Rationality of the exceptional W-algebras ${\mathcal{W}}_k({\mathfrak{sp}}_4,f_{subreg})$ associated with subregular nilpotent elements of ${\mathfrak{sp}}_4$

Abstract

We prove the rationality of the exceptional W-algebras associated with the simple Lie algebra ${\mathfrak{sp}}_4$ and subregular nilpotent elements, proving a new particular case of a conjecture of Kac-Wakimoto. Moreover, we describe the simple ${\mathcal{W}}_k({\mathfrak{sp}}_4,f_{subreg})$-modules and compute their characters. We also explicit the nontrivial action of the component group on the set of these simple modules.

arXiv comment31 pages in English ; in the new version we explicit the nontrivial action of the component group on the set of simple ${\mathcal{W}}_k({\mathfrak{sp}}_4,f_{subreg})$-modules (Sect.8), compute their characters and make a conjecture (Sect.9)

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• Trialities of orthosymplectic $\mathcal{W}$-algebras
• Subregular W-algebras of type A
• Singularities of nilpotent Slodowy slices and collapsing levels of W-algebras
• Orthosymplectic Feigin-Semikhatov duality