Let $\mathfrak{g}$ be a semisimple Lie algebra, $\mathfrak{g}^L$ be its Langlands dual. Feigin--Frenkel duality says
$$
W^k(\mathfrak{g})=W^{k\_L}(\mathfrak{g}^L)
$$
if $r'(k+h^{'})(k\_L+h'\_L)=1$, where $r'$ is the maximum number of edges between two vertices in the Dynkin diagram of $\mathfrak{g}$, $h'$ (resp. $h'\_L$) is the dual Coxeter number of $\mathfrak{g}$ (resp. $\mathfrak{g}\_L$). 


A degeneration of this result when $k\_L$ tends to infinity was used as local input by Beilinson--Drinfeld to establish some results on geometric Langlands (which was globalized using the theory of chiral algebras). Regarding the non-degenerate result, they say "We do not know if this “doubly quantized” picture
can be globalized." Has the non-degenerate result been globalized in some way since then?



Globalizing Feigin--Frenkel duality
The Feigin-Frenkel isomorphism is globalized by the global quantum geometric Langlands conjecture, proposed by Stoyanovsky, and refined by Gaitsgory and his collaborators. See [Gaitsgory's 2016 collection of conjectures](https://arxiv.org/abs/1601.05279), in particular the discussion on page 5.


At irrational level, this appears to be (almost?) a theorem, with a major missing piece worked out earlier this year by [Arakawa and E. Frenkel](https://arxiv.org/abs/1807.01536). However, the language in the introduction and section 2 of that paper does not inspire complete confidence. The key sentence is on page 7:



> 
> According to Gaitsgory [Gai2], a Ran space version of this statement can be used to prove the existence of the quantum geometric Langlands correspondence $\mathbb{L}\_\kappa$ discussed in the Introduction.
> 
> 
> 


and [Gai2] refers to a video of a 2018 lecture rather than a published paper.


