Kac, V. G.; van de Leur, J. W. On classification of superconformal algebras. (English) Zbl 0938.17500 Gates, S. J. jun. (ed.) et al., Strings ’88. Proceedings of a workshop, May 24-28, 1988 at University of Maryland, at College Park, Baltimore, MD, USA. Singapore etc.: World Scientific. 77-106 (1989). From the introduction: It has been known since the early 1970s that there is more than one super extension of the Virasoro algebra. The simplest are the Neveu-Schwarz and Ramond superalgebras, alternatively known as \(N=1\) superconformal algebras. It was realized in the mid-1970s that these Lie superalgebras are the first members of an infinite series, the \(\text{SO}_N\)-superconformal algebras. Furthermore, it was shown that the \(\text{SO}_4\)-superconformal algebra contains yet another example of the \(\text{SU}_2\)-superconformal algebra. At around the same time four series of simple infinite-dimensional Lie superalgebras \(W(M,N)\), \(S(M,N)\), \(H(2M,N)\), \(K(2m-1,N)\) were constructed by V. G. Kac [Adv. Math. 26 No. 1, 8-96 (1977; Zbl 0366.17012)]. For \(N=0\), these become the classical Lie-Cartan series of simple Lie algebras of vector fields on the complex torus, the simplest example being \(\overline{\text{Vir}}=W(1,0)\), the Lie algebra of regular vector fields on \({C}^\times\), called the centerless Virasoro algebra. It was pointed out [ V. G. Kac and I. T. Todorov, Commun. Math. Phys. 102, 337-347 (1985; Zbl 0599.17011,) L. Feigin and D. A. Leites, in: Group theoretical methods in physics, Vols. 1-3 (Zvenigorod, 1982), 264-273 (1983; Zbl 0606.17002); English translation by Harwood Academic Publ., Chur, 623-629 (1985)] that \(K(1,N)\) is nothing else but the \(\text{SO}_N\)-superconformal algebra, and that \(W(1,1)\simeq K(1,2)\). However, as far as we know, it has not been noticed that \(S(1,2)\) is nothing else but the \(\text{SU}_2\)-superconformal algebra. These observations make it plausible that the series \(W(1,N)\), \(S(1,N)\) and \(K(1,N)\), and their ‘variations’ (described in the paper) actually exhaust the list of all superconformal algebras. In the present paper we give an explicit description of the series \(W,S\) and \(K\) (in the spirit of Kac) and classify their central extensions. It turns out that central extensions exist only for small \(N\), hence only a few of our superconformal algebras have nontrivial unitary positive energy representations (since the only such representations of \(\overline{\text{Vir}}\) are trivial). However, the question of which of these algebras have nontrivial modular invariant representations (in the sense of V. G. Kac and M. Wakimoto [ Proc. Natl. Acad. Sci. USA 85, 4956-4960 (1988; Zbl 0652.17010)]) remains an open problem.For the entire collection see [Zbl 0715.53059]. Cited in 2 ReviewsCited in 37 Documents MSC: 17B68 Virasoro and related algebras 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics Citations:Zbl 0366.17012; Zbl 0599.17011; Zbl 0606.17002; Zbl 0652.17010 PDFBibTeX XMLCite \textit{V. G. Kac} and \textit{J. W. van de Leur}, in: Strings '88. Proceedings of a workshop, May 24-28, 1988 at the University of Maryland, at College Park, Baltimore, MD, USA. Singapore etc.: World Scientific. 77--106 (1989; Zbl 0938.17500)