Quantum gravity at a Lifshitz point

PETR HOŘAVA PHYSICAL REVIEW D 79, 084008 (2009)ture—that of a codimension-one foliation [13]. This foliationstructure F is to be viewed as a part of the topologicalstructure of M, before any notion of a Riemannian metricis introduced. The leaves of this foliation are the hypersurfacesof constant time. Coordinate transformationsadapted to the foliation are of the form~x i ¼ ~x i ðx j ;tÞ; ~t ¼ ~tðtÞ: (13)Thus, the transition functions are foliation-preserving diffeomorphisms.We will denote the group of foliationpreservingdiffeomorphisms of M by Diff F ðMÞ. In thelocal adapted coordinate system, the infinitesimal generatorsof Diff F ðMÞ are given byx i ¼ i ðt; xÞ; t ¼ fðtÞ: (14)We will simplify our presentation by further assuming thatthe spacetime foliation is topologically given byM ¼ R ; (15)with all leaves of the foliation topologically equivalent to afixed D-dimensional manifold .Differential geometry of foliations is a well-developedbranch of mathematics, and represents the proper mathematicalsetting for the class of gravity theories studiedhere. We will not review the geometric theory of foliationsin any detail here, instead referring the reader to [14–16].For example, there are two natural classes of functions thatcan be defined on a foliation: In addition to functions thatare allowed to depend on all coordinates, there is a specialclass of functions which take constant values on each leafof the foliation. We will call such functions ‘‘projectable.’’Foliations can be equipped with a Riemannian structure.A Riemannian structure compatible with our codimensiononefoliation of M consists of three objects: g ij , N i , and N,with N a projectable function; both N and N i transform asvectors under the reparametrizations of time. As pointedout above, these fields can be viewed as a decomposition ofa Riemannian metric on M into the metric g ij inducedalong the leaves, the shift variable N i , and the lapse field N.The generators of Diff F ðMÞ act on the fields viag ij ¼ @ i k g jk þ @ j k g ik þ k @ k g ij þ f _g ij ;N i ¼ @ i j N j þ j @ j N i þ _j g ij þ fN _ i þ f N_i ;N ¼ j @ j N þ fN _ þ f N: _(16)In [3], these transformation rules were derived by startingwith the action of spacetime diffeomorphisms on the relativisticmetric in the ADM decomposition, and taking thec !1 limit. We also saw in [3] that N i and N can benaturally interpreted as gauge fields associated with thetime-dependent spatial diffeomophisms and the time reparametrizations,respectively. In particular, since N is thegauge field associated with the time reparametrization fðtÞ,it appears natural to restrict it to be a projectable functionon the spacetime foliation F .If we wish instead to treat N as an arbitrary function ofspacetime, we have essentially two options. First, we canallow an arbitrary spacetime-dependent N as a backgroundfield, but integrate only over space-independent fluctuationsof N in the path integral. As the second option, wewill encounter situations in which N must be allowed to bea general function of spacetime, because it participates inan additional gauge symmetry. When that happens, we willintegrate over the fluctuations of N in the path integral. Anexample of such an extra symmetry is the invariance underanisotropic Weyl transformations discussed in Sec. II C 3below, and in Sec. 5.2 of [3].B. LagrangiansWe formally define our quantum field theory of gravityby a path integral,ZDgij DN i DN expfiSg: (17)Here Dg ij DN i DN denotes the path-integral measurewhose proper treatment involves the Faddeev-Popov gaugefixing of the gauge symmetry Diff F ðMÞ, and S is the mostgeneral action compatible with the requirements of gaugesymmetry (and further restricted by unitarity). As is oftenthe case, this path integral is interpreted as the analyticcontinuation of the theory which has been Wick rotated toimaginary time ¼ it.Our next step is to construct the action S compatible withour symmetry requirements. For simplicity, we will assumethat all global topological effects can be ignored, freelydropping all total derivative terms and not discussing possibleboundary terms in the action. This is equivalent toassuming that our space is compact and its tangentbundle topologically trivial. The refinement of our constructionwhich takes into account global topology andboundary terms is outside of the scope of the present work.1. The kinetic termThe kinetic term in the action will be given by the mostgeneral expression which is (i) quadratic in first timederivatives _g ij of the spatial metric, and (ii) invariant underthe gauge symmetries of foliation-preserving diffeomorphismsDiff F ðMÞ. The object that transforms covariantlyunder Diff F ðMÞ is not _g ij , but instead the second fundamentalformK ij ¼ 12N ð _g ij r i N j r j N i Þ: (18)This tensor measures the extrinsic curvature of the leavesof constant time in the spacetime foliation F . In terms ofK ij and its trace K g ij K ij , the kinetic term is given by084008-4

QUANTUM GRAVITY AT A LIFSHITZ POINT PHYSICAL REVIEW D 79, 084008 (2009)S K ¼ 2 Zdtd D p 2 x ffiffiffi g NðKij K ij K 2 Þ: (19)S V ¼ Z dtd D px ffiffiffi g NV½gij Š; (23)This kinetic term contains two coupling constants: and .The dimension of depends on the spatial dimension D:Since the dimension of the volume element is½dtd D xŠ¼ D z; (20)and each time derivative contributes ½@ t Š¼z, the scalingdimension of is½Š ¼ z D : (21)2As intended, this coupling will be dimensionless in 3 þ 1spacetime dimensions if z ¼ 3.The presence of an additional, dimensionless coupling reflects the fact that each of the two terms in (19) isseparately invariant under Diff F ðMÞ. In other words, therequirement of Diff F ðMÞ symmetry allows the generalizedDe Witt ‘‘metric on the space of metrics’’G ijk‘ ¼ 1 2 ðgik g j‘ þ g i‘ g jk Þ g ij g k‘ (22)to contain a free parameter . It is this generalized De Wittmetric that defines the form quadratic in K ij which appearsin the kinetic term (see [3]).In general relativity, the requirement of invariance underall spacetime diffeomophisms forces ¼ 1. In our theorywith Diff F ðMÞ gauge invariance, represents a dynamicalcoupling constant, susceptible to quantum corrections.It is interesting to note that the kinetic term S K isuniversal, and independent of both the desired value of zand the dimension of spacetime. The only place where thevalue of z shows up in S K is in the scaling dimension of theintegration measure (20), which in turn determines thedimension (21)of. The main difference between theorieswith different z will be in the pieces of the action which areindependent of time derivatives.2. The potentialThe logic of effective field theory suggests that thecomplete action should contain all terms compatible withthe imposed symmetries, which are of dimension equal toor less than the dimension of the kinetic term, ½K ij K ij Š¼2z. In addition to S K , which contains the two independentterms of second order in the time derivatives of the metric,the general action will also contain terms that are independentof time derivatives. Since our framework is fundamentallynonrelativistic, we will refer to all terms in theaction which are independent of the time derivatives (butdo depend on spatial derivatives) simply as the ‘‘potential.’’There is a simple way to construct potential terms invariantunder our gauge symmetry Diff F ðMÞ: Startingwith any scalar function V½g ij Š which depends only onthe metric and its spatial derivatives, the following potentialterm,will be invariant under Diff F ðMÞ.Throughout this paper, our strategy is to focus first onthe potential terms of the same dimension as ½K ij K ij Š,atfirst ignoring all possible relevant terms of lower dimensionsin V. This is equivalent to focusing first on the highenergylimit, where such highest-dimension terms dominate.Once the high-energy behavior of the theory is understood,one can restore the relevant terms, and study theflows of the theory away from the UV fixed point that suchrelevant operators induce in the infrared.With our choice of D ¼ 3 and z ¼ 3, there are manyexamples of terms in V of the same dimension as thekinetic term in (19). Some such terms are quadratic incurvature,r k R ij r k R ij ; r k R ij r i R jk ; RR; R ij R ij ;(24)they will not only add interactions but also modify thepropagator. Other terms, such asR 3 ; R i j Rj k Rk i ; RR ijR ij ; (25)are cubic in curvature, and therefore represent pure interactingterms. Some of the terms of the correct dimensionare related by the Bianchi identity and other symmetries ofthe Riemann tensor, or differ only up to a total derivative.Additional constraints on the possible values of the couplingswill likely follow from the requirements of stabilityand unitarity of the quantum theory. However, the list ofindependent operators appears to be prohibitively large,implying a proliferation of couplings which makes explicitcalculations rather impractical.C. UV theory with detailed balanceIn order to reduce the number of independent couplingconstants, we will impose an additional symmetry on thetheory. The reason for this restriction is purely pragmatic,to limit the proliferation of independent couplings mentionedin the previous paragraph. The way in which thisrestriction will be implemented, however, is very reminiscentof methods used in nonequilibrium critical phenomenaand quantum critical systems. As a result, it isnatural to suspect that there might also be conceptualreasons behind restricting the general class of classicaltheories to conform to this framework in systems withgravity as well.We will require the potential term to be of a special form,S V ¼ 28Zdtd D px ffiffiffi g NE ij G ijk‘ E k‘ ; (26)and will further demand that E ij itself follow from a variationalprinciple,084008-5

PETR HOŘAVA PHYSICAL REVIEW D 79, 084008 (2009)pffiffiffig E ij ¼ W½g k‘Š(27)g ijfor some action W. The two copies of E ij in (26) arecontracted by G ijk‘ , the inverse of the De Witt metric(22). Loosely borrowing terminology from nonequilibriumdynamics, we will say that theories whose potential is ofthe form (26) with (27) for some W satisfy the ‘‘detailedbalance condition.’’In the context of condensed matter, the virtue of thedetailed balance condition is in the simplification of therenormalization properties. Systems which satisfy the detailedbalance condition with some D-dimensional actionW typically exhibit a simpler quantum behavior than ageneric theory in D þ 1 dimensions. Their renormalizationcan be reduced to the simpler renormalization of the associatedtheory described by W, followed by one additionalstep—the renormalization of the relative couplings betweenthe kinetic and potential terms in S. Examples ofthis phenomenon include scalar fields [17] or Yang-Millsgauge theories [9,18]. Investigating the precise circumstancesunder which this ‘‘quantum inheritance principle’’holds for gravity systems will be important for understandingthe quantum properties of gravity models with nonrelativisticvalues of z.Since we are primarily interested in theories which arespatially isotropic, W must be the action of a relativistictheory in Euclidean signature. (Obvious generalizations totheories with additional spatial anisotropies are clearlypossible, but will not be pursued in this paper.) In [3], atheory of gravity in D þ 1 dimensions satisfying the detailedbalance condition was constructed, with W theEinstein-Hilbert actionW ¼ 1 2 WZd D px ffiffiffi g ðR 2W Þ: (28)The potential S V of this theory takes the formS V ¼ 28 4 WZdtd D x ffiffiffipg NR ij 12 Rgij þ W g ij G ijk‘ R k‘ 12 Rgk‘ þ W g k‘ : (29)At short distances, the curvature term in W dominates over W , and the resulting potential S V is quadratic in thecurvature tensor: The theory exhibits anisotropic scalingwith z ¼ 2 in the UV. Turning on W in W leads to lowerdimensionterms in S V which dominate at long distances,and the theory undergoes a classical flow to z ¼ 1 in theIR. The anisotropic scaling in the UV shifts the criticaldimension of this theory, which is now renormalizable bypower counting in 2 þ 1 dimensions. In dimensions higherthan 2 þ 1, the theory with potential (29) is merely a lowenergyeffective field theory, and can be expected to breakdown at the scale set by the dimensionful coupling W .Here we are interested in constructing a theory whichsatisfies detailed balance, and exhibits the short-distancescaling with z ¼ 3 leading to power-counting renormalizabilityin 3 þ 1 dimensions. Therefore, E ij must be of thirdorder in spatial derivatives. As it turns out, there is a uniquecandidate for such an object: the Cotton tensorC ij ¼ " ik‘ r k ðR j 1‘ 4 Rj ‘Þ: (30)This tensor not only exhibits all the required symmetries, italso follows from a variational principle.1. Properties of the Cotton tensorThe Cotton tensor enjoys several symmetry propertieswhich may not be immediately obvious from its definitionin (30):(i) It is symmetric and traceless,C ij ¼ C ji ; g ij C ij ¼ 0: (31)(ii) It is transverse (or covariantly conserved),r i C ij ¼ 0: (32)(iii) It is conformal, with conformal weight 5=2. Moreprecisely, under local spatial Weyl transformationsg ij ! expf2ðxÞgg ij ; (33)it transforms asC ij ! expf 5ðxÞgC ij ; (34)with no terms containing derivatives of ðxÞ.The Cotton tensor plays an important role in geometry.Recall that in dimensions D>3, the property of conformalflatness of a Riemannian metric is equivalent to the vanishingof the Weyl tensor C ijk‘ , defined as the completelytraceless part of the Riemann tensor:1C ijk‘ ¼ R ijk‘D 2 ðg ikR j‘ g i‘ R jk g jk R i‘1þ g j‘ R ik ÞþðD 1ÞðD 2Þ ðg ikg j‘ g i‘ g jk ÞR:(35)In D ¼ 3, however, the Weyl tensor vanishes identically,and another object has to take over the role in the criterionof conformal flatness of 3-manifolds. This object is theCotton tensor, of third order in spatial derivatives.The Cotton tensor also plays an important role in physics.In the initial value problem of the Hamiltonian formulationof general relativity, it is natural to ask what set ofinitial conditions can be freely specified for the metric andits canonical momenta, without violating the constraintpart of Einstein’s equations. It was shown by York [19–21] that the Cotton tensor plays a central role in answeringthis question. The correct initial conditions are set by084008-6

QUANTUM GRAVITY AT A LIFSHITZ POINT PHYSICAL REVIEW D 79, 084008 (2009)specifying the values of two tensors with the symmetries ofthe Cotton tensor: One related to the initial value for theconformal structure of the spatial metric, and the otherspecifying the initial value of the conjugate momenta.For this reason, C ij is often referred to as the ‘‘Cotton-York tensor’’ in the physics literature.Lastly, the Cotton tensor follows from a variationalprinciple, with actionW ¼ 1 Zw 2 ! 3 ð Þ: (36)Here w 2 is a dimensionless coupling, and! 3 ð Þ¼Trð ^ d þ 2 3^ ^ Þ " ijk ð m i‘ @ ‘j km þ 2 3 n i‘ ‘ jm m kn Þd3 x (37)is the gravitational Chern-Simons term, with theChristoffel symbolsijktreated as known functionals ofthe metric g ij , and not as independent variables. Thevariation of (36) with respect to g ij yields the vanishingof the Cotton tensor as the equations of motion.Without any loss of generality, we will assume that thecoupling w 2 is positive; its sign can be changed by flippingthe orientation of the 3-manifold . Unlike in Chern-Simons gauge theories with a compact gauge group, thecoupling constant of Chern-Simons gravity in 2 þ 1 dimensionsis not quantized, as a result of the absence oflarge gauge transformations. In our framework, however,we are only interested in the action of a theory in threedimensions in ‘‘imaginary time,’’ and require that thisEuclidean action be real. This is to be contrasted with theconventional interpretation of the three-dimensional theory,which involves analytic continuation to real time in2 þ 1 dimensions, and imposes slightly different realityconditions on the action.2. z ¼ 3 gravity with detailed balanceHaving reviewed some of the properties of the Cottontensor, we can now write down the full action of our z ¼ 3gravity theory in 3 þ 1 dimensions:S ¼ Z dtd 3 x ffiffiffi p 2g N 2 ðK ijK ij K 2 2 Þ2w 4 C ijC ij¼ Z dtd 3 x ffiffiffi p 2g N 2 ðK ijK ijK 2 Þ 22w 4 r i R jk r i R jk r i R jk r j R ik 18 r iRr i R:(38)As a result of the uniqueness of the Cotton tensor, theaction given in (38) describes the most general z ¼ 3gravity satisfying the detailed balance condition, modulothe possible addition of relevant terms, which will bediscussed in Sec. II E.We can demonstrate that after the Wick rotation toimaginary time, this action can be written—up to a totalderivative—as a sum of squares,S ¼ i Z dd 3 x ffiffiffi p 2g N 2 ðK ijK ij þ K 2 Þþ 22w 4 C ijC ij¼ 2i Z dd 3 x ffiffiffi p 1g N K ij2w 2 C ij 1 G ijk‘ K k‘2w 2 C k‘ ; (39)First, C ij G ijk‘ C k‘ ¼ C ij C ij because C ij is traceless. As tothe cross terms K ij G ijk‘ C k‘ , they can be written as a totalderivative,1 Zdd 3 px ffiffiffi g NKij G ijk‘ Cw 2 k‘¼ 1 Zdd 3 p2w 2 x ffiffiffi g ð _gij r i N j r j N i ÞC ij¼ Z dd 3 px ffiffiffi Wg _g ij þ 1 g ij w 2 r iðN j C ij Þ¼ Z dd x 3 L _ þ 1 w 2 @ pffiffiffiið g Nj C ij Þ ;where we used the transverse property (32)ofC ij , and L isthe Lagrangian density of the action W in (36).Introducing an auxiliary field B ij , it is convenient torewrite the imaginary-time action asS ¼ 2i Z dd 3 x ffiffiffi p 1g N B ij K ij2w 2 C ijB ij G ijk‘ B k‘ : (40)This form of the action, with all terms at least linear in theauxiliary field B ij and with the linear term proportional to agradient flow equation, is symptomatic of theories satisfyingthe detailed balance condition in the context ofcondensed-matter systems, in particular, in the theory ofquantum and dynamical critical phenomena [4,5], stochasticquantization [22,23], and nonequilibrium statisticalmechanics [24].In that condensed-matter context, the property of detailedbalance often has one interesting implication. If aquantum critical system in D þ 1 dimensions satisfies detailedbalance with some W in D dimensions, the partitionfunction of the theory described by W yields a naturalsolution of the Schrödinger equation of the theory in D þ1 dimensions, which plays the role of a candidate groundstatewave function. Similarly, in nonequilibrium statisticalmechanics and dynamical critical phenomena, the correspondingstatement is essentially the Wick rotation of thiscorrespondence to imaginary time: The partition functionof the D-dimensional theory defined by W represents anequilibrium state solution of the dynamical theory withdetailed balance in D þ 1 dimensions.084008-7

QUANTUM GRAVITY AT A LIFSHITZ POINT PHYSICAL REVIEW D 79, 084008 (2009)H ij ¼ ~H ij þ 1 @ i @ j2 ij@ 2 H: (53)D. At the free-field fixed pointThe action (38) of the z ¼ 3 theory with detailed balancecontains three dimensionless coupling constants: , , andw. However, only one of them, w, controls the strength ofinteractions. The noninteracting limit corresponds to sendingw ! 0, while keeping and the ratio ¼ w(47)This choice of variables diagonalizes the equations ofmotion in our gauge. Since the kinetic term is universal,its analysis in the z ¼ 3 theory is identical to that presentedfor z ¼ 2 in Sec. 4.5 of [3]. In our gauge and in terms of thenew variables, the kinetic term takes the formfixed. This limit yields a two-parameter family of free-fieldfixed points, parametrized by and .In preparation for the study of the full interacting theory,it is useful to first investigate the properties of this family offree-field fixed points. The linearization of the z ¼ 3 theoryis performed in exactly the same way as in [3] for theanalogous case of the z ¼ 2 gravity and we will thereforebe relatively brief, referring the reader to [3] for furtherdetails.We expand the theory in small fluctuations h ij , n, and n iaround the flat background,g ij ij þ wh ij ; N 1 þ wn; N i wn i :(48)The reference background is a solution of the equations ofmotion of the z ¼ 3 theory (38). Keeping only quadraticterms in the action, n drops out from the theory. A naturalgauge choice isn i ¼ 0: (49)This fixes most of the Diff F ðMÞ gauge symmetry, leavingtime-independent spatial diffeomorphisms DiffðÞ unfixed.The residual DiffðÞ gauge symmetry can be convenientlyfixed by setting@ i h ij @ j h ¼ 0; (50)where h h ii . Imposing this condition at some fixed timeslice t ¼ t 0 effectively fixes the residual DiffðÞ invariance.The Gauss constraint@ i_ h ij@ j_ h ¼ 0 (51)(which follows from varying the original action with respectto n i ) then ensures that (50) stays valid at all times.In order to diagonalize the linearized equations of motionand read off the dispersion relation of the propagatingmodes implied by our gauge choice (49) and (50), it isconvenient to first redefine the variables by introducingH ij h ij ij h; (52)the gauge condition (50) implies that H ij is transverse. Wethen decompose the transverse tensor H ij into its transversetraceless part ~H ij and its trace H,S K 1 Z dtd 32 2 x ~H _ij_~H ij þ 1 H:2ð1 3Þ _ 2 (54)It would appear that the dependence of the kinetic term ofH on can be absorbed into a rescaling of H, but wechoose not to do so, because it would obscure the geometricorigin of H in the full nonlinear theory.On the other hand, the potential term of the z ¼ 3 theoryreduces toS V 28Zdtd 3 x ~H ij ð@ 2 Þ 3 ~H ij : (55)Because of the conformal properties of the Cotton tensor,the potential term in the Gaussian approximation dependsonly on ~H ij and not on H.As pointed out in [3], the kinetic term (54) indicates thattwo values of play a special role. At ¼ 1=3, the theorybecomes compatible with the local anisotropic Weyl invariancediscussed in Sec. II C 3 above. At that value of ,the scalar mode H is a gauge artifact. The kinetic term forH also appears singular at ¼ 1. As explained in [3], thishappens because at this special value of , the linearizedtheory exhibits an extra gauge invariance, which can beused to eliminate physical excitations of H as well.The transverse traceless tensor ~H ij contains two propagatingphysical polarizations. These gravitons satisfy anonrelativistic gapless dispersion relation,! 2 ¼ 44 ðk2 Þ 3 : (56)For values of outside of the two special values 1 and1=3, the scalar mode H will represent a physical degree offreedom, with its linearized equation of motion givensimply by €H ¼ 0. When the theory is deformed by relevantoperators, the equation of motion for H will contain termswith spatial derivatives up to fourth order, which is notenough to yield a propagator with good ultraviolet properties.It appears that, in order to make the theory powercountingrenormalizable at generic values of not equal to1or1=3, either the scalar mode would have to be eliminatedby an extra gauge symmetry, or super-renormalizableterms which give short-distance spatial dynamics to thescalar mode need to be added to the potential. We willbriefly return to this point in Sec. III B.084008-9

PETR HOŘAVA PHYSICAL REVIEW D 79, 084008 (2009)E. Relevant deformations and the infrared flow to z ¼ 1So far we have concentrated on terms of the highestdimensionterms in S. These terms will dominate the shortdistancedynamics. At long distances, relevant deformationsby operators of lower dimensions will become important,in addition to the renormalization-group (RG)flows of the dimensionless couplings.One could relax the condition of detailed balance, andsimply ask that the action S in 3 þ 1 dimensions be ageneral combination of all marginal and relevant terms.The action of the theory would then take the formS ¼ Z dtd 3 px ffiffiffi gX½O J Š¼6 J O J þ Z dtd 3 px ffiffiffi gX½O A Š

QUANTUM GRAVITY AT A LIFSHITZ POINT PHYSICAL REVIEW D 79, 084008 (2009)Newton constant is given byG N ¼and the effective cosmological constant232c ; (65) ¼ 3 2 W: (66)It is intriguing that the effective speed of light c, theeffective Newton constant G N , and the effective cosmologicalconstant of the low-energy theory all emergefrom the relevant deformations of the deeply nonrelativisticz ¼ 3 theory which dominates at short distances.In theories satisfying the detailed balance condition, thequantum properties of the D þ 1 dimensional theory areusually closely related to the quantum properties of theassociated theory in D dimensions, with action W. Itisinteresting that in our case of 3 þ 1 dimensional gravitytheory with detailed balance, both the Newton constant andthe cosmological constant originate from the couplings inthe action of topologically massive gravity in threeEuclidean dimensions, a theory with excellent ultravioletproperties.2. Soft violations of the detailed balance conditionThere is another possibility that leads to a broaderspectrum of relevant deformations of the z ¼ 3 theory,without completely abandoning the simplifications impliedby the detailed balance condition. Starting with the z ¼ 3theory at short distances, we can add relevant operatorsdirectly to the short-distance action S given in (38),S ! S þ Z dtd 3 px ffiffiffi g ð M 6 þ 4 R þÞ; (67)with M and arbitrary couplings of dimension 1, and‘‘’’ denote other relevant terms with more than twospatial derivatives of the metric.This step will break the detailed balance condition, butonly softly, by relevant operators of lower dimension thanthose appearing in the action at short distances as definedin (38). In the UV, the theory still satisfies detailed balance.At long distances, the theory described by (67) again flowsto z ¼ 1.III. OTHER DIMENSIONS AND VALUES OF zEven though the main focus of the present paper is onthe theory of gravity with z ¼ 3 in 3 þ 1 spacetime dimensions,the ideas are applicable in a broader context.One application of the z ¼ 2 gravity in 2 þ 1 dimensions,as a candidate membrane world-volume theory, was discussedin [3]. Here we take at least a brief look at a list ofother interesting values of z and spacetime dimensions.A. Gravity with z ¼ 4 in 4 þ 1 dimensionsPower-counting renormalizability in 4 þ 1 dimensionsrequires z ¼ 4. Theories with z ¼ 4 satisfying the detailedbalance condition in 4 þ 1 dimensions can be constructedfrom Euclidean gravity actions W quadratic in curvature,familiar from the study of higher-derivative theories in 3 þ1 dimensions. (See, e.g., [2] for a review of higherderivativegravity and supergravity.) As in the case of z ¼3, we begin with first listing all terms of highest order inspatial derivatives, as these are expected to dominate atshort distances, near the hypothetical z ¼ 4 fixed point thatwe are attempting to construct. The four-dimensionalEuclidean action quadratic in curvature is given byW ¼ Z d 4 px ffiffiffi g ðCijk‘ C ijk‘ þ R 2 Þ: (68)This theory has two independent dimensionless couplings and . Modulo topological invariants, this is the mostgeneral four-derivative action for relativistic gravity in fourdimensions. There is no independent R ij R ij term in theaction, becauseZd 4 px ffiffiffi g ðRijk‘ R ijk‘ 4R ij R ij þ R 2 Þ (69)is a topological invariant (measuring the Euler number ofthe spatial slices ), as a consequence of the Gauss-Bonnettheorem in four dimensions.We use W to construct the potential S V of quantumgravity with z ¼ 4 in 4 þ 1 dimensions. The high-energylimit of this theory will again be described byS ¼ S K S V¼ 1 Zdtd 4 x ffiffiffi p 4g N2 2 ðK ijK ij 24K 2 ÞW WGg ijk‘ ; (70)ij g k‘with W now given by (68). is dimensionless, as are thetwo couplings and inherited from W. This action canbe modified by relevant operators, of dimension

PETR HOŘAVA PHYSICAL REVIEW D 79, 084008 (2009)lieved to be asymptotically free [32–34]. As we mentionedin the Introduction, the asymptotic freedom of (68) wouldseem to make this theory an excellent candidate for solvingthe problem of quantum gravity in 3 þ 1 dimensions, wereit not for one persistent flaw: After the Wick rotation to 3 þ1 dimensions, the spectrum of physical states containsghosts which violate unitarity in perturbation theory.Our construction of z ¼ 4 theory in 4 þ 1 dimensionsbenefits from the asymptotic freedom of the fourdimensionalhigher-curvature theory (68), but avoids thepitfall of its perturbative nonunitarity. Indeed, we are onlyinterested in the four-dimensional action W in theEuclidean signature, in order to construct the 4 þ 1 dimensionalaction (70).The only remaining coupling-constant renormalizationin the high-energy limit of the theory in 4 þ 1 dimensionsis the renormalization of . However, is not an independentcoupling associated with interactions; instead, it survivesin the noninteracting limit, and parametrizes a familyof free-field fixed point as and are sent to zero. In thisrespect, the quantum behavior of this theory would be verysimilar to the behavior in quantum critical Yang-Millsstudied in [9], which inherits asymptotic freedom fromrelativistic Yang-Mills in four dimensions.Setting ¼ 0 in (68) and ¼ 1=4 in (70) would lead toa theory which exhibits an additional gauge invariance,acting on the fields asg ij ! expf2ðt; xÞgg ij ; N ! expf4ðt; xÞgN;N i ! expf2ðt; xÞgN i :(71)These are the local anisotropic Weyl transformations withz ¼ 4.B. z ¼ 4 gravity in 3 þ 1 dimensionsIn three dimensions, the action of Euclidean gravityquadratic in the curvature tensor isW ¼ 1 MZd 3 px ffiffiffi g ðRij R ij þ R 2 Þ: (72)As in four dimensions, there are again only two independentterms in W, but for a different reason: When D ¼ 3,the Riemann tensor is determined in terms of the Riccitensor, and the Weyl tensor vanishes identically. The twocouplings M and M= are now dimensionful, of dimension1. In power counting, this makes the theory described by(72) super-renormalizable. When we use W to generate thepotential term S V for z ¼ 4 gravity in 3 þ 1 dimensions,we consequently end up with a theory whose action againhas the form (70), now with W given by (72) and in 3 þ 1dimensions, where it is super-renormalizable by powercounting. As in all the previous examples with variousvalues of z, relevant deformations flow the theory to z ¼1 in the infrared.Such super-renormalizable terms can also be added toour z ¼ 3 theory of gravity described in (38). These termswill give spatial dynamics to the conformal factor of thespatial metric, improving the short-distance properties ofthe propagator for the scalar mode H of the metric, restoringpower-counting renormalizability in the case when H ispresent as a physical field.C. The case of z ¼ 0: Ultralocal gravityIn the Hamiltonian formulation of general relativity, theHamiltonian is given by a sum of constraints,H ¼ Z d D xðNH ? þ N i H i Þ: (73)Notably, the algebra of the Hamiltonian constraintsH ? ðxÞ and H i ðxÞ in general relativity is not a true Liealgebra—in particular, the constraints do not form thenaively expected algebra of spacetime diffeomorphisms.Instead, the structure ‘‘constants’’ of the commutator ofH ? ðxÞ with H ? ðyÞ are field dependent, because theycontain the components of the spatial metric.In [35], an alternative theory of gravity was proposed, inwhich the constraints do form a Lie algebra. In this theory,the commutators of H i with themselves and with H ? arethe same as in general relativity, but the problematic fielddependentcommutator of H ? ðxÞ with H ? ðyÞ is simplyreplaced by zero. This symmetry can be viewed either as acontraction of the symmetries formed by the Hamiltonianconstraints of general relativity, or as a contraction of thealgebra of infinitesimal spacetime diffeomorphisms. Thecontracted symmetry algebra respects a dimension-onefoliation of spacetime by a congruence of timelike curves.This congruence can be used to identify the points of spaceat different times; as a result, the spacetime in this theory ofgravity carries a preferred structure of absolute space.The theory of gravity that realizes this symmetry structureis known as the ‘‘ultralocal theory’’ of gravity. It isinteresting to note that ultralocal gravity fits naturally intoour framework of gravity models with anisotropic scalingand nontrivial dynamical exponents z Þ 1. As shown in[35], the required symmetries force the action of the ultralocaltheory to be of the same form S ¼ S K S V as thetheories considered here, with the potential term containingonly the cosmological constant,S V ¼ Z dtd D px ffiffiffi g ; (74)and no curvature-dependent terms. There is a clear way tointerpret (74) in our framework of gravities with anisotropicscaling: The value of z can be read off as one-half ofthe number of derivatives appearing in S V . This is equivalentto declaring (74) to be of the same dimension as thekinetic term S K . Either way, this approach suggests that theultralocal theory corresponds formally to the limiting caseof z ! 0.084008-12

1N ð@ 2g ij r i N j r j N i Þ2 ffiffiffi Wp Gg ijk‘ ¼ 0; (75)g k‘the full equations of motion are automatically satisfied.While the full equations of motion are of second order intime derivatives and of order 2z in spatial derivatives, thesimpler equation (75) has its degree reduced by half. (Thisargument is reminiscent of the BPS condition in supersymmetrictheories.) A simple class of solutions to (75) canQUANTUM GRAVITY AT A LIFSHITZ POINT PHYSICAL REVIEW D 79, 084008 (2009)Historically, the ultralocal theory of gravity has beenstudied for at least two additional reasons, besides thecontext of [35]:now be obtained by setting N ¼ 1, N i ¼ 0, and takingg ij ¼ g ij ðxÞ to be an arbitrary (-independent) solutionof the equations of motion,(i) Ultralocal gravity was proposed by Isham in [36], inWan attempt to introduce a new formal expansion¼ 0;gparameter into general relativity. In [36], the suggestedij(76)expansion parameter was the coefficient infront of the scalar curvature term in S V , equal to onein the potential of general relativity and set equal tozero in the ultralocal theory.of the D-dimensional theory whose action is W. Clearly,this solution can be trivially continued back to real time,and represents a real static solution of the full theory.In particular, let us assume that the Euclidean action W(ii) Ultralocal theory is relevant for early universe cosmologyin general relativity, because it captures the situation is rather generic, and does not pose a very strongis such that it has the Euclidean AdS D as a solution. Thisdynamics of Friedmann-Robertson-Walker solutionsin the so-called ‘‘velocity dominated’’ earlyrestriction on W. With this assumption, the D þ 1 dimensionaltheory will have a classical solution which is thestages after the big bang, as was first shown by direct product of the time dimension and AdS D ,Belinsky, Khalatnikhov, and Lifshitz [37,38].Unfortunately, the z ! 0 limit is rather singular, and theN ¼ 1; N i ¼ 0;program outlined in (i) was never very successful. As tog ij dx i dx j ¼ d 2 þ sinh 2 d 2 D 1(ii), the embedding of ultralocal gravity into our framework(77)of gravity with anisotropic scaling raises the possi-The boundary of this solution is S D 1 R. The isometries:bility of interpreting the cosmological evolution as a flow,from z Þ 1 in the early universe to z ¼ 1 observed now.It is remarkable that, even though the action of ultralocalof the Euclidean AdS D induce conformal symmetries in theboundary. In addition, there is one more bulk isometry,given by time translations. Thus, the full symmetries aretheory is not invariant under all spacetime diffeomorphisms,the theory exhibits ‘‘general covariance’’ [35,39]:SOðD; 1ÞR: (78)In particular, the number of local symmetry generators perspacetime point is D þ 1, i.e., the same as in generalrelativity.These symmetries suggest that such a gravity theory in thebulk can serve as a possible holographic dual of dynamicalfield theories which are already critical in the static limit.Such problems are often encountered in the theory ofD. Bulk-boundary correspondence in gravity at a dynamical critical phenomena. Starting with a universalityLifshitz pointclass of a static critical system in D 1 spatial dimensions,the time-dependent dynamics of the system in DThe availability of gravity models with nontrivial valuesdimensions can also exhibit criticality, with the characteristicproperty of ‘‘critical slowing down’’ of time-of the dynamical critical exponent z can enhance thespectrum of examples of dualities between gravity in thedependent correlation functions. One given static universalityclass can belong to several different dynamical uni-bulk and field theory on the boundary. This could beparticularly relevant for understanding gravity duals ofversality classes. In particular, one universal characteristicnonrelativistic CFTs.of the dynamics is given by the critical exponent z.After the Wick rotation of the z ¼ 3 theory in 3 þ 1If we study such a dynamical critical system on R D ,dimensions to imaginary time , the action of this theoryit will exhibit the anisotropic scaling symmetry given bywas rewritten in a simple form (40) with the use of anauxiliary field B ij (10), with i ¼ 1; ...D 1. Another possibility is to put. The same rewriting applies to a muchthis system on S D 1 R, with the spatial slices of thebroader class of gravity models which satisfy detailedfoliation given by S D 1 of a fixed radius. On such abalance with some D-dimensional action W, such as thefoliation, the scale symmetry (10) is absent, since it wouldz ¼ 4 models discussed above. Using this formalism, wechange the radius of the sphere. However, the system stillcan find a large class of classical solutions of such theories,exhibits the symmetries of conformal transformations ofsimply by noting that if the following equation holds,S D 1 and time translations. Thus, the conformal symmetriesleft unbroken by the foliation are precisely the bulkisometries (78) of the AdS D R solution of gravity theorywith anisotropic scaling.Following [40,41], a nonrelativistic version of the AdS/CFT correspondence has indeed received a lot of attentionrecently. The focus in this area has been primarily on theCFTs with nontrivial values of z which exhibit conventionalrelativistic gravity duals. It is natural to broaden thisframework, and free the gravity side of the duality of the084008-13

QUANTUM GRAVITY AT A LIFSHITZ POINT PHYSICAL REVIEW D 79, 084008 (2009)(October 2008). I wish to thank the organizers for theirhospitality, and the participants for stimulating discussions.This work has been supported by NSF GrantNo. PHY-0555662, DOE Grant No. DE-AC03-76SF00098, and the Berkeley Center for TheoreticalPhysics.[1] S. Weinberg, in General Relativity. An Einstein CentenarySurvey, edited by S. W. Hawking and W. Israel(Cambridge University Press, Cambridge, England, 1980).[2] E. S. Fradkin and A. A. Tseytlin, Phys. Rep. 119, 233(1985).[3] P. Hořava, arXiv:0812.4287, [J. High Energy Phys. (to bepublished)].[4] P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49,435 (1977).[5] S.-K. Ma, Modern Theory of Critical Phenomena(Benjamin, New York, 1976).[6] S. Sachdev, Quantum Phase Transitions (CambridgeUniversity Press, Cambridge, England, 1999).[7] E. M. Lifshitz, Zh. Eksp. Teor. Fiz. 11, 255 (1941); 11, 269(1941).[8] R. M. Hornreich, M. Luban, and S. Shtrikman, Phys. Rev.Lett. 35, 1678 (1975).[9] P. Hořava, arXiv:0811.2217.[10] E. Ardonne, P. Fendley, and E. Fradkin, Ann. Phys. (N.Y.)310, 493 (2004).[11] P. M. Chaikin and T. C. Lubensky, Principles ofCondensed Matter Physics (Cambridge University Press,Cambridge, England, 1995).[12] The idea that Lorentz invariance might be an emergentsymmetry has a long history, going back at least to thepioneering paper [46].[13] In differential geometry, a codimension-q foliation F on ad-dimensional manifold M is defined as M equippedwith an atlas of coordinate systems ðy a ;x i Þ a ¼ 1; ...q,i ¼ 1; ...d q, such that the transition functions take therestricted form ð~y a ; ~x i Þ¼ð~y a ðy b Þ; ~x i ðy b ;x j ÞÞ. The generaltheory of foliations is reviewed e.g. in [14–16].[14] H. B. Lawson, Jr., Bull. Am. Math. Soc. 80, 369 (1974).[15] C. Godbillon, Feuilletages (Birkhauser, Boston, 1991).[16] I. Moerdijk and J. Mrčun, Introduction to Foliations andLie Groupoids (Cambridge University Press, Cambridge,England, 2003).[17] J. Zinn-Justin, Nucl. Phys. B275, 135 (1986).[18] J. Zinn-Justin and D. Zwanziger, Nucl. Phys. B295, 297(1988).[19] J. W. York, Phys. Rev. Lett. 26, 1656 (1971).[20] J. W. York, Phys. Rev. Lett. 28, 1082 (1972).[21] J. W. York, J. Math. Phys. (N.Y.) 13, 125 (1972).[22] G. Parisi and Y.-S. Wu, Sci. Sin. 24, 483 (1981).[23] M. Namiki, Stochastic Quantization (Springer, New York,1992).[24] M. Le Bellac, F. Mortessagne, and G. Batrouni,Equilibrium and Non-Equilibrium Statistical Thermodynamics(Cambridge University Press, Cambridge,England, 2004).[25] E. Witten, arXiv:gr-qc/0306083.[26] In contrast, for some theories with detailed balance,0 expf W=2g does represent a physical normalizableground-state wave function. Examples include the Lifshitzscalar theory (as discussed, for example, in [10]), and thequantum critical Yang-Mills with z ¼ 2 in 4 þ 1 dimensions[9].[27] S. Deser, R. Jackiw, and S. Templeton, Phys. Rev. Lett. 48,975 (1982).[28] S. Deser, R. Jackiw, and S. Templeton, Ann. Phys. (N.Y.)140, 372 (1982).[29] S. Deser and Z. Yang, Classical Quantum Gravity 7, 1603(1990).[30] B. Keszthelyi and G. Kleppe, Phys. Lett. B 281, 33 (1992).[31] The main difference between (58) and topologically massivegravity stems from the fact that here we are onlyinterested in the Euclidean-signature version of (58), withthe real action W. In topologically massive gravity, theEuclidean action W is interpreted as the Wick rotation ofthe real action from the physical signature 2 þ 1, leadingto a slightly different reality condition on W, with w 2purely imaginary. There has been a recent resurgence ofinterest in topological massive gravity, initiated by [47];see also [48].[32] K. S. Stelle, Phys. Rev. D 16, 953 (1977).[33] E. S. Fradkin and A. A. Tseytlin, Nucl. Phys. B201, 469(1982).[34] E. S. Fradkin and A. A. Tseytlin, Nucl. Phys. B203, 157(1982).[35] C. Teitelboim, in General Relativity and Gravitation,edited by A. Held (Plenum Press, New York, 1980), Vol. 1.[36] C. J. Isham, Proc. R. Soc. A 351, 209 (1976).[37] V. A. Belinsky, I. M. Khalatnikov, and E. M. Lifshitz, Adv.Phys. 19, 525 (1970).[38] V. A. Belinsky, I. M. Khalatnikov, and E. Lifshitz, Adv.Phys. 31, 639 (1982).[39] M. Henneaux, Bull. Soc. Math. Belg. 31, 47 (1979).[40] D. T. Son, Phys. Rev. D 78, 046003 (2008).[41] K. Balasubramanian and J. McGreevy, Phys. Rev. Lett.101, 061601 (2008).[42] C. J. Isham, arXiv:gr-qc/9210011.[43] K. V. Kuchař, in Winnipeg 1991, General Relativity andRelativistic Astrophysics, 1991.[44] This picture might change if we allow sufficiently singularfoliations, for example, if such singularities turn out to berequired for a consistent summation over spacetime topologies.[45] P. W. Anderson, Basic Notions of Condensed MatterPhysics (Addison-Wesley, Reading, MA, 1984).[46] S. Chadha and H. B. Nielsen, Nucl. Phys. B217, 125(1983).[47] W. Li, W. Song, and A. Strominger, J. High Energy Phys.04 (2008) 082.[48] E. Witten, arXiv:0706.3359.084008-15