# Conformal/trace anomaly and index theorem

I am reading the chapters on characteristic classes and the index theorems in Nakahara. It is proven in the text that any chiral or gravitational anomaly $\mathcal{A}$ is given by

$$\mathcal{A}=\int I^1_{2r}$$

with $I^1_{2r}$ given by the descent equation,

$$I_{2r+2}=d I_{2r+1}$$

$$\delta_{\mathrm{gauge}}I_{2r+1}=dI_{2r}^1$$

The various $I$'s are related to the theory of the characteristic classes and Chern-Simons forms. On the other hand, the trace anomaly cannot be written in this way, at least I cannot see how. In $2$ and $4$ dimension we have (see for instance Duff),

$$(T_2)^\mu_\mu=cR$$

$$(T_4)^\mu_\mu=cW^2+aE_4+fF_{\mu\nu}F^{\mu\nu}$$

where $W^2$ is the Weyl tensor squared and $E_4= R_{\mu\nu\rho\sigma}^2-4R_{\mu\nu}^2+R^2$ (with some numerical coefficient) is the Euler density. This got me thinking: is there a geometric way to describe the anomaly, i.e. to write a descent equation and the index theorem for those anomalies?

Any pointers to papers, lecture notes, books or other resources are most welcome!

## 1 Answer

Descent equations for the trace anomaly in arbitrary dimensions are derived in this paper: http://arxiv.org/abs/0706.0340.

Since this article is very technical, I think that in order to get a better understanding of the principle, it would be useful to go through the chapters on gravitational anomalies in Bertlmann's book "Anomalies in Quantum Field theory". Unfortunately, an explicit discussion of the procedure for the Weyl anomaly is not given, it might however be interesting to see how it works for other anomalies.