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The vertex algebra \(M(1)^{+}\) and certain affine vertex algebras of level \(-1\). (English) Zbl 1281.17029

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

MSC:

17B69 Vertex operators; vertex operator algebras and related structures
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B68 Virasoro and related algebras