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Journal of High Energy Physics

Gauging ℤN symmetries of Narain CFTs

Keiichi Ando, Kohki Kawabata, 辰磨 西岡

Published Feb 4, 2026• 0 citations
Publisher

Abstract

A bstract We investigate the gauging of a ℤ N symmetry in lattice conformal field theories (CFTs), also known as Narain CFTs. For prime N , we derive a spin selection rule for operators in a ℤ N charge-twisted sector of a general bosonic CFT. Using this result, we formulate the gauging procedures in lattice CFTs as modifications of the momentum lattices by a lattice vector that specifies a non-anomalous ℤ N symmetry. Applying this formulation to code CFTs, i.e., Narain CFTs constructed from error-correcting codes, we express the torus partition functions of the orbifolded and parafermionized theories in terms of the weight enumerator polynomials of the underlying codes. As an application, we identify a class of codes that yield self-dual bosonic CFTs under the orbifolding by a ℤ N symmetry.

DOI
10.1007/jhep02(2026)058

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A bstract We investigate the gauging of a ℤ N symmetry in lattice conformal field theories (CFTs), also known as Narain CFTs. For prime N , we derive a spin selection rule for operators in a ℤ N charge-twisted sector of a general bosonic CFT. Using this result, we formulate the gauging procedures in lattice CFTs as modifications of the momentum lattices by a lattice vector that specifies a non-anomalous ℤ N symmetry. Applying this formulation to code CFTs, i.e., Narain CFTs constructed from error-correcting codes, we express the torus partition functions of the orbifolded and parafermionized theories in terms of the weight enumerator polynomials of the underlying codes. As an application, we identify a class of codes that yield self-dual bosonic CFTs under the orbifolding by a ℤ N symmetry.
Gauging ZN\mathbb{Z}_NZN​ symmetries of Narain CFTs
We investigate the gauging of a ZN\mathbb{Z}_NZN​ symmetry in lattice conformal field theories (CFTs), also known as Narain CFTs. For prime NNN, we derive a spin selection rule for operators in a ZN\mathbb{Z}_NZN​ charge-twisted sector of a general bosonic CFT. Using this result, we formulate the gauging procedures in lattice CFTs as modifications of the momentum lattices by a lattice vector that specifies a non-anomalous ZN\mathbb{Z}_NZN​ symmetry. Applying this formulation to code CFTs, i.e., Narain CFTs constructed from error-correcting codes, we express the torus partition functions of the orbifolded and parafermionized theories in terms of the weight enumerator polynomials of the underlying codes. As an application, we identify a class of codes that yield self-dual bosonic CFTs under the orbifolding by a ZN\mathbb{Z}_NZN​ symmetry.
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Narain CFTs from quantum codes and their Z2{\mathbb{Z}}_{2}Z2​ gauging
A bstract We investigate the gauging of a Z2{\mathbb{Z}}_{2}Z2​ symmetry in Narain conformal field theories (CFTs) constructed from qudit stabilizer codes. Considering both orbifold and fermionization, we establish a connection between Z2{\mathbb{Z}}_{2}Z2​ gauging procedures and modifications of the momentum lattice by vectors characterizing the Z2{\mathbb{Z}}_{2}Z2​ symmetry. We also provide three-dimensional interpretations of Z2{\mathbb{Z}}_{2}Z2​ gaugings through abelian Chern-Simons theories, which act as symmetry topological field theories.
Quantum subsystem codes, CFTs and their Z2\mathbb{Z}_2Z2​-gaugings
We construct Narain conformal field theories (CFTs) from quantum subsystem codes, a more comprehensive class of quantum error-correcting codes than quantum stabilizer codes, for qudit systems of prime dimensions. The resulting code CFTs exhibit a global Z2\mathbb{Z}_2Z2​ symmetry, enabling us to perform the Z2\mathbb{Z}_2Z2​-gauging to derive their orbifolded and fermionized theories when the symmetry is non-anomalous. We classify a subset of these subsystem code CFTs using weighted oriented graphs and enumerate those with small central charges. Consequently, we identify several bosonic code CFTs self-dual under the Z2\mathbb{Z}_2Z2​-orbifold, new supersymmetric code CFTs, and a few fermionic code CFTs with spontaneously broken supersymmetry.
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 E8 theory, which is based on the root lattice of E8 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.
Abstract Code CFTs are 2d conformal field theories defined by error-correcting codes. Recently, Dymarsky and Shapere generalized the construction of code CFTs to include quantum error-correcting codes. In this paper, we explore this connection at higher genus. We prove that the higher-genus partition functions take the form of polynomials of higher-weight theta functions, and that the higher-genus modular group acts as simple linear transformations on these polynomials. We explain how to solve the modular constraints explicitly, which we do for genus 2. The result is that modular invariance at genus 1 and genus 2 is much more constraining than genus 1 alone. This allows us to drastically reduce the space of possible code CFTs. We also consider a number of examples of “isospectral theories” — CFTs with the same genus 1 partition function — and we find that they have different genus 2 partition functions. Finally, we make connection to some 2d CFTs known from the modular bootstrap. The n = 4 theory conjectured to have the largest possible gap in Virasoro characters, the SO(8) WZW model, is a code CFT, allowing us to give an expression for its genus 2 partition function. We also find some other known CFTs which are not code theories but whose partition functions satisfy the same simple polynomial ansatz as the code theories. This leads us to speculate about the usefulness of the code polynomial form beyond the study of code CFTs.