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.