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

communications in mathematical physics/communications in mathematical physics

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