Rationality and fusion rules of exceptional $mathcal {W}$-algebras

First, we prove the Kac–Wakimoto conjecture on modular invariance of characters of exceptional affine $mathcal {W}$-algebras. In fact more generally we prove modular invariance of characters of all lisse $mathcal {W}$-algebras obtained through Hamiltonian reduction of admissible affine vertex algebras. Second, we prove the rationality of a large subclass of these $mathcal {W}$-algebras, which includes all exceptional $mathcal {W}$-algebras of type $mathcal {A}$ and lisse subregular $mathcal {W}$-algebras in simply laced types. Third, for the latter cases we compute $mathcal {S}$-matrices and fusion rules. Our results provide the first examples of rational $mathcal {W}$-algebras associated with nonprincipal distinguished nilpotent elements, and the corresponding fusion rules are rather mysteriou