Cosets of free field algebras via arc spaces

Abstract: Using the invariant theory of arc spaces, we find minimal strong generating sets for certain cosets of affine vertex algebras inside free field algebras that are related to classical Howe duality. These results have several applications. First, for any vertex algebra $\mathcal V$, we have a surjective homomorphism of differential algebras $$\mathbb{C}[J_{\infty}(X_{\mathcal V})] \rightarrow \text{gr}^F(\mathcal V)$$; equivalently, the singular support of $\mathcal V$ is a closed subscheme of the arc space of the associated scheme $X_{\mathcal V}$. We give many new examples of classically free vertex algebras (i.e., this map is an isomorphism), including $L_k(\mathfrak{sp}_{2n})$ for all positive integers $n$ and $k$. We also give new examples where the kernel of this map is nontrivial but is finitely generated as a differential ideal. Next, we prove a coset realization of the subregular ${\mathcal W}$-algebra of $\mathfrak{sl}_n$ at critical level that was previously conjectured by Creutzig, Gao, and the first author. Finally, we give some new level-rank dualities involving affine vertex superalgebras.