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

TL;DR

This paper constructs fermionic realizations of extended Kac-Moody and Virasoro algebras associated with the two-sphere and two-torus, using regularization techniques to define the generators properly.

Contribution

It introduces explicit fermionic realizations of these extended algebras for higher-dimensional manifolds, extending the usual Kac-Moody and Virasoro frameworks.

Findings

Extensions can be derived from standard algebras for these manifolds.

Fermionic realizations are explicitly constructed.

Regularization via Riemann zeta-function ensures well-defined generators.

Abstract

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.