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 $\mathbf{\Lambda } _{\mathrm{k}}^{\ast }/\mathbf{\Lambda }_{\mathrm{k}}$ 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.