The authors study equivariant twisted de Rham cohomology of a manifold \(M\) with action of a compact Lie group \(G\).
Recall first that in this setting equivariant (but untwisted) de Rham cohomology can be defined in two different ways: either using forms with values in basic Weil-algebra valued forms, or (following Cartan) using forms with values in the symmetric algebra of the dual of the Lie algebra.
Twisted de Rham cohomology, on the other hand, is obtained by modifying the de Rham differential to \(d+H\), where \(H\) is a closed \(3\)-form (the resulting complex is then of course only \(\mathbb Z/2\mathbb Z\)-graded).
The authors define twisted equivariant de Rham cohomology by twisting the Cartan differential above with an \(G\)-invariant 3-form \(H\). They show that there is a canonical H one can use to twist the Weil complex which is quasi-isomorphic. There is one additional modification in the Weil complex, however: the Weil algebra has to be replaced by its completion.
The authors then show how to define a twist-form \(H_B\) on the Borel construction \(M\times_G EG\) (considered as limit of forms defined on manifolds approximating the universal free \(G\)-space \(EG\)) and they show that there is a Chern-Weil isomorphism between \(H\)-twisted equivariant de Rham cohomology of \(M\) and \(H_B\)-twisted de Rham cohomology of \(M\times_G EG\).
As a corollary, the authors conclude that twisted equivariant de Rham cohomology has all the properties of an equivariant cohomology theory (e.g. Mayer-Vietoris).
They finally recall some results about equivariant cohomology of exact Courant algebroids, showing in particular that this can be interpreted as twisted equivariant cohomology of the underlying manifold for a suitable twist. They show how to induce a generalized complex structure on \(M\times_G P\) if \(M\) is generalized complex with a Hamiltonion \(G\)-action and \(P\) is a \(G\)-principal bundle over a base with generalized complex structure.