Let $\sqrt{n}\mathbf{Z}$ be the one dimensional lattice, whose generator has length $2$. Associated to this is a lattice vertex algebra
$$V(\sqrt{2}\mathbf{Z}).$$
We also have the simple quotient of the affine vertex algebra associated to $\mathfrak{sl}_2$ at level $1$:
$$L_1(\mathfrak{sl}_2).$$
A priori lattice vertex algebras and affine vertex algebras have very different flavours. However, one can in fact write down a map using an explicit formula (5.3.6. of Frenkel-Ben Zvi)
$$L_1(\mathfrak{sl}_2)\ \longrightarrow\ V(\sqrt{2}\mathbf{Z})$$
which is an isomorphism. In *Vertex Representations via Finite Groups and
the McKay Correspondence*, Frenkel-Jing-Wang do the same for general $\mathfrak{g}$.

**Question:** Is there a geometric proof of this fact, e.g. construct both vertex algebras as D-modules on the affine Grassmannian and show they're isomorphic there?

Currently all proofs I'm aware of just write down an explicit formula and show that it works.

I am partially hopeful that there might actually be an answer to this because one can e.g. prove the character part of the boson-fermion correspondence (i.e. the Jacobi triple product formula) by working on $\text{Gr}_G$ (you prove it with the BGG resolution, which comes down to working on $\text{Gr}_G$).