Rationality of the Exceptional $$\mathcal {W}$$-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 $$\mathcal {W}$$ -algebras $$\mathcal {W}_k(\mathfrak {g},f)$$ associated with the simple Lie algebra $$\mathfrak {g}=\mathfrak {sp}_{4}$$ and a subregular nilpotent element $$f=f_{subreg}$$ of $$\mathfrak {sp}_{4}$$ , proving a new particular case of a conjecture of Kac–Wakimoto. Moreover, we describe the simple $$\mathcal {W}_k(\mathfrak {g},f)$$ -modules and compute their characters. We also work out the nontrivial action of the component group on the set of simple $$\mathcal {W}_k(\mathfrak {g},f)$$ -modules.