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Supergravities on Branes

Supergravity brane solutions allow for a generalised type of Kaluza-Klein reduction onto brane worldvolumes. The known replacement of a flat worldvolume metric by a Ricci-flat metric constitutes a consistent Kaluza-Klein truncation of the starting higher-dimensional supergravity theory down to a lower-dimensional pure gravity theory. This paper shows how to extend such a brane-worldvolume pure-gravity consistent truncation to that for a full nonlinear supergravity theory for the Type IIB D3-brane and the M-theory/Type IIA M5-brane. The extension of worldvolume supersymmetry is given by the unbroken supersymmetry of the original flat"skeleton"brane. Compatibility with further worldvolume diagonal and transverse vertical dimensional reductions is also shown, providing the brane-worldvolume supergravity embeddings of all descendants of the skeleton D3- and M5-branes. Examples are given of brane-worldvolume supergravity solutions embedded into the corresponding higher-dimensional supergravities.

The idea of a gravitational theory belonging to a Minkowski-signature subspace of a higher dimensional spacetime, i.e. a braneworld gravity theory, has a long history in theoretical physics investigations, and has clearly been of interest for cosmology [1]. In Ref. [2], a categorisation, or "taxonomy" of such braneworld gravities was given for constructions involving noncompact transverse spaces. The conceptually simplest of these categories, "Type I", involves a consistent truncation of the higher-dimensional supergravity to a lower-dimensional theory on the worldvolume. This type will be the main focus of the present paper. It should be contrasted with the generation, not via a consistent truncation, of a lower-dimensional effective theory that is only valid within a certain low-energy range of scales, called "Type III" in Ref. [2]. Examples of the latter include Randall-Sundrum (RS) models [3] involving patching of sections of a higher-dimensional AdS space, or models [4] which naturally generate a mass gap between a discrete L 2 normalisable transverse-space wave function mode and the continuum of delta-function normalisable modes expected from a noncompact transverse space [5].
Type I and Type III constructions have different virtues. Type I constructions, involving a consistent truncation of the higher dimensional theory, give rise to fully nonlinear lower-dimensional worldvolume theories. Type III constructions, not based on a consistent truncation, inherently give only a perturbative realisation of lower-dimensional gravitational physics, but they do generate an intrinsic localisation of the effective gravity theory at least within a limited range of scales [2,6].
The existence of Type I consistent truncations to worldvolume gravities was found in Ref. [7] for domain-wall or more general magnetic braneworlds. These constructions promote the flat Minkowskian worldvolume metric of an initial "vacuum" supergravity brane solution to a generalisation with a worldvolume Ricci-flat metric. Within the context of the original higher dimensional theory, this amounts to making a consistent truncation to the lowerdimensional theory, but just for the worldvolume metric. This gave rise to applications such as braneworld black holes [8].
There can be a lot more to worldvolume physics than just Ricci-flat pure gravity met-rics, however. An indication of this possibility was given in a study [9] of the embedding of Maxwell-Einstein d = 4 supergravity within a D = 5 RS gauged supergravity context, which was then lifted further up to D = 10 Type IIB supergravity using an existing Kaluza-Klein S 5 ansätze [10]. In that case, the unbroken supersymmetry of the vacuum RS construction allowed an extension of a d = 4 metric-only truncation to one involving also a lowerdimensional Maxwell field. This provided a good example of a Kaluza-Klein reduction down to a d = 4 interacting gravity-matter theory with nontrivial dependence on the transverse space -in that case, reduction on the single coordinate transverse to the RS patch surface.
Since an AdS 5 ×S 5 spacetime is the asymptotic near-horizon geometry of a Type IIB D3brane, the construction of Ref. [9] may be taken to suggest a further generalised embedding of D3 worldvolume (i.e. horizon) supergravity into D = 10 Type IIB supergravity. The original D3-brane is a 1 2 -BPS solution, i.e. it possesses sixteen unbroken supercharges. So the suggested worldvolume supergravity theory in the D3 brane case is N = 4, d = 4 supergravity. The original flat Type IIB D3-brane solution should serve as a structural "skeleton" for such a construction, generalising the way that D = 5 AdS space serves in the construction of Ref. [9].
Showing how a fully nonlinear N = 4 supergravity theory embedding on a D3-brane worldvolume works is the first result of the present paper, as presented in Section 2. The resulting KK ansätze is motivated by considering carefully the two asymptotic limits: flat D = 10 space at transverse-space infinity and AdS 5 × S 5 in the near-horizon region, with the result confirmed by showing agreement, after a voluntary truncation to just N = 2 supersymmetry, with the d = 4 structure of Ref. [9]. We follow this up by demonstrating how a wide class of N = 4 worldvolume black-hole solutions is embedded into the Type IIB theory, using the harmonic-map construction of Refs [11][12][13][14].
In Section 3, we present an analogous construction for the M5-brane, with fully nonlinear N = (4, 0), d = 6 worldvolume supergravity. An example of an embedded solution into D = 11 M-theory supergravity in this case is given by a worldvolume multi-charged antiself-dual string solution.
Section 4 shows how further diagonal and vertical dimensional reductions may consistently be combined with the braneworld embeddings, giving the braneworld supergravity embeddings of all descendants of the D3-and M5-branes.

D3-branes with their worldvolume supergravities
The geometry of the 1 2 -BPS solution of D = 10 Type IIB supergravity describing parallel D3-branes is a warped product R 1,3 × W R 6 . As shown in Ref. [7], it is consistent to replace the R 1,3 worldvolume of the D3-branes with a generic, four-dimensional manifold M 4 with a general Ricci-flat metric g µν . In the language of Ref. [2], this is a "Type I" consistent truncation of Type IIB supergravity down to pure, N = 1, d = 4 supergravity. However, since the original flat-worldvolume solution preserves sixteen supercharges, we anticipate the existence of a more general consistent truncation to pure N = 4, d = 4 supergravity.
The Ricci-flat replacement R 1,3 → M 4 would then be just a further consistent truncation of the N = 4 theory to its purely gravitational sector. Our goal here is to construct such a generalisation of flat D3-branes, referred to henceforth as skeleton branes, to ones with full four-dimensional worldvolume N = 4 supergravity. We will then classify the stationary black-hole solutions of the N = 4 theory and, for completeness, will explicitly construct a family of charged black hole solutions. These solutions, when uplifted back into Type IIB supergravity using our embedding, have interpretations as D3-branes with charged black holes on their worldvolume.
where dΩ 2 5 is the metric on the round five-sphere, vol 5 is its associated volume form, and for simplicity, the string coupling constant has been set to one. This solution preserves sixteen rigid supercharges. For this to be a string theory background, the five-form flux must obey the quantisation condition 1 16π 4 for any non-trivial five-cycle C 5 . The only non-trivial five-cycle in (2.1) is C 5 = S 5 , and for it we find 1 16π 4 In terms of the Cartesian coordinates y Λ , Λ ∈ {1, . . . , 6}, on the transverse space R 6 , the volume form of the five-sphere is where r 2 = y Λ y Λ , ǫ 123456 = 1, the Λ indices are raised/lowered by the flat Euclidean metric δ ΛΣ , and * 6 is the Hodge dual with respect to δ ΛΣ . The five-form is then explicitlŷ where vol 4 is the volume form on the Minkowski worldvolume.
The solution (2.1) is the skeleton on which we will embed the pure N = 4, d = 4 supergravity theory. This theory contains one graviton, four gravitini, six U(1) vectors, four dilatini, and one complex scalar, which parametrises the coset space SL(2, R)/SO(2). We will denote the bosonic degrees of freedom by the metric g µν , six one-forms A Λ (1) , where Λ ∈ {1, . . . , 6}, and the complex scalar τ = χ + ie −φ . The Lagrangian describing the bosonic sector of the d = 4 theory is given by 1) . For the rest of this paper, all fields without a hat are taken to be lower-dimensional.
For completeness, we will record below the Bianchi identities and the equations of motion obtained from (2.6): is the SL(2, R) dual of F Λ (2) .

Embedding N = 4 supergravity on D3-branes
To embed the pure N = 4 supergravity theory on the worldvolume of the skeleton D3-branes, we need to generate six Abelian vector fields and a complex scalar that parameterises the coset SL(2, R)/SO (2). These vector fields cannot be obtained through a usual Kaluza-Klein reduction on the transverse R 6 , as the D3-branes are supported by a harmonic function on that R 6 . They also cannot be generated through a Pauli reduction on the transverse S 5 , i.e. the five-sphere boundary of R 6 at infinity, because U(1) 6 is not a subgroup of SO (6).
This means that the six vector fields must appear in the fluxes of the Type IIB theory, and, due to gauge invariance, they must appear through their field strengths. The field strengths must also appear in an SO(6) invariant manner, as the N = 4 theory has a global SO (6) symmetry that rotates them. Furthermore, the D = 10 Type IIB theory already contains a complex scalarτ =Ĉ 0 + ie −Φ that parameterises the SL(2, R)/SO(2) coset. With these facts in mind, we find, after some trial and error, that the embedding of the full d = 4, N = 4 supergravity theory on the worldvolume of skeleton D3-branes is given by (2.13) where F Λ (2) are two-forms on the four-dimensional subspace that we will show correspond to the field strengths of the N = 4 theory, * 4 is the Hodge dual computed with respect to the four-dimensional metric g µν , vol 4 is the volume form associated with g µν , and * 6 is the Hodge dual computed with respect to the flat metric δ ΛΣ on the transverse R 6 .
We will demonstrate that (2.13) solves the Type IIB Bianchi identities and equations of motion, which are presented in Appendix A, provided that the four-dimensional fields (g µν , φ, χ, F Λ (2) ) solve the Bianchi identities and equations of motion of N = 4 supergravity (2.7)-(2.11). Let's begin with the Bianchi identities. The five-form in (2.13) is closed since H is harmonic, so its Bianchi identity (A.1) becomeŝ which is satisfied by our three-form fluxes, as F Λ (2) ∧ * 4 F Σ (2) is symmetric in (Λ, Σ). Next, the Bianchi identities (A.3,A.2) for the three-forms givê as defined in (2.12). These are the Bianchi identities and flux equations of motion of the N = 4 theory.
To evaluate the equations of motion, we record below the Hodge dualŝ where vol 6 is the volume form on R 6 . Beginning with theF (3) equation (A.7), we find where we have used the Bianchi identity dF Λ (2) = 0 and the identity d * 6 dy Λ = 0. Next, identically. For theĤ (3) equation (A.8), a similar calculation shows that it reduces to a 0 = 0 identity using the Bianchi identities and field equations for F Λ (2) . For the RR scalar equation, we have as dH ∧ vol 6 = 0 and d vol 6 = 0, and and we find that the transverse dependence factors out nicely. Similarly, the dilaton equation These are the scalar equations for the N = 4 theory.
Finally, we move to the Einstein equations (A.11). The non-zero components of the Ricci tensor areR be the contribution ofF (5) to the stress tensor. A straightforward calculation shows that the non-zero components ofT 1 M N arê which perfectly cancels the "brane part" of the Ricci tensor. Thus, the Einstein equations reduce to for the worldvolume components, for the mixed components, and for the transverse components. Using the ingredients provided in Appendix B, we find that (2.28) reduces to We have now shown explicitly that (2.13) is a consistent truncation of Type IIB supergravity on a background of N parallel D3-branes to the N = 4 supergravity theory on the four-dimensional worldvolume.

Asymptotic limits
The skeleton D3-brane solution (2.1) is part of a large class of interpolating solitons. At small r, the solution becomes AdS 5 × S 5 , and at large r, it becomes the R 1,9 vacuum. We will examine our embedding of the N = 4 theory obtained in (2.13) at these limits, and find that (2.13) retains the interpolating structure.

The flat limit
As r → ∞, the harmonic function H → 1, and the embedding of the N = 4 theory (2.13) becomes which describes the embedding of the N = 4 theory into the R 1,9 vacuum of Type IIB supergravity. It is easy to check that (2.32) is indeed a solution to the Type IIB equations of motion provided that the four-dimensional fields satisfy the equations of motion of the N = 4 theory.
As an aside, it is interesting to consider compactifying the R 6 in (2.32) by a compact quotient. This is now possible because the R 6 flat-space trivialisation of the harmonic function supporting the original solution allows ∂/∂y Λ to be Killing vectors. The simplest compactification will be the unit six-torus T 6 , where the coordinates y Λ take values in [0, 2π). With this identification, the Page charges associated toĤ (3) andF (3) , corresponding to the charges of the NS5-and D5-branes sourced by these fluxes, are now finite, with and where C 2 and C ′ 2 are, in principle, different two-cycles. We can write this more succinctly as where c 1 denotes the first Chern class. The first Chern classes are quantised, so In string theory, however, we must have Q NS5 , Q D5 ∈ Z. To remedy this, we employ a trombone rescaling of the Type IIB fields (see (A.12)), are integers. We note that the NS5 charge corresponds to the magnetic charges of the N = 4 theory, whereas the D5 charge corresponds to the generalised electric charges.

The horizon limit and N = 2 supergravity
As r → 0, the harmonic function H → c/r 4 with c = 4πN, and the embedding of the N = 4 theory (2.13) becomes where we have used spherical polar coordinates on R 6 , and we recall that dΩ 2 5 and vol 5 are respectively the metric and the volume form on the round five-sphere. For an asymptotically flat metric g µν , (2.39) corresponds to a truncation of Type IIB supergravity on the five-sphere down to an asymptotically AdS 5 spacetime. It can be verified that (2.39) is also an exact solution to the Type IIB equations of motion provided that the four-dimensional fields obey the corresponding equations of motion of the N = 4 theory.
The N = 4 supergravity theory (2.6) in four dimensions admits a consistent truncation to the N = 2 supergravity theory, which contains one graviton, one vector, and two gravitini. The Lagrangian for the bosonic sector of this theory is simply the Einstein-Maxwell Lagrangian, There are multiple ways of getting to the N = 2 theory from the N = 4 theory, but at least two of the vectors of the N = 4 theory must be retained. Keeping only F 5 (2) and F 6 (2) , the truncation is given by In terms of the Type IIB embedding (2.39), we have (2.42) The N = 2 theory was also embedded into the AdS 5 × S 5 near-horizon of parallel D3-branes in Ref.
[9] using a specific system of spherical polar coordinates. We shall now show that our embedding (2.42) is equivalent to that of Ref. [9].
In the absence of the Type IIB scalars, it is convenient to combine the NSNS and RR three-forms into a complex three-formĜ (3) =F (3) + iĤ (3) . The Bianchi identities and equations of motion of the Type IIB theory in terms ofĜ (3) are given in Appendix A.
Consequently, the construction of Ref. [9] represents an N = 2 truncated asymptotic limit as r → 0 of the full D3-brane N = 4 embedding of the present paper.

Stationary solutions and black holes
The N = 4 supergravity theory contains many supersymmetric, stationary black hole solutions. Using our embedding (2.13), these black holes can be uplifted to Type IIB supergravity, where they then have the interpretation of being situated on worldvolumes of D3-branes.
These black holes can be constructed by first reducing the four-dimensional theory along a timelike U(1) isometry to a Euclidean theory in three dimensions, whose bosonic sector consists of a metric coupled to non-linear sigma model with a pseudo-Riemannian target space M S . Following the work of [11][12][13][14], null geodesics on M S correspond to certain classes of black holes. Here, we will perform the timelike Kaluza-Klein reduction explicitly and show that the corresponding target space is M S = SO(8, 2)/SO(6, 2) × SO (2). A class of null geodesics will also be constructed on a submanifold N S = R × SO(2, 1)/SO(1, 1) ⊂ M S for completeness.
We have thus shown that all stationary solutions to the N = 4 supergravity theory are encoded in a three-dimensional Euclidean theory describing gravity minimally coupled to sixteen scalars parametrising the coset space SO(8, 2)/(SO(6, 2) × SO (2)). A special class of these stationary solutions are black hole solutions which correspond to harmonic maps between three-dimensional Euclidean space and the coset space [11][12][13][14]. From (2.57), the equations of motion for the metric h mn and the scalars M are where ∇ is the Levi-Civita connection with respect to h mn . A key observation of [11] was that these equations simplify greatly if the scalars are taken to depend on the three spatial coordinates through a single, harmonic function σ(x). In this case, the scalar equations For illustration, we will now construct a simple null geodesic on the scalar target manifold.
It is consistent to keep only φ 1 , φ 2 and one of the a Λ , say a 1 = a. The metric on the scalar manifold then becomes which is the metric on R × SO(2, 1)/SO(1, 1). The geodesic equations are then where the primes denote derivatives with respect to the harmonic function σ. The general solution for the φ 2 equation is where p 0,1 are constants. With this, the null condition reads which when substituted into the φ 1 (or a) equation, gives Performing the field redefinition u(σ) = e φ 1 (σ)/ √ 2 , this becomes the eigenvalue equation the general solution of which is where q 0,1 are constants. The value of a(σ) can then be obtained through (2.69). In the limit p 1 → 0, which corresponds to consistently truncating out the dynamics of φ 2 , we have

M5-branes with their worldvolume supergravities
The geometry of the 1 2 -BPS solution of M-theory describing parallel M5-branes, or in the present language, skeleton M5-branes, is a warped product R 1,5 × W R 5 . As with the skeleton D3-branes of the previous section, it is perfectly consistent to replace the R 1,5 worldvolume with a generic, six-dimensional manifold M 6 with a Ricci-flat metric g µν . Since the original solution preserves sixteen supercharges and since the Killing spinor is chiral on the sixdimensional worldvolume, we may now anticipate that there is in fact a consistent truncation to pure worldvolume N = (4, 0) supergravity in six dimensions. In this section, we will construct the corresponding embedding of six-dimensional, N = (4, 0) supergravity onto the worldvolume of the skeleton M5-branes. We will then consider anti-self-dual string solutions of the N = (4, 0) theory, and will examine their dynamics when uplifted back into M-theory using the braneworld embedding.

Skeleton M5-branes and d = 6, N = (4, 0) supergravity
The M-theory solution describing N parallel M5-branes in units where the M-theory length scale has l 11 = 1 is given by where i ∈ {1, . . . , 5}, r 2 = y i y i , and * 5 is the Hodge dual with respect to the flat metric on the transverse R 5 . For this to be a valid background for a quantum theory, the four-form flux must obey the quantisation condition, for any non-trivial four-cycle C 4 . The only non-trivial four-cycle in (3.1) is C 4 = S 4 , the asymptotic four-sphere on the transverse R 5 , and for it we find The solution (3.1) is the skeleton on which we will embed the pure, chiral, six-dimensional, N = (4, 0) supergravity. The bosonic sector of this theory contains a graviton and five two-form potentials with anti-self-dual field strengths. Denoting these bosonic degrees of freedom by g µν and B i (2) , the equations of motion of the N = (4, 0) theory are 2) , and * 6 is the Hodge dual with respect to the sixdimensional metric g µν .

Embedding N = (4, 0) supergravity on M5-branes
We find, using similar techniques to the case of the D3-branes, the following embedding of the N = (4, 0) supergravity theory on the worldvolume of the skeleton M5-branes, where we impose the anti-self-duality of G i (3) by hand. Let's begin with the Bianchi identity (C.1). We find where vol 6 is the volume form with respect to the six-dimensional metric g µν , and we used the anti-self-duality of G i (3) . Then, using the closure of G i (3) , we find The Chern-Simons term readŝ , which is symmetric under exchange of (i, j), so the second term vanishes. This leaves the first term, which using the symmetry of the Hodge dual, can be rewritten aŝ identically. Unlike the skeleton M5-branes, the Chern-Simons contribution to the flux equation of motion is non-zero, and is crucial for the consistency of our truncation.
We now turn to the Einstein equations (C.3). The non-zero components of the Ricci tensor areR where R µν is the Ricci tensor of g µν . Using the ingredients in Appendix D, we find that the non-zero components of the stress tensor arê The Einstein equations in the worldvolume directions then reduce to

Asymptotic limits
As is the case for the skeleton D3-branes, skeleton M5-branes are interpolating solitons. At distances far away from the brane, as |y i |→ ∞, the solution (3.1) becomes the R 1,10 vacuum, and close to the brane, |y i |→ 0, the solution becomes AdS 7 × S 4 . We will again examine our embedding (3.6) at these limits, and find that (3.6) retains this interpolating structure.

The flat limit
As |y i |→ ∞, the harmonic function H → 1, and the embedding (3.6) becomes . It can be easily checked that this is a solution to the M-theory equations of motion provided that the six-dimensional fields solve the equations of motion of the N = (4, 0) theory. Similar to the D3-branes, it is useful to compactify the transverse R 5 , which again is possible because ∂/∂y i are Killing vectors due to the trivialisation of the harmonic function. Choosing this compactification to be T 5 , so y i ∈ [0, 2π) for all i ∈ {1, . . . , 5}, we find that the quantisation which is guaranteed as the middle term is just a sum of the quantisation conditions of G i in the six-dimensional N = (4, 0) theory.

Anti-self-dual strings
The N = (4, 0) supergravity theory admits the multi-charged anti-self-dual string solution where |q| 2 = q i q i , and vol 3 is the volume form on the round three-sphere. The charges are normalised so that 1 4π 2 As R → 0, the solution asymptotes to AdS 3 × S 3 , where the three-sphere radius is proportional to |q|, and as R → ∞, the solution asymptotes to R 1,5 .
Using our embedding (3.6), we can construct the anti-self-dual-string-M5 solution, which can be viewed as a configuration of open membranes ending on the skeleton M5-branes.
This solution also exhibits an "asymptotically asymptotically" AdS structure. There is an eleven-dimensional asymptotic, |y i |→ 0, which brings the eleven-dimensional geometry into an M 7 × S 4 structure, where M 7 is asymptotically AdS 7 provided that the six-dimensional metric g µν is asymptotically flat. There is also a six-dimensional asymptotic, R → 0, with which the six-dimensional metric becomes AdS 3 × S 3 . So, our solution encompasses quite a few different AdS solutions of different dimensions. A particular configuration is to take |y i |→ 0 and let the six-dimensional radius R vary. The resulting solution describes a flow between the AdS 7 vacuum and a warped AdS 3 solution.

Diagonal and vertical reductions
Descendants of the D3-and M5-branes can be obtained by diagonal or vertical dimensional reductions [17,18]. In the following, we will illustrate the diagonal and vertical reductions using the M5-brane with its embedded worldvolume supergravity.
The worldvolume supergravity theory is then pure N = 4 supergravity in five dimensions, whose bosonic sector consists of a metric g µν , a scalar φ and six vectors A 0 (1) , A i (1) where i ∈ {1, . . . , 5}. We will now show that this can be obtained by a standard Kaluza-Klein reduction of the M5-brane's worldvolume supergravity, the N = (4, 0) supergravity theory in six dimensions. Recall that the N = (4, 0) supergravity theory contains a graviton and five two-form potentials with anti-self-dual fluxes. The Kaluza-Klein ansätze is where we put hats on the six-dimensional fields, define the SO(1, 1) scalar Σ = e −φ/ √ 6 , and 1) , and have already imposed the anti-self-duality condition onĜ i (3) . With this ansätze, we find that the equations of motion of the N = (4, 0) theory are encoded in the five-dimensional Lagrangian, where F 0 (2) = dA 0 (2) . This is the canonical Lagrangian of the pure N = 4 supergravity theory in five dimensions [19]. We can now employ our embedding (3.6) to embed this N = 4 supergravity theory on the M5-brane worldvolume, Here, * 5 is the Hodge dual with respect to the five-dimensional, worldvolume metric ds 2 5 , and * R 5 is the Hodge dual with respect to the flat metric on the transverse R 5 . This can then be interpreted as the Type IIA solution 3 With this choice of harmonic function, the M5-brane solution with its worldvolume supergravity embedded becomes where * 4 is the Hodge dual with respect to the flat metric on R 4 , and we recall that G a which describes the pure six-dimensional N = (4, 0) supergravity theory embedded on the NS5-brane worldvolume.

Conclusion
In this paper, we have demonstrated how noncompact Kaluza-Klein consistent reductions to corresponding supergravities can be made onto the worldvolumes of D3-branes in D = 10 Type IIB supergravity and of M5-branes in D = 11 M-theory or D = 10 Type IIA supergravity. We also have showed how further diagonal or vertical dimensional reductions can be made consistently with the braneworld embeddings. These results clearly suggest a conjecture that consistent Kaluza-Klein reductions can be made onto the worldvolumes of any brane solutions with unbroken supersymmetry to worldvolume supergravity theories with the type of supersymmetry as the underlying "skeleton" brane. We have constructed the consistent-truncation Kaluza-Klein ansätze for the bosonic sectors of the illustrative cases studied, but extensions to include the corresponding fermionic fields are not expected to pose key difficulties.
Another question is how the braneworld supergravity embeddings that we have found relate to string theory constructions. Clearly, in a perturbative α ′ expansion, the general Weylanomaly beta function relations between string and gravitational field theories [20] gives a leading-order in α ′ consistency condition for closed strings on the embedded supergravity backgrounds we have found. Proceeding to all-orders in α ′ is more delicate. Replacement of flat-space worldvolume conformal blocks with conformal blocks of equivalent central charge or using string-theory dualities are approaches that have been used to generate all-orders consistent superstring backgrounds with "curved" worldvolumes [21]. Further string-theory implications of the supergravity-level embeddings we have found remain as interesting areas for further development.

A Type IIB supergravity conventions
Our conventions are taken from [22]. The bosonic sector of Type IIB supergravity contains the RR formsF (1) ,F (3) ,F (5) , the NSNS three-formĤ (3) , the dilatonΦ and the metric, which will be taken to be the Einstein frame metric. The Bianchi identities are dF for a constant k.
In a configuration whereΦ = 0 andF (1) = 0, it is convenient to package the three-form fluxes into a complex three-formĜ  where the indices on the RHS are raised by either g µν or δ ΛΣ .

(C.4)
where the ten-dimensional Type IIA fields are distinguished with a tilde and are independent of θ. The metricg is the Einstein frame metric of the Type IIA theory. The Bianchi identity ofF (4)  whereF (2) = dÃ (1) . These can be integrated by introducing the NSNS two-form potential B (2) and RR three-form potentialÃ (3) , (2) ,F (4) = dÃ (

D Useful identities (M5)
From (3.6), the non-zero components ofF (4) arê where the i, j, . . . indices are raised by δ ij . Then, where the indices on the RHS are raised by either g µν or δ ij . For the (i, j) components, we used the anti-self-duality of G i (3) , which states that G i