## Abstract

There is a rich connection between classical error-correcting codes, Euclidean lattices, and chiral conformal field theories. Here we show that quantum error-correcting codes, those of the stabilizer type, are related to Lorentzian lattices and non-chiral CFTs. More specifically, real self-dual stabilizer codes can be associated with even self-dual Lorentzian lattices, and thus define Narain CFTs. We dub the resulting theories code CFTs and study their properties. T-duality transformations of a code CFT, at the level of the underlying code, reduce to code equivalences. By means of such equivalences, any stabilizer code can be reduced to a graph code. We can therefore represent code CFTs by graphs. We study code CFTs with small central charge c = n ≤ 12, and find many interesting examples. Among them is a non-chiral E_{8} theory, which is based on the root lattice of E_{8} understood as an even self-dual Lorentzian lattice. By analyzing all graphs with n ≤ 8 nodes we find many pairs and triples of physically distinct isospectral theories. We also construct numerous modular invariant functions satisfying all the basic properties expected of the CFT partition function, yet which are not partition functions of any known CFTs. We consider the ensemble average over all code theories, calculate the corresponding partition function, and discuss its possible holographic interpretation. The paper is written in a self-contained manner, and includes an extensive pedagogical introduction and many explicit examples.

Original language | English |
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Article number | 160 |

Journal | Journal of High Energy Physics |

Volume | 2021 |

Issue number | 3 |

DOIs | |

State | Published - Mar 2021 |

### Bibliographical note

Publisher Copyright:© 2021, The Author(s).

### Funding

Funders | Funder number |
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Horizon 2020 Framework Programme | 677368, 1720374 |

## Keywords

- Conformal Field Theory
- Integrable Field Theories

## ASJC Scopus subject areas

- Nuclear and High Energy Physics