Characteristic Classes for $E_8$ Bundles

2011-11-09T03:09:28Z

Given a principal $E_8$ bundle $P\rightarrow X$ one can take the adjoint representation $\rho :E_8\rightarrow SU(\mathbb C^{248})$ and form the associated vector bundle $V=P\times_{\rho}\mathbb C^{248}$. This vector bundle has an $E_8$ characteristic classes $\lambda (V)\in H^4(X,\mathbb Z)$ obtained by pulling back the generator $\lambda \in H^4(BE_8,\mathbb Z)$ and it has a second Chern class $c_2(V)$ given by pulling back the generator $c_2\in H^4(BSU(248),\mathbb Z)$.

I am looking for a reference for the following fact: $$\lambda (V)=\frac{c_2(V)}{60}.$$

There is the map $B\rho^* : H^4(BSU(248),\mathbb Z)\rightarrow H^4(BE_8, \mathbb Z)$. Since both groups are canonically isomorphic to $\mathbb Z$, the map is determined by a single integer, which is the Dynkin index of $E_8$ and has been computed to be $60$. The above fact essentially follows from this. I am writing a paper and would prefer to just state the fact and then point the reader to whoever first presented a thorough argument filling in all the details. I have seen it mentioned as a footnote on page 68 in this paper by Friedman, Morgan, and Witten and some of the details regarding the Dynkin index are discussed in this paper by Totaro.

Characteristic Classes for $E_8$ Bundles - MathOverflow

Characteristic Classes for $E_8$ Bundles