Fermion realizations of generalized Kac–Moody and Virasoro algebras associated to the two-sphere and the two-torus

Abstract

By using the notion of extension of Kac–Moody algebras for higher-dimensional compact manifolds recently introduced, we show that for the two-torus [Formula: see text] and the two-sphere [Formula: see text], these extensions, as well as extensions of the Virasoro algebra can be obtained naturally from the usual Kac–Moody and Virasoro algebras. Explicit fermionic realizations are proposed. In order to have well-defined generators, beyond the usual normal ordering prescription, we introduce a regulator and regularize infinite sums by means of Riemann [Formula: see text]-function.