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Equivariant line bundle isomorphism classes are classified by the equivariant cohomology group $H^2_{P}(X;\mathbb{Z})$ and let us take $P$ to be finite abelian and $X$ a finite dimensional CW-complex to simplify matters.

I was wondering how to understand these classes in terms of the connection $A$ of the associated $U(1)$-bundle. I know of some examples where it is not the curvature $F_{A}$ but the connection $A$ itself integrated over a subregion (or boundary of said subregion) fixed under the $P$-action to obtain some of these classes.

Is there a general procedure to describe any given class in terms of an integral of the connection $A$ or its curvature $F_{A}$ over some subregion of $X$ and how does this region relate to the $P$-action?

Thanks for reading

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  • $\begingroup$ If you're trying to get the class by integrating anything, you will presumably be working in $H^2_P (X, \mathbb R)$. If $P$ is finite, then this is just the $P$-invariants of the usual cohomology of $\mathbb R$, since the characteristic zero representation theory of finite groups is semisimple, so you just want to integrate the connection normally. $\endgroup$
    – Will Sawin
    Commented Jul 29, 2021 at 22:25
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    $\begingroup$ Dear Will, no, in fact torsion invariants are most of the interesting classes and one can obtain torsion integrating i.e. see Freed's discussion about torsion and equation (3.9). $\endgroup$
    – Time suspect
    Commented Jul 30, 2021 at 14:28

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