Holomorphic vertex operator algebras of small central charge (Q1884533)

The authors consider a basic family of vertex operator algebras. Let \(V\) be a \(C_2\)-cofinite holomorphic vertex operator algebra of CFT-type and assume that the central charge \(c\) of \(V\) is small, that is, \(c \leq 24\). Since \(c\) is a multiple of \(8\) in this case by \textit{Y. Zhu} [J. Am. Math. Soc. 9, 237--302 (1996; Zbl 0854.17034)], \(c\) is in fact \(8\), \(16\), or \(24\). The study of those vertex operator algebras with \(c=24\) was initiated by \textit{A. N. Schellekens} [Commun. Math. Phys. 153, 159--185 (1993; Zbl 0782.17014)]. It was shown that there are \(71\) possibilities of the character of such a vertex operator algebra. In the paper under review the authors take a different approach and obtain the following assertions. (1) If \(c = 8\) or \(16\), then \(V\) is isomorphic to a vertex operator algebra associated with a positive definite even unimodular lattice of rank \(c\). (2) If \(c=24\), then either \(V\) is isomorphic to a vertex operator algebra associated with a Niemeier lattice or the Leech lattice, or else the weight one subspace \(V_1\) is a semi-simple Lie algebra of Lie rank less than \(24\). Moreover, the structure of the Lie algebra \(V_1\) is studied in detail. It is shown that there are only finitely many choices for the structure of \(V_1\). Note that there is only one positive definite even unimodular lattice of rank \(8\), namely \(E_8\), and there are exactly two inequivalent such lattices of rank \(16\). The argument is based on the modular invariance of the character of~\(V\).