Wakimoto construction for double loop algebras and $\zeta$-function regularisation

TL;DR

This paper extends the Wakimoto construction to double loop algebras and demonstrates that divergent sums encountered can be regularized to vanish using $$-function techniques, providing a consistent mathematical framework.

Contribution

It introduces $$-function regularisation into the Wakimoto construction for double loop algebras, resolving divergence issues and advancing the understanding of affine Lie algebra representations.

Findings

Divergent sums in double loop algebra constructions can be regularized to zero.

$$-function regularisation provides a consistent method for handling divergences.

The extended Wakimoto construction applies successfully to double loop algebras.

Abstract

The Feigin-Frenkel homomorphism underpinning the Wakimoto construction realises an affine Lie algebra at critical level in terms of the $\beta\gamma$-system of free fields. It was recently shown that much of the construction also goes through for double loop algebras. However, certain divergent sums appear. In this paper, we show that, suggestively, these sums vanish when one performs $\zeta$-function regularisation.