The commutant of $L_{{\widehat{\mathfrak{sl}}}_2}(n,0)$ in the vertex operator algebra $L_{{\widehat{\mathfrak{sl}}}_2}(1,0{)}^{\otimes n}$

Abstract

We study the commutant $L_{{\widehat{\mathfrak{sl}}}_2}(n,0{)}^c$ of $L_{{\widehat{\mathfrak{sl}}}_2}(n,0)$ in the vertex operator algebra $L_{{\widehat{\mathfrak{sl}}}_2}(1,0{)}^{\otimes n}$, for $n\ge 2$. The main results include a complete classification of all irreducible $L_{{\widehat{\mathfrak{sl}}}_2}(n,0{)}^c$-modules and a proof that $L_{{\widehat{\mathfrak{sl}}}_2}(n,0{)}^c$ is a rational vertex operator algebra. As a consequence, every irreducible $L_{{\widehat{\mathfrak{sl}}}_2}(n,0{)}^c$-module arises from the coset construction as conjectured in \cite{LS}.

28 pages