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

TL;DR

This paper extends vertex operator constructions to generalized Kac-Moody and Virasoro algebras on compact manifolds like the two-sphere and two-torus, providing explicit bosonic realizations and regularization techniques.

Contribution

It introduces explicit bosonic realizations of extended algebras on compact manifolds and develops regularization methods for infinite sums in this context.

Findings

Explicit bosonic realizations on  imes  and 

Regularization of infinite sums using Riemann -function

Extension of vertex operators to new geometric settings

Abstract

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.