Supersymmetric solutions for non-relativistic holography
aa r X i v : . [ h e p - t h ] J a n Imperial/TP/2009/JG/02DESY 09-006
Supersymmetric solutions fornon-relativistic holography
Aristomenis Donos and Jerome P. Gauntlett DESY Theory Group, DESY HamburgNotkestrasse 85, D 22603 Hamburg, Germany Theoretical Physics Group, Blackett Laboratory,Imperial College, London SW7 2AZ, U.K. The Institute for Mathematical Sciences,Imperial College, London SW7 2PE, U.K.
Abstract
We construct families of supersymmetric solutions of type IIB and D = 11supergravity that are invariant under the non-relativistic conformal alge-bra for various values of dynamical exponent z ≥ z ≥
3, respectively.The solutions are based on five- and seven-dimensional Sasaki-Einsteinmanifolds and generalise the known solutions with dynamical exponent z = 4 for the type IIB case and z = 3 for the D = 11 case, respectively. Introduction
There has recently been much interest in finding holographic realisations of systemsinvariant under the non-relativistic conformal algebra starting with the work [1],[2] and discussed further in related work [3]-[32]. Such systems are invariant underGalilean transformations, generated by time and spatial translations, spatial rota-tions, Galilean boosts and a mass operator, which is a central element of the algebra,combined with scale transformations. If x + is the time coordinate, and x denotes d spatial coordinates, the scaling symmetry acts as x → µ x , x + → µ z x + , (1.1)where z is called the dynamical exponent. When z = 2 this non-relativistic conformalsymmetry can be enlarged to an invariance under the Schr¨odinger algebra whichincludes an additional special conformal generator.The solutions found in [1], [2] with d = 2 and z = 2 were subsequently em-bedded into type IIB string theory in [8],[9],[10] and were based on an arbitraryfive-dimensional Sasaki-Einstein manifold, SE . The work of [9] also constructedtype IIB solutions with d = 2 and z = 4 and again these were constructed using anarbitrary SE . It was also shown in [9] that the solutions with z = 2 and z = 4can be obtained from a five dimensional theory with a massive vector field after aKaluza-Klein reduction on the SE space [9]. This procedure was generalised tosolutions of D = 11 supergravity in [31]: using a similar KK reduction on an arbi-trary seven-dimensional Sasaki-Einstein space, SE , solutions with non relativisticconformal symmetry with d = 1 and z = 3 were found.The type IIB solution of [8],[9],[10] with z = 2 do not preserve any supersymmetry[9]. One aim of this note is to show that, by contrast, the type IIB solutions of [9]with z = 4 and the D = 11 solutions of [31] with z = 3 are both supersymmetricand generically preserve two supersymmetries. A second aim is to generalise bothof these supersymmetric solutions to different values of z . We will construct newsupersymmetric solutions using eigenmodes of the Laplacian acting on one-forms onthe SE or SE space. If the eiegenvalue is µ then we obtain type IIB solutions with z = 1 + √ µ and D = 11 solutions with z = 1 + √ µ . This gives rise to typeIIB solutions with z ≥ D = 11 solutions with z ≥
3, respectively. For the caseof S we get solutions with z = 4 , , . . . while for the case of S we get solutions with z = 3 , , , . . . and both of these preserve 8 supersymmetries.Our constructions have some similarities with the construction of type IIB solu-tions in [24] that were based on eigenmodes of the Laplacian acting on scalar functions1n the SE space. Our IIB solutions preserve the same supersymmetry and we showhow our solutions can be superposed with those of [24] while maintaining a scalingsymmetry. An analogous superposition is possible for the D = 11 solutions, whichwe shall also describe. The ansatz for the type IIB solutions we shall consider is given by ds = dr r + r (cid:2) dx + dx − + dx + dx (cid:3) + ds ( SE ) + 2 r Cdx + F = 4 r dx + ∧ dx − ∧ dr ∧ dx ∧ dx + 4 V ol ( SE ) − dx + ∧ (cid:2) ∗ CY dC + d ( r C ) ∧ dx ∧ dx (cid:3) (2.1)where SE is an arbitrary five-dimensional Sasaki-Einstein space and the metric ds ( SE ) is normalised so that the Ricci tensor is equal to four times the metric(i.e. the same normalisation as that of a unit radius five-sphere). Recall that themetric cone over the SE , ds ( CY ) = dr + r ds ( SE ) , (2.2)is Calabi-Yau. The K¨ahler form on the CY is denoted ω ij and the complex structureis defined by J ij = ω ik g kj , where g ij is the Calabi-Yau cone metric. We will definethe one-form η , which is dual to the Reeb vector on SE by η i = − J ij ( d log r ) j . (2.3)The one-form C is a one-form on the CY cone. When C = 0 we have the standard AdS × SE solution of type IIB which, in general, preserves eight supersymmetries(four Poincar´e and four superconformal), corresponding to an N = 1 SCFT in d = 4.More generally, we can deform this solution by choosing C = 0 provided that dC isco-closed on CY : d ∗ CY dC = 0 . (2.4)With this condition, F is closed and in fact it is also sufficient for the type IIBEinstein equations to be satisfied. As we will show these solutions preserve one While this is standard in the physics literature, often in the maths literature J ij = − ω ik g kj . x − → x − − Λ , C → C + d Λ (2.5)for some function Λ on the CY cone. Thus, if dC = 0, we can remove C , at leastlocally, by such a transformation.We will look for solutions where the one-form C has weight λ under the actionof r∂ r . Then it is straightforward to check, following [1] and [2] that our solution isinvariant under non-relativistic conformal transformations with two spatial dimen-sions x , x and dynamical exponent z = 2 + λ . For example the scaling symmetryis acting as in (1.1) combined with r → µ − r , x − → µ − z x − . Following the analysisof closed and co-closed two forms on cones (such as dC ) in appendix A of [33] weconsider solutions constructed from a co-closed one-form β on the SE space that isan eigenmode of the Laplacian ∆ SE = ( d † d + dd † ) SE : C = r λ β, ∆ SE β = µβ, d † β = 0 . (2.6)It is straightforward to check that dC is co-closed providing that µ = λ ( λ + 2). Forour applications we choose the branch λ = − √ µ leading to solutions with z = 1 + p µ . (2.7)A general result valid for any five-dimensional Einstein space, normalised as we have,is that for co-closed 1-forms µ ≥ µ = 8 holds iff the 1-form is dual to a Killingvector (see section 4.3 of [34]). Thus in general our construction leads to solutionswith z ≥ . (2.8)Since all SE manifolds have at least the Reeb Killing vector, dual to the one-form η ,this bound is always saturated. Indeed the solution of [9] with z = 4 is in our class.Specifically it can be obtained by setting C = σr η (and redefining x − → − x − / η is co-closed on SE and is an eigenmode of ∆ SE witheigenvalue µ = 8. Note that for this solution the two-form dC is proportional to theK¨ahler-form of the Calabi-Yau cone: dC = 2 σω .On S the spectrum of ∆ S acting on one-forms is well known and we have µ =( s +1)( s +3) for s = 1 , , . . . (see for example [35] eq (2.20)) leading to λ = s +1 andhence new classes of solutions with z = 4 , , . . . . Note that these solutions come infamilies, transforming in the SO (6) irreps , , , . . . . To obtain similar resultsfor T , one can consult [36]. 3e now discuss a construction that can be used when the spectrum of the Lapla-cian acting on functions is known, but not acting on one-forms. For example, thescalar Laplacian was studied in [40] for the Y p,q metrics [41], but as far as we knowit has not been discussed acting on one-forms. Specifically we construct (1 ,
1) forms dC on the CY cone using scalar functions Φ on the cone as follows. We write C i = J ij ∂ j Φ (2.9)for some function Φ on CY . A short calculation shows that if ∇ CY Φ = α (2.10)for some constant α then dC is co-closed. The two-form dC is a (1 ,
1) form on CY and it is primitive, J ij dC ij = 0, if and only if α = 0. Observe that the solution of[9] with z = 4 fits into this class by taking Φ = − σr / α = − σ , leading to C = σr η .We now consider solutions with α = 0, corresponding to harmonic functions onthe CY cone with dC (1 ,
1) and primitive. We next writeΦ = r λ f (2.11)where f is a function on the SE space satisfying − ∇ SE f = kf (2.12)with k = λ ( λ + 4) (see e.g. [37]). For the solutions of interest we choose the branch λ = − √ k leading to z = √ k . For the special case of the five-spherewe can check with the results that we obtained above. The eigenfunctions f on thefive-sphere are given by spherical harmonics with k = l ( l + 4), l = 1 , , . . . and hence z = l + 2. The l = 1 harmonic appears to violate the bound (2.8). However, it isstraightforward to see that the construction for l = 1 leads to dC = 0 for which C canbe removed by a transformation of the form (2.5). Thus for S we should consider l ≥ z = 4 , , . . . , as above. It is worth pointing out thatfor higher values of l some of the eigenfunctions will also lead to closed C : if weconsider the harmonic function on R given by x i . . . x i l c i ...i l where c is symmetricand traceless then, with J = dx ∧ dx + dx ∧ dx + dx ∧ dx we see that dC = 0 if J [ ij c k ] ji ...i l = 0. Note that in general the one-form C defined in (2.9) has a component in the dr direction, unlikein (2.6). However, locally we can remove it by a transformation of the form (2.5). Also, one candirectly show that the resulting one-form β is co-closed on the SE space. .1 Supersymmetry We introduce the frame e + = rdx + e − = r ( dx − + C ) e = rdx e = rdx e = drre m = e mSE , m = 5 , . . . , e mSE is an orthonormal frame for the SE space. We can write F = B + ∗ B (2.14) B =4 e + ∧ e − ∧ e ∧ e ∧ e − re + ∧ dC ∧ e ∧ e (2.15)where we have chosen ǫ + − = +1. The Killing spinor equation can be written D M ǫ + i /F Γ M ǫ = D M ǫ + i /B Γ M ǫ = 0 . (2.16)We are using the conventions for type IIB supergravity [42][43] as in [44] and inparticular, Γ = Γ + − with the chiral IIB spinors satisfying Γ ǫ = − ǫ .If ǫ are the Killing spinors for the AdS × SE solution, then we find that we mustalso impose that Γ + − ǫ = iǫ Γ + ǫ = 0 . (2.17)The first condition maintains the Poincar´e supersymmetries but breaks all of thesuperconformal supersymmetries (this can be explicitly checked using, for example,the results of [45]). The second condition breaks a further half of these . Thus when dC = 0, we preserve two Poincar´e supersymmetries for a generic SE and this isincreased to eight Poincar´e supersymmetries for S . That we preserve the Poincar´e supersymmetries suggests that we can extend our solutions awayfrom the near horizon limit of the D3-branes. This is indeed the case but we won’t expand uponthat here. The D = 11 solutions The construction of the D = 11 solutions is very similar. We consider the ansatz forD=11 supergravity solutions: ds = dρ ρ + ρ (cid:2) dx + dx − + dx (cid:3) + ds ( SE ) + 2 ρ Cdx + G = − ρ dx + ∧ dx − ∧ dρ ∧ dx + dx + ∧ dx ∧ d ( ρ C ) (3.1)where SE is a seven-dimensional Sasaki-Einstein space and ds ( SE ) is normalisedso that the Ricci tensor is equal to six times the metric (this is the normalisation of aunit radius seven-sphere). It is convenient to change coordinates via ρ = r to bringthe solution to the form ds = dr r + r (cid:2) dx + dx − + dx (cid:3) + ds ( SE ) + 2 r Cdx + G = − r dx + ∧ dx − ∧ dr ∧ dx + dx + ∧ dx ∧ d ( r C ) . (3.2)In these coordinates the cone metric ds CY = dr + r ds ( SE ) (3.3)is a metric on Calabi-Yau four-fold. We will use the same notation for the CY spaceas in the previous section.When the one-form C is zero we have the standard AdS × SE solution of D = 11supergravity that, in general, preserves eight supersymmetries. We again find thatall the equations of motion are solved if C is a one-form on CY and the two-form dC is co-closed d ∗ CY dC = 0 . (3.4)The solutions are again invariant under the transformation (2.5). We will considersolutions where the one-form C has weight λ under the action of r∂ r , correspondingto dynamical exponent z = 2 + λ/
2. As before, using the results in appendix A of[33], we consider solutions constructed from a co-closed one-form β on the SE spacethat is an eigenmode of the Laplacian ∆ SE : C = r λ β, ∆ SE β = µβ, d † β = 0 . (3.5)One can check that dC is co-closed providing that µ = λ ( λ + 4). For our applicationswe choose the branch λ = − √ µ leading to solutions with z = 1 + p µ . (3.6)6 general result valid for any seven-dimensional Einstein space, normalised as wehave, is that for co-closed 1-forms µ ≥
12 and µ = 12 holds iff the 1-form is dual toa Killing vector (see section 4.3 of [34]). Thus in general our construction leads tosolutions with z ≥ SE spaces. Observe that the solutions of[31] with z = 3 fit into this class. Specifically they are obtained by setting C = σr η (after redefining x → x/ x − → − x − / S the spectrum of ∆ S is wellknown and we have µ = s ( s + 6) + 5 for s = 1 , , . . . (see for example [34] eq (7.2.5))leading to λ = 1 + s and hence new classes of solutions with z = 3 , , , . . . . Thesesolutions come in families transforming in the SO
8) irreps , v , v , . . . .Results on the spectrum of the Laplacian on some homogeneous SE spaces can befound in [46],[47],[48].As before we can construct (1 ,
1) co-closed two-forms dC using scalar functions Φon CY We write C i = J ij ∂ j Φ , ∇ CY Φ = α . (3.8)and dC is again primitive if and only if α = 0. The solutions of [31] with z = 3 ariseby taking Φ = σr and α = − σ leading to C = σr η . We now focus on solutionswith α = 0, corresponding to harmonic functions on the CY cone. We takeΦ = r λ f (3.9)where f is a function on the SE space satisfying − ∇ SE f = kf (3.10)with k = λ ( λ + 6). For our applications we choose the branch λ = − √ k leading to solutions with z = + √ k . For example, on the seven-sphere theeigenfunctions f are given by spherical harmonics with k = l ( l + 6) with l = 1 , , . . . and hence z = 2+ l/
2. Excluding the l = 1 harmonic, as it can be removed by a trans-formation of the form (2.5), for S we are left with solutions with z = 3 , / , , . . . ,as above. 7 .1 Supersymmetry We introduce a frame e + = r dx + e − = r ( dx − + C ) e = r dxe = drre m = e mSE , m = 4 , . . . , . (3.11)We thus have G = 6 e + ∧ e − ∧ e ∧ e + r e + ∧ e ∧ dC ∗ G = − V ol ( SE ) + dx + ∗ CY dC (3.12)where we have chosen the orientation ǫ + − .... = +1.The Killing spinor equation can be written as ∇ M ǫ + 1288 [Γ M N N N N − δ N M Γ N N N ] G N N N N ǫ = 0 . (3.13)We are using the conventions for D = 11 supergravity [49] as in [50] and in particularΓ + − = +1.If ǫ are the Killing spinors arising for the AdS × SE solution, then we find thatwe must also impose that Γ + − ǫ = − ǫ Γ + ǫ = 0 . (3.14)The first condition maintains the Poincar´e supersymmetries but breaks all of thesuperconformal supersymmetries. The second condition breaks a further half of these.Thus when dC = 0, we preserve two Poincar´e supersymmetries for a generic SE andthis is increased to eight Poincar´e supersymmetries for S . If AdS × SE is a supersymmetric solution of D = 11 supergravity, then if we “skew-whiff” by reversing the sign of the flux (or equivalently changing the orientation of SE ) then apart from the special case when the SE space is the round S , allsupersymmetry is broken [51]. Despite the lack of supersymmetry, such solutions areknown to be perturbatively stable [51]. Similarly, if we reverse the sign of the fluxin our new solutions (3.2), we will obtain solutions of D = 11 supergravity that willgenerically not preserve any supersymmetry.8 Further Generalisation
We now discuss a further generalisation of the solutions that we have considered so far,preserving the same amount of supersymmetry, which incorporate the constructionof [24]. For type IIB the metric is now given by ds = dr r + r (cid:2) dx + dx − + dx + dx (cid:3) + ds ( SE ) + r (cid:2) Cdx + + h ( dx + ) (cid:3) (4.1)with the five-form unchanged from (2.1). The conditions on the one-form C are asbefore and we demand that h is a harmonic function on the CY cone: ∇ CY h = 0 . (4.2)Choosing h to have weight λ ′ under r∂ r we take h = r λ ′ f ′ , (4.3)where f ′ is an eigenfunction of the Laplacian on SE with eigenvalue k ′ − ∇ SE f ′ = k ′ f ′ (4.4)with k ′ = λ ′ ( λ ′ + 4). If we set C = 0 and choose the branch λ ′ = − √ k ′ thenthese are the solutions constructed in section 5 of [24] and have dynamical exponent z = √ k ′ . As noted in [24] an application of Lichnerowicz’s theorem [52],[53]implies that these solutions have z ≥ / z = 3 / S . Nowif there is a scalar eigenfunction with eigenvalue k ′ and a one-form eigenmode of theLaplacian on SE with eigenvalue µ that satisfy z = √ k ′ = 1 + √ µ thenwe can superpose the solution with h as in (4.3) and the one-form C as in (2.6) andhave a solution with scaling symmetry with this value of z . For example on S , usingthe notation as before, we have k ′ = l ′ ( l ′ + 4), l ′ = 1 , , . . . and µ = ( s + 1)( s + 3), s = 1 , , . . . and hence we must demand that l ′ = 2( s + 2), s = 1 , , . . . , givingsolutions with z = 3 + s .The story for D = 11 is very similar. The metric is now given by ds = dr r + r (cid:2) dx + dx − + dx (cid:3) + ds ( SE ) + r (cid:2) Cdx + + h ( dx + ) (cid:3) (4.5)with the four-form unchanged from (3.2). The conditions on the one-form C are asbefore and we demand that h is a harmonic function on the CY cone: ∇ CY h = 0 . (4.6)9hoosing h to have weight λ ′ under r∂ r we take h = r λ ′ f ′ , (4.7)where f ′ is an eigenfunction of the Laplacian on SE with eigenvalue k ′ − ∇ SE f ′ = k ′ f ′ (4.8)with k ′ = λ ′ ( λ ′ + 6). If we set C = 0 and chose the branch λ ′ = − √ k ′ thenthese solutions have dynamical exponent z = (1 + √ k ′ ). Lichnerowicz’s theorem[52],[53] implies that these solutions have z ≥ / z = 5 / S .If there is a scalar eigenfunction with eigenvalue k ′ and a one-form eignemode of theLaplacian on SE with eigenvalue µ that satisfy z = (1 + √ k ′ ) = 1 + √ µ then we can superpose the solution with h as in (4.7) and the one-form C as in (3.5)and have a solution with scaling symmetry with this value of z . For example on S ,using the notation as before, we have k ′ = l ′ ( l ′ + 6), l ′ = 1 , , . . . and µ = s ( s + 6) + 5, s = 1 , , . . . and hence we must demand that l ′ = 2( s + 3), s = 1 , , . . . , givingsolutions with z = (5 + s ). Acknowledgements
We would like to thank Seok Kim, James Sparks, Oscar Varela and Daniel Waldram.for helpful discussions. JPG is supported by an EPSRC Senior Fellowship and aRoyal Society Wolfson Award.
References [1] 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]].[2] K. Balasubramanian and J. McGreevy, “Gravity duals for non-relativisticCFTs,” Phys. Rev. Lett. , 061601 (2008) [arXiv:0804.4053 [hep-th]].[3] M. Sakaguchi and K. Yoshida, “Super Schrodinger in Super Conformal,”arXiv:0805.2661 [hep-th].[4] W. D. Goldberger, “AdS/CFT duality for non-relativistic field theory,”arXiv:0806.2867 [hep-th]. 105] J. L. B. Barbon and C. A. Fuertes, “On the spectrum of nonrelativisticAdS/CFT,” JHEP , 030 (2008) [arXiv:0806.3244 [hep-th]].[6] M. Sakaguchi and K. Yoshida, “More super Schrodinger algebras frompsu(2,2—4),” JHEP , 049 (2008) [arXiv:0806.3612 [hep-th]].[7] W. Y. Wen, “AdS/NRCFT for the (super) Calogero model,” arXiv:0807.0633[hep-th].[8] C. P. Herzog, M. Rangamani and S. F. Ross, “Heating up Galilean holography,”JHEP , 080 (2008) [arXiv:0807.1099 [hep-th]].[9] J. Maldacena, D. Martelli and Y. Tachikawa, “Comments on string theory back-grounds with non-relativistic conformal symmetry,” JHEP , 072 (2008)[arXiv:0807.1100 [hep-th]].[10] A. Adams, K. Balasubramanian and J. McGreevy, “Hot Spacetimes for ColdAtoms,” JHEP , 059 (2008) [arXiv:0807.1111 [hep-th]].[11] Y. Nakayama, “Index for Non-relativistic Superconformal Field Theories,” JHEP , 083 (2008) [arXiv:0807.3344 [hep-th]].[12] D. Minic and M. Pleimling, “Non-relativistic AdS/CFT and Aging/Gravity Du-ality,” arXiv:0807.3665 [cond-mat.stat-mech].[13] J. W. Chen and W. Y. Wen, “Shear Viscosity of a Non-Relativistic ConformalGas in Two Dimensions,” arXiv:0808.0399 [hep-th].[14] A. V. Galajinsky, “Remark on quantum mechanics with conformal Galilean sym-metry,” Phys. Rev. D , 087701 (2008) [arXiv:0808.1553 [hep-th]].[15] S. Kachru, X. Liu and M. Mulligan, “Gravity Duals of Lifshitz-like Fixed Points,”Phys. Rev. D , 106005 (2008) [arXiv:0808.1725 [hep-th]].[16] S. S. Pal, “Null Melvin Twist to Sakai-Sugimoto model,” arXiv:0808.3042 [hep-th].[17] S. Sekhar Pal, “Towards Gravity solutions of AdS/CMT,” arXiv:0808.3232 [hep-th].[18] S. Pal, “More gravity solutions of AdS/CMT,” arXiv:0809.1756 [hep-th].1119] P. Kovtun and D. Nickel, “Black holes and non-relativistic quantum systems,”arXiv:0809.2020 [hep-th].[20] C. Duval, M. Hassaine and P. A. Horvathy, “The geometry of Schr´odinger sym-metry in gravity background/non-relativistic CFT,” arXiv:0809.3128 [hep-th].[21] S. S. Lee, “A Non-Fermi Liquid from a Charged Black Hole: A Critical FermiBall,” arXiv:0809.3402 [hep-th].[22] D. Yamada, “Thermodynamics of Black Holes in Schroedinger Space,”arXiv:0809.4928 [hep-th].[23] F. L. Lin and S. Y. Wu, “Non-relativistic Holography and Singular Black Hole,”arXiv:0810.0227 [hep-th].[24] S. A. Hartnoll and K. Yoshida, “Families of IIB duals for nonrelativistic CFTs,”arXiv:0810.0298 [hep-th].[25] M. Schvellinger, “Kerr-AdS black holes and non-relativistic conformal QM the-ories in diverse dimensions,” JHEP , 004 (2008) [arXiv:0810.3011 [hep-th]].[26] L. Mazzucato, Y. Oz and S. Theisen, “Non-relativistic Branes,” arXiv:0810.3673[hep-th].[27] M. Rangamani, S. F. Ross, D. T. Son and E. G. Thompson, “Conformal non-relativistic hydrodynamics from gravity,” arXiv:0811.2049 [hep-th].[28] A. Akhavan, M. Alishahiha, A. Davody and A. Vahedi, “Non-relativistic CFTand Semi-classical Strings,” arXiv:0811.3067 [hep-th].[29] A. Adams, A. Maloney, A. Sinha and S. E. Vazquez, “1/N Effects in Non-Relativistic Gauge-Gravity Duality,” arXiv:0812.0166 [hep-th].[30] M. Taylor, “Non-relativistic holography,” arXiv:0812.0530 [hep-th].[31] J. P. Gauntlett, S. Kim, O. Varela and D. Waldram, “Consistent supersymmetricKaluza–Klein truncations with massive modes,” arXiv:0901.0676 [hep-th].[32] S. Pal, “Anisotropic gravity solutions in AdS/CMT,” arXiv:0901.0599 [hep-th].[33] D. Martelli and J. Sparks, “Symmetry-breaking vacua and baryon condensatesin AdS/CFT,” arXiv:0804.3999 [hep-th].1234] M. J. Duff, B. E. W. Nilsson and C. N. Pope, “Kaluza-Klein Supergravity,”Phys. Rept. (1986) 1.[35] H. J. Kim, L. J. Romans and P. van Nieuwenhuizen, “The Mass Spectrum OfChiral N=2 D=10 Supergravity On S**5,” Phys. Rev. D (1985) 389.[36] A. Ceresole, G. Dall’Agata and R. D’Auria, “KK spectroscopy of type IIB su-pergravity on AdS(5) x T(11),” JHEP (1999) 009 [arXiv:hep-th/9907216].[37] J. P. Gauntlett, D. Martelli, J. Sparks and S. T. Yau, “Obstructions to theexistence of Sasaki-Einstein metrics,” Commun. Math. Phys. (2007) 803[arXiv:hep-th/0607080].[38] S. S. Gubser, “Einstein manifolds and conformal field theories,” Phys. Rev. D (1999) 025006 [arXiv:hep-th/9807164].[39] G. W. Gibbons, S. A. Hartnoll and Y. Yasui, “Properties of some five dimensionalEinstein metrics,” Class. Quant. Grav. (2004) 4697 [arXiv:hep-th/0407030].[40] H. Kihara, M. Sakaguchi and Y. Yasui, “Scalar Laplacian on Sasaki-Einsteinmanifolds Y(p,q),” Phys. Lett. B (2005) 288 [arXiv:hep-th/0505259].[41] J. P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, “Sasaki-Einstein metricson S(2) x S(3),” Adv. Theor. Math. Phys. (2004) 711 [arXiv:hep-th/0403002].[42] J. H. Schwarz, “Covariant Field Equations Of Chiral N=2 D=10 Supergravity,”Nucl. Phys. B (1983) 269.[43] P. S. Howe and P. C. West, “The Complete N=2, D=10 Supergravity,” Nucl.Phys. B (1984) 181.[44] J. P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, “SupersymmetricAdS(5) solutions of type IIB supergravity,” Class. Quant. Grav. (2006) 4693[arXiv:hep-th/0510125].[45] H. Lu, C. N. Pope and J. Rahmfeld, “A construction of Killing spinors on S**n,”J. Math. Phys. , 4518 (1999) [arXiv:hep-th/9805151].[46] D. Fabbri, P. Fre, L. Gualtieri and P. Termonia, “M-theory on AdS(4) x M(111):The complete Osp(2—4) x SU(3) x SU(2) spectrum from harmonic analysis,”Nucl. Phys. B (1999) 617 [arXiv:hep-th/9903036].1347] P. Merlatti, “M-theory on AdS(4) x Q(111): The complete Osp(2—4) x SU(2)x SU(2) x SU(2) spectrum from harmonic analysis,” Class. Quant. Grav. (2001) 2797 [arXiv:hep-th/0012159].[48] P. Termonia, “The complete N = 3 Kaluza-Klein spectrum of 11D supergravityon AdS(4) x N(010),” Nucl. Phys. B (2000) 341 [arXiv:hep-th/9909137].[49] E. Cremmer, B. Julia and J. Scherk, “Supergravity theory in 11 dimensions,”Phys. Lett. B (1978) 409.[50] J. P. Gauntlett and S. Pakis, “The geometry of D = 11 Killing spinors,” JHEP (2003) 039 [arXiv:hep-th/0212008].[51] M. J. Duff, B. E. W. Nilsson and C. N. Pope, “The Criterion For VacuumStability In Kaluza-Klein Supergravity,” Phys. Lett. B (1984) 154.[52] A. Lichnerowicz, “G´eometrie des groupes de transformations,” Dunod, Paris,1958.[53] M. Obata, “Certain conditions for a Riemannian manifold to be isometric to asphere,” J. Math. Soc. Japan14