the generality of "c-theorems"

Question

In (I suppose all?) 2d QFTs it is known to us thanks to Zamolodchikov that (something like?) the central charge is decreasing along the renormalization group flow. This precludes anything like closed loops without fixed points.

What is known for general QFTs? In what cases do we know the RG flow is a gradient flow? In what cases does this break down? Why?

Comments

There isn't any RG flow in a CFT, right? By definition a CFT is a fixed point of the RG flow. I thought that Z's theorem is a statement about general 2D QFTs (assuming unitarity and maybe some other stuff like compactness and a gap in operator dimensions). More explicitly, we can write any QFT as a relevant deformation of the UV CFT, and then the theorem says that c decreases from the UV CFT to the IR endpoint.

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Thanks, good point! Is there a short explanation why one might need a gap in the operator dimension spectrum?

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Coincidentally I asked an expert precisely this question this afternoon and got the following answer. Suppose that there wasn't a gap. Then you could get operators arbitrarily close to dimension zero. This is not a good limit; either the CFT is a logarithmic CFT with IR problems (like a free boson), or the theory is nonunitary (since there are no dimension zero operators besides the identity in a unitary theory). I don't find this answer totally compelling but it might give some intuition.

By the way one needs compactness to rule out logarithmic CFTs. For noncompact sigma models, for example, one can get pathological behavior where T_{aa} is nonzero even though the theory is conformally invariant.

Answer 1

First of all a RG flow explicitly breaks conformal invariance. The point is what happens in fixed points. This is the result Zamolodchikov found and indeed $c_{IR} < c_{UV}$. At the fixed point this number $c$ is indeed the central charge of the CFT. Now, more generally an advance to the subject mainly in 4d QFT was made by Intriligator and Wecht using the $a$-maximization where they "almost" proved the corresponding theorem to Zamolodchikov's $c$-theorem of 2d (almost because of some subtleties of their technique, check their paper). In fixed points of the RG in 4d you get a CFT which is characterized by two central charges $a$ and $c$. In all worked out examples these guys found that indeed $a_{IR} < a_{UV}$ but $c$ was not behaving like this. Later it was found by Martelli, Sparks and Yau that the holographic dual of $a$-maximization, a procedure known as $Z$-minimization of cycles in tori, gave the same resuls, thus giving further evidence on the validity of the $a$-maximization results on $a_{IR}<a_{UV}$.  In 2011 Komargodski and Schwimmer proved the $a$-theorem in its generality. Now, people are working a lot on 3d QFT to determine the above there as well using something called $F$-maximization, proposed by Jeffereis ( this is a nice talk by Niarchos). Now various $c$-extremization techniques are applied to various 2d CFTs as well like the (0,2) theory that arises from the low-energy dynamics of D3-branes wrapped on Riemann surfaces and M5-branes wrapped on four-manifolds. Therefore, the status of the $c$-theorems is subject to change daily. As you see most work is done by generalizations of the original idea of Intriligator and Wecth who extremize a parameter that coincides with the central charge of the CFT. From this one can draw various conclusions but the only proofs we have till now, at least that I am aware of, are the ones of Zamolodchikov and Komargodski/Schwimmer.

Comments

These theorems by their own right are already fascinating, still I wonder what applications are there besides the one mentioned in OP's main post?

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The theorems give you partial understanding of the theory regarding its RG flow of course. Also provide information about the $R$-charges and the scaling dimensions of operators as well as tests of the $AdS$/CFT correspondence.

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Thank you for your input!