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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); $i_E:E\hookrightarrow X, i_F:F\hookrightarrow Y$ 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 by Grigory Kondyrev, Artem Prikhodko, but they don't seem to mention abelian localisation.

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    $\begingroup$ What precisely do you mean by loop spaces? for the interpretation of GRR in the Kondyrev-Prikhodko story (see also the related story in arxiv.org/abs/1305.7175 and in arxiv.org/abs/1804.00879) one uses derived loop spaces, meaning maps from a homotopical version of the circle, and the relevant symmetry is not an ordinary $G_m$ action but an action of this homotopical $S^1$. Localization in this context is studied in Preygel's paper tolypreygel.com/papers/note_loop_short.pdf $\endgroup$
    – David Ben-Zvi
    Commented Apr 19, 2020 at 19:05
  • $\begingroup$ @DavidBen-Zvi This paper looks great, thanks! Do you know of anywhere where abelian localisation for more general derived stacks is proven? $\endgroup$
    – Pulcinella
    Commented Apr 24, 2020 at 14:53

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