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.

Topics & Concepts