Conformal invariance from scale invariance in non-linear sigma models

There exists a certain argument that in even dimensions, scale invariant quantum field theories are conformal invariant. We may try to extend the argument in $2n + \epsilon$ dimensions, but the naive extension has a small loophole, which indeed shows an obstruction in non-linear sigma models in $2+\epsilon$ dimensions. Even though it could have failed due to the loophole, we show that scale invariance does imply conformal invariance of non-linear sigma models in $2+\epsilon$ dimension from the seminal work by Perelman 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 conformal symmetry emerges. Empirically, it is typically the case that scale invariance, Poincaré invariance (Euclidean invariance), and unitarity (reflection positivity) give rise to the enhanced conformal symmetry. Some argument supporting this empirical fact exist in even space-time dimensions, in particular two [2] and four dimensions [3] [4][5] [6], but we do not have general argument in odd dimensions, say in three dimensions. 1 In the perturbative regime, the (non-)existence of scale invariant but not conformal field theory is closely related to the gradient nature of the renormalization group flow and the absence of the limit cycle [8] [9][10] [11] [12][13] [14][15] [16]. Again, we do have supporting evidence for the gradient nature of the renormalization group flow in two and four dimensions. A crucial fact here is that the potential function for the gradient flow is given by Weyl anomaly coefficients at the conformal fixed point. They do exist in even dimensions but they do not exist in odd dimensions.
Without a general argument, it may be a natural idea to explore conformal invariance in odd dimensions by using the extrapolation of d = 2n+ǫ dimensions. Such approaches in various field theories are attempted in [2][17] [18]. In this paper, we offer general discussions on how to obtain a gradient flow of the renormalization group beta function in d = 2n + ǫ dimensions once we know that it is a gradient flow in d = 2n dimensions. This typically implies conformal invariance in (perturbative) scale invariant fixed point in d = 2n + ǫ dimensions if any.
We, however, find a small loophole in this argument, which indeed shows an obstruction in non-linear sigma models in d = 2 + ǫ dimensions. The loophole is related to the question if the potential function for the gradient flow is bounded under the presence of the ambiguities in the beta functions. Even though the simple idea could have failed due to the loophole, we can still show that scale invariance does imply conformal invariance of non-linear sigma models in d = 2 + ǫ dimension from the work by Perelman on the Ricci flow [19]. This, on the other hand, suggests that a general argument without a loophole would be quite non-trivial: at least it should directly imply Perelman's theorem on the Ricci flow.

A simple argument and possible loopholes
We study a renormalization group flow of a local quantum field theory with Poincaré invariance. The properties of the renormalization group flow is characterized by the beta functions that appear in the trace of the energy-momentum tensor.
Consider a general structure of the trace of the energy-momentum tensor (in flat space-time) By using identities in a given field theory such as the non-conservation of the vector operator ∂ µ J µ a = f I a O I , it is more convenient to rewrite the right hand side as We will use this scheme to evolve the coupling constant under the renormalization group flow: dg I dt = B I (g). Scale invariance demands β I = 0 while conformal invariance demands B I = 0. If s a ∂ µ J µ a 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 gives a non-trivial consequence by adding further ambiguities in the definition of the beta functions.
In even dimensions d = 2n, there is a general argument that at the scale invariant fixed point all B I (rather than β I ) vanish, and the conformal invariance follows. One such argument is based on the gradient property of the beta functions. It claims that the beta functions are generated by a gradient flow: with respect to a certain potential function a(g), where we assume χ IJ (g) is positive definite. If this is the case, we can show where χ IJ is an inverse of χ IJ . In other words, a(g(t)) is monotonically decreasing along the renormalization group flow. 2 In d = 2n dimensions, a(g) at the conformal fixed point is the Weyl anomaly coefficient which is positive definite. Therefore, if the theory under consideration can be deformed to be gapped, a(g) cannot decrease forever. In the perturbative regime, we can argue that scale invariance demands da dt is (at the worst) constant, but the constant must be zero since a(g) is bounded. Then the positivity of χ IJ demands B I are all zero, implying that the scale invariant fixed points are actually conformal invariant.
In the literature, there have been substantial works on how to implement the above scenario 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 to review 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 = 2n dimensions, and we would like to see if we can extend the above analysis in d = 2n + ǫ dimensions. When we use the dimensional regularization with minimal subtraction, the beta functionsB I in d = 2n + ǫ dimension and that of d = 2n dimensions B I are related byB where k I may depend on the operator under consideration. 3 We also note that this simple relation only holds in a particular renormalization scheme, and we will commit ourselves to such a scheme in the following.
Let us further assume we are working in the perturbative regime so that we may regard the field space metric as a unit matrix χ IJ = δ IJ . Then, if B I is a gradient flow,B I is a gradient flow as wellB 2 Our convention is t = log Λ with cut-off Λ, and large t corresponds to ultraviolet. Throughout the paper, we use the conventional term "monotonically decreasing" along the renormalization group flow, but it actually means monotonically increasing with respect to t. 3 For example, if we consider Yukawa-φ 4 theory in d = 4 − ǫ dimensions, the Yukawa coupling has k = 1/2 while the scalar quartic coupling has k = 1.
whereã(g) = ǫ 2 k I g I g I + a(g). Note that the gradient extension might fail beyond the perturbation theory in which χ IJ can be regarded as a constant. 4 Now we can repeat the analysis in d = 2n dimensions. Ifã(g) were bounded, then we could argueB I = 0 at the scale invariant fixed point and then we would conclude that the fixed point is conformal invariant. Here is, however, a small loophole. In d = 2n dimensions, a(g) has a clear physical meaning such as the Weyl anomaly coefficient and it has a manifest positivity at the conformal fixed point. In d = 2n + ǫ dimension, the precise physical meaning ofã(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 4 theory with the coupling so that in d = 4 + ǫ dimensions (being careful about the sign convention of ǫ), we havẽ This is a gradient flow with respect to the potential We see thatã is monotonically decreasing along the (physical) renormalization group flow.
We also see that at the scale invariant fixed point, we haveB a = 0 with the enhanced conformal invariance. This is a favorable situation in which the monotonicity ofã gives proof of conformal invariance. 4 By perturbation theory, we mean that we are close to a (conformal) fixed point. It does not necessarily mean that we are close to the Gaussian fixed point.

A direct approach
Let us consider the non-linear sigma mode defined by the classical action In two dimensions, it is well-known that the one-loop beta function is given by the Ricci up to the ambiguity of the beta functions that can be added to the right hand side [22]. This ambiguity is associated with the dilaton In 2 + ǫ dimensions, the one-loop beta function becomes (again up to ambiguities) and the condition for scale invariance is for a particular vector field V N (X) on M with the covariant derivative D M . Note that derivative of the metric), so the target space is "physically the same" with or without it. 6 If V M is a gradient vector: then the scale invariant fixed point is conformal invariant because one can always remove it from the above ambiguity of the beta function. In [2], it was directly shown that F = 0 when ǫ = 0 (even without using the ambiguity just mentioned). We would like to show a similar result when ǫ = 0.
Acting D M D N on (13) and using the Bianchi identity as well as (13) again, we obtain 5 We would like to avoid a confusion with the target space Ricci scalar constructed out of G IJ . 6 An interesting application of this vector field can be found in [23].
Here R = G M N R M N is the Ricci scalar. Let us pick a point p such that R takes the minimum value on M. Since D M D M R ≥ 0 and 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 Here R M N − R D g M N is the traceless Ricci tensor. 7 We will show that the right hand side is non-positive when ǫ ≤ 0.
Indeed, the trace of (13)  This nice argument does not apply when ǫ > 0 and we cannot assign a definite sign on the right hand side of (14). If this were literally true, we could conclude V M = 0 even without considering the possibility that it could be a gradient V M = ∂ M F . On the contrary, it is known that when ǫ > 0 there does exist a solution of (13) with non-trivial 8 and this approach must fail. We had to invent a more elaborate argument to show that scale invariance implies conformal invariance when ǫ > 0.

A gradient approach 1
Given success of Zamolodchikov's c-theorem in two dimensions [26], it is somewhat surprising that the explicit form of the monotonically decreasing c-function with the gradient beta functions for the non-linear sigma model was only available after the seminal work by Perelman [19] (see also related works [27]

[28][29][30][31]).
We consider the following D-dimensional target space action 7 The idea that the traceless Ricci tensor is useful here is inspired by the work by Hamilton [24]. 8 The first compact one was discovered by Koiso [25]. We will also see a non-compact example later. and the c-function is defined by the minimum of C[G] = −inf φ S[G, φ] by varying φ that satisfies the normalization condition 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 to impose the normalization condition (17). To make it distinguished, it is sometimes called Perelman's dilaton or minimizer φ m .
This action can be used to derive the monotonic gradient flow of the beta function where Perelman's dilaton φ m is not arbitrary but is fixed from G M N by minimizing S[G, φ].
Remarkably this is identified with the beta function B M N of the metric, and in the particular 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)  In d = 2 + ǫ dimensions, the beta function in the minimal subtraction scheme is given by 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 byC Here, in the first line, we do not vary φ, which is already fixed in computing C[G]. This clearly gives a monotonically decreasing gradeint flow in 2 + ǫ dimensions in a particular scheme where the ambiguity Φ in the beta function is fixed by Perelman's dilaton.
One may ask if this gives the proof that scale invariance implies conformal invariance in 2 + ǫ dimensions. The problem is thatC[G] is monotonically decreasing only for a particular φ m . We also do not know ifC[G] must be a constant at the scale invariant fixed point.
To see the difficulty in an example, let us consider the case with G M N = δ M N . It is somewhat surprising but crucial to notice here that B M N is zero only if we supplement 9 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 solution here is φ = const. This means that even if we have a scale invariant field theory, the c-functionC[G] is monotonically decreasing forever. 10 This is nothing but the loophole we have mentioned in section 2.

A gradient approach 2
In the seminal paper [19], Perelman introduced the other monotonically decreasing functional, which he called the entropy. The direct renormalization group interpretation of Perelman's entropy in non-linear sigma models in two dimension was not obvious, but we find that it has a direct connection with conformal invariance of non-linear sigma models in d = 2 + ǫ dimensions. We will map the problem of finding a scale but not conformal fixed point in the non-linear sigma model in d = 2 + ǫ dimension to the renormalization group flow in two-dimensions. Then we see that the stationarity of Perelman's entropy implies conformal invariance in d = 2 + ǫ dimensions for ǫ > 0.
Let us first map a scale invariant fixed point in d = 2 + ǫ dimensions to a non-trivial renormalization group flow in two dimensions. We will assume ǫ > 0. In d = 2 + ǫ dimensions, scale invariance implies that the metric satisfies Since Ricci tensor is invariant under the rescaling of the metric (i.e. R IJ (G) = R IJ (αG)), near t = 1 the time-dependent metric G M N (t) satisfies the Ricci-flow equation where R M N (t) is the Ricci tensor for G M N (t). This time evolution is nothing but the renormalization group equation in two dimensions. In this way, we have mapped a scale invariant renormalization group fixed point in non-linear sigma models d = 2 + ǫ dimensional to a particular renormalization group flow in two dimensions. 11 We may now want to study the renormalization group flow of G M N (t) in the sense of the auxiliary two-dimensional non-linear sigma model. We expect that it shows the monotonic behavior under the conventional c-function (or its generalization discussed in the previous section), but it is less useful in our setup because the metric typically blows up. At this point, Pelerman introduced the other monotonically decreasing quantity, which he called the entropy. Consider the functional which explicitly depends on t: The claim is that this functional is monotonically decreasing along the renormalization group flow. Note that Zamolodchikov's c-function does not depend on t explicitly so it cannot be identified with the conventional c-function.
We study the time-dependence of this functional under the generalized Ricci flow 12 The direct computation gives 11 The discussion that follows does not explicitly use the fact that ǫ is small, but since we are neglecting the higher terms in the renormalization group beta functions, we effectively assume that ǫ is small. 12 The time-dependence is motivated as follows: we start with the gradient flow dGMN Thus for t > 0, S[t; G M N (t), φ(t)] is monotonically decreasing along the renormalization group flow (i.e. monotonically increasing with respect to t). In particular, if for a particular f . Let us now argue that scale invariant fixed point in 2 + ǫ dimension is conformal invariant from the monotonic properties of the entropy functional. For this purpose, we On the other hand, for generic Ricci flow, we know that the time-dependence is given by (26). When it is stationary, it means for a particular f (t) that corresponds to the minimizer. However, at t = 1 the condition can be rewritten in terms of the original metric G M N as This implies that the vector field V M = ∂ M F is a gradient and the scale invariant fixed point in d + ǫ dimension is conformal invariant.
As we have alluded above, unlike the case with ǫ < 0, we cannot conclude that F is constant. Indeed, the manifold that satisfies the condition (29) is known as a gradient shrinking Ricci soliton (for positive ǫ) and some non-trivial examples are available in the literature (see e.g. [25]). On the other hand, for negative ǫ, it is known as a gradient expanding Ricci soliton, but we have already seen that they must be Einstein manifold and trivial (i.e. F = 0).

Discussions
We have shown that scale invariance implies conformal invariance in non-linear sigma models in d = 2 + ǫ dimensions by using the mathematical result on the Ricci flow by where β 0 is a constant. We may now define the associated beta function in four dimensions from g −2 (t) = ǫtg −2 * . It satisfies the d = 4 dimensional renormalization group equation at one-loop. Note that it was crucial that β 0 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 nonlinear sigma models and gauge theories, the two-loop term (e.g. R M I R I N ) is not invariant under the rescaling of the coupling constant, so the naive mapping does not work. It is therefore an interesting question to show conformal invariance of non-linear sigma models in d = 2 + ǫ dimensions beyond the one-loop approximation.