Fermion realisations of generalised Kac--Moody and Virasoro algebras associated to the two-sphere and the two-torus

Using the notion of extension of Kac-Moody algebras for higher dimensional compact manifolds recently introduced in [1], we show that for the two-torus ${\mathbb{S}}^1\times {\mathbb{S}}^1$ and the two-sphere ${\mathbb{S}}^2$, these extensions, as well as extensions of the Virasoro algebra can be obtained naturally from the usual Kac-Moody and Virasoro algebras. Explicit fermionic realisations are proposed. In order to have well defined generators, beyond the usual normal ordering prescription, we introduce a regulator and regularise infinite sums by means of Riemann $\zeta -$function.