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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!

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Sorry for the comment,i only persue the truth but not mend to comment someone needless to say Atiyah-Bott, so if you have the answer or you 're sure of it ,why not show me the correct.By the way , i also thinks Atiyah-Bott are of the most important celebrates in the 20'!! – HKSHLZW Dec 11 '10 at 1:05
I've taken the liberty of editing the question to make it more readable. – José Figueroa-O'Farrill Dec 11 '10 at 1:34
up vote 4 down vote accepted

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.

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.

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