Spectral Flow and Application to Unitarity of Representations of Minimal W-algebras

Using spectral flow, we provide a proof of [V. G. Kac, P. Moseneder Frajria and P. Papi, Unitarity of minimal W-algebras and their representations II: Ramond sector, Jpn. J. Math.20(2) (2025) 169-281, 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 the same paper). When g=spo(2|2n), F(4), D(2,1;m/n), it is also proven that the unitarity of extremal (=massless) representations of the unitary minimal W-algebra W-min(k)(g) in the Ramond sector is equivalent to the unitarity of extremal representations in the Neveu-Schwarz sector.