Modularity of admissible-level $\mathfrak{sl}_{3}$ minimal models with denominator $2$

TL;DR

This paper investigates the representation theory of certain affine vertex algebras related to sl_3 at specific levels, using inverse quantum Hamiltonian reduction to classify modules and verify fusion rules.

Contribution

It introduces a novel application of inverse quantum Hamiltonian reduction to classify irreducible modules of sl_3 minimal models at denominator 2 and confirms the Verlinde formula predictions.

Findings

Constructed all irreducible modules via inverse reduction.

Derived modular S-transforms for characters.

Verified nonnegative integer fusion multiplicities.

Abstract

We use the newly developed technique of inverse quantum hamiltonian reduction to investigate the representation theory of the simple affine vertex algebra $\mathsf{A}_{2}(\mathsf{u},2)$ associated to $\mathfrak{sl}_{3}$ at level $\mathsf{k} = -3+\frac{\mathsf{u}}{2}$, for $\mathsf{u}\ge3$ odd. Starting from the irreducible modules of the corresponding simple Bershadsky-Polyakov vertex operator algebras, we show that inverse reduction constructs all irreducible lower-bounded weight $\mathsf{A}_{2}(\mathsf{u},2)$-modules. This proceeds by first constructing a complete set of coherent families of fully relaxed highest-weight $\mathsf{A}_{2}(\mathsf{u},2)$-modules and then noting that the reducible members of these families degenerate to give all remaining irreducibles. Using this fully relaxed construction and the degenerations, we deduce modular S-transforms for certain natural generalised characters of these irreducibles and their spectral flows. With this modular data in hand, we verify that the (conjectural) standard Verlinde formula predicts Grothendieck fusion rules with nonnegative-integer multiplicities.