Lattice models from CFT on surfaces with holes II: Cloaking boundary conditions and loop models

In this paper we continue to investigate the lattice models obtained from 2d CFTs via the construction introduced in [arXiv:2112.01563]. On the side of the 2d CFT we consider the cloaking boundary condition relative to a fixed fusion category F of topological line defects. The resulting lattice model realises the topological symmetry F exactly. We compute the state spaces and Boltzmann weights of these lattice model in the example of unitary Virasoro minimal models. We work directly with amplitudes, rather than with normalised correlators, and we provide a careful treatment of the Weyl anomaly factor in terms of the Liouville action. We numerically evaluate the Ising CFT on the torus with one hole and cloaking boundary condition in two channels, and illustrate in this example that the anomaly factors are essential to obtain matching results for the amplitudes. We show that lattice models obtained from Virasoro minimal models at lowest non-trivial cutoff can be exactly mapped to loop models. This provides a first non-trivial check that our lattice models can contain the 2d CFT they were constructed from in their phase diagram, and we propose a condition on the cloaking boundary condition for F under which we expect this to happen in general.