Cosets of affine vertex algebras inside larger structures

TL;DR

This paper characterizes and proves the strong finite generation of certain cosets of affine vertex algebras, providing new insights into their structure and applications such as the rationality of specific superconformal algebras.

Contribution

It offers a general characterization of cosets of affine vertex algebras and establishes their strong finite generation in new cases, with constructive methods and minimal generating sets.

Findings

Characterization of cosets for generic levels k

Proof of strong finite generation in specific cases

New proof of rationality for N=2 superconformal algebra

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 $V^k(\mathfrak{g},B)\rightarrow \mathcal{A}^k$. 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: $\mathcal{A}^k = V^k(\mathfrak{g}',B')$, $\mathcal{A}^k = V^{k-l}(\mathfrak{g}',B') \otimes \mathcal{F}$, and $\mathcal{A}^k = V^{k-l}(\mathfrak{g}',B') \otimes V^{l}(\mathfrak{g}",B")$. Here $\mathfrak{g}'$ and $\mathfrak{g}"$ are finite-dimensional Lie (super)algebras containing $\mathfrak{g}$, equipped with nondegenerate, invariant, (super)symmetric bilinear forms $B'$ and $B"$ which extend $B$, $l \in \mathbb{C}$ is fixed, and $\mathcal{F}$ is a free field algebra admitting a homomorphism $V^l(\mathfrak{g},B) \rightarrow \mathcal{F}$. 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 $k$.