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 $\mathcal{L}(-)$ 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.

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.

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 $\mathcal{L}(-)$ carries the usual rotation action of $\mathbf{G}_m$. 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.

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 $\mathcal{L}(-)$ 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.