Geometric explanation for the $\widehat{\mathfrak{sl}}_2$ free field realisation (alias $L_1(\mathfrak{sl}_2)\to V(\sqrt{2}\mathbf{Z})$)

Question

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 $\mathfrak{g}$.

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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.

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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 $\text{Gr}_G$).

Answer 1

As Pavel Safranov commented, this is done in the paper arxiv.org/abs/0710.5247 by Zhu. I've skimmed the paper and will sketch how I think it works.

Write $\mathcal{L}_G$ for the determinant line bundle on the BD Grassmannian $(\text{Gr}_{G,X^n})$, and $\pi_n:\text{Gr}_{G,X^n}\to \text{Bun}_{X,G}$. The BD Grassmannian for $T$ lies inside the BD Grassmannian for $G$, and $\mathcal{L}_G$ restricts to $\mathcal{L}_T$ (3.3.1).

The factorisation algebra version of the free field map is the dual of restriction $$\pi_{n,\star}(\mathcal{L}_G\vert_{\text{Gr}_{T,X^n}}) ^\vee\ \longrightarrow\ (\pi_{n,\star}\mathcal{L}_G)^\vee$$ where $\star$ is the O-module pushforward. The LHS is the lattice vertex algebra and the RHS is the simple quotient of the affine vertex algebra. 3.3.2 shows this is an isomorphism of factorisation algebras.

Taking global sections in the $\mathfrak{sl}_2$ case gives the isomorphism $$V(\sqrt{2}\mathbf{Z})\ \stackrel{\sim}{\longrightarrow}\ L_1(\mathfrak{sl}_2).$$ One slight surprise is that I had expected the proof to construct a map in the other direction (maybe in a more general setting), then (restricting to our context) show it's an isomorphism.