next consider the linear map Λ : m → k such that where f cb a are the structure constants of the Lie algebra of K, k, in a basis t a . Then the most general linear invariant connection is where ρ a α Ω α is the canonical connection of P (K) = G × α K. For a proof of this and more details, see e.g [29].
The curvature F of Σ is The induced connection on where D is the representation of k on V induced by D.
To construct the most general class of metrics invariant under G on M, let us consider where Π a A are constants, and impose the condition This condition is required for Π to transform covariantly under the right H transformations. Clearly a fibre metric on P (K) × D V will be invariant under the action of G iff For completeness, the invariant forms with values in the tensor product of bundle ⊗ q (G× K V ) satisfy the condition All the above formulae can be easily adapted to the special case where K = H and ρ = 1 h .

Main result: Invariant fields on the spacetime
In the context of homogeneous spaces the metric g AB , connection Λ α A and Π a A and fibre metric γ ab are constants. To construct a metric on E D = G × H V invariant under G and so suitable to model an invariant metric on M in the neighbourhood of an orbit, we allow g AB , Λ α A , Π a A and γ ab to depend on y, i.e. become functions of the fibre V . With this understanding, the metric where D α a b = ρ a α D a a b and L Dα denotes the Lie derivative with respect to the vector fields (27) are the analogues to those in (6) but there is a difference. Here the vector field D α appearing in the Lie derivative L Dα is that of a rotation. Because of this, it is more straightforward to solve (27) instead of (6).
A similar construction works for other fields. In particular an invariant form under the group action of G in the neighbourhood of an orbit G/H is provided that where e a = dy a + Σ a D a a c y c + Π a A ℓ A . A special solution to (27) and (29) can be constructed as follows. In the context of homogeneous spaces, the conditions (17), (18), (23), (24) and (25) determine Λ, γ, Π, g and ω up to some constants. Now if one replaces those constants with invariant functions under the representation D of H on V , the conditions (27) and (29) will be automatically satisfied. This is an assumption made in all examples demonstrated below.
Although the metric proposed in (26) is constructed under some smoothness assumptions for both the group action and the associated underlying manifold, it is a good starting point to investigate solutions to the Einstein equations even in the case that these assumptions are violated. In fact it is expected that many solutions to the field equations that can be constructed using (26) will be singular. In the presence of singularities, the slice and principal orbit theorems may not apply everywhere but locally (26) still can be used to find solutions. Similarly, one can remove some of the compactness assumptions necessary for the validity of the slice and principal orbit theorems and still use (26) to identify some invariant metrics.
The metric (26) describes the invariant geometry of the spacetime in the neighbourhood of an orbit N independent on whether the orbit is principal or special 7 . N is identified with the zero section of E D . For principal orbits, we have already mentioned that as a consequence of the principal orbit theorem E D must be a topologically trivial bundle over G/H. This is not a sufficient. In particular one has to demonstrate that the orbits of G on E D away from the zero section are still N. This can be proven directly by identifying the isotropy group of the orbits of G in a neighbourhood of the zero section in E D and compare it with the isotropy group of G acting on N. If the isotropy subgroups in G are isomorphic up to a conjugation, then this will prove that the orbit N is principal. One can also compute the codimension of the orbits of G in a neighbourhood of the zero section of E D and compare it with that of N in M. For this test, it is equivalent to require that the equations have as many independent solutions as the codimension of N in M, where f is a function defined on a patch of E D and ξ r are the vector fields generated by the action of G on E D .
To see this observe that as a consequence of the Frobenius theorem, the vector field ξ r can be written as in (2). The independent solutions f of (30) are the coordinates z of the spacetime, i.e. the coordinates of the base space B of the foliation.

Some examples
In all examples that we shall investigate below as well as in all applications to AdS backgrounds, we shall assume that the L D terms in the conditions (27) vanish. In the cases that D is the trivial representation, this follows automatically. Otherwise, it is an additional assumption that we use to find solutions. Assuming this, the remaining conditions in (27) are algebraic and can be solved. As it has already been mentioned the end result is that components of the invariant fields will depend on invariant functions of the coordinates y of V under the action of the D representation of H on V .

Invariant geometry on SU(2) and S 2
Before we proceed to apply the formalism developed so far to AdS backgrounds let us consider examples mainly focused on S 2 and S 3 orbits. To describe the most general SU(2) invariant metric on a manifold with S 2 = SU(2)/U(1) orbits, parameterize SU(2) in terms of the Hopf coordinates as where 0 ≤ η ≤ π 2 and 0 ≤ ϑ 1 , ϑ 2 ≤ 2π. Choosing as a basis in su(2) the anti-Hermitian matrices {t 1 = iσ 1 , t 2 = iσ 2 , t 3 = iσ 3 }, where σ 1 , σ 2 , σ 3 are the Pauli matrices, the left invariant 1-forms arê where 2τ = ϑ 2 − ϑ 1 and 2ρ = ϑ 1 + ϑ 2 . In this basis, the left invariant vector fields are Note that [L r , L s ] = −ǫ rs t L t , where r, s, t = 1, 2, 3. Similarly the right-invariant 1-forms arer and the corresponding right-invariant vector fields are where [R r , R s ] = ǫ rs t R t . To give the most general invariant metric on a manifold with S 2 orbits, let us first investigate the invariant geometry on S 2 = SU(2)/U (1). For this assume without loss of generality that the right U(1) action on SU(2) generates the left invariant vector field L 3 . If π : SU(2) → S 2 = SU(2)/U(1) is the standard projection, then clearly π * L 3 = π * ∂ τ = 0, where π * is the push forward map associated to π. The left action of SU(2) on S 2 generates the vector fields where ρ and η are the coordinates of S 2 . As [π * X, π * Y ] = π * [X, Y ], the pushed forward vector fields π * R s , s = 1, 2, 3, satisfy the same Lie algebra bracket relations as those of R s , i.e. their Lie algebra is su(2).

Codimension two S 2 orbits
As irreducible non-trivial real representations D n , n ∈ Z−{0}, of U(1) are two dimensional, let us first assume that S 2 is a codimension 2 orbit. Next consider the associated invariant vector bundle E n = SU(2) × Dn R 2 . The U(1) action on R 2 via the representation D n generates the vector field D n = n(y 1 ∂ ∂y 2 − y 2 ∂ ∂y 1 ) , where (y 1 , y 2 ) are the standard coordinates of R 2 . Note that in radial coordinates y 1 = r cos χ and y 2 = r sin χ on R 2 − {0} Let p : SU(2) × R 2 → E n be the standard projection. It is clear that p * (L 3 − n∂ χ ) = p * (∂ τ + n∂ χ ) = 0. The vector fields generated by the left action of SU(2) on E n , away from the zero section, are given by where (ρ, η, χ, r) are the coordinates of SU(2) × U (1) R 2 with (ρ, η) the coordinates of the base space S 2 . Notice that p * R 1 , p * R 2 , p * R 3 can have a non-trivial component along R 2 . If q : SU(2) × Dn R 2 → S 2 is the standard projection, then it is straightforward to observe that q * p * R s = π * R s , s = 1, 2, 3. According to the slice theorem p * R 1 , p * R 2 , p * R 3 model the general action of SU(2) at a neighbourhood of an S 2 orbit. To construct the most general invariant metric on SU(2) × U (1) R 2 first observe that the canonical connection Ω of SU(2)/U(1) is Ω = s * l3 = − cos(2η)dρ, where the local section 8 s : W ⊂ S 2 → SU(2) is chosen as s(η, ρ) = (η, ρ, 0). The condition on Λ in (18) implies that Λ = 0. Furthermore if in addition n = ±1, (23) also implies that Π = 0. So the only twisting of the fibre coordinates of the fibration is induced by the canonical connection. As S 2 is a symmetric space the invariant metric on S 2 is uniquely specified up to a constant a. Using these, one finds that the invariant metric (26) can be written as where ℓ A = s * lA , A = 1, 2, and a, b 1 , b 2 depend only on the coordinate r (n = ±1, 0). The fibre metric γ decomposes as indicated because of (24). For n = ±1, there is an additional contribution from Π in the metric (26) but this will not be explored here. For n = 0, the vector bundles SU(2) × Dn R 2 are topologically non-trivial and so do not model the neighbourhood of principal S 2 orbits of codimension 2. Alternatively, use (30) and observe that the equations ξ r f = 0, r = 1, 2, 3, for n = 0, have only one independent solution instead of two required for codimension two orbits of SU (2) The independent solution is f = r. This again rules out SU(2)× U (1) R 2 as a neighbourhood for principal S 2 orbits. The metric (40) describes the geometry of a codimension one orbit of SU(2) as approaches a codimension two special S 2 orbit.
On the other hand for n = 0, ξ r f = 0, r = 1, 2, 3, has two independent solutions f = r, χ. The orbit S 2 is principal. Without loss of generality one can consider S 2 as codimension 1 principal orbit as D 0 is the trivial representation of U(1) which is 1dimensional. As S 2 = SU(2)/U(1) is a symmetric space, the invariant metric on S 2 is unique up to an overall scale. The connection Σ a D a vanishes as D n = 0 for n = 0. Next focus on the contribution that comes from Π in the metric (26). h = u(1) acts trivially on the fibre as D 0 = 0 but on the other hand acts with the fundamental 2-dimensional representation on m. As a consequence (23) implies that Π = 0. Thus the most general invariant metric that one can write down is where A, B = 1, 2 and a 2 , b 2 are arbitrary functions of y. This is a warped metric on R × S 2 , where S 2 is the round 2-sphere. The generalization to codimension ≥ 2 principal S 2 orbits is straightforward.
To construct the most general invariant metric on SU(2) × U (1) R 4 first observe that the canonical connection Ω of SU(2)/U(1) is Ω = s * l3 = − cos(2η)dρ, where the local section is chosen as s(η, ρ) = (η, ρ, 0) as in the previous section. The condition on Λ in (18) implies that Λ = 0. Furthermore if in addition n, m = ±1, (23) also implies that Π = 0. So the only twisting of the fibre coordinates of the fibration is induced by the canonical connection. Using this, one finds that the invariant metric (26) in this case can be written as a, b 1 , b 2 , b 3 , b 4 depend on the coordinates r 1 , r 2 and mχ 1 − nχ 2 , and n, m = ±1.
The bundle SU(2) × U (1) R 4 is not a neighbourhood of principal S 2 orbits for n, m = 0. To see this observe that for n, m = 0, the equation ξ r f = 0, r = 1, 2, 3, has only three independent solutions f = r 1 , r 2 , mχ 1 − nχ 2 instead of the four required for codimension four orbits. So the metric (43) describes the geometry of M in a neighbourhood of a special S 2 orbit of codimension 4.

S 3 = SU(2) orbits
As the isotropy group is the identity, H = {e}, both the representations on m and D on V = R k are trivial. As a result there is no contribution in the metric from either the canonical connection Ω or Λ. However, the invariance condition (23) is automatically satisfied and so where Π a A are some constants and ℓ A =l A , A = 1, 2, 3, as given in (32). In turn the invariant metric (26) on E = SU(2) × R k is where now g AB , γ ab and Π a A depend on the coordinates y of the fibre R k . The Killing vector fields are R A given in (35). The metric (45) can admit a larger isometry group provided g AB , γ ab and Π a A are chosen appropriately, i.e. it can also be invariant under the right action of SU(2) generated by L A in (33).
To make a connection with the discussion on Kaluza-Klein theory below as well some global aspects of principal bundles, let us rewrite the metric (45) adapted to a fibration with fibre SU(2), i.e. in the form (3). Indeed This is a local metric on a principal SU(2) fibration with fibre metricĝ AB which depends on the base manifold coordinates y. Principal bundle theory is set up with the patching conditions to act with left transformations on on the typical fibe SU(2). These do not commute with the isometries generated by R A and so the left group action generated by R A does not patch globally. As a result to retain globally an SU(2) invariance, one has to require that the metric (46) is invariant under SU(2) transformations acting on the right generated by L A . In turn this requires the conditions that where is viewed as the principal bundle connection. Therefore the fibre metricĝ AB must be bi-invariant but still can depend on the coordinates y. If in addition the fibre metricĝ AB is taken to be constant, then the metric (46) is the DeWitt ansatz for a Kaluza-Klein vacuum with internal space the group manifold SU(2) which always yields a consistent truncation of the Kaluza-Klein spectrum. This generalizes to all group manifold Kaluza-Klein reductions.
Indeed observe that ξ r h = 0, r = 1, 2, 3, 4, has one solution h = r instead of two required for codimension 2 orbits. Therefore, the metric (53) models the geometry around a special S 3 orbit.
So to model the geometry of principal SU(2) × U (1) U(1) orbits, one should take n = 0. In such a case, one can demonstrate that where e is a constant. Then the invariant metric reads where now a 1 , a 2 , b 1 , b 2 and e are arbitrary functions of χ, r. Note that although for principal orbits the representation D is trivial, the metric above still contains rotation terms and it is not just a warped product type of metric. This example can be easily generalized to SU(2) × U(1)/U(1) p,q , where p, q ∈ Z − {0} and co-prime. In such a case h is generated by pt 3 + qt 0 which in turn generates the vector field pL 3 + q∂ ψ on SU(2) × U(1). Choose m to be spanned by {pt 3 − qt 0 , t 1 , t 2 }. Then the canonical connection is and so where s is any local section of the fibration U(1) → SU(2)×U(1) → SU(2)×U(1)/U(1) p,q . Furthermore, one can demonstrate that where f is a constant. Next consider invariant metrics on (SU(2 × U(1)) × U (1) R 2 . For n = ±p, one can show that Π = 0 as a consequence of (23). Using this the invariant metric on (SU(2 × U(1)) × U (1) R 2 can be written as where a 1 , a 2 , b 1 , b 2 , f depend on r, ℓ A ′ = s * lA ′ , A ′ , B ′ = 1, 2, Ω = 1 2 s * ( 1 pl 3 + 1 q dψ) and Again the metric (59) for p, q, n = 0 models the geometry in a neighbourhood of a special SU(2) × U(1)/U(1) p,q orbit.
For principal orbits, one has to again take n = 0. After an analysis similar to that we have explained above, the invariant metric can be written as in (55), where now As a final example consider SU(2) × SU(2) = × 2 SU(2) acting on a manifold with S 3 = SU(2) × SU (2) SU(2) orbits. The right action of SU(2) on × 2 SU(2) generates the vector fields L α −R α , α = 1, 2, 3, where L α are the left invariant vector fields given in (33) on SU(2) × {e} whileR α are the right-invariant vector fields given in (35) on {e} × SU (2). Therefore L α −R α span the Lie algebra of the isotropy group h = su (2). The × 2 SU(2) action on S 3 = SU(2) × SU (2) SU(2) is generated by π * R r , π * Lr , r = 1, 2, 3, where π : (2). Suppose now that the orbit S 3 has codimension 3. As the principal bundle × 2 SU(2) → SU(2) × SU (2) SU(2) is topologically trivial, all the associated vector bundles of this are topologically trivial and so the associated vector bundle E D = × 2 SU(2) × SU (2) R 3 , where the representation of SU(2) on the typical fibre R 3 is the same as that of the isotropy group SU(2) on m, i.e. D is the standard vector representation. The condition on Λ in (18) can be solved by setting Λ = e1, where e a constant. Similarly, the condition on Π can be solved to yield Π a = f δ a A s * (l A +r A ), where f is a constant. As it is well known there is a single invariant metric on SU(2) × SU (2) SU(2) up to a constant. Promoting these constants to invariant functions on the fibre R 3 under the action of the vector representation of SU(2), one finds that (26) reads where a, b, e, f are functions of the radial coordinate r of R 3 and we have used the local section s : V ⊂ S 3 → × 2 SU(2) with s(g) = (g, e). Thus s * r = 0 and so the connection is Ω α = 1 2 s * (l α −r α ) = 1 2 s * lα and similarly the frame is ℓ A = 1 2 s * (l A +r A ) = 1 2 s * lA , wherê ℓ A are the left-invariant forms (32) on S 3 .
Before we complete the discussion notice that the most general SO(3) invariant metric on R 2 is b 2 1 (r)dr 2 + b 2 2 (r)r 2 ds 2 (S 2 ) and so it is determined by two functions. However one of them can be eliminated using a coordinate transformation of r, e.g. set b 1 = b 2 = b. This is in agreement with the form of the metric in (60) in which the fibre metric depends on one function b.

Applications to AdS backgrounds
Let us next turn to explore some applications in the context of AdS backgrounds. If the R-symmetry is abelian, say U(1), and acts (almost) effectively on the internal space M generating a vector field X, one can always adapt a coordinate t to X, X = ∂ t . As H = {e} and G = U(1), the only contribution along the fibre in the invariant metric (26) is from Π. In particular the metric can be written as where the only restriction on a 2 , γ, Π is that they should be independent of t and otherwise depend on all y coordinates. This metric has the form of (7) and it can be easily transformed to the familiar expression in (3). In what follows, we shall focus on the geometry in the neighbourhood of principal obits.

AdS 6 backgrounds
Since smooth IIB AdS 6 backgrounds with compact internal space are the only ones that have not been classified [12], let us apply our analysis above to this case. The Lie algebra of the R-symmetry group is so(3) = su(2) and the only unresolved case is that with principal orbits S 2 = SU(2)/U(1). The internal space of AdS 6 backgrounds in IIB has dimension 4, the principal orbit S 2 has codimension 2 and so the normal bundle has rank 2. In addition from the results of section 4.1.2, the normal bundle must be associated with the trivial representation of the isotropy group u(1) and so the metric of the internal space is given in equation (41).

AdS 5 backgrounds
Next consider AdS 5 backgrounds. The maximally supersymmetric AdS 5 backgrounds have been classified in [2] and those preserving 24 supersymmetries have been shown to be locally isometric to the maximally supersymmetric ones [3]. AdS 5 backgrounds that preserve 16 and 8 supersymmetries are known to admit a u(2) and u(1) R-symmetry algebras, respectively. The latter have already been dealt with as part of the general analysis of backgrounds with a u(1) symmetry above. It remains to consider the backgrounds preserving 16 supersymmetries. Up to discrete identifications, the homogeneous spaces which admit an (almost) effective SU(2) × U(1) action are SU(2) × U(1) and SU(2) × U(1)/U(1) p,q , where p, q ∈ Z are co-prime, p = 0.
In what follows, let us seek metrics on the internal spaces for AdS 5 backgrounds with SU(2) × U(1) and SU(2) × U(1)/U(1) p,q as principal orbits. In a type II theory in 10 dimensions, the principal orbit SU(2) × U(1) has codimension 1 in the internal space. So the most general metric invariant metric on the internal space is where the metric g AB on SU(2) × U(1), Π and b 2 depend only on y. The metric above can be written as a principal bundle metric. The results in section 4.1.4 obtained for SU (2) can be easily adapted for SU(2) × U(1).
Next turn to investigate the internal spaces with principal SU(2)×U(1)/U(1) p,q orbits. These in type II 10-dimensional theories have codimension 2 in the internal space. A detailed analysis has already been carried out in section 4.1.5. The metric on the internal space is given in (55) for manifolds with a SU(2) × U(1)/U(1) orbit, p = q = 1. For the rest it is again given in (55) after an appropriate definition of ℓ 3 , see discussion in section 4.1.5.

AdS 4 backgrounds
Next consider AdS 4 backgrounds. The internal spaces of those preserving 4 supersymmetries admit no R-symmetries and those preserving 8 supersymmetries admit an so(2) symmetry that we have already investigated. Backgrounds preserving 12 supersymmetries admit an so(3) = su(2) action. There are two kind of orbits, SU(2) and SU(2)/U(1) = S 2 , up to discrete identifications, that can occur admit an (almost) effective SU(2) action. We have already described manifolds with principal SU(2) and SU (2) It remains to investigate AdS 4 backgrounds that preserve 16 supersymmetries. These must admit an (almost) effective so(4) action. The homogeneous spaces which admit an (almost) effective SO(4) action, up to discrete identifications, are SO(4) , SO(4)/SO(2) m,n , SO(4)/(SO(2) × SO (2) where m, n are integers, (relatively prime), which specify the embedding of SO(2) into SO(4). Let us begin with principal SO(4) orbits. Such orbits are of codimension 0 in the internal spaces of 10-dimensional backgrounds and of codimension 1 in 11-dimensional backgrounds. In the former case, the internal manifold is homogeneous. In the latter case, the metric on the internal space can be written as in (63) and the description of the components is the same as in the previous case but now SU(2) × U(1) is replaced by SO(4), see also section 4.1.4.
Next let us explore internal spaces of AdS 4 backgrounds with principal SO(4)/SO(3) = SU(2) × SU (2) SU(2) = S 3 orbits in type II 10-dimensional theories. These have codimension 3 in the internal space. It is clear that from the results of section 4.1.6 that the representation D of the little group SU(2) on R 3 must trivial. It is then straightforward to verify that Π = 0. The invariant metric on the internal space can be written as where b 2 and γ depend only on the coordinates y. The metric g AB = b 2 δ AB is required as the metric on S 3 must be both left-and right-invariant. From the remaining cases, let us consider internal spaces with principal SO(4)/SO(2) m,n , m, n = 0, orbits. These are of codimension 1 in the internal spaces of 10-dimensional AdS 4 backgrounds. The D representation of SO(2) on the fibre is trivial, therefore the canonical connection and Λ do not contribute to the metric. Provided that the isotropy group SO(2) generates the vector field mL 3 − nR 3 on SO(4) ∼ SU(2) × SU(2), the conditions (23) imply that Π = 1 2 e s * 1 where e is a constant and s a local section. After solving the invariance conditions (17), one finds that the invariant metric on the internal space is where ℓ A ′ = s * lA ′ , rÃ ′ = s * rÃ ′ , A ′ ,Ã ′ = 1, 2, ℓ 3 = 1 2 s * 1 ml 3 + 1 nr 3 , and a 1 , a 2 , b 1 , b 2 , e depend on y. A more systematic investigation of AdS backgrounds which will include the remaining invariant fields of the associated theories will be presented elsewhere.

Concluding Remarks
We have provided a systematic way to construct invariant metrics and other invariant fields under the action of a (non-abelian) group G on a manifold M. Such metrics model the invariant geometry around an orbit N in M of the group G either this orbit is principal or special. For this we utilized the geometry of homogeneous spaces, N = G/H, together with the slice and principal orbit theorems. The slice theorem provides a local model of the action of a group in a neighbourhood around an orbit, e.g. provides an expression for the vector fields generated by the group action in a convenient coordinate system. We presented several examples that illustrate the construction mostly focused on S 2 and S 3 orbits. Furthermore, we used our results to construct invariant metrics on the internal space of AdS backgrounds under the action of the R-symmetry group with main focus on the geometry in a neighbourhood of a principal orbit.
As the invariant metric (26) provides a model for the local geometry of M around any orbit either it is principal or special, (26) can also be used to investigate the geometry of the internal spaces of AdS backgrounds that contain special orbits of an R-symmetry group. Combined this with the results we have described in section 4.2 will provide a complete description of the local geometry of internal spaces. Of course, the expression for the metric in (26) solves the kinematic problem. To find a background one also has to solve the field equations of a theory. Nevertheless, the approach proposed is systematic and the problem is further simplified for the supersymmetric backgrounds.
The metric (26) can also be used in the context of Kaluza-Klein theory. For this, one first rewrites it as a fibration with fibre the orbit N, i.e. as in (3). As the vector fields ξ r are isometries, one can gauge these isometries by adding a Paul term, i.e. replace dw I with dw I − A r a ξ I r dz a in (3), where A is a gauge field that depends on the coordinates of the lower dimensional spacetime. Such a metric or a simplification of it can be used as a Kaluza-Klein vacuum. There are several simplifications to (26) that can be made by setting some components of the metric that are allowed to depend on the spacetime coordinates to be constant, i.e. make a selection of the allowed breathing modes. Of course it is not apparent that such an ansatz will lead to consistent truncation to a lower dimensional theory, see e.g. [31] for a recent discussion and references within. Nevertheless (26) can provide a model for the local geometry of a region where the internal space of a compactification degenerates to a lower-dimensional internal space while preserving the number of isometries of the internal space or equivalently the number of gauge fields A of the lower dimensional theory.