Rationality of W-algebras: principal nilpotent cases

Abstract

We prove the rationality of all the minimal series principal W-algebras discovered by Frenkel, Kac and Wakimoto in 1992, thereby giving a new family of rational and C_2-cofinite vertex operator algebras. A key ingredient in our proof is the study of Zhu's algebra of simple W-algebras via the quantized Drinfeld-Sokolov reduction. We show that the functor of taking Zhu's algebra commutes with the reduction functor. Using this general fact we determine the maximal spectrums of the associated graded of Zhu's algebra of all the admissible affine vertex algebras as well.

revised, to appear in Annals of Math

References in corpus (2)

• Two-sided BGG resolutions of admissible representations
• Localization of affine W-algebras

Cited by in corpus (52)

• W-algebras as coset vertex algebras
• Schur-Weyl Duality for Heisenberg Cosets
• Rationality of admissible affine vertex algebras in the category O
• Cosets of affine vertex algebras inside larger structures
• Joseph ideals and lisse minimal W-algebras
• Duality of subregular W-algebras and principal W-superalgebras
• Orbifolds and cosets of minimal $\mathcal{W}$-algebras
• Explicit generators in rectangular affine W-algebras of type A
• Gluing vertex algebras
• Tensor categories of affine Lie algebras beyond admissible levels
• Universal two-parameter ${\mathcal{W}}_{\infty }$-algebra and vertex algebras of type $\mathcal{W}(2,3,\dots ,N)$
• Schur sector of Argyres-Douglas theory and $W$-algebra
• A realisation of the Bershadsky--Polyakov algebras and their relaxed modules
• Universal two-parameter even spin ${\mathcal{W}}_{\infty }$-algebra
• Weight representations of admissible affine vertex algebras
• Testing our understanding of SCFTs: a catalogue of rank-2 $\mathcal{N}$=2 theories in four dimensions
• Cosets of Bershadsky-Polyakov algebras and rational $\mathcal{W}$-algebras of type $A$
• Trialities of orthosymplectic $\mathcal{W}$-algebras
• Conformal blocks from vertex algebras and their connections on ${\overline{\mathcal{M}}}_{g,n}$
• Cosets of the ${\mathcal{W}}^k({\mathfrak{sl}}_4,f_{\text{subreg}})$-algebra
• Haploid algebras in $C^{\ast }$-tensor categories and the Schellekens list
• Representation theory in chiral conformal field theory: from fields to observables
• Positive energy representations of affine vertex algebras
• Classifying relaxed highest-weight modules for admissible-level Bershadsky-Polyakov algebras
• Correspondences of categories for subregular W-algebras and principal W-superalgebras
• Modularity of Bershadsky-Polyakov minimal models
• Subregular W-algebras of type A
• $N=4$ superconformal algebras and diagonal cosets
• Rationality of the exceptional W-algebras ${\mathcal{W}}_k({\mathfrak{sp}}_4,f_{subreg})$ associated with subregular nilpotent elements of ${\mathfrak{sp}}_4$
• Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$
• Fusion and positivity in chiral conformal field theory
• Generalized parafermions of orthogonal type
• Singularities of nilpotent Slodowy slices and collapsing levels of W-algebras
• The ${\mathbb{Z}}_2$-orbifold of the ${\mathcal{W}}_3$-algebra
• On Miura maps for W-superalgebras
• On the structure of W-algebras in type A
• Simplicity of vacuum modules and associated varieties
• Coloured ${\mathfrak{sl}}_r$ invariants of torus knots and characters of ${\mathcal{W}}_r$ algebras
• Ordinary modules for vertex algebras of ${\mathfrak{osp}}_{1|2n}$
• The Orbit Method for Poisson Orders
• Orbifolds of Gaiotto-Rapčák $Y$-algebras
• On the representation theory of the vertex algebra $L_{-5/2}(sl(4))$
• Commutative algebras in Grothendieck-Verdier categories, rigidity, and vertex operator algebras
• Regular vertex operator subalgebras and compressions of intertwining operators
• Twisted chiral algebras of class $\mathcal{S}$ and mixed Feigin-Frenkel gluing
• Cosets from equivariant W-algebras
• Orthosymplectic Feigin-Semikhatov duality
• Generalized finite and affine $W$-algebras in type $A$
• Simple modules for Affine vertex algebras in the minimal nilpotent orbit
• Polynomial energy bounds for type $F_4$ WZW-models
• 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