标题： 谱流对于极小$W$-代数及其表示的单位性的应用 显示英文标题

标题： Spectral flow for minimal $W$-algebras and application to unitarity of their representations

摘要： 使用谱流，我们提供了对[9, 定理9.17]的证明，该定理涉及最小$W$-代数的Ramond扭曲非极端表示的单位性，该证明不依赖于仍为猜想的扭曲量子约化函子的精确性（参见[9]中的猜想9.11）。当$\mathfrak g=spo(2|2n)$，$F(4)$，$D(2,1;\frac{m}{n})$时，也证明了最小$W$-代数$W^k_{\min}(\mathfrak g)$在Ramond扇区中极值（=无质量）表示的单位性等价于Neveu-Schwarz扇区中极值表示的单位性。 显示英文摘要

摘要： Using spectral flow, we provide a proof of [9, Theorem 9.17] on unitarity of Ramond twisted non-extremal representations of minimal $W$-algebras that does not rely on the still conjectural exactness of the twisted quantum reduction functor (see Conjecture 9.11 of [9]). 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 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.