No Conformal Anomaly in Unimodular Gravity
PPreprint typeset in JHEP style - HYPER VERSION
IFT-UAM/CSIC-12-116;FTUAM-12-117
No Conformal Anomaly in Unimodular
Gravity.
Enrique ´Alvarez (cid:63) • and Mario Herrero-Valea • (cid:63) Physics Department. Theory Unit CERN 1211 Gen`eve 23, Switzerland • Instituto de F´ısica Te´orica UAM/CSIC and Departamento de F´ısica Te´oricaUniversidad Aut´onoma de Madrid, E-28049–Madrid, SpainE-mail: [email protected] [email protected]
Abstract:
The conformal invariance of unimodular gravity survives quantum cor-rections, even in the presence of conformal matter. Unimodular gravity can actuallybe understood as a certain truncation of the full Einstein-Hilbert theory, where inthe Einstein frame the metric tensor enjoys unit determinant. Our result is com-patible with the idea that the corresponding restriction in the functional integral isconsistent as well. a r X i v : . [ h e p - t h ] M a r ontents
1. Introduction. 12. A more general scalar-tensor theory 33. ’t Hooft’s approach: effective action after integrating the confor-mal factor. 54. Conformal invariance 75. One-loop computation 96. Conclusions. 13A. Weyl covariant curvature 14B. Conformal anomaly 16 – 1 – . Introduction.
A radical approach towards explaining why (the zero mode of) the vacuum energyseems to violate the equivalence principle (the active cosmological constant problem )is just to eliminate the direct coupling in the action between the potential energyand the gravitational field [1]. This leads to consider unimodular theories, where themetric tensor is constrained to be unimodular g E ≡ (cid:12)(cid:12) det g Eµν (cid:12)(cid:12) = 1 (1.1)in the Einstein frame. This equality only stands in those reference frames obtainedfrom the Einstein one by an area preserving diffeomorphism. Those are by definitionthe ones that enjoy unit jacobian, and g is a singlet under them.We shall represent the absolute value of the determinant of the metric tensor in anarbitrary frame as g instead of | g | in order to simplify the corresponding formu-las. We work in arbitrary dimension n in order to be able to employ dimensionalregularization as needed. The simplest nontrivial such unimodular action [1] reads S U ≡ − M n − (cid:90) d n x R E + S matt = − M n − (cid:90) d n x g n (cid:18) R + ( n − n − n g µν ∇ µ g ∇ ν gg (cid:19) + S matt (1.2)where the n-dimensional Planck mass is related to the n-dimensional Newton constantthrough M n − ≡ πG . (1.3)and S matt is the matter contribution to the action.This theory is conformally (Weyl) invariant under˜ g µν = Ω ( x ) g µν ( x ) (1.4)(the Einstein metric is inert under those) as well as under area preserving (trans-verse) diffeomorphisms, id est, those that enjoy unit jacobian, thereby preservingthe Lebesgue measure. We shall speak always of conformal invariance in the abovesense.The aim of this paper is to explore whether this gauge symmetry is anomalous orsurvives when one loop quantum corrections are taken into account. The result wehave found is that, given the fact that this theory can be thought as a partial gauge– 1 –xed sector of a conformal upgrading of General Relativity, there is no conformalanomaly for unimodular gravity, even when conformal matter is included.Other interesting viewpoints on the cosmological constant from the point of view ofunimodular gravity are [11] [12]. In this last reference Smolin suggested the absenceof conformal anomaly for related theories.We will proceed as follows. First, we will define a more general scalar-tensor theoryby introducing a spurion field σ . This theory is diffeomorphism as well as conformalinvariant and unimodular gravity is no more than a partial gauge fixed sector ofit. This happens to be, also, the same theory that t’Hooft proposed [10] in order tosolve some special issues of black hole complementarity. Consequently, in section 3 wewill explore t’Hooft’s approach in order to obtain the divergent part of the one-loopeffective action of such theory. In section 4, however, we will show how the scalar-tensor action can be written in a more useful and manifestly conformal invariantform and we will use it to easily compute the one-loop gravitational countertermin section 5. Our result is not only consistent with t’Hoofts computations but alsoshows in a very clear way how the anomaly vanishes.– 2 – . A more general scalar-tensor theory It is technically quite complicated to gauge fixing a theory invariant under area pre-serving (transverse) diffeomorphisms only, because the theory is reducible , id est, thecorresponding gauge parameters are not independent. This usually demands a hugeghost sector [9]. In order to avoid these, presumably physically irrelevant intricacies, it proves convenient to introduce a new theory which enjoys diffeomorphism invari-ance and that is such that the unimodular theory is a partial gauge fixing of it. Thisis easily achieved by introducing a compensating field , C ( x ), defined so that g ( x ) C ≡ e n √ ( n − n − σ ( x ) (2.1)transforms as a true scalar. This diffeomorphism invariant theory is still Weyl in-variant provided that e n √ ( n − n − ˜ σ ( x ) = Ω n e n √ ( n − n − σ ( x ) . (2.2)Id est, the composite exponential field has got conformal weight − n . In general, aconformal tensor of conformal weight − λ behaves under conformal transformationsas δT = λ T (2.3)At the linear level, Ω( x ) ≡ ω ( x ), the spurion field σ transforms with the gaugeparameter like a true Goldstone boson does. δσ = (cid:112) ( n − n − ω (2.4)This result conveys the fact that the spurion is nothing else than the dilaton . Theunimodular theory of our interest is recovered when the partial unitary gauge C = 1 (2.5)is chosen. The residual gauge symmetries are then the area preserving diffeomor-phisms as well as Weyl invariance.Let us be quite explicit on this point. Under a diffeomorphism δx µ = ξ µ (2.6)the compensating field behaves as δC = ∂ λ ξ λ C − ξ λ ∂ λ C (2.7)– 3 –hereas it is a singlet under conformal transformations. Under finite transformations C (cid:48) ( x (cid:48) ) = C ( x ) . det ∂x (cid:48) ∂x . (2.8)To reach the gauge C = 1 starting from a non-vanishing C (cid:54) = 0 it is then enough tochoose det (cid:18) ∂x (cid:48) ∂x (cid:19) = 1 C ( x ) (2.9)The gauge C = 1 means that e n √ ( n − n − σ ( x ) = g ( x ) . (2.10)The new action is then written as S ≡ (cid:90) d n x √ g (cid:26) e − (cid:113) n − n − σ (cid:20) − M n − ( R + g µν ∇ µ σ ∇ ν σ ) + 12 g µν ∇ µ Φ ∇ ν Φ (cid:21) − e − n √ ( n − n − σ V (Φ) (cid:27) . (2.11)This action is conformal invariant as well as diffeomorphism invariant with all fieldstransforming as indicated above.The spurion σ ( x ) corresponds to a conformal rescaling of the metric, and behavesas a ghost (its kinetic energy term has got the wrong sign). It is standard, sincethe work [6] to perform its functional integral over imaginary values of the field. Weshall keep an open mind on this issue for the time being.The canonically normalized field is φ g ≡ − √ M n − (cid:114) n − n − e − (cid:113) n − n − σ . (2.12)The old gauge C = 1 now reads φ g + 2 M n − (cid:114) n − n − g − n − n = 0 . (2.13)In terms of φ g the action is S ST = (cid:90) d n x √ g (cid:26) − n − n − R φ g − g µν ∇ µ φ g ∇ ν φ g + n − n − M n − φ g
12 ( ∇ Φ) − ( − nn − (cid:18) n − n − (cid:19) nn − M n φ nn − g V (Φ) (cid:27) . (2.14)– 4 –t is instructive to study in detail how the equations of motion (EM) of the scalar-tensor theory reduce to the unimodular ones in the unitary gauge. Indeed, δS U δg µν = δS ST δg µν + δS ST δφ g δφ g δg µν (cid:12)(cid:12)(cid:12)(cid:12) φ g = − M n − (cid:113) n − n − g − n − n . (2.15)This conveys the fact that the scalar-tensor EM imply the unimodular EM, whereasthe converse assertion is untrue: the unimodular EM do not imply the scalar-tensorones. The unimodular theory is a subsector of the more general scalar tensor theory,as stated in the title of the present section.In this scalar-tensor theory it is possible to go to the Einstein frame through g µν = 2 n − M (cid:18) n − n − (cid:19) n − φ − n − g g Eµν . (2.16)This metric g Eµν is a conformal singlet; that is, it remains invariant under Weyltransformations. In the gauge C = 1 we go round the whole circle and the metric inEinstein’s frame is unimodular, { C = 1 } ⇒ g E = 1 . (2.17)It is also possible here to define, again following [6][10], i log φ g ≡ η. (2.18)In the following we will just work with the gravitational sector, since as we will showlater, inclussion of matter will not change any of our conclusions. Thus, in the restof this text we will forget about the scalar field.
3. ’t Hooft’s approach: effective action after integrating theconformal factor.
The gravitational piece of the previous Lagrangian is identical to the one proposedby ’t Hooft in [10] in order to solve conceptual problems of black holes. For thispurpose it is essential to integrate first over the scalar field in such a way as toget a conformally invariant theory of gravity. The divergent part of the functional– 5 –ntegral over the scalar field can be easily computed after it is conveniently rotatedto imaginary values, as advertised earlier e − (cid:113) n − n − σ ( x ) ≡ iα ( x ) α ∈ R (3.1)and the result of this integration over D α can be expressed in terms of the Weyltensor. Let us remind its origin. The Schouten tensor is defined as A αβ ≡ n − (cid:18) R αβ − n − Rg αβ (cid:19) and the Weyl tensor reads W αβµν ≡ R αβµν + ( A βµ g αν + A αν g βµ − A βν g αµ − A αµ g βν ) . Under conformal transformations, it transforms as a conformal tensor of weight λ = −
1: ˜ W αβµν ≡ e σ W αβµν . (3.2)So its square has got weight λ = 2 in such a way that | g | /n W µνρσ W µνρσ = | g | /n (cid:18) R µνρσ R µνρσ − R µν R µν + 13 R (cid:19) (3.3)is pointwise invariant (but behaves as a true scalar in four dimensions only).The Weyl tensor vanishes identically in low dimension n = 2 and n = 3 and aspace with n ≥ W = 0. In that case, the claim is that thecounterterm reads L div = − √ g π ( n − W µνρσ W µνρσ . (3.4)This functional behavior (barring the coefficient) could have been predicted from thefact that the result theory had to be pointwise conformal invariant. It can also bewritten as L div = √ g π ( n − (cid:18) R µν R µν − R (cid:19) . (3.5)The second expression is easily obtained assuming that the Euler topological invariantvanishes, id est (cid:90) d x √ g (cid:0) R µνρσ R µνρσ − R µν R µν − R (cid:1) = 0 . (3.6)It is perhaps worth remarking that the quantity δ (cid:16) g n W µνρσ W µνρσ (cid:17) = 0 (3.7)– 6 –hich is invariant under area preserving diffeomorphisms only, enjoys pointwise con-formal invariance in any dimension , when the power of the determinant is determinedin order to enhance area preserving diffeomorphisms to the full group of diffeomor-phisms, the resulting expression is conformal invariant only in dimension n = 4 δ ( √ gW µνρσ W µνρσ ) = − − n n nω ( x ) √ g W µνρσ W µνρσ . (3.8)This fact, first noticed by Duff [4] leads to the understanding of the standard con-formal anomaly in dimensional regularization through finite remainders coming fromthe (cid:15) (cid:15) cancellation.
4. Conformal invariance
Instead of working with the scalar-tensor theory in the form we just obtained, let usclarify its physical content by defining the following vector field W µ ≡ n − e − (cid:113) n − n − σ ∇ µ e (cid:113) n − n − σ = 1 (cid:112) ( n − n − ∇ µ σ (4.1)which under conformal transformations behaves as an abelian gauge field W (cid:48) µ = Ω − ∇ µ Ω + W µ . (4.2)This fact encodes a deep meaning, namely that in general we should be always able toconstruct a pointwise invariant conformal theory from a non-invariant one by addinginteractions with this gauge field in a similar way as it is done in a Yang-Mills theoryto implement local invariance under SU ( N ) to the fermionic matter. This is preciselythe situation we have in the unimodular theory (which is more clear when describedthrough this more general scalar-tensor theory), which naively, and forgetting forthe moment the implications of the C = 1 partial gauge fixing, is no more than anupgrading of Einstein-Hilbert theory into a conformal invariant one, so it has to bepossible to rewrite it just as General Relativity coupled to this W µ field.Thus, let us start as usual by defining a gauge covariant derivative by meanings ofthe gauge connection, which upgrades the riemmanian connection toΓ( W ) µνρ = Γ µνρ − δ µν W ρ − δ µρ W ν + g νρ W µ (4.3)– 7 –hich allows us to define a conformal (as well as diffeomorphism) covariant derivativeby D µ T = ∇ Γ( W ) µ T + λW µ T (4.4)where − λ is the conformal weight of the tensor T and ∇ Γ( W ) µ states for the derivativedefined through the Weyl connection Γ( W ).The important fact that arises here is that even if this Weyl connection is not ametric one, all dynamical quantities can however be canonically constructed just bydefining a new metric in such a way that G αβ = e − σ √ ( n − n − g αβ −→ Γ( W ) µνρ [ g αβ ] = Γ µνρ [ G αβ ] (4.5)which enjoys all expected properties.So at this point things are straightforward and we can compute naive Weyl invariant(once proper integration measure is provided) geometrical quantities out of the D µ derivative, such as the Riemman tensor defined by its conmutator, which will berelated to the ones computed just with the usual metric g µν in a fancy way. Weconsign details of those computations to the appendix but just let us recall the finalresult for the Weyl curvature scalar in terms of the usual one together with thespurion field, which is R = R − (cid:114) n − n − ∇ σ − ( ∇ σ ) . (4.6)The success of this construct is that, via an integration by parts, it correspondsexactly with the Lagrangian density of the scalar-tensor theory, so the full actioncan be rewritten in a manifestly Weyl invariant way as S = (cid:90) d n x √ G R = (cid:90) d n x √ g e − (cid:113) n − n − σ (cid:0) R + ( ∇ σ ) (cid:1) (4.7)and this shows clearly how the Weyl invariant scalar-tensor theory is just a com-pletion of the usual Einstein-Hilbert theory in order to have pointwise conformalinvariance through the gauge field W µ .It is also interesting to check what the partial gauge fixing C = 1 means with respectto conformal invariance. From the equation[2.1], we can see that it reduces to just G = 1, which is exactly the unimodularity condition that we also imposed in theEinstein-Hilbert Lagrangian to define the Unimodular theory. This clearly showsthat this theory, at least at the classical level, is no more than a common partially– 8 –auge fixed sector of both General Relativity and Conformal Gravity, correspondingto those physical systems that, maintaining conformal invariance (which implies theimpossibility of adding a cosmological constant term to the Lagrangian) have generalcoordinate transformations invariance reduced to area preserving diffeomorphismsonly.This statement has also another useful implication, which is that when writtenthrough the conformally invariant metric G µν , the background field expansion ofthe action is straightforward and identical to the expansion of the Einstein-HilbertLagrangian with the added step of changing all geometrical quantities by the onesconstructed through W µ . This is easily understood since the covariant structure isthe same in both cases and the only difference is the adding of conformal invariance.
5. One-loop computation
Our goal in this work was to determine whether the conformal invariance of unimod-ular gravity was broken by quantum corrections in the form of a trace anomaly.There is a general issue of consistency here.When computing anomalies, the problem is usually reduced to a theory propagatingin a background (non-dynamical) gravitational field. This gives rise to the computa-tion of determinants that depend upon the background metric. What we are doingin this paper is slightly different, in the sense that we are considering the gravita-tional field as a dynamical entity, and computing its one loop effects. It is a fact thatthe Einstein-Hilbert Lagrangian is non renormalizable. This has been shown to bethe case also for the unimodular variants, as studied in [1]. The consistency of ourapproach is then not guaranteed. The meaning of our result is then rather that noobvious inconsistency appears when considering the theory to one loop order. Thisfact alone is highly nontrivial.As it is explained in Appendix B, the computation of the conformal anomaly canbe reduced to the calculation of the n = d Schwinger-de Witt coefficient in theexpansion of the heat kernel corresponding to the quadratic differential operator ofthe effective action for quantum fluctuations. However, when the expression [4.7] istaken into account, things are easier, since what the heat kernel expansion states isthat the conformal (or trace) anomaly is − (cid:90) d ( vol ) T = λa d (5.1)– 9 –here − λ is the conformal weight of the corresponding second order operator, T ≡ T µν g µν is the trace of the one-loop energy momentum tensor, and a d is certaincoefficient in the expansion of the heat kernel of the operator of quadratic fluctuationsas given in the Appendix, formula (B.8).So if we are dealing with pointwise conformaloperators in our Lagrangian, this vanishes identically and computing the Schwinger-de Witt coefficient is not necessary. And, recalling what we proved before, this isexactly the situation we are dealing with ,so we should expect the conformal anomalyto cancel in this theory. However, let allow us to be more explicit and compute thecounterterm exactly by recalling that the full action of the unimodular theory in thescalar-tensor description was written in a manifestly conformally (Weyl) invariantway, namely S = − M n − (cid:90) d n x √ G R = − M n − (cid:90) d n x √ g e − (cid:113) n − n − σ (cid:0) R + ( ∇ σ ) (cid:1) (5.2)Therefore, performing a background field expansion (which has been discussed insome detail in the second reference of [1]) g µν ≡ ¯ g µν + h µν (5.3) σ ≡ ¯ σ + σ provided with the (often dubbed classical) conformal transformations δ C ¯ g µν = 2 ω ( x )¯ g µν (5.4) δ C h µν = 2 ω ( x ) h µν δ C ¯ σ = (cid:112) ( n − n − ωδ C σ = 0we can reconstruct again the conformal invariant structure, this time at the linearlevel, by expanding all quantities in the same way as we did in section 4, but using thistime the background field, so we will denote everything computed this way by addinga bar over it. The fact that the variation of the dinamical spurion σ vanishes meansthat all the expressions of Weyl invariant geometrical quantities will be identical tothe ones at the non-linear level by just replacing the full field σ by the background one ¯ σ and since all these changes can be encoded, as we showed before, into aconformal rescaling of the metric, this implies that the perturbative expansion ofthis action will match the well-known one of Einstein-Hilbert Lagrangian with just It is worth remarking that doing this, the covariant derivative of the gravitational fluctuation h µν , which is a tensor of conformal weight λ = −
2, transforms as another conformal tensor of thesame weight. – 10 –he corresponding change of metric and operators done at every step. So doing itand taking care of fixing the gauge in a conformally (Weyl) background invariantway , we are done.The background (zeroth order) term reads simply¯ S = − M n − (cid:90) d n x √ ¯ g e − (cid:113) n − n − ¯ σ ¯ R . (5.5)On the other hand, the linear terms that have to cancel in order to ensure absenceof tadpoles are S σ = − M n − (cid:90) d n x √ ¯ ge − (cid:113) n − n − ¯ σ (cid:114) n − n − R σ (5.6) S h = M n − (cid:90) d n x √ ¯ ge − (cid:113) n − n − ¯ σ (cid:114) n − n − (cid:15) µν h µν (5.7)where ¯ (cid:15) µν is the background Einstein tensor and we have performed a convenientpartial integration in the gravitational fluctuation action. The linear equations ofmotion for the background metric are then encoded into these linear terms and read − ¯ R αβ + 12 ¯ R ¯ g αβ = ¯ ∇ α ¯ σ ¯ ∇ β ¯ σ − (cid:0) ¯ ∇ ¯ σ (cid:1) ¯ g αβ . (5.8)The trace of the above implies directly ¯ R = − (cid:0) ¯ ∇ ¯ σ (cid:1) and on the other hand, thegeometrical Bianchi identities demand that0 = ¯ ∇ α (cid:18) − ¯ R αβ + 12 ¯ R ¯ g αβ (cid:19) = ¯ ∇ ¯ σ ¯ ∇ β ¯ σ = 12 (cid:114) n − n − ∇ β ¯ σ (cid:16)(cid:0) ¯ ∇ ¯ σ (cid:1) − ¯ R (cid:17) . (5.9)Altogether they imply ¯ R = (cid:0) ¯ ∇ ¯ σ (cid:1) = ¯ ∇ ¯ σ = 0 = ¯ R , which, as with Einstein equa-tions, is no more than a consequence of the background equations of motion oncewe take account of the substitution of operators by conformal ones that we werediscussing.Finally and as we argued, the second order term has to be the same as in theexpansion of the Einstein-Hilbert Lagrangian, where at each step the substitution¯ g µν → ¯ G αβ = e − √ ( n − n − ¯ σ ¯ g αβ (5.10) The best option, taking into account that our goal is to compute the gravitational one-loopcounterterm, is generalizing the harmonic gauge to ¯ D µ h µν = 0. – 11 –s made, which implies also subtituing all derivatives by the background Weyl invari-ant one ¯ D µ .Thus S h = − M n − (cid:90) d n x (cid:112) ¯ G (cid:20)
14 ¯ D µ H ¯ D µ H −
12 ¯ D µ H ¯ D ρ H µρ + 12 ¯ D µ H µρ ¯ D ν H νρ −−
14 ¯ D µ H νρ ¯ D µ H νρ − ¯ R νβ H βα H να + 12 h ¯ R αβ H αβ − ¯ R (cid:18) H − H αβ H αβ (cid:19)(cid:21) (5.11)where H µν is the graviton fluctuation of the rescaled metric G µν , corresponding to H µν = e − √ ( n − n − ¯ σ (cid:32) h µν − σ ¯ g µν (cid:112) ( n − n − (cid:33) . (5.12)This in turn means that (provided that the corresponding conformal harmonic gaugefixing is used) the counterterm is simply given in terms of the t’Hooft-Veltman [10]counterterm by performing the same operator substitution we were doing formerly S c = 18 π ( n −
4) 20380 (cid:90) d n x (cid:112) ¯ G ¯ R == 18 π ( n −
4) 20380 (cid:90) d n x √ ¯ g e − (cid:113) n − n − ¯ σ (cid:32) ¯ R − (cid:114) n − n − ∇ ¯ σ − (cid:0) ¯ ∇ ¯ σ (cid:1) (cid:33) (5.13)which is manifestly pointwise conformally invariant and also it vanishes on-shell whenbackground equations of motion are taken into account. This is in accord with thenaive fact that the conformal anomaly should vanish owing to the manifest conformalinvariance of the action.The inclusion of non-interacting conformal matter does not change the situation. Forexample, a scalar field interacts with the gravitational field according to S matt ≡ (cid:90) d n x g µνE ∇ µ Φ ∇ ν Φ = (cid:90) d n x g n g µν ∇ µ Φ ∇ ν Φ (5.14)Once embedded in a diffeomorphism invariant theory, the action principle reads S = (cid:90) d n x √ ¯ g e − (cid:113) n − n − ¯ σ
12 ¯ g µν ¯ ∇ µ Φ ¯ ∇ ν Φ (5.15)– 12 –nd given the transformation of ¯ σ , it is plain to check that the operator∆ f ≡ ¯ ∇ µ (cid:18) √ ¯ g e − (cid:113) n − n − ¯ σ ¯ g µν ¯ ∇ µ f (cid:19) (5.16)is conformally invariant.
6. Conclusions.
It has been shown that the conformal invariance of unimodular gravity survives quan-tum corrections, even in the presence of scalar conformal matter. This result is aconsequence of the fact that the corresponding operator governing quadratic fluctu-ations around an arbitrary background is manifestly conformal invariant (vanishingconformal weight).Another way of looking at this result is through the computation of the counterterm,which is quite simply determined from the standard ’t Hooft-Veltman counterterm.This counterterm is Weyl invariant for any dimension, id est, its variation vanishesas opposed to being proportional to n −
4. It actually vanishes on shell, once thebackground equations of motion are used. The fact that the conformal anomalyshould vanish for unimodular gravity was already conjectured by Blas in his Ph.D.thesis work [3].The physical situation is not unlike the gauge current in a vectorlike gauge theory,where it is also quite plain that no anomaly is present.As a general remark, the unimodular theory can be understood as a certain trunca-tion of the full Einstein-Hilbert theory, where in a certain frame (the Einstein frame)the metric tensor is unimodular (with determinant equal to one). Our result is com-patible with the idea that the corresponding restriction at the quantum level (i.e. inthe functional integral) is consistent as well.– 13 – . Weyl covariant curvature
Once the Weyl covariant derivative defined through the gauge field W µ is constructed,geometrical quantities can be computed. To start with, the commutator of two ofsuch derivatives defines a curvature through Ricci’s identity (and is independent ofthe conformal weight of the tensor acted upon, so the in appearance arbitrary term λW µ T does not cause any contradiction and indeed it is needed to ensure that thederivative of the metric vanishes) R µνρσ = R µνρσ − g µρ ( ∇ ν W σ + W ν W σ ) − g µσ ( ∇ ν W σ + W ν W ρ ) (A.1)+ g νρ ( ∇ µ W σ + W µ W σ ) + g νσ ( ∇ µ W ρ + W µ W ρ ) + (cid:0) ∇ λ W λ (cid:1) ( g µσ g νρ − g µρ g νσ ) = R µνρσ + g µρ ( ∇ ν ∇ σ σ + ∇ ν σ ∇ σ σ ) − g µσ ( ∇ ν ∇ σ σ + ∇ ν σ ∇ ρ σ ) − g νρ ( ∇ µ ∇ σ σ + ∇ µ σ ∇ σ σ ) + g νσ ( ∇ µ ∇ ρ σ + ∇ µ σ ∇ ρ σ ) + ( ∇ σ ) ( g µσ g νρ − g µρ g νσ ) . It is easy to realize that, defining a new metric by a conformal rescaling G αβ = e − σ √ ( n − n − g αβ , what we have is R µνρσ = e √ ( n − n − σ R µνρσ (cid:20) g αβ e − √ ( n − n − σ (cid:21) = (cid:18) Gg (cid:19) /n R µνρσ [ G αβ ] (A.2)which corresponds to the usual Riemman tensor that we would compute using themetric G αβ with a prefactor ( G/g ) /n whose origin is to ensure pointwise conformalinvariance. Accordingly R µν = R µν + ( n −
2) ( ∇ µ W ν + W µ W ν ) + g µν (cid:0) ∇ λ W λ − ( n − W λ W λ (cid:1) == R µν − (cid:114) n − n − ∇ µ ∇ ν σ + 1 n − ∇ µ σ ∇ ν σ −− g µν (cid:32) (cid:112) ( n − n − ∇ σ + 1 n − ∇ λ σ ∇ λ σ (cid:33) . And this Ricci tensor has also got a quite simple interpretation R µν = e √ ( n − n − σ R µν (cid:20) g αβ e − √ ( n − n − σ (cid:21) = (cid:18) Gg (cid:19) /n R µν [ G αβ ] (A.3)manifestly conformal invariant under g µν → Ω g µν e − √ ( n − n − σ → Ω − e − √ ( n − n − σ . (A.4)– 14 –rom this, the curvature scalar is straightforward and inherits the same interpretation R = R + 2 ( n − ∇ λ W λ − ( n −
2) ( n − W λ W λ == R − (cid:114) n − n − ∇ σ − ( ∇ σ ) = (cid:18) Gg (cid:19) /n R [ G αβ ] . (A.5)From all this, the Einstein tensor results to be E µν = R µν − Rg µν + 1 n − ∇ µ σ ∇ ν σ + (cid:112) ( n − n − ∇ σg µν + n − n −
1) ( ∇ σ ) g µν . (A.6)Finally, taking into account that the measure √ g e − n √ ( n − n − σ d n x = √ G (cid:18) Gg (cid:19) − /n (A.7)is conformal invariant, the only dimension two pointwise invariant operator is (cid:90) d n x √ ge − n √ ( n − n − σ R = (cid:90) d n x √ g e − (cid:113) n − n − σ (cid:0) R + ( ∇ σ ) (cid:1) (A.8)and after integration by parts, the full action can then be written as S = (cid:90) d n x √ ge − n √ ( n − n − σ R = (cid:90) d n x √ G R (A.9)where the factors G/g cancel exactly and show how dynamics can be obtained fromthe metric G αβ even if it does not encode all information about the nature of theWeyl covariant derivative (explicitely, it knows nothing about the λW µ T term of thederivative).At the linear level, the conformal classical (or background) transformations are δ C ¯ g µν = 2 ω ( x )¯ g µν δ C h µν = 2 ω ( x ) h µν δ C ¯ σ = (cid:112) ( n − n − ωδ C σ = 0 (A.10)and since they vanish for the spurion field fluctuation, this means that all the ge-ometrical construct we just did in this appendix can be redone on the backgroundfield expansion as well just by replacing σ by ¯ σ .– 15 – . Conformal anomaly It is well known that one loop computations are equivalent to the calculation offunctional determinants. One of the simplest definitions of the determinant of anoperator is through the ζ -function technique [8]. We shall follow conventions as in[2]. Given a differential operator of the general form∆ ≡ − D µ D µ + Y (B.1)with D µ ≡ ∂ µ + X µ , we assume that the elliptic operator ∆ enjoys eigenvalues λ n ∆ φ n = λ n φ n (B.2)normalized in such a way that (cid:90) d n x √ g φ i φ j = δ ij . (B.3)Now the heat kernel is formally defined as K ( τ ) ≡ e − τ ∆ (B.4)and its action on functions reads( Kf )( x ) = (cid:90) d ( vol ) y K ( x, y ; τ ) f ( y ) . (B.5)The ultraviolet (UV) behavior is controlled by the short time Schwinger-de Wittexpansion which reads K ( x, y ; τ ) = K ( x, y ; τ ) (cid:88) p =0 b p τ p (B.6)where for instance the flat space kernel reads K ( x, y ; τ ) = 1(4 πτ ) n e − ( x − y )24 τ . (B.7)The integrated quantity Y ( τ, f ) ≡ tr ( Kf ) also enjoys a corresponding short timeexpansion Y ( τ, f ) = (cid:88) k =0 τ k − n a k ( f ) . (B.8)The trace in the preceding formulas involves spacetime integration as well as sumover all finite rank indices. Sometimes one simply writes Y ( τ ) ≡ Y ( τ, zeta function is defined asΓ( s ) ζ ( s ) = (cid:90) ∞ dt t s − Y ( t ) = (cid:88) n λ − sn (B.9)where the second equality is even more formal than the first one.The determinant of the differential operator is then defined [8] asdet ∆ ≡ (cid:89) n λ n ≡ e − ζ (cid:48) (0) (B.10)Now assume that we have a quantum field theory that we dimensionally regularize,id est, we make n = d + (cid:15) , where d is the physical dimension (for example d = 4),then, at the one-loop level, there is a divergent piece in the effective action W ∞ = −
12 log det∆ | ∞ = − µ (cid:15) a d (cid:15) . (B.11)On the other hand, when performing a rigid Weyl transformation on the spacetimemetric (cid:101) g µν = Ω g µν = (1 + 2 ω ) g µν (B.12)the eigenvalues of the operator transform in a definite manner which coincide withthe conformal weight λ of the operator. (cid:102) λ n ≡ Ω − λ λ n . (B.13)Usually the conformal weight is just the mass dimension of the operator in the senseof dimensional analysis.According to Branson [7] a conformal covariant operator D transforms under local (not only rigid) Weyl transformations in such a way that there exist two numbers( a, b ) such that the Weyl rescaled operator is given by (cid:101) Dφ = Ω − b D (Ω a φ ) . (B.14)It follows that that the new eigenfunctions are given by (cid:101) φ n = Ω − a φ n (B.15)and the new eigenvalues by (cid:101) λ n = Ω − b λ n . (B.16)The archetype of such operators is the conformal laplacian∆ c ≡ ∆ − n − n − R (B.17)– 17 –hich is such that ∆ c (cid:16) Ω − n − φ (cid:17) = Ω − n +22 ∆ φ. (B.18)There are no known diffeomorphisms invariant operators built out of the metric alonewith b = 0.In the case of the standard scalar laplacian,∆ ≡ ∇ ≡ √ g ∂ µ ( g µν √ g∂ ν ) (B.19)the conformal weight coindices with its mass dimension, λ = 2.The new zeta function after the Weyl transformation is given in general by (cid:101) ζ ( s ) = Ω Ds ζ ( s ) (B.20)so that the determinant defined through the ζ -function scales asdet (cid:101) ∆ = Ω − λζ (0) det ∆ (B.21)and this modifies correspondingly the effective action (cid:102) W = W + λ ω ζ (0) . (B.22)The energy-momentum tensor is defined in such a way that under a general variationof the metric the variation of the effective action reads δW ≡ (cid:90) d ( vol ) x T µν δg µν (B.23)which in the particular case that this variation is proportional to the metric tensoritself (like in a conformal transformation at the lineal level), δg µν = − ωg µν yieldsthe integrated trece of the energy-momentum tensor δW = − (cid:90) d ( vol ) ωT. (B.24)Conformal invariance in the above sense then means that the energy-momentumtensor must be traceless. When quantum corrections are taken into account, itfollows that − (cid:90) d ( vol ) T = λ ζ (0) . (B.25)It is not difficult to show that ζ (0) ≡ lim s → s (cid:90) ∞ dt t s − Y ( t ) = lim s → s (cid:90) dt t s − Y ( t ) = a d (B.26)– 18 –here n = d is the specific value of the spacetime dimension. The conformal anomalyis usually then written as − (cid:90) d ( vol ) T = λa d . (B.27)The Schwinger-de Witt coefficient corresponding to the physical dimension, n = d precisely coincides with the divergent part of the effective action when computedin dimensional regularization as indicated above. This means that in order to com-pute the one loop conformal anomaly in many cases it is enough to compute thecorresponding counterterm.This argument shows clearly that when the conformal weight of the operator ofinterest vanishes, λ = 0 all eigenvalues remain invariant and there is no conformalanomaly for determinants defined through the zeta function. In our case this willfollow from the manifest Weyl invariance of the construction of the operator at allsteps. This conformal invariance in turn in inherited from the mother theory whichenjoys invariance under area preserving diffeomorphisms only. This is the origin ofthe background dilaton ¯ σ of gravitational origin, essential in our approach.– 19 – cknowledgments We have enjoyed many discussions with Luis Alvarez-Gaum´e. This work has beenpartially supported by the European Union FP7 ITN INVISIBLES (Marie Curie Ac-tions, PITN- GA-2011- 289442)and (HPRN-CT-200-00148) as well as by FPA2009-09017 (DGI del MCyT, Spain) and S2009ESP-1473 (CA Madrid). M.H. acknowl-edges a ”Campus de Excelencia” grant from the Departamento de F´ısica Te´orica ofthe UAM. The authors acknowledge the support of the Spanish MINECOs Centrode Excelencia Severo Ochoa Programme under grant SEV-2012-0249.
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