Conformal Dilaton Gravity: Classical Noninvariance Gives Rise To Quantum Invariance
IIFT-UAM/CSIC-15-112, FTUAM-15-34, FTI-UCM/161
Conformal Dilaton Gravity: Classical Noninvariance Gives Rise To QuantumInvariance
Enrique ´Alvarez,
1, 2, ∗ Sergio Gonz´alez-Mart´ın,
1, 2, † and Carmelo P. Mart´ın ‡ Instituto de F´ısica Te´orica, IFT-UAM/CSIC, Universidad Aut´onoma, 28049 Madrid, Spain Departamento de F´ısica Te´orica, Universidad Aut´onoma de Madrid, 28049 Madrid, Spain Universidad Complutense de Madrid (UCM), Departamento de F´ısica Te´orica I,Facultad de Ciencias F´ısicas, Av. Complutense S/N (Ciudad Univ.), 28040 Madrid, Spain
When quantizing Conformal Dilaton Gravity there is a conformal anomaly which starts at twoloop order. This anomaly stems from evanescent operators on the divergent parts of the effectiveaction. The general form of the finite counterterm which is necessary in order to insure cancellationof the Weyl anomaly to every order in perturbation theory has been determined using only conformalinvariance . Those finite counterterms do not have any inverse power of any mass scale in front ofthem (precisely because of conformal invariance) and then they are not negligible in the low energydeep infrared limit. The general form of the ensuing modifications to the scalar field equation ofmotion has been determined and some physical consequences extracted.
INTRODUCTION
It is well known [ ? ] that an anomalous term in a Wardidentity only qualifies as a true anomaly when there isno local counterterm that can be added to the action insuch a way that it cancels the putative anomalous pieceof the identity. Put it in another way, there is a consis-tency condition, the Wess-Zumino consistency condition[2], which is a reflection on the gauge algebra acting onthe effective action. If an anomalous variation of the ac-tion under a gauge transformation with parameters Λ a appears δWδ Λ a ( x ) ≡ G a ( x ) (1)(Although this formalism has been developed with non-abelian gauge anomalies in mind, it can easily be adaptedto conformal anomalies as well [3]). Then the consistencyconditions read δG a ( x ) δ Λ b ( y ) − δG a ( y ) δ Λ b ( x ) = f abc δ ( x − y ) G c ( x ) (2)True anomalies are then solutions of the consistencyequations which are not themselves variations, that is,that there is no local lagrangian such that G a ( x ) = δCδ Λ a (3)Defining the contraction of the anomaly with the ghosts G ≡ c a G a (4)(the superindex as a reminder of the ghost number) theconsistency relationships can be written in a sophisti-cated way as s G = dα (5) The appearance of a total differential on the second mem-ber is due to the fact that it is only necessary for it tovanish when integrated. The demand that the anomalyis not trivial reads in BRST language, G (cid:54) = sG + dβ (6)In a recent paper [4] (whose notation will be followedhere) we have examined an apparently quasi-trivial the-ory, namely what we have dubbed conformal dilatongravity (CDG). In it the Weyl parameter is upgraded to aSt¨uckelberg field to compensate the Weyl transformationof the Einstein-Hilbert lagrangian.What we have found rests on the mild assumption thatwhat was true at one loop (namely that the on-shell coun-terterm is the Weyl transform of the Einstein-Hilbertcounterterm) remains true to two loops, so that the twoloop counterterm will also be the Weyl transform of theGoroff-Sagnotti [5] one. With this assumption we founda two-loop Weyl anomaly in our trivial theory. Thisanomaly is however trivial in the sense that it can beeliminated by a counterterm, which is not strictly local,involving logarithms of the physical fields.This is a very surprising result. It means that Weyl trans-formations are much less trivial than previously thought.It questions, in particular, the full equivalence of Einsteinframe and Jordan frame.Of course it is possible to take the less rigid point ofview that the counterterms are admissible in spite ofthem being nonlocal. Once this is taken for granted thenone quickly realizes that these counterterms are not sup-pressed by any mass scale (precisely because of confor-mal invariance). They could then legitimately also be a r X i v : . [ h e p - t h ] M a r included in the classical theory. Precisely our aim inthe present paper is to follow the consequences of thisviewpoint and speculate on some of the properties of ahypothetical conformal theory of gravity to all orders inperturbation theory.Conformal invariance for us is exactly the same as Weylinvariance under ˜ g µν ≡ Ω ( x ) g µν ( x )˜ ψ = Ω − λ ψ ψ (7)Where λ ψ is by definition the conformal weight of thematter field ψ . To be specific , we will be interested inthe four-dimensional action of CDG, that is. S CDG = − (cid:90) d n x (cid:112) | g | (cid:18) − φ R − ∇ µ φ ∇ µ φ + g φ (cid:19) (8)which is classically Weyl invariant in a somewhat tauto-logical way, provided the conformal weight of the field φ is λ φ = 1. This graviscalar field φ is none other thanthe Weyl parameter promoted to the range of a physicalfield. This is sometimes called a St¨uckelberg field or elsea compensating field . For a mathematician this is simplya group averaging of sorts.Conformal physics is not very intuitive in that it does notsingle out any scale. For example, we all are used to thethe idea that quantum gravity effects should decouple atenergies much smaller than Planck mass, M p ≡ √ πG ,so that they can be safely ignored in particle physics ex-cept in exotic circumstances.This fails to be true in a conformal theory, as has beenexplicitly shown in [4] (essentially because there is noany preferred scale in them). Conformal theories are alsobizarre in other aspects [6, 7], notably in the absence ofthe usual concept of particle .The existence of a conformal invariant fundamental the-ory including the gravitational interaction is however oneof the Holy Grails of theoretical physics. Such a hypo-thetical theory would be extremely interesting, were itonly as a theoretical model [8]. We are here still in theinitial steps of this quest.When quantizing the theory in background field gauge,as has been done in [4], there is a four dimensional Wardidentity stemming from conformal invariance, namely D W (cid:2) ¯ g, ¯ φ (cid:3) ≡ δδ Ω( x ) W (cid:2) ¯ g, ¯ φ (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) Ω=1 == (cid:18) − g µν δδ ¯ g µν + ¯ φ δδ ¯ φ (cid:19) W [¯ g, ¯ φ ] = 0 (9)where the on shell free energy W [¯ g, ¯ φ ] is the logarithmof the on shell partition function given by the functionalpath integral W [¯ g, ¯ φ ] ≡ − log Z [¯ g, ¯ φ ] (10)This is precisely the Ward identity we claim to be anoma-lous to two-loop order. IMPLEMENTATION OF THE WARD IDENTITYIN PERTURBATION THEORY THROUGHCOUNTERTERMS
It was shown in [4] that modulo wave-function renormal-izations –which are physically irrelevant– the one-loopUV divergence of Conformal Dilaton Gravity is the ap-propriate Weyl transform of the corresponding one-loopUV divergence of General Relativity found by in [9], andthus it vanishes. The appropriate Weyl transformationin question reads ¯ g µν → M p ¯ φ ¯ g µν (11)Furnished with this result and the fact that the classi-cal action of Conformal Dilaton Gravity is obtained fromthat of General Relativity by applying the previous Weyltransformation, one is led to assume that modulo wave-function renormalizations the two-loop UV divergenceof Conformal Dilaton Gravity can be obtained from thetwo-loop counterterm [5] of General Relativity by apply-ing to the latter the Weyl transformation we have justmentioned. Hence, the resulting two-loop counterterm–see [4], for further details– reads W L =2 ∞ [ ¯ φ, ¯ g ] = 1 n − π ) (cid:90) (cid:112) | g | d n x ¯ φ − W ( n )6 [¯ g ] , (12)in the n -dimensional space of Dimensional Regulariza-tion. W ( n )6 [¯ g ], which has conformal weight λ = 6, isdefined in terms of the Weyl tensor W ( n ) α α α α in n dimensions as follows W (4)6 [¯ g ] = W ( n ) α α α α W ( n ) α α α α W ( n ) α α α α (13)The previous results make it natural to speculate thatthe L-loop divergence in Conformal Dilaton Gravity willbe of the form W L ∞ [ ¯ φ, ¯ g ] = 1 n − (cid:90) (cid:112) | g | d n x ¯ φ − L +2 (cid:88) j g j P jL +1 [¯ g ](14)The constants g j are unknown but calculable coefficients,and P j ( L +1) j = 1 . . . N stand for the set of purely gravi-tational terms with conformal weight λ = 2 L +2 (like thetrace of the product of L + 1 Weyl tensors). These termshave mass dimension 2 L + 2, so that the full integrandis dimensionless. An example is the scalar made out of L + 1 Weyl tensors. The complete set of conformal ten-sors is not explicitly known, but this fact is not essentialin our argument.Let us recall that D be given by D = − g µν δδ ¯ g µν + ¯ φ δδ ¯ φ (15)Then, in the n dimensional space of dimensional regular-ization D d ( vol ) ¯ φ − L +2 (cid:88) j g j P jL +1 = − ( n − d ( vol ) ¯ φ − L +2 (cid:88) j g j P jL +1 [¯ g ] (16)This means that the integrand of W L ∞ [ ¯ φ, ¯ g ] is an evanes-cent operator which yields a putative anomaly A L [ ¯ φ, ¯ g ] ≡ − (cid:90) d ( vol ) ¯ φ − L +2 (cid:88) g j P jL +1 [¯ g ] (17)This anomaly-to-be can actually be cancelled by a finite counterterm∆ W L [ ¯ φ, ¯ g ] = (cid:90) d ( vol ) ¯ φ − L +2 log ¯ φ (cid:88) g j P jL +1 [¯ g ] (18)(just because D log ¯ φ = 1). It is to be remarked that theintegrand of the above expression is again dimensionless,so that there is no room for any dimensionful couplingconstant in front.The full modified action of CDG will be of the form W CDG [ ¯ φ, ¯ g ] = S CDG [ ¯ φ, ¯ g ] + ∞ (cid:88) L =1 (cid:0) W LR [ ¯ φ, ¯ g ] + ∆ W L [ ¯ φ, ¯ g ] (cid:1) (19)This modified action obeys D W CDG [ ¯ φ, ¯ g ] = 0 (20) so it qualifies as a conformally invariant one.The finite part of this action (which we propose to recon-sider as a classical action of sorts) reads W class CDG [ ¯ φ, ¯ g ] ≡ − (cid:90) d ( vol ) (cid:20) −
112 ¯ φ ¯ R − ∇ µ ¯ φ ∇ µ ¯ φ + λφ + ∞ (cid:88) L =1 φ − (2 L − log ¯ φ (cid:88) j g j P jL +1 [¯ g µν ] (cid:21) (21)and does obey instead D W class CDG [ ¯ φ, ¯ g ] = (cid:88) L A L [ ¯ φ, ¯ g ] (22) CLASSICAL EFFECTS OF THE FINITEQUANTUM COUNTERTERMS.
The counterterms needed to cancel the putative anoma-lies enjoy two main properties. First of all, they are finite.Besides, and more importantly, there is no small cou-pling constant (id est, no κ , because there is no κ in theoriginal lagrangian) in front of them; there is no reasonwhy they should be negligible compared with the clas-sical Lagrangian. This justifies a consideration of thoseterms already at the classical level. The modified scalarEM reads (suppressing hats on the fields from now on) (cid:50) φ − R φ + 4 λφ −− ∞ (cid:88) L =1 (1 − (2 L −
2) log φ ) φ − L +1 (cid:88) j g j P jL +1 [¯ g µν ] = 0(23)Under a Weyl rescaling the conformal wave operator (cid:50) c ≡ (cid:50) − R behaves as (cid:50) c → Ω − (cid:50) c Ω (24)It is then possible to write the Weyl transform of thescalar EMΩ − (cid:18) (cid:50) − R (cid:19) φ + 4Ω − λφ −− Ω − ∞ (cid:88) L =1 (cid:18) − (2 L −
2) log φ Ω (cid:19) φ − L +1 (cid:88) j g j P jL +1 [¯ g µν ] = 0(25) A one-dimensional toy model
As the general form of the gravitational terms is notknown, a one-dimensional toy model that captures somefeatures of the general setting can be studied. In order todo that, let us assume that (cid:80) j g j P jL +1 [¯ g µν ] ≡ C is just aconstant independent of L . Then the sums over the looporder can be exactly done ∞ (cid:88) L =1 φ − L +1 = φφ − ∞ (cid:88) L =1 (2 L − φ − L +1 = 2 φ ( φ − (27)which are convergent only when (cid:12)(cid:12)(cid:12)(cid:12) φ (cid:12)(cid:12)(cid:12)(cid:12) < φ = 0 is still a solution.Specific analysis are necessary in other situations.In this case the one-dimensional model then would read d f ( x ) dx − Af ( x ) + Bf ( x ) −− C (cid:18) f ( x ) f ( x ) − − f ( x )( f ( x ) − log f ( x ) (cid:19) = 0 (29)Consider, besides, A, B as arbitrary constants (where A is proportional to the Ricci scalar and B to the self-coupling g ).As the non-invariance of the theory comes from the log-arithm, the (toy version of the) conformal case can berecovered in the case C = 0. In this case the equationcan be solved easily by transforming f ( x ) → λf ( x ), thenit reads λ d f ( x ) dx − Aλf ( x ) + λ Bf ( x ) = 0 (30)Setting λ B = 2 (which can be done as long as B (cid:54) = 0)and dividing by λ we find now d f ( x ) dx − Af ( x ) + 2 f ( x ) = 0 (31)finally, calling m = 2 − A the previous equation now reads d f ( x ) dx − (2 − m ) f ( x ) + 2 f ( x ) = 0 (32)which is solved by the Jacobi elliptic function dn( x | m ). ( x ) Figure 1This function is defined in terms of the elliptic integral u = (cid:90) φ dθ (cid:112) − m sin θ (33)where m is called the parameter . Then the function weare dealing with is defined as sn ( u | m ) = sin φ .In order to see explicitly the properties of this solution,it is useful to define the quarter-periods K and iK (cid:48) , inthe following way.Let m (the complementary parameter ) be such that m + m = 1, then K = (cid:90) π dθ (cid:112) − m sin θ = π ∞ (cid:88) n =0 (cid:18) (2 n )!2 n ( n !) (cid:19) m n (34) iK (cid:48) = i (cid:90) π dθ (cid:112) − m sin θ = i π ∞ (cid:88) n =0 (cid:18) (2 n )!2 n ( n !) (cid:19) (1 − m ) n (35)In terms of these, the function has periods 2 K , 4 K +4 iK (cid:48) and 4 iK (cid:48) . As long as we have a real solution here, weare only concerned by 2 K which can be expressed (to thelowest order) in terms of A as2 K = 3 π − π A (36)thus its period is smaller as bigger is the curvature.We consider now the non-conformal case (i.e C (cid:54) = 0) givenby (30). The first thing to notice is that due to thepresence of the logarithm f ( x ) = 0 is no longer a solution.In fact, this is expected as we are working in the brokenphase ( φ (cid:54) = 0). This is the main difference between theconformal and non-conformal case which is also expectedin a more complicated model.Concerning the constants, neither the sign of A or (i.e.the curvature) or C change the shape of the function.Depending on the initial conditions there are two possi-bilities1. If C > f ( x ) = 1, numericintegration yields finite answer even in this case) ( x ) Figure 22. In any other case 3 the solution does not oscillateand goes to zero. ( x ) Figure 3: f (cid:48) (0) > ( x ) Figure 4: f (cid:48) (0) < f ( x ) = ± (cid:114) AB are solutions of the equa-tions of motion. However, this does not happens in thenon-conformal model, where there are not any constantsolutions.Although our toy model is one-dimensional, it is likelythat it embodies some of the characteristics of the fullfledged four-dimensional situation. THE SYMMETRIC PHASE OF CDG
It is not obvious how the symmetric phase of CDG (whichcorresponds in the background field language to ¯ φ = 0)should be understood. The first problem is that thereis no propagator to damp the gravitational fluctuationsin the loop approximation. This has been emphasized inparticular by ’t Hooft [10]. Nevertheless there are someobservations that can be made on general grounds. Thefull partition function can be written as Z [¯ g µν ] ≡ (cid:82) D φ D h µν e − (cid:82) d ( vol ) (cid:20) φ ( ¯ ∇ − R ) φ + O ( φ h,h )+ O ( φ h ) (cid:21) (37)Please note that the integral over the ghosts and auxiliaryfields are implicitly included in the measure D h αβ D φ .Also note that the gauge-fixing conditions that suit ouranalysis should contain no monomial linear in the fields h µν and φ . This way the gauge-fixing terms will containthree or more quantum fields.The one loop contribution only involves the quantumfluctuations of the dilaton. Its divergent part can beeasily computed: Z ( L =1) ∞ [¯ g µν ] = exp (cid:18) − π n − (cid:90) d ( vol ) (cid:18) (cid:16) R µνρσ −− R µν + (cid:50) R (cid:17)(cid:17)(cid:17) == exp (cid:18) − π n − (cid:90) d ( vol ) 1180 (cid:18) W − E + (cid:50) R (cid:19)(cid:19) (38)We have represented by W the square of Weyl tensor W ≡ R µνρσ − R µν + 13 R (39)and by E the four-dimensional Pfaffian (which upon in-tegration yields Euler’s characteristic up to a constant) E ≡ R µνρσ − R µν + R (40)We are not able to perform the loop integrals over thegravitational fluctuations because there is no propaga-tor for the gravitational field. Assuming (and this is anexplicit hypothesis) that this integral makes sense (forexample by discretizing the system) then the partitionfunction can be defined as given by the expression Z ∞ = ∞ (cid:88) L =1 Z L ∞ [¯ g µν ] (41)which is the sum of all higher loop divergent pieces Z L ∞ [¯ g µν ] each of which is a conformal invariant functionalof ¯ g µν . There is only one of those in four dimensions,namely Z ∞ [¯ g µν ] ≡ e − g (cid:82) d ( vol ) W (42)All loop contributions are of the same form, so that wecan represent by g the coefficient of the whole sum. Thisprocedure is formal in more that one way; there is noreason in particular to expect the loop expansion to con-verge or even to be asymptotic. CONCLUSIONS.
The general form of the the finite counterterms which isnecessary to insure cancellation of the Weyl anomaly toevery order in perturbation theory has been determined–under a sensible assumption, using only conformal in-variance. They involve logarithms of the physical scalar,so that they are not local terms sensu stricto .We found it interesting to examine the most broadminded hypothesis in which they are indeed acceptable counterterms. Then two facts immediately come to ourattention. First of all, and in spite of their being loopeffects (and so carrying powers of ¯ h , so to speak) thosefinite counterterms do not have any inverse power of anymass scale in front of them (precisely because of confor-mal invariance) and then they are not negligible in thelow energy deep infrared limit. This might be identifiedin some sense with the classical limit .It is then of interest to consider the classical effects ofthose terms. The most important such is that the statusof the trivial conformal invariant solution φ = 0 (43)changes slightly. This solution is the only compatiblewith the symmetric phase of conformal symmetry.When the space-time is of Petrov type O (that is, Weylflat) then the symmetric configuration is still a solution.When the Weyl tensor does not vanish, then the analysisis more involved.Consider the oversimplified situation in which the purelygravitational contribution is L-independent (cid:88) j g j P jL +1 [¯ g µν ] ≡ G ( x ) (44)Then the scalar equation of motion reduces to (cid:18) (cid:50) − R + g φ (cid:19) φ ++ (cid:18) φ ( φ − log φ − φφ − (cid:19) G ( x ) = 0 (45)In that sense, φ = 0 is still a solution. Specific analysisare necessary in other situations. In order get an ideahow to do that a one-dimensional toy model has beenstudied. A comparison was made between results withouttaking into account the counterterms (this a conformallyinvariant situation which can be exactly solved in termsof Jacobian elliptic functions) and results including termsin our toy model that mimick the said counterterms.It is to be stressed that in spite of the above, those are not the classical equations of motion to be used in the contextof the background field gauge technique to express physi-cal results on shell, Those correspond to the ¯ h = 0 sectoronly; that is without including the corrections studied inthe present paper. The fact that the solution correspond-ing to the symmetric phase is not always admissible inthe present setting has to be interpreted as the fact thatin those cases the counterterms are necessarily singularin the symmetric phase.Finally we conjecture that the form of the symmetricphase of conformal dilaton gravity ought to be propor-tional to the Weyl squared theory. ACKNOWLEDGMENTS
We are grateful for helpful discussions with ManuelAsorey and Mario Herrero-Valea. We also acknowledgeuseful email exchange with Michael Duff as well as dis-cussions with Roman Jackiw and So Young Pi. Part ofthis work was done while E.A. was on the
Aspen In-stitute of Physics and on the Lawrence Berkeley Labo-ratory. We have been partially supported by the Euro-pean Union FP7 ITN INVISIBLES (Marie Curie Actions,PITN- GA-2011- 289442)and (HPRN-CT-200-00148) aswell as by FPA2012-31880 (Spain), FPA2014-54154-P,COST action MP1405 (Quantum Structure of Space-time) and S2009ESP-1473 (CA Madrid). The authorsacknowledge the support of the Spanish MINECO
Cen-tro de Excelencia Severo Ochoa
Programme under grantSEV-2012-0249.
Conformal invariants
Let us summarize here some known facts about conformal(Weyl) invariants. The Schouten tensor is defined as A αβ ≡ n − (cid:18) R αβ − n − Rg αβ (cid:19) (46)It is invariant under rigid Weyl rescaling, that is, it trans-forms under Ω ≡ e σ as˜ A αβ = A αβ − σ αβ −
12 ( ∇ σ ) g αβ (47)The Weyl tensor reads W αβµν ≡ R αβµν +( A βµ g αν + A αν g βµ − A βν g αµ − A αµ g βν )(48)It transforms as a conformal tensor of weight λ = − W αβµν ≡ e σ W αβµν (49)Its square has got scale dimension λ = 2˜ W αβµν ˜ W αβµν = e − σ W αβµν W αβµν (50)in such a way that | g | /n W (51) is pointwise invariant (but behaves as a true scalar infour dimensions only).The Weyl tensor vanishes identically in low dimension n = 2 and n = 3. An space with n ≥ W = 0.The Cotton tensor reads C αβγ ≡ ∇ α A βγ − ∇ β A αγ (52)is a conformal invariant of scaling dimension λ = 0 in n = 3 dimensions (and only there).It is traceless in any dimension. g µν C αµν = ∇ α A − g µν ∇ µ A αν = 12( n − ∇ α R − n − ∇ α R = 0 (53)The Bach tensor reads B µν ≡ ∇ ρ C ρµν + A αβ W αµνβ = ∇ α ∇ δ W αµνδ − R αδ W αµνδ (54)(this fact stems from the second Bianchi identity).The Bach tensor is transverse ∇ β B αβ = 0 (55)and it inherits its tracelessness from the same propertyfor Weyl and Cotton tensors g µν B µν = 0 (56)It is a conformal invariant of scaling dimension λ = 1 infour dimensions only.The variation of the four-dimensional Weyl-squared ac-tion yields precisely the Bach tensor δ (cid:90) | W | d ( vol ) = (cid:90) B µν δg µν d ( vol ) (57)It is of course well known that that there is an extensionof the Laplacian (cid:50) c ≡ (cid:50) − n − n − R (58)that is such that ˜ (cid:50) c = Ω − n +22 (cid:50) c Ω n − (59)On the other hand, the operator (which is a total deriva-tive) (cid:50) ≡ √− g (cid:50) (60)transforms as ˜ (cid:50) = ∂ µ (cid:0) Ω − g µν ∂ ν (cid:1) (61)The quartic Paneitz operator in arbitrary dimension Q ( g ) ≡ (cid:50) + ∇ ν (cid:18) − n − R µν + n − n + 82( n − n − R g µν (cid:19) ∂ µ (62)is conformal invariant in the same sense as the conformalLaplacian is; that is, under˜ g αβ ≡ Ω g αβ (63)transforms as ˜ Q = Ω − n +42 Q Ω n − (64)In four dimensions this gives∆ P ≡ √− g (cid:18) ∆ + 2 ∇ µ (cid:18) R µν − R g µν (cid:19) ∇ ν (cid:19) (65)The Fefferman-Graham (FG) obstruction tensor O µν [11]is a trace-free symmetric two-tensor which has got scalingdimension λ = n −
22 (66)and is divergenceless ∇ λ O µλ = 0 (67)and vanishes for conformally Einstein metrics. It is theDirichlet obstruction to the existence of a formal powerseries solution for the ambient metric associated to agiven conformal structure. For example, the equation R αβ [ g + ] + ng + µν = O (cid:0) x n − log x (cid:1) (68)admits a solution of the form g + µν = 1 x (cid:0) dx + g xµν (cid:1) (69)where g xµν = h xµν + r xµν x n log x (70)Then n c n r µν = O µν (71)where c n ≡ n − ( n/ − n − n ≥ O µν = ∆ n/ − (cid:0) ∆ P µν − ∇ ν ∇ µ P λλ (cid:1) + lots == 13 − n ∆ n/ − ∇ ρ ∇ σ W σµνρ + lots (73)There is also an analogue of the four-dimensional Weylaction [13], namely, the Q-curvature [14], which is nota point wise conformal invariant, but yields neverthelessa conformal invariant under integration on a compactmanifold; in fact [15] (cid:82) Q is a combination of the Eulercharacteristic and the integral of a point wise conformalinvariant.It is related to the conformally invariant n-th power ofthe Laplacian P n and under Weyl ˜ g = e σ g , e nσ ˜ Q = Q + P n σ P n is self-adjoint and annihilated con-stants, the preceding result follows.Consider an asymptotic expansion of the volumeVol g + ( (cid:15) < x < (cid:15) ) = c (cid:15) − n + c (cid:15) − n +2 + . . . + c n − (cid:15) − +L log 1 (cid:15) + O (1)(75)The logarithmic term is related to the integral of the Q-curvature (cid:90) Qdv = ( − n/ n ( n − c n L (76) δ (cid:90) M Q (cid:112) | g | d n x = ( − n n − (cid:90) M (cid:112) | g | d n x O µν δg µν (77) ∗ [email protected] † [email protected] ‡ carmelop@fis.ucm.es[1] S. L. Adler, “Anomalies to all orders,” hep-th/0405040.[2] B. Zumino, “Chiral Anomalies And Differential Geom-etry: Lectures Given At Les Houches, August 1983,”In *Treiman, S.b. ( Ed.) Et Al.: Current Algebra andAnomalies*, 361-391 and Lawrence Berkeley Lab. - LBL-16747 (83,REC.OCT.) 46p[3] N. Boulanger, “General solutions of the Wess-Zuminoconsistency condition for the Weyl anomalies,” JHEP (2007) 069 doi:10.1088/1126-6708/2007/07/069[4] E. Alvarez, M. Herrero-Valea and C. P. Martin, “Confor-mal and non Conformal Dilaton Gravity,” JHEP 1410(2014) 115 [arXiv:1404.0806 [hep-th]]. [5] M. H. Goroff and A. Sagnotti, “The Ultraviolet Behaviorof Einstein Gravity,” Nucl. Phys. B 266 (1986) 709.[6] H. Georgi, “Unparticle physics,” Phys. Rev. Lett. 98(2007) 221601 [hep-ph/0703260].[7] S. Weinberg, “Minimal fields of canonical dimension-ality are free,” Phys. Rev. D 86 (2012) 105015[arXiv:1210.3864 [hep-th]].[8] E. S. Fradkin and A. A. Tseytlin, “Conformal Super-gravity,” Phys. Rept. (1985) 233. doi:10.1016/0370-1573(85)90138-3[9] G. ’t Hooft and M. J. G. Veltman, “One loop divergen-cies in the theory of gravitation,” Annales Poincare Phys.Theor. A 20 (1974) 69.[10] G. ’t. Hooft, “The Conformal Constraint in CanonicalQuantum Gravity,” arXiv:1011.0061 [gr-qc].[11] C. Fefferman and C. R. Graham, “The ambient metric,”arXiv:0710.0919 [math.DG]. C. Fefferman and K. Hirachi, Ambient metric construc-tion of Q-curvature in conformal and CR geometries ,arXiv:math/0303184 Hep :: HepNames :: Institutions ::Conferences :: Jobs :: Experiments :: Journals :: Help[12] C. Robin Graham and Kengo Hirachi,
The ambient ob-struction tensor and Q-curvature , arXiv:math/0405068C. Robin Graham,Ralph Jenne,Lionel J. Mason andGeorge A.J. Sparling,