Abelian localisation says approximately that for a proper equivaraint map $f:X\to Y$ between schemes with a $\mathbf{G}_m$ action, the pushforward on cohomology $f_*\omega$ can be computed by the pushforward on the fixed locus:
$$f_{0*}\frac{i_E^*\omega}{e_T(E/X)}\ =\ \frac{i_F^*(f_*\omega)}{e_T(F/Y)},$$
See [localization and conjectures from string duality (p. 5)][1]; $i_E:E\hookrightarrow X, i_F:F\hookrightarrow Y$ are the fixed loci and $f_0=f\vert_E$. $i_F^*$ is injective so this formula specifies $f_*\omega$ uniquely. The same paper suggests the following question:

> For an arbitrary proper map $g:X\to Y$, can we apply abelian localisation to  $\mathcal{L}X\to\mathcal{L}Y$ (loop spaces) to get Grothendieck Riemann Roch for $g$? 

Here (edit:the derived loop space) $\mathcal{L}(-)$ carries the usual rotation action of $S^1$. Apparently this is obvious  if you can get abelian localisastion to work for loop spaces. How does this ``obvious'' implication work, and *does* abelian localisation work for loop groups?

Another way of answering it might have something to do with the [nice paper][2] by Grigory Kondyrev, Artem Prikhodko, but they don't seem to mention abelian localisation.


  [1]: https://arxiv.org/pdf/math-ph/0701057.pdf
  [2]: https://arxiv.org/abs/1906.00172