Supersymmetric Yang-Mills theory on conformal supergravity backgrounds in ten dimensions

We consider bosonic supersymmetric backgrounds of ten-dimensional conformal supergravity. Up to local conformal isometry, we classify the maximally supersymmetric backgrounds, determine their conformal symmetry superalgebras and show how they arise as near-horizon geometries of certain half-BPS backgrounds or as a plane-wave limit thereof. We then show how to define Yang-Mills theory with rigid supersymmetry on any supersymmetric conformal supergravity background and, in particular, on the maximally supersymmetric backgrounds. We conclude by commenting on a striking resemblance between the supersymmetric backgrounds of ten-dimensional conformal supergravity and those of eleven-dimensional Poincaré supergravity.


Introduction
Supersymmetry multiplets in ten-dimensional spacetime not only underpin the five critical string theories (and their respective low-energy supergravity limits) but also encode the intricate structure of extended supersymmetry in many interesting quantum field theories in lower dimensions. For example, the Yang-Mills supermultiplet in ten dimensions elegantly captures the structure of extended supersymmetry and R-symmetry for gauge couplings in lower dimensions. Of course, in dimensions greater than four, even supersymmetric quantum field theories are not expected to be renormalisable without some kind of nonperturbative UV completion (indeed, this is precisely what string theory aims to provide). Without this completion, they should merely be regarded as low-energy effective field theories.

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In addition to the more familiar (gauged) type I, (Romans 1 ) type IIA and type IIB Poincaré gravity supermultiplets [1][2][3][4][5][6][7][8] associated with critical string theory, there is also a conformal gravity supermultiplet in ten dimensions [9]. This conformal gravity supermultiplet can be gauged and the coupling described in [9] to a Yang-Mills supermultiplet in ten dimensions is reminiscent of the analogous Chapline-Manton [1] coupling for type I supergravity. Unlike the Poincaré supergravity theories in ten dimensions though, this conformal supergravity theory is manifestly off-shell and must be supplemented with some differential constraints in order to render it local. As a supergravity theory, it is therefore somewhat exotic but admits a consistent truncation to type I supergravity and reduces correctly to known extended conformal supergravity theories in both four and five dimensions. There is also a little conceptual deviation from the unextended conformal gravity supermultiplets in lower dimensions which result from gauging one of the conformal superalgebras on Nahm's list [10]. Of course, this is not surprising since there are no conformal superalgebras of the conventional type above dimension six. 2 There do exist more general notions of a conformal superalgebra where the conformal algebra is contained in a less obvious manner. In particular, it was shown in [12] that the Lie superalgebra osp(1|32) can be thought of as a conformal superalgebra for R 9,1 with respect to a particular so(10, 2) < osp(1|32). Alas, it remains unclear though whether conformal supergravity in ten dimensions is somehow related to gauging this osp(1|32).
There is a vast literature on the classification of supersymmetric solutions of supergravity theories in diverse dimensions: that is to say, backgrounds which preserve some amount of rigid supersymmetry and solve the supergravity field equations. Indeed, at least for Poincaré supergravities, it is often the case that the preservation of a sufficient amount of rigid supersymmetry will guarantee that all of the supergravity field equations are satisfied. This typically comes from the so-called integrability conditions which result from iterating the 'Killing spinor' equations imposed by the preservation of supersymmetry.
In recent years, there has been mounting interest in the somewhat broader task of classifying supersymmetric backgrounds of conformal and Poincaré supergravity theories (which need not necessarily solve the field equations, only the integrability conditions). This is motivated primarily by a renewed curiosity in the general structure of quantum field theories with rigid supersymmetry in curved space , for which supersymmetric localisation has substantiated many important exact results and novel holographic applications [18,20,28,31,34,47,52,53,55,57,67,78,85,87,88]. The general strategy for obtaining non-trivial background geometries which support rigid supersymmetry builds on the pioneering work of Festuccia and Seiberg in four dimensions [15]. Given a rigid supermultiplet in flat space, it is often possible to promote it to a local supermultiplet in curved space via an appropriate supergravity coupling. For example, such a coupling JHEP03(2016)087 can be induced holographically in a superconformal field theory in flat space that is dual to a string theory in an asymptotically anti-de Sitter background. A judicious choice of decoupling limit (in which the Planck mass becomes infinite) typically ensures that the dynamics of the gravity supermultiplet are effectively frozen out, leaving only the fixed bosonic supergravity fields as data encoding the geometry of the rigidly supersymmetric curved background.
The aim of this paper is to explore various aspects of bosonic supersymmetric backgrounds of conformal supergravity in ten dimensions and elucidate the structure of the rigid Yang-Mills supermultiplet on these backgrounds. In particular, we will classify the maximally supersymmetric conformal supergravity backgrounds, compute their associated conformal symmetry superalgebras and show how they are related to each other via certain algebraic limits. We will also show how to ascribe to any conformal supergravity background a conformal Killing superalgebra that is generated by its Killing spinors. Paying close attention to the non-trivial Weyl symmetry which acts within this class of conformal supergravity backgrounds, we will see how to recover the subclass of type I supergravity backgrounds and how certain Weyl-transformed versions of the half-BPS string and fivebrane backgrounds of type I supergravity recover, in the near-horizon limit, the maximally supersymmetric conformal supergravity backgrounds of Freund-Rubin type. We will then describe the rigid supersymmetry transformations and invariant lagrangian for the Yang-Mills supermultiplet on any bosonic supersymmetric conformal supergravity background. This will be done both on-shell and in the partially off-shell formalism of [94][95][96]. We conclude with a curious observation that several highly supersymmetric conformal supergravity backgrounds in ten dimensions can be embedded in solutions of eleven-dimensional Poincaré supergravity which preserve twice as much supersymmetry.
This paper is organised as follows. In section 2 we discuss supersymmetric backgrounds of ten-dimensional conformal supergravity. In sections 2.1 and 2.2 we discuss the conformal gravity supermultiplet, the Killing spinor equation and its integrability condition. In section 2.3 we define the notion of a conformal symmetry superalgebra and show that every supersymmetric conformal supergravity background admits a conformal Killing superalgebra, which we define to be the ideal of a conformal symmetry superalgebra that is generated by the Killing spinors of the background. In section 2.4 we classify those conformal supergravity backgrounds preserving maximal supersymmetry. The results mimic those of eleven-dimensional supergravity: besides the (conformally) flat background, we have a pair of Freund-Rubin families and their plane-wave limit. In sections 2.4.1, 2.4.2 and 2.4.3 we work out the conformal symmetry superalgebras of these backgrounds and show in section 2.4.4 that the Killing superalgebra of the plane-wave limit arises as an Inönü-Wigner contraction of the Killing superalgebra of the Freund-Rubin backgrounds. In section 2.4.5 we comment on the maximal superalgebra of the maximally supersymmetric backgrounds, showing that it is isomorphic to osp(1|16) for the Freund-Rubin backgrounds and non-existent for their plane-wave limit. In section 2.5 we discuss some of the half-BPS backgrounds of conformal supergravity and show how the Freund-Rubin maximally supersymmetric backgrounds arise as near-horizon geometries. In section 3 we introduce the on-shell Yang-Mills supermultiplet and write down a supersymmetric lagrangian on JHEP03(2016)087 any supersymmetric conformal supergravity background. We then describe a partially offshell formulation of supersymmetric Yang-Mills theory on any such background. Finally, in section 4 we explore a possible relation between ten-dimensional conformal supergravity and eleven-dimensional Poincaré supergravity suggested by the resemblance between their maximally supersymmetric backgrounds and some of the half-BPS backgrounds of both theories. Appendix A contains our Clifford algebra conventions.

Conformal gravity supermultiplet
The off-shell conformal gravity supermultiplet in ten dimensions was constructed in [9]. The bosonic sector contains a metric g µν , a six-form gauge potential C µ 1 ...µ 6 and an auxiliary scalar φ. The fermionic sector contains a gravitino ψ µ and an auxiliary spinor χ. Both ψ µ and χ are Majorana-Weyl spinor-valued, with opposite chiralities. 3 The bosonic fields g µν and C µ 1 ...µ 6 contribute 44+84 off-shell degrees of freedom, matching the 8×16 offshell degrees of freedom from the fermionic field ψ µ . The fields (g µν , C µ 1 ...µ 6 , φ, ψ µ , χ) are assigned Weyl weights 2, 0, w, 1 2 , − 1 2 . The supersymmetry variations for this theory can be found in equation (3.34) of [9] and must be supplemented with the constraint defined in their equation (3.35). Their 'Q' and 'S' supersymmetry parameters are described by a pair of Majorana-Weyl spinors ǫ and η with opposite chiralities: for definiteness, we shall take ǫ to have positive chirality, i.e., Γǫ = ǫ. A bosonic supersymmetric background of this theory follows by solving the equations obtained by setting to zero the combined 'Q' and 'S' supersymmetry variation of ψ µ and χ, evaluated at ψ µ = 0 and χ = 0.
On a ten-dimensional lorentzian manifold (M, g) equipped with Levi-Civita connection ∇, the equations which follow from this procedure are where K = dC. The constraint in equation (3.36) of [9] follows as an integrability condition from (2.1). Notice that the second equation in (2.1), derived from the supersymmetry variation of χ, is simply a definition of η in terms of the other background data. Substituting this definition into the first equation in (2.1) thus yields the defining condition for a bosonic supersymmetric background.

Supersymmetric backgrounds
Let us now define a more convenient set of background fields to work with:
In terms of this data, the defining condition (2.1) for a bosonic supersymmetric background (M, g, G, H) of conformal supergravity in ten dimensions becomes Under a Weyl transformation g µν → Ω 2 g µν , for some positive function Ω, it follows that Γ µ → ΩΓ µ and ǫ → √ Ωǫ. The condition (2.3) is therefore preserved under any such transformation provided H → Ω 2 H and Φ → Φ + 3 ln Ω. Consequently, performing this transformation with Ω = e −Φ/3 allows one to fix G = 0 in equation (2.3) with H coclosed: The condition (2.3) implies that the 'Dirac current' one-form ξ µ = ǫΓ µ ǫ and the selfdual five-form ζ µνρστ = ǫΓ µνρστ ǫ obey and Taking the (µν) symmetric part of (2.5) implies L ξ g = −2σ ξ g with which shows that ξ is a conformal Killing vector. Furthermore, acting with ∇ σ on (2.6) and using closure of e −2Φ ⋆H and G together with (2.5) and (2.6) on the right hand side implies If H is closed then solutions of (2.3) with Gǫ = 1 2 Hǫ describe bosonic supersymmetric backgrounds of type I supergravity in ten dimensions. In that case, (2.3) reduces to ∇ µ ǫ = 1 8 H µνρ Γ νρ ǫ, ξ is a Killing vector and ι ξ H is closed. Clearly any such background is a special case of (2.3) and so one can always perform a Weyl transformation to obtain a solution of (2.4). However, if the original background had G = 0 then the new supersymmetric background of conformal supergravity solving (2.4) will no longer be a supersymmetric background of type I supergravity since the required Weyl transformation does not preserve the defining conditions dH = 0 and Gǫ = 1 2 Hǫ.

Conformal symmetry superalgebras
Let (M, g, G, H) be a bosonic supersymmetric background of conformal supergravity in ten dimensions. Let C(M, g) denote the Lie algebra of conformal Killing vectors on the ten-dimensional lorentzian manifold (M, g). The Lie subalgebra of homothetic conformal Killing vectors will be written H(M, g) < C(M, g) which contains as an ideal the Lie algebra of Killing vectors K(M, g) ⊳ H(M, g). Now let us ascribe to (M, g, G, H) a Z 2 -graded vector space s = s0 ⊕ s1, with even part s0 ⊂ C(M, g) and odd part s1 = ker D spanned by solutions ǫ of (2.3). We would like to equip s with the structure of a Lie superalgebra. The first step is to define a bracket on s, i.e., a skewsymmetric (in the graded sense) bilinear map [

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The Kosmann-Schwarzbach Lie derivativê along any X ∈ C(M, g) (i.e., L X g = −2σ X g), defines a natural conformally equivariant action of C(M, g) on spinors. It is therefore tempting to define for all X ∈ s0 and ǫ ∈ s1. However, for any ǫ ∈ s1, one finds thatL X ǫ ∈ s1 (i.e., where α X = G X + 3σ X and β X = L X H + 2σ X H, for all X ∈ s0. Under a Weyl transformation (g, G, H) → (Ω 2 g, G + 3 d(ln Ω), Ω 2 H) of the background, for any X ∈ s0, it follows that α X → α X and β X → Ω 2 β X . This implies that the condition (2.14) is Weyl-invariant. If (2.14) is satisfied, the bracket (2.13) solves the [001] Jacobi. Now recall from above (2.7) that any ǫ ∈ s1 has Dirac current ξ ǫ ∈ C(M, g). Moreover, (2.7) and (2.8) are precisely the conditions α ξǫ = 0 and β ξǫ = 0 which, if ξ ǫ ∈ s0 < C(M, g), would ensure that (2.14) is satisfied. With this in mind, let us now define the [s1, s1] bracket such that for all ǫ ∈ s1. Being symmetric bilinear in its entries, the general [s1, s1] bracket follows via the polarisation 1 , for any ǫ, ǫ ′ ∈ s1. Given (2.14), it is straightforward to check that the symmetric bilinear map defined by (2.15) is indeed equivariant with respect to the s0-action defined by (2.13), whence solving the [011] Jacobi. Furthermore, it follows using (2.5) that [ξ ǫ , ǫ] =L ξǫ ǫ = 0 , (2. 16) for all ǫ ∈ s1, so the final [111] Jacobi is satisfied identically. In summary, we have shown that the brackets defined by (2.13) and (2.15) equip s with the structure of Lie superalgebra provided the condition (2.14) is satisfied. Any such Lie superalgebra s with s0 < C(M, g) maximal will be referred to as the conformal symmetry superalgebra of (M, g, G, H). By construction, a conformal symmetry superalgebra s must have [s1, s1] ⊳ s0 < C(M, g). The s1-generated ideal k = [s1, s1] ⊕ s1 of a conformal symmetry superalgebra s will be referred to as the conformal Killing superalgebra of (M, g, G, H). It follows that every bosonic supersymmetric background of conformal supergravity in ten dimensions admits a conformal Killing superalgebra because (2.14) is identically satisfied (as a consequence of (2.7) and (2.8)) for all conformal Killing vectors in [s1, s1]. Of course, because the construction is manifestly Weyl-equivariant, strictly speaking a conformal symmetry superalgebra is ascribed to a conformal class of supersymmetric conformal supergravity backgrounds.
We will not attempt to obtain the general solution of (2.14) though it will be useful to describe what happens for conformal supergravity backgrounds which preserve more than half the maximal amount of supersymmetry. A simple algebraic proof was given JHEP03(2016)087 in ( [97], section 3.3) that any bosonic supersymmetric background of type I supergravity in ten dimensions which preserves more than half the maximal amount of supersymmetry is necessarily locally homogeneous. The same logic implies that any bosonic supersymmetric background of conformal supergravity in ten dimensions which preserves more than half the maximal amount of supersymmetry is necessarily locally conformally homogeneous. In both cases, the trick is to show that, for any given x ∈ M , the values at x of all (conformal) Killing vectors ξ ǫ obtained by 'squaring' supersymmetry parameters ǫ span the tangent space T x M (i.e., the evaluation at x of the squaring map ǫ → ξ ǫ is surjective). Acting with ǫΓ µ on (2.14) implies for all ǫ ∈ s1 and X ∈ s0. Therefore, in this case, the condition (2.17) says that α X must be (locally) constant, for all X ∈ s0. The condition (2.14) then just says that, for any vector field Y , the two-form ι Y β X must annihilate ǫ, for all X ∈ s0 and ǫ ∈ s1. This means that the element ι Y β X ∈ spin(9, 1) ⊂ Cℓ(9, 1) annihilates a linear subspace of ∆ (9,1) + of dimension > 8 and hence by ( [98], appendix B) (see also [99], table 2) it must vanish. Thus, we have shown that demanding (2.14) for all ǫ ∈ s1 with dim s1 > 8 implies for all X ∈ s0, which then trivially implies (2.14), showing that they are equivalent. Now consider the Weyl transformation defined above (2.4), which can be used to eliminate G. This maps a supersymmetric conformal supergravity background (M, g, G, H) with supersymmetry parameter ǫ to another supersymmetric conformal supergravity background (M,g = Ω 2 g,G = 0,H = Ω 2 H) with supersymmetry parameterǫ = √ Ωǫ, where Ω = e −Φ/3 . If the conditions (2.18) are satisfied then for all X ∈ s0. The first condition in (2.19) implies that every conformal Killing vector X with respect to g is homothetic with respect tog (since α X is constant), i.e., C(M, g) = H(M,g). Moreover, since α ξǫ = 0 for all ǫ ∈ s1, every conformal Killing vector in [s1, s1] is a Killing vector with respect tog. In this case, (M, g) being (locally) conformally homogeneous implies that (M,g) is (locally) homogeneous.

Maximally supersymmetric backgrounds
Maximally supersymmetric backgrounds are such that the connection D defined by equation (2.3) is flat. Hence one can determine the maximally supersymmetric backgrounds of conformal supergravity in ten dimensions by solving the flatness equation which results by abstracting ǫ from equation (2.9) and solving the resulting equation for endomorphisms of the spinor bundle. For maximally supersymmetric backgrounds of type I supergravity, the condition Gǫ = 1 2 Hǫ implies G = 0, H = 0 and (2.9) then implies that the Riemann tensor must also vanish. The only maximally supersymmetric background of type I supergravity in ten dimensions is therefore locally isometric to Minkowski space, which is Theorem 4 in [100].

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For maximally supersymmetric backgrounds of conformal supergravity, the flatness equation derived from (2.9) implies The last equation gives rise to two branches of solutions: those with H = 0 and those with H = 0 and hence G = 0. If H = 0 then (2.20) are trivially satisfied and the flatness equation from (2.9) is equivalent to The condition (2.21) just says that the Riemann tensor of the Weyl transformed metric e −2Φ/3 g is zero. In other words, g is conformally flat.
On the other hand, if H = 0, then the third condition in (2.20) implies that G = 0 and the flatness equation derived from (2.9) is equivalent to (2.22) together with the first two conditions in (2.20). The first of those conditions says that H is parallel with respect to the Levi-Civita connection ∇ and, by equation (2.22), so is the Riemann tensor of g. In other words, the background must be locally isometric to a lorentzian symmetric space. Now we shall classify maximally supersymmetric backgrounds of conformal supergravity with G = 0, making use of several key techniques developed in [100].
The second condition in (2.20), written in a more invariant way, is for all vector fields X, Y . This is none other than the family of Plücker quadrics for H (see, e.g., [101], Chapter 1), which is equivalent to H being decomposable; that is, H = α∧β ∧γ, for one-forms α, β, γ. Any background of interest is therefore locally isometric to a tendimensional lorentzian symmetric space M equipped with a parallel decomposable threeform H. These conditions are quite restrictive and solutions are distinguished according to whether the constant ||H|| 2 := 1 6 H µνρ H µνρ is positive, negative or zero. The geometric meaning of this constant has to do with the metric nature of the tangent 3-planes which H defines: they can be either euclidean, lorentzian or degenerate, according to whether ||H|| 2 is positive, negative or zero, respectively. From (2.22), it follows that the constant scalar curvature of g is R = − 1 2 ||H|| 2 . The maximally supersymmetric backgrounds are summarised below (with the scalar curvature of each AdS and S factor denoted in parenthesis).

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The background CW 10 (A) denotes a ten-dimensional Cahen-Wallach lorentzian symmetric space with metric in terms of local coordinates (x ± , x a ). For a general constant symmetric matrix A = (A ab ), it follows that g is conformally flat only if A is proportional to the identity matrix. Clearly this is not the case for the particular A which defines the maximally supersymmetric background in the third item above (unless µ = 0, in which case CW 10 (0) = R 9,1 ). Moreover, the maximally supersymmetric backgrounds with R = 0 in the first two items above are not conformally flat since, in each case, the constant sectional curvatures of the AdS and S factors are not equal and opposite (e.g., see (1.167) in [102]). It follows from ( [103], section 4) that the Freund-Rubin backgrounds AdS 3 × S 7 and AdS 7 × S 3 found above have two distinct plane-wave (or Penrose-Güven) limits up to local isometry. If the geodetic vector of the null geodesic along which we take the limit is tangent to the anti-de Sitter space, then the limit is flat, whereas if the geodetic vector has a nonzero component tangent to the sphere, the limit is isometric to the Cahen-Wallach background we found above. Indeed, the ratio (= 4, in this case) between the two eigenvalues of the symmetric matrix A defining the Cahen-Wallach metric is the square of the ratio (= 2, in this case) of the radii of curvature of the 3-and 7-dimensional factors in the Freund-Rubin geometry. This gives another proof that the Freund-Rubin backgrounds are not conformally flat, since conformal flatness is a hereditary property under the plane-wave limit ( [103], section 3.2), but the CW 10 (A) geometry above is not conformally flat for µ = 0.

Conformal symmetry superalgebras
Let us now investigate how the construction of conformal symmetry superalgebras in section 2.3 plays out for the maximally supersymmetric conformal supergravity backgrounds we have just classified.

s(AdS
The preceding discussion has established that the only conformal Killing vectors for these two geometries are Killing vectors. Moreover, it is not difficult to prove that all such Killing vectors correspond to Killing vectors on the individual AdS and S factors. The supersymmetry condition (2.4) reduces to a pair of Killing spinor equations on the individual AdS and S factors. In our conventions, a spinor ψ on a lorentzian/riemannian spin manifold M is Killing if, for any vector field X on M , it obeys ∇ X ψ = ± κ 2 Xψ, for some real/imaginary constant κ. In the case at hand, the Killing constants are given by 2|R|. Both AdS m and S n can be described via the canonical quadric embedding in (an open subset of) R m−1,2 and R n+1 respectively. Conversely, the flat metrics on both R m−1,2 and R n+1 can be written as (lorentzian and riemannian) cone metrics whose bases form the respective AdS m and S n geometries. This cone construction is particularly useful in describing Killing vectors and Killing spinors on these geometries (see [104] for a review in a similar context). Every Killing vector on the base lifts to a constant two-form on the cone and vice versa. Thus K(AdS m ) ∼ = so(m − 1, 2) ( ∼ = ∧ 2 R m−1,2 as a vector space) and K(S n ) ∼ = so(n + 1) ( ∼ = ∧ 2 R n+1 as a vector space). Every Killing spinor on the base lifts to a constant spinor on the cone and vice versa. More precisely, if both m and n are odd, there is a bijection between Killing spinors on the base and constant chiral spinors on the cone. The Kosmann-Schwarzbach Lie derivative of a Killing spinor along a Killing vector on the base lifts to the obvious Clifford action of a constant two-form on a constant spinor on the cone.
The cleanest way to discuss the explicit structure of s(AdS 3 × S 7 ) and s(AdS 7 × S 3 ) is as particular real forms of the same complex Lie superalgebra s C . The even part of s C is It is a straightforward exercise to check that (2.25) obey the graded Jacobi identities; although the [111] component requires use of the following identities, which hold for all ψ ∈ ∆ (4,C) + and ϕ ∈ ∆ Let ε ∈ ∧ 4 C 4 with ε 1234 = 1 and let γ = −γ 1234 define the chirality matrix for Cℓ(4). It follows that γ AB γ = 1 2 ε ABCD γ CD , so any ψ ∈ ∆ (4,C) + defines a self-dual two-form ψ, γ AB ψ . This implies that the bracket defined by (2.25) of the sp 1 (C) < so 4 (C) spanned by ∧ 2 − C 4 with every other element in s C is zero. The action of the other sp 1 (C) < so 4 (C) (spanned by ∧ 2 + C 4 ) on ∆ (4,C) + just corresponds to the defining representation ∆ C of this sp 1 (C). Excluding the decoupled sp 1 (C) factor from s C leaves a simple complex Lie superalgebra that is isomorphic to osp 8|1 (C) (a.k.a. D(4, 1) in the Kac classification [105]), with even part so 8 (C) ⊕ sp 1 (C) and odd part ∆ The real forms of all complex classical Lie superalgebras in [105] were classified in [106]. Up to isomorphism, the real forms of a given complex classical Lie superalgebra are uniquely determined by the real forms of the complex reductive Lie algebra which constitutes its even part. The even part of s C is s C 0 = so 4 (C) ⊕ so 8 (C) which admits many non-isomorphic real forms. However, of these real forms, only so(2, 2) ⊕ so(8) and so(6, 2) ⊕ so(4) are isomorphic to the Lie algebra of isometries of AdS 3 × S 7 and AdS 7 × S 3 , respectively. It is then straightforward to deduce the associated real forms which describe their conformal symmetry superalgebras s. The pertinent data is summarised in table 1.
We have opted for the more common physics notation to write the real form osp(8|2) rather than its perhaps more logical alias osp 8|1 (R). The notation is that ∆ (p,q) + denotes the positive-chirality spinor representation of so(p, q) when p + q is even. As vector spaces, ∆ with invariant quaternionic structures J 1 and J 2 , their tensor product J 1 ⊗ J 2 defines a real structure on W 1 ⊗ W 2 , where the tensor product is over C. This means that is a real representation which can be identified with the subspace of real elements (i.e., fixed points of the real structure) in W 1 ⊗ W 2 . Note that so(2, 2) ∼ = sl 2 (R) ⊕ sl 2 (R) with ∆ A basis of Killing vectors for this geometry is given by (2.28) Their non-vanishing Lie brackets are as follows The Killing vectors (ξ, q, p) are generic for plane wave geometries and we see from (2.29) that they form a 17-dimensional Lie subalgebra isomorphic to the Heisenberg algebra heis 8 (R). The Killing vectors (J, M ) span the Lie subalgebra so(2) ⊕ so(6) < so(8) which stabilises A.

Contractions
As we have seen, the Cahen-Wallach background is the plane-wave limit of the Freund-Rubin backgrounds. Therefore we might expect, based on what happens in ten-and eleven-dimensional Poincaré supergravities [103,107], that s(CW 10 (A)) is a contraction (in the sense of Inönü-Wigner) of s(AdS 3 × S 7 ) and s(AdS 7 × S 3 ). Indeed, it was precisely that observation in [108] which led to the identification of the maximally supersymmetric plane-wave solutions of eleven-dimensional and IIB supergravities as plane-wave limits of JHEP03(2016)087 the corresponding Freund-Rubin solutions in [109]. We will give the details only for the AdS 3 × S 7 Freund-Rubin background, and leave the similar calculation for AdS 7 × S 3 to the imagination.

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and where, for ε ∈ ∆ (9,1) with λ 2 = µ 6 . (We tacitly assume µ > 0, but in fact the factor µ is inessential and can always be taken to be 1, if nonzero.) It is clear by inspection that Υ t defines a vector space isomorphism for any t = 0. For definiteness, let us take t > 0. We may define a family of Lie brackets [−, −] t on E by transporting the Lie bracket on s via Υ t : which agrees with equation (2.29). Next we consider the bracket [p ′ α , Ψ ′ (ε − )], for ε − ∈ ker Γ − , given by which shows that η − = Γ + ε − for agreement with equation (2.34). Next, we consider the

Maximal superalgebras
In [110] the notion of the maximal superalgebra of a supergravity background was introduced, generalising to non-flat backgrounds the M-algebra of [111]. Given a supergravity background with Killing superalgebra k = k0 ⊕ k1, the maximal superalgebra (should it exist) is defined to be Lie superalgebra m = m0 ⊕ m1, satisfying the following properties 1. m1 = k1 and k0 is a Lie subalgebra of m0; 2. the odd-odd bracket is an isomorphism ⊙ 2 m1 ∼ = m0; and 3. the projection ⊙ 2 m1 → k0 coincides with the odd-odd bracket of k and the restriction to k0 of the bracket m0 ⊗ m1 → m1 is the k-bracket.  [110], it follows from ( [112], appendix A) that any Lie superalgebra satisfying (2) -in particular, the maximal superalgebra of a background -is uniquely JHEP03(2016)087 determined by some ω ∈ ∧ 2 m * 1 m 0 ⊂ ∧ 2 k * 1 k 0 . Indeed, by (2) above, if Q a is a basis for m1, Z ab := [Q a , Q b ] is a basis for m0 and the Lie brackets in this basis are given by where ω ab := ω(Q a , Q b ) and where the second equation follows from the first, using the Jacobi identity and the definition of Z ab . In this section we explore the maximal superalgebras of the maximally supersymmetric backgrounds.
First of all, we show that the Cahen-Wallach background does not admit a maximal superalgebra. The proof is virtually identical to the one for the Cahen-Wallach vacua of eleven-dimensional and type IIB Poincaré supergravities in [110]. As explained in ( [110], section 3), for any maximal superalgebra m, k0 acts trivially on the radical k ⊥ 1 of the skewsymmetric bilinear form ω characterising m. Now, inspecting equation (2.34) we see that ζ = ∂ − acts semisimply on k1 with nonzero eigenvalues, so that k k0 1 = 0. Therefore ω, having trivial radical, must be symplectic and hence, from equation (2.49), it follows that m must have trivial centre. But now notice that ξ = ∂ + acts trivially on m1 and hence it is central in m, thus contradicting the existence of a maximal superalgebra for the Cahen-Wallach background.
Next we discuss the two maximally supersymmetric Freund-Rubin backgrounds AdS 7 × S 3 and AdS 3 × S 7 . Here it is convenient to again think of their Killing superalgebras as different real forms of the same complex Lie superalgebra. At the same time we must make a distinction between the symmetry superalgebra s C and the Killing superalgebra, which is the ideal k C of s C generated by k C 1 . For the Freund-Rubin backgrounds, s C is strictly larger, containing a simple ideal isomorphic to sl 2 (C), which acts trivially on the Killing spinors.
In this case, we have k C 0 ∼ = sl 2 (C) ⊕ so 8 (C) and k C 1 ∼ = ∆ C ⊗ ∆ (8,C) + as an k C 0 -module, where ∆ C is the defining representation of sl 2 (C) and the tensor product is over C. There is precisely one invariant skew-symmetric bilinear form (up to scale) on k C 1 : it is the product of the sl 2 (C)-invariant complex symplectic structure −, − on ∆ C and the so 8 (C)-invariant complex orthogonal structure (−, −) on ∆ (8,C) + . It is clearly nondegenerate, hence complex symplectic. Therefore if k C admits a maximal superalgebra, it has to be the complexification osp(1|16) C of osp(1|16). On the other hand, it is shown in [106] that osp(1|16) is the unique real form of osp(1|16) C , so if 'maximisation' were to commute with complexification, we would conclude that the maximal superalgebras of the Freund-Rubin backgrounds would be isomorphic to osp(1|16). We do not have such a result at our disposal and it is unlikely that such a general result actually exists since not every Lie superalgebra can be 'maximised', as illustrated by the Killing superalgebra of the Cahen-Wallach backgrounds. This means we need to work harder.
We start by showing that the maximal superalgebra of k C is indeed isomorphic to osp(1|16) C , following the construction in [110], mutatis mutandis. First of all, let us de-

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compose the symmetric square of m C 1 into irreducible representations of k C 0 : which defines z C 0 . Except for the z C 0 , this is precisely the odd-odd bracket of k C . It follows from the discussion in ( [110], section 4.3) that the action of k0 on k1, when viewed through the lens of the cone construction, is via the Clifford action of the parallel 2-forms on the cones of AdS p and S q to which the special Killing 1-forms in k0 lift. Therefore we extend this Clifford action to all of m0, which also lift as parallel forms to the cones. This complexifies and gives the following construction of m C ; although we prefer to use a different basis, which unfortunately obscures the embedding of k C 0 into m C 0 . If ψ 1 ⊗ ϕ 1 , ψ 2 ⊗ ϕ 2 ∈ m C 1 , their symplectic inner product is given by We define the following rank-1 endomorphisms of ∆ C and ∆ and we define the odd-odd bracket in m C via By a judicious use of the Fierz identities, the rank-1 endomorphisms above can be expressed in terms of the standard basis for the Clifford algebra in terms of exterior forms, and in this way clarify the embedding k C 0 ⊂ m C 0 , but we have no need to do that. The action of m C 0 on m C 1 is given simply by the Clifford action, which is How about the maximal subalgebras of k(AdS 3 × S 7 ) and k(S 3 × AdS 7 )? Let k be one of these Killing superalgebras. It is a real form of k C , so in particular k1 is the real subspace of k C 1 defined by a k0-invariant conjugation. Now consider k1 as a real subspace of m C 1 . It generates a real subalgebra of m C , which satisfies property (2) of a maximal subalgebra because the restriction to k1 of the odd-odd bracket is an isomorphism onto its image. This means that the brackets are of the form (2.49) with ω being the restriction JHEP03(2016)087 of the complex-symplectic form on m C 1 to the real subspace k1. The other properties for a maximal subalgebra are satisfied because k C is the complexification of k. Therefore we see that k admits a maximal superalgebra, but to identify it we need to understand the restriction of ω to k1. It pays to be a little bit more general.
Let (E, ω) be a complex symplectic vector space. Let g be a Lie algebra, whose complexification g C acts on E preserving ω. Now suppose that c is a g-invariant conjugation on E and let E R be its fixed (real) subspace; that is, E = E R ⊗ R C. Because c is g-invariant, g acts on E R . Now, ω restricts to a real skewsymmetric bilinear form ω R on E R . Since ω is g C -invariant, it is in particular also g-invariant and hence so is ω R . Its radical, therefore, is a g-submodule of E R . Now suppose that E R is irreducible as a g-module. Then the radical of ω R must either be trivial, in which case ω R is a symplectic form, or it must be all of E R , in which case ω R = 0. Now let us apply this to our situation, with the role of (E, ω) played by (k C 1 , ω). We have that k1 is an irreducible module of k0, so that the restriction of ω to k1 is either symplectic or zero. But it cannot be zero, because otherwise k0 would be central and in particular, an abelian Lie algebra. Therefore we conclude that ω restricts to a symplectic form on k1 and hence k1 generates a maximal superalgebra isomorphic to osp(1|16).

F1-string and NS5-brane, Weyl transformations and near-horizon limits
Backgrounds which solve (2.3) for precisely eight linearly independent supersymmetry parameters ǫ are called half-BPS. Two well-known half-BPS backgrounds in ten dimensions are the F1-string [113] and the NS5-brane [114]. They solve (2.3) with dH = 0 and Gǫ = 1 2 Hǫ and thus define half-BPS backgrounds of type I supergravity in ten dimensions. To define them, it is convenient to write g R p,q and vol R p,q for the canonical flat metric and volume form on R p,q .
The F1-string background has metric and three-form given by where e −2Φ is a harmonic function on R 8 so that d(e −2Φ ⋆ H) = 0. For example, thinking of R 8 as a cone over S 7 with radial coordinate r, one can take e −2Φ = 1 + |k 2 | r 6 for some constant k 2 . The supersymmetry parameter is given by where ǫ 0 is a constant positive chirality Majorana-Weyl spinor on R 9,1 . Now consider the Weyl transformation (with Ω = e −Φ/3 ) of the F1-string that defines a solution of (2.4). This is a new half-BPS background of conformal (but not Poincaré) supergravity in ten dimensions. Its 'near-horizon' limit is defined by taking the radial coordinate r → 0, which recovers precisely the maximally supersymmetric AdS 3 × S 7 background obtained in section 2.4 (identifying |k 2 | −1/3 = R/18).
The NS5-brane background has metric and three-form given by

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where e 2Φ is a harmonic function on R 4 so that dH = 0. For example, thinking of R 4 as a cone over S 3 with radial coordinate r, one can take e 2Φ = 1 + |k 6 | r 2 for some constant k 6 . The supersymmetry parameter is given by where ǫ 0 is a constant positive chirality Majorana-Weyl spinor on R 9,1 . The near-horizon limit of the NS5-brane defines a metric on R 5,1 × R + × S 3 and is therefore not conformally equivalent to the maximally supersymmetric AdS 7 × S 3 background obtained section 2.4. However, it is important to stress that any choice of function e 2Φ on R 4 for the NS5-brane defines a half-BPS background of conformal supergravity in ten dimensions. Let us therefore not assume that e 2Φ is harmonic on R 4 and perform the Weyl transformation (with Ω = e −Φ/3 ) to define a solution of (2.4). Now, for this new half-BPS background of conformal supergravity, taking e 2Φ = 1 + |k ′ 6 | r 3 for some constant k ′ 6 (which is not harmonic on R 4 ), one recovers in the near-horizon limit precisely the maximally supersymmetric AdS 7 × S 3 background obtained in section 2.4 (identifying |k ′ 6 | −2/3 = −2R/9).

Yang-Mills supermultiplet
The on-shell Yang-Mills supermultiplet in ten dimensions contains a bosonic gauge field A µ and a fermionic Majorana-Weyl spinor λ (we take λ with positive chirality, i.e., Γλ = λ). Both fields are valued in a real Lie algebra g with invariant inner product (−, −).
Up to boundary terms, the lagrangian [23] for the subclass of bosonic supersymmetric backgrounds of type I supergravity in ten dimensions.) The prefactor e −2Φ acts as an effective gauge coupling in (3.2). For generic backgrounds with H = 0, notice that rigid supersymmetry necessitates both a mass term for λ and a Chern-Simons coupling for the gauge field. Closure of e −2Φ ⋆ H ensures that the Chern-Simons coupling is gauge-invariant.
A novel (partially) off-shell formulation of supersymmetric Yang-Mills theory on R 9,1 was obtained by Berkovits in [94] (see also [95,96]). To match the 16 off-shell fermionic degrees of freedom of λ, the 9 off-shell degrees of freedom of A µ are supplemented by 7 bosonic auxiliary scalar fields Y i (where i = 1, . . . , 7). All fields are g-valued. The supersymmetry parameter ǫ is also supplemented by seven linearly independent bosonic Majorana-Weyl spinors θ i , each with the same positive chirality as ǫ. The index i corresponds to the vector representation of the spin(7) factor in the isotropy algebra spin(7) ⋉ R 8 of ǫ.
Now consider the following supersymmetry variations for the partially off-shell Yang-Mills supermultiplet on a bosonic supersymmetric conformal supergravity background Hλ .

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is invariant under (3.5). It is also manifestly invariant under spin(7) rotations of the auxiliary fields. Moreover, the integral of (3.8) is Weyl-invariant with respect to the aforementioned transformation rules for fields and background data. Of all the bosonic supersymmetric conformal supergravity backgrounds the Yang-Mills supermultiplet above can be defined upon, the maximally supersymmetric AdS 3 × S 7 and AdS 7 × S 3 Freund-Rubin backgrounds classified in section 2.4 are perhaps the most compelling. In particular, it would interesting to explore whether the Yang-Mills supermultiplet on these conformal supergravity backgrounds admits a consistent truncation that would recover one of the theories described in [13,22,24,76,92]. The relevant theories in [13] (or [24]) would follow by dimensionally reducing the on-shell (or partially off-shell) Yang-Mills supermultiplet on R 9,1 to some lower dimension d equal to either 7 or 3, before deforming the resulting supermultiplet in dimension d in such a way that it retains rigid supersymmetry on a curved space admitting the maximum number of real or imaginary Killing spinors, i.e., either AdS d or S d . The deformation involves introducing several nonminimal couplings that do not seem to figure in (3.1) and (3.2), though this discrepancy may be the result of a non-standard reduction along some subset of Killing vectors of S 10−d or AdS 10−d that is necessary for the conformal supergravity background instead of along the obvious translations in R 10−d or R 9−d,1 , as in [13,24]. We leave this question for future work.

Lifting to eleven dimensions
It should not have gone unnoticed that the supersymmetric backgrounds of conformal supergravity in ten dimensions that we have been discussing bear a striking resemblance to supersymmetric backgrounds of Poincaré supergravity in eleven dimensions. For instance, each maximally supersymmetric background obtained in section 2.4 has an obvious maximally supersymmetric counterpart in Theorem 1 of [100]. Moreover, the structure of the half-BPS string and five-brane backgrounds obtained in section 2.5 is virtually identical to that of the well-known half-BPS M2-brane and M5-brane solutions of Poincaré supergravity in eleven dimensions. This empirical evidence hints at an embedding of (at least some) supersymmetric backgrounds of ten-dimensional conformal supergravity in supersymmetric solutions of elevendimensional Poincaré supergravity. Of course, this would be distinct from the well-known Kaluza-Klein reduction along a spacelike Killing vector for supergravity backgrounds in eleven dimensions, yielding backgrounds of type IIA supergravity in ten dimensions. After a brief synopsis of the defining conditions for bosonic supersymmetric backgrounds and solutions of eleven-dimensional Poincaré supergravity, we shall spend the rest of this final section investigating a few different types of embedding for some of the backgrounds of ten-dimensional conformal supergravity that we have already encountered. This will begin with a review of the Kaluza-Klein embedding of supersymmetric backgrounds of type I supergravity. We will then describe a novel 'equatorial' embedding for the maximally supersymmetric Freund-Rubin backgrounds of ten-dimensional conformal supergravity in their eleven-dimensional counterparts. Finally, we will describe the embedding of the half-

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BPS string and five-brane backgrounds of ten-dimensional conformal supergravity and show how to recover the maximally supersymmetric Freund-Rubin backgrounds via delocalisation and near-horizon limits.

Supersymmetric solutions in eleven dimensions
The bosonic fields of Poincaré supergravity in eleven dimensions consist of a metricĝ and a closed four-formF . Following the conventions of [100], a bosonic supersymmetric background is given by a solution of whereǫ is a Majorana spinor in eleven dimensions. Any such background is called a supersymmetric solution if it also obeys the field equationŝ ∧F .

Kaluza-Klein embedding of supersymmetric type I backgrounds
It is well-known that any supersymmetric solution of type IIA supergravity in ten dimensions can be uplifted to a supersymmetric solution of supergravity in eleven dimensions via the 'string-frame' Kaluza-Klein ansatz. This recovers only the subset of supersymmetric solutions of supergravity in eleven dimensions which admit a spacelike Killing vector ξ with L ξF = 0. At least locally, one can write ξ = ∂ z in terms of the eleventh coordinate z. Now consider the following special case of the aforementioned ansatz: in terms of a metric g, a function Φ and a three-form H in ten dimensions. It follows that⋆F = e −2Φ ⋆H. Plugging (4.3) into the second field equation in (4.2) therefore gives d(e −2Φ ⋆H) = 0. It also follows that dH = 0 sinceF is closed. The ansatz (4.3) allows one to define an idempotent element I = e −2Φ/3Γ z which anticommutes with everyΓ µ (where µ is any index M = z). Ifǫ = Iǫ then it can be identified with a positive chirality Majorana-Weyl spinor e −Φ/6 ǫ in ten dimensions. Assuming this to be the case then plugging (4.3) into (4.1) gives Hǫ , defining condition for a bosonic supersymmetric background of type I supergravity in ten dimensions. To summarise, we have shown that any bosonic supersymmetric background of type I supergravity in ten dimensions can be embedded via (4.3) in a bosonic supersymmetric background of Poincaré supergravity in eleven dimensions, obeying (4.1) for someǫ = Iǫ and the second field equation in (4.2). Of course, the projection condition in eleven dimensions is because any background of type I supergravity in ten dimensions can preserve no more than sixteen real supercharges (in contrast with the maximum of thirty two in eleven dimensions).

Embedding of maximally supersymmetric Freund-Rubin backgrounds
Poincaré supergravity in eleven dimensions admits two well-known maximally supersymmetric Freund-Rubin solutions. In terms of the scalar curvatureR ofĝ, they are of the form (The scalar curvature of each AdS and S factor is denoted in parenthesis.) To make our description of the embedding as transparent as possible, let us adopt the following notation. Let g n denote the 'unit radius' metric on either AdS n or S n (i.e., the metric with constant scalar curvature −n(n − 1) for AdS n or n(n − 1) for S n ). Any metric of the form κ 2 g n + λ 2 g m will be assumed to be Lorentzian (i.e., AdS n × S m or S n × AdS m ). Let vol n denote the volume form with respect to g n . Let ψ n denote a Killing spinor with respect to g n , obeying ∇ µ ψ n = ± 1 2 Γ µ ψ n for AdS n or ∇ µ ψ n = ± i 2 Γ µ ψ n for S n . For AdS n (or S n ), ψ n lifts to a constant spinor on the flat cone C(AdS n ) ∼ = R n−1,2 (or C(S n ) ∼ = R n+1 ). We shall refer to ψ n as having unit Killing constant. Rescaling g n by a factor of κ 2 rescales the Killing constant by a factor of κ −1 .
In terms of this notation, the data for the maximally supersymmetric Freund-Rubin solutions of eleven-dimensional Poincaré supergravity is given bŷ g =κ 2 (g 4 + 4g 7 ) andF = 3κ 3 vol 4 , (4.5) while the supersymmetry parameterǫ involves a tensor product of ψ 4 (with Killing constant κ −1 ) and ψ 7 (with Killing constant (2κ) −1 ). The constant Observe that the factors ofκ above are precisely the same as for the homothety noted at the end of section 4.1, hence we can and will fixκ = 1 via the action of a homothety with α =κ −1 .
On the other hand, the data for the maximally supersymmetric Freund-Rubin backgrounds of ten-dimensional conformal supergravity is given by and H = 3κ 2 vol 3 , (4.7)

Branes, delocalisation and near-horizon limits
Two well-known half-BPS solutions of supergravity in eleven dimensions are the M2-brane and the M5-brane.
The M2-brane solution has metric and four-form given bŷ where f is a harmonic function on R 8 so that the second field equation in (4.2) is satisfied (i.e., d⋆F = 0 sinceF ∧F = 0 for (4.13)). The supersymmetry parameter is given bŷ whereǫ 0 is a constant Majorana spinor on R 10,1 .
The M5-brane solution has metric and four-form given bŷ where f is a harmonic function on R 5 so that dF = 0. The supersymmetry parameter is given byǫ where againǫ 0 is a constant Majorana spinor on R 10,1 .
By identifying z with a coordinate on R 5 and f = e 2Φ , one recognises that (4.15) is of the form (4.3). However, ∂ z is a Killing vector only if f is harmonic on the subspace R 4 ⊂ R 5 orthogonal to the z-direction. Making this assumption is known as 'delocalisation' along the z-direction. With respect to these identifications, the data (g, H) in (4.3) for the delocalised M5-brane gives precisely the NS5-brane background of type I supergravity in ten dimensions in (2.57). The near-horizon limit of the delocalised M5-brane (4.15) in eleven dimensions with f = e 2Φ = 1+ |k 6 | r 2 defines a half-BPS background that is conformally equivalent to R 5,1 ×H 2 ×S 3 (cf. R 5,1 ×R + ×S 3 in the near-horizon limit of the NS5-brane in ten dimensions). On the other hand, without delocalisation, the near-horizon limit of (4.15) with f = e 2Φ = 1 + |k ′ 6 | r 3 gives the maximally supersymmetric solution AdS 7 (−7R) × S 4 (8R) withF = 6R vol S 4 (after identifying the constant k ′ 6 such that |k ′ 6 | −2/3 = 2R/3). Of course, without delocalisation, the ansatz (4.3) cannot be used to reduce to ten dimensions. Even so, notice that the data (e −2Φ/3 g, e −2Φ/3 H) in ten dimensions obtained by comparing (4.3) with (4.15) without delocalisation gives precisely the half-BPS background of conformal supergravity with maximally supersymmetric AdS 7 × S 3 near-horizon limit.
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