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.

We construct a new equivariant cohomology theory for a certain class of differential vertex algebras, which we call the chiral equivariant cohomology. A principal example of a differential vertex algebra in this class is the chiral de Rham complex of Malikov-Schechtman-Vaintrob of a manifold with a group action. The main idea in this paper is to synthesize the algebraic approach to classical equivariant cohomology due to H. Cartan 1 , with the theory of differential vertex algebras, by using an appropriate notion of invariant theory. We also construct the vertex algebra analogues of the Mathai-Quillen isomorphism, the Weil and the Cartan models for equivariant cohomology, and the Chern-Weil map. We give interesting cohomology classes in the new theory that have no classical analogues.

Let V be a simple vertex operator algebra containing a rank n Heisenberg vertex algebra H and let C = Com (H,V) be the coset of H in V. Assuming that the representation categories of interest are vertex tensor categories in the sense of Huang, Lepowsky and Zhang, a Schur-Weyl type duality for both simple and indecomposable but reducible modules is proven. Families of vertex algebra extensions of C are found and every simple C-module is shown to be contained in at least one V-module. A corollary of this is that if V is rational and C 2 -cofinite and CFT-type, and Com (C,V) is a rational lattice vertex operator algebra, then so is C. These results are illustrated with many examples and the C 1 -cofiniteness of certain interesting classes of modules is established. is exact, for M ′′ ∼ = (J ⊠ M)/M ′ = 0. But, fusion is right-exact [HLZ, Prop. 4.26], sois exact. However, M ′′ = 0 implies that J −1 ⊠ M ′′ is a non-zero quotient of M, by (1), so we must have J −1 ⊠ M ′′ ∼ = M, as M is simple. Fusing with J now gives J ⊠ M ∼ = M ′′ , so we conclude that M ′ = 0 and that J ⊠ M is simple. The simplicity of J ∼ = J ⊠ V now follows from that of V, completing the proof of (3).To prove (4), note that applying right-exactness to the short exact sequence 0where f is the induced map from J ⊠ M ′ to J ⊠ M that might not be an inclusion. Fusing with J −1 and applying (2.4), we arrive at 5)

Given a finite-dimensional reductive Lie algebra g equipped with a nondegenerate, invariant, symmetric bilinear form B, let V k (g, B) denote the universal affine vertex algebra associated to g and B at level k. Let A k be a vertex (super)algebra admitting a homomorphism V k (g, B) → A k . Under some technical conditions on A k , we characterize the coset Com(V k (g, B), A k ) for generic values of k. We establish the strong finite generation of this coset in full generality in the following cases:Here g ′ and g ′′ are finitedimensional Lie (super)algebras containing g, equipped with nondegenerate, invariant, (super)symmetric bilinear forms B ′ and B ′′ which extend B, l ∈ C is fixed, and F is a free field algebra admitting a homomorphism V l (g, B) → F . Our approach is essentially constructive and leads to minimal strong finite generating sets for many interesting examples. As an application, we give a new proof of the rationality of the simple N = 2 superconformal algebra with c = 3k k+2 for all positive integers k.As a matter of notation, we say that a vertex algebra A is of type W (d 1 , d 2 , . . . ) if it has a minimal strong generating set consisting of one field in each weight d 1 , d 2 , . . . . Affine vertex algebras.Let g be a finite-dimensional, Lie (super)algebra, equipped with a (super)symmetric, invariant bilinear form B. The universal affine vertex (super)algebra V k (g, B) associated to g and B is freely generated by elements X ξ , ξ ∈ g, satisfying the operator product expansions X ξ (z)X η (w) ∼ kB(ξ, η)(z − w) −2 + X [ξ,η] (w)(z − w) −1 .6

Let V be a simple VOA and consider a representation category of V that is a vertex tensor category in the sense of Huang-Lepowsky. In particular, this category is a braided tensor category. Let J be an object in this category that is a simple current of order two of either integer or half-integer conformal dimension. We prove that V ⊕ J is either a VOA or a super VOA. If the representation category of V is in addition ribbon, then the categorical dimension of J decides this parity question. Combining with Carnahan's work, we extend this result to simple currents of arbitrary order. Our next result is a simple sufficient criterion for lifting indecomposable objects that only depends on conformal dimensions. Several examples of simple current extensions that are C2-cofinite and nonrational are then given and induced modules listed.• Which generalized modules 1 of V lift to those of the extension V e ?We will use our answers to these questions to construct three new families of C 2 -cofinite VOAs together with all modules that lift at the end of this work. Our main intention is however to find genuinely new C 2 -cofinite VOAs, that is VOAs that are not directly related to the well-known triplet VOA. For example consider the following diagramHere A k is a family of VOAs with one-dimensional associated variety [A1] containing a Heisenberg sub VOA H and having a simple current J of infinite order. Our picture is that the Heisenberg coset Com (H, A k ) still has a one-dimensional associated variety while the extension E k only has finitely many irreducible objects. Especially Com (H, E k ) is our candidate for new C 2 -cofinite VOAs. A natural example is. . } as it has one-dimensional variety of modules [AdM3]. In [CR1, CR2] it is conjectured that these VOAs allow for infinite order simple current extensions E k , which then would only have finitely many simple modules. These VOAs would be somehow unusual as they would not be of CFT-type, and these VOAs are not our final goal, but rather Com (H, E k ). In the example of k = −1/2 this construction would just yield W(2) and in the case of k = −4/3 it would just give W(3) [Ad1, CRW, R2]. In all other cases we expect new C 2 -cofinite VOAs. Another potential candidate is the Bershadsky-Polyakov algebra whose Heisenberg coset is studied in [ACL]. In a subsequent work, we will thus develop general properties of Heisenberg cosets beyond semi-simplicity [CKLR]. This work will rely on our findings here and will then further be used for interesting examples.In the present work, we will translate results into the corresponding statements in a braided tensor category using the theory of [KO, HKL]. The advantage is that the categorical picture is much better suited for proving properties of the representation category. For us it provides a very nice way to understand the problem of the two questions: Does a module lift to a module of the extended VOA? Is the extension a VOA or a super VOA? The work [KO] assumes categories to be semi-simple and focuses on algebras with trivial twist....