# 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!