An admissible level $\widehat{\mathfrak{osp}}\left(1|2\right)$-model: modular transformations and the Verlinde formula

Abstract

The modular properties of the simple vertex operator superalgebra associated to the affine Kac-Moody superalgebra $\widehat{\mathfrak{osp}}\left(1|2\right)$ at level $-\frac{5}{4}$ are investigated. After classifying the relaxed highest-weight modules over this vertex operator superalgebra, the characters and supercharacters of the simple weight modules are computed and their modular transforms are determined. This leads to a complete list of the Grothendieck fusion rules by way of a continuous superalgebraic analogue of the Verlinde formula. All Grothendieck fusion coefficients are observed to be non-negative integers. These results indicate that the extension to general admissible levels will follow using the same methodology once the classification of relaxed highest-weight modules is completed.

41 pages, 1 figure

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• Cosets, characters and fusion for admissible-level $\mathfrak{osp}(1|2)$ minimal models
• A realisation of the Bershadsky--Polyakov algebras and their relaxed modules
• Modularity of logarithmic parafermion vertex algebras
• Admissible-level ${\mathfrak{sl}}_3$ minimal models
• 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 $\widehat{osp(1,2)}$
• 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 $A$
• Representations of the Nappi--Witten vertex operator algebra
• $A_{2l}^{(2)}$ at level $-l-\frac{1}{2}$
• Fusion rules and rigidity for weight modules over the simple admissible affine $\mathfrak{sl}(2)$ and $\mathcal{N}=2$ superconformal vertex operator superalgebras
• A duality between vertex superalgebras $L_{-3/2}(\mathfrak{osp}(1|2))$ and ${\mathcal{V}}^{(2)}$ and generalizations to logarithmic vertex algebras