Domain walls in topological phases and the Brauer-Picard ring for $\operatorname{Vec}(\mathbb{Z}/p\mathbb{Z})$

TL;DR

This paper develops a diagrammatic method to compute the Brauer-Picard ring for certain fusion categories and interprets bimodule categories as domain walls in topological phases, linking algebraic and physical perspectives.

Contribution

It introduces ladder string diagrams for calculating the relative tensor product of bimodule categories and provides a physical interpretation of bimodule categories as domain walls.

Findings

Computed the Brauer-Picard ring for Vec(Z/pZ)

Established a physical interpretation of bimodule categories as domain walls

Demonstrated how to derive algebraic structures from physical insights

Abstract

We show how to calculate the relative tensor product of bimodule categories (not necessarily invertible) using ladder string diagrams. As an illustrative example, we compute the Brauer-Picard ring for the fusion category $\operatorname{Vec}(\mathbb{Z}/p\mathbb{Z})$. Moreover, we provide a physical interpretation of all indecomposable bimodule categories in terms of domain walls in the associated topological phase. We show how this interpretation can be used to compute the Brauer-Picard ring from a physical perspective.