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

Abstract

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 $A$ and lisse subregular $\mathcal {W}$-algebras in simply laced types. Third, for the latter cases we compute $S$-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 mysterious.