Universal time-dependent deformations of Schrodinger geometry
aa r X i v : . [ h e p - t h ] F e b UCB-PTH-10/03,IPMU10-0021
Universal time-dependent deformations ofSchr¨odinger geometry
Yu Nakayama
Berkeley Center for Theoretical Physics,University of California, Berkeley, CA 94720, USAandInstitute for the Physics and Mathematics of the Universe,University of Tokyo, Kashiwa, Chiba 277-8582, Japan
Abstract
We investigate universal time-dependent exact deformations of Schr¨odinger ge-ometry. We present 1) scale invariant but non-conformal deformation, 2) non-conformal but scale invariant deformation, and 3) both scale and conformal invari-ant deformation. All these solutions are universal in the sense that we could embedthem in any supergravity constructions of the Schr¨odinger invariant geometry. Wegive a field theory interpretation of our time-dependent solutions. In particular,we argue that any time-dependent chemical potential can be treated exactly in ourgravity dual approach.
Introduction
The advent of the AdS/CMP correspondence radically changes the status of the stringtheory, or quantum gravity. The holography is believed to be one of the fundamental prin-ciples of quantum gravity, but so far it has been formidable to acquire any experimentalevidence. First of all, our observable universe is unique, so the holographic approach toour universe (if possible) is restricted to one and the only one particular example. On theother hand, we have investigated the string/gravity dual for the QCD, but again our QCDis unique, so experimental comparison of our holographic theory has been quite limited.The AdS/CMP correspondence has completely changed the situation. We can engineerthe condensed matter system as we like, and as a consequence we may have infinitely manyexperimentally testable holographic setups in principle. We believe that we will be ableto compare various multiverse with condensed matter systems in the near future: we willsoon realize that we are surrounded by quantum universes realized in condensed mattersystems.We are, however, still on the way to the above-mentioned paradise of holographic quan-tum gravities. Unlike the AdS/CFT correspondence and partially successful AdS/QCDcorrespondence, we do not have any concrete (experimentally testable) realizations of con-densed matter systems in terms of quantum gravity. In this sense, we have not reachedeven the standard of AdS/QCD correspondence, where we can at least compare qualitativepredictions with experiments, assuming N = 3 is large enough.In particular, less is known for the field theory dual of the non-relativistic AdS/CFTcorrespondence. The geometry that has the isometry corresponding to the Schr¨odingergroup was first advocated in [3][4], and its supergravity embedding has been discussedin the context of the string theory [7][8][9][10][11][12][13] as well as in the M-theory[12][14][15]. Unfortunately, we do not know the corresponding gauge theories exceptin some specific cases where the theory is supposed to be obtained from the discretelight cone quantization (DLCQ) of the N = 4 super Yang-Mills theory. The DLCQ isnotoriously difficult to study, so in practice, we do not have any calculable Lagrangiandescription of the dual field theory for the non-relativistic AdS/CFT correspondence. CMP stands for Condensed Matter Physics. See [1][2] for recent reviews on the subject. A geometric realization of the Schr¨odinger group was pioneered in the earlier work [5], whose relationto [3][4] was discussed in [6].
1n this paper, as a first step to understand the nature of the non-relativistic AdS/CFTcorrespondence, we study the universal time-dependent deformations of the simplestSchr¨odinger invariant geometry. Our solutions are exact and universal in the sense thatwe can embed them in any supergravity constructions of Schr¨odinger invariant geome-try. We study the field theory interpretations of our deformations, and we claim thatthe field theories dual to the Schr¨odinger invariant geometry should always admit suchtime-dependent exact deformations of the action, and they should also possess the statescorresponding to our time-dependent solutions. The analysis furthermore reveals thatthe dual field theories always include certain operators that are not contained in theminimal operator contents of the Schr¨odinger invariant field theory. In other words, theSchr¨odinger invariant field theories that have a gravity dual predict an existence of theseparticular operators.As a spin-off of our results, we argue that any time-dependent chemical potential canbe treated exactly in our gravity dual approach. The time-dependent chemical potentialturns out to be simply the time-dependent coordinate transformation of the bulk theory.Our exact time-dependent deformations will show PP-wave singularities from the bulkgravitational theory viewpoint. It would be interesting to understand the nature of thesingularity from the field theory perspective. Since we have not succeeded in finding thecomplete dual field theories, we cannot say much about the fate of the singularity fromthe field theory viewpoint. We hope that once the non-relativistic AdS/CFT correspon-dence is much better established, we would be able to attach the resolutions of PP-wavesingularities from the dual non-relativistic field theory viewpoint. We leave this importantissues for the future study.
We begin with the geometry with the Schr¨odinger invariance [3][4]: ds d +3 = − dt z + − dtdξ + dx i + dz z , (2.1)where i = 1 , · · · d . The light-like ξ direction is compactified as ξ ≃ ξ + 2 πR so that thespectrum reproduces the quantization of the particle number in the dual field theory. The2etric (2.1) is the solution of the Einstein equation with the massive (Proca) vector field S = Z d d x i dξdtdz √− g (cid:18) R − Λ − F µν F µν − m A µ A µ (cid:19) . (2.2)where F µν = ∂ µ A ν − ∂ ν A µ . One can show that A = A µ dx µ = − dtz solves the equation ofmotion as well as the Einstein equation, providedΛ = −
12 ( d + 1)( d + 2) , m = 2( d + 2) . (2.3)The geometry has an obvious invariance under the translation in ( t, x i ) as well as theEuclidean rotation in x i . It is also invariant under the Galilean boost x i → x i − v i t , ξ → ξ − v i x i + 12 v t . (2.4)Furthermore, the geometry has the full non-relativistic conformal invariance [16][17]. Thedilatation is generated by t → λ t , x i → λx i , z → λz , ξ → ξ. (2.5)The non-relativistic special conformal transformation is generated by t → t at , x i → x i at , z → z at , ξ → ξ − a x i x i + z at . (2.6)[18] argues that the metric (2.1) is the simplest geometrical realization of the Schr¨odingerinvariance within the coset space construction.The geometry is proposed to be dual to a non-relativistic Schr¨odinger invariant fieldtheory. So far, there is no concrete proposal for what is the precise dual field theorycorresponding to the geometry. It is suggested in [3] that it would be obtained by arelevant deformation of the relativistic CFT dual to the AdS space: δS CFT = J Z d d +2 xO t (2.7)after compactifying a light-cone direction because the GKPW prescription dictates thatthe deformation (2.7) is induced by the Proca field A µ . The relativistic scaling dimensionof O µ is d + 2, where O µ is dual to A µ .One of the main objectives of this paper is to understand the physics of this Schr¨odingerinvariant geometry by introducing the exact time-dependent deformations. In section 2.1, Note that the time t here is a light-cone time x + in the original relativistic AdS/CFT coordinate.
3e first study the scale invariant but non-conformal deformations, and in section 2.2, westudy the conformal but non-scale invariant deformations. The both deformations soundpeculiar from our experience in relativistic field theories: indeed, we would like to arguethat they are special features of time-dependent non-relativistic field theories. In section2.3, we extend our analysis to broader classes of exact time-dependent solutions.
We begin with the scale invariant but non-conformal deformations of the Schr¨odingergeometry. It was shown [19] that this is impossible without breaking further symmetries,so we investigate the solutions that are time-dependent explicitly. The most general scaleinvariant deformations (up to a coordinate transformation) are given by ds d +3 = − C (cid:18) tz (cid:19) dt z + D (cid:18) tz (cid:19) − dtdξ + dx i + dz z (2.8)for the metric and A = A µ dx µ = − E (cid:18) tz (cid:19) dtz , (2.9)for the vector field. It is easy to see that t/z is invariant under the scale transformation(2.5) but not under the non-relativistic special conformal transformation (2.6). We haveused the diffeomorphism invariance to remove the dzdt component of the metric.The ( zt ) component of the Einstein equation tells us D = const, so we set D = 1. Theother equations of motion can be solved exactly by C = 2 + C z t + C (cid:18) z t (cid:19) d/ + ǫ d + 2 d + 3 z d +8 t d +4 E = 1 + ǫ z d +4 t d/ , (2.10)where C , C and ǫ are integration constants.The deformations by C and C do not depend on the background vector field, andthey even exist without introducing the vector field ( e.g. in locally AdS background).Such PP-wave deformations in the AdS space was studied in [20][21][22][23]. The metricis given by ds d +3 = − C z t + C (cid:18) z t (cid:19) d/ ! dt z + − dtdξ + dx i + dz z , (2.11) See also [30][25][26] for related null deformed backgrounds. C is dual to the time-dependentvacuum expectation value (VEV) for a particular component of the energy momentumtensor: h T tt i = C t d/ . (2.12)Later, we will discuss the similar operator has a VEV in the dual Schr¨odinger invariantfield theory.It is sometimes believed that the scale invariance implies the conformal invariance.This is not always true because there is no symmetric reason why this is so [27] a-priori, and indeed there are some known counterexamples [28][29][30]. However, it wasshown that in (1 + 1) dimension, the Poincar´e invariance, unitarity, and the discrete-ness of the spectrum guarantees the equivalence between the conformal invariance andthe scale invariance [31][32]. It is hoped that a similar statement should hold in higherdimensions with a suitable generalization of the assumptions [33][34][35]. In the unitarySch¨odinger invariant field theories, it is conjectured that the scale invariance togetherwith the Galilean invariance, rotation and translational invariance would imply the fullnon-relativistic conformal invariance [19]. It is easy to see that there is no such time-independent deformations in the above simple Sch¨odinger invariant geometry. On theother hand, in the discussion above, the time-translational invariance is explicitly brokenso that the conjecture does not apply.
Now, we will present a more peculiar situation where the solution is invariant under thenon-relativistic conformal transformation but not invariant under the dilatation. Such ageometry is impossible without breaking a further symmetry, and here again, we considerexplicitly time-dependent solutions. The most general non-relativistic conformal invariant Again, note that t here corresponds to the light-cone direction x + in the original relativistic AdS/CFTcoordinate. In higer dimensional supergravity embeddings, there could exist other terms that are scale invariantbut not conformal invariant. We also note that the discussion here specializes in the case with thedynamical critical exponent Z = 2. When Z 6 = 2, the non-relativistic special conformal transformationcannot be constructed in the algebra. See [9][10][36][12] for examples of such geometries. ds d +3 = − C (cid:18) tz (cid:19) dt z + D (cid:18) tz (cid:19) − dtdξ + dx i + dz z (2.13)for the metric and A = − E (cid:18) tz (cid:19) dtz , (2.14)for the vector field. It is easy to see that t/z is invariant under the non-relativistic specialconformal transformation (2.6) but not under the scale transformation (2.5). Again, wehave used the diffeomorphism invariance to remove the dzdt component of the metric.The ( zt ) component of the Einstein equation tells us D = const, so we set D = 1. Theother equations of motion can be solved exactly by C = 2 + C z t + C (cid:16) zt (cid:17) d +4 + ǫ d + 2 d + 3 z d +8 t d +8 E = 1 + ǫ z d +4 t d +4 , (2.15)where C , C and ǫ are integration constants. The deformations by C and C do notdepend on the background vector field, and they even exist without introducing the vectorfield ( e.g. in locally AdS background).Such PP-wave deformations in the AdS space was studied in [20][21][22][23]. Themetric is given by ds d +3 = − (cid:18) C z t + C (cid:16) zt (cid:17) d +4 (cid:19) dt z + − dtdξ + dx i + dz z . (2.16)It was argued [21] that the deformation by C is due to the time-dependent VEV for aparticular component of the energy momentum tensor: h T tt i = C t d +4 . (2.17)The closure of the non-relativistic conformal algebra ( i.e. i [ H, K ] = D : see appendixA for details) demands that the non-relativistic special conformal invariance implies thescale invariance. Thus, it is impossible to break the scale invariance without spoiling thespecial conformal invariance. Here, the explicit time-dependence alleviates the situation,and without the conserved Hamiltonian, the conformal invariance without the dilation ispossible. 6 .3 More general deformations Although the symmetry is more reduced, our deformations in the previous subsectionsare particular cases of more general time-dependent exact solutions of the simplest action(2.2) that allows the Schr¨odinger invariant geometry. The more general time-dependentsolutions are given by ds d +3 = − C ( t, z ) dt z + D ( t, z ) − dtdξ + dx i + dz z (2.18)for the metric and A = − E ( t, z ) dtz , (2.19)where C ( t, z ) = 2 + z c ( t ) + c ( t ) z d +4 + ǫ ( t ) d + 2 d + 3 z d +4) D ( t, z ) = 1 E ( t, z ) = 1 + ǫ ( t ) z d +4 . (2.20) c ( t ), c ( t ) and ǫ ( t ) are arbitrary functions of t .There are several special choices of these functions: • scale invariance: c ( t ) = t , c ( t ) = t d +42 , ǫ ( t ) = t d +4 . • conformal invariance: c ( t ) = t , c ( t ) = t d +4 , ǫ ( t ) = t d +4) • scale and conformal invariance: c ( t ) = δ ( t ).We have discussed the first two choices in the previous subsections. The last one mayneed a comment. The solution contains a delta function and it looks singular, but in ourapproach, this is the only possible solution that is scale invariant and conformal invariantat the same time. As we would like to discuss it later, however, the deformation by c ( t )can be gauged away by a coordinate transformation, so in practice, there is no non-trivialscale and conformal invariant time-dependent deformation of the Schr¨odinger invariantgeometry.The impossibility to obtain a scale and conformal invariant time-dependent solutionmay be regarded as a dual statement of the impossibility to construct a scale invariant7ut non-conformal field theory without breaking any translation invariance. The point isthat the non-relativistic conformal algebra has a non-trivial involution anti-automorphism,which exchanges the Hamiltonain H and the non-relativistic special conformal transforma-tion K (see appendix A for details). Therefore, from the representation theory viewpoint,it is very close to study the theory with no conserved Hamiltonian but invariant underall the other generators including the conformal transformation and to study the theorywith no conformal invariance but invariant under all the other generators including theHamiltonian.Let us briefly discuss the singularity structure of the geometry. First of all, it is notdifficult to see that all the curvature invariants are constant irrespective of the deforma-tions. However, there is a PP-wave singularity at z = 0 as well as t = 0 (or t = ±∞ ) when c ( t ) or ǫ ( t ) becomes infinite as t → t = ±∞ ). In particular, the latter conditionapplies at t = 0 to the scale/conformal invariant deformations discussed in section 2.2and 2.3.Since our deformations vanish as z →
0, the singularity at z = 0 is the same as that ofthe original Sch¨odinger space-time. See for example [10] for the analysis of the singularityat z = 0. On the other hand, the analysis of the PP-wave singularity structure has beendone in the vacuum AdS PP-wave geometry in [20]. Our metric is within the analysisdone in [20] except that our solution does not satisfy the vacuum Einstein equation butrather it has the source term from the Proca field. The PP-wave singularity appearswhenever A = z ∂ z (cid:18) ∂ z ( z C ( t, z )) z (cid:19) (2.21)is diverging, and this is precisely the above condition ( i.e. diverging c ( t ), ǫ ( t ) at t = 0).Again note that the deformation by c ( t ) does not introduce any PP-wave singularity atall. In this section, we will study the field theory interpretation of the exact time-dependentdeformations introduced in the last section. We use the natural generalization of the The apparent singularity due to c ( t ) can be removed by the coordinate transformation. We will givea field theory interpretation of this removal of singularity later in section 3. c ( t ) corresponds to the introduction of the time-dependent chem-ical potential in the action: M Z dtd d xc ( t ) ρ ( t, x ) , (3.1)where ρ ( t, x ) is the particle number density. For instance, in the free Schr¨odinger theorywith the action S = Z dtd d x (cid:18) i Φ ∗ ∂ t Φ − M ∂ i Φ ∗ ∂ i Φ (cid:19) , (3.2)the particle number density is given by ρ = Φ † Φ. This identification was first proposed in[3], and we will confirm the identification by comparing the field theory expectation andthe gravity prediction. In the local Schr¨odinger invariant field theories, such an operatoralways exists because it generates the particle number operator N = R d d xρ that lies inthe non-relativistic conformal algebra (see Appendix A).In the free Schr¨odinger theory, the introduction of this deformation modifies the two-point function as (cid:10) Φ( t , x )Φ † ( t , x ) (cid:11) = 1( t − t ) d/ exp (cid:18) iM ( x − x ) t − t + iM Z t t dtc ( t ) (cid:19) (3.3)up to a proportional factor. It is important to notice that in the simple Schr¨odinger in-variant field theories with Lagrangian description, the time-dependent chemical potential(3.1) can be exactly treated. The introduction of (3.1) is equivalent to the field redefini-tion Φ( t, x i ) → e iM R t dsc ( s ) Φ( t, x i ). We will argue this is also true whenever the theorieshave a gravity dual.To probe the implication of the gravity deformation by c ( t ), let us consider a mini-mally coupled scalar field φ with the gravity: S φ = Z dtdξdzd d x √− g (cid:0) ∂ µ φ ∗ ∂ µ φ − m | φ | (cid:1) . (3.4)The equation of motion for φ is given by ∂ z φ − z ( d + 2) ∂ z φ + (cid:18) iM ∂ t − k − M c ( t ) − m z (cid:19) φ = 0 , (3.5)9here M is the momentum eigenvalue in ξ direction, and k i is that for the x i directions: φ ( z, t, x, ξ ) = φ ( z, t ) e ikx + iMξ . m = m + 4 M is the effective mass of the KK mode inthe light-cone direction ξ . We can use the technique of separation of variables to solvethe scalar equation of motion. It turns out that the superpositions of the wavefunction: φ = z d/ K ν ( pz ) e − iEt + iM R t dsc ( s ) , (3.6)where p = √ k − M E , will give a complete basis of the solution. In practice, this simplymodifies the time-dependence of the scalar field by the phase factor e iM R t dsc ( s ) comparedwith the undeformed c ( t ) = 0 case. Explicitly, the two-point function can be computedas (cid:10) O ( t , x ) O † ( t , x ) (cid:11) = Z dEdpe − iE ( t − t )+ ip ( x − x ) (2 M E − p ) ν e iM R t t dtc ( t ) = C ν ( t − t ) d +22 + ν exp (cid:18) iM ( x − x ) t − t + iM Z t t dtc ( t ) (cid:19) , (3.7)where C ν is a (cut-off dependent) constant. We note the exact agreement with (3.3).This simplicity of the correlation functions actually suggests that the deformationby c ( t ) is only apparent. To see it, we note that the coordinate transformation ξ → ξ − R t dsc ( s ) will lead us to the original metric. It means that the time-dependentchemical potential in the Schr¨odinger invariant theory is always solved by the coordinatetransformation in the dual gravity.We emphasize that even though in the simplest field theory examples, the time-dependent chemical potential can be introduced/removed by the time-dependent fieldredefinition Φ( x, t ) → e iM R t dsc ( s ) Φ( x, t ), it is not at all obvious this is always the case,when the system is strongly coupled and the Schr¨odinger symmetry is rather emergent.We have, on the other hand, showed that it is always possible to introduce the time-dependent chemical potential in the gravity approach. This will further constrain thecandidates of the field theory duals.The interpretation of c ( t ) is given by assigning the VEV to a certain operator in thedual field theory: h T tt i = c ( t ) . (3.8)The scaling dimension of T tt is d +4. This operator does not always exist in non-relativisticSchr¨odinger invariant field theory, but the geometric construction requires the existence.10he reason why we call it T tt is that it is exactly the tt component of the energy-momentumtensor in the DLCQ construction of the Schr¨odinger invariant field theories. In the freeSchr¨odinger theory, it is given by ∂ t Φ † ∂ t Φ.Finally, ǫ ( t ) corresponds to the VEV of a certain (non-universal) operator of the non-relativistic CFT in the first oder approximation: h O t i = ǫ ( t ) . (3.9)The scaling dimension of O t is 2( d + 4). In the DLCQ of the relativistic conformal fieldtheory, we have discussed how O µ was introduced to generate the Schr¨odinger invariantgeometry. Here, the same operator has been given the VEV. Later, we will discuss thesupergravity embedding of our deformations, and there, we will see the origin of theeffective vector field corresponding to O t . Various form fields in supergravity play the roleof the operator.So far, we have discussed the field theory interpretation of the exact time-dependentdeformations of the simplest Schr¨odinger invariant field theory. Now, we would like to re-verse the logic and ask the question: what would we expect for the universal deformationsfrom the field theory viewpoint? The minimal ingredient of the Schr¨odinger invariant fieldtheory admits two universal deformations that preserve the Galilean boost and rotationalinvariance. They are given by the density operator ρ and the trace of the energy momen-tum tensor T ii = 2 T . The latter, however, only changes the normalization of the kineticterm with respect to the potential terms, so the universal deformations are simply givenby adding the chemical potential in agreement with the gravity discussion above.The other two deformations by c ( t ) and ǫ ( t ) are not in the minimal list of the non-relativistic conformal field theory. Thus, the non-relativistic conformal field theory dual tothe simplest Schr¨odinger invariant predicts the extra existence of the universal operators.In later section 4, we will discuss how these universal deformations appear in the generalsupergravity embedding of the Schr¨odinger invariant geometry. It looks peculiar that there is no corresponding (dual) operator insertion in the action. That wouldcorrespond to just the change of scale for t . Equivalently, the universal deformation by R dtd d x H doesthis job, where H = T is the Hamiltonian density. See the discussion in the following. Note that this T is different from T tt introduced above. In the free Schr¨odinger theory, T = m ∂ i Φ † ∂ i Φ. .1 more on the correlation functions We have studied the exact modifications of correlation functions induced by c ( t ). Thedeformations given by c ( t ) and ǫ ( t ) are more subtle than the deformation by c ( t ) interms of the correlation functions. These deformation will only affect the IR behavior ofthe scalar field in the geometry. To understand the situation, let us again consider theminimally coupled scalar field under the c ( t ) deformation. The same analysis applies for ǫ ( t ) deformation. The equation of motion is ∂ z φ − z ( d + 1) ∂ z φ + (cid:18) iM ∂ t − k − M c ( t ) z d +2 − m z (cid:19) φ = 0 . (3.10)The UV behavior ( i.e. z →
0) of the scalar field is the same as in the undeformedcase: φ ∼ e ipx − iwt z d/ (cid:18) I − ν ( | k | z ) + A ( k ) I ν ( | k | z ) I − ν ( | k | ǫ ) + A ( k ) I ν ( | k | ǫ ) (cid:19) , (3.11)where ǫ is the cutoff, k = p − M ω , and ν = p ( d + 2) + 4 m . In the undeformed casewith c ( t ) = 0, the regularity of the Wick-rotated wavefunction at z → ∞ determines A ( k ) so that φ is given by K ν ( | k | z ). Here the equation itself is modified toward z → ∞ and in addition we do not have a good guiding principle to set the boundary conditionthere in our explicitly time-dependent setup.To investigate the behavior z → ∞ and study the boundary condition, we need tosolve the equation toward z → ∞ . Unfortunately, there is no analytic solution of theequation (3.10) except for M = 0. When M = 0, the solution is uniquely given by φ ∼ e ipx − iwt z d/ K ν ( | p | z ) K ν ( | p | ǫ ) . (3.12)The wavefunction with M = 0 corresponds to the zero-norm state of the field theory, andthe two-point functions between such states are not affected at all by the deformation c ( t )(and ǫ ( t ) by repeating the same argument). See [42] for the discussions on the zero-normstates in the Schr¨odinger invariant field theories and their gravity dual.Instead of giving the analytic solutions of (3.10), we can study the power series solu-tion: φ = z d +2 − ∆ ∞ X n =0 a n ( t ) e ikx z n + z ∆ ∞ X n =0 b n ( t ) e ikx z n , (3.13)12here ∆ is related to the conformal dimension of the operator and it is given by ∆ = d +22 + ν = d +2+ √ ( d +2) +4 m . The equations of motion will determine the higher powers a n ( t ) and b n ( t ) from the boundary data a ( t ) and b ( t ). For instance, a ( t ) = − iM ∂ t a ( t ) − k a ( t )4( ν − , b ( t ) = − iM ∂ t b ( t ) − k b ( t )4( ν + 1) , (3.14)and the higher terms are determined recursively. The series coincide with the powerexpansion of the Bessel function, up to the ( d + 2)-th order. At the ( d + 4)-th order, theyshow a deviation: δa d +4 ( t ) = M c ( t )2(∆ − d − d + 4) a ( t ) δb d +4 ( t ) = − M c ( t )2(∆ + 1)( d + 4) b ( t ) . (3.15)Similarly, with the ǫ ( t ) deformation of the metric, the deviation appears at the 2 d + 8-thorder: δa d +8 ( t ) = − M ǫ ( t ) ( d + 2)2( d + 3)( d + 4)(3 d + 10 − a ( t ) δb d +8 ( t ) = − M ǫ ( t ) ( d + 2)2( d + 3)( d + 6)( d + 4 + ∆) b ( t ) (3.16)by assuming that the vector field does not affect the equation of motion for the scalar.The special case of the Kaigorodov space [43][44], where c ( t ) = c = const wasdiscussed in [21]. It was argued that the two-point function does not show any explicit dependence on c up to contact terms, so we can choose the same boundary condition asin the case c ( t ) = 0, which will yield the independence of the two-point function on c .Similarly, when ǫ ( t ) is a constant, the deformation only gives explicit corrections to thecontact terms, so we can choose the same boundary condition as in the case ǫ ( t ) = 0, andwe obtain no dependence on ǫ in the two-point functions.Formally, one can study the perturbative corrections to the two-point functions by us-ing the Witten diagram and computing the perturbative correction from the “interaction” R dtdzd d xc ( t ) zM | φ | . We do not know how to compute the same perturbative correc-tions from the field theory side because the expectation value of T tt alone does not seemto specify the “state” we evaluate the correlation function. Even in the time-independentKaigorodov case, we do not know the precise prescription to compute the correlationfunctions from the field theory side. Further studies are needed in this direction to revealthe nature of c ( t ) and ǫ ( t ) deformations. 13 Supergravity embedding
We have discussed the three independent universal deformations ( i.e. c ( t ), c ( t ) and ǫ ( t )) of the simplest Schr¨odinger geometry given by the metric (2.1), which is supportedby a free Proca field with the definite mass. It would be interesting to see whether suchdeformations can be embedded in the string/M-theory solutions.There are several supergravity solutions that possess the Schr¨odinger isometry with[10][11][12][14][13][15] or without supersymmetries [7][8][9]. The simplest one is to considerthe DLCQ of the AdS space. The compactification of the light-cone direction renders theisometry of the AdS space down to the Schr¨odinger group. In that case, as we havementioned in the previous section, our deformations can also be applied in the DLCQ ofthe AdS space. We, however, prefer the geometry with dt z term as in (2.1).We claim that our deformations are universal in the sense that they can be applied toany (known) solutions with the Schr¨odinger invariance embedded in the supergravity. Inthis sense, our exact time-deformations are universal and the dual field theory interpre-tations should be valid in all the field theory duals of such geometries. In other words,the existence of such exact time-dependent deformations are common features of the dualfield theories.We begin with the deformation given by c ( t ). As we observed, the deformation by c ( t ) is actually locally trivial (up to a possible change of the boundary condition). It issimply induced by the coordinate transformation of the geometry ξ → ξ + R t dsc ( s ).Since this is just a coordinate transformation, there is no obstacle to do it in any su-pergravity embeddings. Obviously we can do the same thing in the DLCQ of the AdScompactification.Now let us consider the non-trivial deformation given by c ( t ). In the AdS compact-ifications of the supergravity, such a deformation, a generalization of the Kaigorodovspace, was studied in [21][30], where the deformation by c ( t ) does not change the otherequations of motion except for the ( tt ) component of the Einstein equations which aresolved by assigning a definite power of z . Similarly, in our case, it can be shown that thedeformation does not change the equations of motion except for the ( tt ) component ofthe Einstein equation in any supergravity compactification. Again, the ( tt ) component14f the Einstein equation is solved by assigning a definite power of z . Thus, the general-ized time-dependent Kaigorodov deformation by c ( t ) gives the exact deformations of theany known supergravity solutions with the Sch¨odinger invariance. We also note that thedeformation by c ( t ) does not change the equations motion for flux with the ansatz thatis compatible with the Sch¨odinger invariance. We will explicitly see the structure belowwhen we discuss a concrete example.The deformations by c ( t ) and c ( t ) only deal with the metric. Now, we consider thedeformation given by the vector field ǫ ( t ). In the typical supergravity embedding, thevector field in the effective d + 2 dimensional compactifications comes from the flux in thesupergravity action. The effective mass term comes from the non-trivial eigenvalue forthe internal Laplacian for the flux. The details will vary with respect to the origin of thevector field, but quite generally we can find the solutions of the supergravity equationsmotion corresponding to the deformation ǫ ( t ) in the following manner.We first investigate the equation of motion for the flux. Thanks to the Schr¨odingerinvariance and our ansatz for the time-dependent flux, g tt component of the metric doesnot affect the flux equation motion at all. Furthermore, the time-dependence in the fluxdoes not affect the equations of motion, so we regard the time-dependence of the flux asif it were constant. This agrees with our feasibility to introduce an arbitrary function ǫ ( t ) in the vector field. The flux equation motion, as a consequence, determines the z dependence of the flux ansatz as in our effective field theory approach in section 2.The introduction of the time-dependence in the flux (as ǫ ( t )) will back-react to thegeometry in the Einstein equation. We note that the backreaction only affects the ( tt )component of the Einstein equation. In addition, the ( tt ) component of the Einsteinequation can be solved by introducing the ( tt ) component of the metric deformationsof order ǫ ( t ). A crucial point here is that this ( tt ) component of the metric, which isPP-wave type, will not affect all the other components of Einstein equations as well asthe flux equations of motion, so by a suitable choice of the g tt deformation, we are ableto solve all the equations motion induced by the flux deformation induced by ǫ ( t ). To make the above argument concrete, let us consider a particular M-theory compactifi-cation of the Schr¨odinger space ( d = 1) based on the Sasaki-Einstein 7-fold. The original15olution [12] is given by ds = 14 (cid:18) q dt z − dtdξz + dx + dz z (cid:19) + ds ( SE ) G = − z dt ∧ dξ ∧ dx ∧ dz + dt ∧ d (cid:16) τ z (cid:17) , (4.1)where τ is a two-form on the Sasaki-Einstein 7-fold. The flux equations of motion demand − ∗ d ∗ dτ = 24 τ , d ∗ τ = 0 , (4.2)where the Hodge star ∗ act on the internal space SE . The ( tt ) component of the Einsteinequation demands D i D i q + 40 q = − | τ | − | dτ | , (4.3)where D i is a covariant derivative on the internal space SE . All the other equationsmotions are solved by (4.1). It is clear that the first terms in G gives the effectivecosmological constant and the second term gives the effective massive vector field whosemass squared is determined by the flux equation of motion (4.2).In particular, for SE = S , the explicit form of τ was constructed in [12] by splittingthe parent CY = R into R = R × R and considering a sum of terms which are (1 , dx i onthe other. In this simplest case, q is a constant, which does not depend on the internalcoordinate, but in general it may depend on the internal coordinate.As discussed above, we can introduce the c ( t ) and c ( t ) deformation by simply re-placing the metric with ds = 14 (cid:20)(cid:0) q − c ( t ) z − c ( t ) z (cid:1) dt z − dtdξz + dx + dz z (cid:21) + ds ( SE ) . (4.4)The only equation of motion affected by the deformation is the ( tt ) component of theEinstein equation, and it determines the power of z appearing in the first term of (4.4).As discussed in the last subsection, we can explicitly see that the introduction of c ( t )and c ( t ) does not change the other equations motion.We now studies the flux deformation. G = − z dt ∧ dξ ∧ dx ∧ dz + dt ∧ d (cid:16) E ( t, z ) τ z (cid:17) . (4.5)16ince we identified the second term in G as the effective massive vector field in theSchr¨odinger geometry, the introduction of E ( t, z ) is natural. The 11-dimensional fluxequations motion d ∗ G + G ∧ G = 0 (4.6)is insensitive to the t dependence in E ( t, z ). The z dependence is uniquely fixed by E ( t, z ) = 1 + ǫ ( t ) z . (4.7)together with (4.2). Finally, the ( tt ) component of the Einstein equation is now sourcedby the new term coming from ǫ ( t ). This will be solved by introducing order ǫ ( t ) term in g tt . The modification of g tt as well as the introduction of ǫ ( t ) in the flux do not affect allthe other equations of motion so that we find the embedding of the solutions within theM-theory compactification.It is clear that the above construction can be repeated in other string/M-theory real-izations of the Schr¨odinger space-time. The only non-trivial part is to identify the effectivemassive vector field in the flux ansatz. In section 2.3, we have studied the time-dependent scale invariant and/or conformallyinvariant deformations of the Schr¨odinger invariant field theory. A similar question can beaddressed in the relativistic AdS/CFT correspondence. From the field theory viewpoint,it seems possible to deform the relativistic CFT so that only the translational invarianceis broken while preserving dilation, special conformal transformation as well as Lorentztransformation. In the Lagrangian description, for instance, we may add the interaction Z d d x ( x ) O ( x ) , (5.1)where O ( x ) is a primary scalar operator of conformal dimension d + 2. At the first order,such deformations are scale and conformal invariant as well as Lorentz invariant, but theybreak the translational invariance due to the explicit x dependence in the interaction.Are there corresponding deformations in the CFT side? Surprisingly it seems verydifficult to introduce such deformations contrary to the naive expectation from the field17heory discussions. We first note that the special conformal transformation acts as ( a =0 , , · · · , d ) δx a = 2( ǫ b x b ) x a − ( z + x b x b ) ǫ a , δz = 2( ǫ b x b ) z , (5.2)so while x /z is invariant under the dilation, it is not invariant under the special conformaltransformation. In particular, any scalar field profile like φ = f ( x /z ) is invariant underthe scale transformation, but not invariant under the special conformal transformation.Indeed, as a simple corollary, there would be no such scalar deformations possible fromthe bulk theory viewpoint.While there is no invariant scalar perturbation, there are possible vector or metricperturbations. Let us consider the vector field profile A = A µ dx µ = K (cid:18) x z (cid:19) dzz + J (cid:18) x z (cid:19) x a dx a z . (5.3)Again because of the non-invariance of x /z under the special conformal transformation,each terms in (5.3) are not invariant under the special conformal transformation whileit is obviously scale invariant. In order for the deformation to be conformally invariant, K ( y ) and J ( y ) should satisfy K ′ ( y )(1 + y ) + J ( y ) = 02 K ( y ) + J ( y )( − y ) = 0 J ′ ( y )(1 + y ) + J = 0 . (5.4)This is an overdetermined system, but it has a solution: K ( y ) = c − y yJ ( y ) = c y , (5.5)where c is an integration constant.Similarly, we can study the following metric perturbations: ds = A (cid:18) x z (cid:19) dz z + B (cid:18) x z (cid:19) dx z + C (cid:18) x z (cid:19) x a dx a dzz + D (cid:18) x z (cid:19) ( x a dx a ) z . (5.6)The invariance under the special conformal transformation demands(1 + y ) A ′ ( y ) + C ( y ) = 018 A ( y ) − B ( y ) + ( y − C ( y ) = 0(1 + y ) B ′ ( y ) = 0(1 + y ) C ′ ( y ) + C ( y ) + 2 D ( y ) = 0 C ( y ) + ( − y ) D ( y ) = 0(1 + y ) D ′ ( y ) + 2 D ( y ) = 0 . (5.7)Again, the equations look overdetermined, but there is a nontrivial solution: A ′ ( y ) = c ′ y − y ) B ′ ( y ) = 0 C ( y ) = c ′ − y (1 + y ) D ( y ) = c ′ y ) (5.8)with another integration constant c ′ .We note that we have not imposed any equations of motion yet. In this sense, the solu-tions are quite restrictive. It is clear that arbitrary actions and equations of motion do notallow such a solution, and in general, there would be no scale and conformal invariant de-formations of the relativistic conformal field theory that break the translational invariancefrom the gravity viewpoint. Again, this may be related to the fact that the representationtheory of the relativistic conformal algebra has a non-trivial involution anti-automorphismreplacing the momentum P µ with the special conformal generator K µ , so it is likely thatthe difficulty to find a scale and conformal invariant but non-translational invariant de-formations may be related to the difficulty to find a scale invariant but non-conformaldeformations of the relativistic conformal field theories. What happens if we hit the singularity of the universe? Is it the end of the universe, or willthe stringy effects remove it? How does the time begin or end? These are fundamentalquestions that should be addressed and hopefully answered in fundamental theories ofquantum gravity. If the holography is one of the most fundamental nature of the gravity,the dual field theory approach would enable us to answer the question.19he AdS/CMP correspondence might give us a novel way to probe the singularitiesof the universe from our lab experiments. Our time-dependent deformations of the non-relativistic AdS/CFT correspondence contain the PP-wave singularity. Since they areuniversal, one may expect that they are realized in, for example, the cold atoms, orunitary fermion system.In this paper, we have also clarified that the distinction between the scale invarianceand the conformal invariance is more manifest in the time-dependent non-relativistic sys-tem. Although in the unitary Poincar´e invariant field theories, or in the unitary Galileaninvariant field theories, these two concepts might be equivalent, we have found otherwisein the time-dependent background from the gravity approach. It would be interesting toconfirm our result from the field theory approach.The subtle relation between the conformal invariance and the scale invariance is afundamental question in theoretical physics that is yet to be solved. The probelm isone of the few good examples that can be shared by string theorists and the condensedmatter physicists. We hope that further collaborations will give a theoretical as well asexperimental clue to this elementary question that lies in the basic foundation of the worldsheet formulation of the string theory, quantum gravity, as well as critical phenomena incondensed matter physics.
Acknowledgements
The author would like to thank Gabriel Wong for the collaboration in the early stage ofthis work. He also thanks Ben Freivogel for the discussion of the PP-wave singularity.The work was supported in part by the National Science Foundation under Grant No.PHY05-55662 and the UC Berkeley Center for Theoretical Physics and World PremierInternational Research Center Initiative (WPI Initiative), MEXT, Japan.
A non-relativistic conformal algebra
We summarize the non-relativistic conformal algebra [16][45] in (1+2) dimension. Thehigher dimensional analogue will be obvious: i [ J, P + ] = − iP + , i [ J, P − ] = + iP − , i [ J, G + ] = − iG + , i [ J, G − ] = + iG − , [ H, G + ] = + P + , i [ H, G − ] = + P − , i [ K, P + ] = − G + , i [ K, P − ] = − G − ,i [ D, P + ] = − P + , i [ D, P − ] = − P − , i [ D, G + ] = + G + , i [ D, G − ] = + G − ,i [ H, D ] = 2
H , i [ H, K ] =
D , i [ D, K ] = 2
K , i [ P + , G − ] = 2 M . (A.1)In our notation, H is the non-relativistic Hamiltonian, P ± = P x ± iP y are the momentum, J is the U(1) angular momentum, D is the dilatation, K is the special conformal transfor-mation, and G ± = G x ± iG y are the Galilean boost generators. Moreover, M is the totalmass generator. The total mass M is related to the particle number by a proportionalfactor: M = mN = m R d d xρ .We note that the non-relativistic superconformal algebra has a grading structure withrespect to the dilatation operator D and can be triangular-decomposed as A + ⊕ A ⊕ A − , (A.2)where A + = { P − , P + , H }A = { J, M, D }A − = { G − , G + , K } . (A.3)We also notice that the non-relativistic conformal algebra has a non-trivial involutionanti-automorphism of the algebra [46] given by w ( J ) = J, w ( P ± ) = G ∓ , w ( G ± ) = P ∓ , w ( H ) = − K,w ( R ) = R, w ( D ) = − D, w ( M ) = − M, w ( K ) = − H . (A.4)This anti-automorphism is essential in the “radial” quantization of non-relativistic con-formal field theories [47].
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