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

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• Liouville Correlation Functions from Four-dimensional Gauge Theories
• Vertex Algebras and Algebraic Curves
• An AdS_3 Dual for Minimal Model CFTs
• Quantum Reduction for Affine Superalgebras
• Finite vs. affine W-algebras
• Higher spin AdS_3 supergravity and its dual CFT
• Braided tensor categories and extensions of vertex operator algebras
• Tensor categories for vertex operator superalgebra extensions
• Rationality of W-algebras: principal nilpotent cases
• Rationality of admissible affine vertex algebras in the category O
• Cosets of affine vertex algebras inside larger structures
• Regularity of fixed-point vertex operator subalgebras
• N=1 supersymmetric higher spin holography on AdS_3
• Abelianizing vertex algebras
• Classical W-algebras and generalized Drinfeld-Sokolov bi-Hamiltonian systems within the theory of Poisson vertex algebras
• Orbifolds and cosets of minimal $\mathcal{W}$-algebras
• Screening operators for W-algebras
• W-Constraints for the Total Descendant Potential of a Simple Singularity
• Quantum Langlands Correspondence
• Vanishing of cohomology associated to quantized Drinfeld-Sokolov reduction
• Rational Vertex Operator Algebras and the Effective Central Charge
• Level-rank duality via tensor categories
• Representation Theory of W-Algebras
• Branching rules for quantum toroidal gl(n)
• Two-sided BGG resolutions of admissible representations
• Finite vs infinite decompositions in conformal embeddings
• Coset Vertex Operator Algebras and $\W$-Algebras
• Level-Rank Duality for Vertex Operator Algebras of types B and D

Cited by in corpus (63)

• Tensor categories for vertex operator superalgebra extensions
• Cosets of affine vertex algebras inside larger structures
• Holography in de Sitter Space via Chern-Simons Gauge Theory
• CFT duals of three-dimensional de Sitter gravity
• Braided tensor categories of admissible modules for affine Lie algebras
• Trialities of $\mathcal{W}$-algebras
• Logarithmic W-algebras and Argyres-Douglas theories at higher rank
• Duality of subregular W-algebras and principal W-superalgebras
• The vertex algebras ${\mathcal{R}}^{(p)}$ and ${\mathcal{V}}^{(p)}$
• Gluing vertex algebras
• Universal two-parameter ${\mathcal{W}}_{\infty }$-algebra and vertex algebras of type $\mathcal{W}(2,3,\dots ,N)$
• Tensor categories of affine Lie algebras beyond admissible levels
• A realisation of the Bershadsky--Polyakov algebras and their relaxed modules
• Universal two-parameter even spin ${\mathcal{W}}_{\infty }$-algebra
• S-duality for the large $N=4$ superconformal algebra
• Quantum Langlands duality of representations of W-algebras
• Rectangular W-(super)algebras and their representations
• Trialities of orthosymplectic $\mathcal{W}$-algebras
• Haploid algebras in $C^{\ast }$-tensor categories and the Schellekens list
• Representation theory in chiral conformal field theory: from fields to observables
• Screening operators and Parabolic inductions for Affine W-algebras (with an appendix by Shigenori Nakatsuka)
• Correspondences of categories for subregular W-algebras and principal W-superalgebras
• Local energy bounds and strong locality in chiral CFT
• $N=4$ superconformal algebras and diagonal cosets
• Three-dimensional de Sitter holography and bulk correlators at late time
• Higher Airy structures, W algebras and topological recursion
• Fusion and positivity in chiral conformal field theory
• Whittaker vectors for $\mathcal{W}$-algebras from topological recursion
• On the Feigin-Tipunin conjecture
• Generalized parafermions of orthogonal type
• Unitary representations of the ${\mathcal{W}}_3$-algebra with $c\ge 2$
• Higher rank FZZ-dualities
• Cosets of free field algebras via arc spaces
• Late-time correlation functions in dS${}_3$/CFT${}_2$ correspondence
• Simplicity of higher rank triplet W-algebras
• On Miura maps for W-superalgebras
• Singular vectors for the $W_N$ algebras
• On rationality for $C_2$-cofinite vertex operator algebras
• Quantum dimensions and irreducible modules of some diagonal coset vertex operator algebras
• Tensor Categories arising from the Virasoro Algebra
• On the structure of W-algebras in type A
• Ordinary modules for vertex algebras of ${\mathfrak{osp}}_{1|2n}$
• Invariant subalgebras of the small $\mathcal{N}=4$ superconformal algebra
• Orbifolds of Gaiotto-Rapčák $Y$-algebras
• Affine Lie algebras and tensor categories
• Hilbert schemes of nonreduced divisors in Calabi-Yau threefolds and W-algebras
• Cosets from equivariant W-algebras
• Regular vertex operator subalgebras and compressions of intertwining operators
• Orthosymplectic Feigin-Semikhatov duality
• Trace functions and fusion rules of diagonal coset vertex operator algebras
• FZZ-triality and large $\mathcal{N}=4$ super Liouville theory
• Vertex Algebras and Commutative Algebras
• Polynomial energy bounds for type $F_4$ WZW-models
• On global generation of vector bundles on the moduli space of curves from representations of vertex operator algebras
• Associated varieties of simple affine VOAs $L_k(s{l}_3)$ and $W$-algebras $W_k(s{l}_3,f)$
• Feigin-Semikhatov conjecture and related topics
• Connecting affine $\mathcal{W}$-algebras: A case study on ${\mathfrak{sl}}_4$
• Conformal nets from minimal W-algebras
• Duality via convolution of W-algebras
• Correlator correspondences for subregular $\mathcal{W}$-algebras and principal $\mathcal{W}$-superalgebras
• The Drinfeld--Sokolov reduction of admissible representations of affine Lie algebras
• Relaxed and logarithmic modules of $\widehat{{\mathfrak{sl}}_3}$
• Generators, spanning sets and existence of twisted modules for a grading-restricted vertex (super)algebra