Rationality and Fusion Rules of Exceptional W-Algebras

First, we prove the Kac-Wakimoto conjecture on modular invariance of characters of exceptional affine W-algebras. In fact more generally we prove modular invariance of characters of all lisse W-algebras obtained through Hamiltonian reduction of admissible affine vertex algebras. Second, we prove the rationality of a large subclass of these W-algebras, which includes all exceptional W-algebras of type A and lisse subregular 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 W-algebras associated with non-principal distinguished nilpotent elements, and the corresponding fusion rules are rather mysterious.