De-Higgsing in eleven-dimensional supergravity on the squashed S 7

In this paper we construct the subset of modes on S 7 that are relevant in the compactification of eleven-dimensional supergravity on a squashed S 7 when restricted to the sector that comprises singlets under the Sp(1)×Sp(2) isometry of the squashed sphere. Some of the properties of these modes, connected to the transition from the round S 7 to the squashed S 7, are analysed in detail. Special features of the Rarita–Schwinger operator, described in earlier work by Buchdahl, are explained and related to properties of the squashed S 7 operator spectrum obtained in previous work by the authors. We then discuss how the singlet modes give rise to supermultiplets in the left-squashed case, the phenomenon of de-Higgsing, and what happens to the AdS4 fields in these supermultiplets under an orientation reversal (‘skew-whiffing’) of the squashed S 7. Finally, we consider the possible choices of boundary conditions that appear for some of these fields in AdS4 in the case of the right-squashed non-supersymmetric compactification, and how these choices may affect the stability of the gravity theory.


Introduction
There has recently been a lot of effort put into deriving the complete spectrum of some non-supersymmetric AdS vacua of maximal supergravity theories in ten and eleven dimensions.One reason for this endeavour is to challenge, or vindicate, conjectures related to the swampland project, in particular the AdS stability conjecture proposed by Ooguri and Vafa [1].This issue becomes especially interesting when discussing vacua that are BF (Breitenlöhner-Freedman) stable but which so far have not been proven unstable despite having no supersymmetry.In fact, for D = 11 supergravity, the squashed seven-sphere vacuum without supersymmetry, the so called right-squashed sphere [2,3], is one of a very small set of such examples 1 .The squashed S 7 spectrum of fields and their SO(2, 3) irrep content has recently been derived in full detail [6][7][8][9] 2 which makes it possible to study also the relation between the round S 7 compactification and its cousins with partially or totally broken supersymmetries, i.e., the left-squashed and right-squashed S 7 vacua with N = 1 and N = 0 supersymmetry, respectively.Some aspects of this relation for scalar and spin-1 2 modes were addressed in [11]. 1 Another case suggested recently is based on a deformed S-fold compactification [4,5]. 2 The spectrum of the squashed seven-sphere was reproduced recently in [10] by entirely different methods.
A number of comments on this relation were made in [6].Here we discuss an additional aspect connected to, in particular, the Rarita-Schwinger operator and properties of its eigenvalue equation on non-Einstein versions of the squashed S 7 .This is, in fact, a wellknown issue which was studied some time ago by Buchdahl [12][13][14].In the present context it becomes especially intriguing since for a large portion of the mode spectrum on S 7 can be traced continuously between the round and the Einstein-squashed vacuum solutions of D = 11 supergravity (see [11]), while Buchdahl's results indicate that this should not be possible in general.
Our results are based on an explicit construction of the entire set of singlet mode functions on the squashed S 7 .Although they constitute only a small subset of the totality of modes in the complete spectrum, the singlet modes are of particular interest because they form a consistent truncation in their own right, and additionally, they have elegant geometrical interpretations, allowing their simple explicit construction.They also exhibit some of the most intriguing features in the relation between the round and the squashed S 7 vacua.The complete singlet sector involves in addition to the singlet Rarita-Schwinger mode relevant for the Buchdahl issue also its superpartners, the "squashing mode," which is the Lichnerowicz operator eigenmode first discussed by Page in [15], and two singlet modes of the first-order 3-form operator Q = * d.Adding to these the scalar and spin- 1  2 singlet modes they make up precisely two Wess-Zumino supermultiplets.The latter two singlet modes are also responsible for the N = 1 supergravity multiplet.The properties of these modes on the non-supersymmetric right-squashed S 7 are also analysed in detail.
The paper is organised as follows.In section 2 we give some basic material needed in the following sections.Section 3 contains the construction of the singlet modes on S 7 and a discussion of some of their properties connected to the transition from the round S 7 compactification to the squashed one in particular the special features of the Rarita-Schwinger operator.Then, in section 4, we explain how the singlet modes give rise to supermultiplets in the left-squashed case and what happens to the AdS 4 fields in these supermultiplets under skew-whiffing.The possible choices of boundary conditions in the right-squashed non-supersymmetric case are identified and possible implications for the issue of stability are discussed.The final section contains a summary and some conclusions.

Preliminaries
We recall that the mass operators for the various towers in the spectrum of AdS 4 supergravity from compactification on M 7 are given by (see table 5 of Phys.Reps.
The internal space M 7 carries an Einstein metric, with Ricci tensor given by R ab = 6m 2 g ab .
The remaining notation here is as follows: The entries in the left-hand column in (2.1) denote the spins of the four-dimensional fields in the Kaluza-Klein towers.The notation of the additional superscripts is explained in detail in [3].The operators appearing in the right-hand column are defined as follows: For vectors, the transversality condition ∇ a V a = 0 is imposed and similarly for 2-forms and 3-forms.Lichnerowicz modes are required to be transverse and traceless and modes of the Rarita-Schwinger operator are required to be gamma-traceless, γ a ψ a = 0.As will be discussed later, consistency also then implies that ∇ a ψ a = 0 and that the metric be Einstein.The Rarita-Schwinger operator then reduces to the Dirac operator acting on vector-spinors, which we denote by / D 3/2 .
We shall be concerned with the spectrum of modes in the round and squashed S 7 vacua of D = 11 supergravity.The family of squashed S 7 metrics, described as an SU (2) bundle over S 4 , is given by

.3)
Here A i = cos 2 1 2 u Σ i , and σ i and Σ i are two sets of left-invariant 1-forms on the group manifold SU (2).They obey When required, we use the vielbein basis e a with e 0 = dµ , where i ranges over the values 1, 2 and 3, and î= ranges over 4, 5 and 6, with 1 = 4, 2 = 5 and 3 = 6.The constant λ is the squashing parameter.The isometry group of the metric is Sp(1) × Sp(2) (which is locally the same as SU (2) × SO(5)) for generic values of λ, enhancing to SO(8) in the case λ = 1, which corresponds to the round S 7 .
The metric (2.3) is Einstein when λ = 1, obeying and when λ = 1 √ 5, with For general values of λ, the Ricci scalar is (2.8) It will be convenient, when calculating the eigenvalues, to rescale the metric so that it has R = 42m 2 for all values of the squashing parameter λ.This will imply that eigenvalues of a linear operator will need to be scaled by the factor Eigenvalues of a second-order operator will need to be scaled by the square of (2.9).Note that after the rescaling of the metric, the round and the squashed Einstein metrics will both depend on the parameter m 2 and obey R ab = 6m 2 g ab . (2.10) 3 Singlet Modes On The Squashed S 7 In this section, we shall collect together a number of previously-known results for singlet modes of the relevant operators on the squashed S 7 whose eigenvalues govern the masses of the corresponding four-dimensional fields in the S 7 compactifications of D = 11 supergravity.Some of our discussion here will also extend beyond that previously presented in the literature.The results we obtain here will then be used in the next section in order to study the mass spectrum in the singlet sector for both the left-squashed and the right-squashed S 7 vacua, and to relate these spectra to the spectrum in the round-sphere vacuum.
The singlet modes we shall consider in this section comprise the singlet mode of the scalar Laplacian ∆ 0 = − (this, of course, is just the trivial constant mode); a singlet in the spectrum of the Lichnerowicz operator ∆ L acting on transverse, traceless 2-index symmetric tensors; two singlets in the spectrum of the first-order operator Q = * d acting on 3-forms; a singlet in the spectrum of the Dirac operator acting on spin-1 2 fields on the S 7 ; and a singlet in spectrum of the Rarita-Schwinger (RS) operator acting on transverse spin- 3  2 fields on the S 7 .In all, these comprise the entire set of singlet modes that are relevant for determining the spectrum of fields in the S 7 compactifications of D = 11 supergravity.
Note that the Hodge-de Rham operators ∆ 1 and ∆ 2 acting on 1-forms and 2-forms have no Sp(1) × Sp(2) singlet modes in the round or squashed S 7 backgrounds.

Scalar singlet modes
The complete spectrum of the scalar Laplacian ∆ 0 = − on the whole family of squashed S 7 metrics was obtained in [11].For our present purposes, it suffices to note that the only singlet in the spectrum of ∆ 0 is the constant mode, with eigenvalue 0.

Lichnerowicz singlet modes
There is only one singlet in the spectrum of the Lichnerowicz operator acting on transverse, traceless symmetric tensor h ab in the squashed S 7 metric, namely the so-called "squashing mode" that corresponds to an infinitesimal variation of the squashing parameter while keeping the volume fixed.In the orthonormal frame defined in eqn (2.5), it corresponds to a tensor whose non-vanishing components (up to an constant scale) are given by It was first discussed in [15].A straightforward calculation shows that in the background of the metric (2.3), it is an eigentensor satisfying ∆ L h ab = σ L h ab with σ L = 7λ 2 .After making the rescaling by two powers of the factor in eqn (2.9), so as to normalise to squashed metrics with R = 42m 2 , we get the Lichnerowicz eigenvalue It should be noted that because the eigenfunctions of the Lichnerowicz operator are defined for all values of the squashing, we can in particular continuously follow them as we pass from the round S 7 to the Einstein-squashed S 7 .For the particular case of the singlet mode (3.1), we see from eqn (3.2) that we have These values accord with those obtained in [15].Note that the Lichnerowicz eigenvalue σ L = 28m 2 on the round S 7 is in accordance with the known round-sphere results, where the lowest level of TT Lichnerowicz modes are in the 300 representation of SO(8), and have eigenvalue 28m 2 (see, for example, [3]).And indeed, as must be the case, the 300 of SO(8) has a singlet in its decomposition under the Sp(1) × Sp(2) subgroup (the isometry group of the squashed S 7 ).
In fact one can straightforwardly see that this Sp(1) × Sp(2) singlet is the unique singlet in the entire spectrum of SO(8) representations for TT eigentensors of the Lichnerowicz operator on the round sphere.This justifies the statement made above that the tensor (3.1) is the unique singlet TT Lichnerowicz mode in the squashed S 7 background.We can construct these singlet modes by considering 3-forms in the squashed sphere metric (2.3) of the form where α and β are constants, ν i ≡ σ i − A i , and which can be solved for σ giving Normalising the squashed sphere so that its Ricci scalar is R = 42m 2 for all λ implies we should scale these eigenvalues by the factor given in (2.9), and so the eigenvalues become Thus for the σ + mode we have This 3-form mode is the singlet in the decomposition of the 840-dimensional (2, 0, 0, 2) SO( 8) representation in the round-sphere vacuum.
continuously followed from the round sphere to any of the squashed sphere family of metrics.
For the the σ − mode we have This 3-form mode is the singlet in the decomposition of the 35-dimensional (0, 0, 2, 0) SO( 8) representation.
Of course, since Q = * d is a first-order operator its eigenvalues will reverse in sign if the orientation of the manifold is reversed.

Singlet modes of the Dirac operator
Turning now to the fermionic modes, we begin with the Dirac operator acting on spinor modes.Modes of the Dirac operator can be followed continuously from the round to the squashed sphere, and so the singlet Dirac modes on the squashed sphere can be counted by counting the number of singlets in the decomposition of the SO(8) Dirac mode representations of the round sphere under the Sp(1) × Sp(2) subgroup.There is in fact a unique such singlet.These and the following features of the Dirac equation can be found in [11].
The Sp(1) × Sp(2) singlet Dirac mode can be constructed by first constructing the Killing spinor η in the Einstein-squashed S 7 .Using the orthonormal frame in eqn (2.5), and making a natural choice for the spin frame, η is a spinor with constant components, satisfying in the λ = 1 √ 5 squashed S 7 (we have rescaled the metric so that R = 42m 2 , as discussed previously).A straightforward calculation shows that the same constant-component spinor η is an eigenspinor of the Dirac operator for arbitrary values of the squashing parameter, and its eigenvalue σ, defined by i Thus in particular, on the round S 7 and on the Einstein-squashed S 7 we have This Dirac mode is the singlet in the decomposition of the 56-dimensional (1, 0, 0, 1) SO( 8) representation of Dirac modes on the round S 7 .

Singlet modes of the Rarita-Schwinger operator
Consider a Rarita-Schwinger mode obeying Multiplying by γ a , we obtain 5i We should impose the gamma-traceless condition so eqn (3.15) then implies that Now instead act on eqn (3.14) with ∇ a .Using the identity it follows after a little algebra that If the metric is Einstein, this just gives the same conclusion that we saw before, namely that as well as the gamma-traceless condition (3.16) we shall also have the transversality condition (3.17).However, if the metric is not Einstein, we obtain the independent algebraic condition5 R ab γ a ψ b = 0 . (3.20) For our present example of the squashed S 7 we have in general, in vielbein components, where α = 0, 1, 2, 3 and î = 4, 5, 6.The constants a and b depend on the squashing parameter λ, and are unequal except when λ 2 = 1 or λ 2 = 1 5 .Thus (3.19), together with the original condition γ a ψ a = 0, implies that for general squashing we must have the two independent algebraic conditions whereas when the metric is Einstein, we have only the condition γ a ψ a = 0. Evidently, then, the Rarita-Schwinger eigenvalue problem is over-constrained if the metric is not Einstein.
Since we need to restrict attention to Einstein metrics when considering the modes of the Rarita-Schwinger operator, this means that we must consider only λ = 1 (the round ).It is easily seen that the SO(8) representations for the RS modes on the round sphere do not include any irreps whose decomposition under Sp(1) × Sp(2) includes any singlets.One might be tempted to conclude that there could therefore be no singlet RS modes in the Einstein-squashed S 7 .However, unlike the situation for all the operators we discussed previously, here we have no continuous path by which we can deform the RS modes on the round sphere to the RS modes on the Einsteinsquashed sphere.In fact, as we shall now show 6 , there does exist a singlet RS mode in the Einstein-squashed S 7 .
To see this, we first note that we can construct a class of RS modes in the Einsteinsquashed S 7 as follows.Let h ab be any transverse-traceless mode of the Lichnerowicz operator: we now consider a spin-3 2 field ψ a , defined by where η is the singlet Killing spinor on the Einstein-squashed S7 , obeying eqn (3.11), and α is some constant to be chosen later. 7Given the Killing spinor equation (3.11) and the fact that h ab is transverse and traceless, it is straightforward to see that ψ a is divergence-free and gamma-traceless, where i / D 3/2 denotes the Dirac operator i γ a ∇ a when acting on vector-spinors.
Substituting the definition of ψ a in eqn (3.24) into eqn (3.26) gives equations that determine the value that must be chosen for α, as well as determining the eigenvalue σ.
After some calculation one finds that eqn (3.14 The two structures h ab γ b η and (∇ c h ab )γ cb η are in general linearly independent, and so equating the coefficients for each implies Solving, this gives Thus in general a Lichnerowicz mode h ab with eigenvalue λ L gives rise to two Rarita-Schwinger modes, with different eigenvalues σ + and σ − , corresponding to the plus or minus sign choices.
If we now apply this construction to the Lichnerowicz squashing-mode of Page, given in eqn (3.1), it turns out that for this particular case the two structures in the two terms in the definition of ψ a in eqn (3.24) are proportional to one another, with Thus in this case, we may simply choose to write the RS mode in the form The RS eigenvalue then corresponds just to the minus sign choice in eqn (3.29), implying, for the Lichnerowicz singlet mode, that This is the eigenvalue of the singlet RS mode on the Einstein-squashed S 7 .
As discussed above, because there is no way to make a continuous deformation between modes of the Rarita-Schwinger operator on the round S 7 and modes on the Einsteinsquashed S 7 , it does not necessarily have to be the case that the RS modes on the Einsteinsquashed S 7 occur in irreps of Sp(1) × Sp(2) that are in 1-1 correspondence with the decomposition of the SO(8) irreps of the round S 7 under the Sp(1) × Sp(2) subgroup.We have exhibited a concrete example of this phenomenon in the construction above.Namely, we have found that there exists a singlet RS mode on the Einstein-squashed S 7 , while on the round S 7 none of the irreps of the RS modes decomposes to include a singlet under . This is very different from the situation for the singlet Dirac mode on the Einstein-squashed S 7 (i.e. the Killing spinor η), which can be traced back to the round S 7 as the singlet in the decomposition of the 56 Dirac modes in the (1, 0, 0, 1) of SO(8) on the round sphere.Unlike the spinor η, which is responsible for the space-invader phenomenon in the squashed vacuum, the singlet RS mode on the Einstein-squashed S 7 is effectively a "mode from nowhere," which has no counterpart on the round sphere.This proves the correctness of this particular aspect of the spectrum construction in [6].
4 The Singlets in the AdS 4 Spectrum In the previous section, we constructed explicitly the complete set of singlet modes of all the mass operators that arise in the analysis of the mass spectrum of the AdS 4 × M 7 vacua of eleven-dimensional supergravity, where M 7 is either the round S 7 or the Einsteinsquashed S 7 .In this section, we shall show explicitly how these give rise to fields and, in the supersymmetric cases, supermultiplets, in the AdS 4 vacuum.
The four-dimensional fields form representations of the SO(2, 3) isometry group of the AdS 4 background metric.They are denoted by D(E 0 , s), where E 0 is the lowest energy eigenvalue and s is the total angular momentum quantum number of the lowest-energy state.The representation is unitary if E 0 ≥ s + 1 for s ≥ 1, and if E 0 ≥ s + 1 2 for s = 0 or s = 1 2 . 8We refer to [3,17] for further details.In [3], the E 0 values of the AdS 4 representations are listed for each spin: It should be noted that in general the eigenvalue spectra of the first-order mass operators (Dirac, Rarita-Schwinger and the 3-form operator Q = * d) will be different if the orientation of the compactifying seven-manifold M 7 is reversed.Specifically, the orientation reversal will reverse the signs of all the eigenvalues of the first-order operators, while the eigenvalues of the second-order mass operators are unaffected.The round S 7 is an exception to the general rule; the eigenvalues of the first-order mass operators on the round S 7 occur always in pairs, with every positive eigenvalue having an equal and opposite negative-eigenvalue partner.By contrast, the eigenvalue spectra associated with first-order operators on a squashed S 7 are asymmetric between positive and negative.This can be seen in the results of the previous section, where the totality of singlet modes in the squashed S 7 background is presented, or by consulting the more general results of [7][8][9].
Because of the asymmetry of the squashed-sphere spectra under orientation reversal, there are two inequivalent squashed S 7 compactifications of eleven-dimensional supergrav-ity, as was shown in [2,3].These are conventionally referred to as the left-squashed compactification and the right-squashed compactification.The left-squashed vacuum has N = 1 supersymmetry, while the right-squashed vacuum is non-supersymmetric, so N = 0.
For the left-squashed vacuum, we see from the results in section 3, from the mass operators in eqn (2.1) and from eqn (4.1) that singlets will give rise to the following Heidenreich-type supermultiplets: The massless supergravity supermultiplet with spin (2, 3  2 ) and two Wess-Zumino multiplets with spin content (0 + , 0 − , 1  2 ), one of which is related to singlet modes for the operators (∆ 0 , Q, i/ D 1/2 ) and the second to the modes of the operators Table 1 summarises the situation for the Sp(1) × Sp(2) singlet sectors of the round S 7 vacuum and the left-squashed S 7 vacuum.In particular, one can see where the singlet modes in the left-squashed vacuum have come from in the round-sphere vacuum.Two features in particular are noteworthy: Firstly, the massless spin- 3  2 field (the gravitino) of the left-squashed vacuum has not originated from one of the eight massless gravitini of the round-sphere vacuum; rather, it has come from the decomposition of the level n = 1 56 c of massive gravitini in the roundsphere vacuum.This is the "space-invaders" phenomenon that was first demonstrated in [2].Of course, a corollary of this is that although the truncation of the full round-sphere spectrum to the singlets under the Sp(1)×Sp(2) subgroup of SO( 8) is a perfectly consistent one, the singlet modes do not fall into supermultiplets in the round-sphere vacuum, because all eight massless gravitini in the round-sphere vacuum are projected out in this singlet truncation.
Secondly, as we showed in [6] and have been further emphasising in this paper, because the massless gravitino in the left-squashed vacuum originated from a massive gravitino in the round-sphere vacuum, there must be an "inverse Higgs", or de-Higgs, phenomenon, in which the helicity- 1  2 states in the massive gravitino in the round-vacuum are ejected and become a genuine spin- 1  2 mode in the left-squashed vacuum that was not seen as a distinct spin- 1  2 mode in the round-sphere vacuum.This is precisely what is seen in the second row of fields listed in table 1, where the Rarita-Schwinger mass operator for the 1 2 (2) modes has a singlet mode on the Einstein-squashed S 7 but it has no singlet mode on the round S 7 .  ), (0 ) ( 2) ).Note that the 1 2 (2) field occurs only in the squashed vacuum, since the Rarita-Schwinger operator / D 3/2 has no Sp(1) × Sp(2) singlet modes on the round sphere.
relevance for the issue of stability of the right-squashed non-supersymmetric S 7 compactification.This theory is well-known to be BF stable and hence, in view of the AdS stability conjecture proposed in [1], it is of some interest to see if there are other decay modes possible.One such is connected to the appearance of marginal operators in the boundary theory [18] (see also, for instance, [19]).Whether or not the modes analysed in this paper lead to AdS fields whose dual operators on the boundary can be used to construct such marginal operators depends on the boundary conditions chosen for the fields in the third column of the second row of table 2. The fields in the third row cannot give rise to any such Of the four options appearing in table 2 for the fields in the second row one can for the first case form marginal triple trace operators dual to three pseudo-scalar fields, P 3 , or to two fermions and one pseudo-scalar, P λλ.Here we denote the fields, as well as their dual single trace operators on the boundary, as (S, P, λ).The second option of boundary conditions in table 2 allows only for a triple trace operator of the kind P 3 while the remaining two options cannot be used for this purpose.
can be followed as a function of the squashing parameter this is probably not the case for the masses since their relation to the operators are not only theory dependent but also valid only in backgrounds that solve the field equations.This also presents a difficulty for following the change in E 0 as a function of the squashing parameter.
We have also discussed the relevance of our results for the issue of stability of the rightsquashed non-supersymmetric S 7 compactification.This theory is BF stable but we found in the previous section that one can form marginal operators in the boundary theory.As is clear from table 2 above, this depends on the boundary conditions that are used for some of the singlet fields after skew-whiffing, some options make it possible to construct marginal triple-trace operator.However, as mentioned at the end of the previous section, by adopting the reasoning of [6], there must exist a fermionic singleton in the right-squashed vacuum, and this excludes two of the four options on the second row of table 2. One of the two options that remain can then not lead to any marginal operators and would provide a theory that could still be completely stable.
In order to check if the marginal operators do indeed generate instabilities one has, however, to compute 1/N corrections to some beta-functions in the boundary theory, which is beyond the scope of the present paper.This would, however, be an interesting topic for a future investigation.

3. 3
Singlet modes of the Q = * d operator on 3-forms 3-form modes of the operator Q = * d exist for all values of the squashing parameter, and therefore we can again count the number of singlet modes by looking at the decomposition of the known SO(8) representations for 3-form modes on the round sphere, and counting the number of singlets in their decomposition under Sp(1) × Sp(2).There are in fact two singlets in total.

± 1 2 (
operators.Recall that if a scalar field hasM 2 in the range −m 2 ≤ M 2 ≤ 3m 2 then both signs in E 0 = 3 2 M/m) 2 +1 are compatible with unitarity, the upper sign corresponding to Dirichlet and the lower sign to Neumann boundary conditions.A similar situation arises for Dirac fields with a mass satisfying |M | ≤ m (m is assumed positive here).

Table 2
contains a complete list of all Sp(1) × Sp(2) singlets fields in SO(2, 3) irreps D(E 0 , s), grouped into N = 1 supermultiplets in the supersymmetric left-squashed vacuum.This table also presents the field content of the non-supersymmetric right-squashed vacuum.We end this section with a few comments on table 2 and then in particular on its