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

Using spectral flow, we provide a proof of [11, Theorem 9.17] on unitarity of Ramond twisted non-extremal representations of unitary minimal [Formula: see text]-algebras that does not rely on the still conjectural exactness of the twisted quantum reduction functor (see Conjecture 9.11 of [11]). When [Formula: see text], [Formula: see text], [Formula: see text], it is also proven that the unitarity of extremal (=massless) representations of the unitary minimal [Formula: see text]-algebra [Formula: see text] in the Ramond sector is equivalent to the unitarity of extremal representations in the Neveu-Schwarz sector.