Precision Spectroscopy and Higher Spin symmetry in the ABJM model

We revisit Kaluza-Klein compactification of 11-d supergravity on S^7/Z_k using group theory techniques that may find application in other flux vacua with internal coset spaces. Among the SO(2) neutral states, we identify marginal deformations and fields that couple to the recently discussed world-sheet instanton of Type IIA on CP^3. We also discuss charged states, dual to monopole operators, and the Z_k projection of the Osp(4|8) singleton and its tensor products. In particular, we show that the doubleton spectrum may account for N=6 higher spin symmetry enhancement in the limit of vanishing 't Hooft coupling in the boundary Chern-Simons theory.


Introduction
The spectrum of Kaluza-Klein (KK) excitations in flux vacua plays an important role both in attempts to embed the Standard Model in String Theory and in the holographic correspondence. In the spirit of holography, the seminal observation of Schwarz's [1] and the subsequent work of Bagger and Lambert [2][3][4][5] and, independently, of Gustavsson [6], motivated Aharony, Bergman, Jafferis and Maldacena (ABJM) [7,8] to propose a duality between superconformal Chern-Simons (C-S) theories in d = 3 dimensions and String / M-theory on AdS 4 .
The duality has been thoroughly tested and extended to cases with lower supersymmetry [9][10][11][12][13][14]. In particular the superconformal index has been matched both in the regime k >> 1 (SO(2) singlets) [15,16] and at finite k [17,18]. A detailed analysis of the (BPS) spectrum and the supermultiplet structure is however still incomplete. Aim of this note is to fill in this gap and perform precision spectroscopy of 11-d supergravity on AdS 4 × S 7 /Z k or, equivalently, Type IIA on AdS 4 ×CP 3 . We will also discuss higher spin symmetry enhancement in the limit of vanishing 't Hooft coupling in the boundary N = 6 Chern-Simons theory.
The plan of the paper is as follows. After reviewing the ABJM model, presenting both bulk and boundary vantage points, we will revisit KK reduction of 11-d supergravity on S 7 [19] and then perform the decomposition of SO (8) into SO(6) × SO(2) so as to derive the KK excitations of N = 6 gauged supergravity [20], including states charged under SO(2) that are expected to be dual to 'monopole' operators on the boundary [7,8]. Since we rely on group theory techniques which are not easily found in the available literature, we try to make this part of the presentation as pedagogical as possible, also in view of applications to other flux vacua with internal coset manifolds G/H. We then compare the resulting bulk spectrum with the spectrum of gauge-invariant operators on the boundary. Finally we compute the partition function of the boundary theory performing an orbifold projection on the parent theory (k = 1, 2 cases) and examine the higher spin content of the theory. Various appendices summarize useful SO (8) and SO(6) group theory formulae.

The ABJM model
The near-horizon geometry of a stack of N M2-branes is AdS 4 × S 7 with N units of F 4 flux along AdS 4 and as many units of its dual F 7 along S 7 [21]. The metric reads for later use, note that L AdS = L/2 with L the radius of S 7 and henceforth the metrics of the subspaces are for unit curvature radii.
ABJM have conjectured that 11-d supergravity on AdS 4 ×S 7 /Z k , corresponding to the near horizon geometry of N M2-branes at a C 4 /Z k singularity, be dual to N = 6 C-S theory in d = 3 with gauge group U(N) k × U(N) −k and opposite CS couplings k 1 = k = −k 2 [7,8].

Supergravity description
The Type IIA solution corresponding to the ABJM model reads where L = 32π 2 N k is the curvature radius in string units. The string coupling, related to the VEV of the dilaton, is given by Thus the perturbative Type IIA description should be valid for L >> 1 and g s << 1 i.e. for N 1/5 << k << N while λ = N/k is the 't Hooft coupling of the boundary CS theory.
The 11-d supergravity approximation should be valid in the double-scaling limit k → ∞, N → ∞ with λ = N/k fixed and large. The CFT description, to which we momentarily turn our attention, should instead be valid when λ << 1, i.e. k >> N. As λ → 0 higher spin symmetry enhancement takes place as we will eventually see.

Boundary CFT description
N = 6 CS theories are conveniently constructed from N = 3 CS theories. The case N = 3 arises in turn from the N = 4 case obtained after dimensional reduction of N ′ = 2 in d = 4. In this way, each vector multiplet includes an N = 2 (i.e. N = 1 ′ in d = 4) chiral multiplet in the adjoint Φ = Φ a t a and couples to various hypers Q andQ in real (reducible) representations. Adding to the 'standard' N = 4 superpotential W =QΦQ (7) the CS term, giving a mass m = g 2 Y M k 4π to the vectors, and a CS superpotential W = k 8π T rΦ 2 (8) breaks N = 4 to N = 3. Integrating out Φ yields The resulting N = 3 theory has no marginal susy preserving deformations [23][24][25].
In the process R-symmetry is reduced to SO(3) ≈ SU (2) for N = 3 from the original SO(4) of N = 4.
The case N = 6 is special. Starting with the N = 3 theory with G = U(N) k × U(N) −k and two pairs of hypers, A r ∈ (N, N * ) and Bṁ ∈ (N * , N) and integrating out Φ 1 and Φ 2 one gets W = 2π k ǫ rs ǫṁṅT r(A r BṁA s Bṅ) Since the manifest 'flavour' symmetry of W under SU(2)×SU(2)×U(1) B does not commute with R-symmetry SO(3) ≈ SU(2) under which A and B form doublets, the full theory has a larger SU(4) ≈ SO(6) symmetry which is the R-symmetry of N = 6. To expose the symmetry it is convenient to define X i = (A 1 , A 2 , B * 1 , B * 2 ) and their conjugate X * i that together transform as 4 +1 + 4 * −1 of SO(6) × SO(2). As we will momentarily see, SO(2) ∼ U(1) acts as a baryonic symmetry. Further (super)symmetry enhancement to N = 8 with SO(8) R-symmetry takes place for k = 1 and k = 2. The former corresponds to compactification on S 7 the latter to S 7 /Z 2 (only 'even' spherical harmonics).
Compactification of Type IIA supergravity on CP 3 was studied in [26]. KK excitations with Q = 0, i.e. neutral wrt SO(2), were identified there. The nonperturbative spectrum, contains various wrapped branes, including D0-branes that are charged wrt SO (2). The latter correspond to 11-d KK modes along the compact circle that can be obtained by a Z k projection of the M-theory compactification on S 7 . The dual to SO(2) charged states are monopole operators on the boundary [8,27,28]. Although the fundamental fields (A r , B˙s) are neutral wrt the diagonal U(1) that couples to µ acts as a baryonic symmetry. The corresponding current, J B = * F + , is conserved thanks to Bianchi identities. Due to the CS coupling k A − ∧ F + , configurations with A + magnetic charge are electrically charged wrt A − . Alternatively one can introduce a Lagrange multiplier τ for dF + = 0 (on-shell kA − = dτ ) and form combinations e inτ that can screen the baryonic charge of matter field composites. In general one can consider magnetic monopoles charged under U(1) N ⊂ U(N) with H = (Q 1 , ..., Q N ). Without loss of generality one can take Q 1 ≥ Q 2 ≥ ... ≥ Q N . Since elementary fields have unit charges and transform in the fundamental of SU(N), these monopole operators correspond to Young Tableaux with kQ i boxes in the i th row. For k = 1, 2 dressing composite vector currents in the 6 ±2 and scalar operators in the 10 ±2 and 10 * ∓2 (with ∆ ± = 1, 2) with charge 2 monopole operators is crucial to the enhancement of supersymmetry to N = 8 with full SO(8) R-symmetry [8]. Monopole and antimonopole operators however appear in the spectrum even when k ≥ 3 and no (super)symmetry enhancement takes place [27,28].
Before concluding this preliminary look, let us note that out of the two U(1) in the boundary CS theory only the Baryonic U(1) B = U(1) − is visible as a global symmetry, whose Z k subgroup is gauged, in the bulk description. The fate of the other U(1) is a sort of Higgs mechanism, under which A M → A µ and C M N P → C µ J ab mix. Only the combination kA µ + NC µ remains massless and couples to U(1) B while the orthogonal combination NA µ − kC µ becomes massive by 'eating' the (pseudo)scalar β from B 2 = βJ . A 5-brane instanton is thus expected to mediate processes in which k D0-branes transform into N D4-branes wrapped around CP 2 ⊂ CP 3 [22].

Compactification on S 7 revisited
For the later use let us briefly review the mass spectrum of the Freund-Rubin solution of d = 11 supergravity on S 7 [19,29,30]. The gravitino field as well as all the fermions are set to zero, the AdS 4 Riemann tensor and the three-form field strength are given by: where ε 0123 = −1. The metric and the three form field with mixed indices vanish: and also F αβγδ (y) = 0 (14) µ, ν, ρ = 0, ..., 3 are d = 4 indices, α, β, γ = 1, ..., 7 are internal indices.
Let us then consider fluctuations around the Freund-Rubin solution. The linearized field equations are obtained by replacing the background fields in the d = 11 field equations by background fields plus arbitrary fluctuations. An elegant and quite general method to determine the complete mass spectrum on any coset manifold relies on generalized harmonic expansion. In our case, one expands the fluctuations in a complete set of spherical harmonics of S 7 = SO(8)/SO (7). The coefficient functions of the spherical harmonics correspond to the physical fields in d = 4. In order to diagonalize the linearized equations it turns out to be convenient to parameterize the fluctuations as follows: In particular the Weyl rescaled spacetime metric appears in (17) so as to put the d = 4 Einstein action in canonical form. The spherical harmonic expansions of the fluctuations of the metric and of the antisymmetric tensor fields are given by: All superscripts N r (r = 1, 7, 21, 27, 35) have infinite range, since they should provide a basis for arbitrary fields on the 7-sphere. The index r specifies the SO(7) representation of the corresponding spherical harmonic. For example, Y N 35 αβγ is in the third rank totally antisymmetric representation of SO(7) with dimension 35, while Y N 27 (αβ) is in the symmetric traceless 27-dimensional representation. Derivatives of Y 's appear in the expansions since any tensor can be decomposed into its transverse and longitudinal parts. After fixing all local symmetries which do not correspond to gauge invariances of the final d = 4 theory and by choosing de Donder type, D α h (αβ) (x, y) = 0, and Lorentz type, D α h αµ (x, y) = 0, conditions the last term in h µα and the last two terms in h (αβ) drop out. To fix the local symmetries of the antisymmetric tensor fields we choose the Lorentz conditions D α A αβγ (x, y) = D α A αβµ (x, y) = D α A αµν (x, y) = 0. As a consequence, also these fields have only transverse harmonics a N 1 µν (x) = a N 7 µ (x) = a N 21 (x) = 0. Substituting the resulting expansions into the d = 11 field equations, the coefficients of each independent spherical harmonic yield the d = 4 field equations.
In the Einstein equation for R µν only Y N 1 spherical harmonics appear without derivatives. Thus there is only one field equation, i.e. one KK tower, for traceless symmetric tensors in AdS 4 .
Examining the Einstein equation for R αβ one can see that the vector fields B N 7 µ are massive and transversal, except for the lowest lying state corresponding to the Killing vectors on S 7 . The spin-0 fields φ N 27 have a mass matrix ∆ y + 12 (∆ y is the Hodge-de Rham operator). By a judicious gauge choice one can eliminate H N 1 µ µ in favour of π N 1 namely H N 1 µ µ = 9 7 π N 1 . Collecting the coefficients of the spherical harmonics Y N 7 α and D α Y N 1 in the Einstein equation for R µα , one finds that the spin-1 spectrum consists of linear combinations of B N 7 µ and C N 7 µ (from a N 7 ρσ ) and that one can eliminate the divergence D µ H N 1 µν in favour of π N 1 , a N 1 ρστ except when Y N 1 is a constant. Similarly, inspecting the equations for p-form field strengths (p = 1, 2, 3, 4), one concludes that field expansions in spherical harmonics can be chosen such that only the first terms in the expansions survive with Y s being transversal and traceless.
In particular, from the three-form field strength equation one finds that a N 1 µνρ = ε µνρλ D λ σ N 1 . This implies that the divergence of H N 1 µν is proportional to a gradient. From the four-form field strength equation one gets an equation for x σ N 1 . Taking the trace of the equations for R µν and R αβ , an equation involving x σ N 1 and x H N 1 µ µ arises. Resolving the mixing between a N 1 µνρ and H N 1 µ µ produces to independent combinations and as many KK towers of scalars.
From the two-form field strength equation one finds D µ a N 7 µν = 0, which implies a N 7 µν = ε ρσ µν D ρ C N 7 σ . Using one of the three-form field strength equations one finds Spin Field After diagonalizing the bosonic field equations one obtains the mass spectrum summarized in Table 2. The resulting bosonic spectrum includes the massless graviton, 28 massless vectors of SO(8), corresponding to a combination of B µ (in h µα ) and C µ (in A µνα ), 35 v scalars (∆ = 1) and 35 s (∆ = 2) pseudoscalars with (ML AdS ) 2 = −2. In the supergravity literature [19,29,30] masses of scalars are often shifted by −R/6 so that (ML AdS ) 2 → (ML AdS ) 2 = (ML AdS ) 2 + 2. The 70 (pseudo)scalars in the N = 8 supergravity multiplet are 'massless' in the sense that (ML AdS ) 2 = 0. Moreover, there are three families of scalars and two families of pseudoscalar excitations. Three of them (0 + 2 , 0 + 3 and 0 − 2 ) contain only states with positive mass square and correspond to irrelevant operators in the dual CFT. The remaining families 0 + 1 and 0 − 1 contain states with positive, zero and negative mass squared corresponding to irrelevant, marginal and relevant operators, respectively.
A similar analysis can be performed for fermionic fluctuations. In Table 2 we summarize the fermionic mass spectrum.
The KK spectrum does not include the states with * for ℓ = −1, since they do not propagate in the bulk but live on the conformal boundary of AdS 4 . They correspond to the singleton representation of Osp(8|4) that consists of 8 v bosons The KK excitations on S 7 can be put in one-to-one correspondence with 'gauge-invariant' composite operators on the boundary. The dictionary for bosonic operators schematically reads: A similar dictionary can be compiled for fermions.

Polynomial representations for SO(8) and U (4)
In order to decompose KK harmonics on S 7 = SO(8)/SO (7) into KK harmonics on CP 3 = U(4)/U(3) × U(1), we will present the construction of arbitrary representations of SO (8) in the space of polynomials of 12 variables. The latter are the coordinates of the subgroup Z SO(8) + generated by the raising operators of SO(8). We will then describe a technique which allows to identify which of the above polynomials correspond to highest weight states of representations of U(4) ⊂ SO (8). The method we use is quite standard in representation theory of Lie groups (see e.g. Chapter 16 of [31]).
It is convenient to start with SO(8, C) defined as the group of 8 × 8 complex matrices which leave invariant the quadratic form X T C (8) X, where X is a complex (column) vector whose components will be enumerated as By definition all matrices g ∈ SO(8) satisfy the condition g T C (8) g = C (8) . Eventually, in order to select the compact real form SO(8) of our interest, one should identify the coordinates X˜i withX i (bar means complex conjugate). A generic SO(8) matrix g can be (uniquely) decomposed as (Gauss decomposition): where ζ ∈ Z − , z ∈ Z + , λ ∈ Λ with Z + (Z − ) being the subgroup of lower (upper) triangular matrices with 1's on the diagonal and Λ is the subgroup of diagonal matrices (Cartan subgroup). Let's set λ = Diag(λ 1 , λ 2 , λ 3 , λ 4 , λ −1 4 , λ −1 3 , λ −1 2 , λ −1 1 ). We will realize the irreducible representations of the group SO(8) on some spaces of functions defined on it. In particular, the role of the highest weight vector will be played by the function : where m 1 ≥ m 2 ≥ m 3 ≥ |m 4 | (m i are either all integers or all half-integers) uniquely characterize the irrep. The eigenvalues λ i can be expressed in terms of the matrix elements of g explicitly: where ∆ 0 = 1 and ∆ p , p = 1, 2, 3, 4 are the diagonal minors Introducing the notation S − = ∆ 3 √ ∆ 4 , S + = √ ∆ 4 (it is easy to see that S +,− polynomially depend on the matrix elements of g) we can rewrite eq. (33) as where 4 and ℓ 4 = m 3 + m 4 are nonnegative integers commonly referred as the Dynkin labels of the irrep. Consider the space R α of all linear combinations of the functions α(gg 0 ), g 0 ∈ SO(8). SO (8) is represented in R α simply by the right multiplication of the argument. As already mentioned the function α(g) plays the role of the highest weight state. For any function f (g) ∈ R α we have f (ζλz) = α(λ)f (z) which shows that to restore its full g-dependence it is sufficient to only know the values the function assumes on the subgroup Z + . This is why actually we get representation on a space of functions of z, in fact polynomials due to the polynomial dependence on g of α(g) mentioned earlier.
There is an elegant way to characterize this space of polynomials. Consider the four raising generators corresponding to the simple roots where E pq denotes the 8 × 8 matrix whose only non-zero entry 1 is at the position (p, q). Denote their left action on R α by D 1 , D 2 , D − , D + . It is not difficult to prove that The key observation is that the same equations are valid also for arbitrary functions f ∈ R α , since they are all generated by α(g) through right multiplications which commute with left multiplications. Below we will use a convenient explicit parametrization of Z + ⊂ SO(8) in terms of two 4 × 4 matrices η and a η = Let us further introduce the 8 × 8 matrices which in 2 × 2 block form read An arbitrary z ∈ Z + can be (uniquely) represented as Left multiplication by raising generators (37) induces infinitesimal motion on the parameters a, η. A straightforward algebra shows that e.g.
Similarly examining the remaining three generators we find Thus any irreducible representation of SO(8) is realized on the space of polynomials of 12 variables a, η subject to the constraints Note that the constant polynomial always satisfies (45) and corresponds to the highest weight state. Considering right multiplication it is not difficult to find explicit expressions for the generators of SO(8) as operators acting on the space of polynomials. For our later proposes let us specify how the diagonal part where Notice that the variable a ij shifts the weights as Consider now the GL(4, C) ⊂ SO(8, C) subgroup whose off-diagonal blocks in 2×2 block notation are zero. This subgroup does not mix the coordinates X i with X˜i and after restriction to the real sector it becomes the subgroup U(4) ⊂ SO (8).
In other words, for the reduction from S 7 to S 7 /Z k or CP 3 ⋉ S 1 we are interested in, the decomposition SO(8) → SO(6) × SO(2) is given by the embedding where (ℓ 1 , ℓ 2 , ℓ 3 , ℓ 4 ) and [k, l, m] denote SO(8) and SO(6) Dynkin labels respectively. As a result, for the Adjoint representation one has for the spinorial representations.
Our goal is to identify the highest weight states of this subgroup inside the space of polynomials of a given representation of SO (8). It is evident from the decomposition (42,41) that the right action by the raising operators of GL(4) subgroup e 1 , e 2 , e − (see eq. (37)) shifts the parameters η and leave the parameters a untouched. Thus, in order to be a highest weight state, a polynomial, besides satisfying the equations (45) should be independent of η. The indicator system for the highest weight states becomes Solving these equations one can fully decompose KK harmonics on S 7 into KK harmonics of CP 3 × S 1 which is our next task.
The CP 3 solution of the d = 10 theory can be obtained from the S 7 solution of the d = 11 theory by Hopf fibration, i.e. keeping only U(1) invariant states [26]. The compactification on CP 3 of the d = 10 theory yields a four dimensional theory with N = 6 supersymmetry and with gauge group SO(6) × SO(2).
The truncation from S 7 to CP 3 ⋉ S 1 cannot be thought of as spontaneous (super)symmetry breaking and one has to really discard the states that are projected out by Z k or SO(2) for k → ∞ even if it acts freely. In particular we will later check that no Higgsing can account for the breaking of SO(8) to SO(6) × SO(2) but rather the coset vectors are dressed with monopole operators and become massive for k = 1, 2 [7,8,22,28,32].
Let us start with the KK towers of bosons. Using the procedure described in the previous section or otherwise, for scalar spherical harmonics with Dynkin labels (ℓ, 0, 0, 0) one finds as independent polynomials {a m 14 | m = 0, ..., ℓ}. Thus the following decomposition holds: where the subscript is the SO(2) charge Q of the appropriate representation.
One obtains the decomposition of the representation (ℓ − 2, 1, 0, 0) from the previous one by shifting ℓ to ℓ − 2. In what follows we will simply omit the decompositions which differ by shifts of the parameter ℓ. , a 13 a n 14 , a 34 a 13 a n 14 , (a 34 a 12 − a 13 a 24 )a 13 a n 14 | m = 0, ..., ℓ − 1, n = 0, ..., ℓ − 2} as independent polynomials. One then finds the following decomposition: The decomposition of the KK towers corresponding to 0 + 1 and 0 + 2 can be found from the decomposition of 2 + via appropriate shifts.
The zero charge spectrum i.e. the states which constitute the KK spectrum of Type IIA supergravity on CP 3 can be easily identified in the above decompositions. For completeness and comparison with the original literature [26], we collect the relevant formulae in an Appendix.

A closer look at the KK spectrum
As already observed, the Z k orbifold projection from S 7 to S 7 /Z k ≈ CP 3 ⋉ S 1 cannot be thought of as spontaneous (super)symmetry breaking. 'Untwisted' states that are projected out do not simply become 'massive' but are rather eliminated from the spectrum. In particular in the large k limit only SO(2) singlets survive. It is amusing to observe that only states with ℓ even on S 7 give rise to neutral states. This suggests that the parent theory could be either a compactification on S 7 or on RP 7 = S 7 /Z 2 . Indeed both lead to SO(8) gauged supergravity corresponding to the 'massless' multiplet Massless scalars, corresponding to marginal operators with ∆ = 3 on the boundary, only appear in higher KK multiplets, i.e. in the 840 ′ = (2, 0, 0, 2) and 1386 = (6, 0, 0, 0). None of these can play the role of Stückelberg field for the 12 coset vectors in the 6 +2 + 6 −2 of SO(8)/SO(6) × SO(2). This means that the massless scalars in the 840 vc (2, 0, 2, 0) cannot account for the 'needed' Stückelberg fields in the 6 +2 + 6 −2 . Yet one can recognize massless scalars neutral under SO(2) that survive in k → ∞ limit and transform nontrivially under SO (6). Turning them on in the bulk, e.g. in domain-wall solutions, should trigger RG flows to theories with lower supersymmetry on the boundary. Once again there are no 6 +2 + 6 −2 . In this case, 'neutral' fields appear in the 300 representation of SO(6).
In the KK spectrum, neutral (wrt to SO (2)) singlets (of SO (6)) appear in the decomposition of 35 s parity odd scalars 0 − 2 with M 2 L 2 AdS = 10 that reads They correspond to boundary operators with dimension ∆ = 5. The only other neutral singlets arise from the SO(8) singlet parity even scalar with M 2 L 2 AdS = 18, i.e. ∆ = 6. Neither ones belongs in the supergravity multiplet 1 . They correspond to the 'stabilized' complexified Kähler deformation J + iB and as such couple to the Type IIA world-sheet instanton recently identified in [33]. Indeed the bosonic action schematically reads S wsi = J + iB = L 2 /α ′ since B = 0 in the ABJM model, while B = l/k with l = 1, ..., k − 1 for the ABJ model involving fractional M2-branes. Effects induced by world-sheet instantons in Type IIA on CP 3 should be dual to the non-perturbative corrections discussed in [34]. It may be worth to observe that in 'ungauged' N = 6 supergravity, arising from freely acting asymmetric orbifolds of Type II superstrings on tori, world-sheet and other asymmetric brane instantons [35,36] should correct R 4 terms very much as in their parents with N = 8 local supersymmetry.
Other non-perturbative effects are induced by E5-brane instantons that should mediate the process of annihilation of k D0-branes into N D4-branes wrapping CP 2 [7,22]. In order to determine the action of such an instanton it is worth recalling that the pseudo-scalar mode B 2 = β(x)J 2 (y) is eaten by the vector field A H µ = kA D4 µ − NA D0 µ that becomes massive. The complete E5-brane instanton action should be S E5 = L 6 /g 2 s (α ′ ) 3 + iβ that indeed shifts under U(1) H gauge transformations and as such can compensate for the 'charge' violation in the above process as in similar cases with unoriented D-brane instantons [37].

Spins
In this section, we would like to discuss the higher spin (HS) extension of N = 6 gauged supergravity. Higher spin extensions of various supergravity theories in AdS 4 have been studied in [38][39][40] but to the best of our knowledge the case of N = 6 has been overlooked.
Let us start by briefly recalling some basic features of higher spin theories in . We will continue and call ∆ the dimension and s or j spin. In 'radial' quantization the 'Hamiltonian' H has eigenvalues ∆.
For later use let us collect here the partition functions of the two singletons that take into account their conformal descendants i.e. non vanishing derivatives. For free bosons such that ∂ 2 X = 0 one has For free fermions ∂Ψ = 0 one has Combining n b = 8 v free bosons and n f = 8 c free fermions one finds the singleton representation of Osp(8|4) ⊃ SO(8) × SO(3, 2), whose Witten index reads One can also keep track of the spin of the states in the spectrum by including a chemical potential y = e iα (y J 3 = e iαJ 3 ) and find is the character of the fundamental representation of the 'Lorentz' group SU (2).
Before switching to higher spins, notice that Z k acts on the singleton simply as with ω = e 2πi/k playing the role of chemical potential or rather fugacity for the SO(2) ≈ U(1) B charge Q commuting with SO(6) R-symmetry. One can introduce another three chemical potentials β i or fugacities x i = e iβ i in order to keep track of the three Cartan's of SO(6) ≈ SU(4). We refrain from doing so here.

Doubleton and higher spin gauge fields
Doubleton representations can be obtained as tensor products of two singletons [45][46][47]. or A consistent truncation, giving rise to minimal HS theories with even spins only, stems from restricting to symmetric tensors for bosons It is reassuring to recognize above the 'massless' states of N = 8 gauged supergravity on AdS 4 . The remaining states with spin s ≤ 2 belong to the 'short' Konishi multiplet and a 'semishort' multiplet with spin ranging from 2 to 6 [48][49][50]. Holography allows to relate AdS compactifications of supergravity and superstring theories to singleton field theories on the 3-d boundary. As a first step, these field theories can be constructed on the boundary of AdS as free superconformal theories. A remarkable property of singletons is that the symmetric product of two super-singletons gives an infinite tower of massless higher spin states. In the limit λ → 0, all higher spin states become massless. After turning on interactions, a pantagruelic Higgs mechanism, named Grande Bouffe in [51][52][53][54], takes place. All but a handful of HS gauge fields become massive after 'eating' lowest spin states. The boundary counterpart of this phenomenon is the appearance of anomalous dimensions for HS currents and their superpartners. One should keep in mind that genuinely massive states are already present in the spectrum at λ → 0 and arise in the product of three and more singletons.
Interacting theories for massless HS gauge fields, thus only describing the doubleton, have been proposed by Vasiliev [43] that capture some aspects of the holographic correspondence in the extremely stringy (high AdS curvature) regime. Only vague glimpses of an interacting theory incorporating the Grande Bouffe have been offered so far [51][52][53][54].
Barring these subtle issues, let us discuss how to perform a Z k projection of the spectrum giving rise to an N = 6 HS supergravity in AdS 4 . In the limit k → ∞ only SO(2) singlets survive where indicated in bold-face are the surviving representations of the SO(6) Rsymmetry. Candidate bosonic HS operators on the boundary in the 1 + 15 of SO(6) are where dots stand for symmetrization and subtraction of the traces and the coefficients of the linear combination are to be chosen appropriately.
At finite k and λ, states with SO(2) charges Q = kn survive. One can exploit orbifold technique to deduce the 'free' spectrum 4 .
The partition function or rather Witten index for the super-singleton of OSp(8|4) reads: the Z k projection reads where with ω = e 2πi/k . Clearly Z Z k = 0 since Σ k−1 r=0 ω r = 0. A non-trivial spectrum arises from the doubleton partition function. Prior to the Z k projection one has for the (graded) symmetric doubleton, giving rise to precisely the spectrum of hs(8|4) discussed above.

Tripletons and higher n-pletons
For higher multipletons one has to resort to Polya theory [51][52][53][54]. Consider a set of 'words' A, B, ... of n 'letters' chosen within the alphabet {a i } with i = 1, ...p. When p → ∞, let us denote by Z 1 (q) the single letter 'partition function'. Let also G be a group action defining the equivalence relation A ∼ B for A = gB with g an element of G ⊂ S n . Elements g ∈ S n can be divided into conjugacy classes In particular, for the cyclic group G = Z n , conjugacy classes are [g] = (d) n/d for each divisor d of n. The number of elements in a given conjugacy class labelled by d is given by Eulers totient function E(d), equal to the number of integers relatively prime to d. For d = 1 one defines E(1) = 1.
For the full symmetric group one has Let us consider the product of three singletons.
There are thus three kinds of tri-pletons.
The totally symmetric tripleton is coded in the partition function where u collectively denotes the 'fugacities' q, y = e iα , ω ≈ t, ....

For the cyclic tripleton one has
For totally anti-symmetric tripletons one finds while for mixed symmetry, incompatible with the cyclicity of the trace, one eventually finds Recalling the singleton partition function where ω = e 2πi/k and χ1 2 (α) = tr 1/2 exp(iαJ 3 ), one eventually finds for the totally symmetric tripleton. Let us analyze the spectrum arising in this case. Except for the 1/2 BPS states, we will consider later on, only 'massive' representations above the unitary bound, whose characters read appear in the decomposition Indeed it is easy to see that no current like (twist τ = 1) fields appear beyond the double-ton, since the twist whenever n X + n Ψ > 2.
For higher multi-pletons the analysis is similar. It is clear that only states with charge Q = ±n, ±(n − 2), ... are present in the n-pleton spectrum. In particular Q = 0 states are only present when n is even as already observed. We defer a detailed analysis to the future. For the time being let us only display the partition functions for the cyclic tetrapleton and for the totally symmetric tetrapleton The Z k projection on n-pletons reads and corresponds to keeping only states with Q = kn i.e. integer multiples of k.

KK excitations
Let us now focus on the KK excitations, which deserve a separate treatment. One can indeed write down the single-particle partition function on S 7 , decompose it into super-characters and identify the SO(2) charge sectors, relevant for the subsequent Z k projection i.e. compactification on CP 3 .
After some algebra, putting t = 1, one finds a factor (1 − q) 2 cancels between numerator and denominator meaning that not only n b = n f and the sum with ∆ 1 vanishes but also the sum with ∆ 2 should vanish. This should be related to the absence of quantum corrections to the negative vacuum energy, i.e. cosmological constant in the bulk.
The 1/2 BPS partition function is given by The simplicity of the result is due to 'miraculous' cancellations between bosonic and fermionic operators with the same scaling dimensions in different KK multiplets i.e. with different ℓ's. This does not happen in AdS 5 /CF T 4 holography, whereby (protected) bosonic operator have integer dimensions and (protected) fermionic operators have half-integer dimensions [41,42,[51][52][53][54][55].
In order to perform the Z k projection it is useful to decompose into SO(2) charge sectors according to where P 2 (t) = 10t +2 + 15 + 10t −2 Depending on the choice of k one can recognize the surviving 1/2 BPS states as those with Q = kn. In formulae one has to replace t with ω r and sum over r = 0, ..., k − 1.

Conclusions
We have re-analyzed the KK spectrum of d = 11 supergravity on S 7 and S 7 /Z k . The latter includes monopole operators dual to charged states in Type IIA on CP 3 . To this end we have presented some group theoretic methods for the decomposition of the SO(8) into SO(6) × SO(2) valid also for other cosets [56][57][58] where resolution of the mixing among various fluctuations should be possible on the basis of symmetry arguments. In particular, massless vectors associated to Killing vectors in generic flux vacua with isometries have been recently discussed in [59].
We have then considered higher spin symmetry enhancement. We have displayed the partition functions for singletons, doubletons and tripletons and discussed in details higher spin fields and 1/2 BPS states corresponding to KK excitations of N = 6 gauged supergravity. It would be worth pursuing the analysis to higher n-pletons and to cases with lower supersymmetry, yet based on internal coset manifolds. (2 + ℓ 1 + ℓ 2 )(2 + ℓ 2 + ℓ 3 )(2 + ℓ 2 + ℓ 4 ) Specific cases (KK harmonics)

sentations
The generating function for multiplicities of the scalar spherical harmonics on S 7 is given by The coefficient of q ℓ gives the dimension of the SO(8) representation with Dynkin label (ℓ, 0, 0, 0).