Perturbing AdS$_6 \times_w S^4$: linearised equations and spin-2 spectrum

We initiate the analysis of the Kaluza--Klein mass spectrum of massive IIA supergravity on the warped AdS$_6 \times_w S^4$ background, by deriving the linearised equations of motion of bosonic and fermionic fluctuations, and determining the mass spectrum of those of spin-2. The spin-2 modes are given in terms of hypergeometric functions and a careful analysis of their boundary conditions uncovers the existence of two branches of mass spectra, bounded from below. The modes that saturate the bounds belong to short multiplets which we identify in the representation theory of the $\mathfrak{f}(4)$ symmetry superalgebra of the AdS$_6 \times_w S^4$ solution.


Introduction
Gauge field theories in five dimensions are non-renormalizable, however string theory predicts the existence of strongly coupled superconformal field theories for certain gauge groups and matter content [1,2]. Such an example appears in type I' string theory from a system of D4-branes probing an O8-plane with a stack of D8-branes on top of it. This system admits a supergravity description in massive IIA supergravity [3,4], and in the near-horizon limit the geometry becomes a warped product of six-dimensional anti-de Sitter spacetime AdS 6 , and a four-sphere S 4 . The warp factor is singular at the equator of S 4 , due the presence of the O8-plane, and the internal space is therefore actually a hemisphere. The near-horizon background has an exceptional f(4) symmetry superalgebra, which is the unique superconformal algebra in five dimensions. The bosonic subalgebra of f(4) is so(2, 5) ⊕ su (2), with so(2, 5) realised as the isometry algebra of AdS 6 and the (R-symmetry algebra) su (2) as an isometry of S 4 . The dual superconformal field theory arises as the UV fixed point of N = 1 supersymmetric USp(2N) Yang-Mills theory coupled to N f < 8 hypermultiplets in the fundamental representation, and one hypermultiplet in the antisymmetric representation. 1 In view of the AdS/CFT correspondence, there is the motivation to study this supergravity background in order to learn about the dual field theory. Examples include the calculation of the holographic entanglement entropy [6], and the action of probe branes [7,8]. Another such study is that of the Kaluza-Klein mass spectrum, which corresponds to the spectrum of the dual field theory operators. Complete Kaluza-Klein mass spectra have been obtained for anti-de Sitter compactifications whose geometry is a direct product, and the internal space a coset space, typically a sphere; for example [9,10,11,12]. In this note we progress towards obtaining the Kaluza-Klein mass spectrum of a warped compactification, by completing the task of obtaining the linearised equations of motion for small fluctuations around the background. Furthermore, we analyse the spectrum of massive spin-2 particles or gravitons in AdS 6 , uncovering some interesting features 2 .
The analysis of the spectrum is complicated by the presence of the warp factor as it modifies the differential operators which determine it. These are differential operators on the internal manifold, in the present case a four-sphere, and turn out to be warped versions of the Laplace operator on S 4 . Hence the standard spherical harmonic analysis is not readily available, a fact that is also due to the presence of the singularity at the equator. In the case of the spin-2 modes we reduce the problem to solving a hypergeometric ordinary differential equation with the mass spectrum determined by imposing appropriate boundary conditions. Although a rather modest task compared to the analysis of the full spectrum, it already reveals some interesting features: we find two branches of spin-2 mass spectra one of which is rather exceptional in that a certain derivative of the modes is singular. Both branches are bounded from below, and the bound is saturated by modes belonging to short multiplets which we have identified in the work of [18,19] on representations of the f(4) superconformal algebra 3 .
The remainder of this note is as follows. In section 2 we briefly review massive IIA supergravity and its AdS 6 × w S 4 solution. In section 3 we present the linearised equations of motion for fluctuations around this solution. In section 4 we determine the mass spectrum of fluctuations of spin-2. We end in section 5 with a discussion and comments on future work. Certain technical details are included in appendices.

Massive IIA supergravity and its AdS 6 solution
In this section we collect the equations of motion of massive IIA supergravity and review the AdS 6 × w S 4 solution [3].

Equations of motion
Massive IIA supergravity 4 in ten dimensions consists of the following bosonic fields: the metric g, the dilaton Φ, the field strengths H 3 , F 2 , F 4 , with the subscript denoting their form rank, and the constant "Romans mass" F 0 . The field strengths satisfy the Bianchi identities and are given in terms of the potentials B 2 , A 1 and A 3 by The equations of motion of the bosonic fields are: 5 In the above R M N is the Ricci tensor and ǫ M 1 M 2 ···M 10 the totally antisymmetric tensor.
where R is the Ricci scalar.

The AdS 6 × w S 4 solution
The AdS 6 × w S 4 solution of massive IIA supergravity was found [3] by considering the near-horizon limit of a system of D4-D8-branes in the presence of an O8 orientifold plane. In this background (we use˚above a field to denote its background value) the metric is a warped product of AdS 6 and S 4 given by 6 ds 2 10 = e 2A(y) 9 4 ds 2 AdS 6 (x) + ds 2 S 4 (y) . (2.11) Here x, y denote the external and internal coordinates respectively, and the line elements on AdS 6 and S 4 are of unit radius. In what follows we will use the external metric g µν , and internal metric g mn defined by The warp factor is The remaining non-zero fields of the solution are the dilaton and the 4-form field strength given by: 14) The coordinate θ lies in the interval [0, π/2], and at θ = π/2, where the warp factor diverges, the geometry has a boundary corresponding to the location of the O8-plane. The internal space is therefore more accurately a hemisphere HS 4 with an S 3 boundary at θ = π/2.

Linearised equations of motion
In this section we consider small fluctuations around the AdS 6 × w S 4 solution outlined in the previous section, determine the equations of motion linearised in fluctuations, and reorganise them as field equations for massive free fields propagating in AdS 6 . We perturb the bosonic fields around their background values as: 1) 6 We work in the Einstein frame. An overall scale related to the "trombone symmetry" of the equations of motion has been set to one. It can be reinstated by ds 2 10 → L 2 ds 2 10 , and similarly for the fermionic fields: The Bianchi identities (2.1) allow us to introduce potentials b 2 , a 1 and a 3 such that and it is in terms of these potentials that we will write the equations of motion.
In what follows all geometric quantities, in particular covariant derivatives, and contractions are with respect to the g µν and g mn metrics defined by (2.12). In order to keep the equations covariant we will not substitute for the value of the function A(θ), given by (2.13). Also, where it occurs, we will replace the background dilaton by its equivalent valueΦ = −20A. The Laplace-de Rham operators acting on 0-, 1-, 2-and 3-forms on AdS 6 are defined as where R µν = − 20 9 g µν and R µκνλ = 4 9 (g µλ g κν − g µν g κλ ) are the Ricci and Riemann tensor of the AdS 6 metric g µν . Finally, we will introduce the following notation for the warped Laplace operators on S 4 which appear in the linearised equations: where ∆ S 3 is the S 3 Laplace-Beltrami operator.

Bosonic sector Einstein equation
We start with the Einstein equation (2.3) which splits into three subequations: where A comma denotes partial differentiation e.g. A ,m := ∂ m A and ǫ mnpq is the Levi-Civita tensor.

Dilaton equation
Next we linearise the alternate form of the dilaton equation (2.8): The Ricci tensors of g µν and g mn are R µν = − 20 9 g µν and R mn = 3g mn . Furthermore, the expression ∇ m ∇ n A + 24A ,m A ,n evaluates to Hence, we can recast the linearised dilaton equation in a simpler form: (3.10)

Spin-2 mass spectrum
In this section we look at the the spectrum of massive gravitons or spin-2 particles propagating in AdS 6 . These are the transverse and traceless parts of the metric fluctuation h µν , which we will denote by h tt µν : From the linearised Einstein equation we see that it satisfies where recall L (1) h tt µν := e −8A ∇ m e 8A ∇ m h tt µν . Taking into account the fact that the anti-de Sitter metric has radius 3 2 , we recognize the above equation as the equation of motion of a massive graviton of mass squared M 2 , given by the eigenvalues of L (1) : L (1) h tt µν = − 4 9 M 2 h tt µν . We proceed to solve the eigenvalue problem by factorizing h tt µν as and further expanding Υ in terms of S 3 scalar spherical harmonics: where Y ℓ are S 3 spherical harmonics of eigenvalue −ℓ(ℓ + 2). The differential operator L (1) takes the form where a prime denotes differentiation with respect to θ. We now make a change of variables to z = sin 2 θ , z ∈ [0, 1] (4.7) and the ODE becomes the hypergeometric differential equation (henceforth dropping the ℓ subscript): In order to define a space of admissible solutions, we will recast the hypergeometric equation in a Sturm-Liouville form: and λ := We introduce the weighted inner product and impose boundary conditions such that two eigenfunctions f 1 , f 2 of distinct eigenvalues λ 1 , λ 2 are orthogonal. We compute and hence we will impose Given the above condition we can derive a bound on the mass spectrum, as follows: 16) and hence λ ≥ 0 or Returning to the hypergeometric equation, near z = 0, the solution is a linear combination of 2 F 1 (a, b; c; z) and (1 − z) 1−c 2 F 1 (a − c + 1, b − c + 1; 2 − c; z), but since 2 − c = −ℓ ∈ Z ≤ 0, the latter needs to be replaced by a more complicated expression 7 which is however singular at z = 0 and thus we discard it. We conclude: 1 (a, b; c; z) . (4.18) In order to check regularity near z = 1 we employ the identity Since c − a − b = 1/3, we conclude that f is regular at z = 1, with A similar check for f ′ = C ab c 2 F 1 (a + 1, b + 1; c + 1; z) shows that it is singular at z = 1: Given that f and f ′ are constant at z = 0, we have pf ′ f 0 = 0. On the other hand Notice that the singularity of f ′ at z = 1 is of the same order as the zero of p, so they cancel. We would like to make lim z→1 pf ′ f vanish. There are two ways to do so: A. a = −j ∈ Z ≤0 in which case f is a polynomial and f ′ is regular at z = 1. Imposing so, we derive the mass spectrum: f becomes proportional to the Jacobi polynomial P in which case lim z→1 f = 0 and f ′ is singular at z = 1. Imposing so, we derive the mass spectrum: f becomes proportional to (1 − z) 1/3 P (1/3,ℓ+1) j (2z − 1).
We will refer to the above two branches of the mass spectrum as branch A and branch B. One might be sceptical about branch B, since in that case f ′ is singular, but as we will see the representation theory of the f(4) superconformal algebra supports its existence.
In particular, the Kaluza-Klein excitations should organize in multiplets of f(4), the symmetry of the AdS 6 × w S 4 solution, that include states of highest spin 2. The states lying at the bottom of the spectrum, at j = 0, are expected to belong to short multiplets with masses determined by their su(2) R R-symmetry charge (spin), which for states corresponding to S 3 harmonics Y ℓ is ℓ/2 9 .
According to the AdS/CFT dictionary, the scaling dimension ∆ of the operator dual to a bulk graviton excitation is given by the relation (4.25) First note that via this relation the bound (4.17) for the mass spectrum maps to a unitarity bound for the dimension of the dual field theory operators: ∆ ≥ 3 2 ℓ + 5. Furthermore, (4.25) gives the following dimensions for the two branches: branch A: ∆ = 3 2 ℓ + 3j + 5 , branch B: ∆ = 3 2 ℓ + 3j + 6 . We thus expect short multiplets of f(4) with a state of (highest) spin 2 and dimension ∆ = 3 2 ℓ + 5 for branch A and dimension ∆ = 3 2 ℓ + 6 for branch B. This is indeed the case as shown in the work of [18,19]: in [18] the corresponding multiplets are B[0, 0; k] and A[0, 0; k] in Table 1 respectively, and in [19] they are B 2 and A 4 in Table 22, given explicitly in section 4.7 there.
The dual five-dimensional superconformal field theory has no Lagrangian description, but arises as the UV fixed point of a Yang-Mills theory coupled to hypermultiplets. In particular the gauge group is USp(2N), and the matter content comprises N f < 8 hypermultiplets in the fundamental representation and one hypermultiplet in the antisymmetric representation. We can employ the fields of this theory for a schematic description of the operators dual to the graviton modes. The operator dual to the massless graviton is of course the stress-energy tensor T µν with dimension ∆ = 5. For branch A, we can construct operators O A µν by multiplying T µν with the scalars in the hypermultiplets which transform as a doublet under SU(2) R and have dimension ∆ = 3 2 .
In particular we have: where A I ab is the hypermultiplet in the antisymmetric representation of USp(2N), with I an SU(2) R index and a, b USp(2N) gauge indices. The trace Tr refers to the contraction of the gauge indices of A which are contracted with J ab , the USp(2N) invariant antisymmetric tensor, while ǫ IJ is the SU(2) R one. The description of the operators dual to the graviton modes of branch B, in terms of the fields of the IR theory is less clear, if possible. We expect these operators to have the form where O is a scalar operator of dimension one and R-charge zero. However, O doesn't admit a straightforward representation by the IR fields. A potential candidate for O would be the scalar in the vector multiplet, however the latter is not a representation of the superconformal algebra f(4).

Conclusions
In this note we have taken a first step towards obtaining the Kaluza-Klein mass spectrum of massive IIA supergravity on warped AdS 6 × w S 4 . In particular, we have derived the linearised equations of motion for fluctuations (bosonic and fermionic) around the background and determined the mass spectrum of the spin-2 ones. By a careful analysis of the boudary conditions of the latter at the singularity of the background solution, we have uncovered the existence of two branches of mass spectra. These are bounded from below and the excitations that saturate the bound belong to short supermultiplets, which we have identified from the representation theory of the symmetry algebra of the solution. For one of the two branches we have provided an effective description of the dual field theory operators, in terms of the fields of the theory which at the UV gives rise to the strongly coupled supersymmetric fixed point. For the second branch we lack such a description, and it would be interesting to investigate more the nature of these spin-2 operators. The next step in this endeavour is to determine the mass spectrum for the rest of the fluctuations. This is a challenging task as the warped nature of the background complicates the equations of motion, and the harmonic expansion of the modes on the internal manifold. A convenient gauge for the modes has to be chosen, in which the equations of motion simplify, and the form of the latter suggest a warped generalization of the transverse gauge that is usually used for Kaluza-Klein theories on spheres. Ulti-mately, we expect that the mass spectrum will be determined by the eigenvalue problem of the warped Laplace operators L (k) , defined in (3.5). As was the case for the spin-2 modes, for which k = 1, this eigenvalue problem can be mapped to a hypergeometric differential equation; see appendix B.