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

TL;DR

The paper develops a unified tube-algebra framework to classify excitations in Dijkgraaf-Witten gauge theories across dimensions, culminating in the twisted quantum triple for (3+1)d. It extends Ocneanu-type tube algebra methods to higher dimensions and encodes fusion and braiding via explicit comultiplication and R-matrices, with a detailed representation theory for point-like and loop-like excitations. A central innovation is the loop-groupoid (and iterated loop-groupoid) lifting, which expresses higher-dimensional tube algebras as lifted lower-dimensional models and clarifies the role of transgressed cocycles in twisting. The work also situates these constructions in a category-theoretic setting (loop groupoids, delooping, 2-categories), enabling a principled route to generalize to other boundaries and higher gauge theories, with potential applications to gapped boundaries and topological quantum computation.

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 (1)

• Figure 1: Coherence relation of the $2 \leftrightharpoons 3$ Pachner moves. Given a complex obtained as the gluing of three 3-simplices, two different sequences of $2 \leftrightharpoons 3$ Pachner moves result in the same complex obtained as the gluing of six 3-simplices. Each arrow is decorated by the amplitude of the corresponding move obtained as the evaluation of $\omega$ on the 4-simplex whose boundary provides the 3-simplices involved in the corresponding $2 \leftrightharpoons 3$ move. This coherence relation is satisfied if $\omega$ is a 4-cocycle.

Theorems & Definitions (15)

• Definition 2.1: Group cohomology
• Definition 2.2: Pinched interval
• Definition 5.1: Category
• Definition 5.2: Monoidal category
• Example 5.1: Category of $G$-graded vector spaces
• Definition 5.3: Braided monoidal category
• Definition 5.4: 2-vector spaces
• Definition 5.5: Groupoid
• Definition 5.6: Classifying space of a groupoid
• Definition 5.7: Delooping of a group