Algebraic constructions of code lattices in Narain conformal field theories

TL;DR

This paper explores algebraic lattice constructions in Narain conformal field theories, providing explicit methods for building relevant lattices and analyzing their inclusion relations and structural properties.

Contribution

It introduces new algebraic constructions of code-related lattices in Narain CFTs, including explicit methods for various ranks and dimensions.

Findings

Explicit constructions of lattices for rank r=d=1 and higher dimensions.

Analysis of inclusion relations among the lattices based on discriminant groups.

Discussion of structural features and potential generalizations.

Abstract

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.