W-algebras as coset vertex algebras

Abstract

We prove the long-standing conjecture on the coset construction of the minimal series principal $W$-algebras of $ADE$ types in full generality. We do this by first establishing Feigin's conjecture on the coset realization of the universal principal $W$-algebras, which are not necessarily simple. As consequences, the unitarity of the "discrete series" of principal $W$-algebras is established, a second coset realization of rational and unitary $W$-algebras of type $A$ and $D$ are given and the rationality of Kazama-Suzuki coset vertex superalgebras is derived.

Minor corrections and typos fixed. Proposition 3.4 is strengthened, which simplifies the proofs of Lemma 5.2 and Lemma 8.1. Final version to appear in Invent. Math

References in corpus (2)

• Braided tensor categories and extensions of vertex operator algebras
• Coset Vertex Operator Algebras and $\W$-Algebras

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