Consistent massive truncations of IIB supergravity on Sasaki-Einstein manifolds
aa r X i v : . [ h e p - t h ] A p r MCTP-10-14
Consistent massive truncations of IIB supergravity onSasaki-Einstein manifolds
James T. Liu, ∗ Phillip Szepietowski, † and Zhichen Zhao ‡ Michigan Center for Theoretical Physics, Randall Laboratory of Physics,The University of Michigan, Ann Arbor, MI 48109–1040, USA
Abstract
Recent work on holographic superconductivity and gravitational duals of systems with non-relativistic conformal symmetry have made use of consistent truncations of D = 10 and D = 11supergravity retaining some massive modes in the Kaluza-Klein tower. In this paper we focus onreductions of IIB supergravity to five dimensions on a Sasaki-Einstein manifold, and extend theseprevious truncations to encompass the entire bosonic sector of gauged D = 5, N = 2 supergravitycoupled to massive multiplets up to the second Kaluza-Klein level. We conjecture that a necessarycondition for the consistency of massive truncations is to only retain the lowest modes in themassive trajectories of the Kaluza-Klein mode decomposition of the original fields. This is anextension of the well-known result that consistent truncations may be obtained by restricting tothe singlet sector of the internal symmetry group. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] . INTRODUCTION Recent developments in AdS/CFT have expanded the scope of applications from therealm of strongly coupled relativistic gauge theories to various condensed matter systemswhose dynamics are expected to be described by a strongly coupled theory. These includesystems with behavior governed by a quantum critical point [1, 2], as well as cold atomsand similar systems exhibiting non-relativistic conformal symmetry [3, 4]. Much current at-tention is also directed towards holographic descriptions of superfluids and superconductors[5–8].The main feature used in the construction of a dual model of superconductivity is theexistence of a charged scalar field in the dual AdS background [6, 8]. Turning on tempera-ture and non-zero chemical potential corresponds to working with a charged black hole inAdS. Then, as the temperature is lowered, the charged scalar develops an instability andcondenses, so that the black hole develops scalar hair . This condensate breaks the U(1)symmetry, and is a sign of superconductivity (in the case where the U(1) is “weakly gauged”on the boundary).The basic model dual to a 2+1 dimensional superconductor is simply that of a chargedscalar coupled to a Maxwell field and gravity, and may be described by a Lagrangian of theform L = R + 6 L − F µν − | ∂ µ ψ − iqA µ ψ | − m | ψ | . (1)The properties of the system may then be studied for various values of mass m and charge q . While this is a perfectly acceptable framework, a more complete understanding demandsthat this somewhat phenomenological Lagrangian be embedded in a more complete theorysuch as string theory, or at least its supergravity limit. For AdS duals of 2+1 dimensionalsuperconductors, this was examined at the linearized level in [12], and embedded into D = 11supergravity at the full non-linear level in [13–15] for the case m L = − q = 2.Similarly, a IIB supergravity model for an AdS dual to 3+1 dimensional superconductorswas constructed in [16] with m L = − q = 2.The AdS model of [13–15] and the AdS model of [16] are based on Kaluza-Klein trun- Recent models have generalized this construction to encompass both p-wave [9, 10] and d-wave [11]condensates. q = 2 charged scalar arises from the massive level of the Kaluza-Klein truncation. Thisappears to go against the standard lore of consistent truncations, where it was thought thattruncations keeping only a finite number of massive modes would necessarily be inconsis-tent. A heuristic argument is that states in the Kaluza-Klein tower carry charges underthe internal symmetry, and hence would couple at the non-linear level to source higher andhigher states, all the way up the Kaluza-Klein tower. This hints that one way to obtain aconsistent truncation is simply to truncate to singlets of the internal symmetry group, andindeed such a construction is consistent. An example of this is a standard torus reduction,where only zero modes on the torus are kept. On the other hand, sphere reductions to max-imal gauged supergravities in D = 4, 5 and 7 do not follow this rule, as they are expectedto be consistent, even though some of the lower-dimensional fields (such as the non-abeliangraviphotons) are charged under the R -symmetry. In fact, the issue of Kaluza-Klein con-sistency is not yet fully resolved, and often must be treated on a case by base basis. Thishas led us to explore the squashed Sasaki-Einstein compactifications to see if additionalconsistent massive truncations may be found.In addition to embedding holographic models of superconductivity into string theory,several groups have demonstrated the embedding of dual non-relativistic CFT backgroundsinto string theory [17–19]. These geometries where originally constructed from a toy modelof a massive vector field coupled to gravity with a negative cosmological constant [3, 4] ofthe form (given here for a deformation of AdS ): L = R + 12 L − F µν − m A µ , (2)with mass related to the scaling exponent z according to m L = z ( z + 2). The z = 2 and z = 4 models ( m L = 8 and m L = 24, respectively) were subsequently realized withinIIB supergravity in terms of consistent truncations retaining a massive vector (along withpossibly other fields as well) [17–19]. These results have further opened up the possibilityof obtaining large classes of consistent truncations retaining massive modes of various spin.3 . Consistent massive truncations of IIB supergravity For the most part, the massive consistent truncations used in the study of AdS/condensedmatter systems have not been supersymmetric . Nevertheless this has motivated us to inves-tigate the possibility of obtaining new supersymmetric massive truncations of IIB supergrav-ity. In particular, we are mainly interested in reducing IIB supergravity on a Sasaki-Einsteinmanifold to obtain gauged supergravity in D = 5 coupled to possibly massive supermulti-plets.Following the construction of D = 11 supergravity [20] and the realization that it admitsan AdS × S vacuum solution [21], it was soon postulated that the Kaluza-Klein reductionon the sphere would give rise to gauged N = 8 supergravity at the “massless” Kaluza-Kleinlevel [22–24]. This notion was reinforced by a linearized Kaluza-Klein mode analysis demon-strating that the full spectrum of Kaluza-Klein excitations falls into supermultiplets of the D = 4, N = 8 superalgebra OSp(4 |
8) [25–27]. However, demonstrating full consistencyof the non-linear reduction to gauged N = 8 supergravity has remained elusive. Never-theless, all indications are that the reduction is consistent [28], and this has in fact beendemonstrated for the related case of reducing to D = 7 on S [29, 30].The story is similar for the case of IIB supergravity reduced on S . A linearized Kaluza-Klein mode analysis demonstrates that the spectrum of Kaluza-Klein excitations falls intocomplete supermultiplets of the D = 5, N = 8 superalgebra SU(2 , | N = 8 supergravity multiplet [31, 32]. In this case, onlypartial results are known about the full non-linear reduction to gauged supergravity, butthere is strong evidence for its consistency [33–36].More generally, it was conjectured in [37, 38] and [39], that, for any supergravity re-duction, it is always possible to consistently truncate to the supermultiplet containing themassless graviton. This is a non-trivial statement, as the truncation must satisfy rather re-strictive consistency conditions related to the gauge symmetries generated by the isometriesof the internal manifold [40, 41]. This conjecture has recently been shown to be true forSasaki-Einstein reductions of IIB supergravity on SE [42] and D = 11 supergravity on SE [39], yielding minimal D = 5, N = 2 and D = 4, N = 2 gauged supergravity, respectively The massive truncation given in [13] is supersymmetric, although the connection to a holographic super-conductor was done through the non-supersymmetric skew-whiffed case. × SU(2) vector multiplets that naturally arise in the compactification ofIIB supergravity on T , .For many of the above reasons, it has often been a challenge to explore consistent super-symmetric truncations, even at the massless Kaluza-Klein level. However, bosonic trunca-tions retaining massive breathing and squashing modes [45] have been known to be consistentfor some time. In this case, consistency is guaranteed by retaining only singlets under the in-ternal symmetry group SU(4) × U(1) for the squashed S or SU(3) × U(1) for the squashed S .The supersymmetry of background solutions involving the breathing and squashing modeswas explored in [46], where it was further conjectured that a supersymmetric consistenttruncation could be found that retains the full breathing/squashing supermultiplet.Although this massive consistent truncation conjecture was made for squashed spherecompactifications, it naturally generalizes to compactification on more general internalspaces, such as Sasaki-Einstein spaces. For D = 11 supergravity compactified on a squashed S , written as U(1) bundled over CP , truncation of the N = 8 Kaluza-Klein spectrum toSU(4) singlets under the decomposition SO(8) ⊃ SU(4) × U(1) yields the N = 2 supergravitymultiplet n = 0 : D (2 , = D (3 , + D ( , ) − + D ( , ) + D (2 , , (3)at the massless ( n = 0) Kaluza-Klein level. No SU(4) singlets survive at the first ( n =1) massive Kaluza-Klein level, and the breathing and squashing modes finally make theirappearance at the second ( n = 2) Kaluza-Klein level in a massive vector multiplet [46] n = 2 : D (4 , = D (5 , + D ( , ) − + D ( , ) + D ( , ) − + D ( , ) + D (4 , + D (5 , + D (5 , − + D (5 , + D (6 , . (4) The OSp(4 |
2) super-representations D ( E , s ) q and SO(2,3) representations D ( E , s ) q are labeled by energy E , spin s and U(1) charge q under OSp(4 | ⊃ SO(2 , × U(1) ⊃ SO(2) × SO(3) × U(1). S by SE amounts to replacing CP by an appropriate Kahler-Einstein base B .In this case, the internal isometry is generically reduced from SU(4) × U(1). Nevertheless,the notion of truncating to SU(4) singlets may simply be replaced by the prescription oftruncating to zero modes on the base B . This procedure was in fact done in [13], whichconstructed the non-linear Kaluza-Klein reduction for all the bosonic fields contained in theabove supermultiplets (3) and (4) and furthermore verified the N = 2 supersymmetry.For the case of IIB supergravity compactified on SE , it is straightforward to general-ize the squashed S conjecture of [46]. In this case, however, the Kaluza-Klein spectrum ismore involved, and is given in Table I. A curious feature shows up here in that an additionalLH+RH chiral matter multiplet shows up at the ‘massless’ Kaluza-Klein level. The E = 4scalar in this multiplet corresponds to the IIB axi-dilaton, while the additional E = 3charged scalar is precisely the charged scalar constructed in the holographic model of [16].At the higher Kaluza-Klein levels, the breathing and squashing mode scalars correspondto the E = 8 and E = 6 scalars in the massive vector multiplet. In addition, consistenttruncations involving the E = 5 ( m L = 8) doublet of vectors in the semi-long LH+RHmassive gravitino multiplet and the E = 7 ( m L = 24) vector in the massive vector mul-tiplet were constructed in [17–19] in the context of investigating non-relativistic conformalbackgrounds in string theory.What we have seen so far is that massive consistent truncations of IIB supergravity havebeen obtained keeping various subsets of the bosonic fields identified in Table I. The goal ofthis paper is to construct a complete non-linear Kaluza-Klein reduction of IIB supergravityon SE retaining all the bosonic fields in the multiplets up to the n = 2 level. This com-plements the massive Kaluza-Klein truncation of D = 11 supergravity [13], and providesanother example of a consistent truncation retaining the breathing mode supermultiplet.We proceed in Section II with the Sasaki-Einstein reduction of IIB supergravity. Then inSection III we connect the full non-linear reduction with the linearized Kaluza-Klein analysisof [31, 32] and show how the bosonic fields in Table I are related to the original IIB fields.In Section IV we relate the complete non-linear reduction to previous results by performingadditional truncations to a subset of active fields. Finally, we conclude in Section V withsome further speculation on massive consistent truncations of supergravity.While this work was being completed we became aware of [47–49] which independentlyworked out the massive consistent truncation of IIB supergravity on SE .6 Multiplet SU (2 , | SO (2 , × U (1)0 supergraviton D (3 , , ) D (4 , , + D (3 , , ) − + D (3 , , + D (3 , , ) D (3 , , D (3 , , + D (3 , , + D (4 , , D (3 , , − D (3 , , ) − + D (3 , , − + D (4 , , D (4 , , ) D (5 , , + D (5 , , ) + D (5 , , + D (6 , , + D (4 , , ) + D (5 , , ) − D (4 , , − D (5 , , ) − + D (5 , , ) + D (5 , , − + D (6 , , + D (4 , , − + D (5 , , D (6 , , D (7 , , ) + D (6 , , − + D (6 , , ) + D (7 , , ) − + D (7 , , + D (6 , , + D (7 , , − + D (7 , , + D (8 , , TABLE I: The truncated Kaluza-Klein spectrum of IIB supergravity on squashed S [46], orequivalently on SE . Here n denotes the Kaluza-Klein level. The consistent truncation is expectedto terminate at level n = 2 with the breathing mode supermultiplet. II. SASAKI-EINSTEIN REDUCTION OF IIB SUPERGRAVITY
The bosonic field content of IIB supergravity consists of the NSNS fields ( g MN , B MN , φ )and the RR potentials ( C , C , C ). Because of the self-dual field strength F +5 = dC , it isnot possible to write down a covariant action. However, we may take a bosonic Lagrangianof the form L IIB = R ∗ − τ dτ ∧ ∗ d ¯ τ − M ij F i ∧ ∗ F j − e F ∧ ∗ e F − ǫ ij C ∧ F i ∧ F j , (5)where self-duality e F = ∗ e F is to be imposed by hand after deriving the equations of motion.We have given the Lagrangian in an SL(2, R ) invariant form where τ = C + ie − φ , M = 1 τ | τ | − τ − τ , (6)and where F i = dB i , B i = B C , e F = dC + ǫ ij B i ∧ dB j . (7)7he equations of motion following from (5) and the self-duality of e F are d e F = ǫ ij F i ∧ F j , e F = ∗ e F ,d ( M ij ∗ F j ) = − ǫ ij e F ∧ F j ,d ∗ dττ + i dτ ∧ ∗ dττ = − i τ G ∧ ∗ G , (8)and the Einstein equation (in Ricci form) R MN = 12 τ ∂ ( M τ ∂ N ) ¯ τ + 14 M ij (cid:18) F iMP Q F j P QN − g MN F iP QR F j P QR (cid:19) + 14 · e F MP QRS e F N P QRS . (9)In the above we have introduced the complex three-form G = F − τ F . If desired, thisallows us to rewrite the three-form equation of motion as d ∗ G = − i dτ τ ∧ ∗ ( G + ¯ G ) + i e F ∧ G . (10) A. The reduction ansatz
Before writing out the reduction ansatz, we note a few key features of Sasaki-Einsteinmanifolds. A Sasaki-Einstein manifold has a preferred U(1) isometry related to the Reebvector. This allows us to write the metric as a U(1) fibration over a Kahler-Einstein base
Bds ( SE ) = ds ( B ) + ( dψ + A ) , (11)where d A = 2 J with J the Kahler form on B . Moreover, B admits an SU(2) structuredefined by the (1,1) and (2,0) forms J and Ω satisfying J ∧ Ω = 0 , Ω ∧ ¯Ω = 2 J ∧ J = 4 ∗ , ∗ J = J, ∗ Ω = Ω , (12)as well as dJ = 0 , d Ω = 3 i ( dψ + A ) ∧ Ω . (13)Note that we are taking the ‘unit radius’ Einstein condition R ij = 4 g ij on the Sasaki-Einsteinmanifold, which corresponds to R ab = 6 g ab on the Kahler-Einstein base.For the reduction, we write down the most general decomposition of the bosonic IIB fieldsconsistent with the isometries of B . For the metric, we take ds = e A ds + e B ds ( B ) + e C ( η + A ) , (14)8here η = dψ + A . Since A gauges the U(1) isometry, it will be related to the D = 5graviphoton. Note, however, that the graviphoton receives additional contributions fromthe five-form.The three-form and five-form field strengths can be expanded in a basis of invarianttensors on B . For the three-forms, we work with the potentials B i = b i + b i ∧ ( η + A ) + b i Ω + ¯ b i ¯Ω . (15)The scalars b i are complex, while the remaining fields are real. Note that we do not includea term of the form e b i J in the ansatz, as this field will act simply as a St¨ u ckelburg field inthe five-dimensional theory. In particular, it does not give rise to any new dynamics in theequations of motion as it can be repackaged as a total derivative plus terms which wouldsimply shift b i and b i ,2 e b i J = d ( e b i ∧ ( η + A )) − d e b i ∧ ( η + A ) − e b i F . (16)Taking F i = dB i gives F i = ( db i − b i ∧ F ) + db i ∧ ( η + A ) − b i ∧ J + Db i ∧ Ω + D ¯ b i ∧ ¯Ω+3 ib i Ω ∧ ( η + A ) − i ¯ b i ¯Ω ∧ ( η + A ) , (17)where D is the U(1) gauge covariant derivative Db i = db i − iA b i . (18)For convenience, we write this as F i = g i + g i ∧ ( η + A ) + g i ∧ J + f i ∧ Ω + ¯ f i ∧ ¯Ω + f i ∧ Ω ∧ ( η + A ) + ¯ f i ∧ ¯Ω ∧ ( η + A ) , (19)where our notation is such that the g i ’s are real and the f i ’s are complex.For the self-dual five-form, we take e F = (1+ ∗ )[(4+ φ ) ∗ ∧ ( η + A )+ A ∧∗ p ∧ J ∧ ( η + A )+ q ∧ Ω ∧ ( η + A )+ ¯ q ∧ ¯Ω ∧ ( η + A )] , (20)where ∗ B . Note that we have pulledout a constant background component e F = 4(1 + ∗ )vol( SE ) , (21)9hich sets up the Freund-Rubin compactification . The two-forms q are complex, while theother fields are real. For later convenience, we take the explicit 10-dimensional dual in themetric (14) to obtain e F = (4 + φ ) ∗ ∧ ( η + A ) + A ∧ ∗ p ∧ J ∧ ( η + A ) + q ∧ Ω ∧ ( η + A )+¯ q ∧ ¯Ω ∧ ( η + A ) + e A − B − C (4 + φ ) ∗ − e A − B + C ∗ A ∧ ( η + A )+ e A − C ∗ p ∧ J + e A − C ∗ q ∧ Ω + e A − C ∗ ¯ q ∧ ¯Ω , (22)where ∗ now denotes the Hodge dual in the D = 5 spacetime. B. Reduction of the equations of motion
In order to obtain the reduction, it is now simply a matter of inserting the above decom-positions into the IIB equations of motion. The e F equation yields d ( e A − C ∗ p ) = 2 e A − B + C ∗ A − p ∧ F + ǫ ij g i ∧ g j ,Dq = 3 ie A − C ∗ q + ǫ ij ( f i ∧ g j − f i g j ) , (23)along with the constraints φ = − i ǫ ij ( f i ¯ f j − ¯ f i f j ) ,p = ǫ ij g i ∧ g j − d [ A + A + i ǫ ij ( f i ¯ f j − ¯ f i f j )] . (24)The implication of this is that e F gives rise to two physical D = 5 fields, namely a massivevector A and a complex antisymmetric tensor q satisfying an odd-dimensional self-dualityequation and with m = 9. The mass of A is not directly apparent from (23) as it mixeswith A from the metric to yield the massless graviphoton as well as a m = 24 massivevector. For simplicity, we have assumed a unit radius ( L = 1) compactification. F i equation yields D ( e A + C M ij ∗ f j ) = − ie A − C M ij f j ∗ ǫ ij [(4 + φ ) e A − B − C f j ∗ − q ∧ g j + e A − C ∗ q ∧ g j + e A − B + C ∗ A ∧ f j ] ,d ( e A +4 B − C M ij ∗ g j ) = M ij [ e − A +4 B + C ∗ g j ∧ F + 4 e A + C ∗ g j ]+ ǫ ij [ − e A − C ∗ p ∧ g j − A ∧ g j − e A − C ( ∗ q ∧ ¯ f j + ∗ ¯ q ∧ f j )] ,d ( e − A +4 B + C M ij ∗ g j ) = ǫ ij [ − (4 + φ ) g j + A ∧ g j − p ∧ g j − q ∧ ¯ f j + ¯ q ∧ f j )+4 e A − C ( ¯ f j ∗ q + f j ∗ ¯ q )] . (25)These correspond to a pair of charged scalars f i , a pair of m = 8 massive vectors g i and apair of massive antisymmetric tensors b i .The ten-dimensional Einstein equation (9) reduces to a five-dimensional Einstein equa-tion, as well as the equations of motion for the breathing and squashing modes B and C and the graviphoton A . In particular, in the natural vielbein basis, the frame componentsof the ten-dimensional Ricci tensor corresponding to the reduction (14) are given by R αβ = e − A [ R αβ − ∇ α ∇ β (3 A + 4 B + C ) − η αβ ∂ γ A∂ γ (3 A + 4 B + C ) − η αβ (cid:3) A +3 ∂ α A∂ β A − ∂ α B∂ β B − ∂ α C∂ β C + 4( ∂ α A∂ β B + ∂ α B∂ β A )+( ∂ α A∂ β C + ∂ α C∂ β A )] − e C − A F αγ F βγ , R ab = δ ab [6 e − B − e C − B − e − A ( (cid:3) B + ∂ γ B∂ γ (3 A + 4 B + C ))] , R = 4 e C − B + e C − A F γδ F γδ − e − A ( (cid:3) C + ∂ γ C∂ γ (3 A + 4 B + C )) , R α = e C − A [ ∇ γ F αγ + F αγ ∂ γ ( A + 4 B + 3 C )] . (26)The α and β indices correspond to the D = 5 spacetime, while a and b correspond tothe Kahler-Einstein base B and 9 corresponds to the U(1) fiber direction. The covariantderivatives and frame indices on the right hand side of these quantities are with respect tothe D = 5 metric. In order to reduce to the D = 5 Einstein frame metric, we now choose3 A + 4 B + C = 0, or A = − B − C. (27)For convenience, we will retain A in the expressions below. However, it is not independent,and should always be thought of as a shorthand for (27).Equating the ten-dimensional Ricci tensor (26) to the stress tensor formed out of F i and11 F of (19) and (22), we obtain the D = 5 Einstein equation R αβ = η αβ ( − e A − B + 4 e A +3 C + e A (4 + φ ) ) + ∂ α B∂ β B + ∂ ( α B∂ β ) C + ∂ α C∂ β C + τ ∂ ( α τ ∂ β ) ¯ τ + e C − A ( F αγ F βγ − η αβ F γδ F γδ ) + e − B A α A β + e A − C [( p αγ p β γ − η αβ p γδ p γδ ) + 4( q ( αγ ¯ q β ) γ − η αβ q γδ ¯ q γδ )]+ M ij [ e A − C η αβ ( f i ¯ f j + ¯ f i f j ) + e − A − C ( g iαγ g j γβ − η αβ g iγδ g j γδ )+ e − A ( g iαγδ g j γδβ − η αβ g iγδǫ g j γδǫ ) + e − B ( g iα g jβ + 2( f iα ¯ f jβ + ¯ f iα f jβ ))] , (28)as well as the B , C and A equations of motion d ∗ dB = [6 e A − B − e A +3 C − e A (4 + φ ) ] ∗ − e − B A ∧ ∗ A + M ij [ e − A − C g i ∧ ∗ g j + e − A g i ∧ ∗ g j − e A − C ( f i ¯ f j + ¯ f i f j ) ∗ − e − B ( g i ∧ ∗ g j + 2( f i ∧ ∗ ¯ f j + ¯ f i ∧ ∗ f j ))] ,d ∗ dC = [4 e A +3 C − e A (4 + φ ) ] ∗ e C − A F ∧ ∗ F + e − B A ∧ ∗ A − e A − C ( p ∧ ∗ p + 4 q ∧ ∗ ¯ q ) + M ij [ − e − A − C g i ∧ ∗ g j + e − A g i ∧ ∗ g j − e A − C ( f i ¯ f j + ¯ f i f j ) ∗ e − B ( g i ∧ ∗ g j + 2( f i ∧ ∗ ¯ f j + ¯ f i ∧ ∗ f j ))] ,d ( e C − A ∗ F ) = (4 + φ ) e − B ∗ A − p ∧ p − q ∧ ¯ q + M ij [4 e − B ∗ ( f i ¯ f j + ¯ f i f j ) + e − A ∗ g i ∧ g j ] . (29)Note that, in order to obtain the D = 5 Einstein equation, we had to shift the reductionof R αβ an appropriate combination of R ab and R in order to remove the η αβ (cid:3) A component in the first line of (26).The IIB equations of motion thus reduce to (23), (25), (28) and (29) as well as the axi-dilaton equation, which we have not written down explicitly, but which will be shown to beconsistent below. C. The effective five-dimensional Lagrangian
We now wish to construct an effective D = 5 Lagrangian which reproduces the aboveequations of motion. This may be done by noting that the D = 5 Einstein equation (28)12rises naturally from a Lagrangian of the form L = R ∗ e A − B − e A +3 C − e A (4 + φ ) ) ∗ − dB ∧ ∗ dB − dB ∧ ∗ dC − dC ∧ ∗ dC − τ dτ ∧ ∗ d ¯ τ − e C − A F ∧ ∗ F − e − B A ∧ ∗ A − e A − C ( p ∧ ∗ p + 4 q ∧ ∗ ¯ q ) + M ij [ − e A − C ( f i ¯ f j + ¯ f i f j ) ∗ − e − A − C g i ∧ ∗ g j − e − A g i ∧ ∗ g j − e − B ( g i ∧ ∗ g j + 2( f i ∧ ∗ ¯ f j + ¯ f i ∧ ∗ f j ))]+ L CS . (30)We have included a Chern-Simons piece L CS which cannot be determined from the Einsteinequation.It is now possible to verify that (30) reproduces all the terms in the equations of motion(23), (25) and (29) involving the metric ( ie the Hodge *). The remaining terms may beobtained from the addition of the topological piece L CS = i ( q ∧ d ¯ q − ¯ q ∧ dq ) − A ∧ q ∧ ¯ q + 2 ǫ ij b i ∧ db j + i [(¯ q − i ǫ ij ¯ f i g j ) ∧ ǫ kl ( f k ∧ g l − f k g l ) − ( q + i ǫ ij f i g j ) ∧ ǫ kl ( ¯ f k ∧ g l − ¯ f k g l )] − A ∧ ( p − ǫ ij g i ∧ g j ) ∧ ( p − ǫ kl g k ∧ g l ) − A + i ǫ ij ( f i ¯ f j − ¯ f i f j )] ∧ ǫ kl ( g k ∧ g l − g k ∧ g l ∧ F ) . (31)Here we recall the definitions f i = 3 ib i , f i = Db i , g i = − b i , g i = db i , g i = db i − b i ∧ F , (32)implicit in (17) and (19). Furthermore, φ and p are given by (24). Note that, while A ismassive, and does not have a gauge invariance associated with it, it is natural to make theshift A → A ′ − i ǫ ij ( f i ¯ f j − ¯ f i f j ) , (33)so that p = ǫ ij g i ∧ g j − F − F ′ , (34)where F ′ = d A ′ .We now turn to the axi-dilaton equation obtained from (30). Since τ only shows up inthe kinetic term and in M ij , we see that the τ equation of motion obtained from the D = 5Lagrangian reproduces that obtained from the original IIB Lagrangian. This is because thequantity in the square brackets multiplying M ij in (30) is the straightforward reduction of − F i ∧ ∗ F j in the original IIB Lagrangian (5).13 II. MATCHING THE LINEARIZED KALUZA-KLEIN ANALYSIS
The complete D = 5 Lagrangian, as given by (30) and (31), is somewhat opaque. Thus inthis section, we demonstrate that it in fact contains the fields corresponding to the Kaluza-Klein mass spectrum noted in Table I. To do this, it is sufficient to look at the linearizedlevel. We first note that the effective D = 5 fields are the complex scalars ( τ, b i ), real scalars( B, C ), one-form potentials ( A , b i , A ), pair of real two-forms ( b i ), the complex two-form( q ), and of course the metric ( g µν ). The D = 5 equations of motion (23), (25) and (29)may be linearized on the matter fields to obtain the set d ∗ db i = (9 δ ij + 12 i N ij ) b j ∗ ,d ∗ db i = − ∗ b i ,d ∗ db i = − N ij db j ,dq = 3 i ∗ q ,d ∗ F = 4 ∗ A , d ∗ F + d ∗ F = − ∗ A ,d ∗ dB = 4(7 B + C ) ∗ , d ∗ dC = 16( B + C ) ∗ . (35)Here we have introduced N = M − ǫ = 1 τ − τ −| τ | τ , (36)with eigenvalues + i and − i , corresponding to eigenvectors (cid:16) τ (cid:17) T and (cid:16) τ (cid:17) T , respectively.The first equation in (35) then decomposes into a pair of equations for the complex scalars b m = − and b m =210 with masses m = − m = 21 according to b i = τ b m = − + τ b m =210 . (37)The second equation is that of an SL(2, R ) doublet of real vectors b i with mass m = 8.The third equation can be converted to an odd-dimensional self-duality equation [50] db i =4 N ij ∗ b j , for a doublet of antisymmetric tensors b i with mass m = 16. The fourth equationis already in odd-dimensional self-duality form, and shows that the complex antisymmetrictensor q has mass m = 9.The vector equations can be diagonalized d ∗ ( F + F ) = 0 , d ∗ F = − ∗ A , (38)14o identify the massless graviphoton A + A and the massive m = 24 vector A . Finallythe B and C equations may be diagonalized to identify the m = 32 breathing and m = 12squashing modes d ∗ dρ = 32 ρ ∗ , d ∗ dσ = 12 σ ∗ , (39)where B = ρ + σ, C = ρ − σ. (40)It is now possible to see how the above linearized modes are organized into N = 2supermultiplets. As shown in Table I, at the zeroth Kaluza-Klein level, we have the gravitonsupermultiplet D (3 , , ) = D (4 , , + D (3 , , ) − + D (3 , , + D (3 , , ) , (41)with bosonic fields being the graviton g µν and the massless graviphoton A + A . Still atthe zeroth level, there is also a LH+RH chiral multiplet D (3 , , = D (3 , , + D (3 , , + D (4 , , , D (3 , , − = D (3 , , ) − + D (3 , , − + D (4 , , . (42)The charged E = 3 scalar corresponds to the m = − b m = − , while the complex E = 4 scalar is the axi-dilaton τ .At the first Kaluza-Klein level, we have a semi-long LH+RH massive gravitino multiplet D (4 , , ) = D (5 , , + D (5 , , ) + D (5 , , + D (6 , , + D (4 , , ) + D (5 , , ) − , D (4 , , − = D (5 , , ) − + D (5 , , ) + D (5 , , − + D (6 , , + D (4 , , − + D (5 , , . (43)The bosonic field content is an SL(2, R ) doublet of m = 8 ( E = 5) vectors b i , a charged m = 9 ( E = 5) anti-symmetric tensor q , and a doublet of m = 16 ( E = 6) anti-symmetric tensors b i .At the second Kaluza-Klein level, we have a massive vector multiplet D (6 , , = D (7 , , ) + D (6 , , − + D (6 , , ) + D (7 , , ) − + D (7 , , + D (6 , , + D (7 , , − + D (7 , , + D (8 , , . (44)15 Multiplet State Field0 supergraviton D (4 , , g µν D (3 , , ) A + A D (3 , , ± b m = − D (4 , , + D (4 , , τ D (5 , , ) + D (5 , , ) b i D (5 , , + D (5 , , − q D (6 , , + D (6 , , b i D (7 , , ) A D (6 , , σD (7 , , ± b m =210 D (8 , , ρ TABLE II: Identification of the bosonic states in the Kaluza-Klein spectrum with the linearizedmodes in the reduction.
The massive E = 7 vector is the m = 24 mode A . The real E = 6 and E = 8 scalars arethe m = 12 squashing and m = 32 breathing modes, σ and ρ , respectively. The charged E = 7 scalar is b m =210 with m = 21. This identification of the linearized fields with theKaluza-Klein modes is shown in Table II.For the case of IIB supergravity on S , is interesting to note that these fields lie atthe lowest level of the massive trajectories in the Kaluza-Klein mode decomposition of the D = 10 fields [31, 32]. We note that the massive Kaluza-Klein tower is built out of scalar,vector and tensor harmonics on S , and the lowest harmonics generally have simple behavioron the internal sphere coordinates. For example, the lowest scalar harmonic is the constantmode on the sphere, while the lowest vector harmonics generate the Killing vectors on thesphere. It is presumably the simplicity of the lowest harmonics that allows the truncationto be consistent, even at the non-linear level.Although the harmonics on SE are more involved (see e.g. [51] for the case of T , ),it is clear that the decomposition (15) and (22) of the D = 10 fields in terms of invarianttensors on SE is equivalent to the truncation to the lowest harmonics on the sphere. Thisappears to be an essential feature guaranteeing the consistency of the massive truncation,16nd hence we do not expect to be able to keep any additional multiplets in the Kaluza-Kleintower beyond the n = 2 level. IV. FURTHER TRUNCATIONS
In order to make a connection with previous results on massive consistent truncationsof IIB supergravity, we note that the semi-long LH+RH massive gravitino multiplet at thefirst Kaluza-Klein level may be truncated out by setting b i = 0 , b i = 0 , q = 0 . (45)It is easy to see that this truncation is consistent, since the respective equations of motionfor q in (23) and g i and g i in (25) are trivially satisfied in this case. The resulting D = 5Lagrangian takes the form L = R ∗ e A − B − e A +3 C − e A (4 + φ ) ) ∗ − dB ∧ ∗ dB − dB ∧ ∗ dC − dC ∧ ∗ dC − τ dτ ∧ ∗ d ¯ τ − e C − A F ∧ ∗ F − e A − C ( F + F ′ ) ∧ ∗ ( F + F ′ ) − e − B [ A ′ − i ǫ ij ( f i ¯ f j − ¯ f i f j )] ∧ ∗ [ A ′ − i ǫ ij ( f i ¯ f j − ¯ f i f j )] − M ij [ e A − C ( f i ¯ f j + ¯ f i f j ) ∗ e − B ( f i ∧ ∗ ¯ f j + ¯ f i ∧ ∗ f j )] − A ∧ ( F + F ′ ) ∧ ( F + F ′ ) , (46)where f i = 3 ib i , f i = Db i , φ = − i ǫ ij ( f i ¯ f j − ¯ f i f j ) . (47)A further truncation to the massless N = 2 supergravity sector may be obtained bysetting b i = 0 , B = 0 , C = 0 , A = 0 , (48)along with taking a constant background for the axi-dilaton, τ = τ . This leaves only g µν and A , and yields the standard Lagrangian for the bosonic fields of minimal gauged supergravity L = R ∗ g ∗ − F ∧ ∗ F − √ A ∧ F ∧ F , (49)where we have rescaled the graviphoton, A → √ A , so that it has a canonical kinetic term,and where we have restored the dimensionful gauged supergravity coupling g .17 . Truncation to the zeroth Kaluza-Klein level Beyond the truncation to minimal supergravity discussed above, the first nontrivial trun-cation involves keeping only the lowest Kaluza-Klein level fields { τ, b m = − } dynamical. Inwhat follows we will denote b m = − simply as b so that ( b , b ) = ( b, τ b ). This truncation isnot as simple as setting all other fields to zero, as the equations of motion demand certainconstraints to be satisfied. For this case we start with the Lagrangian (46), obtained bysetting b i = b i = q = 0. We then impose the constraints b m =210 = 0 , e B = e − C = 1 − τ | b | , A = − iτ ( bD ¯ b − ¯ bDb ) + 4 | b | dτ . (50)These in turn imply that φ = − τ | b | , p = − dA . (51)To guarantee consistency, we have to check four constraints from the equations of motion(the B , C , f i , and combined Maxwell Equation). They are all verified to hold identically,and hence the truncation to the supergravity plus the LH+RH chiral multiplet is consistent.The Lagrangian is given by L = R ∗ (cid:2) − τ | b | ) e − B − e − B − e − B (4 + φ ) (cid:3) ∗ − dB ∧ ∗ dB − F ∧ ∗ F − e − B A ∧ ∗ A − e − B τ Db ∧ ∗ D ¯ b − ie − B (¯ bDb ∧ ∗ d ¯ τ − bD ¯ b ∧ ∗ dτ ) − τ (1 + 8 e − B τ | b | ) dτ ∧ ∗ d ¯ τ − A ∧ F ∧ F . (52)This expression can be simplified by defining λ ≡ τ | b | , giving L = R ∗ − λ )(1 − λ ) ∗ − dλ ∧ ∗ dλ − λ ) − (1 + λ ) dτ ∧ ∗ d ¯ τ − λ ) τ − F ∧ ∗ F − A ∧ ∗ A − λ ) − τ Db ∧ ∗ D ¯ b − λ − i − λ (¯ bDb ∧ ∗ d ¯ τ − bD ¯ b ∧ ∗ dτ ) − A ∧ F ∧ F . (53)If we further truncate the model by setting τ = ie − φ = ig − s , which is consistent withthe equation of motion for τ given in (8), this reproduces the model used in [16] to describea holographic superconductor using a m = − q = 2 charged scalar. If we denote b = √ g s f e iθ , the truncated Lagrangian reads L = R ∗ − F ∧ ∗ F − A ∧ F ∧ F +12 (1 − f )(1 − f ) ∗ − df ∧ ∗ df (1 − f ) − f ( dθ − A ) ∧ ∗ ( dθ − A )(1 − f ) . (54)A further redefinition f = tanh η then reproduces the Lagrangian given in [16].18 . Truncation to the second Kaluza-Klein level Starting with the Lagrangian (46) with b i = b i = q = 0, it is possible to retain the b m =210 scalar by setting b m = − = 0. In this case, we first let b = ¯ τ b and define b = √ g s he iξ ,so that ( h, ξ ) describe the m = 21 scalar. Again, the scalar equations of motion lead toconstraints, and in particular the first equation in (25) yields the equation of motion for h and ξ as well as d ( e A + C ∗ dτ ) + ie A + C τ dτ ∧ ∗ dτ = 0 . (55)This is simply the τ equation of motion without any sources, and the simplest thing todo is to set τ to be constant, τ = ie − φ = ig − s . The remaining field content is then { g µν , A , ρ, σ, b m =210 , A } , corresponding to the supergravity multiplet coupled to the massivevector multiplet. It is now straightforward to complete the truncation, and the Lagrangianis given by L = R ∗ (cid:0) e − ρ − σ − e − ρ − σ − e − ρ (1 + 6 h ) (cid:1) ∗ − dρ ∧ ∗ dρ − dσ ∧ ∗ dσ − e ρ − σ F ∧ ∗ F − e − ρ +2 σ ( F + F ′ ) ∧ ∗ ( F + F ′ ) − e − ρ − σ ( A ′ + 8 h Γ) ∧ ∗ ( A ′ + 8 h Γ) − A ∧ ( F + F ′ ) ∧ ( F + F ′ ) − (cid:0) e − ρ − σ dh ∧ ∗ dh + e − ρ − σ h Γ ∧ ∗ Γ + e − ρ +2 σ h ∗ (cid:1) , (56)where we have defined Γ = dξ − A .We can further truncate this by removing the m = 21 scalar ( i.e. by setting h = ξ = 0),giving the Lagrangian L = R ∗ e − ρ − σ − e − ρ − σ − e − ρ ) ∗ − dρ ∧ ∗ dρ − dσ ∧ ∗ dσ − e ρ − σ F ∧ ∗ F − e − ρ +2 σ ( F + F ′ ) ∧ ∗ ( F + F ′ ) − e − ρ − σ A ′ ∧ ∗ A ′ − A ∧ ( F + F ′ ) ∧ ( F + F ′ ) , (57)which corresponds to the m = 24 massive vector field truncation of [18]. C. Non-supersymmetric truncations
All the truncations we have listed so far have field content which fills the bosonic sector ofAdS supermultiplets and so are presumably supersymmetric truncations. It is also useful to19onsider truncations which contain dynamical fields belonging to different supermultiplets,without keeping the entire multiplet. In this sense these truncations are not supersymmet-ric, although they are perfectly consistent truncations and solutions of the ten-dimensionalequations of motion. For these truncations, we start with the complete Lagrangian given in(30) and (31).
1. Massive vector field
The first non-supersymmetric truncation we will discuss involves keeping the m = 8vector field, b i , and has already been noted in [18]. The field content in this truncationconsists of one component of b i (denoted b ), τ , ρ , σ and g µν . Note that the graviphoton isturned off here so that even at the lowest level this cannot be supersymmetric. Furthermore,by keeping only one component of b i , the τ equation of motion demands that we must set τ = 0. With this field content, the D = 10 constraints (24) are trivially satisfied with φ = 0 and p = 0, and the Lagrangian (30) becomes [18] L = R ∗ e − ρ − σ − e − ρ − σ − e − ρ ) ∗ − dρ ∧ ∗ dρ − dσ ∧ ∗ dσ − τ dτ ∧ ∗ dτ − τ e ρ +4 σ db ∧ ∗ db − τ e − ρ − σ b ∧ ∗ b . (58)
2. Complex massive anti-symmetric tensor
We can also truncate to theories containing the m = 9 complex anti-symmetric tensorfield q . The field content here is given by, q , A , B , C , τ , g µν and A . The D = 10constraints become φ = 0 and p = − dA − d A . All the other equations of motion areeither satisfied by setting the rest of the fields to zero or can be derived from the Lagrangian L = R ∗ e − ρ − σ − e − ρ − σ − e − ρ ) ∗ − dρ ∧ ∗ dρ − dσ ∧ ∗ dσ − e ρ − σ F ∧ ∗ F − e − ρ +2 σ ( p ∧ ∗ p + 4 q ∧ ∗ ¯ q ) − τ dτ ∧ ∗ d ¯ τ − e − ρ − σ A ∧ ∗ A + i ( q ∧ d ¯ q − ¯ q ∧ dq ) − A ∧ p ∧ p − A ∧ q ∧ ¯ q . (59)Note that it is consistent to further truncate to a constant axi-dilaton τ = τ .20 . Real massive anti-symmetric tensor Along similar lines to the case above for a massive vector field, we can set A = 0 andmake a truncation including the m = 16 real anti-symmetric tensor doublet b i by keepingonly the graviton coupled to b i , τ , ρ and σ. Again, the equations of motion for the otherfields are trivially satisfied and the constraints are also trivial φ = 0 and p = 0. Thisleaves the Lagrangian L = R ∗ e − ρ − σ − e − ρ − σ − e − ρ ) ∗ − dρ ∧ ∗ dρ − dσ ∧ ∗ dσ − τ dτ ∧ ∗ d ¯ τ − e ρ M ij db i ∧ ∗ db j + 2 ǫ ij b i ∧ ∗ db j . (60)As in the previous truncation it is consistent to further truncate to τ = τ . V. DISCUSSION
In the above, we have examined massive reductions of 10-dimensional IIB supergravityon Sasaki-Einstein manifolds. By utilizing the structure of SE , we have given a generaldecomposition of the IIB fields based on the invariant tensors associated with the internalmanifold. The field content obtained in five-dimensions completes the bosonic sector ofvarious AdS supermultiplets, and in particular they fill out the lowest Kaluza-Klein towerup to the breathing mode supermultiplet. This proves, at least at the level of the bosonicfields, the conjecture raised in [46] that a consistent massive truncation may be obtainedby truncating to the singlet sector on the Kahler-Einstein base B (which is CP for thesquashed S ) and further restricting to the level of the breathing mode multiplet and below.As suggested at the end of Section III, it is this truncation to constant modes on the base B that ensures the consistency of the reduction. In a sense, this is a generalization of theold consistency criterion of restricting to singlets of the internal isometry group, except thathere restricting to singlets of an appropriate subgroup turned out to be sufficient. For thisreason, we believe it is not that the breathing mode is special in itself which allows for aconsistent truncation retaining its supermultiplet, but rather that in the examples given hereand in [13], the breathing mode superpartners so happen to be the lowest harmonics in theirrespective Kaluza-Klein towers. It is an unusual feature of Kaluza-Klein compactifications oncurved internal spaces that states originating from different levels of the harmonic expansion21ay combine into a single supermultiplet. Thus, while the breathing mode is always thelowest state in its tower (being a constant mode on the internal space), its superpartners maycarry excitations on the internal space. This does not occur for the N = 2 compactificationof IIB supergravity on SE (nor does it for D = 11 supergravity on SE ). However, inextended theories, such as IIB supergravity on the round S , the superpartners will involvenon-trivial harmonics. In particular, the N = 8 superpartners to the breathing mode includea massive spin-2 excitation of the graviton involving the second harmonic (d-waves) on thesphere. Thus we believe it to be unlikely that an N = 8 massive truncation with thebreathing mode multiplet will be consistent.Consistent truncations of the type discussed here have recently been of particular interestin the growing literature on AdS/CFT applications to condensed matter systems. Untilrecently a strictly phenomenological approach has been taken in this area. In these systemsthe inclusion of a scalar condensate is required in the gravity theory to source an operatorwhose expectation value acts as an order parameter describing superconductor/superfluidphase transitions in the strongly coupled system. In the phenomenological approach, theorigin of this scalar and its properties have not been of immediate interest; rather the generalbehavior was determined and many interesting similarities to real condensed matter systemshave been noted. However, this approach lacks strong theoretical control in that systems aredescribed by a set of free parameters which can be tuned to provide the property of interest.Recently there has been some work to embed these models in UV complete theories, wherethe parameters are no longer free but are determined by the underlying features of thetheory, such as an origin in string theory. The discussion here has put these reductions intoa more general framework and gives further examples of UV complete systems whose dualsmay have useful applications in the AdS/CMT correspondence.Given that the fields in these truncations fall into specific supermultiplets it is an obviousand relevant question to discuss their fermionic partners. This would involve reducing thesupersymmetry variations and fermion equations in ten-dimensions down to five-dimensionsand determining the complete supersymmetric action of these truncations. This is also ofinterest in terms of AdS/CMT where there has been much interest in describing fermionbehavior in condensed matter systems such as the Fermi-liquid theory using the holographiccorrespondence. In particular, the full supersymmetric action could give us examples ofspecific interactions studied in these systems coupling scalar condensates to the fermionic22xcitations [52–54]. We leave the study of the fermionic modes and connections to condensedmatter systems to future work. Acknowledgments
We wish to thank I. Bah, M. Duff, J. Gauntlett and D. Vaman for stimulating discussions.This work was supported in part by the US Department of Energy under grant DE-FG02-95ER40899. [1] C. P. Herzog, P. Kovtun, S. Sachdev and D. T. Son,
Quantum critical transport, duality, andM-theory , Phys. Rev. D , 085020 (2007) [arXiv:hep-th/0701036].[2] S. A. Hartnoll, P. K. Kovtun, M. Muller and S. Sachdev, Theory of the Nernst effect nearquantum phase transitions in condensed matter, and in dyonic black holes , Phys. Rev. B ,144502 (2007) [arXiv:0706.3215 [cond-mat.str-el]].[3] D. T. Son, Toward an AdS/cold atoms correspondence: a geometric realization of theSchroedinger symmetry , Phys. Rev. D , 046003 (2008) [arXiv:0804.3972 [hep-th]].[4] K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs , Phys. Rev.Lett. , 061601 (2008) [arXiv:0804.4053 [hep-th]].[5] S. S. Gubser,
Breaking an Abelian gauge symmetry near a black hole horizon , Phys. Rev. D , 065034 (2008) [arXiv:0801.2977 [hep-th]].[6] S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, Building a Holographic Superconductor ,Phys. Rev. Lett. , 031601 (2008) [arXiv:0803.3295 [hep-th]].[7] C. P. Herzog, P. K. Kovtun and D. T. Son,
Holographic model of superfluidity , Phys. Rev. D , 066002 (2009) [arXiv:0809.4870 [hep-th]].[8] S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, Holographic Superconductors , JHEP ,015 (2008) [arXiv:0810.1563 [hep-th]].[9] S. S. Gubser and S. S. Pufu,
The gravity dual of a p-wave superconductor , JHEP , 033(2008) [arXiv:0805.2960 [hep-th]].[10] M. M. Roberts and S. A. Hartnoll,
Pseudogap and time reversal breaking in a holographicsuperconductor , JHEP , 035 (2008) [arXiv:0805.3898 [hep-th]].
11] J. W. Chen, Y. J. Kao, D. Maity, W. Y. Wen and C. P. Yeh,
Towards A Holographic Modelof D-Wave Superconductors , arXiv:1003.2991 [hep-th].[12] F. Denef and S. A. Hartnoll,
Landscape of superconducting membranes , Phys. Rev. D ,126008 (2009) [arXiv:0901.1160 [hep-th]].[13] J. P. Gauntlett, S. Kim, O. Varela and D. Waldram, Consistent supersymmetric Kaluza–Kleintruncations with massive modes , JHEP , 102 (2009) [arXiv:0901.0676 [hep-th]].[14] J. P. Gauntlett, J. Sonner and T. Wiseman,
Holographic superconductivity in M-Theory ,Phys. Rev. Lett. , 151601 (2009) [arXiv:0907.3796 [hep-th]].[15] J. P. Gauntlett, J. Sonner and T. Wiseman,
Quantum Criticality and Holographic Supercon-ductors in M-theory , JHEP , 060 (2010) [arXiv:0912.0512 [hep-th]].[16] S. S. Gubser, C. P. Herzog, S. S. Pufu and T. Tesileanu,
Superconductors from Superstrings ,Phys. Rev. Lett. , 141601 (2009) [arXiv:0907.3510 [hep-th]].[17] C. P. Herzog, M. Rangamani and S. F. Ross,
Heating up Galilean holography , JHEP ,080 (2008) [arXiv:0807.1099 [hep-th]].[18] J. Maldacena, D. Martelli and Y. Tachikawa,
Comments on string theory backgrounds withnon-relativistic conformal symmetry , JHEP , 072 (2008) [arXiv:0807.1100 [hep-th]].[19] A. Adams, K. Balasubramanian and J. McGreevy,
Hot Spacetimes for Cold Atoms , JHEP , 059 (2008) [arXiv:0807.1111 [hep-th]].[20] E. Cremmer, B. Julia and J. Scherk,
Supergravity theory in 11 dimensions , Phys. Lett. B ,409 (1978).[21] P. G. O. Freund and M. A. Rubin, Dynamics Of Dimensional Reduction , Phys. Lett. B ,233 (1980).[22] M. J. Duff and D. J. Toms, Kaluza-Klein Kounterterms , in J. Ellis and S. Ferrara eds.,
Unification of the fundamental particle interactions, II (Plenum, New York, 1983).[23] M. J. Duff,
Ultraviolet Divergences In Extended Supergravity , in S. Ferrara and J.G. Taylor,eds.,
Supergravity ’81: Proceedings of the 1st School on Supergravity held on 22 April–6 May1981 at the International Centre for Theoretical Physics, Trieste, Italy (Cambridge UniversityPress, Cambridge, 1982).[24] M. J. Duff and C. N. Pope,
Kaluza-Klein Supergravity And The Seven Sphere , in S. Ferrara,J.G. Taylor and P. van Nieuwenhuizen, eds.,
Supersymmetry and supergravity ’82: proceedingsof the Trieste September 1982 school (World Scientific, Singapore, 1983)
25] B. Biran, A. Casher, F. Englert, M. Rooman and P. Spindel,
The Fluctuating Seven SphereIn Eleven-Dimensional Supergravity , Phys. Lett. B , 179 (1984).[26] E. Sezgin,
The Spectrum Of The Eleven-Dimensional Supergravity Compactified On TheRound Seven Sphere , Phys. Lett. B , 57 (1984).[27] A. Casher, F. Englert, H. Nicolai and M. Rooman,
The Mass Spectrum Of Supergravity OnThe Round Seven Sphere , Nucl. Phys. B , 173 (1984).[28] B. de Wit and H. Nicolai,
The Consistency of the S Truncation in D = 11 Supergravity ,Nucl. Phys. B , 211 (1987).[29] H. Nastase, D. Vaman and P. van Nieuwenhuizen,
Consistent nonlinear K K reduction of 11dsupergravity on AdS × S and self-duality in odd dimensions , Phys. Lett. B , 96 (1999)[arXiv:hep-th/9905075].[30] H. Nastase, D. Vaman and P. van Nieuwenhuizen, Consistency of the AdS × S reduc-tion and the origin of self-duality in odd dimensions , Nucl. Phys. B , 179 (2000)[arXiv:hep-th/9911238].[31] M. Gunaydin and N. Marcus, The spectrum of the S compactification of the chiral N = 2 , D = 10 supergravity and the unitary supermultiplets of U(2,2/4) , Class. Quant. Grav. , L11(1985).[32] H. J. Kim, L. J. Romans and P. van Nieuwenhuizen, Mass spectrum of chiral ten-dimensional N = 2 supergravity on S , Phys. Rev. D , 389 (1985).[33] A. Khavaev, K. Pilch and N. P. Warner, New vacua of gauged N = 8 supergravity in fivedimensions , Phys. Lett. B , 14 (2000) [arXiv:hep-th/9812035].[34] M. Cvetic et al. , Embedding AdS black holes in ten and eleven dimensions , Nucl. Phys. B , 96 (1999) [arXiv:hep-th/9903214].[35] H. Lu, C. N. Pope and T. A. Tran,
Five-dimensional N = 4 , SU (2) × U (1) gauged supergravityfrom type IIB , Phys. Lett. B , 261 (2000) [arXiv:hep-th/9909203].[36] M. Cvetic, H. Lu, C. N. Pope, A. Sadrzadeh and T. A. Tran, Consistent SO(6) reduction oftype IIB supergravity on S , Nucl. Phys. B , 275 (2000) [arXiv:hep-th/0003103].[37] C. N. Pope, Consistency Of Truncations In Kaluza-Klein , in T. Goldman and M.M. Nieto,eds.,
Proceedings of the Santa Fe meeting: First Annual Meeting (new series) of the Divisionof Particles and Fields of the American Physics Society , (World Scientific, Philadelphia, 1985).[38] M. J. Duff and C. N. Pope,
Consistent Truncations In Kaluza-Klein Theories , Nucl. Phys. B (1985) 355.[39] J. P. Gauntlett and O. Varela, Consistent Kaluza-Klein Reductions for General Supersym-metric AdS Solutions , Phys. Rev. D , 126007 (2007) [arXiv:0707.2315 [hep-th]].[40] M. J. Duff, B. E. W. Nilsson, C. N. Pope and N. P. Warner, On The Consistency Of TheKaluza-Klein Ansatz , Phys. Lett. B , 90 (1984).[41] P. Hoxha, R. R. Martinez-Acosta and C. N. Pope,
Kaluza-Klein consistency, Killing vectors,and Kaehler spaces , Class. Quant. Grav. , 4207 (2000) [arXiv:hep-th/0005172].[42] A. Buchel and J. T. Liu, Gauged supergravity from type IIB string theory on Y p,q manifolds ,Nucl. Phys. B , 93 (2007) [arXiv:hep-th/0608002].[43] J. P. Gauntlett, E. O Colgain and O. Varela, Properties of some conformal field theories withM-theory duals , JHEP , 049 (2007) [arXiv:hep-th/0611219].[44] J. P. Gauntlett and O. Varela, D = 5 SU (2) × U (1) Gauged Supergravity from D = 11 Supergravity , JHEP , 083 (2008) [arXiv:0712.3560 [hep-th]].[45] M. S. Bremer, M. J. Duff, H. Lu, C. N. Pope and K. S. Stelle,
Instanton cosmology and domainwalls from M-theory and string theory , Nucl. Phys. B , 321 (1999) [arXiv:hep-th/9807051].[46] J. T. Liu and H. Sati,
Breathing mode compactifications and supersymmetry of the brane-world , Nucl. Phys. B , 116 (2001) [arXiv:hep-th/0009184].[47] D. Cassani, G. Dall’Agata and A. F. Faedo,
Type IIB supergravity on squashed Sasaki-Einsteinmanifolds , arXiv:1003.4283 [hep-th].[48] J. P. Gauntlett and O. Varela,
Universal Kaluza-Klein reductions of type IIB to N = 4 supergravity in five dimensions , arXiv:1003.5642 [hep-th].[49] K. Skenderis, M. Taylor and D. Tsimpis, A consistent truncation of IIB supergravity onmanifolds admitting a Sasaki-Einstein structure , arXiv:1003.5657 [hep-th].[50] P. K. Townsend, K. Pilch and P. van Nieuwenhuizen,
Selfduality In Odd Dimensions , Phys.Lett. , 38 (1984) [Addendum-ibid. , 443 (1984)].[51] A. Ceresole, G. Dall’Agata, R. D’Auria and S. Ferrara,
Spectrum of type IIB supergrav-ity on AdS × T : Predictions on N = 1 SCFT’s , Phys. Rev. D , 066001 (2000)[arXiv:hep-th/9905226].[52] J. W. Chen, Y. J. Kao and W. Y. Wen, Peak-Dip-Hump from Holographic Superconductivity ,arXiv:0911.2821 [hep-th].[53] T. Faulkner, G. T. Horowitz, J. McGreevy, M. M. Roberts and D. Vegh,
Photoemission experiments’ on holographic superconductors , arXiv:0911.3402 [hep-th].[54] S. S. Gubser, F. D. Rocha and P. Talavera, Normalizable fermion modes in a holographicsuperconductor , arXiv:0911.3632 [hep-th]., arXiv:0911.3632 [hep-th].