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.