Boundary algebras of the Kitaev Quantum Double model

TL;DR

This paper proves local topological order axioms for Kitaev's Quantum Double model, identifies boundary nets with fusion categorical nets, and explores their connections to SPT phases and higher-dimensional models.

Contribution

It establishes the LTO axioms for the Quantum Double model, characterizes boundary nets with fusion categories, and links boundary algebras to SPT phases and higher-dimensional cases.

Findings

Proved LTO axioms for Kitaev's Quantum Double model.

Identified boundary nets with fusion categorical nets.

Computed boundary algebras for (3+1)D Quantum Double model.

Abstract

The recent article [arXiv:2307.12552] gave local topological order (LTO) axioms for a quantum spin system, showed they held in Kitaev's Toric Code and in Levin-Wen string net models, and gave a bulk boundary correspondence to describe bulk excitations in terms of the boundary net of algebras. In this article, we prove the LTO axioms for Kitaev's Quantum Double model for a finite group $G$. We identify the boundary nets of algebras with fusion categorical nets associated to $(\mathsf{Hilb}(G),\mathbb{C}[G])$ or $(\mathsf{Rep}(G),\mathbb{C}^G)$ depending on whether the boundary cut is rough or smooth respectively. This allows us to make connections to work of Ogata on the type of the cone von Neumann algebras in the algebraic quantum field theory approach to topological superselection sectors. We show that the boundary algebras can also be calculated from a trivial $G$-symmetry protected topological phase ($G$-SPT), and that the gauging map preserves the boundary algebras. Finally, we compute the boundary algebras for the (3+1)D Quantum Double model associated to an abelian group.