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.

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

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• A realisation of the Bershadsky--Polyakov algebras and their relaxed modules
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• Unitary and non-unitary $N=2$ minimal models
• Admissible-level ${\mathfrak{sl}}_3$ minimal models
• Positive energy representations of affine vertex algebras
• Anyonic Topological Order in Twisted Equivariant Differential (TED) K-Theory
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• Classifying relaxed highest-weight modules for admissible-level Bershadsky-Polyakov algebras
• Realizations of simple affine vertex algebras and their modules: the cases $\widehat{sl(2)}$ and $\widehat{osp(1,2)}$
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• 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 $L_{-5/2}(sl(4))$
• Relaxed and logarithmic modules of $\widehat{{\mathfrak{sl}}_3}$
• A duality between vertex superalgebras $L_{-3/2}(\mathfrak{osp}(1|2))$ and ${\mathcal{V}}^{(2)}$ and generalizations to logarithmic vertex algebras
• Simple modules for Affine vertex algebras in the minimal nilpotent orbit
• Feigin-Semikhatov conjecture and related topics