Rationality and Fusion Rules of Exceptional W-Algebras

Abstract

First, we prove the Kac-Wakimoto conjecture on modular invariance of characters of exceptional affine W-algebras. In fact more generally we prove modular invariance of characters of all lisse W-algebras obtained through Hamiltonian reduction of admissible affine vertex algebras. Second, we prove the rationality of a large subclass of these W-algebras, which includes all exceptional W-algebras of type A and lisse subregular W-algebras in simply laced types. Third, for the latter cases we compute S-matrices and fusion rules. Our results provide the first examples of rational W-algebras associated with non-principal distinguished nilpotent elements, and the corresponding fusion rules are rather mysterious.

arXiv comment39 pages

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Cited by in corpus (13)

• Trialities of $\mathcal{W}$-algebras
• Duality of subregular W-algebras and principal W-superalgebras
• A realisation of the Bershadsky--Polyakov algebras and their relaxed modules
• Admissible-level ${\mathfrak{sl}}_3$ minimal models
• Modularity of Bershadsky-Polyakov minimal models
• Correspondences of categories for subregular W-algebras and principal W-superalgebras
• Subregular W-algebras of type A
• Rationality of the exceptional W-algebras ${\mathcal{W}}_k({\mathfrak{sp}}_4,f_{subreg})$ associated with subregular nilpotent elements of ${\mathfrak{sp}}_4$
• Simplicity of vacuum modules and associated varieties
• On rationality for $C_2$-cofinite vertex operator algebras
• On the representation theory of the vertex algebra $L_{-5/2}(sl(4))$
• 4d $\mathcal{N}=2$ SCFTs and lisse W-algebras
• Vertex Algebras and Commutative Algebras