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

We pursue our study of generalised 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 realisations of the semi-direct product of the extension of Kac-Moody and Virasoro algebras on ${\mathbb{S}}^1\times {\mathbb{S}}^1$ and ${\mathbb{S}}^2$, respectively. As for the fermionic realisation previously constructed, in order to have well defined algebras, we introduce, beyond the usual normal ordering prescription, a regulator and regularise infinite sums by means of the Riemann $\zeta$-function.