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 $L_{ C_{ \ell} ^{ (1)}}(-\Lambda_0)$.
Categories: math.QA math-ph math.MP math.RT