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 affine vertex algebras inside free field that are related to classical Howe duality. These results have several applications. First, any algebra ${{\mathcal {V}}}$, a surjective homomorphism differential $\mathbb {C}[J_{\infty }(X_{{{\mathcal {V}}}})] \rightarrow \text {gr}^{F}({{\mathcal {V}}})$; equivalently, singular support {V}}}$ is closed subscheme space associated scheme $X_{{{\mathcal {V}}}}$. We give many new examples classically (i.e., this map an isomorphism), including $L_{k}({{\mathfrak {s}}}{{\mathfrak {p}}}_{2n})$ all positive integers $n$ and $k$. also where kernel nontrivial but finitely generated as ideal. Next, prove coset realization subregular {W}}}$-algebra ${{\mathfrak {l}}}_{n}$ at critical level was previously conjectured by Creutzig, Gao, 1st author. Finally, some level-rank dualities involving superalgebras.