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. 1 this difficulty by establishing the following assertion that has been conjectured by B. Feigin (cf. [FJMM16]).Main Theorem 2 (Theorem 8.7). Let g be simply laced, k + h ∨ ∈ Q 0 , and define ℓ ∈ C by the formula (1). We have the vertex algebra isomorphismMoreover, W ℓ (g) and V k+1 (g) form a dual pair in V k (g)⊗L 1 (g) if k is generic.The advantage of replacing W ℓ (g) by the universal W -algebra W ℓ (g) lies in the fact that one can use the description of W ℓ (g) in terms of screening operators, at least for a generic ℓ. Using such a description, we are able to establish the statement of Main Theorem 2 for deformable families [CL19] of W ℓ (g) and (V k (g)⊗L 1 (g)) g [t] , see Section 8 for the details. The main tool here is a property of the semi-regular bimodule obtained in [Ara14], see Proposition 3.4.

Quantum dimensions and irreducible modules of some diagonal coset vertex operator algebras

“…We shall prove that the coset vertex operator algebra C(L E 8 (k + 2, 0), L E 8 (k, 0) ⊗ L E 8 (2, 0)) is rational and C 2 -cofinite. To prove this result, we need the following important result which was obtained in [4]. (1, 0)) is strongly regular.…”

Rectangular W-algebras, extended higher spin gravity and dual coset CFTs

“…This is important as the coset realization of the A-series is precisely the dual theory for the ordinary bosonic higher spin gravity correspondence of Gaberdiel and Gopakumar [10]. Moreover the proof of [41] consists of finding a kernel of screenings realization of the coset theory and this step generalizes and some generalizations are currently work in progress. Now, the bosonic higher spin algebra of Gaberdiel and Gopakumar is of type 2, 3, 4, .…”

“…We need three axes, e.g., x 1 , x 2 , x 3 to express the plane partitions, and the invariance under the rotation of axes corresponds to the triality relation. It has been known for a long time [74,75], see [41,Theorem 13.1] for a proof, that the coset…”

Fusion categories for affine vertex algebras at admissible levels

“…First of all, we of course need the existence of vertex tensor category structure proven in [12]. Secondly, we need a relation of L ℓ (g) to a better understood family of vertex operator algebras and this relation is given by the coset realization of principal Walgebras of simply-laced Lie algebras recently proven in joint work with Arakawa and Linshaw [15]. Thirdly we need the theory of vertex algebra extensions developed in collaboration with Kanade and McRae [16].…”