Consistent supersymmetric Kaluza--Klein truncations with massive modes
aa r X i v : . [ h e p - t h ] M a r Imperial/TP/2009/JG/01
Consistent supersymmetric Kaluza–Kleintruncations with massive modes
Jerome P. Gauntlett, Seok Kim, Oscar Varela and Daniel Waldram
Theoretical Physics Group, Blackett Laboratory,Imperial College, London SW7 2AZ, U.K. and
The Institute for Mathematical Sciences,Imperial College, London SW7 2PE, U.K.
Abstract
We construct consistent Kaluza–Klein reductions of D = 11 supergravityto four dimensions using an arbitrary seven-dimensional Sasaki–Einsteinmanifold. At the level of bosonic fields, we extend the known reduction,which leads to minimal N = 2 gauged supergravity, to also include amultiplet of massive fields, containing the breathing mode of the Sasaki–Einstein space, and still consistent with N = 2 supersymmetry. In thecontext of flux compactifications, the Sasaki–Einstein reductions are gen-eralizations of type IIA SU (3)-structure reductions which include bothmetric and form-field flux and lead to a massive universal tensor multi-plet. We carry out a similar analysis for an arbitrary weak G manifoldleading to an N = 1 supergravity with massive fields. The straightforwardextension of our results to the case of the seven-sphere would imply thatthere is a four-dimensional Lagrangian with N = 8 supersymmetry con-taining both massless and massive spin two fields. We use our results toconstruct solutions of M-theory with non-relativistic conformal symmetry. Introduction
It is now understood that there are very general situations in which one can performconsistent Kaluza–Klein (KK) reductions of supergravity theories. Starting with anysupersymmetric solution of D = 10 or D = 11 supergravity that is the warped prod-uct of an AdS d +1 space with an internal space M , it was conjectured [1] (see also [2])that one can always consistently reduce on the space M to obtain a gauged super-gravity theory in d + 1 dimensions, incorporating only the fields of the supermultipletcontaining the metric. In the dual SCFT these fields are dual to the superconformalcurrent multiplet which includes the energy momentum tensor and R symmetry cur-rents. This conjecture has now been proven to be true for a number of general classesof AdS solutions [3][4][1][5].One simple class of examples consists of
AdS × SE solutions of type IIB su-pergravity, dual to N = 1 SCFTs in d = 4, and AdS × SE solutions of D = 11supergravity, dual to N = 2 SCFTs in d = 3, where SE n is an n -dimensional Sasaki–Einstein manifold. In the former case it is known that one can reduce type IIBsupergravity on a SE space to get minimal N = 1 gauged supergravity in D = 5[3]. Similarly, one can reduce D = 11 supergravity on a SE space to get minimal N = 2 gauged supergravity in D = 4 [1]. In both cases, the bosonic fields in the lowerdimensional supergravity theory are massless, consisting of the metric and the gaugefield, dual to the energy momentum tensor and the R -symmetry current, respectively.For the special cases when SE = S or SE = S these truncations were shown tobe consistent in [6] and [7], respectively. For these special cases, it is expected orknown that there are more general consistent truncations to the maximal gauged su-pergravities in five dimensions (for various partial results see [8][9][10][11]) and fourdimensions [12], respectively.Interestingly, it has recently been shown that the consistent KK reduction of typeIIB on a SE space of [3] can be generalised to also include some massive bosonicfields [13]. The bosonic fields included massive gauge fields as well as massive scalars.One of these massive scalars arises from the breathing mode of the SE . Viewing the SE space, locally, as a U (1) fibration over a four-dimensional K¨ahler-Einstein base,the other massive scalar arises from the mode that squashes the size of the fibre withrespect to the size of the base. This work thus extends earlier work on including suchbreathing and squashing modes for the special case of the five-sphere in [14][15].In order to understand this in more detail, here we will study similar extensions ofthe KK reductions of D = 11 supergravity on a SE space. For the special case of the1even-sphere some results on KK reductions involving the breathing and squashingmodes appear in [14][15]. In this paper we shall show that one can also generalise theKK reduction of [1] to include massive fields: at the level of the bosonic fields we willshow that there is a consistent KK reduction that includes the massless graviton su-permultiplet as well as the massive supermultiplet that contains the breathing mode.In the off-shell four-dimensional N = 2 theory, in addition to the gravity multiplet,the action contains a tensor multiplet together with a single vector multiplet whichacts as a St¨uckelberg field to give mass to the tensor multiplet. We show that onecan also dualize to get an action containing a massive vector multiplet with a gaugedhypermultiplet acting as the St¨uckelberg field. This gives a simple example of themechanism first observed in [16] and then analyzed in [17, 18, 19, 20].We note that our truncation also has a natural interpretation in terms of flux com-pactifications. Viewing the SE manifold locally as a U (1) fibration over a K¨ahler–Einstein manifold, KE , one can reduce from M-theory to type IIA. The truncationthen has the structure of a IIA reduction on a six-dimensional SU (3) structure man-ifold [21]. The tensor and vector multiplets in the N = 2 action correspond to theuniversal tensor multiplet which contains the dilaton, the NS two-form B and a com-plex scalar arising from a RR potential parallel to the (3 ,
0) form on KE , and theuniversal vector multiplet containing a vector and scalars that arises from scaling thecomplexified K¨ahler form. The presence of the background four-form flux, and the“metric fluxes” coming from the twisting of the U (1) fibration and the fact that the(3 ,
0) form on KE is not closed lead to a gauging of the four-dimensional theory.This is complementary to the model discussed in [22] which had a similar structurebut considered different intrinsic torsion in the SU (3) structure. Note that since ourtruncation is consistent there are no approximations in analysing which KK modesshould be kept in the four-dimensional theory.A simple modification of our ansatz leads to an analogous result for a consistentKK reduction of D = 11 supergravity on seven-dimensional manifolds M with weak G holonomy. Recall that such manifolds can be used to construct AdS × M solu-tions that are dual to N = 1 superconformal field theories in d = 3. The conjectureof [1] is rather trivial for this case since it just says that there should be a consistentKK reduction to pure N = 1 supergravity. Here, however, we will see that this can beextended to include the massive N = 1 chiral multiplet that contains the breathingmode of M . The consistent KK truncation that we construct is compatible with thegeneral low-energy KK analysis of D = 11 supergravity reduced on manifolds withweak G structure that was analysed in [23].2iven these results, it is plausible that for AdS × M solutions with any amount ofsupersymmetry 1 ≤ N ≤ N = 8supersymmetry, arising from reduction on S , is that the supermultiplet containingthe breathing mode now contains massive spin-2 fields. Thus if our conjecture iscorrect the consistent KK reduction would lead to a four-dimensional interactingtheory with both massless and massive spin 2 fields, which has been widely thoughtnot to exist. We will return to this point in the discussion section later.A similar result could also hold for reductions of type IIB on S to maximally su-persymmetric theories in five spacetime dimensions containing the massless gravitonsupermultiplet and the massive breathing mode supermultiplet, which again containsmassive spin 2 fields. What is much more certain, however, is that for reductions on SE one can extend the ansatz of [13] to be consistent with N = 1 supersymmetry[24].A principal motivation for constructing consistent KK reductions is that theyprovide powerful methods to construct explicit solutions. Starting with the work of[25][26] there has been some recent interest in constructing solutions of string/M-theory that possess a non-relativistic conformal symmetry. In [13] the KK reductionson SE spaces were used to construct such solutions and examples with dynamicalexponent z = 4 and also z = 2, and hence possessing an enlarged Schr¨odinger sym-metry, were found. The solutions with z = 2 were independently found in [27][28].Here we shall construct similar solutions in D = 11 supergravity for arbitrary SE spaces that exhibit a non-relativistic conformal symmetry with dynamical exponent z = 3.Our presentation will focus on supersymmetric AdS × SE solutions. It is wellknown that for each supersymmetric solution there is a “skew -whiffed” solutionobtained by reversing the sign of the four-form flux, or equivalently changing theorientation on the SE [29]. Apart from the special case of the round S the skew-whiffed solution does not preserve any supersymmetry, but is known to be perturba-tively stable in supergravity [29]. We will show that for the skew-whiffed solutionsthere is also a consistent truncation on the SE space to the bosonic fields of a four-dimensional N = 2 gauged supergravity theory with an AdS vacuum that upliftsto the skew-whiffed solution. Our action is a non-linear extension of one of thoseconsidered recently in [30] in the context of solutions corresponding to holographicsuperconductivity [31][32][33] and offers the possibility of finding exact embeddings3f such solutions into D = 11 supergravity. D = 11 supergravity reduced on SE AdS × SE solutions of D = 11supergravity given by ds = ds ( AdS ) + ds ( SE ) G = vol( AdS ) (2.1)where ds ( AdS ) is the standard unit-radius metric on AdS and the Sasaki–Einsteinmetric ds ( SE ) is normalised so that the Ricci tensor is six times the metric (as fora unit-radius round seven-sphere). The SE space has a globally defined one-form η that is dual to the Reeb Killing vector, and locally we can write ds ( SE ) = ds ( KE ) + η ⊗ η (2.2)where ds ( KE ) is a local K¨ahler-Einstein metric with positive curvature, normalisedso that the Ricci tensor is eight times the metric. On SE there is also a globallydefined two-form J and a (3 , ds ( KE ) respectively and satisfy Ω ∧ Ω ∗ = − iJ / dη = 2 J ,d
Ω = 4 iη ∧ Ω . (2.3)Our conventions for D = 11 supergravity are as in [34]. For completeness, in ap-pendix A we show in detail that given these conventions, together with those for theSasaki–Einstein structure, the solution (2.1) is indeed supersymmetric. We now investigate consistent Kaluza–Klein reductions using this class of solutions.Our ansatz for the metric of D = 11 supergravity is given by ds = ds + e U ds ( KE ) + e V ( η + A ) ⊗ ( η + A ) , (2.4)where ds is an arbitrary metric on a four-dimensional spacetime, U and V are scalarfields and A is a one-form defined on the four-dimensional space. For the four-form4e take G = f vol + H ∧ ( η + A ) + H ∧ J + H ∧ J ∧ ( η + A )+ 2 hJ ∧ J + √ χ ∧ Ω + χ ( η + A ) ∧ Ω + c.c.] , (2.5)where f and h are real scalars, H p , p = 1 , ,
3, are real p -forms, χ is a complexone-form, χ is a complex scalar on the four-dimensional spacetime and “c.c.” denotescomplex conjugate.Notice that this ansatz incorporates all of the constant bosonic modes that arisefrom the G -structure tensors ( η, J, Ω). It generalises the ansatz considered in [1],as we shall discuss in section 3.1. Together the two scalar fields U and V containthe “breathing mode” of the SE space and the “squashing mode” that scales thefibre direction with respect to the local KE space, as we will discuss more explicitlybelow. It is also worth observing that if η, J, Ω instead satisfied dη = dJ = d Ω = 0 thisansatz would be the same ansatz that one would use to reduce D = 11 supergravityon S × CY , keeping the universal N = 2 vector multiplet, with scalars coming fromthe volume mode of the Calabi–Yau, and the universal hypermultiplet. In particular,we should expect that same off-shell supermultiplet degrees of freedom to appear inour four-dimensional theory.We now substitute this ansatz into the equations of motion of D = 11 super-gravity. We will simply summarise the main results here. More details can be foundin appendix B. By analysing the Bianchi identities and the equations of motion forthe four-form, we find that the dynamical degrees of freedom turn out to be thefour-dimensional fields g µν , B , B , A , U, V, h and χ with H = dB H = dB + 2 B + hF F = dA (2.6)Furthermore we find that H = dh and χ = − i Dχ , where Dχ ≡ dχ − iA χ (2.7)and f = 6 e − U − V (1 + h + | χ | ) . (2.8)Note that the expression for f comes from solving (B.12) and incorporates a conve-nient integration constant which fixes the radius of the AdS vacuum and also ensures5hat the reduced D = 4 theory includes the supersymmetric AdS × SE solution(2.1). The expression for the four-form can be tidied up a little to read G = 6 e − U − V (cid:0) h + | χ | (cid:1) vol + H ∧ ( η + A ) + H ∧ J + dh ∧ J ∧ ( η + A ) + 2 hJ ∧ J + √ (cid:2) χ ( η + A ) ∧ Ω − i Dχ ∧ Ω + c.c. (cid:3) . (2.9)We find that all dependence on the internal SE space drops out of the D = 11equations of motion and we are left with equations of motion for the four-dimensionalfields which are written in appendix B. Thus the ansatz (2.4), (2.9) defines a con-sistent KK truncation. The equations of motion can be derived from the followingfour-dimensional action: S = Z d x √− ge U + V h R + 30( ∇ U ) + 12 ∇ U · ∇ V − e − U − V ( ∇ h ) − e − U | Dχ | − e V F µν F µν − e − V H µνρ H µνρ − e − U H µν H µν + 48 e − U − e − U +2 V − h e − U − (cid:0) h + | χ | (cid:1) e − U − V − e − U − V | χ | i + Z h − hH ∧ H + 3 h H ∧ F − h F ∧ F + 6 A ∧ H − i H ∧ ( χ ∗ Dχ − χDχ ∗ ) i . (2.10)It is also helpful to write this with respect to the Einstein-frame metric g E ≡ e U + V g and we find S = Z d x √− g E h R E − ∇ U ) − ( ∇ V ) − ∇ U · ∇ V − e − U − V ( ∇ h ) − e − U | Dχ | − e U +3 V F µν F µν − e U H µνρ H µνρ − e U + V H µν H µν + 48 e − U − V − e − U + V − h e − U − V − (cid:0) h + | χ | (cid:1) e − U − V − e − U − V | χ | i + Z h − hH ∧ H + 3 h H ∧ F − h F ∧ F + 6 A ∧ H − i H ∧ ( χ ∗ Dχ − χDχ ∗ ) i . (2.11) When we set H = H = F = U = V = h = χ = 0, and thus f = 6, the equationsof motion are solved by taking the four-dimensional metric to be ds ( AdS ). This“vacuum solution” uplifts to give the AdS × SE solution given in (2.1). We can6ork out the masses of the other fields, considered as perturbations about this vacuumsolution, by analysing the quadratic terms in the Lagrangian (2.11). One immediatelydeduces that the scalar fields h and χ have m h = 40 and m χ = 40. One can diagonalisethe terms involving the scalar fields U and V by writing U = − u + vV = 6 u + v (2.12)and we find that m u = 16 and m v = 72. Note that in terms of u and v our KK ansatzfor the metric (2.4) can be written ds = e − v/ ds E + e v/ (cid:2) e − u ds ( KE ) + e u ( η + A ) ⊗ ( η + A ) (cid:3) (2.13)and we can identify the scalar field v as the “breathing mode” and u as the “squashingmode” that squashes the size of the fibre with respect to the size of KE , preservingthe volume of the SE space.The quadratic action for the fields A , B , B (setting U = V = h = χ = 0) is Z − F ∧ ∗ F + H ∧ ∗ H − ( dB + 2 B ) ∧ ∗ ( dB + 2 B ) + 6 A ∧ H . (2.14)If one ignores the final term, we see that this has the standard form for a masslessgauge field A and a massive two-form B with B acting as a St¨uckelberg field.However the presence of the final term means that the fields are not properly diag-onalized. To find the mass eigenstates, it is helpful to regard H ′ ≡ dB as a basicfield by introducing a Lagrange multiplier one-form ˜ B and adding a term Z B ∧ dH ′ (2.15)to the action: indeed integrating out ˜ B brings one back to the original quadraticaction. Integrating out H ′ instead, we find H ′ = − ∗ ˜ H − B , where ˜ H ≡ d ˜ B ,and after substitution one obtains the dualised action Z − F ∧ ∗ F + H ∧ ∗ H − ˜ H ∧ ∗ ˜ H + 6 H ∧ ( ˜ B − A ) . (2.16)Continuing we now introduce A = (cid:0) A + 3 ˜ B (cid:1) , B = √ (cid:0) A − ˜ B (cid:1) , (2.17)so that the action can be written Z − d A ∧ ∗ d A + H ∧ ∗ H − d B ∧ ∗ d B − √ H ∧ B . (2.18)7learly A is a massless vector field. The action for the one-form B and the two-form B appears, for instance, in [35]. It can be viewed as describing either a massivevector or a massive two-form field, which are well-known to be equivalent (see forexample [36][37]), with m = 48. For instance, if one further dualises B , one obtainsthe standard St¨uckelberg form for a massive two-form. Alternatively one can dualisethe two-form B to obtain a pseudoscalar a . This is achieved by adding Z adH (2.19)to the action. Integrating out H , we find that H = − ∗ ( da − √ B ) and get theaction for a massive vector field B Z − d A ∧ ∗ d A − d B ∧ ∗ d B + (cid:0) da − √ B (cid:1) ∧ ∗ (cid:0) da − √ B (cid:1) . (2.20)In this form, we see that a is a standard St¨uckelberg scalar field: using the cor-responding gauge symmetry to set a = 0 reveals that B is indeed massive with m = 48.It is interesting to determine the scaling dimensions of the operators in the dualSCFT that correspond to the modes we are considering. The massless vector field, A , has ∆ = 2 and the massless graviton has ∆ = 3. For the scalar fields, using theformula ∆ = ± √ m (2.21)we deduce that the scaling dimensions of u, h, χ and v are given by∆ u = 4 , ∆ h = ∆ χ = 5 , ∆ v = 6 (2.22)Finally, for the massive vector field with m = 48, defined by the fields B and B ,we can use the formula for a massive p -form,∆ = ± p (3 − p ) + m (2.23)to deduce that the dual operator has ∆ = 5.We will show in the next section that the fields that we have retained are thebosonic fields of an N = 2 supergravity theory. In particular they form the bosonicfields of unitary irreducible representations of Osp (2 | M (3 ,
2) for which the supermultiplet structure wasanalysed in detail in [38]. The massless graviton and the massless gauge field that8e have kept are the bosonic fields of the massless graviton multiplet, whose fieldcontent is summarised in table 8 of [38]. By analysing the results of [38] we find that the remaining massive fields are the bosonic fields of a long vector multiplet withfield content as in table 3 of [38] with E = 4, y = 0. Note in particular that with y = 0 the only bosonic modes with non-zero R -charge (“hypercharge”) are the twoscalar fields with ∆ = 5. These correspond to the χ fields which indeed have non-zero R -charge since the (3 , R -charge. N = 2 supersymmetry We now show that the Lagrangian (2.11) is the bosonic part of an N = 2 super-symmetric theory. As formulated it contains, in addition to the N = 2 supergravitymultiplet, a massive two-form and five scalar fields. The appearance of supersymmet-ric theories with a massive two-form in dimensional reductions with non-trivial fluxeswas first observed in [16]. In terms of supermultiplets the two-form and three scalarsshould form a tensor multiplet, while the St¨uckelberg gauge field and the remainingtwo-scalars form a vector multiplet. The general couplings of such N = 2 theoriesare discussed in [17, 18, 19] (see also [20] for the N = 1 analogue). For the case inhand, it should be possible to dualize to a massive vector multiplet and a conventional(gauged) hypermultiplet.As we have noted, our Sasaki–Einstein reduction can also be viewed as a fluxcompactification of type IIA supergravity. In particular, if instead of a reductionon a Sasaki–Einstein manifold we were considering a reduction on S × X where X is a Calabi–Yau threefold, the fields U , χ and the scalar dual of B would param-eterize a universal hypermultiplet. Similarly, 2 U + V and h would be the scalarsfor the universal vector multiplet related to rescaling the metric on the Calabi–Yauspace. The kinetic terms of these fields should be unchanged by going to the Sasaki–Einstein reduction, so our expectation is that the action (2.11) can be rewritten asa gauged universal hypermultiplet coupled to a single universal vector multiplet. Inthe following we will show how this structure arises.Let us first identify the structure before dualizing. The generic form for the cou-pling of vector and tensor multiplets has been discussed in some detail for instancein [39]. We note that in identifying with the theory of general N = 2 gauged su-pergravities as summarised in appendix B, we should multiply the overall action in More specifically, our modes are obtained in (3.19) and (3.20) of [38] with M = M = 0, andhence J = 0, consistent with the fact that the modes are singlets with respect to the SU (3) flavoursymmetry of this specific Sasaki–Einstein manifold. /
2, which we will do in this section only. We can write (half)the action (2.11) as S = Z d x √− g E (cid:0) R E − V (cid:1) + S V + S H (2.24)where S V = 12 Z √− g E h − ( ∇ (2 U + V )) − (cid:0) e − U − V (cid:1) ( ∇ h ) − e U +3 V F µν F µν − e U + V H µν H µν i + 12 Z − hH ∧ H + 3 h H ∧ F − h F ∧ F , (2.25)while S H = 12 Z d x √− g E (cid:2) − ( ∇ (6 U )) − e U H µνρ H µνρ − e − U | Dχ | (cid:3) + 12 Z A ∧ H − i H ∧ ( χ ∗ Dχ − χDχ ∗ ) , (2.26)and V = − e − U − V + 3 e − U + V + 12 h e − U − V + 12 | χ | e − U − V + 9 (cid:0) h + | χ | (cid:1) e − U − V . (2.27)If we ignore the B term in the definition of H in (2.6) we see that S V canbe written in the form of an ungauged vector multiplet action, as summarized inappendix C, as follows. Introducing τ = h + ie U + V we define X I = (1 , τ ) and thegauge fields F I = √ ( F , − dB ) with I = 0 ,
1. One then finds the N IJ matrix isgiven by N IJ = τ ( τ − h ) 3 hτ hτ − τ + h ) ! (2.28)together with the corresponding holomorphic prepotential F ( X ) = − ( X ) X , (2.29)giving the K¨ahler potential K V = − log (cid:0) i ¯ X I F I − iX I ¯ F (cid:1) = − log i ( τ − ¯ τ ) . (2.30)This is the standard form that arises from flux compactification on a SU (3) structuremanifold [16], with a single K¨ahler modulus. We also note that, if we ignore thecoupling to the vector multiplets, one can dualize the two-form B to get a pseudo-scalar a by adding the term (2.19) to S H giving e U H = −∗ (cid:2) da − i ( χ ∗ dχ − χdχ ∗ ) (cid:3) .10hen identifying ρ = 4 e U , σ = 4 a and ξ = √ χ , we get the standard metric (C.9)on the universal hypermultiplet space.To make the full dualization from the massive tensor multiplet ( e U , B , χ, ¯ χ ) toa massive vector multiplet, one must first dualise the field B by adding the term(2.15) to the action and then integrating out H ′ . We now find that H ′ + 2 B + hF = 14 h + e U +2 V h h ( ˜ H + h F ) − e U + V ∗ ( ˜ H + h F ) i (2.31)where ˜ H ≡ d ˜ B as before, and after substitution one finds a dual action containinggauge fields A , ˜ B , B . One can then dualise the two-form B to obtain a pseudo-scalar by adding the term (2.19). After integrating out H we now find e U H = − ∗ h da − A − ˜ B ) − i ( χ ∗ Dχ − χDχ ∗ ) i . (2.32)After these dualisations the new expressions for S V and S H are S V = 12 Z d x √− g E h − (cid:0) ∇ (2 U + V ) (cid:1) − (cid:0) e − U − V (cid:1) ( ∇ h ) i + 12 Z h Im( τ + h ) − (cid:0) ˜ H + h F (cid:1) ∧ ∗ (cid:0) ˜ H + h F (cid:1) + Re( τ + h ) − (cid:0) ˜ H + h F (cid:1) ∧ (cid:0) ˜ H + h F (cid:1) − (cid:0) e U + V (cid:1) F ∧ ∗ F − h ˜ H ∧ F − h F ∧ F i (2.33)and S H = − Z d x √− g E (cid:20)(cid:0) ∇ (6 U ) (cid:1) + 3( e − U ) (cid:12)(cid:12) dχ − iA χ (cid:12)(cid:12) + (cid:0) e − U (cid:1) (cid:16) ∇ a − A − ˜ B ) − i ( χ ∗ Dχ − χDχ ∗ ) (cid:17) (cid:21) . (2.34)We now compare S V with the general gauged N = 2 action (C.1) given in ap-pendix C. If we identify the gauge fields ˜ F I = ( F , − ˜ H ) and introduce new homo-geneous coordinates ˜ X I = (1 , τ ), we find the gauge kinetic matrix in (C.1) is givenby ˜ N IJ = 12( τ + h ) − τ ¯ τ hτ hτ ! , (2.35)and, since we have dualized the gauge fields, there is a new holomorphic prepotential˜ F = q ˜ X ( ˜ X ) . (2.36)This indeed correctly reproduces (2.35) and leads to the K¨ahler potential˜ K V = − log i ( τ − ¯ τ ) + log 2 , (2.37)11hich agrees with (2.30) up to a (constant) K¨ahler transformation. Notice that( ˜ X I , ˜ F I ) and ˜ N IJ can be obtained from ( X I , F I ) and N IJ by a symplectic transfor-mation (C.5) with A BC D ! = √ − , (2.38)together with a rescaling of the ˜ X I homogeneous coordinates by 1 / √
2. Note that,up to a normalization and as expected given we are dualizing B , the matrix (2.38)simply exchanges the electric and magnetic gauge fields for F .Now we consider S H . Identifying ρ = 4 e U , σ = 4 a , ξ = √ χ we see that it indeedmatches the universal hypermultiplet form given in (C.9). In appendix C we havelabelled these coordinates q u , u = 1 , . . . ,
4. From the terms Dq u = dq u − k uI A I in(C.1) we see that gauging is along Killing vectors k = 6 ∂ a + 4 i ( χ∂ χ − ¯ χ∂ ¯ χ ) = 24 ∂ σ − i ( ξ∂ ξ − ¯ ξ∂ ¯ ξ ) ,k = 6 ∂ a = 24 ∂ σ . (2.39)Given the formulae in appendix C for the quaternionic geometry it is straightforwardto calculate that for the Killing vector k = ∂ σ we have the Killing prepotential P σ = i/ ρ − i/ ρ ! (2.40)and for k = iξ∂ ξ − i ¯ ξ∂ ¯ ξ we have P ξ = i (1 − ρ − ξ ¯ ξ ) − iξρ − / − i ¯ ξρ − / − i (1 − ρ − ξ ¯ ξ ) ! . (2.41)The Killing prepotentials P I , corresponding to (2.39), are therefore P = 24 P σ − P ξ , P = 24 P σ . (2.42)Finally substituting these expressions into the general form (C.8) for the potential V we reproduce (2.27). This completes our demonstration that our action is the bosonicaction of an N = 2 supergravity theory In this section we observe that there are some additional consistent truncations in-corporated in our KK ansatz (2.4), (2.9), compatible with the general equations of12otion contained in appendix B. We begin by observing that it is consistent to setthe complex scalar field χ = 0. This is not surprising as this is the only field inthe ansatz that carries non-zero R -charge. The resulting equations of motion can beobtained from an action obtained by setting χ = 0 in (2.11). It is also consistent to set U = V = h = χ = H = 0, f = 6 and H = − ∗ F . Thissets all of the massive fields to zero and we then find that the equations of motioncome from a Lagrangian given by S = Z d x √− g [ R − F µν F µν + 24] (3.1)This is the consistent KK reduction on a SE to the massless fields of N = 2 D = 4gauged supergravity that was discussed in [1].It is interesting to ask whether this truncation to minimal gauged supergravitycan be extended to just include the breathing mode scalar v . However, if we take h = χ = H = 0, H = − ∗ F with U = V = v/ f = 6 e − v/ we find thatconsistency requires v = 0 in addition. We next observe that it is possible to consistently truncate to just the scalar fieldsplus the metric by setting H = H = F = 0 and Im χ = 0. The resulting equationsof motion follow from an action which can be obtained by substituting this truncationdirectly into the general action (2.11): S = Z d x √− g E h R E − ∇ u ) − ( ∇ v ) − e − u − v ( ∇ h ) − e u − v ( ∇ χ R ) + 48 e u − v − e u − v − h e u − v − h + χ ) e − v − e − u − v χ i (3.2)and we have switched from U and V to u and v via (2.12) and χ R = Re χ .In fact, it is also consistent to further set χ R = 0 or h = 0, or both, and theequations of motion are those that are obtained by substituting into the action (3.2).Note that for the case of the seven-sphere setting χ R = h = 0, which just maintainsthe breathing and squashing mode scalars, was also considered in [14]. Indeed if we Note that in section 2.2 of [14] they also consider the truncation with, in the language of this χ R = h = 0, v = − ˜ ϕ/ √ u = ˜ φ/ √
21 into (3.2) we obtain resultsequivalent to (2.20) and (2.21) of [14].A different, further consistent truncation is achieved by setting u = 0 and χ R = √ h in (3.2). This truncation generically breaks supersymmetry down to N = 1, aswe will see in the next subsection. G case As we have just noted, it is consistent to set H = H = F = 0, u = 0 (or,equivalently, U = V ) and χ = √ h . The resulting equations of motion can beobtained from an action which can be obtained from (3.2) and reads: S = Z d x √− g E h R E − ( ∇ v ) − e − v ( ∇ h ) + 42 e − v − e − v h − (cid:0) h (cid:1) e − v i . (3.3)Note that expanding about the AdS vacuum we find m v = 72, m h = 40 and hence∆ v = 6, ∆ h = 5.It is interesting to observe that for this truncation, the KK ansatz (2.4), (2.9) forthe D = 11 fields can be written ds = ds + e v/ ds ( SE ) G = f vol + dh ∧ ϕ + 4 h ∗ ϕ (3.4)where f = 2 e − v/ (3 + 7 h ) and we have introduced the quantities ϕ = J ∧ η + Im Ω ∗ ϕ = J ∧ J + η ∧ Re Ω (3.5)that satisfy dϕ = 4 ∗ ϕ (3.6)Interpreting ϕ as a G structure on the seven-dimensional space SE , the condi-tion (3.6) is equivalent to weak G holonomy (i.e. that the cone over the space has Spin (7) holonomy). One can then generalize by replacing SE with an arbitraryspace M with weak G holonomy and the ansatz (3.4) still gives a consistent trun-cation. One would expect such a truncation to have N = 1 supersymmetry, with paper, h = χ = H = 0. However, this is not a consistent truncation: equation (B.4) implies that H = 0 and then (B.9) implies that F = 0. N = 1 supermultiplet and the breathing mode in a massive N = 1 chiral multiplet. In fact, introducing a complex scalar φ = e v + ih , the N = 1supersymmetry of the action (3.3) can be explicitly exhibited by rewriting it in termsof a K¨ahler metric g φ ¯ φ = ∂ φ ∂ ¯ φ K with K¨ahler potential K = − φ + ¯ φ ), and asuperpotential W = 4 √ φ − S = Z d x √− g E h R E − g φ ¯ φ ∂ µ φ∂ µ ¯ φ − e K (cid:16) g φ ¯ φ | D φ W | − | W | (cid:17)i (3.7)where D φ W = ∂ φ W + ( ∂ φ K ) W . It is worthwhile noting that starting with the KKansatz (3.4) this superpotential can be derived from the general expression for theform of the superpotential in KK reductions on manifolds with G structure that wasobtained in [23]. In contrast to [23], here we have also shown that this particular KKreduction is a consistent KK truncation.We could go one step further and also set h = 0. We then get the consistent KKtruncation that is valid for any Einstein seven-manifold, where one keeps only themetric and the breathing mode scalar. The action is given by S = Z d x √− g E (cid:2) R E − ( ∇ v ) + 42 e − v − e − v (cid:3) (3.8)and we see that m v = 72 and hence ∆ u = 6. The action (3.8) was first obtained in[14] in the context of the seven-sphere. Specifically (3.8) can be obtained from (2.6),(2.7) of [14] by setting φ = −√ v , c = 6, R = 42. We can also consider a truncation to a metric and a massive vector field. We now set U = V = h = χ = 0, f = 6, H = ∗ F and H = 8 ∗ A . We now find that providedwe restrict to configurations that satisfy F ∧ F = F ∧ ∗ F = A ∧ ∗ A = 0 , (3.9)the equations of motion can be written d ∗ F = 48 ∗ A ,R µν = − g µν + F µρ F ν ρ + 32 A µ A ν = − g µν + (cid:0) F µρ F νρ − g µν F ρσ F ρσ (cid:1) + 32 ( A µ A ν − g µν A ρ A ρ ) (3.10)where we have used (3.9) to get the last line. These equations of motion come fromthe Lagrangian S = Z d x √− g (cid:2) R − F µν F µν − A µ A µ + 24 (cid:3) (3.11)15hich describes a metric coupled to a massive vector field with m = 48, providedthat we impose the conditions (3.9) by hand. We will return to this truncation toconstruct solutions of D = 11 supergravity in the next section. As an application we construct solutions of D = 11 supergravity by constructingsolutions to the four-dimensional equations given in (3.9) and (3.10). We considerthe ansatz given by ds = − α ρ k ( dx + ) + dρ ρ + ρ (cid:0) − dx + dx − + dx (cid:1) A = cρ k dx + (4.1)We find that k = 3 with c = α solves all the equations as does k = − c =15 α /
8. We can now uplift these solutions to D = 11 by setting U = V = h = χ = 0, f = 6, H = ∗ F and H = 8 ∗ A and substituting into (2.4) and (2.9). Writingthis out explicitly for the k = 3 case case we obtain ds = − α ρ ( dx + ) + dρ ρ + ρ (cid:0) − dx + dx − + dx (cid:1) + ds ( KE ) + ( η + αρ dx + ) G = ρ dx + ∧ dx − ∧ dρ ∧ dx + α dx + ∧ dx ∧ d ( ρ η ) (4.2)This solution is in close analogy to the solutions considered in [13] and has a non-relativistic conformal symmetry with dynamical exponent z = 3 i.e. is invariant underGalilean transformations generated by time and spatial translations, Galilean boosts,a central mass operator, and scale transformations . This solution is supersymmetric,generically preserving two supersymmetries, as explained in [40]. Recall that for each
AdS × M Freund–Rubin solution there is another “skew-whiffed”solution [29] which can be obtained by reversing the sign of the flux (or equivalentlychanging the orientation of M ). With the exception of the special case where M is Recall that only for dynamical exponent z = 2 can the algebra be enlarged to include anadditional special conformal generator. Also note that the k = − z = − S , at most only one of the two solutions is supersymmetric. For example,if we reverse the sign of the flux in the supersymmetric AdS × SE solution (2.1) weobtain another AdS × SE solution of D = 11 supergravity given by ds = ds ( AdS ) + ds ( SE ) G = − vol( AdS ) (5.1)which does not preserve any supersymmetry (provided SE is not S ).By a very small modification of the truncation discussed in section 2.1 above, wecan obtain a second consistent truncation on SE to a D = 4 theory that containsthe skew-whiffed solution. In particular, we solve (B.12) by now setting f = 6 e − U − V ( − h + | χ | ) (5.2)where we have changed the sign of the constant factor (when U = V = h = χ = 0).The rest of the analysis essentially goes through unchanged but the sign propagatesinto the D = 4 action in two places. In (2.11) (1 + h + | χ | ) → ( − h + | χ | ) and 6 A ∧ H → − A ∧ H . The AdS vacuum solution of this theory now uplifts tothe non-supersymmetric skew-whiffed solution. The mass spectrum for this vacuumcan be easily calculated and the only difference from section 2.2 is that now m χ = − m h = − ± = 1 ,
2. As expected this bosonicmass spectrum is inconsistent with a vacuum preserving N = 2 D = 4 supersymmetrysince it does not match the bosonic Osp (2 |
4) multiplet structure.Despite the fact that the skew-whiffed vacuum is not supersymmetric the D = 4action has the bosonic content consistent with N = 2 supersymmetry. The analysisof section 2.3 goes through essentially unchanged, but the sign change in the D = 4action, 6 A ∧ H → − A ∧ H , means that the gauging is now along Killing vectorsgiven by k = − ∂ a + 4 i ( χ∂ χ − ¯ χ∂ ¯ χ ) = − ∂ σ − i ( ξ∂ ξ − ¯ ξ∂ ¯ ξ ) ,k = 6 ∂ a = 24 ∂ σ . (5.3)The corresponding Killing prepotentials P I are then P = − P σ − P ξ , P = 24 P σ . (5.4)Substituting these expressions into the general form (C.8) for the potential V wereproduce (2.27) after the change (1 + h + | χ | ) → ( − h + | χ | ) . For a general SE the AdS vacuum spontaneously breaks the N = 2 supersymmetry of the actionwith f = − Here we are assuming that the truncation at the level of the bosonic fields can be extended toinclude the fermions.
17s we have already noted, for the special case that SE is the round S , thecorresponding AdS × S solutions are supersymmetric for either sign of the flux. Itis interesting to observe that while the AdS vacuum of the truncated theory with f = 6 contains modes that fall into OSp (2 |
4) multiplets, this is not the case for the
AdS vacuum of the theory with f = −
6, despite the fact that the uplifted solutionis (maximally) supersymmetric. In particular, while the f = 6 theory retains an N = 2 breathing mode multiplet together with the supergravity multiplet, in the f = − h and χ fields are no longer part ofthe breathing multiplet but instead are part of the N = 8 graviton supermultiplet.Nonetheless this leads to a consistent truncation. This is a novel and interestingphenomenon that would be worth investigating further, including from the dual SCFTpoint of view.Many of the additional truncations of the N = 2 theory that we considered insection 3 have similar analogues in the skew-whiffed theory with only some minorobvious sign changes required. For example, the D = 4 action that contains thenon-supersymmetric skew-whiffed weak G case can be written in a manifestly N =1 language and we find that the only difference is that the superpotential W =4 √ φ − → √ φ + 3). For the reduction to the massive vector field, weshould now set U = V = h = χ = 0, f = − H = − ∗ F and H = − ∗ A .These sign changes mean that when we uplift the solution (4.1) to D = 11 we obtainthe solution (4.2) but with the sign of the four-form flux reversed. Note however, asnoticed in [30], it is no longer possible to truncate to the field content of minimalgauged supergravity as in section 3.1.Recently KK reductions of AdS × SE solutions were considered at the linearisedlevel [30] and it was shown that, for the skew-whiffed solution, the modes correspond-ing to the massless gauge-field A and the complex scalar χ lead to a D = 4 theorythat exhibits holographic superconductivity. Indeed, at the linearised level, in ouranalysis we can set U = V = h = H = 0 with F = ± ∗ H where the upper(lower) sign corresponds to the supersymmetric (skew-whiffed) truncation. Writing A = A / χ = p / φ , the linearised action is given by S = Z d x √− g h R + 24 − F µν F µν − | Dφ | − m | φ | i (5.5)with Dφ = dφ − i A φ and m = 40 , − M = 2, L = 1 / q = 2and g = 1 in their equation (1). In particular, for the skew-whiffed solution, thereare solutions of this linearised theory corresponding to holographic superconductors.18ur generalised, non-linear and consistently truncated action for the skew-whiffedsolutions thus provides an ideal set up to extend the work of [30] to obtain analogousexact solutions of D = 11 supergravity. In this paper we have considered consistent truncations on Freund–Rubin back-grounds, keeping the breathing mode and with varying degrees of supersymmetry.We have shown that for
AdS × M solutions of D = 11 supergravity where M isan Einstein space, it is always consistent to truncate the KK spectrum to the gravi-ton plus the breathing mode, which is dual to an operator in the dual CFT with∆ = 6. For AdS × M solutions with N = 1 and N = 2 supersymmetry, where M has weak G holonomy or is a Sasaki–Einstein seven manifold, respectively, we havealso shown that it is consistent to truncate to the massless graviton supermultipletcombined with the supermultiplet containing the massive breathing mode. In bothcases, the KK ansatz contains the constant KK modes associated with the weak G or the Sasaki–Einstein structure.Moving to AdS × M solutions with N = 3 supersymmetry, where M is tri-Sasakian, it is natural to expect that a similar story unfolds. Recall that a tri-Sasakianmanifold has an SO (3) group of isometries corresponding to SO (3) R -symmetry. Bywriting down a KK ansatz that incorporates the constant modes associated with thetri-Sasaki structure we strongly suspect that it will be possible to obtain a consistentKK truncation with N = 3 supersymmetry. Such a truncation would retain the fieldsof the massless graviton supermultiplet (table 3 of [41]) which consist of the gravitonand the SO (3) vector fields, and the breathing mode supermultiplet, which now sitsin a long gravitino multiplet (table 2 of [41] with J = 0) consisting of six massivevectors, transforming in two spin-one representations of SO (3), four scalars in thespin-zero representation, and ten scalars transforming in two spin-two representations.Following this pattern one is led to consider the maximally supersymmetric AdS × S solution with N = 8 supersymmetry. It is again natural to conjecture that thereis an analogous consistent KK truncation that extends the one containing just the N = 8 graviton supermultiplet [12], i.e. N = 8 SO (8) gauged supergravity, to alsoinclude the N = 8 supermultiplet containing the breathing mode. Using the resultsof [42] or [43] we conclude that the bosonic fields of this supermultiplet consist ofscalars in the v , ′ s , , s and irreps of SO (8), where the singlet is thebreathing mode, vectors in the v , and irreps and massive spin-two fields in19he v irrep. A particularly interesting feature is the appearance of massive spin-twofields in addition to the graviton. This is remarkable since some general argumentshave been put forward, for instance in [44], that it is not possible to have consistenttheories of a finite number of massive and massless spin-two fields. However, forinstance, the group theory arguments in [44], as for conventional N = 8 SO (8)supergravity, are not directly applicable here, and furthermore we are led to a theorywith a very particular matter content, which suggests a picture where consistencyarises from particular conspiracies among the fields, and perhaps depending cruciallyon the existence of an AdS vacuum. If this putative theory exists, it may also notbe possible to further truncate the theory while keeping massive spin-two fields. Itis worth pointing out that unlike the cases we have studied in this paper, and thetri-Sasakian case mentioned above, it is much less clear how to directly construct theKK truncation ansatz for this case.Let us now return to the
AdS × M solutions of type IIB supergravity where M isEinstein. Once again there is a consistent KK truncation that keeps the graviton andthe breathing mode which is now dual to an operator with ∆ = 8. If M is Sasaki–Einstein then it is possible to generalise the ans¨atze of [3] and of [13] to obtain aconsistent KK truncation that includes the bosonic fields of the N = 1 gravitonmultiplet plus the breathing mode multiplet. We will report on the details of this in[24].For the special case when M = S we are led to conjecture that there is a con-sistent truncation to the massless graviton supermultiplet, i.e. the fields of maximal SO (6) gauged supergravity, combined with the massive breathing mode multipletwhose field content can be obtained from [45]: the bosonic fields consist of scalars inthe , C , C , , C , irreps of SU (4), where the breathing mode is againthe singlet, vectors in the , C , irreps, two-forms in the C , C , C irrepsand massive spin-two fields in the irrep. Note that for this case the operator dualto the breathing mode has been argued to be dual to an operator in N = 4 superYang-Mills theory of the form T rF + . . . , where here F is the N = 4 Yang-Millsfield strength, and it has been argued that its detailed form can be obtained fromexpanding the Dirac–Born–Infeld action for the D3-brane [46][47][48].In a similar spirit we can consider AdS × S solutions of D = 11 supergravity.There is a known consistent truncation [14] that keeps the graviton and the breathingmode which is now dual to an operator with scaling dimension ∆ = 12. If this can beextended to include the full N = 8 supermultiplets then there would be a consistentKK truncation extending the known one to maximal SO (5) gauged supergravity2049][50] to also include the breathing mode supermultiplet. The field content of thislatter multiplet can be found in [51] (based on the results of [52] [53]): we find scalarsin the , and irreps of SO (5), where the singlet is the breathing mode, vectorsin the and irreps, three-forms satisfying self-dual equations in the and irreps, two-forms in the irrep and massive spin-two fields in the irrep.It would also be interesting to see if similar results can be obtained for classes ofsupersymmetric AdS solutions outside of the Freund-Rubin class that we have beenconsidering so far. The KK truncations to the massless graviton supermultiplets forthe class of N = 2 and N = 1 AdS solutions of D = 11 supergravity classified in [54]and [55] were presented in [5] and [4], respectively. Similarly, the KK truncations forthe class of N = 2 AdS solutions of D = 11 supergravity and the class of N = 1 AdS solutions of type IIB which were classified in [56] and [57], respectively, werepresented in [1]. It would be interesting to extend these KK truncations to alsoinclude breathing mode multiplets.Finally, it would be desirable to have an argument from the SCFT side of thecorrespondence as to why the KK truncations containing both the graviton multipletsand the massive breathing mode multiplets are consistent. Acknowledgements
We would like to thank Frederik Denef, Mike Duff, Ami Hanany, Sean Hartnoll, ChrisHull, Mukund Rangamani, James Sparks, Kelly Stelle, Arkady Tseytlin and TobyWiseman for helpful discussions. JPG is supported by an EPSRC Senior Fellowshipand a Royal Society Wolfson Award. OV is supported by a Spanish Government’sMEC-FECYT postdoctoral fellowship, and partially through MEC grant FIS2008-1980.
A Supersymmetry of the
AdS × SE Solution
In this appendix we show that the solution given by ds = ds ( AdS ) + ds ( SE ) ,G = 6 vol = vol( AdS ) , (A.1)is supersymmetric given our set of conventions. These are, for D = 11 supergravity,the conventions given in [34], the structure on SE is defined by the forms η , J and21 satisfying dη = 2 J ,d
Ω = 4 iη ∧ Ω , vol( SE ) = η ∧ J = η ∧ i Ω ∧ Ω ∗ , (A.2)and the D = 11 volume form is ǫ = vol ∧ vol( SE ).It will be sufficient to focus on the Poincar´e supersymmetries. To do so, we startby rewriting the solution in terms of a Calabi–Yau fourfold cone metric. We introducecoordinates for the AdS space ds ( AdS ) = 14 (cid:18) dρ ρ + ρ η µν d ¯ ξ µ d ¯ ξ ν (cid:19) = dr r + r η µν dξ µ dξ ν (A.3)with ρ = r , ¯ ξ µ = 2 ξ µ and µ = 0 , ,
2, and define the four-dimensional volume formvol = r dξ ∧ dξ ∧ dξ ∧ dr . The D = 11 solution can then be recast in the form ds = H − / η µν dξ µ dξ ν + H / ds ( C ) ,G = dξ ∧ dξ ∧ dξ ∧ d ( H − ) (A.4)where we have introduced the cone metric over the SE space, ds ( C ) = dr + r ds ( SE ) , (A.5)and H = r − is harmonic on C . The eleven-dimensional volume form is then ǫ = H / dξ ∧ dξ ∧ dξ ∧ vol( C ) where vol( C ) = r dr ∧ vol( SE ).The Sasaki–Einstein structure (A.2) defines a unique Calabi–Yau structure on thecone given by the SU (4) invariant tensors J CY = rdr ∧ η + r J , Ω CY = r ( dr + irη ) ∧ Ω , (A.6)determined by requiring the closure of J CY and Ω CY to be equivalent to dη = 2 J and d Ω = 4 iη ∧ Ω. In particular, we then findvol( C ) = J CY = Ω CY ∧ Ω ∗ CY . (A.7)We now turn to the supersymmetry. We introduce a D = 11 orthonormal frame: e µ = H − / dξ µ , e a +2 = H / g a , a = 1 , . . . , , (A.8)where g a is an orthonormal frame for the cone metric. Following the conventionsof [34], by definition ǫ = e ∧ e ∧ · · · ∧ e and sovol( C ) = g ∧ g ∧ · · · ∧ g . (A.9)22e can then decompose the D = 11 gamma-matrices asΓ µ = τ µ ⊗ γ (8) , Γ a +2 = ⊗ γ a , a = 1 , . . . , τ = 1 and where γ (8) = γ γ . . . γ is the chirality operator in D = 8. The D =11 supersymmetry equations given in [34] are satisfied by a solution of the form (A.4)provided the supersymmetry transformation parameter satisfies the gamma-matrixprojection condition Γ ǫ = ǫ ⇔ Γ ... ǫ = ǫ . (A.11)More precisely, there are Poincar´e Killing spinors of the form ǫ = H − / α ⊗ β , (A.12)where α is a constant two-component Majorana spinor in D = 3 and β is a 16-component Majorana–Weyl spinor in D = 8 satisfying ∇ a β = 0 , γ (8) β = β . (A.13)For there to be two independent solutions β ( i ) with i = 1 ,
2, the cone metric must beCalabi–Yau. In particular, the β ( i ) can be chosen to be orthogonal and the Calabi–Yau structure J CY and Ω CY can be written as bilinears in β ( i ) . Specifically, one canchoose a frame { g a } and spinor projections exactly as in appendix B of [58] such that J CY = g + g + g + g , Ω CY = ( g + ig ) ∧ ( g + ig ) ∧ ( g + ig ) ∧ ( g + ig ) . (A.14)Crucially, from (A.9), we see these satisfy the orientation relation (A.7). Thus theCalabi–Yau structure (A.6) on the cone C defined by the Sasaki–Einstein struc-ture (A.2) is indeed of the type required for the solution to be supersymmetric.Note that if one takes the skew-whiffed solution where G = − vol( AdS ), su-persymmetry would then imply γ (8) β = − β . This would in turn require a Calabi–Yau structure ( J ′ CY , Ω ′ CY ) on C satisfying vol( C ) = − J ′ CY = − Ω ′ ∧ Ω ′ ⋆ . Thestructure defined by the Sasaki–Einstein manifold is not of this type, and hence theskew-whiffed solution is generically not supersymmetric. B Details on the KK reduction
As discussed in the main text, our ansatz for the metric of D = 11 supergravity isgiven by ds = ds + e U ds ( KE ) + e V ( η + A ) ⊗ ( η + A ) (B.1)23hile for the four-form we consider G = f vol + H ∧ ( η + A ) + H ∧ J + H ∧ J ∧ ( η + A ) + 2 hJ ∧ J + √ χ ∧ Ω + χ ( η + A ) ∧ Ω + c.c.] . (B.2)For the D = 11 volume-form we choose ǫ = e U + V vol ∧ vol( KE ) ∧ η , where vol isthe D = 4 volume form. In both D = 11 and D = 4 we use a mostly plus signatureconvention.We now substitute this ansatz into the equations of motion of D = 11 supergravity.The Bianchi identity dG = 0 is satisfied provided dH = 0 , (B.3) dH = 2 H + H ∧ F , (B.4) H = dh , (B.5) χ = − i Dχ , (B.6)where F ≡ dA , Dχ ≡ dχ − iA χ and we note that (B.3) follows from (B.4) and(B.5). Note that using (B.5) and (B.6) we can write the four-form as G = f vol + H ∧ ( η + A ) + H ∧ J + dh ∧ J ∧ ( η + A ) + 2 hJ ∧ J + √ (cid:2) χ ( η + A ) ∧ Ω − i Dχ ∧ Ω + c.c. (cid:3) . (B.7)We solve equations (B.3) and (B.4) by introducing potentials B and B via H = dB ,H = dB + 2 B + hF . (B.8)Similarly the equation of motion for the four-form, d ∗ G + G ∧ G = 0, is alsosatisfied if d (cid:0) e U − V ∗ H (cid:1) − e U + V f F + 6 e U + V ∗ H + 12 hH + 3 i Dχ ∧ Dχ ∗ = 0 (B.9) d (cid:0) e U + V ∗ H (cid:1) + 2 dh ∧ H + 4 hH = 0 (B.10) d (cid:0) e U − V ∗ dh (cid:1) + e U + V ∗ H ∧ F + H ∧ H + 4 h (cid:0) f + 4 e − U + V (cid:1) vol = 0 (B.11) d [ e U + V f − h + | χ | )] = 0 (B.12) D (cid:0) e V ∗ Dχ (cid:1) + iH ∧ Dχ + 4 χ ( f + 4 e − V )vol = 0 . (B.13)One can show that (B.10) can be obtained by acting with d on (B.9). We can solve(B.12) by setting f = 6 e − U − V ( ± h + | χ | ) (B.14)24here the constant factor of ± U = V = h = χ = 0) is chosen as a convenientnormalisation. The upper sign corresponds to reducing to a D = 4 theory thatcontains the supesymmetric AdS × SE solution of D = 11 supergravity while thelower sign corresponds to the skew-whiffed AdS × SE solution, which genericallydoesn’t preserve any supersymmetry.Finally we consider the D = 11 Einstein equations: R AB = G AC C C G BC C C − g AB G C C C C G C C C C . (B.15)To calculate the Ricci tensor for the D = 11 metric we use the orthonormal frame¯ e α = e α , α = 0 , , , , ¯ e i = e U e i , i = 1 , . . . , , ¯ e = e V ˆ e ≡ e V ( η + A ) . (B.16)We then observe that the corresponding spin connection can be written¯ ω αβ = ω αβ − e V F αβ ˆ e ¯ ω αi = − e U ∂ α U e i ¯ ω α = − e V ∂ α V ˆ e − e V F αβ e β ¯ ω ij = ω ij − e V − U J ij ˆ e ¯ ω i = − e V − U J ij e j (B.17)After some computation we find that the components of the Ricci tensor, ¯ R AB , aregiven by¯ R αβ = R αβ − ∇ β ∇ α U + ∂ α U ∂ β U ) − ( ∇ β ∇ α V + ∂ α V ∂ β V ) − e V F αγ F β γ ¯ R αi = 0¯ R α = − e − V − U ∇ γ (cid:0) e V +6 U F γα (cid:1) ¯ R ij = δ ij (cid:2) e − U − e V − U − ∇ γ ∇ γ U − ∂ γ U ∂ γ U − ∂ γ U ∂ γ V (cid:3) ¯ R i = 0¯ R = 6 e V − U − ∇ γ ∇ γ V − ∂ γ U ∂ γ V − ∂ γ V ∂ γ V + e V F αβ F αβ (B.18)Using these results we find that the D = 11 Einstein equations (B.15) reduce to25he following four equations in D = 4: R αβ = 6 ( ∇ β ∇ α U + ∂ α U ∂ β U ) + ( ∇ β ∇ α V + ∂ α V ∂ β V )+ e − U − V (cid:0) ∇ α h ∇ β h − η αβ ∇ λ h ∇ λ h (cid:1) + e − U (cid:2) ( D α χ )( D β χ ∗ ) + ( D β χ )( D α χ ∗ ) − η αβ ( D γ χ )( D γ χ ∗ ) (cid:3) − η αβ (cid:0) e − U h + f + 4 e − U − V | χ | (cid:1) + e V F αγ F βγ + e − V (cid:0) H αλµ H βλµ − η αβ H λµν H λµν (cid:1) + e − U (cid:0) H αλ H βλ − η αβ H λµ H λµ (cid:1) (B.19) ∇ γ (cid:0) e V +6 U F γα (cid:1) = e U + V f ǫ αβγδ H βγδ + 3 e U + V H αβ ∇ β h + 6 ie V [ χ ∗ D α χ − χD α χ ∗ ] (B.20) ∇ γ ∇ γ U + 6 ∂ γ U ∂ γ U + ∂ γ U ∂ γ V + e − U ( D γ χ )( D γ χ ∗ ) − e − V H αβγ H αβγ − e − U + 2 e V − U + 8 e − U h + f + 4 e − U − V | χ | = 0 (B.21) ∇ γ ∇ γ V + 6 ∂ γ U ∂ γ V + ∂ γ V ∂ γ V + e − U − V ∇ λ h ∇ λ h − e − U ( D γ χ )( D γ χ ∗ ) − e V − U − e − U h + f + 16 e − U − V | χ | − e V F αβ F αβ + e − V H αβγ H αβγ − e − U H αβ H αβ = 0 (B.22)All of the dependence on the internal SE space has dropped out. In particularany solution to the D = 4 field equations (B.3)–(B.6), (B.9)–(B.13), (B.19)–(B.22)gives rise to an exact solution to the equations of motion of D = 11 supergravity.Thus the KK ansatz (B.1), (B.7) is consistent. C N = 2 supergravity The bosonic part of the general gauged N = 2 supergravity action coupled to vectorand hypermultiplets is given by [59, 16] S = Z R ∗ g i ¯ j Dt i ∧ ∗ D ¯ t j + h uv Dq u ∧ ∗ Dq v + Im N IJ F I ∧ ∗ F J + Re N IJ F I ∧ F J − V . (C.1)Here t i , i = 1 , . . . , n V are the complex scalar fields in the n V vector multiplets pa-rameterizing a special K¨ahler manifold with metric g i ¯ j , while q u , u = 1 , . . . , n H , arethe real scalar fields in the n H hypermultiplets parameterizing a quaternionic mani-fold with metric h uv . The two-forms F I = dA I with I = 0 , , . . . , n V are the gaugefield strengths for the vector multiplet and graviphoton potentials A I . In the gaugedtheory D µ t i = ∂ µ t i − k iI A Iµ , D µ q u = ∂ µ q u − k uI A Iµ , (C.2)26here k iI and k uI are Killing vectors on the special K¨ahler and quaternionic manifolds.For the theories appearing in this paper k iI = 0.The metric on the special K¨ahler manifold and the gauge kinetic terms can both bewritten in terms of a holomorphic prepotential F ( X ) where X I ( t ) are homogeneouscoordinates on the manifold and which is a homogeneous function of degree two.Explicitly the K¨ahler potential and N IJ matrix are given by K V = − log (cid:0) i ¯ X I F I − iX I ¯ F I (cid:1) , N IJ = ¯ F IJ + 2 i (Im F IK )(Im F JL ) X K X L (Im F AB ) X A X B , (C.3)with F I = ∂ I F and F IJ = ∂ I ∂ J F . Under symplectic transformations acting onthe gauge fields F I and the generalised duals G I = ∂ L /∂A I , where L is the scalarLagrangian for the supergravity action (C.1), one has F I G I ! ˜ F I ˜ G I ! = A BC D ! F I G I ! , (C.4)where A T D − C T B = 1, A T C = C T A and B T D = D T B . The ( X I , F I ) coordinatesand N IJ then transform as X I F I ! ˜ X I ˜ F I ! = A BC D ! X I F I ! , N 7→ ˜ N = ( C + D N )( A + B N ) − . (C.5)The quaternionic manifold has SU (2) × Sp (2 n H ) special holonomy, so, as in forexample [60], one can introduce vielbeins V Aα where A = 1 , α = 1 , . . . , n H suchthat h uv = V Aαu V Bβv ǫ AB C αβ where ǫ = − C αβ is the constant symplectic formfor Sp ( n H ). This defines SU (2) and Sp ( n V ) connections via dV Aa + ω AB ∧ V Ba +∆ ab ∧ V Ab = 0. The triplet of K¨ahler forms can then be written as K = K x (cid:0) − i σ x (cid:1) = − ( dω + ω ∧ ω ) , (C.6)where σ x with x = 1 , , J x ) uv = h uw ( K x ) wv then satisfy the quaternion algebra.Given the Killing vectors k uI one can then introduce triplets of Killing prepotentials P I = P xI (cid:0) − i σ x (cid:1) satisfying i k I K = dP + [ ω, P ] . (C.7)If the gauging is only in the hypermultiplet sector then the potential V in the ac-tion (C.1) is given by V = e K V X I ¯ X J (4 h uv k uI k vJ ) − (cid:0) (Im N ) − IJ + 4 e K V X I ¯ X J (cid:1) P xI P xJ . (C.8)27t is well-known that the universal hypermultiplet parameterizes a SU (2 , /U (2)coset. One can identify the particular quaternionic geometry as follows [61]. Themetric h uv can be written as h uv dq u dq v = 14 ρ dρ + 14 ρ (cid:2) dσ − i ( ξd ¯ ξ − ¯ ξdξ ) (cid:3) + 1 ρ dξd ¯ ξ , (C.9)which has Ricci tensor equal to minus six times the metric. Introducing the one-forms α = dξ √ ρ , β = 12 ρ (cid:0) dρ + idσ + ξd ¯ ξ − ¯ ξdξ (cid:1) (C.10)one can write V Aα = 1 √ α ¯ ββ − ¯ α ! Aα (C.11)and h uv = ǫ αβ C AB V Aαu V Bβv with the constant symplectic form C having components C = 1. We also find ω AB = ( β − ¯ β ) − α ¯ α − ( β − ¯ β ) ! AB , ∆ αβ = − ( β − ¯ β ) 00 ( β − ¯ β ) ! αβ . (C.12)and K AB = ( α ∧ ¯ α − β ∧ ¯ β ) α ∧ ¯ ββ ∧ ¯ α − ( α ∧ ¯ α − β ∧ ¯ β ) ! AB . (C.13) References [1] J. P. Gauntlett and O. Varela, “Consistent Kaluza–Klein Reductions forGeneral Supersymmetric AdS Solutions,” Phys. Rev. D (2007) 126007[arXiv:0707.2315 [hep-th]].[2] M. J. Duff and C. N. Pope, “Consistent Truncations In Kaluza–Klein Theories,”Nucl. Phys. B (1985) 355.[3] A. Buchel and J. T. Liu, “Gauged supergravity from type IIB string theory on Y ( p, q ) manifolds,” Nucl. Phys. B (2007) 93 [arXiv:hep-th/0608002].[4] J. P. Gauntlett, E. O Colgain and O. Varela, “Properties of some conformal fieldtheories with M-theory duals,” JHEP (2007) 049 [arXiv:hep-th/0611219].[5] J. P. Gauntlett and O. Varela, “ D = 5, SU (2) × U (1) Gauged Supergravity from D = 11 Supergravity,” JHEP (2008) 083 [arXiv:0712.3560 [hep-th]].286] T. T. Tsikas, “Consistent Truncations Of Chiral N = 2 D = 10 SupergravityOn The Round Five Sphere,” Class. Quant. Grav. (1986) 733.[7] M. J. Duff, B. E. W. Nilsson, C. N. Pope and N. P. Warner, “On The ConsistencyOf The Kaluza–Klein Ansatz,” Phys. Lett. B (1984) 90.[8] M. Cvetic et al. , “Embedding AdS black holes in ten and eleven dimensions,”Nucl. Phys. B (1999) 96 [arXiv:hep-th/9903214].[9] H. Lu, C. N. Pope and T. A. Tran, “Five-dimensional N = 4, SU (2) × U (1) gauged supergravity from type IIB,” Phys. Lett. B (2000) 261[arXiv:hep-th/9909203].[10] M. Cvetic, H. Lu, C. N. Pope, A. Sadrzadeh and T. A. Tran, “Consistent SO (6) reduction of type IIB supergravity on S ,” Nucl. Phys. B (2000)275 [arXiv:hep-th/0003103].[11] A. Khavaev, K. Pilch and N. P. Warner, “New vacua of gauged N = 8 super-gravity in five dimensions,” Phys. Lett. B (2000) 14 [arXiv:hep-th/9812035].[12] B. de Wit and H. Nicolai, “The Consistency of the S Truncation in D = 11Supergravity,” Nucl. Phys. B (1987) 211.[13] J. Maldacena, D. Martelli and Y. Tachikawa, “Comments on string theory back-grounds with non-relativistic conformal symmetry,” JHEP (2008) 072[arXiv:0807.1100 [hep-th]].[14] M. S. Bremer, M. J. Duff, H. Lu, C. N. Pope and K. S. Stelle, “Instantoncosmology and domain walls from M-theory and string theory,” Nucl. Phys. B (1999) 321 [arXiv:hep-th/9807051].[15] J. T. Liu and H. Sati, “Breathing mode compactifications and supersymmetryof the brane-world,” Nucl. Phys. B (2001) 116 [arXiv:hep-th/0009184].[16] J. Louis and A. Micu, “Type II theories compactified on Calabi-Yau three-folds in the presence of background fluxes,” Nucl. Phys. B , 395 (2002)[arXiv:hep-th/0202168].[17] U. Theis and S. Vandoren, “N = 2 supersymmetric scalar-tensor couplings,”JHEP , 042 (2003) [arXiv:hep-th/0303048].2918] G. Dall’Agata, R. D’Auria, L. Sommovigo and S. Vaula, “D = 4, N = 2 gaugedsupergravity in the presence of tensor multiplets,” Nucl. Phys. B , 243 (2004)[arXiv:hep-th/0312210],R. D’Auria, L. Sommovigo and S. Vaula, “N = 2 supergravity Lagrangian cou-pled to tensor multiplets with electric and magnetic fluxes,” JHEP , 028(2004) [arXiv:hep-th/0409097].[19] S. M. Kuzenko, “On massive tensor multiplets,” JHEP , 041 (2005)[arXiv:hep-th/0412190],S. M. Kuzenko, “On N = 2 supergravity and projective superspace: Dual for-mulations,” Nucl. Phys. B , 135 (2009) [arXiv:0807.3381 [hep-th]].[20] R. D’Auria and S. Ferrara, “Dyonic masses from conformal field strengths in D even dimensions,” Phys. Lett. B , 211 (2005) [arXiv:hep-th/0410051],J. Louis and W. Schulgin, “Massive tensor multiplets in N = 1 supersymmetry,”Fortsch. Phys. , 235 (2005) [arXiv:hep-th/0410149],U. Theis, “Masses and dualities in extended Freedman-Townsend models,” Phys.Lett. B , 402 (2005) [arXiv:hep-th/0412177].[21] For a review see M. Grana, “Flux compactifications in string theory: A compre-hensive review,” Phys. Rept. , 91 (2006) [arXiv:hep-th/0509003].[22] O. Aharony, M. Berkooz, J. Louis and A. Micu, “Non-Abelian structures incompactifications of M-theory on seven-manifolds with SU (3) structure,” JHEP , 108 (2008) [arXiv:0806.1051 [hep-th]].[23] T. House and A. Micu, “M-theory compactifications on manifolds with G struc-ture,” Class. Quant. Grav. (2005) 1709 [arXiv:hep-th/0412006].[24] J.P. Gauntlett, S. Kim, O. Varela and D. Waldram, to appear.[25] D. T. Son, “Toward an AdS/cold atoms correspondence: a geometric realizationof the Schroedinger symmetry,” Phys. Rev. D (2008) 046003 [arXiv:0804.3972[hep-th]].[26] K. Balasubramanian and J. McGreevy, “Gravity duals for non-relativisticCFTs,” Phys. Rev. Lett. (2008) 061601 [arXiv:0804.4053 [hep-th]].[27] C. P. Herzog, M. Rangamani and S. F. Ross, “Heating up Galilean holography,”arXiv:0807.1099 [hep-th]. 3028] A. Adams, K. Balasubramanian and J. McGreevy, “Hot Spacetimes for ColdAtoms,” arXiv:0807.1111 [hep-th].[29] M. J. Duff, B. E. W. Nilsson and C. N. Pope, “The Criterion For VacuumStability In Kaluza-Klein Supergravity,” Phys. Lett. B (1984) 154.[30] F. Denef and S. A. Hartnoll, “Landscape of superconducting membranes,”arXiv:0901.1160 [hep-th].[31] S. S. Gubser, “Breaking an Abelian gauge symmetry near a black hole horizon,”arXiv:0801.2977 [hep-th].[32] S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, “Building a Holographic Su-perconductor,” Phys. Rev. Lett. (2008) 031601 [arXiv:0803.3295 [hep-th]].[33] S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, “Holographic Superconduc-tors,” JHEP (2008) 015 [arXiv:0810.1563 [hep-th]].[34] J. P. Gauntlett and S. Pakis, “The geometry of D = 11 Killing spinors,” JHEP (2003) 039 [arXiv:hep-th/0212008].[35] J. A. Minahan and R. C. Warner, “Stuckelberg Revisited,” UFIFT PreprintHEP-89-15, 1989.[36] Y. Takahashi and R. Palmer, “Gauge-independent formulation of a massive fieldwith spin one,” Phys. Rev. D , 2974 (1970).[37] F. Quevedo and C. A. Trugenberger, “Phases of antisymmetric tensor field the-ories,” Nucl. Phys. B , 143 (1997) [arXiv:hep-th/9604196].[38] D. Fabbri, P. Fre, L. Gualtieri and P. Termonia, “M-theory on AdS × M : Thecomplete Osp (2 | × SU (3) × SU (2) spectrum from harmonic analysis,” Nucl.Phys. B (1999) 617 [arXiv:hep-th/9903036].[39] M. Gunaydin, S. McReynolds and M. Zagermann, “The R-map and the couplingof N = 2 tensor multiplets in 5 and 4 dimensions,” JHEP , 168 (2006)[arXiv:hep-th/0511025].[40] A. Donos and J. P. Gauntlett, “Supersymmetric solutions for non-relativisticholography,” arXiv:0901.0818 [hep-th].3141] P. Fre’, L. Gualtieri and P. Termonia, “The structure of N = 3 multiplets in AdS and the complete Osp (3 | × SU (3) spectrum of M-theory on AdS × N (0 , , , 27 (1999) [arXiv:hep-th/9909188].[42] A. Casher, F. Englert, H. Nicolai and M. Rooman, “The Mass Spectrum OfSupergravity On The Round Seven Sphere,” Nucl. Phys. B (1984) 173.[43] M. J. Duff, B. E. W. Nilsson and C. N. Pope, “Kaluza-Klein Supergravity,”Phys. Rept. (1986) 1.[44] M. J. Duff, C. N. Pope and K. S. Stelle, “Consistent interacting massive spin-2requires an infinity of states,” Phys. Lett. B (1989) 386.[45] H. J. Kim, L. J. Romans and P. van Nieuwenhuizen, “The Mass Spectrum OfChiral N = 2 D = 10 Supergravity On S ,” Phys. Rev. D (1985) 389.[46] S. S. Gubser, A. Hashimoto, I. R. Klebanov and M. Krasnitz, “Scalar absorptionand the breaking of the world volume conformal invariance,” Nucl. Phys. B (1998) 393 [arXiv:hep-th/9803023].[47] K. A. Intriligator, “Maximally supersymmetric RG flows and AdS duality,” Nucl.Phys. B (2000) 99 [arXiv:hep-th/9909082].[48] U. H. Danielsson, A. Guijosa, M. Kruczenski and B. Sundborg, “D3-brane holog-raphy,” JHEP (2000) 028 [arXiv:hep-th/0004187].[49] H. Nastase, D. Vaman and P. van Nieuwenhuizen, “Consistent nonlinear KKreduction of 11d supergravity on AdS × S and self-duality in odd dimensions,”Phys. Lett. B (1999) 96 [arXiv:hep-th/9905075].[50] H. Nastase, D. Vaman and P. van Nieuwenhuizen, “Consistency of the AdS × S reduction and the origin of self-duality in odd dimensions,” Nucl. Phys. B (2000) 179 [arXiv:hep-th/9911238].[51] R. G. Leigh and M. Rozali, “The large N limit of the (2,0) superconformal fieldtheory,” Phys. Lett. B (1998) 311 [arXiv:hep-th/9803068].[52] P. van Nieuwenhuizen, “The Complete Mass Spectrum Of D = 11 SupergravityCompactified On S And A General Mass Formula For Arbitrary Cosets M ,”Class. Quant. Grav. (1985) 1. 3253] M. Gunaydin, P. van Nieuwenhuizen and N. P. Warner, “General ConstructionOf The Unitary Representations Of Anti-De Sitter Superalgebras And The Spec-trum Of The S Compactification Of Eleven-Dimensional Supergravity,” Nucl.Phys. B (1985) 63.[54] H. Lin, O. Lunin and J. M. Maldacena, “Bubbling AdS space and 1/2 BPSgeometries,” JHEP (2004) 025 [arXiv:hep-th/0409174].[55] J. P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, “Supersym-metric
AdS solutions of M-theory,” Class. Quant. Grav. (2004) 4335[arXiv:hep-th/0402153].[56] J. P. Gauntlett, O. A. P. Mac Conamhna, T. Mateos and D. Wal-dram, “AdS spacetimes from wrapped M5 branes,” JHEP (2006) 053[arXiv:hep-th/0605146].[57] J. P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, “Supersymmetric AdS solutions of type IIB supergravity,” Class. Quant. Grav. (2006) 4693[arXiv:hep-th/0510125].[58] J. P. Gauntlett, D. Martelli and D. Waldram, “Superstrings with intrinsic tor-sion,” Phys. Rev. D (2004) 086002 [arXiv:hep-th/0302158].[59] For a complete review see L. Andrianopoli, M. Bertolini, A. Ceresole, R. D’Auria,S. Ferrara, P. Fre and T. Magri, “ N = 2 supergravity and N = 2 super Yang-Mills theory on general scalar manifolds: Symplectic covariance, gaugings andthe momentum map,” J. Geom. Phys. , 111 (1997) [arXiv:hep-th/9605032].[60] A. Lukas, B. A. Ovrut, K. S. Stelle and D. Waldram, “Heterotic M-theory infive dimensions,” Nucl. Phys. B (1999) 246 [arXiv:hep-th/9806051].[61] A. Strominger, “Loop corrections to the universal hypermultiplet,” Phys. Lett.B421