Tube algebras, excitations statistics and compactification in gauge models of topological phases

TL;DR

This paper generalizes the tube algebra approach to classify and analyze loop-like excitations in higher-dimensional topological gauge models, extending known 2D results to 3D and beyond, and explores dimensional reduction via compactification.

Contribution

It introduces a generalized tube algebra framework applicable in any dimension and applies it to derive the algebraic structure of loop excitations in 3D topological models.

Findings

Derived the twisted quantum triple algebra for 3D excitations.

Established a correspondence between irreducible representations and loop-like excitations.

Explained dimensional reduction using loop-groupoids and compactification.

Abstract

We consider lattice Hamiltonian realizations of ($d$+1)-dimensional Dijkgraaf-Witten theory. In (2+1)d, it is well-known that the Hamiltonian yields point-like excitations classified by irreducible representations of the twisted quantum double. This can be confirmed using a tube algebra approach. In this paper, we propose a generalization of this strategy that is valid in any dimensions. We then apply the tube algebra approach to derive the algebraic structure of loop-like excitations in (3+1)d, namely the twisted quantum triple. The irreducible representations of the twisted quantum triple algebra correspond to the simple loop-like excitations of the model. Similarly to its (2+1)d counterpart, the twisted quantum triple comes equipped with a compatible comultiplication map and an $R$-matrix that encode the fusion and the braiding statistics of the loop-like excitations, respectively. Moreover, we explain using the language of loop-groupoids how a model defined on a manifold that is $n$-times compactified can be expressed in terms of another model in $n$-lower dimensions. This can in turn be used to recast higher-dimensional tube algebras in terms of lower dimensional analogues.

Figures