$\mathbb{Z}_N$ Duality and Parafermions Revisited

TL;DR

This paper explores the dualities and boundary states in two-dimensional theories with non-anomalous a_Nb_Nb symmetry, generalizing known a_2b_2b cases to parafermions and their topological properties.

Contribution

It extends the understanding of a_2b_2b dualities to a_Nb_Nb theories, introducing a comprehensive framework for boundary states and operators in parafermionization.

Findings

Identified operators defining topological boundary states for a_Nb_Nb symmetry

Analyzed algebraic properties of boundary states and operators

Mapped the a_Nb_Nb duality web

Abstract

Given a two-dimensional bosonic theory with a non-anomalous $\mathbb{Z}_2$ symmetry, the orbifolding and fermionization can be understood holographically using three-dimensional BF theory with level $2$. From a Hamiltonian perspective, the information of dualities is encoded in a topological boundary state which is defined as an eigenstate of certain Wilson loop operators (anyons) in the bulk. We generalize this story to two-dimensional theories with non-anomalous $\mathbb{Z}_N$ symmetry, focusing on parafermionization. We find the generic operators defining different topological boundary states including orbifolding and parafermionization with $\mathbb{Z}_N$ or subgroups of $\mathbb{Z}_N$, and discuss their algebraic properties as well as the $\mathbb{Z}_N$ duality web.