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

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.