Relaxed highest-weight modules I: rank $1$ cases

Abstract

Relaxed highest-weight modules play a central role in the study of many important vertex operator (super)algebras and their associated (logarithmic) conformal field theories, including the admissible-level affine models. Indeed, their structure and their (super)characters together form the crucial input data for the standard module formalism that describes the modular transformations and Grothendieck fusion rules of such theories. In this article, character formulae are proved for relaxed highest-weight modules over the simple admissible-level affine vertex operator superalgebras associated to ${\mathfrak{sl}}_2$ and $\mathfrak{osp}(1|2)$. Moreover, the structures of these modules are specified completely. This proves several conjectural statements in the literature for ${\mathfrak{sl}}_2$, at arbitrary admissible levels, and for $\mathfrak{osp}(1|2)$ at level $-\frac{5}{4}$. For other admissible levels, the $\mathfrak{osp}(1|2)$ results are believed to be new.

Minor revision, 28 pages, 1 figure, to appear in Comm. Math. Phys

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Cited by in corpus (25)

• Logarithmic W-algebras and Argyres-Douglas theories at higher rank
• On fusion rules and intertwining operators for the Weyl vertex algebra
• Cosets, characters and fusion for admissible-level $\mathfrak{osp}(1|2)$ minimal models
• Tensor categories of affine Lie algebras beyond admissible levels
• A realisation of the Bershadsky--Polyakov algebras and their relaxed modules
• Unitary and non-unitary $N=2$ minimal models
• Admissible-level ${\mathfrak{sl}}_3$ minimal models
• Positive energy representations of affine vertex algebras
• Classifying relaxed highest-weight modules for admissible-level Bershadsky-Polyakov algebras
• Anyonic Topological Order in Twisted Equivariant Differential (TED) K-Theory
• Anyonic Defect Branes and Conformal Blocks in Twisted Equivariant Differential (TED) K-theory
• Realizations of simple affine vertex algebras and their modules: the cases $\widehat{sl(2)}$ and $\widehat{osp(1,2)}$
• Modularity of Bershadsky-Polyakov minimal models
• Admissible level $\mathfrak{osp}(1|2)$ minimal models and their relaxed highest weight modules
• Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$
• On conformal field theories based on Takiff superalgebras
• Representations of the Nappi--Witten vertex operator algebra
• On irreducibility of modules of Whittaker type for cyclic orbifold vertex algebra
• On resolution of highest weight modules over the $\mathcal{N}=2$ superconformal algebra
• Staggered modules of $N=2$ superconformal minimal models
• On the representation theory of the vertex algebra $L_{-5/2}(sl(4))$
• Feigin-Semikhatov conjecture and related topics
• Relaxed and logarithmic modules of $\widehat{{\mathfrak{sl}}_3}$
• Simple modules for Affine vertex algebras in the minimal nilpotent orbit
• A duality between vertex superalgebras $L_{-3/2}(\mathfrak{osp}(1|2))$ and ${\mathcal{V}}^{(2)}$ and generalizations to logarithmic vertex algebras