The Vertex Algebra $M(1)^+$ and Certain Affine Vertex Algebras of Level -1

TL;DR

This paper constructs a realization of the vertex operator algebra $M(1)^+$ as a coset in affine vertex algebras at level -1, explores embeddings between different affine algebras, and analyzes related subalgebras.

Contribution

It provides a new coset realization of $M(1)^+$, establishes embeddings between affine vertex algebras, and investigates their subalgebra structures.

Findings

Realization of $M(1)^+$ as a coset in affine vertex algebras at level -1

Embedding of $L_{C_{ ext{ell}}}(- ext{Lambda}_0)$ into $L_{A_{2 ext{ell}-1}}(- ext{Lambda}_0)$

Analysis of coset subalgebras within $L_{C_{ ext{ell}}}(- ext{Lambda}_0)$

Abstract

We give a coset realization of the vertex operator algebra $M(1)^+$ with central charge $\ell$. We realize $M(1)^+$ as a commutant of certain affine vertex algebras of level -1 in the vertex algebra $L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}\Lambda_0) \otimes L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}\Lambda_0)$. We show that the simple vertex algebra $L_{C_{\ell} ^{(1)}}(-\Lambda_0)$ can be (conformally) embedded into $L_{A_{2 \ell -1} ^{(1)}} (-\Lambda_0)$ and find the corresponding decomposition. We also study certain coset subalgebras inside $L_{C_{\ell} ^{(1)}}(-\Lambda_0)$.