# Proof of the general expression for anomaly in a CFT and its partition function

• I think the statement is that for any dimensional CFT the following is true,

$$\langle T^{\mu}_\mu \rangle = \sum B_n I_n - 2(-1)^{d/2}AE_d,$$

where $E_d$ is the "Euler density" and $I_n$ are the independent "Weyl invariants of weight $-d$".

(...I am not sure of the definition of the geometric quantities coming on the R.H.S and I wonder if the notion of the "Euler density" and Weyl invariants" are related to the ideas of the Weyl tensor and the Euler tensor..)

• For the special case of $3+1$ CFTs I am told that there is a relationship between the effective action and these anomaly coefficients as,

$W = \frac {a_0}{\epsilon^4} + \frac {a_1}{\epsilon^2} + a_2 ln (\epsilon) + w(g)$

where I guess $\epsilon$ is the UV regulator, $W$ is I guess the connected functional/free energy defined as $W = - ln (Z)$ ($Z$ being the partition function), $w(g)$ is the UV finite part dependent on the metric $g$ such that $w(\lambda^2 g) = w(g) - a_2ln(\lambda)$.

The most important and universal part of this answer is apparently that $a_2$ is universal and has the following properties that,

(1) $a_2(e^{-2\omega}g) = a_2(g)$

(2) $a_2 = AE_4 + BI_4$ where $E_4 = \frac{1}{64}\int (R_{\alpha \beta \mu \nu}R^{\alpha \beta \mu \nu} - 4R_{\mu \nu}R^{\mu \nu} + R^2)$ and $I_4 = -\frac{1}{64}\int (R_{\alpha \beta \mu \nu}R^{\alpha \beta \mu \nu} - 2R_{\mu \nu}R^{\mu \nu} + \frac{R^2}{3})$

Thus one says that the universal part of the free energy of a $3+1$ CFT is given by the integrated conformal anomaly.

I would like to know what is the proof of the above and its generalization to higher dimensions (if known!).

It seems very mysterious to me that anomalies should determine the universal parts of a partition function!

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–  Dimensio1n0 Aug 25 '13 at 9:12
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## 1 Answer

The answer to your question is contained in the recent book by Spyros Alexakis `The Decomposition of Global Conformal Invariants'' published last year by Princeton University Press. He answers the Deser-Schwimmer conjecture which (as far as I can understand your question) is the broader context of what you are asking.

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So the Deser-Schwimmer paper is a conjecture? I thought that the first equation I wrote down for $\langle T_\mu ^\mu \rangle$ is a well-established result in QFT! Can you give any other online reference (pedagogic/review papers?) along these lines - which will help understand this? –  user6818 Sep 2 '13 at 21:31
It was one of those conjectures which everyone (in physics) assumed to be obviously true, I guess, but giving a rigorous proof turned out to be a prodigious feat. The formula you wrote down is a special case and may well have been known beforehand. Alexakis' book has a lot of exposition and pointers to previous work, so that is probably the place to start. Much or all of this appears in papers of his posted on the arXiv too. –  Rafe Mazzeo Sep 4 '13 at 2:52
Do you have any specific paper of Spyros in mind which will work like a pedagogic introduction? It would be great if you can may be help with a more basic question along these lines, mathoverflow.net/questions/146332/… –  user6818 Oct 29 '13 at 19:15
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