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

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.