# Computing Defects Associated to Bounded Domain Wall Structures: The $\operatorname{Vec}(\mathbb{Z}/p\mathbb{Z})$ Case

## TL;DR

This paper introduces an algorithm to compute defects in domain wall structures for topological phases, demonstrating its application to the Vec(Z/pZ) case and generalizing Levin-Wen models with domain walls and defects.

## Contribution

The paper presents a novel algorithm for computing compound defects in domain wall structures and applies it to show triviality of the bimodule associator for Vec(Z/pZ) domain walls.

## Key findings

- The domain wall structure algorithm effectively computes compound defects.
- The bimodule associator is trivial for all Vec(Z/pZ) domain walls.
- Ground states of generalized Levin-Wen models can be computed using this algorithm.

## Abstract

A domain wall structure consists of a planar graph with faces labeled by fusion categories/topological phases. Edges are labeled by bimodules/domain walls. When the vertices are labeled by point defects we get a compound defect. We present an algorithm, called the domain wall structure algorithm, for computing the compound defect. We apply this algorithm to show that the \emph{bimodule associator}, related to the $O_3$ obstruction of [Etingof et al., Quantum Topol. 1, 209 (2010), arXiv:0909.3140], is trivial for all domain walls of $\operatorname{Vec}(\mathbb{Z}/p\mathbb{Z})$. In the language of this paper, the ground states of the Levin-Wen model are compound defects. We use this to define a generalization of the Levin-Wen model with domain walls and point defects. The domain wall structure algorithm can be used to compute the ground states of these generalized Levin-Wen type models.