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

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2112.01563.pdf

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Brehm_2022_J._Phys._A _Math._Theor._55_235001.pdf

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##### Citation

Brehm, E. M., & Runkel, I. (2022). Lattice models from CFT on surfaces with holes
I: Torus partition function via two lattice cells.* Journal of physics A,* *55*(23):
235001. doi:10.1088/1751-8121/ac6a91.

Cite as: https://hdl.handle.net/21.11116/0000-0009-B498-0

##### 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.

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.