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JHEP04(2009)102 = 2 N April 27, 2009 March 10, 2009 tifications : February 2, 2009 : : = 11 supergravity to , Published Accepted D Received leads to minimal = 2 supersymmetry. In the -theory with non-relativistic d imply that there is a four- 10.1088/1126-6708/2009/04/102 analysis for an arbitrary weak nd form-field flux and lead to ki-Einstein manifold. At the N fields, containing the breathing h ctions are generalizations of type doi: College, , Published by IOP Publishing for SISSA [email protected] , = 8 supersymmetry containing both massless and massive [email protected] = 1 supergravity with massive fields. The straightforward N , N 0901.0676 AdS-CFT Correspondence, Supergravity Models, Flux compac We construct consistent Kaluza-Klein reductions of [email protected] manifold leading to an SISSA 2009 2 London SW7 2PE, U.K. E-mail: London SW7 2AZ, U.K. The Institute for Mathematical Sciences, Imperial College Theoretical Physics Group, Blackett Laboratory, Imperial [email protected] c ArXiv ePrint: conformal symmetry. Keywords: spin two fields. We use our results to construct solutions of M dimensional Lagrangian with extension of our results to the case of the seven-sphere woul context of flux compactifications, the Sasaki-EinsteinIIA redu SU(3)-structure reductions whicha include massive both universal metric tensor a multiplet.G We carry out a similar

Abstract: Jerome P. Gauntlett, Seok Kim, Oscar Varela and Daniel Waldram gauged supergravity, to also includemode a of multiplet of the massive Sasaki-Einstein space, and still consistent wit Consistent supersymmetric Kaluza-Klein truncations with massive modes four dimensions using anlevel of arbitrary bosonic seven-dimensional Sasa fields, we extend the known reduction, which JHEP04(2009)102 4 6 8 1 4 11 12 12 13 16 18 20 23 11 14 14 + 1 d space to 5 SE ]) that one can 2 ]. 5 – 3 , 1 = 11 supergravity, dual ravity on a n which one can perform D ons [ ] (see also [ ent multiplet which includes 1 plet containing the metric. In heories. Starting with any su- solutions of type IIB supergravity, 5 SE solutions of × 7 5 -dimensional Sasaki-Einstein manifold. In n SE solution 7 AdS × 7 to obtain a gauged supergravity theory in – 1 – 4 is an , it was conjectured [ M SE = 11 supergravity that is the warped product of an n symmetry currents. This conjecture has now been AdS M × D R SE 4 AdS = 10 or = 4, and d D = 3, where case d 2 G = 2 supersymmetry = 1 SCFTs in N space with an internal space N = 2 supergravity = 11 supergravity reduced on SE = 2 SCFTs in +1 d One simple class of examples consists of N 3.1 Minimal gauged3.2 supergravity Scalars 3.3 The weak 3.4 Massive vector D 2.1 The consistent2.2 Kaluza-Klein reduction Masses and2.3 dual operators N AdS proven to be true for a number of general classes of AdS soluti 1 Introduction It is nowconsistent understood Kaluza-Klein that (KK) there reductionspersymmetric of are solution supergravity of very t general situations i B Details on the KK reduction C 5 Skew-whiffing 6 Discussion A Supersymmetry of the 4 Solutions 3 Some further truncations Contents 1 Introduction 2 always consistently reduce on the space dimensions, incorporating only thethe fields dual SCFT of these the fieldsthe supermulti are dual energy to the momentum superconformal tensor curr and to dual to the former case it is known that one can reduce type IIB superg JHEP04(2009)102 ] ]. 1 13 ]. ]. In = 11 these 1 20 7 ds [ D – S n has the 17 = = 4 [ ]. The ten- ]) and four ] which had 7 21 11 D space, locally, 22 – 5 SE 8 or SE KK reduction of [ 5 S = n in terms of flux com- similar extensions of the will show that there is a 5 sive scalars. One of these ontains a tensor multiplet ction containing a massive ng from the twisting of the spect to the size of the base. gravity theory are massless, nt KK reduction of type IIB y momentum tensor and the ing and squashing modes for tuklegfield.Stückelberg This gives SE l discussed in [ t truncations to the maximal cture manifold [ d form. Kähler The presence g and squashing modes appear al results see [ n n the off-shell four-dimensional n supermultiplet as well as the mations in analysing which KK kelberg field to give mass to the n base, the other massive scalar . Viewing the ion in the SU(3) structure. Note 5 ] and then analyzed in [ is not closed lead to a gauging of the 16 ]. Similarly, one can reduce SE 6 3 ], respectively. For these special cases, 7 space. For the special case of the seven- KE and a complex scalar arising from a RR = 2 gauged supergravity in 7 = 5 [ B N SE ] and [ D , and the universal vector multiplet containing 6 ]. 6 – 2 – 15 , KE 0) form on 14 , = 2 action correspond to the universal tensor multiplet manifold locally as a U(1) fibration over a Kähler-Einstein N 7 0) form on SE , space to get minimal 7 ] can be generalised to also include some massive bosonic fiel = 11 supergravity on a 3 SE D = 1 gauged supergravity in , one can reduce from M-theory to type IIA. The truncation the ], respectively. 6 N 12 space of [ KE ]. In this paper we shall show that one can also generalise the 5 15 , SE Interestingly, it has recently been shown that the consiste We note that our truncation also has a natural interpretatio In order to understand this in more detail, here we will study 14 = 2 theory, in addition to the gravity multiplet, the action c -symmetry current, respectively. For the special cases whe potential parallel to the (3 as a U(1) fibration overarises a from the four-dimensional mode Kähler-Einstei thatThis squashes work the thus size extends of earlier thethe fibre work special with on case re including of such the breath five-sphere in [ of the background four-form flux,U(1) and fibration and the the “metric fact fluxes” that the comi (3 a vector and scalars that arises from scaling the complexifie manifold, which contains the dilaton, the NS two-form on a get minimal to include massive fields:consistent KK at reduction the that includesmassive level supermultiplet the of that massless the contains gravito the bosonicN breathing mode. fields we I pactifications. Viewing the sor and vector multiplets in the a similar structure butthat considered since different our intrinsic truncation tors ismodes consistent should there be are kept no in approxi the four-dimensional theory. supergravity on a The bosonic fields includedmassive massive scalars arises gauge from fields the as breathing mode well of as the mas KK reductions of it is expected or known that there are more general consisten both cases, the bosonicconsisting fields of the in metric the andR lower the dimensional gauge super field, dual to the energ gauged supergravities in fivedimensions dimensions [ (for various parti structure of a IIA reduction on a six-dimensional SU(3) stru four-dimensional theory. This is complementary to the mode sphere some results on KKin reductions involving [ the breathin together with a single vectortensor multiplet which multiplet. acts We as show a thatvector Stüc one multiplet can with also a dualizea gauged to simple hypermultiplet get example acting an of as a the the mechanism first observed in [ truncations were shown to be consistent in [ JHEP04(2009)102 7 ]. ow = 1 29 SE [ = 2, = 11 7 ]. z holon- N ] there D one can 2 23 26 SE 5 G ]. We will , 29 25 SE ] is rather trivial ]. spaces were used 1 solutions that are 5 24 7 = 4 and also = 8 supersymmetry, to maximally super- with weak z M SE 5 7 lt for a consistent KK N S is a non-linear extension × M turn to this point in the 4 solutions. It is well known eductions on ctions is that they provide . The consistent KK trun- 7 7 onent hich again contains massive in supergravity [ stent truncation on the ermultiplet, preserving all of the massless graviton super- ended to include the massive re found. The solutions with th massless and massive spin solutions with any amount of M energy KK analysis of AdS tring/M-theory that possess a he orientation on the SE ed” solution obtained by revers- 7 ent KK reduction to pure rect the consistent KK reduction g with the work of [ × the case of M 4 × 4 AdS = 2 gauged supergravity theory with an = 1 supersymmetry [ structure that was analysed in [ = 3. The conjecture of [ 2 AdS N N d G the skew-whiffed solution does not preserve 7 ] the KK reductions on ]. Here we shall construct similar solutions in S – 3 – 13 spaces that exhibit a non-relativistic conformal 28 ] in the context of solutions corresponding to holo- 7 , = 3. 30 27 z SE ] and offers the possibility of finding exact embeddings of 33 – , is that the supermultiplet containing the breathing mode n 31 8 there is a consistent KK truncation that includes both the 7 S ≤ ] to be consistent with = 11 supergravity. 13 N D ≤ = 11 supergravity on seven-dimensional manifolds D = 1 superconformal field theories in N vacuum that uplifts to the skew-whiffed solution. Our action 4 Our presentation will focus on supersymmetric A principal motivation for constructing consistent KK redu Given these results, it is plausible that for A simple modification of our ansatz leads to an analogous resu A similar result could also hold for reductions of type IIB on = 11 supergravity for arbitrary = 1 chiral multiplet that contains the breathing mode of = 2 were independently found in [ supergravity reduced on manifolds with weak dual to omy. Recall that such manifolds can be used to construct supersymmetry 1 symmetry with dynamical exponent supergravity. Here, however, we willN see that this can be ext cation that we construct is compatible with the general low- graviton supermultiplet and the massive breathing mode sup the supersymmetry. A particularly interesting feature for to construct such solutions and examples with dynamical exp graphic superconductivity [ Apart from the specialany case supersymmetry, but of is the knownshow round to that be perturbatively for stable the skew-whiffed solutions there is also a consi such solutions into D for this case since it just says that there should be a consist powerful methods to constructhas explicit been solutions. some Startin recentnon-relativistic interest conformal in symmetry. constructing In solutions [ of s reduction of that for each supersymmetric solution thereing is the a “skew-whiff sign of the four-form flux, or equivalently changing t of one of those considered recently in [ AdS and hence possessing anz enlarged symmetry, Schrödinger we arising from reduction on space to the bosonic fields of a four-dimensional extend the ansatz of [ symmetric theories in fivemultiplet spacetime and dimensions the containing massivespin breathing 2 mode fields. supermultiplet, What w is much more certain, however, is that for r contains massive spin-2 fields. Thus ifwould our lead conjecture to is cor a2 four-dimensional fields, interacting which theory has withdiscussion been bo section widely later. thought not to exist. We will re JHEP04(2009)102 A res (2.2) (2.3) (2.5) (2.4) = 11 super- .c.” denotes ], as we shall is a complex D 1 are scalar fields 1 e, normalised so ) χ V 1 that is dual to the , A , our-form we take η ) c (as for a unit-radius 1 + and A η U ( + -forms, solutions of ∧ p η ( 7 J Ω + c.c.] and the Sasaki-Einstein metric ⊗ h those for the Sasaki-Einstein ) ∧ ng this class of solutions. Our η ∧ there is also a globally defined 7 SE 4 ) 1 ) 1 7 ⊗ 1 3!. The Sasaki-Einstein structure × H A SE / η A ]. For completeness, in appendix 4 ( 3 AdS + SE 2 + + 34 . iJ J η )+ ds 3, are real 8 η ( AdS 6 ( , ∧ Ω V − 7 2 χ 2 2 , ) (2.1) ∧ )+ e 4 4 KE = H ( J , iη 2 ∗ of the constant bosonic modes that arise from Ω+ )+ = 1 Ω – 4 – )+ 6 ds AdS AdS ∧ 1 ( = 2 all ∧ p 1 2 A χ Ω = 4 KE vol( , )= ds dη ( d + p 7 3 [ 2 4 8 3 1 η H ( √ ds SE = = ( U ∧ + 2 2 2 4 space has a globally defined one-form 3 e J G Ω). It generalises the ansatz considered in [ ds 7 ds H + ∧ 2 4 + = 11 supergravity is given by ) is indeed supersymmetric. SE 4 η, J, hJ ds D 2.1 = 11 supergravity are as in [ vol 0)-form Ω that locally define the and Kähler complex structu = +2 , f 2 D = ds are real scalars, 4 ) is the standard unit-radius metric on ) is a local metric Kähler-Einstein with positive curvatur G 4 h 6 is a complex scalar on the four-dimensional spacetime and “c and a (3 ) respectively and satisfy Ω 6 KE χ AdS is an arbitrary metric on a four-dimensional spacetime, J ( and ( 2 2 4 2 ) is normalised so that the Ricci tensor is six times the metri is a one-form defined on the four-dimensional space. For the f = 11 supergravity reduced on SE KE 7 f ds ( ds ds 1 -structure tensors ( 2 A Notice that this ansatz incorporates D SE G ds ( 2 two-form that the Ricci tensor is eight times the metric. On on implies that Reeb Killing vector, and locally we can write where Our conventions for round seven-sphere). The Our starting point is the class of supersymmetric ds where structure, the solution ( we show in detail that given these conventions, together wit 2 ansatz for the metric of 2.1 The consistentWe Kaluza-Klein now reduction investigate consistent Kaluza-Klein reductions usi gravity given by where and one-form, complex conjugate. the where JHEP04(2009)102 . . V B = 2 +2 ) (2.7) (2.9) (2.8) (2.6) ∗ U 2 (2.10) . | 4 N ). The − e Dχ χDχ 6 | 2.1 = 11 equa- U 2 − − 6 | rth observing D − χ U | e 2 Dχ Ω + c.c. V 2 3 − ∗ n with respect to 2 e χ ∧ − = 11 supergravity. − ( U 2 solution ( 6 ur-dimensional fields ∧ ) D + 48 Dχ − 3 7 h e i 4 H ∇ e four-dimensional fields µν ( i 24 and also ensures that the J SE − ion for the four-form, we 4 f the Calabi-Yau, and the 3 H V ∧ − 2 can be found in appendix Ω × µν 2 − − wing four-dimensional action: contain the “breathing mode” . V 4 , keeping the universal ∧ 3 ) defines a consistent KK trun- U 2 H . H 3 4 ) V − H ) and incorporates a convenient U the same off-shell supermultiplet 1 ) − 4 2.9 U 2 e ∧ 2 CY | A AdS − )+ 12 3 2 e 1 χ 1 and ), ( space drops out of the | + × − B.12 hF 3 4 χ A A e − , where 7 1 η 1 U + 2 − 2.4 ( + S + 2 V χ 2 + 6 iA 2 SE Dχ η h | 2 4 ( µνρ i B 4 χ ·∇ 3 F | − ∧ H − √ ∧ U 3 + 2 + (1 + 2 = – 5 – 2 1 dχ 1 ∇ µνρ + 2 Ω = 0 this ansatz would be the same ansatz that H V F 1 d h − 3 H + J ≡ χ dB dB dA U h 4 V = 6 ∧ 2 + 12 1+ − = = = − − 2 vol Dχ e 2 e and ) dJ 3 2 2 hJ F F 2 1 U 18 H H 12 | = = 6 ∧ dh comes from solving ( . Thus the ansatz ( ∇ χ − | − 2 f B ) + 2 dη = f with 1 U H + µν 8 1 = 11 supergravity on 2 A 2 χ F − h + 30( H h e D + 2 µν R η h F + 3 and ( 2 1+ V 24 V 2 ∧ H e + V J 1 4 U ∧ − 6 − − 2 ∧ U 6 space, as we will discuss more explicitly below. It is also wo ge , U, V, h − 1 dh 6 hH − e 3 = 4 theory includes the supersymmetric + space and the “squashing mode” that scales the fibre directio √ Ω instead satisfied , A − x 1 7 KE = 6 4 D d 4 ,B Z η, J, SE 2 G Z + We now substitute this ansatz into the equations of motion of We find that all dependence on the internal ,B = µν S tions of motion andwhich are we written are in appendix left with equations of motion for th vector multiplet, with scalarsuniversal coming hypermultiplet. from In the particular, we volumedegrees should of mode expect freedom o to appear in our four-dimensional theory We will simply summarise the main results here. More details one would use to reduce and discuss in section 3.1. Together the two scalar fields expression for the four-form can be tidied up a little to read By analysing thefind Bianchi that identities and theg the dynamical equations degrees of of mot freedom turn out to be the fo integration constant which fixesreduced the radius of the AdS vacuum cation. The equations of motion can be derived from the follo Note that the expression for of the that if the local Furthermore we find that JHEP04(2009)102 . i 2 | 2 | and χ ) | g Dχ ∗ (2.14) (2.13) (2.12) (2.11) (2.15) | V V 3 U + 6 − U − U V χDχ . 6 e e − 12 3 2 3 − U − 8 ≡ H e − = 0) is − , preserving the 2 ) E Dχ ∧ e 6 ) 24 χ 1 ∗ g as the “squashing 1 h our KK ansatz for χ A − = A ( ∇ u v KE ( + 48 + V h ∧ 3 V η 3 2 − = µν ( y diagonalized. To find ) + 6 and − U H 2 form for a massless gauge H ze of U ⊗ i V u 18 ). This “vacuum solution” 4 4 B 3 ) 4 µν − − 1 = e e − H A 2 + 2 2 3 uum solution, by analysing the 3 frame metric U V AdS 1 =6, the equations of motion are + 2 + − H ( as a basic field by introducing a | f 2 η U χ V ∧ 1 ( dB 2 | ). We can work out the masses of ( e u ds 1 1 4 + 4 3 dB ·∇ 12 A 2.1 ′ 2 v e 2 ∧ ∗ (setting v − U h 1 3 ) ≡ 3 1 +6 2 2 ∇ dH 2 ′ 2 )+ + 6 B acting as a field. Stückelberg However the + µνρ 6 F ,B ∧ 1+ H u 1 1 u − 1 H ∧ − 2 + 2 ˜ B 2 – 6 – B KE ) ,B =0, and thus 18 1 ( 3 1 F µνρ = = 6 2 V ). One immediately deduces that the scalar fields χ 3 − A H as the “breathing mode” and h dB Z ∇ ds = U V ( ( U with V u v h − 2 3 3 2.11 2 2 − 12 2 2 e − U = 72. Note that in terms of = − F B − e and adding a term 2 v 14 solution given in ( 1 2 3 V 12 = 40. One can diagonalise the terms involving the scalar ∧ 1 ) − 3 m 7 ˜ e H 2 = 2 χ B U − v/ 2 2 H U h m ∇ SE e 2 ∧ ∗ µν = h 24 3 + F × 24( F H − 4 2 E µν +3 − 1 2 = 2 F V = 16 and ds E 2 + V H 3 + space. 2 u AdS R H U 2 v/ = 40 and ∧ +3 7 m 7 10 F = 2 h 2 U E − − 3 6 by writing g ) can be written e e m e SE ∧ ∗ H 6 hH − 4 1 V = 2 3 2.4 √ − − 2 F − x 1 2 have 4 and a massive two-form ds and d − 1 χ Z U A Z Z + The quadratic action for the fields = and S the metric ( mode” that squashes the size of the fibre with respect to the si solved by taking the four-dimensional metric to be It is also helpful to write this with respect to the Einstein- If one ignores thefield final term, we see that this has the standard presence of the finalthe term mass means eigenstates, that it the is helpful fields to are regard not properl and we can identify the scalar field and we find that the other fields, considered asquadratic perturbations terms about in this the vac h Lagrangian ( fields we find 2.2 Masses andWhen dual we operators set uplifts to give the volume of the Lagrange multiplier one-form JHEP04(2009)102 2 ]), B ding 37 (2.21) (2.20) (2.17) (2.23) (2.18) (2.19) (2.16) , , we can 36 2 B , and after to obtain a ,has∆=2 1 1 . 2 ˜ . . B A B d and 1 ) 1 1 1 B = 48. ≡ A 3 B ∧B 2 ) and get the action 2 √ − 3 and the two-form 1 m 4 ˜ 1 H H B 1 ˜ 3 3 B − B ( √ √ g the formula 4 ∧ 4 da operators in the dual SCFT he two-form t (see for example [ 3 s vector field, − − = 6 (2.22) H 1 v , where ∧ ∗ 2 da B 2 ∆ ( d 1 + 6 B m 2 2 B are given by , , 2 3 ∧ ∗ + ˜ − ∗ H v − 1 , one obtains the standard Stückelberg 1 2 m √ 1 , ) 1 ˜ 2 = 4 B B ˜ p B ∧ ∗ ˜ B d 3 H 2 and − − 2 9+ 2 3 1 = 5 H ˜ 1 − + 3 H χ p da − is indeed massive with A 3 2 1 − ∗ = 48, defined by the fields 2 1 (3 3 adH 1 A 2 1 2 3 − – 7 – B H =∆ = p ± 2 u, h, χ brings one back to the original quadratic action. Z 3 m √ 1 2 + 2 1 h 3 ′ 2 2 1 H 1 ∧ ∗ ∆ ˜ = = ± H B B 3 d 1 1 3 2 ∧ ∗ H ∆= B 3 A 2 1 ∧ ∗ , we find that , H 3 1 + -form, 1 2 ∆= H B 1 p = 4 d + A ]. It can be viewed as describing either a massive vector or a 1 2 is a standard scalar Stückelberg field: using the correspon u 2 1 d 35 F B a ∆ = 0 reveals that − ∧ ∗ 1 a ∧ ∗ 1 A 2 A d instead, we find F d ′ 2 2 1 1 2 ∧ ∗ H − − 1 . This is achieved by adding A Z a Z d 2 1 is a massless vector field. The action for the one-form 1 − = 48. For instance, if one further dualises A 2 Z m It is interesting to determine the scaling dimensions of the use the formula for a massive and the massless graviton has ∆ = 3. For the scalar fields, usin that correspond to the modes we are considering. The massles gauge symmetry to set form for a massive two-form. Alternatively one can dualise t pseudoscalar appears, for instance, in [ we deduce that the scaling dimensions of Integrating out In this form, we see that to the action: indeed integrating out so that the action can be written for a massive vector field massive two-form field, whichwith are well-known to be equivalen Clearly substitution one obtains the dualised action to the action. Integrating out Continuing we now introduce Finally, for the massive vector field with JHEP04(2009)102 ted ]. By ] (see (2.24) ]. The alize to 38 19 fields of 38 – ]. We note 17 2, which we = 0. Note in 39 / = 0, and hence 0 2 led to a single y tail in [ M = = 4, = 2 supersymmetric 1 -charge. ained are the bosonic 0 M N s should form a tensor R E -charge (“hypercharge”) fields which indeed have iewed as a flux compacti- R ) as H is a Calabi-Yau threefold, ersymmetric theories with χ ur expectation is that the pt are the bosonic fields of S eric form for the coupling of terms of these fields should ] with arised in table 8 of [ ial fluxes was first observed X rmultiplet. + ) by a factor of 1 ] with ing two-scalars form a vector 38 2.11 il for instance in [ w this structure arises. = 2 supergravity multiplet, a ar, we can consider the special V 38 S 2.24 N of a reduction on a Sasaki-Einstein ) of [ respect to the SU(3) flavour symmetry of + where X V × ) carries non-zero = 2 gauged supergravities as summarised − 1 = 2 theories are discussed in [ E S N 2.9 4). The KK modes we have kept are present R | N 2 1 – 8 – would parameterize a universal hypermultiplet. (2 2 ) is the bosonic part of an E B g Osp − that the remaining massive fields are the bosonic fields 2.11 √ 1 x 4 0)-form Ω in ( d , Z ] we find would be the scalars for the universal vector multiplet rela = S 38 h = 0 the only bosonic modes with non-zero 0 y and = 1 analogue). For the case in hand, it should be possible to du V N = 2 supergravity theory. In particular they form the bosonic and the scalar dual of + χ supersymmetry 2) for which the supermultiplet structure was analysed in de N ) can be rewritten as a gauged universal hypermultiplet coup , , U -charge since the (3 (3 U R = 2 2.11 M ] for the N ]. In terms of supermultiplets the two-form and three scalar 20 We will show in the next section that the fields that we have ret As we have noted, our Sasaki-Einstein reduction can also be v Let us first identify the structure before dualizing. The gen More specifically, our modes are obtained in (3.19) and (3.20 16 1 = 0, consistent with the fact that the modes are singlets with this specific Sasaki-Einstein manifold. in appendix B, we should multiply the overall action in ( theory. As formulated it contains, in addition to the will do in this section only. We can write (half) the action ( Similarly, 2 J particular that with fields of an massive two-form and five scalara fields. massive two-form The appearance inin of dimensional [ sup reductions with non-triv to deduce that the dual operator has ∆ = 5. 2.3 We now show that the Lagrangian ( the fields are the two scalarnon-zero fields with ∆ = 5. These correspond to the fication of type IIA supergravity. Inmanifold particular, if we instead were considering a reduction on that in identifying with the theory of general massless graviton and thethe massless gauge massless field graviton that multiplet,analysing we whose the have field results ke of content [ is summ multiplet, while the gauge Stückelberg multiplet. field and the The remain generalalso [ couplings of such a massive vector multiplet and a conventional (gauged) hype universal vector multiplet. In the following we will showvector ho and tensor multiplets has been discussed in some deta of a long vector multiplet with field content as in table 3 of [ to rescaling the metricbe on unchanged the Calabi-Yau by space. goingaction The to ( kinetic the Sasaki-Einstein reduction, so o unitary irreducible representations of for any Sasaki-Einstein seven-manifold and so, in particul case of JHEP04(2009)102 ) 1 ) to dB (2.30) (2.29) (2.31) (2.25) (2.26) (2.28) (2.27) ) to a − ¯ , χ 2.15 2 by adding F ( , χ, 2 , as follows. a | 2 i 2 1 ) C √ 2 V ,B Dχ F 3 | = U 2 can be written in − 2 6 re the coupling to U h I ) e 6 U ) on the universal V h . Then identifying F − + S 12 e µν ∇ ) 2 − . , ( et ( C.9 3 2 ∗ e ˜ H 2 3 H 2 2 ) | ( F − µν ¯ on on a SU(3) structure τ χ ∗ V χdχ | ∧ H − − V 2 µνρ ed in appendix − V U by adding the term ( + τ 2 F ! + ( H 3 1 U ) + 12 − i U , 2 dχ h h e 2 B ∗ e ) V ) we see that e µνρ ∗ χ − − + to get a pseudo-scalar log 4 3 2 3 − ( H hτ U 2 i 2.6 τ 2 ) 4 − 3 , U − F 2 14 − B 3 3( χDχ 12 − F ) 2 = − in ( ∧ 0 e 2 e µν 1 − )) 2 2 2 − h X ) and the gauge fields 1 F . X ¯ da h 12 V F H H ( ) 3 + I 2 µν ,τ V h − Dχ + − 3 2 h F 3 ∗ ˜ 2 − iX + 12 V H U χ – 9 – − ∗ U ( − )) ( = (1 hτ + 3 V − )= h +3 (2 18 3 τ U I 2 ∧ 2 + I = U ( . We now find that − X ∇ matrix is given by h 6 U ′ 3 2 2 ( H e (6 F X ( 3 e τ I 2 10 3 2 V H H F ∇ 4 1 ∧ IJ H ¯ , we get the standard metric ( i ( −

X 2 4 2 − +2 3 e χ i U − | N 2 1 3¯ U = χ 12 4 | − − E hH e √ e 1 + 3 g 3 3 IJ log + V + − − E H we define N 2 = − − g 2 h √ ∧ V U h ξ Z − = 8 1 4 + giving 1 Z 2 − √ term in the definition of U A V 1+ e 2 x 2 1 = + 6 2 H 4 K and ie 2 24 d S B = Z a − +9 + hF 2 Z 1 V h = 1 2 S + + ) to = 4 = 2 V = τ ], with a single modulus. Kähler We also note that, if we igno B σ 1. One then finds the 2.19 , H , 16 S + 2 U 6 ′ = 0 2 e I H To make the full dualization from the massive tensor multipl If we ignore the = 4 the term ( the vector multiplets, one can dualize the two-form massive vector multiplet, one must first dualise the field This is themanifold standard [ form that arises from flux compactificati giving the potential Kähler where the action and then integrating out hypermultiplet space. Introducing while with the form of an ungauged vector multiplet action, as summariz and together with the corresponding holomorphic prepotential ρ JHEP04(2009)102 . ) I C ˜ F , I (2.38) (2.35) (2.36) (2.33) (2.34) (2.32) (2.37) ) with ˜ X . ) simply C.5 2 2.38 aining gauge ) ∗ 2 2 ) F . 2. Note that, up h ∧ √ χDχ ∇ 2 2 ) / ( 2 ∗ ) given in appendix F − F 2 F 3 2 2 h h V are h C.1 Dχ − χDχ , the matrix ( − + omorphic prepotential ∗ + H 1 U 2 2 χ 2 − 2 S B ( ˜ , F 2 , − H ˜ i

H , e 4 3 ∧ χ !      Dχ 1 2 ∧ and ∗ 3 1 2 3 − . ˜ to obtain a pseudo-scalar by adding 0 3 ∧ ∗ H χ hτ iA ) 1 V 2 − ) is given by ( 3 2 h 1 − 4 . S i F F = 2 action ( 3 2 ˜ 1 2 4 2 B + log 2 B 3 ¯ 2 τ 3 − F C.1 3 ) h 3 − ) 2 2 3 N 0 1 − ) τ − hτ h 2 by a symplectic transformation ( ¯ τ V ˜ + dχ 1 3 ) 1 − X

F 1 2 ( ) + + A − IJ ˜ 0

˜      B U 2 H ) and introduce new homogeneous coordinates U 6( ˜ τ ∧ ∗ 6 N ) 2 ˜ X 2 ( H − 1 − 2 i h we now ﬁnd (2 ˜ – 10 – − √ e H 1 − q F 1 3 ∇ ) a + 3 1 − A − = ) and leads to the potential K¨ahler log h = H , ) τ ∇ ) and 3 2 homogeneous coordinates by 1 2 6( V h + 3( I ! − ˜ + 2( F F + − I 2 F 2 − 2.35 τ + U = ˜ , 2 ) X = I τ = ( E U e V da A B U C D 6 g Re( X I ˜ IJ − K 2 1 ˜ − (6 ˜

2 3 Im( F e N √ ∇ 3 2 − + − ∗ x + 4 with the general gauged = d E ) up to a (constant) transformation. Kähler Notice that ( Z V g 3 2 Z 1 S − H 1 2 + 2.30 U √ x 12 = as before, and after substitution one finds a dual action cont 4 e d . One can then dualise the two-form 1 2 V ). After integrating out ˜ B S Z d ,B 4 1 ), we find the gauge kinetic matrix in ( 1 2 2.19 ≡ ˜ − B can be obtained from ( 2 , ,τ 1 ˜ = H IJ A ˜ H N We now compare After these dualisations the new expressions for = (1 S I This indeed correctly reproduces ( ˜ X together with a rescaling of the and, since we have dualized the gauge fields, there is a new hol the term ( where If we identify the gauge fields to a normalization andexchanges as the expected electric and given magnetic we gauge are fields dualizing for and fields and which agrees with ( JHEP04(2009)102 x we = 0 = 0, V v 3 (2.42) (2.39) (2.41) (2.40) H = . This sets = 4 gauged sonic action 2 χ ) we see that z that carries F D rom an action = C.1 we have labelled h − ∗ = 2 C in ( = , N we see that it indeed I 2 ) ¯ ξ A H χ u I ) for the potential ¯ 3¯ ξ∂ ons of motion come from k sistent truncations incor- . auged supergravity can be √ − − C.8 . ξ ! u = ) σ ¯ ξ ξ∂ ξ dq P ( ξ rm ( = 6 and 2 i , + 24] (3.1) 1 / ). In appendix 4 = a − 1 f ρ µν u − − = 24 ! C.9 F ), are therefore ρ = 4 − σ . However, if we take 1 Dq 4 iξρ ∂ v we find that consistency requires µν = 0, σ 0 (1 − i/ F 3 i 2 3 , 2.39 − v/ U − H − 7 6 to the massless fields of ρ , P e ) ) = 24 − R 4 7 = we have the Killing prepotential 0 ¯ [ ¯ ξ e ξ χ 2 ξ i/ g P – 11 – σ / χ 1 = 4 ¯ 4 1 χ∂ SE ∂

− − = 6 − = ρ ρ − √ − = = f ), compatible with the general equations of motion ¯ x ξρ h . ]. σ χ − i k σ 4 1 for the quaternionic geometry it is straightforward to P σ d 2.9 P − = ). χ∂ (1 ∂ C ( i 2 i Z 4. From the terms V ), ( 3 and

= 24 2.11 = = = 24 0 + 4 v/ 2.4 , corresponding to ( = S P I a a ,..., U ξ = ∂ ∂ P P . Identifying . We begin by observing that it is consistent to set the comple we have V H = 1 B = 6 = 6 ¯ ξ = 0 in ( S u = 1 0 , ¯ χ ξ∂ k k i u U q − ). This completes our demonstration that our action is the bo ξ = 0. This is not surprising as this is the only field in the ansat with iξ∂ 2 2.27 χ -charge. The resulting equations of motion can be obtained f F = R = 2 supergravity theory k − ∗ N It is interesting to ask whether this truncation to minimal g Now we consider = 2 reproduce ( non-zero obtained by setting Finally substituting these expressions into the general fo Given the formulae in appendix The Killing prepotentials of an 3 Some further truncations In this sectionporated we in observe our that KKcontained there ansatz in are appendix ( somescalar additional field con extended to just include the breathing mode scalar 3.1 Minimal gaugedIt supergravity is also consistent to set calculate that for the Killing vector these coordinates gauging is along Killing vectors all of the massive fieldsa Lagrangian to given zero by and we then find that the equati This is the consistent KK reduction on a matches the universal hypermultiplet form given in ( H and for supergravity that was discussed in [ in addition. JHEP04(2009)102 ]. h = v 3 3 2 14 v √ tion = 6, − (3.4) (3.6) (3.3) (3.2) u v 2 = . = 0 and e ) for the his paper, 3 R = 0, 2 R v χ H 7 χ 2.9 h +48 − v 2 5 e ) = − 2 ), ( R u . 6 R 2 χ 2.4 − χ h χ ∇ = 1, as we will see e ( = 0 and ). Note that for the v 24 2 N u = 0 (or, equivalently, 3+7 − 3.2 ) implies that = Re o − u u = 40 and hence ∆ 6 2 v R 2 h e 7 B.4 − χ 3 2 tting − his truncation directly into m 2 K ansatz ( e to just scalar fields plus the = 0, h ction ( 2 − v ) 2 2 5 ϕ ) 2 R F − ) and 7 h χ e = 0, or both, and the equations of = 72, ∗ = ∇ ) 2 v + h 56 ( h 7 2 v 2.12 2 Re Ω (3.5) m − 2 h H v − SE + 4 ∧ 3 ( u = = 0. The resulting equations of motion 8 2 η ϕ − ϕ via ( e =0or − 3 7 χ ∧ ds + e v ∗ 3 R H ]. Indeed if we substitute 3 2 +ImΩ J 18(1 + χ v/ dh +42 14 η 2 − ∧ 2 − = 4 and e + – 12 – ) 2 ∧ v J ) 4 h u 5 + in [ v 2 1 J dϕ = 0, which just maintains the breathing and squash- − ∇ 2 4 2 vol ∇ ( u to vacuum we find ( 8 h v = = f ds 2 e 2 7 4 V 2 = 0 and Im − = ϕ ϕ ) we obtain results equivalent to (2.20) and (2.21) of [ = = h e 7 − 2 ) and reads: R ∗ 7 2 2 4 2 24 F AdS χ 3.2 ) and G − 3.2 ds ] they also consider the truncation with, in the language of t u − 2 = ) U v 14 ∇ v 2 ) and we have introduced the quantities 3 2 − ∇ H h 42( u ( ): = 0. 7 2 21 into ( 16 − = . The resulting equations of motion can be obtained from an ac case 2 e E √ h − 3 F 6 2 2.11 2 3 R E 2 − (3 + 7 H G √ 3 ˜ R φ/ E v/ = g 7 = E − g − χ u e = 0. However, this is not a consistent truncation: equation ( − √ 2 x √ 4 = 2 H x ) implies that d 4 ) and f ). This truncation generically breaks supersymmetry down t 7 and d = Z B.9 V √ A different, further consistent truncation is achieved by se In fact, it is also consistent to further set It is interesting to observe that, for this truncation, the K χ Z = 5. 3.2 Note that in section 2.2 of [ = 2 = 11 fields can be written = ˜ = h = ϕ/ S S U follow from an actionthe which general can action be ( obtained by substituting t As we have just noted, it is consistent to set motion are those that are obtained by substituting into the a ing mode scalars, was also considered in the next subsection. 3.3 The weak metric by setting 3.2 Scalars We next observe that it is possible to consistently truncate where h case of the seven-sphere setting D in ( that satisfy which can be obtained from ( Note that expanding about the then ( and we have switched from ∆ − JHEP04(2009)102 ] - ) 2 ]. G 14 ] in 23 3.6 (3.9) (3.8) (3.7) 14 (3.10) (3.11) provided we s a consistent ) with weak i ρ = 1 supermul- 2 | A ρ M N W ) can be explicitly | A 3 , the condition ( µν 3.3 v 7 with potential Kähler g − 7 tor field. We now set − 2 | − sion for the form of the e SE K ¯ ν φ W 18 + 24 ∂ , A φ φ µ µ − ) was first obtained in [ ∂ D A | A v here one keeps only the metric ¯ = 0 µ φ 3 3), as = . We now find that 3.8 φ − 1 1 A ¯ g − φ e A A φ structure that was obtained in [ 2 32 g + 32 ( ∗ φ 2 K − ∧ ∗ e + 42 G = 0. We then get the consistent KK 2 1 2(7 2 µν ρσ = 8 ) with an arbitrary space A h ) can be obtained from (2.6), (2.7) of [ − √ F v F 3 7 = 1 chiral multiplet. In fact, introducing a ¯ φ = ∇ H µν ρσ 3.8 µ ( 2 = 4 F F N SE ν 2 7 F 1 3 φ∂ A µν – 13 – W µ and µ − g − ∂ ∧ ∗ = 6. The action ( 8 1 A 2 ¯ E φ 2 v R F φ R F − g ∗ 2 ρ g + 32 = 3 1 = 1 supersymmetry of the action ( ν E − − ρ g 2 F = 42. ν = N F √ . It is worthwhile noting that starting with the KK ansatz − E F 7 µρ x 2 ) still gives a consistent truncation. One would expect such R ∧ √ F 4 R W µρ h x H 2 d ) F 4 3.4 E F , the d K holonomy (i.e. that the cone over the space has Spin(7) holon 2 3 3 2 g Z φ ) to get the last line. These equations of motion come from the ih = 6, 2 = 6, = 1 supersymmetry, with the metric lying in an − ∂ structure on the seven-dimensional space Z + + , = c + G √ f 3.9 1 2 N , = S v x µν µν + ( = 72 and hence ∆ A v G e 4 g g S 7 2 v d ∗ ), and a superpotential W = 12 12 √ ], here we have also shown that this particular KK reduction i = 0, ¯ m φ φ Z − − φ ∂ − as a χ 23 + = = 48 = = = ϕ = φ = 2 S φ µν h F W R φ ∗ = 7 log( d D − V We could go one step further and also set ) this superpotential can be derived from the general expres = = 3.4 truncation that is valid for any Einstein seven-manifold, w and the breathing mode scalar. The action is given by KK truncation. tiplet and the breathingcomplex mode scalar in a massive ( restrict to configurations that satisfy the equations of motion can be written and we see that Interpreting where we have used ( U 3.4 Massive vector We can also consider a truncation to a metric and a massive vec by setting Lagrangian is equivalent to weak exhibited by rewriting it in terms of a metric Kähler superpotential in KK reductions onIn manifolds contrast to with [ the context of the seven-sphere. Specifically ( a truncation to have where omy). One can then generalize by replacing holonomy and the ansatz ( K JHEP04(2009)102 8. / 2 = 6, (4.1) α 2 ) f + = 15 exist for the dx 3 2 is the round = 0, c 7 αρ χ ivalently chang- M + = η ] and has a non- 4 with h ) we obtain another ). Writing this out = 48, provided that ). 13 2 − 7 2 example, if we reverse = ) + ( 2.1 2.9 S dx = 6 m V k + ]. h d in [ = 3 i.e. is invariant under nslations, Galilean boosts, KE − = ( 40 is not 2 z ) (4.2) ) dx ) and ( η U 7 7 ds 4 + in [ solution ( ρ 2.4 This solution is supersymmetric, ). We consider the ansatz given by ( + SE dx 7 SE d ( − 2 3 2 ∧ 3.10 SE ds 2 dx 4 ρ dx × ) (5.1) + 4 4 ∧ )+ + − 4 = 11 supergravity by constructing solutions + 4 solution has a non-relativistic conformal symmetry ) and ( 2 2 ρ − dx AdS AdS D 4 dρ 3.9 + AdS = – 14 – = 11 by setting α dx ( = 2 can the algebra be enlarged to include an additional + k 2 vol( dx 2 1 z 2 D 8 3 − ) ds + + 4 1 − 2 4 dx ρ dx Freund-Rubin solution there is another “skew-whiffed” = = ( and substituting into ( solves all the equations as does ∧ k 7 2 + 4 + 2 2 1 ) by hand. We will return to this truncation to construct ρ 2 2 G M ds α dρ dx 2 A ρ k 4 dρ α 3.9 4 and is singular. We note analogous singular solutions also ∧ × ∗ = = 11 supergravity given by − cρ − 4 2 − ). With the exception of the special case where + c D 7 = 2 = = dx ) = 8 z M 2 1 AdS + ∧ 3 ]. A ds = 3 case case we obtain + dx H 13 ( k 6 dx ρ 2 = 3 with = 11 supergravity in the next section. 2 ρ and k α solution of 3 D 16 ] which can be obtained by reversing the sign of the flux (or equ − 7 F 29 ∗ = = SE 3 1 2 G × ds 4 = Recall that only for dynamical exponent 3 , at most only one of the two solutions is supersymmetric. For 2 7 This solution is in close analogy to the solutions considere with dynamical exponent to the four-dimensional equations given in ( special conformal generator. Also note that the theory considered in [ We can now uplift these solutions to which does not preserve any supersymmetry (provided which describes a metric coupled to a massive vector field wit 4 Solutions As an application we construct solutions of generically preserving two supersymmetries, as explained 5 Skew-whiffing Recall that for each AdS explicitly for the relativistic conformal symmetryGalilean with dynamical transformations exponent generateda by central time mass and operator, spatial and scale tra transformations. ing the orientation of solutions of solution [ We find that we impose the conditions ( S H the sign of the flux in the supersymmetric JHEP04(2009)102 s ) 8 6. − the − and = 6 = 4 (5.4) (5.3) 7 , the = 0). = 2 = f 7 D 2 h vacuum S 2 f χ SE | 4 we repro- m χ above, we | = = 4 action, V + h AdS D 2.1 2 h = 8 and − , metric the V 1+ ) ction = 2 breathing mode 6 theory the modes ¯ = ξ − = − is the round 2 χ ¯ N ξ∂ ome minor obvious sign . For a general 7 2 m U = − → of the action with 2 ξ . w | 4 r this vacuum can be easily SE f 2 but the sign propagates into χ ξ∂ σ | omenon that would be worth = 4 theory that contains the f view. ( 2 n change in the P | i ) (5.2) ) for the potential + 2 4 χ sonic fields can be extended to include D | | 2 χ − h | C.8 + = 24 σ 2 + 1 ∂ to a h = 2 supersymmetry. The analysis of = 4 supersymmetry since it does not ) by now setting 2 1+ = 2 theory that we considered in sec- 24 7 h − N − D N 1+ SE B.12 1+ , P → ) vacuum of the truncated theory with )= ξ − = 6 theory retains an are then 2 ¯ = 2 ( χ P 2. As expected this bosonic mass spectrum is 4 = 2 supersymmetry I f – 15 – V 4 2 , | ¯ χ∂ 2.11 P − N N χ − U | − AdS 6 = 1 σ − + χ P 4) multiplets, this is not the case for the vacuum solution of this theory now uplifts to the non- e ± | 2 . 4 χ∂ 24 h (2 σ ( i − = 6 fields are no longer part of the breathing multiplet but ∂ f 1+ AdS = χ + 4 = 8 graviton supermultiplet. Nonetheless this leads to a OSp 0 6, despite the fact that the uplifted solution is (maximally a = 24 solutions are supersymmetric for either sign of the flux. It i P ∂ − 4) multiplet structure. a N | 6 and 7 ∂ , means that the gauging is now along Killing vectors given by . The S − (2 3 = 3 h H × H = 6 = f 4 Osp ∧ 1 0 ∧ k k 1 1 A A AdS 6 6 ) after the change →− →− 3 3 = 4 action in two places. In ( H have similar analogues in the skew-whiffed theory with only s 2.27 H vacuum spontaneously breaks the 3 ∧ D 4 Many of the additional truncations of the By a very small modification of the truncation discussed in se As we have already noted, for the special case that ∧ Despite the fact that the skew-whiffed vacuum is not supersym Here we are assuming that the truncation at the level of the bo 1 1 4 A A consistent truncation. Thisinvestigating is further, a including novel from and the dual interesting SCFT phen point o tion corresponding to the instead are part of the multiplet together with the supergravity multiplet, in the duce ( supersymmetric. In particular, while the the fermions. skew-whiffed solution. In particular, we solve ( can obtain a second consistent truncation on contains modes that fall into where we have changed the sign of the constant factor (when The rest of thethe analysis essentially goes through unchanged of the theory with AdS corresponding interesting to observe that while the 6 supersymmetric skew-whiffed solution. Thecalculated mass and spectrum the fo only difference from section 2.2 is that no Substituting these expressions into the general form ( The corresponding Killing prepotentials section 2.3 goes through essentially unchanged, but the sig 6 match the bosonic action has the bosonic content consistent with corresponding to operators with ∆ inconsistent with a vacuum preserving JHEP04(2009)102 = 0, (5.5) = 11 + 3). 2 χ D =2 and φ q = 2(7 4, h / √ solutions with 4 = is tri-Sasakian, , the linearised =1 7 φ 7 2 3 V → M / L M es corresponding to 2 = × 2 3) | 4 p φ U − | =2, 2 = 4 theory that exhibits = 2 2 t is an Einstein space, it is φ m AdS χ stent KK truncation with = 1 language and we find D reund-Rubin backgrounds, M 7 symmetry. We have shown stant KK modes associated ith the tri-Sasaki structure n plus the breathing mode, solution, there are solutions hat a tri-Sasakian manifold onsistent to truncate to the − 2, 2(7 M 1. N 2 / ultiplet containing the massive | -symmetry. By writing down a √ erconductors. Our generalised, whiffed solutions thus provides A holonomy or is a Sasaki-Einstein level, in our analysis we can set R 2 Dφ = = 4 ) but with the sign of the four-form G 1 lead to a − | A 4.2 W χ µν F ], upon setting 8 for the supersymmetric and skew-whiffed µν solutions were considered at the linearised ], it is no longer possible to truncate to the 30 − F has weak 7 = 3 supersymmetry, where , 30 4 1 7 . These sign changes mean that when we uplift – 16 – ] to obtain analogous exact solutions of where the upper (lower) sign corresponds to the N 1 SE M − = 4 action that contains the non-supersymmetric 2 = 40 A 30 × 2 ∗ H 4 D m 8 + 24 = 11 supergravity where − ± ∗ R D AdS = and = g 3 φ 2 − H F and the complex scalar A √ i solutions with 1 x 2 7 4 A and d − case can be written in a manifestly = 11 we obtain the solution ( M 2 2 Z × solutions of dφ D F G 4 7 ∗ = or the Sasaki-Einstein structure. = = 0 with 1 3 2 M 3 S ) to AdS − G Dφ H × = 2 supersymmetry, where , 4.1 4 = = A N 2 d h H AdS ] and it was shown that, for the skew-whiffed solution, the mod = = 6, 30 F V − Moving to Recently KK reductions of = 1 and = = =1 in their equation (1). In particular, for the skew-whiffed f it is natural tohas an expect SO(3) that group a of isometries similar corresponding story to unfolds. SO(3) Recall t KK ansatz that incorporateswe the strongly constant modes suspect associated that w it will be possible to obtain a consi always consistent to truncatewhich the is KK dual spectrum to to an the operator gravito in the dual CFT with ∆ = 6. For seven manifold, respectively, wemassless have graviton also supermultiplet combined shown with thatbreathing the mode. superm it is In both c with cases, the the KK weak ansatz contains the con supergravity. 6 Discussion In this paperkeeping we the have breathing considered mode consistentthat and truncations for with varying on degrees F of super N with level [ case, respectively. This is in agreement with [ changes required. For example, the flux reversed. Note however,field as content noticed of in minimal [ gauged supergravity as in section 3. the massless gauge-field action is given by the solution ( For the reduction to the massive vector field, we should now se that the only difference is that the superpotential g skew-whiffed weak of this linearised theorynon-linear corresponding and to consistently holographic truncatedan sup action ideal for set the up skew- to extend the work of [ holographic superconductivity. Indeed, atU the linearised supersymmetric (skew-whiffed) truncation. Writing JHEP04(2009)102 , , , , is ′ s × v 84 5 4 ]. = 8 ] we re is 175 , 840 M 567 48 ], that 43 C , N , where – pin-one AdS v 44 20 46 . . . , ] or [ etailed form 294 + C 42 4 tors in the 126 , T rF = 11 supergravity. ], as for conventional = 12. If this can be t, vectors in the 105 is Sasaki-Einstein then D f the massless graviton 44 upersymmetric 5 results of [ ing mode has been argued ontaining just the M d furthermore we are led to rd, for instance in [ ng gravitino multiplet (table s in [ t keeps the graviton and the ion to the graviton. This is resentation, and ten scalars s, it may also not be possible tiplet whose field content can e tri-Sasakian case mentioned rhaps depending crucially on runcation ansatz for this case. n for the D3-brane [ ts a picture where consistency wo fields. It is worth pointing er of massive and massless spin- ist of scalars in the irrep. A particularly interesting v he fields of maximal SO(6) gauged solutions of 35 ]. ] to obtain a consistent KK truncation 4 24 S 13 irreps and massive spin-two fields in the × C 7 ] that keeps the graviton and the breathing 50 solutions of type IIB supergravity where , 14 – 17 – AdS = 1 graviton multiplet plus the breathing mode ] and of [ C 5 3 we are led to conjecture that there is a consistent N M 5 45 = 8 SO(8) gauged supergravity, to also include the S , × C 5 = 6 N 5 = 4 super Yang-Mills theory of the form M AdS ]) which consist of the graviton and the SO(3) vector fields, N ], i.e. 41 12 ]: the bosonic fields consist of scalars in the = 8 supersymmetry. It is again natural to conjecture that the 45 = 0) consisting of six massive vectors, transforming in two s N 0 = 4 Yang-Mills field strength, and it has been argued that its d J irreps of SO(8), where the singlet is the breathing mode, vec 1 N irreps and massive spin-two fields in the irreps, two-forms in the and 28 ] with s irreps of SU(4), where the breathing mode is again the single is the 41 15 1 35 F and Let us now return to the In a similar spirit we can consider For the special case when Following this pattern one is led to consider the maximally s , , , solution with irrep. Note that for this case the operator dual to the breath = 8 supermultiplet containing the breathing mode. Using the = 3 supersymmetry. Such a truncation would retain the fields o = 8 SO(8) supergravity, are not directly applicable here, an C C 7 here it is possible to generalise the of ansätze [ to be dual to an operator in can be obtained from expanding the Dirac-Born-Infeld actio Einstein. Once again therebreathing is mode a which is consistent now KK dual truncation to an tha operator with ∆ = 8. If 20 multiplet. We will report on the details of this in [ supermultiplet (table 3 of [ a theory with a veryarises particular matter from content, particular which sugges conspiraciesthe among existence the of an fields, AdSto and vacuum. further pe If truncate this the putativeout theory theory that exist while unlike the keeping cases massive weabove, spin-t have it studied is in much this less paper, clear and how th to directly construct the KK t N mode which is now dual to an operator with scaling dimension ∆ that includes the bosonic fields of the conclude that the bosonic fields of this supermultiplet cons N and the breathing mode supermultiplet,2 which now of sits in [ arepresentations lo of SO(3),transforming four in two scalars spin-two representations. in the spin-zero rep it is not possible to havetwo consistent fields. theories of However, a for finiteN instance, numb the group theory argument 64 There is a known consistent truncation [ S truncation to the massless gravitonsupergravity, supermultiplet, combined i.e. with t the massive breathingbe mode obtained mul from [ 300 10 graviton supermultiplet [ feature is the appearanceremarkable since of some massive general spin-two arguments fields have been in put addit forwa an analogous consistent KK truncation that extends the one c 350 JHEP04(2009)102 4 1 = 2 ] to irrep (A.2) (A.1) AdS and 50 irreps, N , 35 35 10 49 = 2 , = 11 super- 55 N and D ] were presented 81 s of 55 ] for FT side of the corre- 34 -FECYT postdoctoral upergravity [ ean Hartnoll, Chris Hull, ] and [ ]. It would be interesting aviton multiplets and the lin and Toby Wiseman for and Ω satisfying 1 54 , solutions of type IIB which btained for classes of super- e multiplets. lowship and a Royal Society at of [ tiplets for the class of ∗ ntent of this latter multiplet J 5 tors in the hat we have been considering , , Ω ) η ∧ irreps, two-forms in the 7 AdS , 5 Ω ) ). SE i 8 7 4 ( 2 = 1 ∧ ]): we find scalars in the and SE ds η AdS N solution 53 30 = , 7 vol( )+ , 3 vol( 4 52 ∧ J Ω 8 3 4 SE 1 3! ∧ – 18 – = AdS × ( ∧ 4 iη J , 2 4 η = vol irrep. ds ǫ = 2 4 1 is defined by the forms 14 ) = Ω = 4 AdS 7 7 = 6 vol = dη d = 11 supergravity classified in [ = 8 supermultiplets then there would be a consistent KK 2 4 SE SE ], respectively, were presented in [ G D ds N 57 vol( ] and [ ] (based on the results of [ 56 solutions of 51 = 11 volume form is 5 = 11 supergravity and the class of D ], respectively. Similarly, the KK truncations for the clas D 4 AdS = 1 ] and [ N It would also be interesting to see if similar results can be o Finally, it would be desirable to have an argument from the SC 5 and that the truncation extending the known one to maximal SO(5) gauged s symmetric AdS solutions outside ofso the far. Freund-Rubin The class KK t truncations to the massless graviton supermul and massive spin-two fields in the is supersymmetric given our set of conventions. These are th gravity, that the structure on extended to include the full in [ also include the breathingcan mode be supermultiplet. found The in field [ co to extend these KK truncations to also include breathing mod spondence as to whymassive the breathing KK mode truncations multiplets containing are both consistent. the gr Acknowledgments We would like to thank FrederikMukund Denef, Mike Rangamani, Duff, James Ami Sparks, Hanany, S Kellyhelpful discussions. Stelle, Arkady JPG Tseyt is supportedWolfson by Award. an OV EPSRC is Senior Fel supportedfellowship, and by partially a through Spanish MEC Government’s MEC grant FIS2008-1980. A Supersymmetry of the solutions of were classified in [ irreps of SO(5), where the singlet is the breathing mode, vec three-forms satisfying self-dual equations in the In this appendix we show that the solution given by and JHEP04(2009)102 ) ], ∧ 0 34 and A.4 dξ (A.8) (A.7) (A.9) (A.6) (A.3) (A.5) 3 J / 1 H = 2 = 8. The = dη ǫ D ν nventions of [ dξ µ , 8 8 (A.10) dξ s. To do so, we start µν η , 4 ,..., ,..., one metric. We introduce ) r . 8 = 11 orthonormal frame: + = 1 C = 1 ( , . ∗ CY 2 2 2 D ) Ω 8 , r space, 7 dr to be equivalent to g ds ) (A.4) 7 ∧ , a Ω 3 1 / = a − SE 1 ∧ ( J , g CY SE ). CY 2 ) 2 H 6 H 7 ( / r ν Ω 1 d ds ¯ ξ + ∧···∧ irη 2 + 1 d SE H 16 ∧ r 2 ν is the chirality operator in µ η + g ¯ 2 and Ω ] are satisfied by a solution of the form ( ξ = + dξ = 8 d ∧ ∧ vol( µ dξ 2 34 dr ( 1 µν +2 – 19 – CY ∧ 3 4 ∧ dξ g CY dr a η J ...γ r rdr 2, and define the four-dimensional volume form 2 J 1 , µν 2 ) defines a unique Calabi-Yau structure on the cone ρ dr 1 1 η 4! 7 γ = 11 solution can then be recast in the form = = dξ )= , )= 3 1 r + , 8 8 / γ ∧ and so A.2 2 D 2 C C , e , a 2 0 CY CY − (8) )= ( = 0 ρ a µ 10 )= 8 J γ 2 = dρ Ω 8 γ e H dξ = 11 gamma-matrices as C . The eleven-dimensional volume form is then vol( dξ µ ⊗ ds C 8 3 ⊗ = = (8) . The D 4 1 / µ C 1 γ τ 1 2 4 vol( dr − = = and space G ds )= ∧ ∧···∧ H µ 4 4 µ 2 1 +2 Γ ξ a e = dξ Γ µ ∧ AdS AdS e ∧ ( 0 = 2 ) where vol( 2 e 1 8 µ C ds ¯ = dξ ξ 4 1 is harmonic on ǫ , ∧ 2 6 vol( Ω. In particular, we then find 0 r = 1 and where − is an orthonormal frame for the cone metric. Following the co ∧ ∧ r dξ 2 5 = a iη r = g 012 dξ τ ρ = It will be sufficient to focus on the Poincarésupersymmetrie We now turn to the supersymmetry. We introduce a The Sasaki-Einstein structure ( H ∧ 4 = 11 supersymmetry equations given in [ 1 Ω = 4 where by definition given by the SU(4) invariant tensors by rewriting the solutioncoordinates for in the terms of a Calabi-Yau fourfold c We can then decompose the d determined by requiring the closure of with vol where we have introduced the cone metric over the with D and dξ JHEP04(2009)102 ) A.2 (B.3) (B.2) is the (A.11) (A.14) (A.12) (A.13) 4 J ), supersym- 4 ). Thus the ∧ hJ A.7 . AdS is a 16-component , where vol = 11 supergravity. ) η ) (B.1) β 8 1 D vol( ) + 2 ∧ ig A 1 3 8 ) ] such that 6 + A + − sfies the gamma-matrix 7 58 η . The structure defined by + g he skew-whiffed solution is = ( ⋆ ( KE metric. ′ 2, the cone metric must be η = 11 supergravity is given by = 3 and 4 , ( ⊗ Ω ∧ G ) . Specifically, one can choose a ∧ D ) D vol( ∧ ) ǫ . 1 orm 6 i ′ = 1 ( J ∧ A ig = β Ω β . 4 i ∧ ǫ + . 1 + 16 1 = η 10 vol 5 ( − β , H g β ... V ( V with ⊗ + 34 2 = + (8) e ∧ ) Γ U α J i 6 ) , ( 6 4 e 4 ′ CY / β ∧ Ω + c.c.] 1 )+ 78 defined by the Sasaki-Einstein structure ( = 4 we use a mostly plus signature convention. J ig 2 6 g = − 1 , ∧ 4! ⇔ 8 – 20 – H , γ D + ǫ ) H + C − 1 3 KE = 0 g ( ǫ = A 56 )+ ( = 0 2 . This would in turn require a Calabi-Yau structure g 3 can be chosen to be orthogonal and the Calabi-Yau 1 ǫ ) = + = β ∧ β ds A ) 8 + i a ǫ η ) − dH ( U ( C 2 + 2 ∇ β 34 χ e = 11 and 012 = ig g η ( Γ + β D + + = 8 satisfying ∧ 2 4 Ω+ 1 (8) 3 12 ) on the cone = 0 is satisfied provided g D ds γ ∧ g can be written as bilinears in H 4 1 = A.6 = ( = χ + dG 2 ), we see these satisfy the orientation relation ( CY 4 3 [ ds satisfying vol( CY CY √ vol A.9 J 8 Ω f + C and Ω = 4 and spinor projections exactly as in appendix B of [ CY = 11 volume-form we choose ) on G J } is a constant two-component Majorana spinor in a D ′ CY g α { Ω , Note that if one takes the skew-whiffed solution where We now substitute this ansatz into the equations of motion of = 4 volume form. In both ′ CY J frame structure Calabi-Yau. In particular, the For there to be two independent solutions Majorana-Weyl spinor in the Sasaki-Einstein manifold isgenerically not not supersymmetric. of this type, and henceB t Details on theAs KK discussed reduction in the main text, our ansatz for the metric of More precisely, there are PoincaréKilling spinors of the f projection condition provided the supersymmetry transformation parameter sati Crucially, from ( is indeed of the type required for the solution to be supersym Calabi-Yau structure ( where while for the four-form we consider ( metry would then imply D The Bianchi identity For the JHEP04(2009)102 ) ). B.5 (B.7) (B.4) (B.6) (B.5) (B.8) B.12 (B.13) (B.15) (B.16) . ) and ( J ∧ =0 (B.11) =0 (B.10) =0 =0 (B.9) B.4 . via ∗ 4 3 4 )] =0 (B.12) 4 hJ , , 1 2 C | 3 3 6 vol B hH Dχ χ , )vol C | = 0, is also satisfied if 2 2 V ∧ ) + 2 ). We can solve ( , V C 4 1 + + 4 − 1 1 + and e , ,..., G 2 2 A C 4 = 4 theory that contains U B.9 Dχ h 2 2 ∧ H G i + 4 − 4 2 B + 4 D 6( = 0 = 1 3 e ∧ η C G ( f on ( 3 − 2 1 ( + ) follows from ( ) (B.14) C = 0) is chosen as a convenient d dh ∧ 2 + 4 2 χ f . 2 + | . C χ 4 V J f , 1 χ B.3 | 2 + + 2 G C hH 2 + 4 ∧ = U 4 h F + 11 6 hF 2 G h e ∗ 2 dh ∧ [ Dχ H d h + + 12 + 4 d 1 = 11 supergravity while the lower sign = + AB ∗ ∧ 2 2 2 Ω + c.c. g H 3 J B V H H 1+ D V ∧ solution, which generically does not preserve + + 1 ∗ ∧ ∧ ± iH Dχ , 144 U 7 3 ( = + 2 2 , 2 i 2 V 4 . + – 21 – V e H Dχ 2 1 + − H H − dh , U i SE − 4 ) U 3 = 11 metric we use the orthonormal frame 1 d 2 U dB dB + C × = = 2 = 6 e − 2 )+ A 2 Dχ and we note that ( D 4 − 1 1 2 1 C = = e F Ω 1 ∗ + χ A 2 3 χ H + 6 ) by introducing potentials C solution of 1 ∧ ∧ η dH V 2 H H B ( + AdS e ) = 6 2 7 4 6 (when 1 iA V ) we can write the four-form as B.4 η = 11 Einstein equations: fF H f e 4 ( G ± A D 3 SE V ∗ D ∧ − B.6 ≡ C + , i + 2 V 3 × 7 i U η C , α + 6 ˆ e e ) can be obtained by acting with dχ ( H 1 4 ) and ( e U α U V χ 2 + e e e ≡ AC e − 4 4 ) and ( B.3 3 B.10 AdS = = = + 3 G i √ vol 7 Dχ α ¯ e ¯ B.5 H e ¯ e 1 , f + 12 1 ∗ dh = ∗ = V dA 4 − V U G ≡ − AB 6 U e 2 R 2 F e d d Finally we consider the where One can show that ( We solve equations ( by setting Note that using ( To calculate the Ricci tensor for the Similarly the equation of motion for the four-form, normalisation. The upper sign corresponds to reducing to a where the constant factor of the supesymmetric any supersymmetry. corresponds to the skew-whiffed JHEP04(2009)102 ] ∗ χ (B.19) (B.18) (B.17) (B.20) ) α ∗ γ χ β γ χD F = 0 (B.22) = 0 (B.21) γ D β − αγ ) , are given by 2 2 | | )( F ∗ F αβ χ ) reduce to the χ χ χ χ αβγ | | V AB α αγ γ H γ 2 ¯ V V H R F e D 2 2 D D αβ ∗ ( 2 1 V − − αβ B.15 2 )( χ H αβγ [ U U e ) F αβ χ − 6 6 U V 2 1 η H γ V V 4 ) − − γ αβ 3 2 β e e V − D ie V + 2 F e ( β i tensor, − − U∂ V 4 1 e written U V ∂ e 2 + 4 γ 6 ) + 6 2 α e h + 16 | ∗ ∂ 2 V ∂ − − 1 ∂ λ h 4 1 36 χ 2 χ β e α f | λµν β − e α ∂ f 2 1 ∇ 6 1 + V + β − ∇ H h 6 1 αβγ 2 D U α + ) λµ λ − V + γ − V ∗ )( αβ H α + F 2 γ h ∇ U H λµν V χ 7 7 χ 6 2 V H λ h ˆ ∇ α ˆ γ e e U∂ β H e λµ − h αβ β αβγ V U γ ij ∇ V ∂ ∇ e D η 2 1 8 D U αβ + ∂ H γ αβ h J ∇ H β 8 3 1 − )( 6 U ∂ η λ F U e V − − j 2 ∇ αβ χ + 4 2 1 9 2 e e V ( e − ∇ − 7 η i − γ 2 j − ) + ( 2 − 8 ˆ e i V ) + ( 1 6 e ∗ h e − + 8 = 11 Einstein equations ( − f D V 2 U V J β V Ue ( 1 2 2 − χ + 3 U γ ) 6 1 γ − 1 U − e α – 22 – α U β 18 U β λµ D 4 ∇ U U U ∇ 6 ∂ − ∂ − λ β 4 4 + h − D − γ β γα − βγδ U∂ β + V U V − − α U∂ 2 V H e γ )( e e e αβ ij e 2 F V H α h H ∂ 4 1 ∇ χ U∂ e 2 αβ ω − − ω − 4 ∂ U 6 −∇ α + e α αλµ αλ U F + 6 ∂ + +6 = = = = = 8 D − U V H H + 2 V αβγδ ( 4 − V 2 − αβ 7 7 V γ + ij i U 3 αi e − V U α − γ αβ F fǫ e ¯ α 2 ¯ ω ω γ ¯ = 4: ω V ¯ U V U U U ω ¯ ω V V 2 − V ∂ 6 2 4 4 2 ∇ α ∇ γ + e e U∂ γ D − − − − e β γ αβ 2 8 γ ∂ U ∇ e e e e ∇ 1 4 η 6 ∂ ∇ β − 3 1 3 4 4 2 2 3 2 e U − + − 6 ∇ + 1 6 + + + + −∇ − U − V 2 6 ( U V U γ = = 6 ( − 4 2 γ e − − − 8 U∂ e V α αβ U∂ γ 2 2 1 γ αβ γ R e ∂ ij F ∂ δ − R U + 6 = = 0 = 6 = 0 = = + 6 +6 7 7 ij V i αi 77 V α αβ U γ ¯ ¯ 3 ¯ ¯ R R ¯ ¯ R R γ R e R ∇ ∇ γ γ γ Using these results we find that the ∇ ∇ ∇ We then observe that the corresponding spin connection can b following four equations in After some computation we find that the components of the Ricc JHEP04(2009)102 V . (C.2) (C.3) (C.5) (C.4) (C.1) − J F ith metric ∧ ) gives rise I , are the real F manifolds. For H IJ B.22 n N 4 ) coordinates and )–( I ) are homogeneous tic terms can both t F Re ( , I 1 2 I ,..., B.19 X , , X + . I µ J vector multiplets parame- L = 1 ), ( 1 A F , ion of degree two. Explicitly X u − u V I , ) k , n K ! ∧ ∗ B B.13 is the scalar Lagrangian for the u I I ! N I . The ( X − q I I ) where X ) F G L B u F )–( B F = 11 supergravity. Thus the KK A X , X q T JL ( + IJ µ are the gauge field strengths for the X F ! D B.9 D ∂ ) F I N A ! ¯ V F = , while )( = AB I ), ( j ¯ Im , where i u A B C D F space has dropped out. In particular any N )(Im I g D q 1 2 . In the gauged theory A B C D iX

7 µ I T D B.6 IK

+ ,...,n − A B (Im = v + F /∂A 1 = 2 supergravity action coupled to vector and )–( I SE = , L – 23 – ! C F Dq ∂ I I N I ! (Im B.3 and = 0. ˜ ˜ I ¯ I = 0 F G , D i = X = ( ˜ i I ∧ ∗ ˜ i F I µ X A

I I k u ˜ T A N

+ 2 G i I ] 7→ C . Under symplectic transformations acting on the gauge k Dq log IJ 7→ 59 F with ! ¯ = uv − − F , J I I matrix are given by ! h i I N 7→ ∂ t I F G C I = = 16 I µ + IJ T ∂ F

dA j X ∂ V ¯ t IJ A N

= K ), one has = = D N are the complex scalar fields in the i I hypermultiplets parameterizing a quaternionic manifold w t IJ ∧ ∗ C.1 µ F V = 1, F i H D = 4 field equations ( n ) is consistent. B Dt are Killing vectors on the special and Kähler quaternionic T j ¯ and D i u I C g B.7 ,...,n k F − I 1+ ), ( ∂ = 1 ∗ D and and the generalised duals i = T = 2 supergravity R B.1 i I I , I 2 1 A k i F t F N then transform as All of the dependence on the internal The metric on the special manifold Kähler and the gauge kine Z . The two-forms IJ = uv ansatz ( solution to the scalar fields in the where vector multiplet and graviphoton potentials the theories appearing in this paper h The bosonic part of the general gauged C supergravity action ( with terizing a special manifold Kähler with metric to an exact solution to the equations of motion of S fields hypermultiplets is given by [ where N Here be written in terms of a holomorphic prepotential coordinates on the manifold andthe which potential is Kähler and a homogeneous funct JHEP04(2009)102 ) res can = 0. C.1 such (C.9) (C.8) (C.7) (C.6) uv (C.10) (C.13) (C.12) (C.11) Ab H h V ∧ U(2) coset. b . . / a β x ,...,n J 1) α , P +∆ x ! I = 1 ) ¯ ξ , P in the action ( ¯ Ba . β having components α V V B − J ]. The metric dξd ∧ C A ¯ ρ 1 β X 61 B ( ! I A 3 4 + ) , ω X ¯ ¯ β ξdξ 2 ) 0 troducing the one-forms 2 and ) erizes a SU(2 V + , ¯ β ) ω ∧ otential − K llows [ e ¯ Aa ξ β − ∧ ¯ 0 ξdξ ¯ = 1 β β ω ξd − ( dV + 4 . − ∧ 4 3 is the constant symplectic form for Aα ¯ α + A + ] ¯ ξ ) special holonomy, so, as in for ex- IJ α ! I − ∧ 1 αβ H ¯ ξd dω α − ¯

α ( idσ ( β C n ) ( i − ω, P 2 1 1 2 = + N where − − − β β α + [ Sp(2 dρ α ) I dσ (Im =

Aα ¯ 1 and β ∆ × 1 2 ρ 2 – 24 – dP V − 1 2 1 ∧ 2 x ρ 1 √ , = β 4 = σ ¯ ) connections via = α − B i 2 = − V ) A + K 12 ∧ v − J ǫ I n ¯ 2 α ! k k β Aα one can then introduce triplets of Killing prepotentials i ) u I ∧ dρ x ¯ V β k u I 2 α , β k K ρ 1 ( uv − with the constant symplectic form α ρ 4 where 2 1 h = β dξ − √ ( (4 =

Bβ 1 4 αβ v K J v = 3 are the Pauli matrices. The corresponding complex structu V C = − ¯ , X α dq 2 ) I B Aα u AB , ¯ u then satisfy the quaternion algebra. β A ǫ X V dq − V K ¯ α wv Bβ = 1 v satisfying uv ) K AB β e V x h ( x C 4 1 x K = Aα αβ σ ( u

ǫ i 2 V V uw with ], one can introduce vielbeins − = = h = x 60 B x I = σ ). This defines SU(2) and Sp( uv uv A P v h h H ω u Given the Killing vectors It is well-known that the universal hypermultiplet paramet The quaternionic manifold has SU(2) =1. We also find ) n = x 12 I J ( P where One can identify the particular quaternionic geometry as fo which has Ricci tensor equal to minus six times the metric. In one can write The triplet of forms Kähler can then be written as be written as If the gauging is only in the hypermultiplet sector then the p is given by Sp( ample [ and that and C JHEP04(2009)102 ]; etric ]. ] ]. , SPIRES ] ) e presence ] [ ]. SPIRES = 11 SPIRES p, q reduction of ( ] [ ] [ D Y ]. supergravity , gauged supergravity , , ]. SPIRES SO(6) ]. gauged supergravity in hep-th/9807051 [ ] [ = 11 ]. supergravity in five U(1) ]. SPIRES D =2 hep-th/0409097 × [ ] [ ]. hep-th/0312210 SPIRES =8 [ N ]. 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