Has anyone read Atiyah and Bott's famous paper "The moment map and equivariant cohomology"?


I have some trouble with the equaations appearing between equations (4.18) and (4.19) in page 13 of the original paper. The paper claims
$$D\lambda a = D(a-i(X)a\theta) = da-i(X)da \theta + i(X)au,$$
whereas I think that it should be
$$D\lambda a=D(a-i(X)a\theta)=da+i(X)da\theta + i(X)au,$$
where the minus sign is due to $\mathcal{L}(X)a=i(X)da+di(X)a = 0$.


Can anyone tell me what it should be?


Thank you!



could you help me understand an equation in a paper by Atiyah & Bott?
I think that the map $\lambda$ should be defined as follows:
$$\lambda a = a - \theta i(X)a.$$


In this way,
$$D\lambda a = D( a - \theta i(X)a) = da + u i(X)a + \theta di(X)a$$
and using that $di(X) a + i(X) da = 0$ and that $\lambda(i(X) a) = i(X) a$, one finds
$$D\lambda a = \lambda( da + i(X)a u ),$$
which is (4.19) in Atiyah-Bott.




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By the way, the generalisation of $\lambda$ to non-abelian $\mathfrak{g}$ is the following. If $\theta^\alpha$, $e\_\alpha$ and $u^\alpha$ are as in the paper, then
$$\lambda b = \prod\_\alpha (1 - \theta^\alpha i(e\_\alpha) )b.$$
This is known as *minimal coupling* in the Physics literature.


