Admissible-level \(\mathfrak{sl}(3)\) minimal models

Classifies the irreducible weight modules of the admissible-level \(\mathfrak{sl}_3\) minimal model vertex operator algebras, providing the input needed for the standard module formalism. At level \(-\tfrac{3}{2}\) this gives modular transformations and conjectural Grothendieck fusion rules, the first application of the formalism to a genuinely rank-two example.

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 \(\hat{\mathfrak{sl}}_3\) 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 \(k=-\frac32\). We expect that the methodology developed here will apply in much greater generality.

Recommended citation: K. Kawasetsu, D. Ridout and S. Wood, Lett. Math. Phys. 112, 96 (2022)
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