About the general expression of trace anomaly and CFT partition functions

Question

I have put up a question here, https://mathoverflow.net/questions/139685/proof-of-the-general-expression-for-anomaly-in-a-cft-and-its-partition-function

Here I am putting up a slightly different version of that question,

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• 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..)

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• Also a similar sounding statement that I see is as in equation 15 (page 5) of http://arxiv.org/abs/hep-th/9806087

  One can see the stark similarity between that the equation referred to in the linked paper and the statement of trace anomaly that I have typed in the first bullet point.

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I can't find a reference to the derivation of these results and/or the relationship between these statements. It would be great if someone can help with this.

Answer 1

The original reference for the derivation of this result is http://arxiv.org/abs/hep-th/9302047.

A brief answer to your questions is:

• the Euler density in generic number of dimensions is a diffeo-invariant functional of the metric, built out of the curvature. In dimension 2d it is a polynomial of degree d in the Riemann tensor. Its defining property -which fixes uniquely its form, up to a conventional overall coefficient- is that its integral on the whole manifold gives a number which is actually independent of the metric (i.e. a topological invariant), called the Euler characteristic.

• the Weyl invariants are diffeo-invariant functionals of the metric, built out of a particular combination of the curvature which is the Weyl tensor. Since the Weyl tensor is not affected by Weyl rescaling of the metrics, the Weyl invariants (as the name suggests) are not only invariant under diffeomorphisms, but also under Weyl rescaling of the metric. In dimension 2d they are polynomials of degree d of the Weyl tensor.