Kaluza-Klein Spectrometry from Exceptional Field Theory
aa r X i v : . [ h e p - t h ] N ov HU-EP-20/24
Kaluza-Klein Spectrometryfrom Exceptional Field Theory
Emanuel Malek a , Henning Samtleben b a Institut f¨ur Physik, Humboldt-Universit¨at zu Berlin,IRIS Geb¨aude, Zum Großen Windkanal 6, 12489 Berlin, Germany b Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS,Laboratoire de Physique, F-69342 Lyon, France
Abstract
Exceptional field theories yield duality-covariant formulations of higher-dimensional su-pergravity. They have proven to be an efficient tool for the construction of consistenttruncations around various background geometries. In this paper, we demonstrate howthe formalism can moreover be turned into a powerful tool for computing the Kaluza-Klein mass spectra around these backgrounds. Most of these geometries have little to noremaining symmetries and their spectra are accessible to standard methods only in selectedsubsectors. The present formalism not only grants access to the full Kaluza-Klein spectrabut also provides the scheme to identify the resulting mass eigenstates in higher dimensions.As a first illustration, we rederive in compact form the mass spectrum of IIB supergravityon S . We further discuss the application of our formalism to determine the mass spectraof higher Kaluza-Klein multiplets around the warped geometries corresponding to someprominent N = 2 and N = 0 AdS vacua in maximal supergravity. [email protected] [email protected] ontents ExFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 E
ExFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Generalised Scherk-Schwarz reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.1 Truncation Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.2 Generalised Leibniz parallelisability . . . . . . . . . . . . . . . . . . . . . . . . 8 , N = 8, SO(6) vacuum: IIB on S . . . . . . . . . . . . . . . . . . . . . . 235.1.2 AdS , N = 2, U(2) vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.2 Vacua of 4-dimensional SO(8) gauged SUGRA . . . . . . . . . . . . . . . . . . . . . . 335.2.1 AdS , N = 2, U(3) vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.2.2 AdS , N = 0, SO(4) vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Whenever a higher-dimensional theory is compactified, towers of infinitely many massive fields arisein the lower-dimensional theory. These Kaluza-Klein towers are the lower-dimensional signature ofthe compactification space and often play a crucial role in the compactified theory. For example, inphenomenological models arising out of string theory, these Kaluza-Klein towers would correspondto massive particles but may also indicate potential instabilities of the background. On the otherhand, in the AdS/CFT correspondence, the masses of the Kaluza-Klein towers are mapped to theconformal dimensions of operators in strongly-coupled CFTs, that cannot be computed directly exceptfor protected operators. Despite the universality and importance of Kaluza-Klein towers, calculatingtheir masses is an exceedingly difficult undertaking. Indeed, obtaining the Kaluza-Klein spectrum1f supergravity compactifications has hitherto only been possible for coset spaces, while on generalbackgrounds this has only been achieved for the spin-2 towers.This paper is a detailed account of the results of [1]. There we announced a new method basedon Exceptional Field Theory (ExFT), which allows us to compute the full Kaluza-Klein spectrumfor any vacuum of a maximal gauged supergravity arising from a consistent truncation of 10- or11-dimensional supergravity. This includes vacua with few or no (super-)symmetries, whose Kaluza-Klein spectra were previously inaccessible. ExFT is a duality-covariant reformulation of maximal10-/11-dimensional supergravity, which unifies fluxes and gravitational degrees of freedom. Since theKaluza-Klein fluctuations mix between the flux and gravitational sectors of supergravity, this makesExFT a natural formulation within which to study this problem.Indeed, as we develop here, we can build on the efficient ExFT description of consistent truncationsto maximal gauged supergravity [2–7] to obtain a remarkably simple expression for the Kaluza-Kleinfluctuations around any vacuum of the lower-dimensional gauged supergravity. The fluctuation Ansatztakes the form of the lower-dimensional supergravity multiplet, making up the consistent truncation,tensored with the scalar harmonics of the maximally symmetric point of the lower-dimensional su-pergravity. As a result, the Ansatz is non-linear in the fields of the lower-dimensional supergravitymultiplet. Due to this non-linearity, it is straightforward to compute the Kaluza-Klein spectrum for any vacuum of the lower-dimensional supergravity arising from the consistent truncation.There are several benefits to our approach: • The fluctuations of all supergravity fields are parametrised in terms of a common set of “scalarharmonics”. In contrast, in the traditional approach, fields in different Lorentz representationsrequire different harmonics. • The scalar harmonics are computed at the maximally symmetric point of the lower-dimensionalsupergravity, even if we are interested in another vacuum of the lower-dimensional supergravitywith a much smaller symmetry group. • As a consequence, we can, for the first time, compute the Kaluza-Klein spectrum around vacuawith few or no (super-)symmetries, including non-supersymmetric vacua, such as the prominentnon-supersymmetric SO(3) × SO(3) AdS vacuum of 11-dimensional supergravity [8]. • The states of every BPS multiplet live in the same Kaluza-Klein level, making the identificationof supermultiplets in 10/11 dimensions considerably easier than using the traditional approach,where BPS multiplets are scattered amongst different Kaluza-Klein levels. • Using the dictionary between the ExFT and the original supergravity variables, it is straight-forward to identify the higher-dimensional origin of the resulting mass eigenstates.The paper is structured as follows. We begin with a review of the relevant aspects of ExFT insection 2. In section 3, we then describe how to efficiently parametrise the Kaluza-Klein fluctuationsin ExFT. In section 4, we show how this leads to compact expressions for the mass matrices of theKaluza-Klein towers, including the vector and the scalar fields. We next demonstrate the power of theformalism by applying it to several prominent AdS vacua of 10- and 11-dimensional supergravity in2ection 5. In particular, we show how our approach leads to a very efficient computation of the Kaluza-Klein spectrum of AdS × S and the identification of the mass eigenstates within IIB supergravity.We then elaborate on the results announced in [1], by • computing the spectrum of the first Kaluza-Klein level above the N = 8 supergravity of theSU(2) × U(1)-invariant AdS vacuum of IIB supergravity [9] that is dual to the Leigh-StrasslerCFT [10], • giving the full bosonic Kaluza-Klein spectrum of the SU(3) × U(1)-invariant AdS vacuum of11-dimensional supergravity [11], dual to a quadratic deformation of ABJM, • reviewing the computation of [12] of the Kaluza-Klein spectrum of the non-supersymmetricSO(3) × SO(3)-invariant AdS vacuum of 11-dimensional supergravity, and the appearance oftachyonic scalars at higher Kaluza-Klein levels.Finally, we conclude with a summary of our results and outlook on further problems to be tackled insection 6. In this section, we briefly review the structure of the relevant exceptional field theories (ExFTs), basedon the exceptional groups E and E , respectively. We refer to [13–16] for further details. Theseare the duality-covariant formulations of maximal supergravity in ten and eleven dimensions, tailoredto describe compactifications to D = 5 and D = 4 dimensions, respectively. ExFT
The Lagrangian of E exceptional field theory (ExFT) is modelled after maximal five-dimensionalsupergravity [17, 18]. Its bosonic field content is given by (cid:8) g µν , M MN , A µM , B µν M (cid:9) , µ, ν = 0 , . . . , , M = 1 , . . . , , (2.1)and combines a 5 × g µν with an ‘internal’ 27 ×
27 generalised metric M MN , thelatter parametrizing the coset space E / USp(8) . Therefore, the generalised metric can be expressedin terms of a generalised vielbein M MN = E M M E N N δ MN , (2.2)where the generalised vielbein, E M M , is an E -valued matrix. Vector and tensor fields A µM and B µν M are labelled by an index M in the (anti-)fundamental representation of E . These arethe fields of maximal five-dimensional supergravity; however, all of them are still living on the fullhigher-dimensional spacetime. The complete bosonic Lagrangian reads L ExFT6 = p | g | (cid:16) b R + 124 g µν D µ M MN D ν M MN − M MN F µνM F µν N + p | g | − L top − V ( g, M ) (cid:17) . (2.3)3t is invariant under generalised internal diffeomorphisms whose action on the scalar matrix M MN has the generic form [19, 20] δ Λ M MN = L Λ M MN = Λ K ∂ K M MN + 2 α d ∂ L Λ K P K LP ( M M N ) P . (2.4)Here, P K LP M is the projector on the adjoint representation of the duality group E d ( d ) , which for E takes the explicit form P M N K L = 118 δ N M δ LK + 16 δ N K δ LM − d NLR d MKR , (2.5)in terms of the totally symmetric cubic E -invariant tensor d KMN . The constant α d in (2.4) isdetermined by closure of the diffeomorphism algebra and is equal to α = 6 for E . The scalarfields in the Lagrangian (2.3) couple via a gauged sigma model on the coset space E / USp(8) .Accordingly, M MN denotes the matrix inverse to M MN , and the covariant derivatives are defined as D µ M MN = ( ∂ µ − L A µ ) M MN , (2.6)corresponding to the action of (2.4).The Einstein-Hilbert term is constructed from the modified Ricci scalar b R , constructed from theexternal metric g µν in the standard way upon covariantising derivatives under internal diffeomorphisms ∂ µ g νρ → ∂ µ g νρ − A µK ∂ K g νρ . The non-Abelian field strengths in (2.3) are given by F µν N = 2 ∂ [ µ A ν ] N − A [ µK ∂ K A ν ] N + 10 d NKR d P LR A [ µP ∂ K A ν ] L + 10 d NKL ∂ K B µν L , (2.7)with a St¨uckelberg-type coupling to the two-form tensors B µν N . In turn, the topological term L top is defined via its derivative d L top ∝ d MNK F M ∧ F N ∧ F K − d MNK H M ∧ ∂ N H K , (2.8)in terms of the field strengths F µν M and H µνρ M = 3 D [ µ B νρ ] M + . . . , with the ellipses denotingChern-Simons type couplings whose explicit form will not be relevant for this paper. Finally, thepotential term V ( g, M ) in (2.3) is built from bilinears in internal derivatives and reads V ( g, M ) = − α d M MN ∂ M M KL ∂ N M KL + 12 M MN ∂ M M KL ∂ L M NK − g − ∂ M g ∂ N M MN − M MN g − ∂ M g g − ∂ N g − M MN ∂ M g µν ∂ N g µν . (2.9)In the formulation (2.3), the internal coordinates are embedded into the 27-dimensional represen-tation of E with derivatives denoted as ∂ M . Gauge invariance of the action requires the so-calledsection constraint, expressed as a condition bilinear in internal derivatives d KMN ∂ M Φ ∂ N Φ = 0 , (2.10)for any couple of fields { Φ , Φ } . The section constraint (2.10) can be solved by breaking E according to E ⊃ SL(6) × SL(2) ⊃ SL(6) × GL(1) , −→ (6 ,
2) + (15 , −→ +1 + 15 ′ + 6 − , (2.11)4nd restricting the coordinate dependence of all fields to the first six coordinates. Upon this choice,the Lagrangian (2.3) becomes equivalent to full eleven-dimensional supergravity. In turn, type IIBsupergravity is recovered upon choosing a second inequivalent solution of the section constraint basedon the group decompositionE ⊃ SL(5) × SL(2) × GL(1)
IIB , −→ (5 , +4 + (5 ′ , +1 + (10 , − + (1 , − , (2.12)and restricting internal coordinate dependence to the first five coordinates.The explicit map of the ExFT fields (2.1) into the fields of ten- and eleven-dimensional supergravityhas been worked out in [14,16]. Here, we just note that the internal part g mn of the higher-dimensionalmetric can be straightforwardly identified within the components of the matrix M MN according to M MN ∂ M ⊗ ∂ N = (det g ) − / g mn ∂ m ⊗ ∂ n , (2.13)where indices m, n label the derivatives along the physical coordinates embedded into the ∂ M accordingto (2.11) and (2.12), respectively. ExFT
The structure of E exceptional field theory (ExFT) closely parallels the previous constructionmodulo a few technical distinctions. The construction of this theory is based on a split of coordinatesinto four external and 56 internal coordinates, the latter constrained by the section constraintΩ MK ( t α ) K N ∂ M Φ ∂ N Φ = 0 = Ω MN ∂ M Φ ∂ N Φ , α = 1 , . . . , (2.14)where Ω MK and ( t α ) M N denote the symplectic invariant tensor and the 133 generators of E , respec-tively. The two inequivalent solutions of the section constraint (2.14) restrict the internal coordinatedependence of the fields to the six and seven internal coordinates of IIB and D = 11 supergravity,respectively. The bosonic field content of E ExFT is given by (cid:8) g µν , M MN , A µM , B µν α , B µν M (cid:9) , µ, ν = 0 , . . . , , M = 1 , . . . , , (2.15)where the ‘internal’ 56 ×
56 metric M MN now parametrizes the coset space E / SU(8) , and can thusalso be expressed in terms of a generalised vielbein M MN = E M M E N N δ MN , (2.16)where the generalised vielbein, E M M , is now an E -valued matrix. Moreover, apart from two-forms B µν α in the adjoint representation of E , the theory features covariantly constrained two forms B µν M , subject to algebraic constraints which parallel the structure of (2.14)0 = Ω MK ( t α ) K N B µν M ∂ N Φ = Ω MK ( t α ) K N B µν M B ρσ N , α = 1 , . . . . (2.17)The dynamics of E ExFT is most compactly described by a pseudo-Lagrangian L ExFT7 = p | g | (cid:16) b R + 148 g µν D µ M MN D ν M MN − M MN F µνM F µν N + p | g | − L top − V ( g, M ) (cid:17) , (2.18)5mended by the twisted self-duality equation F µν M = − p | g | ε µνρσ Ω MN M NK F ρσ K , (2.19)for the non-abelian vector field strengths F µν M ≡ ∂ [ µ A ν ] M − A [ µK ∂ K A ν ] M − (cid:0)
24 ( t α ) MK ( t α ) NL − Ω MK Ω NL (cid:1) A [ µN ∂ K A ν ] L −
12 ( t α ) MN ∂ N B µν α −
12 Ω MN B µν N . (2.20)The various terms in (2.18) are defined in complete analogy to (2.3) above. In particular, covariantderivatives are defined as in (2.6) where now L Λ refers to generalised internal diffeomorphisms (2.4)for the group E with α = 12, and the projector onto the adjoint representation expressed as P K M LN = 124 δ KM δ LN + 112 δ LM δ KN + ( t α ) MN ( t α ) KL −
124 Ω MN Ω KL . (2.21)The topological term is defined via d L top ∝
24 ( t α ) M N F M ∧ ∂ N H α + F M ∧ H M , (2.22)in terms of vector and tensor field strengths, while the potential term is still of the universal form(2.9), now with α = 12 . In analogy with (2.13), the internal part of the higher-dimensional metriccan be identified among the components of M MN as M MN ∂ M ⊗ ∂ N = (det g ) − / g mn ∂ m ⊗ ∂ n . (2.23)The field equations derived from (2.3) and (2.18) reproduce the field equations of D = 11 and IIBsupergravity, depending on the choice of solution of the section constraint. Moreover, massive IIAsupergravity can be reproduced upon further deformation of the gauge structures [21, 22]. One of the powerful applications of the ExFT framework is the description of consistent truncationsof higher-dimensional supergravities [5–7], i.e. truncations to lower-dimensional supergravities suchthat any solution of the lower-dimensional field equations can be uplifted to a solution of the higher-dimensional field equations. Here, we focus on consistent truncations to maximal supergravities whosefield content is precisely of the form (2.1) and (2.15), respectively, i.e. mirrors the ExFT variables,with fields depending only on the external coordinates.
In terms of the ExFT variables, a consistent truncation to D = 5 and D = 4 dimensions, respectively,is described by a reduction Ansatz which on the vector fields takes the form A µM ( x, y ) = U N M ( y ) A µN ( x ) , (2.24)6actorizing the dependence on internal and external coordinates into an (E d ( d ) × R + )-valued twistmatrix U depending on the internal coordinates and the gauge fields A µN of the lower-dimensionalmaximal supergravity. Similarly, external and internal metrics reduce as g µν ( x, y ) = ρ − ( y ) g µν ( x ) , M MN ( x, y ) = U M K ( y ) U N L ( y ) M KL ( x ) , (2.25)respectively, upon decomposing the twist matrix according to U M N ≡ ρ − ( U − ) M N , (2.26)into a unimodular matrix U − ∈ E d ( d ) , and a scale factor ρ . Finally, the reduction Ansatz for thetwo-form tensor fields takes the formE : B µν M ( x, y ) = ρ − ( y ) U M N ( y ) B µν N ( x ) , E : ( B µν α ( x, y ) = ρ − ( y ) U αβ ( y ) B µν β ( x ) , B µν M ( x, y ) = − ρ − ( y ) ( U − ) SP ( y ) ∂ M U P R ( y )( t α ) RS B µν α ( x ) , (2.27)in E ExFT and E
ExFT, respectively. Here, U αβ denotes the twist matrix evaluated in theadjoint representation of E . Consistency of the truncation Ansatz (2.24)–(2.27) is encoded in a setof differential equations on the twist matrix which take the universal form (cid:2) Γ MN K (cid:3) R d = − γ d X MN K , Γ MN M = (1 − D ) ρ − ∂ N ρ , (2.28)in terms of the algebra valued currentsΓ MN K ≡ ( U − ) N L ∂ M U LK , ∂ M ≡ U M N ∂ N . (2.29)Here, γ d are normalization constants given by γ = , γ = , for E ExFT and E
ExFT, respec-tively. X MN K denotes the constant embedding tensor characterizing the lower-dimensional theory.The projection [ . . . ] R refers to the projection of the rank three tensor Γ MN K onto the irreduciblerepresentation of E d ( d ) in which the embedding tensor transforms. For the theories discussed in thispaper, these are R = and R = .Let us note that, using the explicit form of the projectors in (2.28) (which can, for example, befound in [23]), the first of the consistency relations (2.28) can be explicitly spelled out as − X MN K = − α d P LP N K Γ P M L + α d ( D − P M LN K Γ P LP + Γ
MN K , (2.30)which will be useful in the following. For the E case, we point out two more useful relations5 Γ KL [ M d N ] KL = − X KLM d NKL , Γ KL ( M d N ) KL = −
12 Γ
KLK d LMN , (2.31)which are obtained from the contraction of (2.30) with the d -tensor and from the E invariance ofthe d -tensor, respectively. 7very twist matrix solving equations (2.28) defines a consistent truncation via the reduction Ansatz(2.24)–(2.27), such that the higher-dimensional field equations factor into products of twist matricesand the lower-dimensional field equations. For later use, let us also give the explicit form of the scalarpotential induced in the lower-dimensional gauged supergravities as functions of the embedding tensor X MN K [18, 24] V sugra = 12 α d M MN X MP R (cid:0) X NRP + γ d X NT S M P T M RS (cid:1) . (2.32)Let us also recall, that in a given AdS vacuum the relation between AdS length and cosmologicalconstant Λ is given by L = − ( D − D − − ( D − D − V sugra . (2.33) For the purposes of computing the Kaluza-Klein spectrum, it is useful to view the consistent truncationin a slightly different way. In particular, the twist matrix U ∈ E d ( d ) defines an ( x -independent)generalised vielbein for the generalised metric, as in equations (2.2), (2.16), i.e. M MN = ∆ MN = U M M U N N δ MN , (2.34)and thus fully defines the internal part of the background, i.e. the internal metric and fully internal p -form field strengths. Moreover, for a consistent truncation, the twist matrix is globally well-defined.Thus, the generalised frame fields U M M , defined using ρ as in (2.26), defines a collection of nowhere-vanishing generalised vector fields, and the background is called generalised parallelisable, analogousto ordinary parallelise spaces. However, generalised parallelise spaces need not be parallelisable in theordinary sense, but more generally form coset spaces [25, 6, 26].Finally, it is useful to rephrase the consistency equations (2.28) in terms of the global frame, U M M ,as L U M U N = X MN K U K , (2.35)with the action L of generalised diffeomorphisms defined by (2.6) together with a canonical weightterm. Spaces admitting such a generalised frame field are called generalised Leibniz parallelisablespaces and have several important properties. For example, (2.35) immediately implies that thevector fields, K M , contained in the generalised frame fields (2.26) according to K M m ∂ m = U M M ∂ M , (2.36)generate the gauge algebra specified by the embedding tensor, i.e.[ K M , K N ] = X MN P K P , (2.37)where [ , ] denotes the ordinary Lie bracket. Moreover, the vector fields K M generating the compactpart of the gauge group are necessarily Killing vector fields of the background metric that leave the8uxes invariant. This is clear from the expression of the internal Riemannian metric, which can beeasily read off from (2.34) and is given by g mn = K M m K N n δ MN . (2.38)So far we have only discussed the twist matrix U M M , i.e. the background geometry and fluxesaround which we define the consistent truncation. However, the consistent truncation Ansatz (2.25)implies that every space within the truncation is generalised Leibniz parallelisable. To see this, intro-duce a vielbein for the lower-dimensional gauged supergravity (SUGRA) scalar matrix M MN , i.e. M MN ( x ) = V M A ( x ) V N B ( x ) δ AB . (2.39)Now we can define a generalised frame field for every internal space obtained by the consistent trun-cation by dressing the generalised frame field U M M with the scalar vielbein ( V − ) AM , U AM ( x, y ) = ( V − ) AM ( x ) U M M ( y ) , (2.40)and, equivalently, a generalised vielbein, which entirely encodes the geometry and fluxes, E M A ( x, y ) = U M M ( y ) V M A ( x ) , (2.41)such that the generalised metric, (2.2) and (2.16), M MN ( x, y ) = E M A ( x, y ) E N B ( x, y ) δ AB = U M M ( y ) U N N ( y ) M MN ( x ) , (2.42)takes exactly the form of the truncation Ansatz (2.25). Note that the scale factor ρ , as in (2.26),remains unchanged throughout the consistent truncation. Here, and throughout, we will always usethe A, B indices to denote objects that are dressed by the scalar vielbein ( V − ) AM .Since ( V − ) AM only depends on the external coordinates x , the generalised Lie derivative of thedressed generalised frame fields gives rise to the dressed embedding tensor, often called the T -tensorin the gauged SUGRA literature, L U A U B = X ABC U C , (2.43)with X ABC = ( V − ) AM ( V − ) BN V P C X MN P . (2.44)The properties discussed previously now immediately transfer to any background obtain by the con-sistent truncation. For example, the vector fields making up the dressed generalised frame fields K A = ( V − ) AM K M generate the dressed gauge algebra[ K A , K B ] = X ABC K C . (2.45)In particular, consider some particular vacuum of the lower-dimensional gauged SUGRA theory thatwe are interested in, specified by the scalar matrix M MN = ∆ MN = V M A V N B δ AB . (2.46)The Riemannian metric at this point of the scalar potential can be compactly expressed as g mn = K Am K Bn δ AB = K M m K N n ∆ MN , (2.47)with ∆ MN ∆ NP = δ P M . Equation (2.47) shows how the scalar matrix at the vacuum M MN = ∆ MN deforms the internal geometry. Similar expressions can be derived for the fluxes, see for example [6,27],but are typically lengthier so that we will not give them here.9 Fluctuation Ansatz
We will now show that ExFT leads to a particularly nice description of the linearised fluctuationsaround a given 10-/11-dimensional background that corresponds to a solution of maximal gaugedSUGRA. As we will see in the following, the natural ExFT formulation of these linearised fluctuationsleads to a remarkably compact Kaluza-Klein mass matrices for such a background.
We begin by describing general linear fluctuations around a fixed ExFT background with vanishing A µM , B µν M . Such a background is just described by a non-trivial generalised metric M MN = ∆ MN , (3.1)and an external metric ˚ g µν . The linear fluctuations of the external metric are straightforward andgiven by g µν = ρ − (˚ g µν ( x ) + h µν ( x, y )) , (3.2)where ρ − is required to give the ExFT metric, g µν , the right weight, just as in the generalised Scherk-Schwarz Ansatz (2.25). For the vector and 2-form fields, A µM and B µν M , we will use the fact that theconsistent truncation defines a generalised parallelisation for any background within the truncationvia the dressed generalised vielbein ( U − ) AM in (2.41), as discussed in section 2.3.2. In particular,this implies that the matrices ( U − ) AM , seen as a collection of 27 (in the case of E ) or 56 (in thecase of E ) vector fields, provide a well-defined basis of the generalised tangent bundle. Moreover,the generalised vielbein induces a basis for generalised bundles of any representation of the exceptionalgroup. For example, in the case of E , the U M A provide a well-defined basis for the -dimensionalbundle in which the two-forms B µν,M live. As a result, we can expand any A µM and B µν M in termsof the basis defined by the background generalised vielbein U AM , i.e. A µM = ρ − ( U − ) AM (cid:0) A KK (cid:1) µA ( x, y ) , B µν M = ρ − U M A (cid:0) B KK (cid:1) µν A ( x, y ) . (3.3)Finally, we turn to the scalar sector, described by the generalised vielbein, E M A , parametrisingthe coset space E d ( d ) /H d ( d ) . Since E M A is an E d ( d ) element, a linear fluctuation of the scalar fields isdescribed by an element of the Lie algebra j AB ∈ e d ( d ) , with δ E M A = 12 E M B j BA ( x, y ) . (3.4)However, the fluctuations belonging to h d ( d ) are unphysical, so that we should take j AB ∈ e d ( d ) ⊖ h d ( d ) .This implies that j AB = j BA , (3.5)where j AB = j AC δ BC . (3.6)In turn, for the generalised metric (2.2), (2.16), linearised fluctuations are given by M MN = U M A U N B (cid:0) δ AB + j AB ( x, y ) (cid:1) = ∆ MN ( y ) + U M A U N B j AB ( x, y ) . (3.7)10 .2 Harmonics To determine the Kaluza-Klein masses, we now need to expand the fluctuations h µν , (cid:0) A KK (cid:1) µA , (cid:0) B KK (cid:1) µν A and j AB in terms of a complete basis of fields on the internal manifold. One benefitof our approach is already visible. In the ExFT Ansatz, all the linear fluctuations are scalar fields onthe internal manifold, such that we only need to find a complete basis of scalar functions, Y Σ , on theinternal manifold. All the tensorial structure of the fluctuations is taken care of by the generalisedvielbein, U AM , in the fluctuation Ansatz (3.3) and (3.7).We must now choose a good basis of functions Y Σ to obtain the Kaluza-Klein spectrum. Since thetopology of the compactification is the same for any solution of the lower-dimensional gauged SUGRA,we can choose Y Σ to form representations of the largest symmetry group possible, G max , whichwould correspond to the maximally symmetric point of the gauged SUGRA. Note that this maximallysymmetric point must not even correspond to a vacuum of the theory, i.e. it need not satisfy theequations of motion. Using the ExFT methods, we can choose any internal space corresponding to someconfiguration of scalar fields of the lower-dimensional supergravity, even if this scalar configurationdoes not correspond to a minimum of the potential. For example, for the 4-dimensional N = 8, SO(8)theory, the maximally symmetric point would be the S compactification, and we can choose Y Σ toform representations of G max = SO(8) even if we are interested in another solution of the N = 8,SO(8) theory which breaks the SO(8) symmetry. As we will show, this choice of Y Σ allows us toefficiently compute the Kaluza-Klein spectrum.The complete basis of functions Y Σ must form a representation of the maximal symmetry group.Typically, the consistent truncation is also defined around the maximally symmetric point, such thatthe generalised frame fields U M , used to construct the consistent truncation (2.25), define the maxi-mally symmetric point. Therefore, we have L U M Y Σ = U M M ∂ M Y Σ = K M m ∂ m Y Σ = − T M ΣΩ Y Ω , (3.8)where K M are the vector fields making up the generalised frame fields, as in section 2.3.2. Since the U M generate the Lie algebra of G max via (2.35), the matrices T M ΣΩ , defined by (3.8), correspondto the generators of G max in the representation of the complete basis of functions Y Σ . Using thecommutator of generalised Lie derivatives, it is straightforward to show that the generators T M ΣΩ satisfy the algebra [ T M , T N ] = X [ MN ] P T P , (3.9)where X MN P is the embedding tensor of the lower-dimensional gauged SUGRA, as in (2.35).In this paper, we will restrict ourselves to theories with compact G max1 , such that the matrices T M are antisymmetric T M, ΣΩ = − T M, ΩΣ , (3.10)and harmonic indices Σ , Ω are raised and lowered with δ ΣΩ . As we explain in 4.1, the completebasis of functions Y Σ necessarily correspond to the scalar harmonics of the maximally symmetriccompactification. Therefore, we will often refer to Y Σ as the harmonics. Note that even if G max is compact, the gauge group of the gauged supergravity may be non-compact. An exampleof this would be the D = 4, N = 8 ISO(7) gauged supergravity, where G max = SO(7). U M M by the scalarmatrix V , U AM = ( V − ) AM U M M . (3.11)As a result, the generalised vielbein of the background we are interested in has a simple action on thescalar harmonics Y Σ of the maximally symmetric point, given by L U A Y Σ = − T A ΣΩ Y Ω , (3.12)where T A ΣΩ = ( V − ) AM T M ΣΩ , (3.13)are the generators of G max dressed by the scalar vielbein V . Their commutator is given by the dressedembedding tensor (2.44) [ T A , T B ] = X [ AB ] C T C . (3.14)For our Kaluza-Klein Ansatz, we now expand the linear fluctuations of the scalar fields, j AB , interms of the scalar harmonics. This gives M MN = U M A U N B (cid:16) δ AB + X Σ Y Σ j AB, Σ ( x ) (cid:17) , A µM = ρ − ( U − ) AM X Σ Y Σ A µA, Σ ( x ) , B µν M = ρ − U M A X Σ Y Σ B µν A, Σ ( x ) ,g µν = ρ − (cid:16) ˚ g µν ( x ) + X Σ Y Σ h µν, Σ ( x ) (cid:17) , (3.15)with the sum running over scalar harmonics. From now onwards, we will drop the explicit summationsymbol over the scalar harmonics and use the Einstein summation convention instead. As we will see,with this Ansatz for the fluctuations, equations (3.12) and (2.43) are all the differential informationwe need to complete determine the Kaluza-Klein spectrum. In this section, we linearise the field equations in exceptional field theory with the fluctuation Ansatz(3.15) in order to derive general formulas for the Kaluza-Klein mass spectrum around the backgrounddefined by the generalised metric M MN = ∆ MN ⇐⇒ M MN = ∆ MN = V M A V N A . (4.1)As a general rule of notation, when using the flat basis introduced in (2.40), we raise, lower, andcontract flat indices with δ AB . 12 .1 Spin-2 Let us start with the spin-2 sector, for which the computation of the Kaluza-Klein spectrum is themost straightforward. The mass spectrum in this sector is also accessible by other universal methodsand can be traced back to computing the eigenmodes of a higher-dimensional wave operator dependingonly on the background geometry [28–32]. We will explicitly match this result to our approach below.In the ExFT formulation of supergravity, the mass terms for the spin-2 fluctuations descend fromthe universal couplings of the external metric g µν within L pot , L mass ,g = 14 p | g | (cid:0) M MN ∂ M g µν ∂ N g µν + M MN g − ∂ M g ∂ N g (cid:1) , (4.2)c.f. (2.9). With the explicit fluctuation Ansatz (3.15) and the action of internal derivatives on theharmonics Y Σ expressed in terms of the T M matrix according to (3.8) above, the Lagrangian (4.2)gives rise to L mass ,g −→ − ρ − D Y Λ Y Γ ∆ MN T M, ΛΣ T N, ΓΩ h µν, Σ h µν Ω + . . . , (4.3)where the ellipses refer to terms carrying traces and divergences of h µν which play their role in theexplicit realization of the spin-2 Higgs effect but do not contribute to the final mass matrix. The latteris read off from (4.3) after comparing the normalization to the linearised Einstein-Hilbert term from(2.3), (2.18): M ΣΩ = − ∆ MN (cid:0) T M T N (cid:1) ΣΩ = − (cid:0) T A T A (cid:1) ΣΩ , (4.4)in the flat basis introduced in (2.40).The full system of differential equations for the spin-2 modes also includes the couplings of thesemodes to the spin-1 fluctuations via the connection terms in the Einstein-Hilbert term b R and to thespin-0 fluctuations via the respective third terms in the ExFT potential (2.9). Upon gauge fixing,they account for the transfer of degrees of freedom from the massless vector and scalar fluctuationsto building the massive spin-2 modes [38–40]. Rather than working out these couplings in detail,the most direct analysis of their contribution proceeds by spelling out the relevant gauge symmetries.Linearising external diffeomorphisms upon expanding their gauge parameter in accordance with thefluctuation Ansatz (3.15) as ξ µ = P Σ ξ µ Σ Y Σ induces the action δ ξ h µν, Σ = 2 ∂ ( µ ξ ν ) , Σ , δ ξ A µM, Σ = T M, ΣΩ ξ µ Ω , (4.5)which can be used as a shift symmetry to explicitly eliminate those vector field fluctuations whichcouple to the spin-2 fluctuations at the quadratic level. Next, we turn to the scalar fields to identifythe corresponding Goldstone modes here. With the gauge transformations on the scalar matrix givenby (2.4) above, let us project these transformations onto those which at the linearised level yield shiftsymmetries to the scalar fluctuations. I.e. we set M MN to its background value ∆ MN and expand Λ M into harmonics according to the Ansatz (3.15) for the corresponding gauge fields. After going to the This has e.g. been further exploited in [33–37]. δj AB, Σ = h Λ C, Σ Γ CAB + α d D − CDC P DEBA Λ E, Σ − α d Γ CDE P C EBA Λ D, Σ + α d T C, ΣΩ P C DBA Λ D, Ω + α d T C, ΣΩ P C DAB Λ D, Ω + ( A ↔ B ) i coset = h − Λ C, Σ X CAB − Λ C, Σ X CBA + α d Λ A, Ω T B, ΣΩ + α d Λ B, Ω T A, ΣΩ i coset , (4.6)where we have used the projector relations (2.30) from above. In addition, the r.h.s. of (4.6) isunderstood under projection onto the symmetric coset valued index pairs ( AB ), c.f. (3.4), (3.5) above.These gauge transformations combine the standard Higgs effect (giving mass to the spin-1 vectorfields) with the transformations eliminating the Goldstone scalars for the massive spin-2 modes. Toidentify the latter, it is sufficient to evaluate (4.6) for the gauge parameters corresponding to the vectorfields transforming under (4.5). Combining these two formulas, we find that the scalars affected bythe spin-2 Higgs mechanism are those transforming under δj AB, Σ = Π AB, ΣΩ Λ Ω , (4.7)with a gauge parameter Λ Ω , and the tensor Π defined asΠ AB, ΣΩ = (cid:2) − X CAB T C, ΣΩ + α d (cid:0) T A T B (cid:1) ΣΩ (cid:3) coset , (4.8)where again the projection on the r.h.s. refers to projection of the AB indices onto the symmetriccoset valued index pairs ( AB ).To sum up, the full system of differential equations for the spin-2 modes also includes their couplingsto the spin-1 fluctuations singled out by (4.5) and the spin-0 fluctuations defined by (4.7). Propergauge fixing will eliminate the lower spin modes in favour of the massive spin-2 excitations. Thisdoes not alter the result (4.4) for the spin-2 mass matrix, but will have to be taken into account inthe computation of the spin-1 and spin-0 mass spectra, where these Goldstone modes will have to beexplicitly eliminated before calculating the spectrum.Let us finally compare the mass matrix (4.4) to the general analysis of [32]. There, it has beenshown that upon compactification from ten dimensions around a warped background metric ds = e A ( y ) ¯ g µν ( x ) dx µ dx ν + ˆ g mn ( y ) dy m dy n , (4.9)the mass spectrum of the spin-2 fluctuations is encoded in the following Laplace equation on theinternal space ✷ spin2 ψ ≡ e (2 − D ) A | ˆ g | − / ∂ m (cid:16) | ˆ g | / ˆ g mn e DA ∂ n ψ (cid:17) = − m ψ , (4.10)where D is the number of external dimensions: µ = 0 , . . . , D − In particular, this spectrum onlydepends on the internal background geometry.Let us compare (4.10) to the spin-2 mass matrix (4.4) obtained in our framework. For the compact-ifications described by ExFT, the internal background metric, ˆ g mn , is embedded into the generalisedmetric, M MN , according to the universal relationˆ g mn ∂ m ⊗ ∂ n = | ˆ g | / ( D − M MN ∂ M ⊗ ∂ N , (4.11) Strictly speaking, reference [32] gives the result for D = 4, but it straightforwardly generalizes to arbitrary D . y m into the ExFT coordinates, c.f. (2.13), (2.23).Similarly, the ExFT embedding of the external metric together with the reduction Ansatz (2.25) yieldsthe identification e A ( y ) = ρ − | ˆ g | − / ( D − . (4.12)As a result, the Laplacian on the internal manifold can be rewritten as ✷ y ψ = | ˆ g | − / ∂ M (cid:16) | ˆ g | D/ (2( D − M MN ∂ N ψ (cid:17) = e ( D − A ( y ) K Am ∂ m (cid:16) e − DA ( y ) K An ∂ n ψ (cid:17) , (4.13)where we have used the reduction Ansatz (2.25) for the internal metric as well as the identification(2.36) of the Killing vector fields within the Scherk-Schwarz twist matrix.For the operator ✷ spin2 defined in (4.10), we thus find the explicit action ✷ spin2 ψ = K Am ∂ m (cid:0) K An ∂ n ψ (cid:1) , (4.14)in terms of the Killing vector fields. Combining this with (3.8), we find the action on the Y Σ as ✷ spin2 Y Σ = (cid:0) T A T A (cid:1) ΣΩ Y Ω = − M ΣΩ Y Ω , (4.15)showing agreement of the general result [32] with the mass matrix (4.4) in ExFT compactifications.Moreover, this also shows that the Y Σ are harmonics of the Laplacian (4.10), hence our nomenclaturefor the Y Σ . In [37], a similar form of the spin-2 mass matrix has been proposed for reductions to D = 4 maximal supergravity. Let us move on to the mass spectrum of antisymmetric tensor fields. Their appearance is specific tocompactifications to D = 5 dimensions, described by E ExFT. Their field equation in ExFT isobtained from (2.3) by variation w.r.t. the tensor fields B µν M which leads to the first-order dualityequation d P ML ∂ L e M MN F µνN + √ ε µνρστ H ρστM ! = 0 , (4.16)with e = p | g | . To linear order in the fields, the field strength F µν M is given by F µν M −→ ∂ [ µ A ν ] M + 10 d MNK ∂ K B µν N , (4.17)where the last term is responsible for creating the tensor masses in (4.16). The latter are thus encodedin the differential operator d MNK ∂ K . Its action on tensor fields obeying the reduction Ansatz (3.15)is computed as U M A d MNK ∂ K B µνN = U M A d MNK ∂ K (cid:0) ρ − U N B Y Σ (cid:1) B µνB, Σ = − ρ − (cid:0) Z AB δ ΩΣ − d ABC T C, ΩΣ (cid:1) Y Ω B µν B, Σ ≡ ρ − √ Y Ω M A Ω ,B Σ B µν B, Σ , (4.18)15here we have used (2.31), and (2.28) and moreover defined the constant antisymmetric tensor Z AB = 2 d CDA X CDB , (4.19)that encodes the complete information on the embedding tensor in D = 5 dimensions.The result of this computation is the antisymmetric mass matrix M A Σ ,B Ω = 1 √ (cid:0) − Z AB δ ΣΩ + 10 d ABC T C, ΣΩ (cid:1) . (4.20)The first term arises precisely as in the Scherk-Schwarz reduction to D = 5 dimensions, the secondterm captures the effect of internal derivatives acting on the harmonics. Plugging back (4.20) into theduality equation (4.16), we find at linear order0 = d P ML ∂ L (cid:16) ρ − U M A Y Σ (cid:16) ∂ [ µ A ν ] A, Σ + √ M A Σ ,B Ω B µνB, Ω + √ ε µνρστ ∂ ρ B στ A, Σ (cid:17)(cid:17) . (4.21)The ∂ [ µ A ν ] A, Σ terms can be gauge fixed and amount to the spin-1 Goldstone modes absorbed into themassive tensor fields. This results in the five-dimensional first-order equation3 ∂ [ µ B νρ ] A, Σ = 12 ε µνρστ M A Σ ,B Ω B στ B, Ω , (4.22)describing topologically massive tensor fluctuations with the mass matrix (4.20).Let us finally point out that according to (4.21) the entire first-order equation of the tensor fieldsis yet hit with another mass operator d P ML ∂ L . Repeating the same calculation for this action showsthat the final first-order equation is given by contracting (4.22) with another mass matrix (4.20). Inother words, zero eigenmodes of the mass matrix M A Σ ,B Ω are in fact not part of the physical spectrumas the corresponding modes among the B µν A, Σ are projected out from all field equations. In E d ( d ) ExFT, the field equations obtained by varying the Lagrangian w.r.t. the vector fields are ofYang-Mills type (for d < ∇ ν (cid:0) M MN F νµ N (cid:1) = I µ EH M + I µ sc M + I µ top M , (4.23)where the currents on the r.h.s. denote the contributions from the Einstein-Hilbert term, the scalarkinetic term, and the topological term, respectively. As we have discussed in section 4.1 above, uponlinearisation the contributions from I µ EH M only contribute to Higgsing the spin-2 modes and have noimpact on the masses of the physical spin-1 fluctuations. We will deal with these contributions atthe very end by projecting the vector mass matrix on the physical sector invariant under translations(4.5). The contributions I µ top M from the topological term are in general of higher order in the fieldsand drop out after linearisation. The notable exception is E ExFT, where according to (2.22) thiscurrent carries a contribution dual to the field strengths H α , H M , which by virtue of the derivative ofthe twisted self-duality equation (2.19) together with the Bianchi identity gives rise to a contributionwhich equals the l.h.s. of (4.23) up to sign. 16pon linearisation, we will thus extract the vector mass matrix from the r.h.s. of the universalequation ∆ MN ∇ ν F νµ N = J µ sc M (cid:12)(cid:12)(cid:12) lin , (4.24)where the current J µ sc M is defined from variation of the scalar kinetic term e δ A µM J µ sc M = δ A (cid:18) α d e D µ M MN D µ M MN (cid:19) = − e δ A µM (cid:18) α d ( J M ) K L J µLK + e − ∂ N (cid:0) eJ µM N (cid:1)(cid:19) , (4.25)with the currents ( J L ) N M = M NK ∂ L M KM , J µ N M = M NK D µ M KM . (4.26)To linear order in the fluctuations in (4.25), the internal current only contributes its backgroundvalue, which in the flat basis reads J A,BC −→ − (cid:16) Γ ABC + Γ
AC B (cid:17) , (4.27)with Γ ABC from (2.29). The external current J µ N M carries vector and scalar fluctuations. However,the latter do not contribute to the vector masses but ensure the proper absorption of the scalarGoldstone modes for realizing the spin-1 Higgs mechanism. The vector fluctuations arise from theconnection (2.6) within J µ N M and split into terms which due to (2.35) carry the embedding tensor X ABC together with the contributions from the harmonics following from (3.8). In the flat basis(2.40), these take the form J µ AB (cid:12)(cid:12)(cid:12) lin = (cid:0) − (cid:0) X CAB + X CBA (cid:1) A µC, Σ + α d (cid:0) P ABC D + P BAC D (cid:1) T D, ΣΩ A µC, Ω (cid:1) Y Σ . (4.28)In (4.25), this current also appears under internal derivative according to − e − ∂ N (cid:16) eJ µM N (cid:12)(cid:12)(cid:12) lin (cid:17) = ρ U M A ˚ g µν (cid:18) D − BC B J νAC − Γ CAB J ν BC − ∂ B (cid:0) J ν AB (cid:1)(cid:19) (cid:12)(cid:12)(cid:12) lin( . ) = − ρ α d U M A ˚ g µν (cid:0)(cid:0) X ABC + Γ
ABC (cid:1) J νC B + α d ∂ B (cid:0) J νAB (cid:1)(cid:1) (cid:12)(cid:12)(cid:12) lin . With these explicit expressions, the two terms on the r.h.s. of (4.25) combine into J µ sc M (cid:12)(cid:12)(cid:12) lin = − ρ α d U M A ˚ g µν (cid:0) X ABC J µC B + α d ∂ B (cid:0) J µAB (cid:1)(cid:1) (cid:12)(cid:12)(cid:12) lin . (4.29)Consistently, all Γ ABC terms have dropped out and only the terms carrying the constant embeddingtensor X ABC as well as the matrices T A survive.Putting everything together, we can write the linearised D -dimensional vector field equation (4.24)as ∂ ν ∂ ν A µ A, Σ − ∂ ν ∂ µ A ν A, Σ = M A Σ ,B Ω A µ B, Ω , (4.30)17here the mass matrix results from collecting all the resulting terms in (4.29) and takes the form M A Σ ,B Ω = 1 α d X ADC (cid:0) X BC D + X BDC (cid:1) δ ΣΩ + (cid:0) X BAC + X BC A − X ABC − X AC B (cid:1) T C, ΣΩ − α d (cid:0) P AC BD + P CABD (cid:1) (cid:0) T C T D (cid:1) ΣΩ . (4.31)The first term of this mass matrix reproduces the known vector mass matrix within D -dimensionalsupergravity. In particular, it vanishes for compact generators X KLN in accordance with the masslessvectors from the supergravity multiplet associated with the unbroken gauge symmetries. The resultholds for, both, E
ExFT and E
ExFT.In a final step, we need to project out by hand the spin-1 Goldstone modes absorbed into themassive spin-2 fields. To this end, we have to project the vector fluctuations to the subsector thatremains invariant under the corresponding translations (4.5). In contrast, the spin-1 modes absorbedinto massive tensor modes according to the discussion after (4.21) above, appear as zero eigenvaluesof the mass matrix (4.31) and can thus easily be identified.
It remains to work out the scalar mass spectrum for the fluctuation Ansatz presented above. Linearis-ing the theories (2.3), (2.18), the scalar field equations also contain spin-2 and spin-1 contributionsimplementing the corresponding Higgs mechanisms. These have no impact on the masses of the phys-ical scalars. We will deal with these contributions at the end by applying an overall projection to theresulting mass matrix. Ignoring vector and metric fluctuations, the scalar masses are obtained fromthe linearised field equation ✷ M MN (cid:12)(cid:12)(cid:12) lin = h − J M,K L J N,LK − α d J K,M L J L,N K + ∆ QL J K,QK J L,M P ∆ P N + 2 ( D − α d ∂ K ρρ − J M,N K − ( D − ∂ K ρρ − ∆ KL J L,M P ∆ P N + ρ ∆ KL ∂ K (cid:0) ρ − J L,M P (cid:1) ∆ P N − ρ α d ∂ K (cid:0) ρ − J M,N K (cid:1) − D α d ρ − ∂ M ρ ∂ N ρ + 2 D α d ρ − ∂ M ∂ N ρ i coset , lin , (4.32)with the current J M,N K from (4.26). In addition, the r.h.s. is understood as being projected ontosymmetric coset values index pairs
M N , c.f. (3.4), (3.5) above.The computation is considerably more laborious than the preceding calculations for the tensor andthe spin-1 sector. The latter analyses were facilitated by the manifest covariance of the field equationsunder generalised diffeomorphisms which, together with the generalised parallelisability (2.35), allowedfor a compact derivation of the corresponding mass matrices in terms of the embedding tensor andthe matrix T A . The scalar field equation, in contrast, is not manifestly invariant under generaliseddiffeomorphisms. As a consequence, it is lengthier to arrange the numerous contributions resultingfrom (4.32) until the dependence on the internal coordinates factors out.We may, however, exploit the known structures from gauged supergravity to reduce the compu-tation to a few relevant terms. As for the vector mass matrix (4.31), the contributions to the scalarmass matrix from (4.32) can be organised into (schematically) M = XX + X T + T T , (4.33)18ccording to if internal derivatives hit the twist matrices, U , or the harmonics, T , in the fluctuationAnsatz (3.15).The XX terms in (4.33) do not act on the harmonics and by construction coincide with the massformula from gauged supergravity for the lowest multiplet. We can thus directly extract these termsfrom the variation of the D -dimensional supergravity potentials (2.32) and only focus on the remainingterms.To this end, we expand the currents J M,N K to linear order in the fluctuations, which in the flatbasis (2.40) takes the form J D,AB = − n Γ DAB + Γ
DBA + ∂ D j AB + Γ DBE j AE − Γ DEA j EB o , (4.34)extending (4.27). Next, we expand their derivatives in the flat basis and obtain ∇ C J D,AB ≡ ρ − ( U − ) C L ( U − ) DK ( U − ) AM U N B ∂ L (cid:0) ρ − J K,M N (cid:1) = − ∂ C Γ DAB − ∂ C Γ DBA − Γ CAG Γ DGB − Γ CAG Γ DBG − Γ CDG Γ GAB − Γ CDG Γ GBA + Γ
CGB Γ DAG + Γ
CGB Γ DGA − ∂ C Γ DBF j AF + ∂ C Γ DF A j F B − (cid:0) Γ CAF Γ DBG + Γ
CGB Γ DF A (cid:1) j F G − (cid:0) Γ CDG Γ GBF − Γ CGB Γ DGF (cid:1) j AF + (cid:0) Γ CDG Γ GF A + Γ
CAG Γ DF G (cid:1) j F B − Γ DBF ∂ C j AF + Γ DF A ∂ C j F B − Γ CAG ∂ D j GB + Γ CGB ∂ D j AG − Γ CDG ∂ G j AB − ∂ C ∂ D j AB . (4.35)Putting everything together, the r.h.s. of (4.32) vanishes on the background ( j AB →
0) as aconsequence of the fact that we are linearising the theory around a stationary point of the gaugedsupergravity potential. The terms carrying j AB will precisely recombine into the XX contributionsin (4.33) which we can extract from the gauged supergravity describing the lowest multiplet in theabsence of higher fluctuations. The unknown terms in (4.33) thus exclusively descend from derivativeterms, such that we can restrict the above expansions to ∂ C j AB J D,AB = − Γ DAB − Γ DBA − ∂ D j AB + . . . , ∇ C J D,AB = − Γ DBF ∂ C j AF + Γ DF A ∂ C j F B − Γ CAG ∂ D j GB + Γ CGB ∂ D j AG − Γ CDG ∂ G j AB − ∂ C ∂ D j AB + . . . , (4.36)with the ellipses denoting the terms that do not contribute to the X T + T T terms in (4.33). Puttingthis back into (4.32), we are left with˚ ✷ j AB = h − AC D ∂ B j DC − α d Γ CDA ∂ D j BC + 2 Γ DAB ∂ C j DC − CBD ∂ C j AD + 2 Γ CDB ∂ C j AD + 2 α d Γ ACD ∂ C j BD − α d Γ ADB ∂ C j DC + 2 α d Γ CBD ∂ A j DC + 2 α d ∂ C ∂ A j BC − ∂ C ∂ C j AB i coset + “ XX j ” . (4.37)Still, the r.h.s. is projected onto coset valued index pairs ( AB ) .19n a final step, we now expand j AB into harmonics according to (3.15), such that the action ofinternal derivatives can be expressed by the T A matrix. We also make the coset projection manifestby contracting the entire fluctuation equation with another coset-valued fluctuation, such that we findthe Lagrangian quadratic in fluctuations L scalar − fluc ∝ j AB, Σ ˚ ✷ j AB, Σ − AC D T B, ΩΣ j AB, Σ j DC, Ω − α d Γ ACB T D, ΩΣ j AB, Σ j CD, Ω − CAB T C, ΩΣ j AD, Σ j BD, Ω − α d Γ BC A T C, ΩΣ j AD, Σ j BD, Ω − α d T A, ΩΛ T B, ΛΣ j AD, Σ j BD, Ω + T C, ΩΛ T C, ΛΣ j AB, Σ j AB, Ω + “ XXj j ” . (4.38)The resulting couplings may be further simplified upon repeated use of the projector property(2.30) together with j AB, Σ = P ABC D j CD, Σ , T F , ΣΩ j AE, Σ j BE, Ω = T F , ΣΩ P ABC D j DE, Σ j CE, Ω . (4.39)The first of these relations reflects the algebra-valuedness of the fluctuations while the second one isa consequence of the closure of the commutator on the algebra. As a consequence, all Γ ABC in (4.38)can be eliminated in favour of the constant embedding tensor X ABC , as required for consistency.Restoring the
XXj j terms as obtained from variation of the gauged supergravity potential (2.32),the full scalar mass matrix finally reads j AB, Σ M AB Σ ,CD Ω j CD, Ω = X AEF X BF E j AD, Σ j BD, Σ + γ d (cid:0) X AEF X BEF + X EAF X EBF + X EF A X EF B (cid:1) j AD, Σ j BD, Σ + 2 γ d (cid:0) X AC E X BDE − X AEC X BED − X EAC X EBD (cid:1) j AB, Σ j CD, Σ − X AC D T B, ΩΣ j AB, Σ j CD, Ω − X CAB T C, ΩΣ j AD, Σ j BD, Ω + 2 α d T A, ΩΛ T B, ΛΣ j AD, Σ j BD, Ω − T C, ΩΛ T C, ΛΣ j AB, Σ j AB, Ω . (4.40)From the mass spectrum obtained by diagonalising this matrix, we still need to project out theGoldstone modes that render mass to the spin-1 and spin-2 fluctuations, as anticipated at the beginningof this section. As usual, the Goldstone modes absorbed by the massive spin-1 fields appear with zeroeigenvalue in (4.40) and are thus easily identified. The Goldstone modes absorbed into the massivespin-2 fields in contrast need to be projected out explicitly. Following the discussion of section 4.1above, this can be implemented by projecting the mass matrix (4.40) onto those fields that are leftinvariant under the shift transformations (4.7). We have in the previous section worked out general mass formulas (4.4), (4.20), (4.31), (4.40), for thecomplete bosonic Kaluza-Klein spectrum around any vacuum lying within a consistent truncation tomaximal supergravity. After diagonalising the mass matrices, the corresponding mass eigenstates areidentified within ExFT via the fluctuation Ansatz (3.15) and can be uplifted to higher dimensionsusing the dictionary between the ExFT and the original supergravity variables.In this section, we illustrate these formulas by various examples in four and five dimensions.20 .1 Vacua of 5-dimensional
SO(6) gauged SUGRA
In this section, we will apply our mass formulae to the Kaluza-Klein spectra of two vacua of the 5-dimensional SO(6) gauged supergravity [41]. This N = 8 supergravity can be obtained by a consistenttruncation of IIB supergravity on S [6, 7, 27] and contains various interesting vacua, including the N = 8 AdS × S solution of IIB supergravity and the N = 2 SU(2) × U(1)-invariant AdS vacuum [9]dual to the Leigh-Strassler CFT [10]. We will use the example of the AdS × S vacuum to demonstrateour formalism, showing that it allows for a compact identification of the BPS multiplets and the IIBfields sourcing the fluctuations. Next, we show, using the example of the N = 2 SU(2) × U(1) vacuum,that our mass formulae also allow to compute the Kaluza-Klein spectrum of vacua for which this wasnot possible before.Let us begin by setting up our notation for the consistent truncation of IIB supergravity to the5-dimensional SO(6) gauged supergravity. To do this, we use the SL(6) × SL(2) basis of E
ExFT,in which the fundamental representation of E decomposes into −→ (15 , ⊕ (6 ′ , , (cid:8) A M (cid:9) −→ n A ab , A aα o , a = 1 , . . . , , α = 1 , . (5.1)In this basis, the d -symbol takes the form d KMN = ( d abcα,dβ = √ δ abcd ε αβ ,d ab,cd,ef = √ ε abcdef . (5.2)The consistent truncation of IIB supergravity to the SO(6) gauged maximal supergravity of [41]can be described as a generalised Scherk-Schwarz reduction within E ExFT in the sense discussedin section 2.3 above, with twist matrices U M M constructed from the elementary sphere harmonics on S Y a Y a = 1 . (5.3)Specifically, the twist matrices are constructed as SL(6) ⊂ E group matrices, given in terms of theround S metric ˚ g mn = ∂ m Y a ∂ n Y a , and the vector field ˚ ζ n defined by ˚ ∇ n ˚ ζ n = 1 by( U − ) a ˆ m = (cid:8) ( U − ) a , ( U − ) am (cid:9) = ˚ ω / n ˚ ω − Y a , ˚ g mn ∂ n Y a + 4 ˚ ζ m Y a o , (5.4)where we have introduced the SL(6) index ˆ m = 0 , . . . ,
5. The weight factor is given by ρ = ˚ ω − / in terms of the metric determinant ˚ ω = det˚ g mn . For computing the Kaluza-Klein masses, weare particularly interested in the vector components, K M , of the generalised parallelisable framecorresponding to the E twist matrices. These are given by [6, 27] K M = ( K ab = v ab , K aα = 0 , (5.5)where v abm = −√ g mn (cid:0) ∂ m Y [ a (cid:1) Y b ] , (5.6)21re the SO(6) Killing vectors of the round S .The resulting D = 5 theory is described by an embedding tensor X MN P = ( X ab,cdef = 2 √ δ [ e [ a δ b ][ c δ f ] d ] ,X abcαdβ = −√ δ c [ a δ b ] d δ αβ , (5.7) ⇐⇒ Z MN = ( Z ab,cd = 0 ,Z aα,bβ = √ ε αβ δ ab , (5.8)with Z MN defined in (4.19).Finally, we need to choose a complete basis of scalar functions in which to expand the ExFT fieldsvia the Kaluza-Klein Ansatz (3.15). As discussed in section 3.2, it is most convenient to choose thecomplete basis of functions as representations of the maximally symmetric point of the consistenttruncation, which in this case corresponds to the round S . Therefore, we will expand the ExFT fieldsin terms of the scalar harmonics on the round S , which are given by polynomials in the elementary S harmonics (5.3) as (cid:8) Y Σ (cid:9) = { , Y a , Y a a , . . . , Y a ...a n , . . . } , (5.9)where our notation Y a ...a n ≡ Y (( a . . . Y a n )) denotes traceless symmetrisation in the elementaryharmonics. The index Σ thus runs over the tower of symmetric traceless vector representations [ n, , Y a ...a n harmonics as the level n representation.For the mass formulae, we need to compute the action of the vectors K M , defined by the generalisedparallelisation (5.5), on the scalar harmonics Y Σ . By construction, the S Killing vector fields have alinear action on the harmonics, which is block-diagonal level by level and according to (3.8) defines thematrices T M as the SO(6) generators in the symmetric [ n, ,
0] representation. In our conventions, these take the explicit form T M,c ...c n d ...d n = n T M, (( c (( d δ c d . . . δ c n )) d n )) , (5.11)where double parentheses again denote traceless symmetrisation, and the action on the elementaryharmonics is given by T M,cd = ( T ab,cd = √ δ c [ a δ b ] d , T aαcd = 0 . (5.12)We can now straightforwardly apply our mass formulae (4.4), (4.20), (4.31), (4.40) to compute thespectrum of Kaluza-Klein modes around any vacuum of the SO(6) gauged supergravity. All we haveto do is dress the embedding tensor (5.7), (5.8) and the T -matrix (5.12) by the scalar vielbein, V M A ,corresponding to the vacuum we are interested in. Our summation convention for the harmonic indices Σ , Ω is such that A Σ B Σ = A B + A a B a + A a a B a a + . . . + A a ...a n B a ...a n + . . . . (5.10) .1.1 AdS , N = 8 , SO(6) vacuum: IIB on S In this section, we recompute the Kaluza-Klein spectrum around the maximally supersymmetricAdS × S solution of IIB supergravity. This background sits as an N = 8 vacuum within a con-sistent truncation to the D = 5 SO(6) gauged maximal supergravity of [41], which can be describedwithin ExFT, it is thus amenable to our formalism. Originally, the Kaluza-Klein spectrum on thisbackground has been determined in [42, 43] by linearising the IIB field equations and exploiting therepresentation structure of the underlying supergroup SU(2 , | × S vacuum corresponds to the stationary point at the origin M MN = ∆ MN = δ MN ofthe scalar potential (2.32). Thus, we can choose the scalar vielbein as V M A = δ M A . We recall, thatin the flat basis, indices are raised, lowered, and contracted with δ AB which in the index split (5.1) isexpressed in terms of δ ab and δ αβ , respectively. The value of the scalar potential at this point is given by V sugra (cid:12)(cid:12)(cid:12) = −
12 = ⇒ L AdS = 1 . (5.13)In the original formulation of type IIB supergravity, the computation of the Kaluza-Klein spectrumaround this background requires to expand all fields into the corresponding sphere harmonics. Forexample, a ten-dimensional scalar field gives rise to a tower of D = 5 scalar fields φ ( x, y ) = X Σ Y Σ ( y ) ϕ Σ ( x ) , (5.14)according to the tower of scalar harmonics Y Σ on the round S .On the other hand, in the traditional formulation, ten-dimensional fields with non-trivial transfor-mation under the Lorentz group on S in general give rise to several towers of harmonics which arebuilt from products of the elementary harmonics (5.3) and their derivatives. These can be classifiedand determined by group theoretical methods [44]. E.g. the internal part of the ten-dimensional metricgives rise to an expansion g mn ( x, y ) = X Σ Y Σ mn ( y ) g Σ ( x ) , (5.15)with the harmonics Y Σ mn now filling three towers of SO(6) representations built from the differentirreducible components of Y a a ,a ...a n mn ≡ ( ∂ m Y a )( ∂ n Y a ) Y a ...a n . (5.16)In our approach, as discussed in section 3.2, we expand all fields in only the scalar harmonics Y Σ ,and the non-trivial Lorentz structure of the Kaluza-Klein fluctuations will arise entirely from multi-plying the twist matrices appearing in the fluctuation Ansatz (3.15). We will demonstrate explicitlyhow this occurs in the following. We denote both, ‘curved’ SL(6) × SL(2) indices and ‘flat’ SO(6) × SO(2) indices by a, b and α, β . .1.1.1 Spin-2 fluctuations We recall from (3.15) that the spin-2 fluctuations directly organise into the scalar harmonics Y Σ .We immediately obtain their mass spectrum from the expression (4.4) for the mass matrix. Withthe T -matrix given by (5.11)–(5.12), this matrix is (up to normalization) nothing but the quadraticSO(6) Casimir operator, whose eigenvalue on the [ n, ,
0] symmetric vector representation is given by M a ...a n ,b ...b n = n ( n + 4) δ (( a ...a n ))(( b ...b n )) . (5.17)With the conformal dimension of spin-2 fields given by ∆ = 2 + q m L , this gives rise to∆ = 4 + n . (5.18) According to the fluctuation Ansatz (3.15), the tensor field fluctuations combine into the tensorproduct of the fundamental representation (5.1) with the tower of scalar harmonics (5.9). We mayexplicitly spell out the fluctuation coefficients as (cid:8) B µν A, Σ (cid:9) = { B µν ab,c ...c n , B µν aα,c ...c n } . (5.19)At level n they fall into SO(6) × SO(2) representations B µν ab,c ...c n ∈ [ n, , ⊕ [ n, , ⊕ [ n − , , ⊕ [ n − , , ⊕ [ n − , , ,B µν aα,c ...c n ∈ [ n + 1 , , ± ⊕ [ n − , , ± ⊕ [ n − , , ± , (5.20)where we label these representations as [ n , n , n ] j by SO(6) Dynkin weights n i and SO(2) charge j .In terms of the SO(6) vector indices, the different SO(6) representations correspond to the symmetri-sations [ n, ,
0] : . . . , [ n, ,
1] : . . . , [ n, , ⊕ [ n, ,
2] : . . . . (5.21)Summing over all levels, we thus find for the full spectrum (cid:16) [0 , , ⊕ [1 , , ± (cid:17) ⊗ ∞ X n =0 [ n, ,
0] = ∞ X n =0 (2 · [ n, , + [ n, , + [ n, , + [ n + 1 , , ) ⊕ [0 , , ± ⊕ ∞ X n =0 (cid:16) [ n, , ± + 2 · [ n + 1 , , ± (cid:17) . (5.22)Recall, however, from the discussion in section 4.2 that within towers, only tensors of non-vanishingmass are part of the physical spectrum. 24e may now evaluate the action of the mass matrix (4.20) onto the components (5.19). Recallthat the tensor mass matrix is antisymmetric and thus has imaginary eigenvalues. Using the explicitexpressions for d ABC , Z AB , T A from (5.2), (5.8), and (5.12), above, we obtain ( M B ) ab,c ...c n = − n ε abcdef B cd,e (( c ...c n − δ c n )) f , ( M B ) aα,c ...c n = − ( n + 1) ε αβ B ( a | β | c ...c n ) + n ε αβ B dβ,d (( c ...c n − δ c n )) a . (5.23)The first equation shows that among the B ab,c ...c n , the only components carrying non-vanishing masscorrespond to the [ n, , ⊕ [ n, ,
0] representation, antisymmetric in three indices. To compute thecorresponding eigenvalue, we explicitly parametrise B ab,c ...c n as B ab,c ...c n = t ( ± ) ab (( c ,c ...c n )) , (5.24)in terms of a tensor t ( ± ) abc,d ...d n − , (anti-)self-dual in the first three indices t ( ± ) abc,d ...d n − = ± i ε abcdef t ( ± ) def,d ...d n − , (5.25)and traceless under any contraction. The action of the mass matrix (5.23) then yields( M B ) ab,c ...c n = − ε abcde (( x t ( ± ) cde c ...c n )) −
12 ( n − ε abcde (( c t ( ± ) cd c ,c ...c n )) e = ± i ( n + 2) t ( ± ) ab (( c ,c ...c n )) = ± i ( n + 2) B ab,c ...c n , (5.26)where we have used that t [ abc,d ] d ...d n − = 0, as a consequence of (5.25) and tracelessness.Next, we turn to the second equation of (5.23). Its r.h.s. shows that the action of M on B aα,c ...c n isvanishing on the [ n, ,
1] representation and has eigenvalues ± i ( n + 1) on the traceless B µν (( aαc ...c n )) .It remains to compute the eigenvalue on the trace part of B aα,c ...c n . To this end, we explicitlyparametrize the trace fluctuations as B aα,c ...c n = δ a (( c T c ...c n )) ,α = δ a ( c T c ...c n ) ,α − n −
12 ( n + 1) δ ( c c T c ...c n ) a,α , (5.27)in terms of a tensor T , traceless in its SO(6) indices. The latter relates to the trace of B as B α,ac ...c n = ( n + 2)( n + 3) n ( n + 1) T c ...c n ,α . (5.28)The action (5.23) then becomes( M B ) aα,c ...c n = − ( n + 1) ε αβ δ ( ac T c ...c n ) ,β + n − ε αβ δ ( c c T c ...c n a ) ,β + ( n + 5) ε αβ δ a ( c T c ...c n ) ,β − n − n + 1 ε αβ δ a ( c T c ...c n ) ,β = ( n + 3) ε αβ (cid:16) δ a ( c T c ...c n ) ,β − ( n − n + 1) δ ( c c T c ...c n ) a,β (cid:17) , (5.29)25uctuation representation m L ∆ B µνab,c c ...c n [ n − , , ⊕ [ n − , , ( n + 2) n + 4 B µν (( aαc ...c n )) [ n + 1 , , ± ( n + 1) n + 3 B µν bα,bc ...c n [ n − , , ± ( n + 3) n + 5 Table 1:
Masses of tensor fluctuations at level n . The conformal dimension is given by ∆ = 2 + | m | L AdS . with eigenvalue ± i ( n + 3) .We summarize the result for all non-vanishing tensor masses at level n in Table 1. It shows thatat level n the tensor spectrum contains three different representations which all come with differentmasses. In particular, the representations [ n + 1 , , ± and [ n − , , ± directly correspond to masseigenstates. In contrast, computing the Kaluza-Klein spectrum in terms of the original IIB variablesrequires diagonalisation of a coupled system of equations mixing components of different higher-dimensional fields [42]. The fluctuation Ansatz (3.15) precisely solves this diagonalisation problem:the mass eigenstates organise according to the scalar tower of harmonics and mix into the IIB fieldsupon multiplication with the twist matrix U M A . Let us make this explicit. Table 1 shows that thesame representation [ k, , ± appears twice within the massive tensor fluctuations as b + c ...c k ,α ≡ B bα,bc ...c k ,b − c ...c k ,α ≡ B (( c αc ...c k )) = B ( c αc ...c k ) − k −
12 ( k + 1) B dαd ( c ...c k − δ c k − c k ) , (5.30)at levels n = k + 1, and n = k −
1, respectively, for which we read off the mass eigenvalues ( k + 4) ,and k , respectively. This precisely reproduces the result of [42].To identify the higher-dimensional origin of the mass eigenstates, we need to combine this resultwith the dictionary between the ExFT fields and the fields of IIB supergravity [16,27]. For the originalIIB 2-form C µν α and in combination with the fluctuation Ansatz (3.15), this gives rise to an expansion C µνα = Y a ∞ X n =0 Y c ...c n B µνaα,c ...c n ( x ) , (5.31)where the Y a prefactor descends from the twist matrix U M A , and the terms under the sum correspondto the scalar tower of harmonics which fall into mass eigenstates. Expanding the product of harmonicsin (5.31) according to Y a Y c ...c n = Y ac ...c n + n n + 2) δ a (( c Y c ...c n )) , (5.32)we find for the expansion of the IIB 2-form C µνα = ∞ X n =0 (cid:16) Y ac ...c n b − µν ac ...c n ,α + n n + 2) Y c ...c n b + µνc ...c n ,α (cid:17) = 16 b + µν α + ∞ X k =1 Y c ...c k (cid:18) b − µν c ...c k ,α + k + 12 ( k + 3) b + µνc ...c k ,α (cid:19) , (5.33) For the sake of readability, here and in most of the following formulas of this subsection, we omit the space-timeindices µν which are irrelevant for the diagonalisation problem. C µν klmnα , gives rise to its expansion into different linear combinations of the same objects(5.30) according to C µν lmnpα = ∞ X k =1 ˚ ω lmnpq ∂ q Y c Y c ...c k (cid:18) b − µνc c ...c k ,α − k ( k + 1)2 ( k + 2) ( k + 3) b + µν c ...c k ,α (cid:19) + 4 ˚ ω lmnpq ˚ ζ q C µν α , (5.34)where again the mixing of different mass eigenstates originates from multiplying out the harmonicsfrom the twist matrix and the scalar tower of harmonics. We now perform the corresponding computation for the vector spectrum by evaluating the mass matrix(4.31). According to the fluctuation Ansatz (3.15), the vector fluctuations organise into the sameSO(6) × SO(2) representations as the tensor fluctuations, which we explicitly denote as A µab,c ...c n ∈ [ n, , ⊕ [ n, , ⊕ [ n − , , ⊕ [ n − , , ⊕ [ n − , , ,A µaα,c ...c n ∈ [ n + 1 , , ± ⊕ [ n − , , ± ⊕ [ n − , , ± . For the S background, the general vector mass matrix (4.31) simplifies drastically since the generators X A are compact: X ABC = − X AC B . As a consequence, the action of the mass matrix on the vectorfluctuations reduces to( M A ) A Σ = − (cid:0) P AC BD + P C ABD (cid:1) T D, ΛΩ T C, ΣΛ A B Ω + 83 T A, ΣΛ T B, ΛΩ A B Ω . (5.35)Evaluating the r.h.s., we find for the adjoint projector (2.5) with (5.2) P aαcdbβ ef + P cdaαbβ ef = 16 (cid:0) δ ab δ c [ e δ f ] d + 2 δ b [ c δ d ][ e δ f ] a (cid:1) δ αβ − ε abcdef ε αβ , P abef cdgh + P ef abcdgh = 19 δ efab δ ghcd + 16 δ abgh δ cdef + 16 δ efgh δ cdab − δ efghabcd − δ abghefcd . (5.36)Moreover, the product of T matrices takes the explicit form T cd,c ...c n Λ T ef, ΛΩ A Ω = − n ( n − δ [ c (( c A c ...c n − d ][ e δ f ] c n )) + n A f (( c ...c n − δ c n )) ecd − n A e (( c ...c n − δ c n )) f cd . (5.37)Evaluating (5.35) on the components (5.35), we then find after some computation( M A ) ab,c ...c n = 2 n A ab,c ,c ...c n + 2 n A [ a (( c ,c ...c n )) b ] + 4 n ( n − A d (( c ,c ...c n − d [ a δ b ] c n )) − n δ [ a (( c A b ] dc ...c n )) d , ( M A ) aα,c ...c n = n ( n + 3) A aα,c ...c n − n ( n + 3) A (( c | α | ,c ...c n )) a − n ( n − A bα,b (( c ...c n − δ c n )) a . (5.38)27he second equation shows that for the traceless part in A aα,c ...c n only the [ n − , ,
1] contributioncarries a mass whereas the fully symmetric part in the [ n + 1 , ,
0] remains massless. Indeed, the latterstates are absorbed as Goldstone modes into the corresponding massive tensor excitations, c.f. Table 1.To determine the mass eigenvalue of the [ n − , ,
1] vectors, we evaluate the second equation of (5.38)for components satisfying A ( a | α | ,c ...c n ) = 0 together with tracelessness and obtain( M A ) aα,c ...c n = ( n + 1) ( n + 3) A aα,c c ...c n . (5.39)It remains to compute the masses of the trace modes A bα,bc ...c n . Since these states serve as Gold-stone modes for the corresponding [ n − , ,
0] massive tensors of Table 1, they must appear massless.As a consistency check of our formulas, this can indeed explicitly be verified upon parametrizing thefluctuations as A aα,c ...c n = δ a (( c T c ...c n )) ,α = δ a ( c T c ...c n ) ,α − n −
12 ( n + 1) δ ( c c T c ...c n ) a,α , (5.40)with a traceless tensor T , just as (5.27) above, and evaluating the action (5.38).We now turn to the first equation of (5.38). Its first line shows that within the traceless part of A ab,c ,c ...c n , the [ n − , , ⊕ [ n − , ,
2] representations remain massless as required by consistency(they are the Goldstone modes for the corresponding massive tensors, c.f. Table 1). In turn, wecan compute the mass of the remaining [ n, ,
1] representation by parametrising the correspondingfluctuations as A ab,c ,c ...c n = t [ a,b ] c ...c n , (5.41)with a tensor t , traceless, and symmetric in its last n + 1 indices. The action (5.38) then takes theform ( M A ) ab,c ...c n = 2 nA ab,c c ...c n + 2 n A [ a (( c ,c ...c n )) b ] = n ( n + 2) t [ a,b ] c ...c n = n ( n + 2) A ab,c ...c n . (5.42)Finally, we compute the action (5.38) on the trace parts of A µab,c ...Cc n by parametrising these as A µab,c ...c n = δ a (( c T c ...c n )) ,b − δ b (( c T c ...c n )) ,a , ⇒ A µab,ac ...c n = 5 + 4 n + n n ( n + 1) T c ...c n ,b + ( n − n ( n + 1) T b ( c ...c n − ,c n ) , (5.43)in terms of a trace-free tensor T c ...c n ,a , symmetric in its first n − n − , ,
1] representation, we further impose that T ( C ...C n ,A ) = 0 and explicit evaluationof (5.38) after some computation turns into( M A ) ab,c ...c n = (2 + n ) (4 + n ) A ab,c ...c n . (5.44)The [ n, ,
0] representation is described by (5.43) with a fully symmetric T c ...c n ,a , however the massfor this representation is irrelevant as the corresponding modes are the ones absorbed into the massivespin-2 excitations. As discussed in section 4.1, they have to projected out from the physical spectrum.28uctuation representation m L ∆ A µa (( b,c c ...c n )) [ n, , n ( n + 2) n + 3 A µab,c c ...c n b [ n − , , ( n + 2)( n + 4) n + 5 A µαa,c ...c n [ n − , , ± ( n + 1)( n + 3) n + 4 Table 2:
Masses of vector fluctuations at level n . The conformal dimension is given by ∆ = 2+ p m L . We summarize the result for the massive vector fluctuations in Table 2. At level n the vectorspectrum contains three different mass eigenstates in different representations. Summing over alllevels, the representation [ k − , , appears twice within the massive vector fluctuations as a a,c ...c k + ≡ A ab,c ...c k b (cid:12)(cid:12) [ k − , , ,a a,c ...c k − ≡ A a (( c ,c ...c k )) (cid:12)(cid:12) [ k − , , , (5.45)at levels n = k +1, and n = k −
1, respectively, for which we read off the mass eigenvalues ( k +3)( k +5),and k −
1, respectively. This precisely reproduces the result of [42] (c.f. their equation (2.27)).To identify the higher-dimensional origin of these mass eigenstates, we again appeal to the dic-tionary between the ExFT fields and the fields of IIB supergravity [16, 27]. The vector fluctuations A ab,c ...c n descend from the off-diagonal part A µm of the 10D metric and components of the 4-form as A µm ( x, y ) = √ ∂ m Y a Y b ∞ X n =0 Y c ...c n A ab,c ...c n µ ( x ) ,A µ klm ( x, y ) = 12 ˚ ω klmpq ∂ p Y a ∂ q Y b ∞ X n =0 Y c ...c n A ab,c ...c n µ ( x ) . (5.46)Again, the sum corresponds to the tower of scalar harmonics while the prefactor comes from the twistmatrix ( U − ) AM in (3.15). A computation analogous to the one for the tensor fields in section 5.1.1.2,expanding the products of harmonics and rearranging the terms in the tower, yields A µm ( x, y ) = √ ∞ X n =0 (cid:18) ∂ m Y a Y bc ...c n a a,bc ...c n − + n n + 2) ∂ m Y a Y c ...c n a a,c ...c n + (cid:19) , = √ ∂ m Y a a a + + √ ∞ X k =1 ∂ m Y a Y c ...c k (cid:18) a a,c ...c k − + k + 12 ( k + 3) a a,c ...c k + (cid:19) , (5.47)and A µ mpq ( x, y ) = ˚ ω mpqrs ∞ X n =0 (cid:16) n + 1 n + 2 ∂ r Y a ∂ s Y b Y c ...c n a a,bc ...c n − − n ( n − n + 2) ∂ r Y a ∂ s Y c Y c ...c n a a,c ...c n + (cid:17) = ˚ ω mpqrs ∞ X k =1 ∂ r Y a ∂ s Y c ...c k (cid:18) k + 1 a a,c ...c k − − k + 12( k + 3) a a,c ...c k + (cid:19) , (5.48)showing precisely how the mass eigenstates get entangled within the higher-dimensional fields. Again,this reproduces the results from [42]. 29uctuation representation m L ∆ φ (( ab,c ...c n )) [ n + 2 , , n − n + 2 φ ab,abc ...c n − [ n − , , ( n + 2)( n + 6) n + 6 φ αβ,c ...c n [ n, , ± n ( n + 4) n + 4 φ ab,c ...c n [ n − , , n ( n + 4) n + 4 φ abcα,c ...c n [ n, , + ⊕ [ n, , − ( n − n + 3) n + 3 φ abdα,dcc ...c n − [ n − , , − ⊕ [ n − , , + ( n + 1)( n + 5) n + 5 Table 3:
Masses of scalar fluctuations at level n . The conformal dimension is given by ∆ = 2 + p m L . Let us finally sketch how to obtain the scalar mass spectrum in this example. According to the abovediscussion, at level n the scalar fluctuations are described by tensoring the coset valued fluctuations(3.5) from the lowest multiplet with the symmetric vector representation [ n, , j AB, Ω = j ab,cd, Ω = 2 δ a [ c φ d ] b, Ω ,j ab,cα, Ω = φ abcCα, Ω ,j aα,bβ, Ω = φ ab, Ω δ αβ + δ ab φ αβ, Ω , (5.49)in terms of tensors φ ab, Ω , φ αβ, Ω , φ abcα, Ω , constrained by φ [ ab ] , Ω = 0 , φ aa, Ω = 0 , φ [ αβ ] , Ω = 0 , φ αα, Ω = 0 ,φ abcα, Ω = ε abcdef ε αβ φ defβ, Ω . (5.50)Evaluating the tensor product with the harmonics, the scalar fluctuations at level n organise into therepresentations φ ab,c ...c n ∈ [ n + 2 , , ⊕ [ n, , ⊕ [ n − , , ⊕ [ n, , ⊕ [ n − , , ⊕ [ n − , , ,φ αβ,c ...c n ∈ [ n, , ± ,φ abcα,c ...c n ∈ [ n − , , ± ⊕ [ n, , + ⊕ [ n, , − ⊕ [ n − , , − ⊕ [ n − , , + . (5.51)From the previous results, we know that these modes still contain the unphysical Goldstone modes[ n − , , ⊕ [ n − , , ± ⊕ [ n, , ⊕ [ n, , , (5.52)of which the first three are absorbed by the massive vectors and appear with zero mass eigenvalue,whereas the last one is absorbed into the massive spin-2 fields and must be projected out by hand.It remains to evaluate the mass matrix (4.40) on these fluctuations. The calculation is analogous to(although somewhat more lengthy than) the ones presented above for the tensor and vector fields. Wesummarize the result for the various representations in Table 3.30 n + 2 [ n + 2 , n + [ n + 1 , ) + [ n + 1 , n + 3 [ n, n, n + 1 , n + 1 , n,
12 12 ) n + [ n, ) + [ n − , ) + [ n,
0) + [ n − ,
0) + [ n,
1) + [ n, ) n + 4 2 · [ n, n − , n − , n − , · [ n − ,
12 12 ) + [ n, n + [ n − , ) + [ n − , ) + [ n − ,
0) + [ n − ,
0) + [ n − ,
1) + [ n − , ) n + 5 [ n − , n − , n − , n − , n − ,
12 12 ) n + [ n − , ) + [ n − , n + 6 [ n − , Table 4: -BPS multiplets of SU(2 , |
4) in SO(6) × SO(4) notation [ n , n , n ]( j , j ) with Dynkin labels n i ,and ( j , j ) denoting the spins of SO(4) ∼ SU(2) × SU(2).
In the previous sections, we have determined the mass spectrum around the AdS × S background.With the fluctuation Ansatz (3.15) all mass matrices are block-diagonal level by level. With theAnsatz (3.15) for the ExFT variables, internal derivatives act via the combination (2.35) acting onthe twist matrices and (3.8) acting on the tower of harmonics. The latter action is realised by thematrices (5.11), such that the resulting field equations do not mix fluctuations over different SO(6)representations Σ. This is in contrast with the structure in the original IIB variables: after evaluatingthe products of the sphere harmonics Y Σ with the twist matrices in (3.15), fluctuations of the originalIIB fields combine linear combinations of different mass eigenstates as illustrated in (5.33), (5.34) forthe tensors and in (5.47), (5.48) for the vector fields.The same structure underlies the ExFT supersymmetry transformations [45]. As a result, allfluctuations associated with a fixed SO(6) representation Σ = [ n, ,
0] in the towers of (3.15) combineinto a single -BPS multiplet BPS[ n ]. Indeed, the mass spectrum from Tables 1–3 precisely matchesthe bosonic field content of the -BPS multiplet BPS[ n ] which we list in Table 4. The Ansatz (3.15)illustrates the fact that (except for its masses) the representation content of the full Kaluza-Kleinspectrum around a maximally symmetric vacuum such as AdS × S is obtained by tensoring the zeromodes of the torus reduction with the tower of scalar harmonics [46]. AdS , N = 2 , U(2) vacuum
In the previous section, we have worked out the Kaluza-Klein spectrum around the AdS × S back-ground corresponding to the maximally symmetric stationary point of the D = 5 SO(6) gaugedmaximal supergravity of [41]. While this analysis reproduces the known results [42, 43] for the spherespectrum, our formalism allows us to address far more complicated backgrounds which are hardly ac-cessible to standard methods. As an illustration, let us consider another stationary point in the samescalar potential which breaks supersymmetry down to N = 2 and preserves only SU(2) × U(1) of the This is not in contradiction with the fact that the BPS multiplet BPS[ n ] itself does not factorize. It is only afterimposing the explicit form of the mass matrices that the degrees of freedom are distributed among the different fields,such that for example only some of the tensor fields within the product (5.22) actually become part of the physicalspectrum. N = 1 IR superconformal fixed point of the deformation of N = 4 super-Yang-Mills by amass term for one of the three adjoint hypermultiplets [10,48]. The holographic renormalisation group(RG) flow connecting this solution to the AdS × S background has been constructed and studiedin [49].Within the D = 5 supergravity of [41], one may compute the mass spectrum around this back-ground for the fields sitting within the lowest N = 8 multiplet which at the N = 2 point decomposesinto various supermultiplets of the remaining background isometry supergroup SU(2 , | ⊗ SU(2).Organizing these multiplets according to their (external) SU(2) spin, this results in [49][ ] : D A A (3; , ; 0) ⊕ D LB (3; 0 ,
0; +2) C ⊕ D LA (3; 0 , ; 0) C ⊕ D LL (1 + √
7; 0 ,
0; 0) , [ ] : D LB ( ; ,
0; + ) C ⊕ D LA ( ; ,
0; + ) C , [ ] : D A A (2; 0 ,
0; 0) ⊕ D LB ( ; 0 ,
0; +1) C , (5.53)where we follow the notation of [50] and denote SU(2 , |
1) supermultiplets by D (∆; j , j ; r ) with thearguments referring to the conformal dimension, SU(2) ⊗ SU(2) spin and R -charge of the highest weightstate, respectively. Complex multiplets D (∆ , j , j ; r ) C come in pairs D (∆ , j , j ; r ) ⊕ D (∆ , j , j ; − r ). D LL denotes the generic long multiplet, while the notation for the shortening patterns A ℓ , B for shortand semi-short multiplets follows [51].In our fluctuation Ansatz (3.15) and the mass formulas worked out in section 4, the result (5.53)corresponds to the lowest term in the harmonics expansion, i.e. to evaluating the mass matrices onthe one-dimensional space spanned by constant harmonics Y Σ=0 = 1, with T A = 0 . In this formalismit is then straightforward to extend the result to higher levels of the Kaluza-Klein spectrum. As anillustration, let us give the result at level n = 1, again with multiplets organised according to theirexternal SU(2) spin [ ] : 2 · D LA ( ; 0 , ; +1) C ⊕ D LA ( ; , ; +1) C ⊕ D LL ( ; ,
0; +1) C ⊕ D LL (1 + √ ; 0 ,
0; +1) C ⊕ D LL (1 + √ ; 0 ,
0; +1) C , [ ] : D LB ( ; 0 ,
0; + ) C ⊕ D LA ( ; 0 ,
0; + ) C ⊕ D LL ( ; ,
0; + ) C ⊕ D LL ( ; 0 , ; + ) C ⊕ D LL (1 + √ ; , ; + ) C ⊕ D LL (1 + √ ; 0 ,
0; + ) C , [ ] : 2 · D LL (1 + √
7; 0 ,
0; 0) ⊕ D LL (1 + √
7; 0 , ; 0) C ⊕ D LA ( ; ,
0; +1) C ⊕ D LB (3; ,
0; +2) C , [ ] : D LB ( ; 0 ,
0; + ) C ⊕ D LA ( ; 0 ,
0; + ) C . A similar analysis can be performed at the higher Kaluza-Klein levels and be explicitly checked againstthe CFT results [52]. The last multiplet in the list (5.54) has been missing in equation (29) of [1], where this result was first given. .2 Vacua of 4-dimensional SO(8) gauged SUGRA
We can similarly apply our mass matrices to vacua of 4-dimensional gauged supergravity, such as theSO(8)-gauged SUGRA [53] arising from the consistent truncation of 11-dimensional supergravity on S [54]. The SO(8)-gauged SUGRA contains several interesting vacua from a holographic perspective.These include the maximally supersymmetric AdS vacuum a N = 2 AdS vacuum with SU(3) × U(1) R symmetry [55, 56], and a non-supersymmetric AdS vacuum with SO(3) × SO(3) symmetry [55, 56].Using the consistent truncation of 11-dimensional supergravity [54] all these vacua uplift to AdSsolutions of 11-dimensional supergravity.We can use our mass formulae to compute the Kaluza-Klein spectrum around these various 11-dimensional supergravity solutions. For the maximally supersymmetric AdS vacuum, correspondingto the 11-dimensional AdS × S solution, the Kaluza-Klein spectrum can be computed, followingthe steps shown in section 5.1.1 for the AdS × S solution of IIB, to recover the known spectrum ofAdS × S . Since the computation is analogous to that covered in detail in section 5.1.1, we will notrepeat it here. Instead, we will, in the following, show how our technique can be used to compute themass spectrum of the Kaluza-Klein towers of the SU(3) × U(1) R -invariant AdS [11], as well as theSO(3) × SO(3)-invariant AdS [8] vacua of 11-dimensional supergravity.To compute the Kaluza-Klein spectra of these vacua, let us set up our notation for the SO(8) gaugedsupergravity. This is best described using the SL(8) ⊂ E subgroup under which the fundamental representation of E decomposes as −→ ⊕ ′ , (cid:8) A M (cid:9) −→ n A ab , A ab o , a = 1 , . . . , . (5.54)The embedding tensor of the SO(8) gauged SUGRA is given by X MN P = X ab,cdef = − X abef cd = 2 √ δ [ e [ a δ b ][ c δ f ] d ] ,X abcd,ef = 0 ,X abN P = 0 . (5.55)The consistent truncation of 11-dimensional SUGRA to the SO(8) gauged SUGRA can be describedby a generalised Scherk-Schwarz truncation within E ExFT [6, 7], as discussed in section 2.3. Justlike for the consistent truncation of IIB supergravity on S , the twist matrices U M A can be constructedusing the elementary sphere harmonics, Y a , on S , which are just the embedding coordinates of S ⊂ R and thus satisfy Y a Y a = 1 . (5.56)For the masses of the Kaluza-Klein spectrum, we only need to know the vector components, K M , ofthe corresponding generalised parallelisable frame which are given by [6] K M = ( K ab = v ab , K ab = 0 , (5.57)where v abm = −√ g mn (cid:0) ∂ n Y [ a (cid:1) Y b ] , (5.58)33ith ˚ g the round metric on S , are the SO(8) Killing vectors of the round S .To compute the Kaluza-Klein spectrum of any vacuum of the SO(8) gauged SUGRA, we need tochoose a basis of scalar harmonics in which we expand the fields according to (3.2), (3.3) and (3.7).As discussed in section 3.2, we can simply choose the scalar harmonics of the maximally symmetricpoint, which in this case is the round S . These are given, just as in section 5.1.1, by the symmetrictraceless polynomials in the elementary sphere harmonics Y a , i.e. (cid:8) Y Σ (cid:9) = n , Y a , Y ab , . . . , Y a ...a n , . . . o , (5.59)where Y a ...a n ≡ Y (( a . . . Y a n )) denotes traceless symmetrisation in the elementary harmonics. Theindex Σ thus runs over the tower of symmetric vector representations [ n, , ,
0] of SO(8) .In order to evaluate the mass formulae, we need to compute the action of the vectors K A , definedby the generalised parallelisation (5.57), on the scalar harmonics Y Σ . For the S , these are the Killingvectors (5.58) which, like for the S , have a linear action on the harmonics given by the generators ofSO(8) in the [ n, , ,
0] representation, T M,c ...c n d ...d n = n T M, (( c (( d δ d c . . . δ d n )) c n )) , (5.60)in terms of the action on the elementary harmonics, given by T M,cd = ( T ab,cd = √ δ c [ a δ b ] d , T abcd = 0 . (5.61)It is now straightforward to apply our mass formulae to compute the Kaluza-Klein spectrum aroundany vacuum of the SO(8) gauged supergravity. All that is left to do is to dress the embedding tensor(5.55) and the linear action on the harmonics (5.61) by the 4-dimensional scalar matrix correspondingto the vacuum of interest and apply (4.4), (4.31), (4.40). AdS , N = 2 , U(3) vacuum
We will now apply our formalism to compute the Kaluza-Klein spectrum of the 11-dimensional N = 2SU(3) × U(1)-invariant AdS vacuum of 11-dimensional supergravity [11], and which can be upliftedfrom a vacuum [55, 56] of 4-dimensional N = 8 SO(8) gauged SUGRA. This 11-dimensional AdSvacuum [11] is similar in several respects to the AdS × S solution dual to the Leigh-Strassler CFTdiscussed in section 5.1.2. The 3-dimensional N = 2 CFT dual is obtained by deforming the N = 8ABJM CFT via a mass term for a single chiral supermultiplet and flowing to the IR. The correspondingholographic RG flow connecting the AdS × S solution to this N = 2 SU(3) × U(1) R vacuum hasbeen constructed in [11].Some aspects of the Kaluza-Klein spectrum of this SU(3) × U(1) R vacuum have also already beenanalysed. Due to the lack of computational techniques until now, these analyses have been limited tothe pattern of supermultiplets [57] and the spin-2 Kaluza-Klein spectrum [31]. Here we will use ourKaluza-Klein spectrometry to determine the full bosonic Kaluza-Klein spectrum of this 11-dimensionalAdS vacuum.Using our mass matrices (4.4), (4.31) and (4.40), we can compute the entire bosonic Kaluza-Kleinspectrum of this AdS vacuum of 11-dimensional supergravity. In fact, because the mass matrices are34uadratic in U(3) generators and all fields organise themselves into supermultiplets, we can extrapolatethe entire mass spectrum from the first three Kaluza-Klein levels alone. We find the following energies, E , for the graviton (GRAV), vector (VEC) and gravitino (GINO) supermultiplets with SU(3) × U(1) R representation [ p, q ] r appearing at Kaluza-Klein level n :GRAV: E = 12 + s
94 + 12 n ( n + 6) − C p,q + 12 (cid:18) r + 23 ( q − p ) (cid:19) , GINO: E = 12 + r
72 + 12 n ( n + 6) − C p,q + 12 r , VEC: E = 12 + r
174 + 12 n ( n + 6) − C p,q + 12 r , (5.62)where C p,q is the eigenvalue of the representation [ p, q ] under the quadratic Casimir operator, i.e. C p,q = 13 (cid:0) p + q + p q (cid:1) + p + q . (5.63)Since the hypermultiplets (HYP) are necessarily short, their energies are fixed by the BPS bound butcan be written similarly to the other multiplets as E = 12 + r
174 + 12 n ( n + 6) − C p,q + 12 r . (5.64)The U(3) representations of the supermultiplets appearing at a given level n can be computed bytensoring the n = 0 fields with the scalar harmonics and arranging these into supermultiplets. Forexample, the graviton supermultiplets appear at level n in the representationsGRAV: [ p, q ] p − q + a − b , (5.65)where p, q, a, b ∈ Z + are all positive integers satisfying n = p + q + a + b . The result for all super-multiplets appearing at levels n ≤ n with U(3) representation [57]SGRAV: [0 , ± n , SGINO: [ n + 1 , ( n +1) / ⊕ [0 , n + 1] − ( n +1) / , SVEC: [ n + 1 , n/ ⊕ [1 , n + 1] − n/ , HYP: [ n + 2 , ( n +2) / ⊕ [0 , n + 2] − ( n +2) / . (5.66)For these representations, our mass formulae (5.62) exactly reproduce the BPS bound for the shortmultiplets: SGRAV: E = | r | + 2 = n + 2 , SGINO: E = | r | + 32 = 116 + n , SVEC: E = | r | + 1 = n + 33 , HYP: E = | r | = n + 23 . (5.67) For the supermultiplets, we follow the notation of [57]. n ≤ [0 ,
0] [0 ,
1] [0 , SGINO (cid:0) (cid:1) − HYP (cid:0) (cid:1) − LVEC (cid:16) + √ (cid:17) [1 ,
0] [1 , (cid:0) (cid:1) + MVEC (1) [2 , (cid:0) (cid:1) + Table 5:
Energies of the multiplets of the N = 2 CFT dual to the U(3) AdS vacuum at level n = 0. Werepresent the energy E and U(1) R-charge r of a multiplet in the [ p, q ] representation of SU(3) as( E ) r . [0 ,
0] [0 ,
1] [0 ,
2] [0 , ± LGRAV (cid:16) + √ (cid:17) − SGINO (cid:0) (cid:1) − HYP (1) − LVEC (cid:16) + √ (cid:17) ± LGINO (cid:0) (cid:1) + LVEC (cid:0) (cid:1) + LVEC (cid:16) + √ (cid:17) − [1 ,
0] [1 ,
1] [1 , (cid:16) + √ (cid:17) + LGINO 2 × (cid:0) + √ (cid:1) SVEC (cid:0) (cid:1) − LGINO (cid:0) (cid:1) − LVEC (cid:16) + √ (cid:17) + [2 ,
0] [2 , (cid:0) (cid:1) + SVEC (cid:0) (cid:1) + LVEC (cid:0) (cid:1) − [3 , +1 Table 6:
Energies of the multiplets of the N = 2 CFT dual to the U(3) AdS vacuum at level n = 1.We represent the energy E and U(1) R-charge r of a multiplet appearing m times in the [ p, q ]representation of SU(3) as m × ( E ) r . ,
0] [0 ,
1] [0 ,
2] [0 ,
3] [0 , (cid:16) + √ (cid:17) conj. to [1 ,
0] conj. to [2 ,
0] conj. to [3 ,
0] conj. to [4 , ± LVEC (cid:16) + √ (cid:17) ± , 2 × (4) [1 ,
0] [1 ,
1] [1 ,
2] [1 , (cid:16) + √ (cid:17) − , (cid:16) + √ (cid:17) + LGRAV (3) conj. to [2 ,
1] conj. to [3 , (cid:0) (cid:1) − , 2 × (cid:16) + √ (cid:17) + LGINO 2 × (cid:0) + 2 √ (cid:1) ± LVEC (cid:16) + √ (cid:17) − , (cid:16) + √ (cid:17) + LVEC 2 × (cid:16) + √ (cid:17) [2 ,
0] [2 ,
1] [2 , (cid:16) + √ (cid:17) + LGINO 2 × (cid:16) + √ (cid:17) + LVEC (cid:16) + q (cid:17) LGINO (cid:0) (cid:1) − LVEC (cid:16) + √ (cid:17) − LVEC (cid:16) + √ (cid:17) − , 2 × (cid:0) (cid:1) + [3 ,
0] [3 , (cid:0) (cid:1) +1 SVEC (cid:0) (cid:1) + LVEC (cid:16) + q (cid:17) [4 , (cid:0) (cid:1) + T a b l e : E n e r g i e s o f t h e m u l t i p l e t s o f t h e N = C F T du a l t o t h e U ( ) A dS v a c uu m a t l e v e l n = . W e r e p r e s e n tt h ee n e r g y E a nd U ( ) R - c h a r g e r o f a m u l t i p l e t a pp e a r i n g m t i m e s i n t h e [ p , q ] r e p r e s e n t a t i o n o f S U ( ) a s m × ( E ) r . .2.2 AdS , N = 0 , SO(4) vacuum
The SO(8) gauged SUGRA also contains a prominent non-supersymmetric AdS vacuum with SO(3) × SO(3) symmetry [55, 56], whose uplift to 11-dimensional supergravity was constructed in [8]. Intrigu-ingly, this vacuum is stable within the N = 8 4-dimensional supergravity, with all scalar fields abovethe Breitenlohner-Freedman (BF) bound. It was long hoped that the AdS vacuum would also bestable within 11-dimensional supergravity, but since the AdS vacuum is not supersymmetric and hasfew symmetries, computing its Kaluza-Klein spectrum has remained elusive.However, using the technique laid out here, we can exploit the fact that this AdS vacuum arisesby deforming AdS × S by modes living within the SO(8) consistent truncation. As a result, it isstraightforward to compute the bosonic Kaluza-Klein spectrum using our mass formulae (4.4), (4.31)and (4.40), as was done in [12] up to level 6 above the 4-dimensional N = 8 supergravity. TheKaluza-Klein spectrum displays the curious feature that the masses of the Kaluza-Klein modes doesnot increase monotonically with the level n . Instead, even though the Kaluza-Klein scalars at levels0 and 1 are stable, the Kaluza-Klein tower contains tachyonic scalar fields at levels 2 and higherwhose masses violate the BF bound. Therefore, the techniques developed here show that this non-supersymmetric AdS vacuum is unstable within 11-dimensional supergravity, lending further evidenceto the Swampland Conjecture [58] that all non-supersymmetric AdS vacua of string theory must beunstable.Specifically, the scalar mass matrix (4.40) at level 0 yields the following mass eigenvalues( , ) : {− .
714 (2) , . } , ( , ) : {− .
714 (2) , − . , . , . } , (5.68)normalised in units of the inverse AdS length square, where ( j , j ) denotes the SO(4) ∼ SU(2) ⊗ SU(2)representations , and where the states with mass m L = − .
714 appear with multiplicity 2 . Thisreproduces the result of [59] and shows that within N = 8 supergravity, all scalar masses lie abovethe BF bound m L = − .
25 .Evaluating the mass matrix at level n = 1, we obtain the masses( , ) : {− . , − . , − . , − . , . , . , . , . , . , . , . , . } , ( , ) ⊕ ( , ) : {− . , − . , . , . , . , . , . , . } , ( , ) : {− . , − . , − . , . , . , . , . , . , . , . } , (5.69) In [12], we have used the notation ( + , + ) for these representations. n = 2, the mass eigenvalues are given by( , ) : {− . , − . , − . , . , . , . , . , . , . , . , . , . , . , . , . , . , . , . , . } , ( , ) : {− . , − . , − . , . (3) , . , . , . , . , . , . , . , . , . , . , . , . , . , . , . , . , . , . , . , . , . , . , . , . , . , . , . , . , . , . } , ( , ) : {− . , − . , . , . , . , . , . , . , . , . , . , . , . , . , . } , ( , ) ⊕ ( , ) : {− . , − . , . , . , . , . , . , . , . , . , . , . , . , . , . , . , . } , ( , ) ⊕ ( , ) : {− . , − . , . , . , . , . , . , . , . , . , . , . , . , . } , ( , ) ⊕ ( , ) : {− . , − . , . , . , . , . , . , . , . , . , . , . , . , . , . , . , . } , (5.70)and include a number of tachyonic modes m L < − .
25 . Similarly, tachyonic modes are found atthe higher Kaluza-Klein levels [12] .Moreover, the result (5.70) for the Kaluza-Klein modes at level 2 shows 27 physical massless scalarfields (i.e. massless scalars not eaten by massive vector or graviton fields), which transform in the3 · ( , ) of the SO(3) × SO(3) symmetry group. These scalars are thus infinitesimal moduli whichbreak the SO(3) × SO(3) symmetry. If these AdS -preserving deformations can be integrated up tofinite moduli, then this would give rise to a family of non-supersymmetric AdS vacua of 11-dimensionalsupergravity with symmetries smaller than SO(3) × SO(3).
In this paper, we have shown how the formalism of exceptional field theory can be used as a powerfultool for the computation of the complete Kaluza-Klein mass spectra around vacua that lie withinconsistent truncations. In particular, the method applies to deformed backgrounds that may have littleor no isometries left, as well as to non-supersymmetric backgrounds. We have derived the explicitform of the mass matrices (4.4), (4.20), (4.31), (4.40), for compactifications to D = 4 and D = 5dimensions, that are described within E and E ExFT, respectively. They are given in terms ofthe embedding tensor characterizing the consistent truncation to the lowest multiplet, together withthe (dressed) action on the scalar harmonics associated with the maximally symmetric point withinthis consistent truncation. In terms of the ExFT variables, the fluctuations are described by a simpleproduct Ansatz (3.15) between the Scherk-Schwarz twist matrices and the tower of scalar fluctuations.39ranslating this back into the original supergravity variables allows us to straightforwardly identifythe resulting mass eigenstates in higher dimensions.We have illustrated the formalism in various examples. First, we have re-derived the full bosonicKaluza-Klein spectrum around the maximal symmetric AdS × S solution of IIB supergravity, findingagreement with the classic results of [42,43]. Next, we have applied the method to compute the higherKaluza-Klein levels around some prominent AdS vacua with less supersymmetry in D = 5 and D = 4dimensions. This provides valuable information for various holographic dualities and for the stabilityanalysis of non-supersymmetric vacua. Although in this paper, we have restricted the analysis to thebosonic mass spectrum, it is clear that the fermionic mass spectrum can be computed in completeanalogy based on the structures of supersymmetric ExFT [60, 45]. Also, while we have restricted ourexamples to AdS vacua which are of particular interest in the holographic context, the method andthe explicit mass matrices likewise apply for Minkowski and dS vacua.There are many further potential applications of the methods presented in this paper. Some recentand rather exhaustive scans of the potentials of maximal SO(8) gauged supergravity in D = 4 [61] andSO(6) gauged supergravity in D = 5 [62, 63] have revealed a plethora of AdS vacua, most of whichpreserve very few bosonic (and super-)symmetries. Our analysis of the Kaluza-Klein spectrum can beapplied to all of these. Likewise, our method applies to vacua within other maximal supergravities,such as the D = 4, ISO(7) gauged supergravity which describes the consistent truncation of massiveIIA supergravity [64] on S and exhibits a rich vacuum structure [65]. In this case, the maximallysymmetric point, which is used to construct the scalar harmonics, would be the round S , eventhough this is not a vacuum of the theory. Another interesting gauging is the D = 4 SUGRA with[SO(1 , × SO(6)] ⋉ R gauge group whose potential carries numerous interesting AdS vacua [66, 67]with IIB origin [68]. The analysis of their Kaluza-Klein spectra will require a proper treatment of thenon-compact gauge group generator whose associated non-compact direction will have to undergo aproper S-folding in order to extract a discrete spectrum of harmonics. For the spin-2 spectrum, thiswas analysed, for example, in [37].We have derived in this paper the explicit mass matrices for E and E ExFT. However, thefluctuation Ansatz (3.15) is universal and allows to work out the mass matrices for other exceptionalfield theories, giving rise to the Kaluza-Klein spectra in compactifications to other dimensions. Itwould also be very interesting to extend the formalism to vacua sitting in consistent truncations thatpreserve a lower number of supersymmetries building on the constructions of [69, 70].
Acknowledgements
We would like to thank D. Andriot, N. Bobev, C. Eloy, A. Guarino, M. Gutperle, O. Hohm, G. Larios,H. Nicolai, C. Nunez, B. Robinson, E. Sezgin, A. Tomasiello, J. van Muiden, O. Varela, D. Waldram,and N. Warner for useful discussions and comments. We acknowledge the Mainz Institute for Theo-retical Physics (MITP) of the Cluster of Excellence PRISMA+ (Project ID 39083149) for hospitalitywhile this work was initiated. EM is supported by the Deutsche Forschungsgemeinschaft (DFG, Ger-man Research Foundation) via the Emmy Noether program “Exploring the landscape of string theoryflux vacua using exceptional field theory” (project number 426510644).40 eferences [1] E. Malek and H. Samtleben,
Kaluza-Klein spectrometry for supergravity , Phys. Rev. Lett. (2020) 101601 [ ].[2] G. Aldazabal, W. Baron, D. Marqu´es and C. N´u˜nez,
The effective action of double field theory , JHEP (2011) 052 [ ].[3] D. Geissb¨uhler,
Double field theory and N = 4 gauged supergravity , JHEP (2011) 116[ ].[4] M. Gra˜na and D. Marqu´es,
Gauged double field theory , JHEP (2012) 020 [ ].[5] D.S. Berman, E.T. Musaev and D.C. Thompson,
Duality invariant M-theory: Gaugedsupergravities and Scherk-Schwarz reductions , JHEP (2012) 174 [ ].[6] K. Lee, C. Strickland-Constable and D. Waldram,
Spheres, generalised parallelisability andconsistent truncations , Fortsch. Phys. (2017) 1700048 [ ].[7] O. Hohm and H. Samtleben, Consistent Kaluza-Klein truncations via exceptional field theory , JHEP (2015) 131 [ ].[8] H. Godazgar, M. Godazgar, O. Kr¨uger, H. Nicolai and K. Pilch,
An SO(3) × SO(3) invariantsolution of D = 11 supergravity , JHEP (2015) 056 [ ].[9] K. Pilch and N.P. Warner, A new supersymmetric compactification of chiral IIB supergravity , Phys. Lett.
B487 (2000) 22 [ hep-th/0002192 ].[10] R.G. Leigh and M.J. Strassler,
Exactly marginal operators and duality in four-dimensional N = 1 supersymmetric gauge theory , Nucl. Phys.
B447 (1995) 95 [ hep-th/9503121 ].[11] R. Corrado, K. Pilch and N.P. Warner, An N = 2 supersymmetric membrane flow , Nucl. Phys.
B629 (2002) 74 [ hep-th/0107220 ].[12] E. Malek, H. Nicolai and H. Samtleben,
Tachyonic Kaluza-Klein modes and the AdS swamplandconjecture , JHEP (2020) 159 [ ].[13] O. Hohm and H. Samtleben, Exceptional form of D = 11 supergravity , Phys. Rev. Lett. (2013) 231601 [ ].[14] O. Hohm and H. Samtleben,
Exceptional field theory I: E covariant form of M-theory andtype IIB , Phys.Rev.
D89 (2014) 066016 [ ].[15] O. Hohm and H. Samtleben,
Exceptional field theory II: E , Phys.Rev.
D89 (2014) 066017[ ].[16] A. Baguet, O. Hohm and H. Samtleben, E exceptional field theory: Review and embedding oftype IIB , PoS
CORFU2014 (2015) 133 [ ].4117] E. Cremmer,
Supergravities in 5 dimensions , in
Superspace and supergravity : proceedings ,S. Hawking and M. Rocek., eds., Cambridge Univ. Press, 1980.[18] B. de Wit, H. Samtleben and M. Trigiante,
The maximal D = 5 supergravities , Nucl. Phys.
B716 (2005) 215 [ hep-th/0412173 ].[19] A. Coimbra, C. Strickland-Constable and D. Waldram, E d ( d ) × R + generalised geometry,connections and M theory , JHEP (2014) 054 [ ].[20] D.S. Berman, M. Cederwall, A. Kleinschmidt and D.C. Thompson,
The gauge structure ofgeneralised diffeomorphisms , JHEP (2013) 064 [ ].[21] F. Ciceri, A. Guarino and G. Inverso,
The exceptional story of massive IIA supergravity , JHEP (2016) 154 [ ].[22] D. Cassani, O. de Felice, M. Petrini, C. Strickland-Constable and D. Waldram, Exceptionalgeneralised geometry for massive IIA and consistent reductions , JHEP (2016) 074[ ].[23] B. de Wit, H. Samtleben and M. Trigiante, On Lagrangians and gaugings of maximalsupergravities , Nucl. Phys.
B655 (2003) 93 [ hep-th/0212239 ].[24] B. de Wit, H. Samtleben and M. Trigiante,
The maximal D = 4 supergravities , JHEP (2007) 049 [ ].[25] M. Gra˜na, R. Minasian, M. Petrini and D. Waldram, T-duality, generalized geometry andnon-geometric backgrounds , JHEP (2009) 075 [ ].[26] G. Inverso,
Generalised Scherk-Schwarz reductions from gauged supergravity , JHEP (2017) 124 [ ].[27] A. Baguet, O. Hohm and H. Samtleben, Consistent type IIB reductions to maximal 5Dsupergravity , Phys. Rev.
D92 (2015) 065004 [ ].[28] N.R. Constable and R.C. Myers,
Spin two glueballs, positive energy theorems and the AdS /CFT correspondence , JHEP (1999) 037 [ hep-th/9908175 ].[29] R.C. Brower, S.D. Mathur and C.-I. Tan, Discrete spectrum of the graviton in the AdS blackhole background , Nucl. Phys.
B574 (2000) 219 [ hep-th/9908196 ].[30] C. Csaki, J. Erlich, T.J. Hollowood and Y. Shirman,
Universal aspects of gravity localized onthick branes , Nucl. Phys.
B581 (2000) 309 [ hep-th/0001033 ].[31] I.R. Klebanov, S.S. Pufu and F.D. Rocha,
The squashed, stretched, and warped gets perturbed , JHEP (2009) 019 [ ].[32] C. Bachas and J. Estes, Spin-2 spectrum of defect theories , JHEP (2011) 005 [ ].4233] J.-M. Richard, R. Terrisse and D. Tsimpis, On the spin-2 Kaluza-Klein spectrum of
AdS × S ( B ), JHEP (2014) 144 [ ].[34] A. Passias and A. Tomasiello, Spin-2 spectrum of six-dimensional field theories , JHEP (2016) 050 [ ].[35] Y. Pang, J. Rong and O. Varela, Spectrum universality properties of holographic Chern-Simonstheories , JHEP (2018) 061 [ ].[36] M. Gutperle, C.F. Uhlemann and O. Varela, Massive spin 2 excitations in
AdS × S warpedspacetimes , JHEP (2018) 091 [ ].[37] K. Dimmitt, G. Larios, P. Ntokos and O. Varela, Universal properties of Kaluza-Klein gravitons , JHEP (2020) 039 [ ].[38] L. Dolan and M. Duff, Kac-Moody symmetries of Kaluza-Klein theories , Phys. Rev. Lett. (1984) 14.[39] Y. Cho and S. Zoh, Explicit construction of massive spin two fields in Kaluza-Klein theory , Phys.Rev.
D46 (1992) 2290.[40] O. Hohm,
On the infinite-dimensional spin-2 symmetries in Kaluza-Klein theories , Phys.Rev.
D73 (2006) 044003 [ hep-th/0511165 ].[41] M. G¨unaydin, L.J. Romans and N.P. Warner,
Compact and noncompact gauged supergravitytheories in five dimensions , Nucl. Phys.
B272 (1986) 598.[42] H.J. Kim, L.J. Romans and P. van Nieuwenhuizen,
The mass spectrum of chiral ten-dimensional N = 2 supergravity on S , Phys. Rev.
D32 (1985) 389.[43] M. G¨unaydin and N. Marcus,
The spectrum of the S compactification of the chiral N = 2 , D = 10 supergravity and the unitary supermultiplets of U (2 , / Class. Quant. Grav. (1985) L11.[44] A. Salam and J.A. Strathdee, On Kaluza-Klein theory , Annals Phys. (1982) 316.[45] E. Musaev and H. Samtleben,
Fermions and supersymmetry in E exceptional field theory , JHEP (2015) 027 [ ].[46] M. Bianchi, J.F. Morales and H. Samtleben,
On stringy AdS × S and higher spin holography , JHEP (2003) 062 [ hep-th/0305052 ].[47] A. Khavaev, K. Pilch and N.P. Warner, New vacua of gauged N = 8 supergravity infive-dimensions , Phys.Lett.
B487 (2000) 14 [ hep-th/9812035 ].[48] A. Karch, D. L¨ust and A. Miemiec,
New N = 1 superconformal field theories and theirsupergravity description , Phys. Lett.
B454 (1999) 265 [ hep-th/9901041 ].4349] D. Freedman, S. Gubser, K. Pilch and N. Warner,
Renormalization group flows from holographysupersymmetry and a c -theorem , Adv. Theor. Math. Phys. (1999) 363 [ hep-th/9904017 ].[50] M. Flato and C. Fronsdal, Representations of conformal supersymmetry , Lett. Math. Phys. (1984) 159.[51] C. Cordova, T.T. Dumitrescu and K. Intriligator, Multiplets of superconformal symmetry indiverse dimensions , JHEP (2019) 163 [ ].[52] N. Bobev, E. Malek, B. Robinson, H. Samtleben and J. van Muiden, work in progress.[53] B. de Wit and H. Nicolai, N = 8 supergravity , Nucl. Phys.
B208 (1982) 323.[54] B. de Wit, H. Nicolai and N. Warner,
The embedding of gauged N = 8 supergravity into d = 11 supergravity , Nucl.Phys.
B255 (1985) 29.[55] N.P. Warner,
Some properties of the scalar potential in gauged supergravity theories , Nucl. Phys.
B231 (1984) 250.[56] N.P. Warner,
Some new extrema of the scalar potential of gauged N = 8 supergravity , Phys. Lett.
B128 (1983) 169.[57] I. Klebanov, T. Klose and A. Murugan,
AdS /CFT squashed, stretched and warped , JHEP (2009) 140 [ ].[58] H. Ooguri and C. Vafa, Non-supersymmetric AdS and the swampland , Adv. Theor. Math. Phys. (2017) 1787 [ ].[59] T. Fischbacher, K. Pilch and N.P. Warner, New supersymmetric and stable, non-supersymmetricphases in supergravity and holographic field theory , .[60] H. Godazgar, M. Godazgar, O. Hohm, H. Nicolai and H. Samtleben, Supersymmetric E exceptional field theory , JHEP (2014) 044 [ ].[61] I.M. Comsa, M. Firsching and T. Fischbacher,
SO(8) supergravity and the magic of machinelearning , JHEP (2019) 057 [ ].[62] C. Krishnan, V. Mohan and S. Ray, Machine learning N = 8 , D = 5 gauged supergravity , Fortsch. Phys. (2020) 2000027 [ ].[63] N. Bobev, T. Fischbacher, F.F. Gautason and K. Pilch, A cornucopia of AdS vacua , JHEP (2020) 240 [ ].[64] A. Guarino, D.L. Jafferis and O. Varela, String theory origin of dyonic N = 8 supergravity andits Chern-Simons duals , Phys. Rev. Lett. (2015) 091601 [ ].[65] A. Guarino and O. Varela,
Dyonic ISO(7) supergravity and the duality hierarchy , JHEP (2016) 079 [ ]. 4466] G. Dall’Agata and G. Inverso, On the vacua of N = 8 gauged supergravity in 4 dimensions , Nucl.Phys.
B859 (2012) 70 [ ].[67] A. Guarino, C. Sterckx and M. Trigiante, N = 2 supersymmetric S-folds , JHEP (2020) 050[ ].[68] G. Inverso, H. Samtleben and M. Trigiante, Type II supergravity origin of dyonic gaugings , Phys. Rev.
D95 (2017) 066020 [ ].[69] E. Malek,
Half-maximal supersymmetry from exceptional field theory , Fortsch. Phys. (2017) 1700061 [ ].[70] D. Cassani, G. Josse, M. Petrini and D. Waldram, Systematics of consistent truncations fromgeneralised geometry , JHEP (2019) 017 [1907.06730