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

Rutwig Campoamor-Stursberg, Michel Rausch de Traubenberg

Published 2022-11-08Version 1

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.