Vertex operator for generalized Kac–Moody algebras associated to the two-sphere and the two-torus

Abstract

We pursue our study of generalized Kac–Moody and Virasoro algebras defined on compact homogeneous manifolds. Extending the well-known vertex operator in the case of the two-torus or the two-sphere, we obtain explicit bosonic realizations of the semi-direct product of the extension of Kac–Moody and Virasoro algebras on [Formula: see text] and [Formula: see text], respectively. As for the fermionic realization previously constructed, in order to have well defined algebras, we introduce, beyond the usual normal ordering prescription, a regulator and regularize infinite sums by means of the Riemann [Formula: see text]-function.