3. Virasoro algebra and super-Virasoro algebras

Abstract

Starting with the fundamental paper [57] Belavin-Polyakov-Zamolodchikov (BPZ) there has been great interest in the two-dimensional conformal theories [245, 195]. Although the underlying Virasoro algebra was introduced a long time ago by Gelfand-Fuchs [266] in mathematics1 and independently, by Virasoro [561] in physics, was extensively used in dual string theories (cf. e. g. [338]) the lack of a developed representation theory obviously hindered its further applications. It is no wonder that the BPZ paper appeared after the papers of Kac [352] and of Feigin-Fuchs [214]. Even a decade long work on the Liouville theory by Gervais-Neveu was better understood and reshaped (cf. [271], and also [376]). The above developments inevitably made contact with the Kac-Moody symmetry approach to current algebras [570, 134, 124]. That contact was initiated by Segal [533] who showed that every highest weight module (HWM) of a Kac-Moody algebra may be extended to the semi-direct product with the Virasoro algebra. Consequently, the elements of the latter can be realized as bilinear combinations (normally ordered) of elements of the Kac-Moody algebras (i. e. as elements of the latter universal enveloping algebras) [243, 533]. In the physical literature such constructions can be traced back to [548] but were reintroduced in the study of completely integrable models by Sato and others [528]. In the current algebra context they were mentioned by Polyakov [495] and used in full force in [392] (see also [277, 554]). In this chapter, first, following [155] and [156] we give the multiplet classification of all reducible HWMs over the Virasoro and N = 1 super-Virasoro algebras. Along with a simple explicit parametrization of all reducible Verma modules, we show all the possible embedding maps between them. These embedding maps correspond one-to-one to all possible singular vectors of the reducible Verma modules. We recall that any singular vector becomes a homogeneous polynomial and, in a function space realization, it becomes a linear differential operator of an order equal to the degree of the corresponding polynomial. These degrees are given explicitly for the singular vectors corresponding to all noncomposition embedding maps and, thus, for all singular vectors. Further, following [157] we give the characters of all irreducible highest weight modules over the Virasoro algebra and N = 1 super- Virasoro algebras (Neveu-Schwarz superalgebra and Ramond superalgebra). These are given in an explicit and unified form for all three (super-)algebras incorporating all previously known results. Using the character formulae we introduce a Weyl group for Virasoro and N = 1 super-Virasoro algebras [164]. Further, following [109] we present formulae for singular vectors c < 1 Fock modules over the Virasoro algebras. These we present in terms of Schur polynomials generalizing the c = 1 expressions of Goldstone. We reproduce the known formulae for the singular vectors in (1, n), (m, 1) modules and give new formulae in the cases (2, n), (m, 2), (3, 3). Furthermore, we present the characters of the unitarizable highest weight modules over the N = 2 superconformal algebras following mainly [79] and [158]. Finally, following [181] and [182] we consider modular invariants for theta-functions with characteristics and the twisted N = 2 superconformal and su(2) Kac-Moody algebras and the classification of the corresponding modular invariant partition functions.