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Unitary representations of the Virasoro and super-Virasoro algebras. (English) Zbl 0588.17014

Any unitary highest weight representation of the Virasoro algebra is determined by a pair of real numbers (c,h); unitarity implies c0, h0. When c1 it is easy to find a corresponding representation. When c<1 it was shown by D. Friedan, Z. Qiu and S. Shenker [Vertex operators in mathematics and physics, Publ., Math. Sci. Res. Inst. 3, 419-449 (1985; Zbl 0559.58010)] that c belongs to an infinite discrete set and for each such c, h can take only a finite number of values. In previous papers the authors developed a method to obtain representations corresponding to some values of c<1. In the present paper they show that the method in fact provides all the possible unitary highest weight representations of the Virasoro algebra, thus completing its classification.

By a similar method they complete the classification of the unitary highest weight representations of the super-Virasoro algebras.

Reviewer: F.Levstein

MSC:
17B65Infinite-dimensional Lie (super)algebras
17B10Representations of Lie algebras, algebraic theory
58E05Abstract critical point theory
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