# 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.

## Key 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.