# On the correspondence between boundary and bulk lattice models and (logarithmic) conformal field theories

## TL;DR

This paper investigates the relationship between boundary and bulk properties in lattice models, using braid translation in Temperley-Lieb algebra-based models to connect open and periodic chains, with implications for conformal field theories.

## Contribution

It demonstrates how braid translation relates open and periodic lattice models for minimal CFTs, and explores its limitations in logarithmic CFTs, providing new insights into algebraic structures and continuum limits.

## Key findings

- Braid translation maps open to periodic models in minimal CFTs.
- In logarithmic CFTs, braid translation does not produce the expected modules.
- Structural differences in modules affect the continuum limit behavior.

## Abstract

The relationship between bulk and boundary properties is one of the founding features of (Rational) Conformal Field Theory. Our goal in this paper is to explore the possibility of having an equivalent relationship in the context of lattice models. We focus on models based on the Temperley-Lieb algebra, and use the concept of braid translation, which is a natural way to close an open spin chain by adding an interaction between the first and last spins using braiding to bring them next to each other. The interaction thus obtained is in general non-local, but has the key feature that it is expressed solely in terms of the algebra for the open spin chain - the ordinary Temperley-Lieb algebra and its blob algebra generalization. This is in contrast with the usual periodic spin chains which involve only local interactions, and are described by the periodic TL algebra. We show that for the Restricted Solid-On-Solid models, which are known to be described by minimal unitary CFTs in the continuum limit, the braid translation in fact does provide the ordinary periodic model starting from the open model with fixed boundary conditions on the two sides of the strip. This statement has a precise mathematical formulation, which is a pull-back map between irreducible modules of, respectively, the blob algebra and the affine TL algebra. We then turn to the same kind of analysis for two models whose continuum limits are Logarithmic CFTs - the alternating gl(1|1) and sl(2|1) spin chains. We find that the result for minimal models does not hold any longer: braid translation of the relevant TL modules does not give rise to the modules known to be present in the periodic chains. In the gl(1|1) case, the content in terms of the irreducibles is the same, as well as the spectrum, but the detailed structure (like logarithmic coupling) is profoundly different. This carries over to the continuum limit.