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Gapped boundaries and string-like excitations in (3+1)d gauge models of topological phases

Alex Bullivant, Clement Delcamp

TL;DR

<3-5 sentence high-level summary>: The paper develops a comprehensive tube-algebra framework to classify gapped-boundary excitations in (3+1)d Dijkgraaf-Witten gauge models and reveals a dimensional-reduction link to (2+1)d boundary phenomena. It provides explicit tube-algebra computations, relating (3+1)d structures to relative groupoid algebras and introduces a rich higher-categorical formulation: gapped boundaries correspond to separable/pseudo-algebra objects in Vec^{α}_{G} and Vec^{T}(π) respectively, with excitations realized as modules and bimodules, organized by comultiplication and 6j-symbols. The work connects boundary physics to the centre/monoidal-bicategory structures identified for higher gauge theories, offering a concrete computational route and a conceptual bridge between (2+1)d and (3+1)d topological orders. It also outlines pathways to broader contexts such as domain walls, fracton models, and extended TQFTs, indicating a robust framework for higher-dimensional defect TQFTs and quantum computation applications.

Abstract

We study lattice Hamiltonian realisations of (3+1)d Dijkgraaf-Witten theory with gapped boundaries. In addition to the bulk loop-like excitations, the Hamiltonian yields bulk dyonic string-like excitations that terminate at gapped boundaries. Using a tube algebra approach, we classify such excitations and derive the corresponding representation theory. Via a dimensional reduction argument, we relate this tube algebra to that describing (2+1)d boundary point-like excitations at interfaces between two gapped boundaries. Such point-like excitations are well known to be encoded into a bicategory of module categories over the input fusion category. Exploiting this correspondence, we define a bicategory that encodes the string-like excitations ending at gapped boundaries, showing that it is a sub-bicategory of the centre of the input bicategory of group-graded 2-vector spaces. In the process, we explain how gapped boundaries in (3+1)d can be labelled by so-called pseudo-algebra objects over this input bicategory.

Gapped boundaries and string-like excitations in (3+1)d gauge models of topological phases

TL;DR

<3-5 sentence high-level summary>: The paper develops a comprehensive tube-algebra framework to classify gapped-boundary excitations in (3+1)d Dijkgraaf-Witten gauge models and reveals a dimensional-reduction link to (2+1)d boundary phenomena. It provides explicit tube-algebra computations, relating (3+1)d structures to relative groupoid algebras and introduces a rich higher-categorical formulation: gapped boundaries correspond to separable/pseudo-algebra objects in Vec^{α}_{G} and Vec^{T}(π) respectively, with excitations realized as modules and bimodules, organized by comultiplication and 6j-symbols. The work connects boundary physics to the centre/monoidal-bicategory structures identified for higher gauge theories, offering a concrete computational route and a conceptual bridge between (2+1)d and (3+1)d topological orders. It also outlines pathways to broader contexts such as domain walls, fracton models, and extended TQFTs, indicating a robust framework for higher-dimensional defect TQFTs and quantum computation applications.

Abstract

We study lattice Hamiltonian realisations of (3+1)d Dijkgraaf-Witten theory with gapped boundaries. In addition to the bulk loop-like excitations, the Hamiltonian yields bulk dyonic string-like excitations that terminate at gapped boundaries. Using a tube algebra approach, we classify such excitations and derive the corresponding representation theory. Via a dimensional reduction argument, we relate this tube algebra to that describing (2+1)d boundary point-like excitations at interfaces between two gapped boundaries. Such point-like excitations are well known to be encoded into a bicategory of module categories over the input fusion category. Exploiting this correspondence, we define a bicategory that encodes the string-like excitations ending at gapped boundaries, showing that it is a sub-bicategory of the centre of the input bicategory of group-graded 2-vector spaces. In the process, we explain how gapped boundaries in (3+1)d can be labelled by so-called pseudo-algebra objects over this input bicategory.

Paper Structure

This paper contains 40 sections, 1 theorem, 325 equations.

Table of Contents

  1. Introduction
  2. Dijkgraaf-Witten Hamiltonian Model
  3. Partition function
  4. Hamiltonian realisation of Dijkgraaf-Witten theory
  5. Gapped boundary partition function
  6. Hamiltonian model in the presence of gapped boundaries
  7. Tube algebra for gapped boundary excitations in (2+1)d
  8. Definition
  9. Computation of the tube algebra
  10. Groupoid algebra
  11. Tube algebra for gapped boundary excitations in (3+1)d
  12. Definition
  13. Computation of the tube algebra
  14. Relative groupoid algebra
  15. Representation theory and elementary gapped boundary excitations
  16. ...and 25 more sections

Key Result

Proposition 6.1

There exists an equivalence of bicategories between $\mathsf{sAlg}(\mathsf{Vec}^\alpha_\mathcal{G})$ and $\mathsf{MOD}(\mathsf{Vec}^\alpha_\mathcal{G})$ by sending separable algebra objects in $\mathsf{Vec}^\alpha_\mathcal{G}$ to their category of (right) modules in $\mathsf{Vec}^\alpha_\mathcal{G}$

Theorems & Definitions (29)

  • Definition 3.1: Groupoids
  • Example 3.1: Delooping of a group
  • Definition 3.2: Groupoid cohomology
  • Definition 3.3: Normalised cocycles
  • Definition 3.4: Twisted groupoid algebra
  • Definition 4.1: Loop groupoid
  • Definition 4.2: Relative groupoid
  • Definition 4.3
  • Definition 6.1: Bicategory
  • Definition 6.2: Category of $\mathcal{G}$-graded vector spaces
  • ...and 19 more