TQFT gravity dual to the orbifold groupoid of Narain code CFTs over $\mathbb{Z}_4$

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.