Unifying error-correcting code/Narain CFT correspondences via lattices over integers of cyclotomic fields

TL;DR

This paper unifies the construction of Narain conformal field theories with quantum error-correcting codes over cyclotomic integer rings, generalizing known lattice constructions and connecting them to Lie algebra root and weight lattices.

Contribution

It introduces a generalized code-lattice construction for Narain CFTs over cyclotomic fields, extending previous methods and linking QECCs to Lie algebra lattices.

Findings

Constructed Narain CFTs from codes over cyclotomic integer rings.

Unified description of QECCs and Narain CFTs via lattice constructions.

Extended to non-prime q and related to Lie algebra root and weight lattices.

Abstract

We identify Narain conformal field theories (CFTs) that correspond to code lattices for quantum error-correcting codes (QECC) over integers of cyclotomic fields $Q(\zeta_p)$ $(\zeta_p=e^{\frac{2\pi i}p})$ for general prime $p\geq 3$. This code-lattice construction is a generalization of more familiar ones such as Construction A${}_C$ for ternary codes and (after the generalization stated below) Construction A for binary codes, containing them as special cases. This code-lattice construction is redescribed in terms of root and weight lattices of Lie algebras, which allows to construct lattices for codes over rings $Z_q$ with non-prime $q$. Corresponding Narain CFTs are found for codes embedded into quotient rings of root and weight lattices of $ADE$ series, except $E_8$ and $D_k$ with $k$ even. In a sense, this provides a unified description of the relationship between various QECCs over $F_p$ (or $Z_q$) and Narain CFTs. A further extension on constructing the $E_8$ lattice from codes over the Mordell-Weil groups of extremal rational elliptic surfaces is also briefly discussed.