Spectral flow and application to unitarity of representations of minimal $W$-algebras

Abstract

Using spectral flow, we provide a proof of [11, Theorem 9.17] on unitarity of Ramond twisted non-extremal representations of unitary minimal $W$-algebras that does not rely on the still conjectural exactness of the twisted quantum reduction functor (see Conjecture 9.11 of [11]). When $\mathfrak g = spo(2|2n)$, $F (4$), $D(2, 1; \frac{m}{n})$, it is also proven that the unitarity of extremal (=massless) representations of the unitary minimal $W$-algebra $W^k_{\min}(\mathfrak g)$ in the Ramond sector is equivalent to the unitarity of extremal representations in the Neveu-Schwarz sector.