Coulomb Gas Partition Function of a Layered Loop Model

We consider a two-dimensional bi-layered loop model with a certain interlayer coupling and study its spectrum on a torus. Each layer consists of an $O(n)$ model on a honeycomb lattice with periodic boundary conditions; these layers are stacked such that the links of the lattice intersect each other. A complex Boltzmann weight $\lambda$ with unit modulus is assigned to each intersection of two loops each from each layer. The model is reduced to an inhomogeneous vertex model at a special point of parameters. The continuum partition function is represented, based on the idea of the Coulomb gas, by a path integral over two compact bosonic fields. The modular invariance of the partition function follows naturally. Further, because of the topological nature of the interlayer coupling, the fluctuation of loops decomposes into a local and a global part. The existence of the latter leads to a sum over all the pairs of torus knots, which can be Poisson ressummed by the M\"{o}bius inversion formula. This reveals the operator content of the theory. The multiplicity of each operator is explicitly given by a combination of two Ramanujan sums. We calculate each scaling dimension as a function of $\lambda$. We present the flow of dimensions which connects the decoupled-$O(1)$ models at $\lambda=1$ and the layered-$O(1)$ model with the non-trivial coupling $\lambda=-1$. The lower spectrum in the latter model is related to that of a known coset model.