A lot of important work in quantum field theory involves Borel equivariant cohomology of certain geometric objects, usually with the goal of computing integrals over some complicated moduli stack. In algebraic topology, however, people nowadays tend to be more interested in genuine equivariant cohomology (with Mackey functors and whatnot), which is generally understood to carry more information (and which includes Borel cohomology as a special case). I have seen hints in a couple places (e.g. the nLab page on equivariant elliptic cohomology) that some people believe that genuine equivariant cohomology is really the "right" way to do QFT. The only motivating example I've seen, though, is of heterotic strings with some particular background field and associated gauge group, which involves a different action than the ones that show up more generally. (These being Witten's circle action on the loop manifold which gives rise to the elliptic genus by applying localization in equivariant K-theory, and actions of things like the groupoid of conformal symmetries which describe "ambient" structure of the manifold.)
What I'm wondering is, is it "natural" or in any way useful (from the physical or mathematical perspective) to try enhancing the Borel cohomology used in these cases to genuine equivariant cohomology? Like, for example, taking the Borel $S^1$-equivariant elliptic cohomology which gives rise to the Witten genus and replacing it with genuine $S^1$-equivariant elliptic cohomology, or analogous replacements with K-theory or de Rham cohomology. And, moreover, how does this alter the theory and general procedure used for computing these things (e.g. Borel localization)?
