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

Abstract

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.

Figures (14)

• Figure 1: Constructing the lattice model from a CFT: cut a regular lattice of holes into the torus $T$ (a triangular lattice in this example), with distance $d$ and radius $R$; perform a sum over intermediate boundary states at the dashed lines; treat a (truncated) basis of boundary states as states of the lattice model assigned to the edges of the dual lattice (a hexagonal lattice in this example), and the disc amplitudes arising after cutting along the dashed lines as Boltzmann weights assigned to the vertices of the dual lattice.
• Figure 2: a) Definition of the $\mathcal{F}$-cloaking boundary condition $\gamma_{\mathcal{F}}^b(\delta)$ in terms of the cloaking defect. b) The cloaking boundary condition is transparent to topological defects $x \in \mathcal{F}$.
• Figure 3: Example of regular lattices on the torus with (a) triangular cells and (b) square cells, together with holes of radius $R$ at each vertex, equipped with the cloaking boundary condition $\gamma_\mathcal{F}^b(\delta)$. Figures (c) and (d) show the corresponding elementary cells, whose amplitudes $W_{d,R}(\kappa_1,\dots)$ define the vertex weights of the lattice model.
• Figure 4: a) Example of a circle packing on the sphere determined by a triangulation, together with some of the holes of radii $R_v = \rho r_v$ cut into the sphere. b) Example of an elementary cell given by a geodesic triangle with clipped edges as dictated by the radii $r_v$ of the circles in the circle packing and the ratio $\rho = R_v/r_v$.
• Figure 5: The unit disc $D$ with field insertion points and local coordinates. The corresponding amplitude defines the pairing $(\zeta,\xi)_{ab}$.

Theorems & Definitions (2)

• proof
• proof