Rationality and fusion rules of exceptional \(\mathcal{W}\)-algebras (Q6172688)

Summary: 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.