Algebraic Realisation of the Zamolodchikov Metric in Narain Theories

We revisit Narain conformal field theories from an algebraic perspective based on finite dimensional Lie algebras $\mathbf{g}$ and representations ${\mathcal{R}}_{\mathbf{g}}$, and show how the root and weight lattices can encode the momenta and subsequently the partition functions of Narain theories. In this framework, we construct a realisation of the Zamolodchikov metric of the moduli space ${\mathcal{M}}_{\mathbf{g}}$ in terms of Lie algebraic data namely the Cartan matrix K${}_{\mathbf{g}}$ and its inverse K${}_{\mathbf{g}}^{-1}$. Properties regarding the ensemble averaging of these CFTs and their holographic dual are also derived. Additionally, we discuss possible generalisations to NCFTs having dis-symmetric central charges $({\mathrm{c}}_L,{\mathrm{c}}_R)=(\mathrm{s},$ with $s>r$ and highlight further features of the partition function Z${}_{\mathbf{g}}^{(r,r)}$.