For both equivariant de Rham cohomology and equivariant K-theory (in the "naive" or Borel sense), we have localization formulae which allow us to compute this cohomology in terms of the fixed points. Both of these formulae are fundamental in quantum field theory, because they allow us to compute path integrals over "spaces with symmetry", or more precisely over complicated moduli stacks which are realized as quotients by certain certain gauge groups.
I was recently writing up some talk notes, and while describing the philosophy of localization in QFT, I realized that these two theorems only describe it for field theories of dimension $0$ (de Rham) and $1$ (K-theory). In particular, I haven't seen any such result for TMF, which would presumably facilitate computations in 2-dimensional conformal field theories (if you believe Stolz and Teichner). The closest thing I could find was a reference to it in nLab claiming Lurie "hints" at it in his survey of elliptic cohomology—a hint which I wasn't able to pick up on.
So my question is this. Given a Borel equivariant cohomology theory, is there a general notion of "localization" (proven or unproven)? And, if so, when is it known to hold?
