Admissible-level ${\mathfrak{sl}}_3$ minimal models

Abstract

The first part of this work uses the algorithm recently detailed in arXiv:1906.02935 to classify the irreducible weight modules of the minimal model vertex operator algebra $L_k({\mathfrak{sl}}_3)$, when the level $k$ is admissible. These are naturally described in terms of families parametrised by up to two complex numbers. We also determine the action of the relevant group of automorphisms of ${\widehat{\mathfrak{sl}}}_3$ on their isomorphism classes and compute explicitly the decomposition into irreducibles when a given family's parameters are permitted to take certain limiting values. Along with certain character formulae, previously established in arXiv:2003.10148, these results form the input data required by the standard module formalism to consistently compute modular transformations and, assuming the validity of a natural conjecture, the Grothendieck fusion coefficients of the admissible-level ${\mathfrak{sl}}_3$ minimal models. The second part of this work applies the standard module formalism to compute these explicitly when $k=-\frac{3}{2}$. We expect that the methodology developed here will apply in much greater generality.

34 pages, 5 figures; v2: 37 pages, 5 figures, updated refs, added explanations and discussed relationship with other interesting VOAs

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Cited by in corpus (5)

• Modularity of Bershadsky-Polyakov minimal models
• Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$
• On the structure of W-algebras in type A
• Fusion rules and rigidity for weight modules over the simple admissible affine $\mathfrak{sl}(2)$ and $\mathcal{N}=2$ superconformal vertex operator superalgebras
• Associated varieties of simple affine VOAs $L_k(s{l}_3)$ and $W$-algebras $W_k(s{l}_3,f)$