Year: 2026
Type: preprint
Abstract: 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 $\mathbb{Z}_4$ discriminant group. For two hyperbolic pairs, this gives twenty-two even self-dual gluing codes, which fall into seven inequivalent theories at $c=\bar c=6$. 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 $\mathbb{Z}_2$, $\mathbb{Z}_4$ and $\mathbb{Z}_2\times\mathbb{Z}_2$, 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.
Source: arXiv (Cornell University)
Authors: Sammani, Rajae, Sebbar, Abdellah
Institution: University of Ottawa
Language: English
Subfield: Nuclear and High Energy Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223272607