Cosets of affine vertex algebras inside larger structures

Given a finite-dimensional reductive Lie algebra $\mathfrak{g}$ equipped with a nondegenerate, invariant, symmetric bilinear form $B$, let $V^k(\mathfrak{g},B)$ denote the universal affine vertex algebra associated to $\mathfrak{g}$ and $B$ at level $k$. Let ${\mathcal{A}}^k$ be a vertex (super)algebra admitting a homomorphism $V^k(\mathfrak{g},B)\rightarrow $\mathcal{A}$^k$\mathrm{.}Undersometechnicalconditionson$$\mathcal{A}$^k$, we characterize the coset $\text{Com}(V^k(\mathfrak{g},B),{\mathcal{A}}^k)$ for generic values of $k$. We establish the strong finite generation of this coset in full generality in the following cases: $$\mathcal{A}$^k = V^k(\mathfrak{g}‘,B’)$,$$\mathcal{A}$^k = V^{k-l}(\mathfrak{g}‘,B’) \otimes $\mathcal{F}$$,and$$\mathcal{A}$^k = V^{k-l}(\mathfrak{g}‘,B’) \otimes V^{l}(\mathfrak{g}“,B”)$\mathrm{.}Here$\mathfrak{g}'$and$\mathfrak{g}"$ are finite-dimensional Lie (super)algebras containing $\mathfrak{g}$, equipped with nondegenerate, invariant, (super)symmetric bilinear forms $B^{\mathrm{\prime }}$ and $B\mathrm{"}$ which extend $B$, $l\in \mathbb{C}$ is fixed, and $\mathcal{F}$ is a free field algebra admitting a homomorphism $V^l(\mathfrak{g},B)\to \mathcal{F}$. Our approach is essentially constructive and leads to minimal strong finite generating sets for many interesting examples. As an application, we give a new proof of the rationality of the simple $N=2$ superconformal algebra with $c=\frac{3k}{k+2}$ for all positive integers $k$.