RG flow of a Calabi-Yau sigma model inside a given Kähler class

Question

Let X be a complex manifold with $c_{1}(X)=0$. Let g be a Kähler metric on X. Let us consider the N=(2,2) 2d sigma model of target (X,g). The metric g can be thought as a coupling constant of the sigma model. Classically, the sigma model is conformally invariant but it is not true at the quantum level, there is in general a non-trivial beta function for g. The hypothesis $c_{1}(X)=0$ implies by a non-renormalization theorem that there is no renormalization of the Kähler class k = [g]  of g. But under the RG flow, g can still vary in the infinite dimensional space of metrics of given Kähler class k. Using the Yau theorem on the existence of Ricci-flat metrics and correcting this metric order by order in perturbation theory (Nemeschansky, Sen: http://ccdb5fs.kek.jp/cgi-bin/img/allpdf?198605369), it is possible to show that in a given Kähler class, there exists a unique metric where the beta function vanishes, i.e. where the sigma model is conformally invariant. This is generally the useful result to construct N=(2,2) 2d CFT from a sigma-model.

Still, we can ask, what is the dynamic of the sigma model if one starts with any metric in the given Kähler class? By general RG flow, I would expect to have a CFT fixed point in the UV and a CFT fixed point in the IR. But there is a unique CFT point in a given Kähler class and by Zamolochikov's theorem, it is not possible to have the same UV and IR CFT if we have something not a CFT between the two. What is the solution of this apparent puzzle?

Maybe it is simply that the metric g is no longer a good variable in one of the two limits UV-IR but I don't know what it would be. In particular, in a fixed Kähler class, I see no natural "limit" : X has always the same volume...

Comments

Since the theory is strongly coupled in one of these limits (depending on the sign of the curvature), is there any reason to expect that that fixed point is even the same sort of theory?

Answer 1

Recall that under RG flow, the c-theorem states that the central charge is a monotonically decreasing function. So if you have two CFT's connected by RG flow, the following can happen.

1. The central charge remains unchanged. Then, one expects that the two CFT's are connected by a "line" of CFT's obtained by deforming (any) one of the CFT's by a truly marginal operator. Such a situation does happen in the CFT associated with the nonlinear sigma model (NLSM) with target space, a Calabi-Yau threefold. There are two kinds of deformations -- those that deform the Kahler class and those that change the complex structure. The precise numbers of these are given by the Hodge numbers, $h_{1,1}$ and $h_{2,1}$, respectively of the Calabi-Yau threefold. (See Brian Greene's TASI lectures)
2. The second situation is when the central charges are different. This occurs when we consider a CFT perturbed by a relevant operator. This is best illustrated by the LG-CY correspondence discussed originally by Greene-Vafa-Warner. One views the full super potential as a marginal perturbation. The intermediate points in the flow are not conformal. At one  the LG end, one has a free field theory which for the quintic three fold has five scalar fields (chiral multiplets) and has $\hat{c}=5$ (one chiral multiplet contributes $\hat{c}=1$) while  the CY3 CFT is obtained as the fixed point obtained after turning on a quintic super potential has $\hat{c}=3$. 

The analysis of Nemeschansky-Sen can be thought of as one in the neighbourhood of the CY3 fixed point. So the key for the LG-CY3 correspondence to work is that the RG flow falls into the correct Kahler class of the CY3.  There might be a discussion of this in Lerche-Vafa-Warner.

Comments

I don't think that what you have written in the case 2 is correct: in the LG/CY correspondence, we have a moduli space of CFTs with some end identified with the moduli space of CFT sigma models of case 1 and some end identified with LG models. In the quintic case, the CFT sigma model has $\hat{c}=3$ (because 3 complex dimensions) and the LG point is a (orbifold of) a tensor product of 5 minimal models with quintic potential, each with $\hat{c} = 1 - 2/5=3/5$ so the total $\hat{c}$ is 3. So I think the LG/CY correspondence is an example of case 1: two CFTs of same central charge related by a family of CFTs. An example of case 2 is the correspondence LG/ sigma model on X with X not Calabi-Yau. There the RG flow acts non-trivially on the Kähler class and the central charges are different between the UV and the IR. Exemple: X complex projective space of dimension n, $\hat{c}=n$ in the UV, in the IR: n+1 massive vacua so the IR limit CFT is trivial $\hat{c}=0$.

In fact, my question is on the CY case. As you write, there is a family of CFT parametrized by the Kähler classes and my question is about the RG flow dynamic of the metric inside a given Kähler class.  

---

There is a huge difference between having a superpotential and not having one. An arbitrarily small coefficient will do the job. The R-charge assignment that you are giving knows about the superpotential and I am aware of that. What I have in mind is best illustrated by considering the LG model for a minimal model (with (2,2) supersymmetry). I can deform a single chiral field by turning on a superpotential $\phi^{k+2}$ for $k=1,2,\ldots$. My point is that in all these cases, one has $\hat{c}=1$ before switching on the superpotential and then the final value is determined by the precise value of $k$ that we choose. Of course, once you know which chosen value of $k$, the R-charge assignment is $2/(k+2)$.  This is the way Zamalodchikov would think of it. You turn on a relevant operator (corresponding to $\phi^{k+2}$) and then it flows to the $k$-th minimal model in the IR.

---

Thanks for your comment. Now I understand what you wanted to say in case 2. I just had a problem with the terminology "LG/CY correspondence". What you describe is the RG flow from a free theory to a LG model in the IR by turning on a superpotential. What I would call the LG/CY correspondence is the fact that this LG CFT is related by a marginal deformation to a CY sigma model CFT.

---

I will further edit my answer to remove the confusion. Thanks. 

Answer 2

I will try to give an answer to my question, which is basically an extension of the last paragraph of the question and of the comment of Ryan Thorngren.

I will limit myself to a one-loop study. To this order, the RG flow is the « Ricci flow », $\frac{dg_{ij}}{dt} = - R_{ij}$ where $R_{ij}$ is the Ricci curvature and the variable $t$ is something like - log of the energy scale. I choose this variable only by convenience: the direction of the RG flow from the UV to the IR is the same that the positive direction of the « time » $t$.

This is true for any $X$.

For a metric with positive Ricci curvature, the manifold $X$ is large in the UV, the sigma model is asymptotically free, $X$ shrinks under the RG flow and becomes strongly coupled in the IR. The perturbative sigma model description breaks down when $X$ becomes of size of order $\sqrt{\alpha'}$.

For a metric with negative Ricci curvature, the manifold $X$ is large in the IR, the sigma model is a good description in the IR. It is strongly coupled on the UV, it is not clear if the theory exists in the UV.

For a metric with zero Ricci curvature, we have a fixed point of the RG flow and the sigma model defines a well-defined CFT.

My question can be reformulated as : what happens for a metric which is a small perturbation of a Ricci flat metric, small in particular in the sense of having the same Kähler class. Naively, what happens is not clear because the Ricci curvature of such a metric is neither positive or negative, it is not of fixed sign, the Ricci curvature has fluctuations of both signs around zero. This apparent difficulty was basically the reason for my question. So now the question is: how small fluctuations of the metric evolve under the RG flow? Are there smooth out or are there amplified?

I think that the key point is the remark that $R_{ij}$ is roughly (in correct coordinates and up to non-linear terms) minus the Laplacian of g.To find the Laplacian is not surprizing because the Ricci curvature is by defintion roughly the trace of second derivatives in the metrics. The key point is the minus sign. It means that up to non-linear terms, the RG flow is roughly$\frac{dg_{ij}}{dt} = \Delta g_{ij}$ i.e the heat equation for the metric. This implies that under the RG flow, the fluctuations will be smooth out.

So a small fluctuation of the Ricci flat metric will flow in the IR to the Ricci flat fixed point. In the given Kähler class, the Ricci flat fixed point is the only fixed point and it is an attractive point: all the trajectories converge to this point in the IR.

Toward the UV, the RG flow will have exaclty the inverse behavior: if one tries to go to the UV, the fluctuations will be amplified (as for a heat equation with time inversed). If we begin with a random fluctuation and go to the UV, the total size of $X$ does not change (the Kähler class is unchanged) but the metric on $X$ will fluctuate more and more drastically and apparently chaotically. The perturbative sigma model description will break down when the typical size of the fluctuations will become of the order $\sqrt{\alpha'}$ and it is not clear if there is some definition of the theory in the UV.

Comments

This is a historical comment to explain what Nemeschansky-Sen was about. It was not about the Ricci-Flat CY3 metrics!  It was originally thought that the one-loop Ricci flatness condition was true to all orders if you had ``enough'' worldsheet supersymmetry. Then, it was discovered that there are indeed corrections to Ricci-flatness at four-loop. In other words, the Ricci-flat Calabi-Yau metric is not a solution to  the vanishing of the string beta functions. The paper of Nemeschansky-Sen explains how it is possible to correct the Ricci-flat metric (with fixed Kahler class) and make it a solution to the vanishing of the string beta functions.  The only explicit computation (that I know of) of how this works was done, for non-compact CY3's with know Ricci-Flat metrics, in http://arxiv.org/abs/hep-th/0311018. ;

---

Thanks for your comment. I know that, I have written in the question that the CFT point is obtained by taking the Ricci flat metric and by "correcting this metric order by order in perturbation theory", which is possible by Nemeschansky-Sen. In my answer, I have written "I will limit myself to a one-loop study" because it is the dominant term for most of the metrics and even if the higher order corrections shift the CFT point away from the Ricci flat metric, the general dynamic of the RG flow around this point should still be the same. Thanks for the arxiv paper, which I did not know about and which answers a question I asked myself: is there an explicit example of the Nemeshansky-Sen procedure?

---

@40227 I have a question for you. For the CY3 case, shouldn't one consider Kahler-Ricci flow rather than Ricci flow? Does the chaotic behaviour that you mention  hold for Kahler-Ricci flow?

---

The Kähler-Ricci flow is simply the Ricci flow starting from a Kähler metric. One can show that the Ricci flow preserves the Kähler condition. So the Kähler-Ricci flow is the restriction of the Ricci flow to the space of Kähler metrics. In particular, something which is true for the Ricci flow is trivially true for the Kähler-Ricci flow.

---

That can't be true in such generality. Chaotic behaviour is not for a single "trajectory" -- so it could happen that restricting to Kahler manifolds could improve the behaviour. For instance, the Kahlerity condition is important for the Nemeschansky-Sen result.

---

I am not saying that the Ricci (or Kähler-Ricci) flow has a chaotic behaviour. The only place in my answer where I use "chaotically" refers to the metric on X and not to the flow itself. The only thing I claim is that under the Ricci flow (and so also under the Kähler-Ricci flow), the fluctuations of the metric are smoothed out and that under the time inversed Ricci flow (and so also under the Kähler-Ricci flow), the fluctuations of the metrics are amplified. It is true that the Kähler-Ricci flow, by contrast with the Ricci flow, has very special properties, essentially because we can rewrite the evolution equation in terms of the Kähler potential, but I don't think that it matter for what I am saying (in what I am saying, I can replace "metric" by "Kähler potential" if one wishes but I don't have the impress that it is useful).