Conformal invariance from scale invariance in non-linear sigma models
aa r X i v : . [ h e p - t h ] J un RUP-20-20
Conformal invariance from scale invariance in non-linear sigmamodels
Yu NakayamaDepartment of Physics, Rikkyo University, Toshima, Tokyo 171-8501, Japan
Abstract
There exists a certain argument that in even dimensions, scale invariant quantumfield theories are conformal invariant. We may try to extend the argument in 2 n + ǫ dimensions, but the naive extension has a small loophole, which indeed shows anobstruction in non-linear sigma models in 2 + ǫ dimensions. Even though it couldhave failed due to the loophole, we show that scale invariance does imply conformalinvariance of non-linear sigma models in 2 + ǫ dimension from the seminal work byPerelman on the Ricci flow. Introduction
The advent of conformal bootstrap approaches to critical phenomena (e.g [1] for a review)raises a renewed interest in understanding about under which conditions the conformalsymmetry emerges. Empirically, it is typically the case that scale invariance, Poincar´einvariance (Euclidean invariance), and unitarity (reflection positivity) give rise to theenhanced conformal symmetry. Some argument supporting this empirical fact exist ineven space-time dimensions, in particular two [2] and four dimensions [3][4][5][6], but wedo not have general argument in odd dimensions, say in three dimensions. In the perturbative regime, the (non-)existence of scale invariant but not conformalfield theory is closely related to the gradient nature of the renormalization group flowand the absence of the limit cycle [8][9][10][11][12][13][14][15][16]. Again, we do havesupporting evidence for the gradient nature of the renormalization group flow in two andfour dimensions. A crucial fact here is that the potential function for the gradient flow isgiven by Weyl anomaly coefficients at the conformal fixed point. They do exist in evendimensions but they do not exist in odd dimensions.Without a general argument, it may be a natural idea to explore conformal invariancein odd dimensions by using the extrapolation of d = 2 n + ǫ dimensions. Such approaches invarious field theories are attempted in [2][17][18]. In this paper, we offer general discussionson how to obtain a gradient flow of the renormalization group beta function in d = 2 n + ǫ dimensions once we know that it is a gradient flow in d = 2 n dimensions. This typicallyimplies conformal invariance in (perturbative) scale invariant fixed point in d = 2 n + ǫ dimensions if any.We, however, find a small loophole in this argument, which indeed shows an obstruc-tion in non-linear sigma models in d = 2 + ǫ dimensions. The loophole is related to thequestion if the potential function for the gradient flow is bounded under the presence ofthe ambiguities in the beta functions. Even though the simple idea could have failed dueto the loophole, we can still show that scale invariance does imply conformal invariance ofnon-linear sigma models in d = 2 + ǫ dimension from the work by Perelman on the Ricciflow [19]. This, on the other hand, suggests that a general argument without a loopholewould be quite non-trivial: at least it should directly imply Perelman’s theorem on the Indeed, we do have an example of scale invariant but not conformal invariant field theories such as afree U (1) gauge theory in three dimensions [7], so making the condition more precise is imperative. We study a renormalization group flow of a local quantum field theory with Poincar´einvariance. The properties of the renormalization group flow is characterized by the betafunctions that appear in the trace of the energy-momentum tensor.Consider a general structure of the trace of the energy-momentum tensor (in flatspace-time) T µµ = β I O I + s a ∂ µ J a + τ i ✷ Φ i . (1)By using identities in a given field theory such as the non-conservation of the vectoroperator ∂ µ J µa = f Ia O I , it is more convenient to rewrite the right hand side as T µµ = B I O I . (2)We will use this scheme to evolve the coupling constant under the renormalization groupflow: dg I dt = B I ( g ). Scale invariance demands β I = 0 while conformal invariance demands B I = 0. If s a ∂ µ J µa is non-zero, it is sometimes called the Virial current. In most situations,one may remove τ i by adding local counterterms to the action, but sometimes it givesa non-trivial consequence by adding further ambiguities in the definition of the betafunctions.In even dimensions d = 2 n , there is a general argument that at the scale invariantfixed point all B I (rather than β I ) vanish, and the conformal invariance follows. One suchargument is based on the gradient property of the beta functions. It claims that the betafunctions are generated by a gradient flow: dg I dt = B I = χ IJ ∂a∂g J , (3)with respect to a certain potential function a ( g ), where we assume χ IJ ( g ) is positivedefinite. If this is the case, we can show dadt = B I ∂a∂g I = B I χ IJ B J ≥ , (4)2here χ IJ is an inverse of χ IJ . In other words, a ( g ( t )) is monotonically decreasing alongthe renormalization group flow. In d = 2 n dimensions, a ( g ) at the conformal fixed point is the Weyl anomaly coefficientwhich is positive definite. Therefore, if the theory under consideration can be deformedto be gapped, a ( g ) cannot decrease forever. In the perturbative regime, we can arguethat scale invariance demands dadt is (at the worst) constant, but the constant must bezero since a ( g ) is bounded. Then the positivity of χ IJ demands B I are all zero, implyingthat the scale invariant fixed points are actually conformal invariant.In the literature, there have been substantial works on how to implement the abovescenario in a concrete manner. We also realize that there are various subtle issues (e.g. if χ IJ can remain positive definite beyond the perturbation theories). We are not going toreview such issues, but we refer [20] for a review.In this paper, we simply assume the gradient flow nature of the beta functions in d = 2 n dimensions, and we would like to see if we can extend the above analysis in d = 2 n + ǫ dimensions. When we use the dimensional regularization with minimal subtraction, thebeta functions ˜ B I in d = 2 n + ǫ dimension and that of d = 2 n dimensions B I are relatedby ˜ B I = ǫk I + B I , (5)where k I may depend on the operator under consideration. We also note that this simplerelation only holds in a particular renormalization scheme, and we will commit ourselvesto such a scheme in the following.Let us further assume we are working in the perturbative regime so that we may regardthe field space metric as a unit matrix χ IJ = δ IJ . Then, if B I is a gradient flow, ˜ B I is agradient flow as well ˜ B I = χ IJ ∂ ˜ a∂g J , (6) Our convention is t = log Λ with cut-off Λ, and large t corresponds to ultraviolet. Throughout thepaper, we use the conventional term “monotonically decreasing” along the renormalization group flow,but it actually means monotonically increasing with respect to t . For example, if we consider Yukawa- φ theory in d = 4 − ǫ dimensions, the Yukawa coupling has k = 1 / k = 1. a ( g ) = ǫ k I g I g I + a ( g ). Note that the gradient extension might fail beyond theperturbation theory in which χ IJ can be regarded as a constant. Now we can repeat the analysis in d = 2 n dimensions. If ˜ a ( g ) were bounded, then wecould argue ˜ B I = 0 at the scale invariant fixed point and then we would conclude thatthe fixed point is conformal invariant. Here is, however, a small loophole. In d = 2 n dimensions, a ( g ) has a clear physical meaning such as the Weyl anomaly coefficient andit has a manifest positivity at the conformal fixed point. In d = 2 n + ǫ dimension, theprecise physical meaning of ˜ a ( g ) is unclear at this point and it could be unbounded.Let us take a look at an example. In d = 4 dimensions, the φ theory with the couplingconstant λ abcd φ a φ b φ c φ d has the beta function B abcd = 116 π ( λ abef λ efcd + λ acef λ efbd + λ adef λ efbc ) (7)so that in d = 4 + ǫ dimensions (being careful about the sign convention of ǫ ), we have˜ B abcd = + ǫλ abcd + 116 π ( λ abef λ efcd + λ acef λ efbd + λ adef λ efbc ) . (8)This is a gradient flow with respect to the potential˜ a = + ǫ λ abcd λ abcd + 116 π λ abcd λ cdef λ efab . (9)We see that ˜ a is monotonically decreasing along the (physical) renormalization group flow.We also see that at the scale invariant fixed point, we have ˜ B a = 0 with the enhancedconformal invariance. This is a favorable situation in which the monotonicity of ˜ a givesproof of conformal invariance. d = 2 + ǫ dimensions It is widely believed that the infrared renormalization group fixed point of the scalar φ theory in d = 4 + ǫ dimensions (with negative ǫ ) and the ultraviolet renormalization groupfixed point of non-linear sigma models in d = 2 + ǫ dimensions are in the same universalityclass if we extrapolate them to three dimensions. Since we have seen that the fixed pointsof φ theories are conformal invariant in the d = 4 + ǫ dimensions, we expect that thefixed points of the non-linear sigma models in d = 2 + ǫ dimensions are also conformalinvariant. By perturbation theory, we mean that we are close to a (conformal) fixed point. It does not necessarilymean that we are close to the Gaussian fixed point. .1 A direct approach Let us consider the non-linear sigma mode defined by the classical action S = Z d d xG MN ∂ µ X M ∂ µ X N (10)whose target space M is a D -dimensional compact manifold with the metric G MN ( X ).In two dimensions, it is well-known that the one-loop beta function is given by the Riccitensor R MN ( X ) constructed out of G MN ( X ) B MN = dG MN dt = R MN (11)up to the ambiguity of the beta functions that can be added to the right hand side(i.e. D M ∂ N Φ( X ) + D N ∂ M Φ( X )) [21][22]. This ambiguity is associated with the dilatoncoupling R (2) ( x )Φ( X ) or improvement of the energy-momentum tensor. Here R (2) ( x ) isthe curvature of the d dimensional “world-sheet”. In 2 + ǫ dimensions, the one-loop beta function becomes (again up to ambiguities)˜ B MN = − ǫG MN + R MN (12)and the condition for scale invariance is ǫG MN = R MN + D M V N + D N V M (13)for a particular vector field V N ( X ) on M with the covariant derivative D M . Note thatthe term D M V N + D N V M is the diffeomorphism induced by the vector field V M (i.e. Liederivative of the metric), so the target space is “physically the same” with or without it. If V M is a gradient vector: V M = ∂ M F ( X ) for a certain scalar function F ( X ) on M ,then the scale invariant fixed point is conformal invariant because one can always removeit from the above ambiguity of the beta function. In [2], it was directly shown that F = 0when ǫ = 0 (even without using the ambiguity just mentioned). We would like to show asimilar result when ǫ = 0.Acting D M D N on (13) and using the Bianchi identity as well as (13) again, we obtain D M D M R + V M D M R = − R MN R MN + 2 ǫR . (14) We would like to avoid a confusion with the target space Ricci scalar constructed out of G IJ . An interesting application of this vector field can be found in [23]. R = G MN R MN is the Ricci scalar. Let us pick a point p such that R takes theminimum value on M . Since D M D M R ≥ D M R = 0 at p , the left hand side of (14)is non-negative. On the other hand, the right hand side can be rewritten as − R MN R MN + 2 ǫR = − R MN − RD g MN )( R MN − RD g MN ) − R ( RD − ǫ ) . (15)Here R MN − RD g MN is the traceless Ricci tensor. We will show that the right hand sideis non-positive when ǫ ≤ R d D x √ G RD = ǫ R d D x √ G , so ǫ is given by themean curvature (divided by D ). However, the minimum of the curvature is smaller thanits mean, so R ( p ) D ≤ ǫ ≤
0. Thus the right hand side of (15) is a sum of two non-positiveterms, and the both must vanish. It means that R = Dǫ is a global constant, and R MN = RD g MN = ǫg MN , showing V M = 0. As we have promised F = 0, and the scaleinvariant fixed point is conformal invariant. The target space is what is called the Einsteinmanifold.This nice argument does not apply when ǫ > V M = 0even without considering the possibility that it could be a gradient V M = ∂ M F . On thecontrary, it is known that when ǫ > V M = ∂ M F , and this approach must fail. We had to invent a more elaborate argumentto show that scale invariance implies conformal invariance when ǫ > Given success of Zamolodchikov’s c -theorem in two dimensions [26], it is somewhat sur-prising that the explicit form of the monotonically decreasing c -function with the gradientbeta functions for the non-linear sigma model was only available after the seminal workby Perelman [19] (see also related works [27][28][29][30][31]).We consider the following D -dimensional target space action S [ G, φ ] = Z d D X √ Ge − φ ( R + 4 ∂ M φ∂ M φ ) (16) The idea that the traceless Ricci tensor is useful here is inspired by the work by Hamilton [24]. The first compact one was discovered by Koiso [25]. We will also see a non-compact example later. c -function is defined by the minimum of C [ G ] = − inf φ S [ G, φ ] by varying φ thatsatisfies the normalization condition Z d D X √ Ge − φ = 1 . (17)The target space action (16) is closely related to the effective action of the string theory.There φ is identified with a string dilaton and unconstrained, but here it is important toimpose the normalization condition (17). To make it distinguished, it is sometimes calledPerelman’s dilaton or minimizer φ m .This action can be used to derive the monotonic gradient flow of the beta function G IM G JN e φ m √ G δC [ G ] δG IJ = R MN + D M ∂ N φ m + D N ∂ M φ m , (18)where Perelman’s dilaton φ m is not arbitrary but is fixed from G MN by minimizing S [ G, φ ].Remarkably this is identified with the beta function B MN of the metric, and in the par-ticular scheme the renormalization group flow is generated by a gradient flow.Let us now argue scale invariance implies conformal invariance. In two dimensions,we see that at the scale invariant fixed point (18) must vanish to guarantee dC [ G ] dt = 0,and it directly implies the conformal invariance. Actually, repeating the argument in theprevious subsection, we can further prove φ m = const.In d = 2 + ǫ dimensions, the beta function in the minimal subtraction scheme is givenby B MN = − ǫg MN + R MN + D M ∂ N Φ + D M ∂ N Φ . (19)Here Φ( X ) is an arbitrary scalar function on M .Now, as we discussed in section 2 we may introduce the c -function in d = 2 + ǫ dimensions by ˜ C [ G ] = − ǫ Z d D xe − φ m √ G + C [ G ] . (20)Here, in the first line, we do not vary φ , which is already fixed in computing C [ G ]. Thisclearly gives a monotonically decreasing gradeint flow in 2 + ǫ dimensions G IM G JN e φ m √ G δ ˜ C [ G ] δG IJ = − ǫG MN + R MN + D M ∂ N φ m + D N ∂ M φ m , (21)in a particular scheme where the ambiguity Φ in the beta function is fixed by Perelman’sdilaton. 7ne may ask if this gives the proof that scale invariance implies conformal invariancein 2 + ǫ dimensions. The problem is that ˜ C [ G ] is monotonically decreasing only for aparticular φ m . We also do not know if ˜ C [ G ] must be a constant at the scale invariantfixed point.To see the difficulty in an example, let us consider the case with G MN = δ MN . It issomewhat surprising but crucial to notice here that B MN is zero only if we supplementnon-trivial Φ = ǫ δ MN X M X N . On the other hand, when we consider the flow from (21),the Perelman’s dilaton φ m is essentially derived in two dimensions so the obvious solutionhere is φ = const. This means that even if we have a scale invariant field theory, the c -function ˜ C [ G ] is monotonically decreasing forever. This is nothing but the loopholewe have mentioned in section 2.
In the seminal paper [19], Perelman introduced the other monotonically decreasing func-tional, which he called the entropy. The direct renormalization group interpretation ofPerelman’s entropy in non-linear sigma models in two dimension was not obvious, but wefind that it has a direct connection with conformal invariance of non-linear sigma modelsin d = 2 + ǫ dimensions. We will map the problem of finding a scale but not conformalfixed point in the non-linear sigma model in d = 2 + ǫ dimension to the renormalizationgroup flow in two-dimensions. Then we see that the stationarity of Perelman’s entropyimplies conformal invariance in d = 2 + ǫ dimensions for ǫ > d = 2 + ǫ dimensions to a non-trivialrenormalization group flow in two dimensions. We will assume ǫ >
0. In d = 2 + ǫ dimensions, scale invariance implies that the metric satisfies ǫG MN = R MN + D M V N + D N V M (22)for a certain vector field V N . Let us now define the time-dependent metric G MN ( t ) for t > G MN that satisfies (22): G MN ( t ) = ǫtφ ∗ V ( G MN ( x )), where thepullback φ ∗ V is induced by the diffeomorphism ˜ x M = x M − ǫ − log( t ) V M . In mathematics literature, it is known as the Gaussian Ricci soliton. Indeed it is given by − e − Dǫt V and the would-be fixed point is a singular metric of G MN = 0. R IJ ( G ) = R IJ ( αG )), near t = 1 the time-dependent metric G MN ( t ) satisfies the Ricci-flow equation dG MN ( t ) dt = R MN ( t ) , (23)where R MN ( t ) is the Ricci tensor for G MN ( t ). This time evolution is nothing but therenormalization group equation in two dimensions. In this way, we have mapped a scaleinvariant renormalization group fixed point in non-linear sigma models d = 2 + ǫ dimen-sional to a particular renormalization group flow in two dimensions. We may now want to study the renormalization group flow of G MN ( t ) in the senseof the auxiliary two-dimensional non-linear sigma model. We expect that it shows themonotonic behavior under the conventional c -function (or its generalization discussed inthe previous section), but it is less useful in our setup because the metric typically blowsup. At this point, Pelerman introduced the other monotonically decreasing quantity,which he called the entropy. Consider the functional which explicitly depends on t : S [ t ; G MN ( t ) , φ ( t )]= − Z d D X p G ( t ) (cid:0) t (4 ∂ M φ ( t ) ∂ M φ ( t ) + R ( t )) + 2 φ ( t ) − D (cid:1) (4 πt ) − D e − φ ( t ) . (24)The claim is that this functional is monotonically decreasing along the renormalizationgroup flow. Note that Zamolodchikov’s c -function does not depend on t explicitly so itcannot be identified with the conventional c -function.We study the time-dependence of this functional under the generalized Ricci flow dG MN ( t ) dt = R MN ( t ) dφ ( t ) dt = 12 ✷ φ − ∂ M φ∂ M φ + R − D t . (25)The direct computation gives dS [ t ; G MN ( t ) , φ ( t )] dt The discussion that follows does not explicitly use the fact that ǫ is small, but since we are neglectingthe higher terms in the renormalization group beta functions, we effectively assume that ǫ is small. The time-dependence is motivated as follows: we start with the gradient flow dG MN dt = R MN + D M ∂ N φ + D N ∂ M φ under the fixed measure √ G (4 πt ) − D e − φ . The time-dependence of dφ ( t ) dt = ✷ φ + R − D t is induced from the time-independence of the measure. Then we supplement the diffeomorphismof V M = D M φ to make them the Ricci flow as in (24). Z d D x p G ( t ) t (cid:18) R MN ( t ) + 2 D M D N φ ( t ) − t G MN ( t ) (cid:19) (4 πt ) − D e − φ . (26)Thus for t > S [ t ; G MN ( t ) , φ ( t )] is monotonically decreasing along the renormaliza-tion group flow (i.e. monotonically increasing with respect to t ). In particular, if S [ t ; G MN ( t ) , φ ( t )] is stationary, it satisfies R MN ( t ) + 2 D M D N f ( t ) − t G MN ( t ) = 0 (27)for a particular f .We emphasize here that the fixed point of S [ t ; G MN ( t ) , φ ( t )] does not correspond to therenormalization group fixed point of two-dimensional non-linear sigma models. Rather,it is related to a conformal invariant fixed point of non-linear sigma models in d = 2 + ǫ as we will explain.Let us now argue that scale invariant fixed point in 2 + ǫ dimension is conformal in-variant from the monotonic properties of the entropy functional. For this purpose, wemaximize S [ t, G MN ( t ) , φ ( t )] over φ ( t ) under the condition √ G (4 πt ) − D e − φ is fixed. Theresulting ¯ S [ t, G MN ( t )] = sup φ S [ t, G MN ( t ) , φ ( t )] is also monotonically decreasing alongthe renormalization group flow. Now we note that S [ t, G MN ( t )] is invariant under simul-taneous scale change of G MN ( t ) and t (i.e. ( G MN , t ) → α ( G MN , t )). We also note that¯ S [ t, G MN ( t )] is invariant under the diffeomorphism on G MN ( t ) thanks to the minimizationover φ ( t ).Due to these two properties of ¯ S [ t, G MN ( t )], for the Ricci-flow solution induced fromthe scale invariant fixed point in d = 2 + ǫ dimensions, we find that ¯ S [ t, G MN ( t )] isa constant near t = 1 since the time evolution of G MN ( t ) is generated by the scaletransformation and the diffeomorphism.On the other hand, for generic Ricci flow, we know that the time-dependence is givenby (26). When it is stationary, it means R MN ( t ) + 2 D M D N f ( t ) − t G MN ( t ) = 0 , (28)for a particular f ( t ) that corresponds to the minimizer. However, at t = 1 the conditioncan be rewritten in terms of the original metric G MN as ǫG MN = R MN + 2 D M D N F . (29)10his implies that the vector field V M = ∂ M F is a gradient and the scale invariant fixedpoint in d + ǫ dimension is conformal invariant.As we have alluded above, unlike the case with ǫ <
0, we cannot conclude that F isconstant. Indeed, the manifold that satisfies the condition (29) is known as a gradientshrinking Ricci soliton (for positive ǫ ) and some non-trivial examples are available in theliterature (see e.g. [25]). On the other hand, for negative ǫ , it is known as a gradientexpanding Ricci soliton, but we have already seen that they must be Einstein manifoldand trivial (i.e. F = 0). We have shown that scale invariance implies conformal invariance in non-linear sigmamodels in d = 2 + ǫ dimensions by using the mathematical result on the Ricci flow byPerelman. The monotonicity of Perelman’s entropy along the renormalization group flowplays a crucial role, but it is not directly related to the renormalization group c -functionin two dimensions because it explicitly depends on time. It is not the renormalizationgroup c -function in d = 2+ ǫ dimensions either because it is only defined for scale invarianttheories. It, however, indicates whether the fixed point in d = 2+ ǫ dimensions is conformalinvariant or merely scale invariant.It would be interesting to see if a similar function exists in other field theories thannon-linear sigma models at one-loop. In particular, Perelman’s idea to map the scaleinvariant fixed point in d = 2 + ǫ dimension to the non-trivial renormalization group flowin two-dimension is not conventional in physics but may be of potential significance.For the success of the mapping, it was crucial that the Ricci tensor is invariant underthe rescaling of the metric. The similar thing may happen in one-loop gauge theories in d = 4 + ǫ dimensions. Suppose they are at the renormalization group fixed point0 = − ǫg − + β , (30)where β is a constant. We may now define the associated beta function in four dimensionsfrom g − ( t ) = ǫtg − ∗ . It satisfies the d = 4 dimensional renormalization group equation dg − ( t ) dt = β (31)11t one-loop. Note that it was crucial that β is a constant and does not depend on g .Of course, at this point, we do not know if the analog of Perelman’s entropy exists.Also, we admit that the mapping may not work at the higher loop order. Both in non-linear sigma models and gauge theories, the two-loop term (e.g. R MI R IN ) is not invariantunder the rescaling of the coupling constant, so the naive mapping does not work. It istherefore an interesting question to show conformal invariance of non-linear sigma modelsin d = 2 + ǫ dimensions beyond the one-loop approximation. Acknowledgements
This work is in part supported by JSPS KAKENHI Grant Number 17K14301.
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