Quantum gravity at a Lifshitz point

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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
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Quantum gravity at a Lifshitz point

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