Algebraic constructions of code lattices in Narain conformal field theories

We give new results on the structure and representations of the three lattices ${\mathbf{\Lambda }}_{\mathrm{k}},{\mathbf{\Lambda }}_{\mathrm{k}\mathcal{C}},{\mathbf{\Lambda }}_{\mathrm{k}}^{\ast }$ relevant to code CFTs realizing Narain conformal field theories. In this construction, ${\mathbf{\Lambda }}_{\mathrm{k}}^{\ast }$ denotes the dual of the even lattice ${\mathbf{\Lambda }}_{\mathrm{k}}$ and ${\mathbf{\Lambda }}_{\mathrm{k}\mathcal{C}}$ is an even self-dual intermediate lattice with a (d,d) signature. We study the inclusion relations ${\mathbf{\Lambda }}_{\mathrm{k}}\subset {\mathbf{\Lambda }}_{\mathrm{k}\mathcal{C}}\subset {\mathbf{\Lambda }}_{\mathrm{k}}^{\ast }$ characterized by the discriminant group ${\mathbf{\Lambda }}_{\mathrm{k}}^{\ast }\mathrm{/}{\mathbf{\Lambda }}_{\mathrm{k}}$ isomorphic to ${\mathbb{Z}}_{\mathrm{k}}$ and provide explicit constructions of these ${\mathbb{R}}^{(\mathrm{r}d,\mathrm{r}d)}$ lattices first for rank $\mathrm{r}=d=1$ and then for higher dimensional Lie algebras with $\mathrm{r}=d>1$. Additional structural features and generalisations are also discussed.