TQFT gravity dual to the orbifold groupoid of Narain code CFTs over ℤ4
Abstract
In a companion paper arXiv:2610.26610, we classified the Narain conformal field theories obtained by gluing copies of the A3 root and weight lattices along their ℤ4 discriminant group. For two hyperbolic pairs, this gives twenty-two even self-dual gluing codes, which fall into seven inequivalent theories at c=c̅=6. In this paper, we develop the physical interpretation of this family and its connections to three-dimensional topological field theory. We show that all twenty-two codes are topological boundary conditions of a single abelian Chern-Simons theory. Its invertible symmetries, which include fractional B-field shifts, organize the boundaries into three orbits: free, mixed and torsion. Boundaries in different orbits are related only by gauging. We determine all gaugings of order at most four between the seven theories, by ℤ2, ℤ4 and ℤ2×ℤ2, with explicit twisted partition functions, and follow their effect on the spectrum and the current algebra. We then sum the Chern-Simons partition function over handlebodies at genus one and prove a finite Siegel-Weil identity, valid for any number of pairs. For two pairs, the handlebody sum is the uniform average over the free theories. Recovering the average over all twenty-two theories requires one additional contribution, a solid torus with the torsion lines along its core. For three or more pairs, we find that the handlebody sum is not a positive combination of code boundary states.
Related
BibTeX
@misc{sammani2026tqft,
title = {{TQFT gravity dual to the orbifold groupoid of Narain code CFTs over $\mathbbZ_4$}},
author = {Rajae Sammani and Abdellah Sebbar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30525},
eprint = {2610.30525},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30525},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - TQFT gravity dual to the orbifold groupoid of Narain code CFTs over ℤ_4
AU - Rajae Sammani
AU - Abdellah Sebbar
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.30525
UR - https://arxiv.org/abs/2610.30525
AB - In a companion paper arXiv:2610.26610, we classified the Narain conformal field theories obtained by gluing copies of the A_3 root and weight lattices along their ℤ_4 discriminant group. For two hyperbolic pairs, this gives twenty-two even self-dual gluing codes, which fall into seven inequivalent theories at c=c̅=6. In this paper, we develop the physical interpretation of this family and its connections to three-dimensional topological field theory. We show that all twenty-two codes are topological boundary conditions of a single abelian Chern-Simons theory. Its invertible symmetries, which include fractional B-field shifts, organize the boundaries into three orbits: free, mixed and torsion. Boundaries in different orbits are related only by gauging. We determine all gaugings of order at most four between the seven theories, by ℤ_2, ℤ_4 and ℤ_2×ℤ_2, with explicit twisted partition functions, and follow their effect on the spectrum and the current algebra. We then sum the Chern-Simons partition function over handlebodies at genus one and prove a finite Siegel-Weil identity, valid for any number of pairs. For two pairs, the handlebody sum is the uniform average over the free theories. Recovering the average over all twenty-two theories requires one additional contribution, a solid torus with the torsion lines along its core. For three or more pairs, we find that the handlebody sum is not a positive combination of code boundary states.
ER -