Lattice models from CFT on surfaces with holes I: Torus partition function via two lattice cells

TL;DR

This paper introduces a family of lattice models derived from rational conformal field theories on tori with holes, capturing topological and boundary effects, and demonstrating good numerical agreement with CFT predictions.

Contribution

It constructs a novel lattice model framework from CFT on surfaces with holes, incorporating topological line defects and boundary conditions, and provides an explicit recursive evaluation method.

Findings

Model recovers CFT amplitude at zero radius

Topological symmetry is exactly realized in the lattice model

Numerical results show good agreement with CFT at intermediate radii

Abstract

We construct a one-parameter family of lattice models starting from a two-dimensional rational conformal field theory on a torus with a regular lattice of holes, each of which is equipped with a conformal boundary condition. The lattice model is obtained by cutting the surface into triangles with clipped-off edges using open channel factorisation. The parameter is given by the hole radius. At finite radius, high energy states are suppressed and the model is effectively finite. In the zero-radius limit, it recovers the CFT amplitude exactly. In the touching hole limit, one obtains a topological field theory. If one chooses a special conformal boundary condition which we call "cloaking boundary condition", then for each value of the radius the fusion category of topological line defects of the CFT is contained in the lattice model. The fact that the full topological symmetry of the initial CFT is realised exactly is a key feature of our lattice models. We provide an explicit recursive procedure to evaluate the interaction vertex on arbitrary states. As an example, we study the lattice model obtained from the Ising CFT on a torus with one hole, decomposed into two lattice cells. We numerically compare the truncated lattice model to the CFT expression obtained from expanding the boundary state in terms of the hole radius and we find good agreement at intermediate values of the radius.