W-algebras as coset vertex algebras

TL;DR

This paper proves the conjecture that minimal series principal W-algebras of ADE types can be constructed as cosets, confirming longstanding hypotheses and establishing new realizations and properties of these algebraic structures.

Contribution

It proves the conjecture on the coset construction of minimal series principal W-algebras of ADE types and confirms Feigin's conjecture on universal W-algebras, advancing understanding of their structure.

Findings

Confirmed the coset construction conjecture for minimal series principal W-algebras.

Established unitarity of the discrete series of principal W-algebras.

Provided new coset realizations for rational and unitary W-algebras of types A and D.

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.