Unitary representations of Ramond and Neveu-Schwarz superalgebras
Let \(W=\mathfrak{der}[t^{-1},t]\) be the Witt algebra, \(\mathrm{Vir}=W\oplus \mathbb Cz\), its nontrivial central extension, the Virasoro algebra. Let \({\mathfrak h}=\langle L_ 0=t\,d/dt,z\rangle\) be the Cartan, or better say maximal toral, subalgebra in \(\mathrm{Vir}\). All \(\mathfrak h\)-diagonalizable representations with highest weight of \(\mathrm{Vir}\) were described by \textit{B. L. Feigin} and \textit{D. B. Fuks} [Funkts. Anal. Prilozh. 16, No. 2, 47--63 (1982; Zbl 0505.58031); ibid. 17, No. 3, 91--92 (1983; Zbl 0526.17010); Lect. Notes Math. 1060, 230--245 (1984; Zbl 0549.17010)] wherefrom a description of unitary representations, later rediscovered, follows easily (Theorem 2). The main result of the paper (Theorem 3) is the proof of an analogue of Theorem 2 for Neveu-Schwarz (NS) and Ramond (R) superalgebras. NS and R are, respectively, central extensions of the Lie superalgebras of contact vector fields on a supermanifold associated with the trivial (resp. Möbius) rank 1 bundle over a circle. In other words, NS and R are central extensions of the Lie superalgebras preserving contact structures given, respectively, by the forms \(dt+\theta d\theta\) and \(dt+t\theta d\theta.\) Remark. Theorem 3 is by no means the final result. Representations of Lie superalgebras might have odd parameters; as shown by \textit{V. Serganova} [Funkts. Anal. Prilozh. 17, No. 3, 46--54 (1983; Zbl 0545.17001)] there are at least three types of unitarity for Lie superalgebras and the authors have considered just one of them; even if we confine ourselves to the considered type and disregard odd parameters the considered representations are induced from a 1-dimensional representation of the maximal ``torus'' \(\mathfrak h\), whereas for R the superspace of vacuum vectors is generally \((1,1)\)-dimensional.
Lie superalgebras of contact vector fields
scientific article; zbMATH DE number 4107066