Cosets of affine vertex algebras inside larger structures

Abstract

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 $V^k(\mathfrak{g},B)\to {\mathcal{A}}^k$. Under some technical conditions on ${\mathcal{A}}^k$, 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: ${\mathcal{A}}^k=V^k({\mathfrak{g}}^{\prime },B^{\prime })$, ${\mathcal{A}}^k=V^{k-l}({\mathfrak{g}}^{\prime },B^{\prime })\otimes \mathcal{F}$, and ${\mathcal{A}}^k=V^{k-l}({\mathfrak{g}}^{\prime },B^{\prime })\otimes V^l(\mathfrak{g}",B")$. Here ${\mathfrak{g}}^{\prime }$ and $\mathfrak{g}"$ are finite-dimensional Lie (super)algebras containing $\mathfrak{g}$, equipped with nondegenerate, invariant, (super)symmetric bilinear forms $B^{\prime }$ and $B"$ 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 $k$.

Some errors corrected, final version

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