Can we change c in four-dimensional CFTs by exactly marginal deformations?
aa r X i v : . [ h e p - t h ] F e b RUP-17-3
Can we change c in four-dimensional CFTs by exactly marginaldeformations? Yu NakayamaDepartment of Physics, Rikkyo University, Toshima, Tokyo 171-8501, Japan
Abstract
There is no known obstructions, but we have not been aware of any concrete ex-amples, either. The Wess-Zumino consistency condition for the conformal anomalysays that a cannot change but does not say anything about c . In supersymmetricmodels, both a and c are determined from the triangle t’Hooft anomalies and the uni-tarity demands that both must be fixed, so the unitary supersymmetric conformalfield theories do not admit such a possibility. Given this field theory situation, weconstruct an effective AdS/CFT model without supersymmetry in which c changesunder exactly marginal deformations. Introduction
Counting degrees of freedom in quantum field theories is the first step for the classificationof the entire landscape of our theory space. It is long known that in two-dimensionalconformal field theories, the central charge c of the Virasoro algebra[ L m , L n ] = ( m − n ) L m + n + c
12 ( m − m ) δ m + n, (1)plays such a role. The central charge c may also be regarded as a coefficient of the Weylanomaly: the trace of the energy-momentum tensor does not vanish T ii = c π R (2)in the curved background. In unitary conformal field theories with a discrete conformalspectrum, the central charge c monotonically decreases along the renormalization groupflow and it stays constant along the exactly marginal deformations [1]. Because of thisproperty, the central charge c is a good candidate for representing the degrees of freedomin conformal field theories, and indeed, the first thing to know in the classification of theconformal field theories is the central charge c .In four-dimensions, it took more than two decades to completely settle the issue [2][3].In four-dimensional conformal field theories, there are two independent terms in the Weylanomaly: schematically we have T ii = − a Euler + c Weyl (3)and it eventually turned out that one of them called “ a ” plays the similar role as thecentral charge c in two-dimensions. Historically, however, it even took some years thatthe other combinations of a and c are not good candidates for representing the degreesof freedom. The first counterexample in which c increases along the renormalizationgroup flow appeared in the context of the supersymmetric gauge theories, in which non-perturbative determination of a and c is possible [4].The Weyl anomaly coefficient a has a nice property that it cannot depend on theexactly marginal deformations of any conformal field theories in four-dimensions. Thus,from the beginning it is a good candidate for representing the degrees of freedom as thecentral charge c in two-dimensions. However, the question has been still open if we canactually change c along exactly marginal deformations. If it does, then it cannot be a1andidate for representing the degrees of freedom. In fact, there is no known obstruc-tions, but we have not been aware of any concrete examples, either. The Wess-Zuminoconsistency condition for the conformal anomaly says that a cannot change but does notsay anything about c [5].The aim of this paper is to study a possibility to change c along exactly marginaldeformations. We eventually construct such a model in the effective AdS/CFT corre-spondence. Therefore, as an effective theory of gravity, we open up a possibility for amodel with varying c along exactly marginal deformations. Then we may further ask whyit is more difficult to find such a situation in quantum field theories, and this is anotheraim of this paper.To conclude the introduction, we should mention that the proof that a is monotonicallydecreasing along the renormalization group flow appeared in [3], and from that viewpointalone, there remains a less direct motivation to study the properties of c . Nevertheless, wethink it is an interesting tension between “what is allowed must happen in physics” and“we do not know any concrete examples”. Even beyond the theoretical curiosity, study-ing properties of the energy-momentum tensor is certainly one of the most fundamentalaspects of conformal field theories (e.g. [6][7]), and it is important to understand thembetter, in particular without supersymmetry where we have less control over them.The organization of the paper is as follows. We are going to set up a debate if wecan change c in four-dimensional CFTs by exactly marginal deformations. In section 2,we present the field theory argument with a pro and a con. In section 3, we continuethe argument from the holographic perspective with a pro and a con. In section 4, weconclude the debate with further discussions. Statement:
There must be a model in which c changes under exactly marginal deformations. Aswe will show, the most powerful constraint on the Weyl anomaly coming from the Wess-Zumino consistency conditions for the local renormalization group transformation allowssuch a possibility, so from the totalitarian principle of theoretical physics [8] “Everything ot forbidden is compulsory” dictates such a model should exist. We first show that there is a non-perturbative field theory argument that in four-dimensional conformal field theories, the Weyl anomaly coefficient a cannot depend onexactly marginal coupling constants while c may depend on them. The argument wasgiven in [5] as a Wess-Zumino consistency condition for the Weyl transformation with thespace-dependent coupling constant. This is also known as a local renormalization groupanalysis, and we will see that not only space-time dependent metric but also space-timedependent coupling constant plays a crucial role.Let λ ( x ) be a space-time dependent exactly marginal coupling constant of a conformalfield theory, which means that if we put the theory on the flat Minkowski space-time g ij ( x ) = η ij with the space-time independent coupling constant λ ( x ) = λ , the theory isconformal invariant (for a certain range of λ , which we will not bother in the followingdiscussions). Even though the theory is conformal invariant on the flat Minkowski spacewith a space-time independent coupling constant, the variation of the partition functionalunder the Weyl transformation in the curved background with the space-time dependentcoupling constant may not vanish due to the Weyl anomaly: under the infinitesimalWeyl variation of the metric: δ σ g ij ( x ) = 2 σ ( x ) g ij ( x ), we have a variation of the partitionfunctional Z [ g ij , g ] δ σ log Z [ g ij ( x ) , λ ( x )] = Z d x p | g | σ ( x ) (cid:0) a ( λ )Euler − c ( λ )Weyl + · · · (cid:1) , (4)where · · · represents the Weyl anomaly coming from the space-time dependent couplingconstant such as ✷ λ ( x ) ✷ λ ( x ), which, for now, we are not interested in.The Weyl anomaly must satisfy an integrability condition. To see this, we note thatthe Weyl variation is Abelian δ σ δ ˜ σ log Z [ g ij ( x ) , λ ( x )] − δ ˜ σ δ σ log Z [ g ij ( x ) , λ ( x )] = 0 , (5)and this gives a non-trivial constraint on the form of the Weyl anomaly. For example,the pure R term in the Weyl anomaly (i.e. T ii = bR ) is not allowed just from thisintegrability condition [9]: δ σ δ ˜ σ log Z [ g ij ( x ) , λ ( x )] − δ ˜ σ δ σ log Z [ g ij ( x ) , λ ( x )] = Z d x p | g | bR (˜ σ ✷ σ − σ ✷ ˜ σ ) . (6)Let us see what this constraint dictates about the properties of a ( λ ) and c ( λ ). Firstof all, we note that the Weyl squared term is invariant under the Weyl transformation3 σ (cid:16)p | g | Weyl (cid:17) = 0, so there is no constraint on the coefficient c ( λ ) from (5). In partic-ular, there is no obstruction for c to depend on the (exactly marginal) coupling constant λ . On the other hand, the Weyl variation of the Euler density is non-trivial: under theinfinitesimal Weyl variation (i.e. δg ij = 2 σg ij ), it transforms as δ σ ( p | g | Euler) = 8 p | g | (cid:18) R ij − R g ij (cid:19) D i ∂ j σ . (7)Therefore the Wess-Zumino consistency condition demands[ δ ˜ σ , δ σ ] log Z [ g ij ( x ) , λ ( x )] = Z d x p | g | (cid:18) a ( λ ) (cid:18) R ij − R g ij (cid:19) ( σD i ∂ j ˜ σ − ˜ σD i ∂ j σ ) + · · · (8)However, when a ( λ ) is a function of space-time dependent coupling constants such that a ( λ ) is a non-trivial function of x i , one cannot do the integration by part and the righthand side does not vanish while the left hand side should, leading to a contradiction. Onlywhen a is a genuine constant, one may perform the integration by part with the Bianchiidentity to show that it is consistent. This means that the Weyl anomaly coefficient a cannot depend on the exactly marginal coupling constant. We emphasize that thisconclusion only comes from the consideration of space-time dependent coupling constantwith local renormalization group analysis.In order to make this argument complete, we have to further show that the omit-ted terms coming from the space-time dependent coupling constants do not give rise tothe term (cid:0) R ij − R g ij (cid:1) ( σD i ∂ j ˜ σ − ˜ σD i ∂ j σ ) for a possible cancellation, and we can eas-ily see this is indeed the case. It is essentially because the other terms already containthe derivatives of λ ( x ) and the Weyl variation cannot generate the Einstein tensor oncethe beta function for the coupling constant λ vanishes. For example, the variation of R d x p | g | σ ( x ) ✷ λ ( x ) ✷ λ ( x ) is R d x p | g | σ ( x ) ∂ i ˜ σ∂ i λ ✷ λ ( x ) and cannot be used to cancelthe above variation from the Euler density whose variation contains the Einstein tensor. The more details can be found in [5], but we do not need it for the rest of our discussionsbecause our focus is c rather than a . When λ has a non-vanishing beta function, one may be able to cancel it from the terms such as R d x p | g | R ij ∂ i λ ( x ) ∂ j λ ( x ), and this observation is a starting point to discuss a -theorem from the localrenormalization group analysis, but it is another story.
4e therefore conclude that from the totalitarian principle of physics there must exista model in which c changes under exactly marginal deformations. Statement:
There are no known examples in which c changes under exactly marginal deformations.First of all, there is a fine-tuning problem to have exactly marginal deformations, and if weovercome this difficulty with the supersymmetry, then the structure of the supersymmetricfield theory together with unitarity demands that c cannot change under exactly marginaldeformations. Thus the situations presented in section 2.1 are just a rice-cake in thepicture. The conclusion made in section 2.1 is that from the kinematic structure of the effectiveaction alone, there is no obstruction that the Weyl anomaly coefficient c , unlike a , dependson exactly marginal deformations of conformal field theories. As far as we know, however,there have been no concrete field theory examples that show this property.The question whether we can change the Weyl anomaly coefficient c in four-dimensionalCFTs by exactly marginal deformations, first of all, relies on the possibility that we haveexactly marginal deformations. This itself is highly non-trivial. Typically the beta func-tions do not vanish and we need either symmetry reasoning or controlling parameterswith fine-tunings to ensure it. Otherwise it would be quite accidental.Only after setting up concrete ways to realize exactly marginal deformations, we shouldlook for the change of c under the exactly marginal deformations. The best known strategyto obtain exactly marginal deformations in four-dimensional conformal field theories is toimpose the supersymmetry. The non-renormalization theorem allows a possibility to haveexactly marginal deformations in supersymmetric field theories. Then the second naturalquestion to be asked is whether we can change c in four-dimensional superconformal fieldtheories by exactly marginal deformations.Let us look at the Wess-Zumino consistency conditions for the super Weyl anomaly[10][11]. The term containing c is uplifted to δ σ log Z = Z d xd θ (cid:0) Σ κ ( λ ) W αβγ W αβγ + c . c (cid:1) , (9) A Japanese proverb corresponding to a pie in the sky or just wishful thinking. κ ( λ ) = a − c ( λ ) with λ being a chiralbackground superfield (corresponding to an exactly marginal deformation), and W αβγ isthe Weyl superfield. Note that we already know that a cannot depend on the exactlymarginal deformations.This alone does not give us a constraint on κ ( λ ) from the Wess-Zumino consistencycondition because the Weyl superfield squared is super Weyl invariant. However, if weexpand it in terms of the component, the Weyl superfield contains a field strength thatcouples with the superconformal R-current, and together with the term containing a Weylanomaly coefficient, we may relate the R-current t’Hooft anomaly to a and c : a = 916(8 π ) (cid:0) R − Tr R (cid:1) c = 916(8 π ) (cid:18) R −
53 Tr R (cid:19) . (10)Here Tr means the t’Hooft triangle anomaly, or more precisely, in the conformal fieldtheory language, it is computed by a particular three-point functions among three R-currents or one R-current and two energy-momentum tensor. This relation tells that inorder to change c anomaly along the exactly marginal deformations, we need to changethe R-current t’Hooft anomaly while keeping the particular combination a fixed. Further-more, there is a so-called a -maximization procedure [12] to determine the superconformal R symmetry: the superconformal R-symmetry is the R-symmetry such that the aboveexpression for a is maximized.This makes the situation already difficult but we admit that kinematically it is stillnot impossible. For example, let us consider the case with U (1) symmetry whose chargesare denoted by x , y , z . Take a particular R-charge constraint x − y + z = 1, and the trial a -function a = − (cid:0) ( x − y + z )(( x − y ) + z ) + ( x − y + z ) ( x + y ) (cid:1) + 3( x + y )= − (cid:0) (1 − z ) + z (cid:1) (11)and we see that the a -maximization determines a at z = , but the theory still has oneparameter moduli, say in y direction (with x = + y ). On the other hand, the same This example is just for illustration and does not represent a unitary quantum field theory, if any.See below. c = − (cid:0) ( x − y + z )(( x − y ) + z ) + ( x − y + z ) ( x + y ) (cid:1) + 5( x + y )= − (cid:0) (1 − z ) + z (cid:1) + 2(1 + 2 y − z ) (12)and it depends on the moduli y that was not fixed by the a -maximization.This model or any similar kind, however, does not arise in unitary supersymmetric fieldtheories for the following reason. The unitarity demands that two-point functions of theconserved current operators must be positive definite, but the superconformal symmetrydemands that the same two-point functions are proportional to the Hessian of the a -function at its extremum [13][12] . It therefore means that if we have a zero eigenvalue inthe Hessian, the model cannot be realized in unitary quantum field theories.In the above example, the moduli direction corresponds to the zero eigenvalue of theHessian. More generally, if the a -function has a flat moduli direction, then the currenttwo-point functions are always degenerate, which can be realized only in non-unitarytheories. Therefore, there is no unitary superconformal field theory in which c changesalong the exactly marginal deformations.We therefore conclude that since the natural mechanism to allow exactly marginaldeformations cannot allow c to change in unitary theories, the models in which c changealong exactly marginal deformations do not exist unless we resort to some sort of fine-tunings that do not rely on the supersymmetry. Statement:
We do present a model in which c changes along exactly marginal deformations withinthe effective AdS/CFT correspondence. We allow certain fine-tunings of the effectivepotential, but it is theoretically allowed with the controlling parameters of /N . This givesus a concrete example of conformal field theories with varying c under exactly marginaldeformations. We admit that realizing exactly marginal deformations without a symmetry requiresfine-tunings. This leads to a possibility of controlling the renormalization group with7n extra parameter, and one way is to use the large N limit by regarding 1 /N as such.Then one may be able to obtain exactly marginal deformations without invoking quantumcorrections, and one can legitimately set up the question if we can construct conformalfield theories in which c changes along exactly marginal deformations. Furthermore, large N theories may have a holographic description under the AdS/CFT correspondence, andwe will pursue this possibility below.As a model of the AdS/CFT correspondence, we study an effective higher derivativegravity coupled with a “massless” scalar field φ given by the bulk action S = Z d x p | G | (cid:18) κ R − Λ + 12 κ ∂ µ φ∂ µ φ + φ ( αR + βR µν R µν + γR µνρσ R µνρσ ) (cid:19) . (13)As we will show, in order to assure that φ is a moduli field, we should demand 10 α + 2 β + γ = 0. We introduce the Fefferman-Graham expansions of the metric and the scalar field ds = G µν dx µ dx ν = l dρ ρ + g ij dx i dx j ρg ij = g (0) ij + ρg (1) ij + ρ g (2) ij + · · · φ = φ (0) + ρφ (1) + ρ φ (2) + · · · (14)and put the boundary at ρ = ǫ , which plays a role of UV cut-off in the dual conformalfield theory. Our goal is to compute the holographic Weyl anomaly [17] from this model.By using the boundary counterterms to cancel the UV power divergence, we may onlyfocus on the logarithmic terms with respect to ǫ in the action: S ∼ S log log ǫ . They maybe further expanded in powers of φ : S log = − Z d x q | g (0) | (cid:18) L + φ (0) L + φ (1) L + φ (2) L + 2 lκ φ + lκ g ij (0) ∂ i φ (0) ∂ j φ (1) − l κ g ik (0) g jl (0) g (1) kl ∂ i φ (0) ∂ j φ (0) + l κ g kl (0) g (1) kl g ij (0) ∂ i φ (0) ∂ j φ (0) (cid:19) . (15) Indeed, within perturbative quantum field theories, the non-trivial corrections to c with respect togauge/Yukawa/scalar couplings have been discussed in [14][15], and if the beta functions for these couplingconstants vanish, then generically c changes along the exactly marginal deformations. We would like tothank H. Osborn for the contribution to the debate. See appendix A for a review of the field theoryanalysis in pertrurbation theory. A different combination of the higher derivative gravity that does generate a potential for φ wasanalyzed in [16]. L = (cid:18) − lκ − l Λ2 (cid:19) g ij (0) g (2) ij − (cid:18) lκ (cid:19) g (1) ij R ij (0) + (cid:18) l κ (cid:19) R (0) g ij (0) g (1) ij + (cid:18) lκ + l Λ4 (cid:19) g ij (0) g kl (0) g (1) ik g (1) jl + (cid:18) − lκ − l Λ8 (cid:19) ( g ij (0) g (1) ij ) (16)and L = l ( αR + βR ij (0) R (0) ij + γR ijkl (0) R (0) ijkl )+ (cid:18) αl + 8 βl + 4 γl (cid:19) g ij (0) g (2) ij + (cid:18) αl + 12 βl + 12 γl (cid:19) g (1) ij R ij (0) + (cid:18) − αl − βl − γl (cid:19) R (0) g ij (0) g (1) ij + (cid:18) αl + 8 βl + 10 γl (cid:19) g ij (0) g kl (0) g (1) ik g (1) jl + (cid:18) αl + 2 βl + γl (cid:19) ( g ij (0) g (1) ij ) (17)as studied in [18].As mentioned above, we have to assure that φ is a moduli field. For this purpose, wenotice L = α l + β l + γ l , (18)and see that φ (2) appears only in this term. Requiring that the variation of φ (2) vanishthen demands 10 α + 2 β + γ = 0 as advocated. It is equivalent to the statement thatthere is no potential term for the scalar field φ in the AdS space-time. Otherwise, theassumption that φ is a moduli field does not hold and the corresponding coupling constantruns along the renormalization group flow due to the non-zero beta function.After setting 10 α + 2 β + γ = 0 the variation under g (2) ij determines l from Λ ( <
0) as − κ = l , (19)which is not modified by the value of the moduli field φ . As we will see, this determinesthe value of the Weyl anomaly coefficient a , and it corresponds to the fact that a doesnot change under the exactly marginal deformations.Finally, we have L = (cid:18) − αl − βl − γl (cid:19) R (0) + (cid:18) − αl − βl − γl (cid:19) g ij (0) g (1) ij . (20)9emarkably, L vanishes once we set 10 α + 2 β + γ = 0. There seems to exist a typo in eq(27) of [18] so that our expression is different in the second term of (20) but after correctingit, we have this property, which is important for the consistency of the holographic Weylanomaly as we will see.Our next task is to derive the equations of motion to determine φ (1) and g (1) ij in termsof φ (0) and g (0) ij and evaluate the on-shell action. The calculation is greatly simplifiedbecause L = 0. Because of the absence of the additional mixing from L , the variationof φ essentially gives the same expression as studied in [18]. Substituting the solutionback into the on-shell action, we obtain the holographic Weyl anomaly T ii = − a Euler + c ( λ )Weyl + a (cid:18)
12 ( ✷ λ ) + R ij ∂ i λ∂ j λ − Rg ij ∂ i λ∂ j λ + 23 ( g ij ∂ i λ∂ j λ ) (cid:19) . (21)where a = l κ c ( λ ) = l κ − l α + 2 β − γ λ (22)with the condition 10 α + 2 β + γ = 0. Here λ is an exactly marginal deformation dual tothe bulk field φ .This solution corresponds to dual conformal field theories with an exactly marginaldeformation under which the Weyl anomaly coefficient c changes. As in the field theoryanalysis, we see that the Weyl anomaly coefficient a cannot depend on the exactly marginaldeformations but c may.Therefore, we conclude that as long as we may be able to fine-tune the potential of themoduli field φ , we are able to construct a higher derivative holographic model in which c changes along the exactly marginal deformations. Statement: If we did not correct the typo mentioned above, the expression we would get has an independent R term which is inconsistent as mentioned in section 2.2. There are several different (and sometimes more convenient but eventually equivalent) ways to derivethe holographic Weyl anomaly in particular in higher derivative gravities. See e.g. [19][20][21][22]. ot all effective theories of gravity are consistent. If we look at the higher derivativesupergravity, again the unitarity demands that one cannot construct a model in which theholographic c changes along the exactly marginal deformations. The supersymmetric generalization of the O ( R ) term with the field dependent cou-pling constant was first discussed in [23] S = Z d x p | G | (cid:18) R − N IJ F Iµν F Jµν + N IJ ∂ µ φ I ∂ µ φ J + ǫ µνρστ c I A Iµ R αβνρ R σταβ + c IJK ǫ µνρστ A Iµ ∂ ν A Jρ ∂ σ A Kτ ++ c I φ I (cid:18) R − R µν R µν + R µνρσ R µνρσ (cid:19) + V ( φ ) (cid:19) + · · · (23)Here φ I is a vector multiplet scalar (whose partner is A Iµ ), and the supergravity model isparameterized by Chern-Simons couplings c IJK , c I and the gauging parameter P I .The potential V ( φ ) consists of two terms. First, we have the D-term constraint c IJK φ I φ J φ K = 1 , (24)corresponding to the very special structure. We also have the potential terms from thegauging that dictates we have to minimize P = P I φ I (25)under the constraint (24).Suppose we have a supersymmetric solution with the AdS vacuum, then the potentialterm gives a condition that we have to minimize P = P I φ I under the constraint c IJK φ I φ I φ K = 1 , (26)which is equivalent to the a -maximization condition [12][23]. To see this, we define t I = φ I /P , and then it is equivalent to maximize a = c IJK t I t J t K under the condition P I t I = 1.In order to find a situation in which c changes along the exactly marginal deformations, wefirst demand that the potential has a flat direction ∂V∂φ I = 0 and the corresponding Chern-Simons coupling c I is non-zero. Then, as discussed in the previous subsection, we canchange c along the flat moduli direction while a is fixed. It is important to recognize thatthe combination of the higher derivative terms that appear in (23) satisfies the condition10 α + 2 β + γ = 0 discussed in section 3.1. 11rom the kinematical consideration alone, it is not impossible to fine-tune c IJK and P I so that the supergravity action has a flat direction and holographic c function changesalong such a direction with non-zero c I in that direction. However, as in the field the-ory analysis in section 2.2, a further requirement of unitarity makes it unphysical. Thesupersymmetry dictates that the coefficient c IJK not only determines the potential, butalso the kinetic term. The gauge kinetic term is determined [23] from N IJ = − c IJK φ K and if we have a flat direction in the potential, the gauge kinetic function is degenerateand unitarity is violated. Obviously this is the gravity dual description of the unitarityobstruction discussed in section 2.2.Finally, when we actually have a flat direction in the moduli space of a unitary super-conformal field theories, then they belong to a hypermultiplet [24] and the above analysisdoes not apply. We do not know any higher derivative coupling between the hypermulti-plet and R terms in the supergravity, The field theory argument of the a maximizationand the t’Hooft anomaly consideration presumably makes it impossible.We should further notice that even without supersymmetry, the naive model discussedin section 2.2 has an issue of unitarity. The unitarity demands that the central charge c must be bounded below. The positivity of the two-point function demands c > c > a . If we take the naive modeldiscussed in section 2, for larger values of | φ | one may violate this bound. To avoid thisissue one has to introduce a certain cutoff for the range of possible φ . We, therefore, conclude that in the supergravity in which we may control the potentialbetter as in some string compactifications, it is not possible to construct a model withvarying c along the exactly marginal directions without violating the unitarity. The above discussions reveal that indeed we may be able to change c under exactlymarginal deformations in fine-tuned effective holographic models. Note that the fine-tuning here is not for the change of c ; rather it is for obtaining exactly marginal defor- Within the effective holography, this is not difficult. One may just introduce the non-trivial fielddependent kinetic term for the scalar field φ . c along exactly marginaldeformations is demonstrated within the effective holography with higher derivative cor-rections. Here, to conclude the paper, we are going to address two further questions: (1)was it trivial to construct the model with the desired property once we allow fine-tuningsin the effective AdS/CFT correspondence? (2) what was the prospect of fine-tuningsbeyond the effective holography?About the first point, we would like to emphasize that it is a non-trivial question to askeven within the effective AdS/CFT correspondence if we can actually realize various termsin the Weyl anomaly once they are allowed by the Wess-Zumino consistency condition.For example, at the conformal fixed point, the Pontryagin density in the Weyl anomaly T ii = ǫ abcd R abij R ij cd (27)has never been realized either in holography or field theories [27][28] even though it satisfiesthe Wess-Zumino consistency condition. There are other terms like T ii = C IJK ǫ ijk ∂ i λ I ∂ j λ J ∂ k λ K (28)in three dimensions [30][31] or T ii = B IJ ǫ ij ∂ i λ I ∂ j λ J (29)in two-dimensions that were not realized in holography as well as in field theories eventhough they satisfy the Wess-Zumino consistency conditions.With this respect, it is non-trivial for us to be able to construct a model in which c changes along exactly marginal deformations in particular because the leading Einsteingravity cannot afford it. Conversely it is interesting to see if there is field theory analysisthat is not captured by the Wess-Zumino consistency conditions that do not allow thepeculiar Weyl anomaly like the above. Indeed, in a recent paper [32], they pointed out anobstruction to construct the Weyl anomaly (29) from the viewpoint of the (ultra)localityin correlation functions. The argument there does not rely on the unitarity, so the natureof the varying c under exactly marginal deformations is quite different from that. The discussions here assume the conformal invariance. If the theory is only scale invariant withoutconformal invariance, we may construct such terms. See e.g. [29] for a review. c under exactly marginaldeformations. Alternatively, if we may prove that such conformal field theories are forbid-den from the field theory argument, it gives a strong constraint on the possible AdS/CFTcorrespondence. Acknowledgements
The author would like to thank Z. Komargodski for his kind hospitality and discussionsduring the author’s visit to Weizmann Institute where this work was initiated. The authorwould like to thank S. Nojiri for conversions on the typo in [18]. He would like to thankY. Tachikawa for explanations on his papers about the higher derivative supergravity.
A Leading order field theory analysis
For illustrative purposes, let us consider the SU ( N c ) Yang-Mills theory coupled with N f Dirac fermions in the fundamental representations. Perturbative computation at theone-loop order gives the beta function for the gauge coupling constant β g = dgd log µ = − g π (11 N c − N f ) + O ( g ) (30)One may choose the matter contents so that the leading order beta function vanishes,which makes the gauge coupling constant “moduli-like” in the one-loop approximation.With the same order, the Weyl anomaly coefficients a and c have been computed (seee.g. [15] and references therein): a = 6290(8 π ) ( N c −
1) + 1190(8 π ) N f N c + O ( g ) c = 1230(8 π ) ( N c −
1) + 630(8 π ) N f N c − π ) ( N c − N c − N f ) g π + O ( g ) . (31)14his perturbative computation is in agreement with the general argument in the maintext: along the (exactly) marginal deformations (i.e. g in this case with 11 N c − N f = 0), a does not change, but c may change. Note that vanishing of the leading order betafunction 11 N c − N f = 0 does not make the leading order correction to c vanish, whichis proportional to N c − N f . In this example, we do not have a control over the higherorder corrections in the beta function, but we may imagine a more elaborate example inwhich it may be done and can be thought of as a field theory realization of the holographicmodel discussed in the main text.
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