SIGMA 8 (2012), 040, 16 pages      arXiv:1006.1752      https://doi.org/10.3842/SIGMA.2012.040

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

Dražen Adamović and Ozren Perše
Faculty of Science, Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia

Received March 09, 2012, in final form July 01, 2012; Published online July 08, 2012

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 $L_{C_{\ell} ^{(1)}}(-\Lambda_0)$.

Key words: vertex operator algebra; affine Kac-Moody algebra; coset vertex algebra; conformal embedding; $\mathcal{W}$-algebra.

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