Unitarizable highest weight modules over the twisted sector of the \(N=2\) super Virasoro algebra
In this paper, the author classifies unitarizable highest weight modules over the twisted sector of the \(N=2\) Virasoro super-algebra \(\mathfrak{g}\). The problem has been considered before by \textit{W. Boucher}, \textit{D. Friedan} and \textit{A. Kent} in [Phys. Lett. B 172, 316--322 (1986; Zbl 1174.81319)], where the authors gave without proof a necessary condition for unitarizability of a Verma module over \(\mathfrak{g}\). Here the author gives a proof of this result, and shows by explicit construction that the condition is also sufficient. More precisely, there are two families of unitarizable Verma modules over \(\mathfrak{g}\). The \textit{minimal unitary series} is realised via an action of \(\mathfrak{g}\) on simple integrable highest weight modules over the affine Lie algebra \(\widehat{\mathfrak{sl}_2}\). For the other family a direct proof of unitarizability is given using the determinant formula from [\textit{K. Iohara} and \textit{Y. Koga}, J. Funct. Anal. 214, No. 2, 450--518 (2004; Zbl 1155.17309)].
twisted sector of the \(N=2\) Virasoro super-algebra
scientific article; zbMATH DE number 5298319