On the squashed seven-sphere operator spectrum

We derive major parts of the eigenvalue spectrum of the operators on the squashed seven-sphere that appear in the compactification of eleven-dimensional supergravity. These spectra determine the mass spectrum of the fields in $AdS_4$ and are important for the corresponding ${\mathcal N} =1$ supermultiplet structure. This work is a continuation of the work in [1] where the complete spectrum of irreducible isometry representations of the fields in $AdS_4$ was derived for this compactification. Some comments are also made concerning the $G_2$ holonomy and its implications on the structure of the operator equations on the squashed seven-sphere.

1 Introduction The purpose of this paper is to continue the work in [1] and derive the eigenvalue spectrum of some of the operators that determine the masses and supermultiplet structure of the entire Kaluza-Klein spectrum of the squashed seven-sphere compactification of elevendimensional supergravity. The list of previously known eigenvalue spectra, containing ∆ 0 , / D 1/2 and ∆ 1 , is in this paper extended by those of ∆ 2 , ∆ 3 and ∆ L while / D 3/2 remains to be done. Note that parts of the spectrum of the Lichnerowicz operator ∆ L are derived below but they were quoted already in [2] 1 .
With the intention to keep this paper as brief as possible we refer the reader to the review [2] for a Kaluza-Klein background on the problem and for some of the necessary details and conventions needed in the derivations below. For the full structure of irreducible isometry representations on AdS 4 stemming from the squashed seven-sphere compactification we refer to [1]. The latter paper summarises in a few pages the most relevant information from [2] in particular several tables that are spread out in various chapters of [2]. In this spirit we present below just the most crucial formulas that are needed to define the problem and derive the spectra. In the Appendix we collect a number of useful octonionic identities and other formulas, as well as some for our purposes crucial Weyl tensor calculations.
The coset description of the squashed seven-sphere is where, in order to define the denominator, Sp 2 is split into Sp A 1 × Sp B 1 and Sp B+C 1 is the diagonal subgroup of Sp B 1 and Sp C 1 . The G irreps specifying the mode functions Y of a general Fourier expansion on G/H are denoted (p, q; r) [2] and the entire Kaluza-Klein irrep spectrum is derived and tabulated in terms of cross diagrams in [1]. The present paper will fill in some gaps in our knowledge of the eigenvalue spectrum of the relevant operators on the squashed seven-sphere but some are still missing. The remaining issues that we need to address to complete the eigenvalue spectra will be elaborated upon in a forthcoming publication [3].
The interest in deriving complete spectra in various Kaluza-Klein compactifications has recently been revived due to the discovery of new powerful versions of the embedding tensor technique. However, these methods can be applied directly only when the vacuum is an extremum of a maximal gauged supergravity theory in AdS 4 which can be lifted to ten or eleven dimensions, see for instance [4,5] and references therein 2 . The point we want to emphasise here is that, due to the space invaders scenario [2,6], the squashed seven-sphere solution is not of this kind and it therefore seems clear that other methods are required to obtain the AdS 4 spectrum in this case.
There are also potential applications of this work in the context of the swampland program 3 , see for instance [7][8][9] and references therein. This particular connection will be addressed elsewhere.
In the next section we very briefly review the background of the problem and the method that is used in this paper to derive the eigenvalue spectrum of operators on coset manifolds. The method is explained in many places, e.g. [2,10], and in Section 3 we first apply it to obtain the spectra of the spin 0 and 1 operators giving straightforwardly the well known results cited in [2]. Already when applied to the spin 1/2 Dirac operator complications arise which get further pronounced when we subsequently turn to more and more complicated operators. A summary of the obtained eigenvalues is provided in the Conclusions, together with comments on some of the remaining issues. Some technical aspects needed in the derivations below are explained in the Appendix.

Compactification on the seven-sphere
The Fourier expansion technique that we will apply is, following the general strategy explained in [10] and summarised in [2], based on the coset master equation for the Spin(n) covariant derivative on a n-dimensional coset space G/H: (2.1) The mode functions Y have two suppressed indices: One corresponding to the Spin(n) tangent space irrep of the field that is being Fourier expanded 4 and one that specifies the mode, that is a G irrep. This equation is derived in [2] from group elements of the isometry group G. In the conventions used there the Lie algebra of the group G has generators T A , satisfying [T A , T B ] = f AB C T C , which are divided into Tā of the subgroup H and T a in the complement of H in G. Thus, if the Lie algebra of G is reductive it splits as follows: The indices a, b, .. are vector indices in the tangent space of G/H and ∇ a in (2.1) is an Spin(n) covariant derivative with a torsion free spin connection while the Σ ab are 2 We are grateful to Oscar Varela for a clarification on this issue. 3 We are grateful to M.J. Duff for raising some interesting aspects of this question that eventually led to this work. 4 Note, however, that the whole matrix of the G group element is involved in this equation. For symmetric coset spaces the coset master equations thus reads ∇ a Y = −T a Y and the eigenvalue spectrum for the operators on G/H are rather easily derived. The complications arising in the squashed seven-sphere case therefore stem entirely from the structure constant dependent term in (2.1). As first shown in [11] for the squashed seven-sphere they are given by the structure constants of the octonions a abc . In the normalisation used in [2] they read Since octonions will play a key role in the rest of this paper we have listed some useful octonionic identities in the Appendix. Furthermore, in the last expression for these structure constants we have introduced the scale parameter m arising from the ansatz F µνρσ = 3mǫ µνρσ in the compactification of eleven-dimensional supergravity. This is useful since we are dealing with dimensionful quantities, like the covariant derivatives and Hodge-de Rham operators. The conventions used in [2] correspond to setting We will assume m > 0 and also use m = 3 2 √ 5 as for f abc above. The squashed seven-sphere discussed in this work has the orientation that gives rise to N = 1 supersymmetry in AdS 4 after compactification. This fact will be used below when we discuss the corresponding supermultiplets and the SO(3, 2) irreps entering these multiplets.
With this preparation we can readily attack the eigenvalue problems ∆ p Y p = κ 2 p Y p where the Hodge-de Rham 5 operator on p-forms is defined as ∆ p = dδ + δd. Here d is the exterior derivative and δ = (−1) p ⋆ d⋆ its adjoint. These operators act on forms according to where brackets here and in the following are weighted antisymmetrisations, Y [a 1 ...ap] = 1 p! (Y a 1 ...ap + permutations with sign). The explicit form of ∆ p can for all operators considered in this paper be expressed using the Riemann tensor, R abcd , and the d'Alembertian ≡ ∇ a ∇ a . On the squashed seven-sphere, which is an Einstein space with R = 42m 2 , where W abcd is the Weyl tensor given in the Appendix. The main use of the coset master equation (2.1) will be to replace derivatives by group-theoretical algebraic data. In particular, for the squashed sphere, by squaring (2.1) we obtain an expression for which can then be used to replace by algebraic data plus terms linear in ∇ a . This will be clear below.

Eigenvalue spectra on the squashed seven-sphere
In this section we start by briefly reviewing some of the results previously obtained with the coset space techniques mentioned above. These results are then generalised to the more complicated operators for which we present a number of new eigenvalues.
3.1 Review of the method applied to forms of rank 0 and 1 The method outlined above becomes rather trivial when applied to 0-forms. In this case we want to solve the eigenvalue equation where the positive semi-definite Hodge-de Rham operator on 0-forms is ∆ 0 = − . Thus, following [2], the spectrum is obtained directly by squaring the coset master equation (2.1): where the two second order Casimir operators for and Here (p, q) are Dynkin labels for Sp 2 irreps and r, s, t those for the irreps of the various Sp 1 groups that occur in the squashed seven-sphere coset description. The whole problem of deriving the eigenvalue spectrum of harmonics is reduced to determining first which H irreps are contained in the Spin(7) irrep of Y (trivial for 0forms) and then finding all G irreps that when split into H irreps contain any of the H irreps found in the Spin(7) irrep of Y. Some partial results on the spectrum of G irreps are obtained in [2,12] by breaking up the spectrum on the round sphere according to Spin(8) → Sp 2 × Sp 1 while the entire spectrum of all operators is derived directly from the squashed sphere coset in [1] using the coset method just described.
For 0-forms C H = 0 and the spectrum therefore becomes κ 2 0 = C G . However, as explained above, we should introduce the scale parameter m. In view of this we can use m 2 = 9 20 to write the eigenvalues as follows: 6 As an example how this result is used let us consider the spectrum in the graviton sector 7 : In units of m the SO(3, 2) irrep defining energy E 0 (spin) is [14][15][16] E 0 (2 + ) =  6 We will in the rest of this paper display eigenvalues either as κ 2 p or, equivalently, as ∆p. 7 As in [1] we will use the notation of [13] which includes the parity of the field, as in, e.g., 2 + .
where we have used the fact that the mass-square operator for spin 2 fields in AdS 4 is M 2 2 = ∆ 0 . We will occasionally refer to results like this as being of square-root form, here a √ 81-form. The reason for this is that all the fields in a supermultiplet must be of the same square-root form.
We now turn to the 1-form case and review the result and the derivation in [2] 8 . In this case the structure constant term in the coset master equation comes in and complicates the calculation somewhat. As for the 0-form modes we write out the ∆ 1 eigenvalue equation explicitly: We want to use the square of the coset master equation to eliminate the term: Inserting Using the octonionic identities ∇ a a bcd = mc abcd and a abe a cd e = 2δ cd ab + c ab cd (see the Ap- Here we have also used the fact that the irrep 7 of SO(7) splits into (1, 1) and that C H = 12 5 for both of these H irreps. So eliminating the term from the ∆ 1 and G/H equations above gives In order to extract the eigenvalues κ 2 1 from this equation we will have to square it. Let us define the operator DY a ≡ a abc ∇ b Y c . Taking the square of D then goes as follows: (3.12) Using identities from the Appendix this equation becomes Here we have also used the Ricci identity on 1-forms [∇ a , ∇ b ]Y c = R abc d Y d and the fact that the Riemann tensor term in the computation of D 2 above drops due to its contraction with c eabc . Then since the last term in the equation for D 2 Y a above is just ∆ 1 , the equation can be written If we now use DY a = 3 2m (κ 2 1 − C G )Y a and D 2 Y a = ( 3 2m (κ 2 1 − C G )) 2 Y a we get the final result for the 1-form eigenvalues The last form of the answer will be useful later. 8 For a different derivation see [17].

Spin 1/2 by squaring
Before turning our attention to forms of rank two and three, and after that second rank symmetric tensors, we will check that our methods are able to produce the known result for the Dirac operator acting on spin 1/2 modes. The spectrum of / D 1/2 ≡ −i / ∇ was derived in [12] by a different method based on the fibre bundle description of the squashed sevensphere. The virtue of the method in [12] is that one can follow the eigenvalues as one turns the squashing parameter from the round sphere value to the squashed Einstein space value. This nice feature is unfortunately lacking for the methods used in this paper.
To apply our present techniques we start by squaring the Dirac operator in order to obtain a situation that is similar to the one for the Hodge-de Rham operators on p-forms.
It is now possible to use the G 2 structure to split the tangent space spinor irrep into G 2 irreps as 8 → 7 ⊕ 1, or in terms of indices A = (a, 8). Then by defining two spinors η and η a we can expand a general Dirac spinor as follows where V a is a vector field and f a scalar field on the seven-sphere. While η is the standard Killing spinor with components η B = δ 8 B , η a is defined by η a ≡ −iΓ a η which implies the its explicit form is (η a ) B = δ aB . These spinors are linearly independent and satisfy from which we obtain the equations Consider now the coset master equation for Dirac spinor modes. Squaring it gives So eliminating the term and inserting R = 42m 2 we find that the equation we need to solve reads In order to get the last term on the LHS in a nice form we introduce the projection operators for the G 2 split ψ = ψ 1 + ψ 7 corresponding to 8 → 1 ⊕ 7. They are (see Appendix for more details) In terms of these projectors the sum of the last two terms on the LHS above becomes Then using the octonionic version of the seven-dimensional gamma matrices given in the Appendix, some algebra gives Inserting these results into (3.23) we get an equation that can be hit by the projectors P s 1 and P s 7 leading to the two equations and To solve these equations we either have f = 0 or f = 0. In the former case the first equation gives ∇ a V a = 0 which implies that the second equation for V a can be analysed in exactly the same way as for the 1-forms discussed previously. Not surprisingly the result is also of √ 49-form and reads Turning to the latter case, that is f = 0, we start by taking the divergence of the second equation and use the fact that f = − 20m 2 9 C G f to find Now the system of two equations for the functions f and ∇ a V a has the solutions These two sets of eigenvalues are consistent with the known result from [12]. However, in that paper there is no sign ambiguity since there one obtains λ instead of λ 2 . This problem is easily eliminated by applying our present result to the spin 2 + supermultiplet.
This completes the review. We now turn to 2-forms where the results are new.

2-forms
In terms of the Hodge-de Rham operator on 2-forms, the equation to be solved is Using (2.6) for the Riemann tensor gives the following equation for the box operator As for the previous cases the next step is to express the box operator in terms of algebraic objects on the coset manifold. The master coset formula (2.1) for 2-forms reads This derivative is "G 2 " in the sense that it satisfies∇ a a bcd = 0 and∇ a c bcde = 0. This step is not crucial here but we will find more implications later of the presence of the G 2 holonomy so we will have reason to return to this derivative then. (For more details about G 2 in this context, see for instance [18].) To get an algebraic expression for the box operator we can now square the master coset equation, This equation can also be written as Finally, to get the equation that needs to be solved to find the eigenvalues we just eliminate the box operator from (3.33) and (3.37). Using the projectors defined in the Appendix gives the following useful form of the resulting equation which, again, is not entirely algebraic due to the appearance of the operator∇ a in the last term. This also happened in the case of the 1-form where it was easy to handle by a squaring procedure. This trick is a bit harder to apply in the case of 2-forms (as well as for the other operators to be discussed later) as will become clear shortly. Compared to the 1-form case there is also a new feature here namely the presence of the Weyl tensor in one of the terms which will cause additional complications. The approach used in this paper to deal with the Weyl tensor terms, in this and the other cases discussed below, is explained in the Appendix. Thus there are two new issues when trying to solve the 2-form equation (3.38): The Weyl tensor term and the∇ term which now tends to lead to symmetrised derivatives when squared. To deal with the former issue we recall how T a T a can be expressed in terms of the Casimir operators for the groups involved: T a T a = −(C G − C H ). This, however, has implications for the how the spectrum is organised in terms of towers. To see this we use the tangent space decomposition SO (7 which makes it possible to read off the relevant decompositions directly from the McKay and Patera tables [19]. In the case of the 2-form the composition reads, see [2] or the summary in [1], This means that the towers will be tabulated according to their H irrep which will thus obscure the G 2 structure of the spectrum. As is explained in the Appendix this problem is automatically eliminated once the Weyl tensor term is analysed and the result combined with the result from the C H term. In fact, adding these two terms gives 24 5 Y ab = 32 3 m 2 Y ab for all the H irreps in the G 2 irrep 14. Recall that the corresponding answer for the irrep 7 is 12 5 = 16 3 m 2 . Using this insight from the Appendix, namely that the sum C H + W eyl has a common value on all H irreps in the decomposition of each G 2 irrep arising from the SO(7) representation 21, it becomes possible to split the 2-form equation into two by acting on it with the projectors P 7 and P 14 and insert the respective values of C H + W eyl. We find, using the definitions Y Note that the last term in the second equation contains Y a and thus seems to mix with the first equation. However, the structure of the derivative terms as a 2 by 2 matrix shows that this term will have no role in determining the eigenvalues for the 2-form as will be clear below. To proceed we write these two equations as Consider first the possibility to take the square of the first equation above, the one for Y a . However, although this will give rise to a calculation quite similar to the one for the 1-form, there is one important difference. While the 1-form is transverse (i.e., divergence free) this is not the case for Y a coming from the 2-form. Therefore one needs to analyse the two equations for the 2-form in stages: So, when Y a = 0 the equation left to solve is the 14-equation with the derivative term set to zero. This equation therefore gives the eigenvalues According to the characterisation mention above, this result leads to an energy E 0 of the √ 9-form. This fits nicely with a supermultiplet containing spins (1 + , 1/2, 1/2, 0 + ) with masses related to the operators (∆ 2 , / D 3/2 , / D 3/2 , ∆ L ) since we know from Ref. [198] in [2] that ∆ L also has an eigenvalue leading to a √ 9-form. One can check that the corresponding values of E 0 work out as they should in relation to supersymmetry. This eigenvalue of ∆ L leading to the √ 9-form will be derived in detail later in this section.
We now turn to the cases with Y a = 0, namely ∇ a Y a = 0 and ∇ a Y a = 0. Clearly, when ∇ a Y a = 0 the calculation will follow the one for 1-forms very closely. Indeed, the result is also of √ 49-form and reads Although being of √ 49-form, this 2-form eigenvalue formula differs from the one for 1-forms given in [2], and also derived above, The differences are partly compensated for by the different relations between the Hodge-de Rham operators and the respective M 2 operators [2]: If the ∆ 1 eigenvalues are inserted into this formula for M 2 (1 − ) we see that the last term, that is −4m 2 in ∆ 1 , is really required to have a chance for supersymmetry to work in a supermultiplet containing both 1 + and 1 − fields. In fact, the field content of the spin 3/2 supermultiplets has this property and requires E 0 (1 + ) = E 0 (1 − )± 1. To verify this relation we need the energy formula for spin 1 unitary SO(3, 2) irreps This expression gives the energy values where the first ± sign refers to the two towers of spin 1 −(1), (2) fields in AdS 4 supergravity theory. Comparing these to one finds (using also the spectrum of the spin 3/2 fields in these supermultiplets) that it is possible to eliminate the sign ambiguities in the spectra of ∆ 1 and ∆ 2 . For more details see [3]. Finally, when ∇ a Y a = 0 we take the divergence of the 7 projected part of the equation and use a abc ∇ a ∇ b Y c = 0 to obtain a simple equation giving the eigenvalue This completes the analysis of the 2-form eigenvalue spectrum. A list of the obtained eigenvalues can be found in the Conclusions.

3-forms
We now turn to 3-forms leaving the discussion of the Lichnerowicz modes to the next subsection. The reason for doing the analysis in this order is that we will obtain some equations in the 3-form case that can be applied also for Lichnerowicz. The equation for 3-forms that we wish to solve reads Inserting the Riemann and Ricci tensors of the squashed seven-sphere this equation becomes Squaring the coset master equation for 3-forms gives where for later convenience we have used the G 2 derivativẽ Note that this implies that the divergencẽ Eliminating the box-operator term from the above equations gives (3.58) To proceed we decompose the 3-form into G 2 irreps as 35 = 1 ⊕ 7 ⊕ 27. By defining Y ≡ a abc Y abc and Y a ≡ c abcd Y bcd we can split the 3-form as follows Apart from this there are two more problematic terms, the c abcd -term and the derivative term that will force us to square also this equation. However, before doing that we will follow the strategy in the 2-form case and start by splitting the equation into G 2 pieces. The singlet term 1 is obtained by contracting all three indices by a abc which gives Cleaning up the structure constant factors this equation becomes Turning to the 7 part, we find after some algebra When we now come to the last part, the 27, it will require some new steps that will relate it to the metric and the Lichnerowicz equation. This connection is in fact already indicated by the last term in the 7 equation just discussed: cda . This expression suggests that the 2-index tensor a acd Y (27) bcd will be useful as we now elaborate upon (see [18] for a closely related discussion). Let us define a 2-index tensor from the 27 part of the 3-form byỸ where we use a tilde to avoid confusing it with the antisymmetric 2-form discussed previously. Clearly, the symmetric and traceless part ofỸ ab is also in the irrep 27. However, a nice and indeed very useful fact about this definition ofỸ ab is that it is automatically symmetric and traceless. The tracelessness follows immediately from the identity a abc c abcd = 0, or, using the 3-form projectors in the Appendix, from P 1 P 27 = 0, while the vanishing of its antisymmetric part follows from a abcỸbc = 0 and c abcdỸcd = 0 9 . These two results can be checked as follows: a abcỸbc = a abc a bde Y Clearly, also a abc (P 27 ) abc ef g = 0 again showing that the trace ofỸ ab vanishes.
An alternative, and perhaps more direct, way to see thatỸ ab is symmetric and traceless is to just insert the expression for the P 27 projector which gives  9 Note that these two conditions can be combined to the identity since δ cd ab = 1 2 (a abe a cde − c ab cd ). 10 Here one may use the fact that Y The decomposition of the 3-form therefore takes the following simple form in terms of Y, Y a ,Ỹ ab , respectively in irreps 1, 7, 27, With these preliminary results at hand it is now a rather straightforward matter to project the equation (3.58) onto its 27 component. One finds (3.71) Again we can replace the second bracket by its common eigenvalue on all H irreps in the G 2 irrep 27 as explained in the Appendix, that is with 28 5 = 112m 2 9 . We now insert this into the 27 equation and sum up what we have found so far: cda appearing in the 7 equation can be replaced by terms containing only Y and Y a . This relation is derived from the gauge condition ∇ a Y abc = 0 as follows: Contracting this equation with a dbc and using ∇ a a bcd = mc abcd we get However, since c abcd projects onto Y a we can use the relations 77) to rewrite the above 3-form gauge condition as This equation will be used when we summarise the system of equations to be solved below. Finally, we show that the last bracket in the 27 equation will not play a role in the analysis of the spectrum. This can be seen by writing the three equations above in matrix form as The eigenvalues must be the roots of the equation det(X − κ 2 3 1) = 0, where X is the matrix appearing above. Since the element E does not enter this equation we can proceed and solve the three equations in two steps, first the two coupled equations for Y and Y a 1 : and then the single equation involving onlyỸ ab 27 : We start by solving the first two coupled equations. There are two distinct cases: either 1) Y = 0 or 2) Y = 0. Case 1): Setting Y = 0 in the 1 equation gives ∇ a Y a = 0 which implies that the 7 equation is of exactly of the same type as the equation solved previously for 1-forms, but with different coefficients. However, one has to pay attention to the fact that the eigenvalues dealt with here are for 3-forms, not 1-forms, so solving the present equation for Y a involves some new steps. This fact will become clear directly when squaring the operator (DY ) a ≡ a abc ∇ b Y c : Since the c-term vanishes this equation simplifies to At this point the Y a term must be related to the 3-form eigenvalue κ 2 3 . Using Y = 0 and ∇ a Y abc = 0, we find that where we have also used c abcd W e Inserting the expression for (DY ) a from the eigenvalue equation 7 above we find which has the following two solutions (inserting 1 = 20 9 m 2 in front of C G ) These eigenvalues are obtained for ∆ 3 = Q 2 so it is gratifying to find its eigenvalues come out as a square. Case 2): Now we turn to the second case where Y = 0. Then either ∇ a Y a = 0 or ∇ a Y a = 0. The former case gives a simple equation for the eigenvalue directly from the 1 equation. However, taking the divergence of the 7 equation tells us that Y = 0 so the 1 equation reduces to which is the eigenvalue of the singlet constant mode Y abc = a abc as is easily verified. So let us turn to the latter case with ∇ a Y a = 0. Taking the divergence of the 7 equation gives Inserting this equation back into the 1 equation, recalling that Y = −C G Y , gives This equation has the following solutions (inserting again 1 = 20 9 m 2 in front of C G ): Note that the single eigenvalue found above, κ 2 3 = 16m 2 , belongs to the plus branch. We now turn to the 27 part of the 3-form equation Clearly, the last two terms in this expression contain a symmetric combination of the two G 2 covariant derivatives. Such terms can be a real obstacle to carrying through the calculation but we will see below that there is a trick that can be used to eliminate this issue. Before applying this trick we simplify the other terms to get We now address the four terms on the RHS of this equation starting with last one. The trick used to deal with the symmetric combination of derivatives in this term is to reintroduce the 3-form via Y The first term in the last expression is just the G 2 covariant divergence of the 3-form which can be seen to satisfy∇ a Y where in the first equality we have used the fact thatỸ ab is symmetric and in the second that ∇ a Y (1) abc = 0. Thus ∇ a Y abc = 0 is implies∇ aỸ ab = 0 which gives the following much simpler form of the expression above a e(a c a |f |b) (3.99) Inserting this result into the above expression for (DDỸ ) ab we get The other two terms both involve the commutator of two covariant derivatives (sincẽ ∇ aỸ ab = 0). So we need to compute Thus, all single derivative terms cancel except one. Noting also that the two non-derivative terms in the middle of the bracket cancel the commutator becomes Simplifying the non-derivative terms finally gives Since we are interested in expressing the right hand side in terms of the D operator defined above using∇ a we rewrite the last equation as To get the final expression that will be useful here we replace the commutator on the right hand side by the Riemann tensor. This gives The second term in the DDỸ we need is the one with a contraction of the commutator with the c symbol: Inserting the two results in (3.107) and (3.108) into (3.100) we find Replacing the G 2 covariant box with Casimirs gives Then from the Appendix we know that the last bracket gives 112m 2 9 which implies the amazingly simple equation In view of the eigenvalue equation (3.93), we may express the "solution" to the last equation as Then by replacing (DỸ ) ab with the expression coming from the 3-form (3.93), we find Since κ 2 3 is the eigenvalue of ∆ 3 = Q 2 this must be a square. Indeed, it can be written (3.114)

Lichnerowicz
When we now turn to the transverse and traceless metric modes h ab on the squashed sevensphere we can take advantage of the results obtained in the previous case of the 3-form.
To see how to do this let us write out the Lichnerowicz equation explicitly The coset master equation reads in this case which when squared gives Eliminating the box operator from the above ∆ L and G/H equations and using T c T c = −(C H − C H ) gives where we have used the G 2 covariant derivativẽ As in the previous cases we now use the result for the Weyl tensor term from the Appendix, i.e., that C H + 2W acting on the different H irreps in 27 gives the universal value 28 5 = 112 5 m 2 . This gives us the rather simple equation There is one crucial difference between the Lichnerowicz modes and the 3-form modes analysed in the last subsection: Transversality of the metric modes does not imply that the associated 3-form Y abc To deal with the last term we apply the D operator yet another time. To simplify the computation we introduce the notation Y ab ≡∇ e Y eab = ∇ e Y eab where ∇ a Y ab = 0 and define (DY ) ab ≡ 2a cd(a∇ c Y |d|b) = 2a cd(a ∇ c Y |d|b) . It is then immediately found that Expanding out the nontrivial last term on the right side gives We start by analysing the first term in (3.123). When the first covariant derivative passes a ef d we need to compute In the second equality we have used the antisymmetry of Y ab and the fact that the 2-form projector P 7 vanishes when acting on the Weyl tensor (which implies that c ab cd ∝ δ cd ab ). The last equality is true due to the antisymmetry of Y ab together with the symmetrisation (ab). When the covariant derivative acts on a ef g we find We then turn to the second term in (3.123). Here the covariant derivative can act in two different ways. First, moving the derivative past a ef b gives the expression The action of the antisymmetrized covariant derivatives will give two terms which are essentially identical and we only show how to deal with the first one. Using the squashed sphere Riemann tensor it follows that where we have used that a ef b W ce f g = − 1 2 a ef b W cg ef = 0 (see the Appendix). The other term vanishes in exactly the same way. Finally, we have a term coming from the covariant derivative hitting a ef b . After some algebra, this term becomes and so a cd (a a ef b) ∇ c ∇ e Y f d = 2m(DY ) ab . Putting these results together we find that Having calculated the action of D on (DY ) ab we can apply D on (3.121) and then subtracting the previous equation to eliminate the (DY ) ab terms. From this procedure we find the third order equation "Solving" for (Dh) ab and plugging the result back into (3.120) gives the following three different eigenvalues:

Conclusions
Let us summarise what we know so far about the spectrum of operators on the squashed seven-sphere including the new results for ∆ 2 and ∆ 3 obtained in this paper. The previously known eigenvalues for ∆ 0 , / D 1/2 [12], ∆ 1 [17], and ∆ L [2] are while the new ones obtained in this paper are Here it is appropriate to make some comments on the limitations of the obtained results. First, we have so far no results for the eigenvalues of the spin 3/2 operator / D 3/2 , although some can easily be extracted from supersymmetry. Secondly, for ∆ 2 and ∆ L we seem to lack some eigenvalues. This is indicated by the degeneracies in the cross diagrams for these two operators derived in [1], as well as by supersymmetry. For ∆ 3 there may already be too many available eigenvalues but if one is supposed to pick only one sign when removing the square (as done for / D 1/2 ) one is instead lacking two eigenvalues. These problematic features are partly due to the fact that although we have extensive knowledge of the eigenvalue spectra from the list above, the method applied here does not provide direct information how to associate these eigenvalues with the cross diagrams of [1]. These and other issues will be elaborated upon in a forthcoming publication [3].
The results of this paper clearly demonstrate the important role of weak G 2 holonomy when solving the eigenvalue equations. There are, however, deeper issues in the context of holonomy and string/M theory that might be interesting to study in relation to the squashed seven-sphere, for instance the notion of generalised holonomy discussed in [20].
project. We are also grateful to Joel Karlsson for some useful comments on the manuscript. The work of S.E. is partially supported by the Knut and Alice Wallenberg Foundation under the grant: "Exact Results in Gauge and String Theories", Dnr KAW 2015.0083.
Sometimes it is convenient to write the gamma matrices Γ a in seven dimensions in terms of the structure constants defined above and a Killing spinor η satisfying Note that all seven gamma matrices are antisymmetric and imaginary. With these definitions one can check thatη Γ abc η = −ia abc as stipulated above. However, this calculation requires some octonionic structure constant identities that are seldom given in the literature. For the convenience of the reader we list the identities used in this paper below: a abe a cde = 2δ cd ab + c ab cd , (A.9) c abc g c def g + a abc a def = 6δ def abc + 9δ These identities may be verified directly by inserting the actual values of the constants, or by using the above expressions in terms of the Killing spinor η and the gamma matrices. The latter approach requires the Fierz formula Γ a ηηΓ a = 1 − ηη. (A.12) Various contractions of indices in these identities lead directly to a number of additional identities that are used frequently in the main text. A quite extensive list of identities is presented in [18].
Using the Killing spinor equation we furthermore find the derivative identities The purpose of the projection operators in this case is to split the SO(7) spinor representation into two G 2 irreps as follows 8 → 1 ⊕ 7: While they trivially sum to the unit matrix, the fact that they both square to themselves and are orthogonal to each other require some algebra to show. In fact, all these properties follow directly if we show that (P s 1 ) 2 = P s 1 . This is done as follows The key calculation in the last step is to expand Γ abcd Γ ef gh in the gamma basis Γ (n) , n = 0, ...., 7, in seven dimensions: However, when contracted with c abcd c ef gh the first and third terms vanish since they are antisymmetric under interchange of the two index sets abcd and ef gh. The remaining two terms are easily computed using the two identities for c abcd : c abcd c ef cd = 8δ ab ef + 2c ab ef and c abcd c abcd = 168. (P s 1 ) 2 = P s 1 then follows directly.

C The role of the Weyl tensor
The Weyl tensor of the squashed seven-sphere is given in [2]. The explicit expression given there can be written in a more compact form in terms of 't Hooft symbols. By setting α = (0, 1, 2, 3) = (0, i) and introducing the 't Hooft symbols by the Weyl tensor can be explicitly expressed as follows The demonstration that the squashed seven-sphere has holonomy G 2 relies on the fact that defining W ab ≡ 1 4 W ab cd Γ cd we find W 0i = 1 5 (Γ 0i + 1 2 ǫ ijk Γĵk), (C.4) W ij = 1 5 (Γ ij + Γîĵ), (C.5) while the remaining components are not independent but instead given by Thus only 14 components are linearly independent which was shown in [2] to result in G 2 holonomy. An immediate consequence, as can be easily verified using the explicit expressions, is that a abc W bc de = 0 . (C.10) To obtain identities of this kind it is convenient to express the octonionic structure constants in terms of 't Hooft symbols. The non-zero components are then (recall that a = (α,î) and i = (1,2,3) = (4, 5, 6)): a abc : a αβk = −η k αβ , aîĵk = ǫ ijk , (C.11) c abcd : c αβγδ = ǫ αβγδ =η k where ǫ αβγδ is antisymmetric with ǫ 0123 = 1.
In our present context we will for instance use these results to express the 2-form modes Y (14) ab in a way that makes its projections onto the H irreps clear, i.e., the decomposition To this end we will parametrise the Spin(4) × SU (2) subgroup of G = Sp 2 × Sp C 1 using Γ a = (Γ α , Γî) : where σ α ≡ (−i1, σ i ),σ α ≡ (i1, σ i ). (C.15) Note the dot notation on the indices in the second tensor factor. This reflects the fact that we are using the diagonal subgroup of Sp B 1 and Sp C 1 where the former is a subgroup of Sp 2 when split into Sp A 1 × Sp B 1 according to the first tensor factor. The 't Hooft symbols (η m αβ ,η m αβ ) introduced above can also be defined from the Pauli matrices by σ αβ ≡ iη m αβ σ m ,σ αβ ≡ iη m αβ σ m . (C.16)