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Representations of the Loop Braid Group and Aharonov-Bohm like effects in discrete (3+1)-dimensional higher gauge theory

Alex Bullivant, João Faria Martins, Paul Martin

TL;DR

This work shows that LB_n representations arise from Aharonov-Bohm–like effects in finite 2-group higher gauge theory and introduces W-bikoids as a minimal categorification of biracks to realize these representations. By constructing W-bikoids from crossed modules (finite 2-groups) and their associated groupoid algebras, the authors derive unitary representations of the loop braid group from holonomy data, offering a higher-dimensional analogue of Bais’ flux metamorphosis and hinting at a higher quantum group framework. The approach unifies AB-phase intuition in D^2 and D^3 settings with categorical and graphical tools (bikoids, WB_n/VB_n representations, and cross-module groupoids) to produce a robust representation theory for loop excitations in 3+1D topological phases. The results suggest new invariants and computational frameworks for knotted surfaces and higher gauge phenomena in topological quantum matter. The methods pave the way for deeper connections between higher gauge theory, category theory, and quantum topology in 3+1 dimensions.

Abstract

We show that representations of the loop braid group arise from Aharonov-Bohm like effects in finite 2-group (3+1)-dimensional topological higher gauge theory. For this we introduce a minimal categorification of biracks, which we call W-bikoids (welded bikoids). Our main example of W-bikoids arises from finite 2-groups, realised as crossed modules of groups. Given a W-bikoid, and hence a groupoid of symmetries, we construct a family of unitary representations of the loop braid group derived from representations of the groupoid algebra. We thus give a candidate for higher Bais' flux metamorphosis, and hence also a version of a `higher quantum group'.

Representations of the Loop Braid Group and Aharonov-Bohm like effects in discrete (3+1)-dimensional higher gauge theory

TL;DR

This work shows that LB_n representations arise from Aharonov-Bohm–like effects in finite 2-group higher gauge theory and introduces W-bikoids as a minimal categorification of biracks to realize these representations. By constructing W-bikoids from crossed modules (finite 2-groups) and their associated groupoid algebras, the authors derive unitary representations of the loop braid group from holonomy data, offering a higher-dimensional analogue of Bais’ flux metamorphosis and hinting at a higher quantum group framework. The approach unifies AB-phase intuition in D^2 and D^3 settings with categorical and graphical tools (bikoids, WB_n/VB_n representations, and cross-module groupoids) to produce a robust representation theory for loop excitations in 3+1D topological phases. The results suggest new invariants and computational frameworks for knotted surfaces and higher gauge phenomena in topological quantum matter. The methods pave the way for deeper connections between higher gauge theory, category theory, and quantum topology in 3+1 dimensions.

Abstract

We show that representations of the loop braid group arise from Aharonov-Bohm like effects in finite 2-group (3+1)-dimensional topological higher gauge theory. For this we introduce a minimal categorification of biracks, which we call W-bikoids (welded bikoids). Our main example of W-bikoids arises from finite 2-groups, realised as crossed modules of groups. Given a W-bikoid, and hence a groupoid of symmetries, we construct a family of unitary representations of the loop braid group derived from representations of the groupoid algebra. We thus give a candidate for higher Bais' flux metamorphosis, and hence also a version of a `higher quantum group'.

Paper Structure

This paper contains 27 sections, 21 theorems, 192 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Insights from topological gauge theory and higher gauge theory
  3. Finite group gauge theory in $D^2$, quantum doubles and braid groups
  4. Higher gauge theory with a finite 2-group: preliminaries
  5. Higher gauge theory with a finite 2-group and invariants of loop braids
  6. Preliminaries
  7. General conventions for groupoids
  8. Groupoid algebras and their representations
  9. The virtual braid group and the welded braid group
  10. The loop braid group: isomorphism with the welded braid group
  11. Bikoids and welded bikoids (W-bikoid s)
  12. The wreath product groupoid $\Gamma^{\wr n}=\Gamma^{n} \rtimes \Sigma_{n}$ of a groupoid $\Gamma$
  13. Graphical calculus for $\Gamma^n \rtimes \Sigma_n=\Gamma^{\wr n}$
  14. Conventions and notation for biracks and welded biracks (W-biracks)
  15. Bikoid s: definition
  16. ...and 12 more sections

Key Result

Theorem 17

There exists an isomorphism: This isomorphism sends a group generator twoperpectives$g\in {\rm WB}_n$ of the form $S_a^+{{[n]}}$ or $V_a{{[n]}}$, to the pair-isotopy class of the homeomorphism $f^g=\phi_{t=1}^g\colon (D^3,C_n) \to (D^3,C_n)$ at the end of the isotopies $\Phi^g=(t\in [0,1]\mapsto \phi_t^g\in {\rm Homeo}(D^3,\emp

Figures (8)

  • Figure 1: A generic configuration of four particles $p_1,p_2,p_3,p_4$ in the disk $D^2$.
  • Figure 2: A movement of particles passing non-generic configurations. Note $\gamma_{\,\,2}^{\, \,t}$ is homotopic to $\phi_2^t$, for all $t$.
  • Figure 3: Construction of a torus-map $P_c\colon [0,1]^2 \to D^3$ in $D^3$, associated to a loop-particle $c$.
  • Figure 4: A generic configuration of loop-particles $c_1$ and $c_2$.
  • Figure 5: A generator $g \in {\rm WB}_{n}$ gives rise to an isotopy $\Phi^{g}=(t\in [0,1]\mapsto \phi_{\,t}^{\,g}\in {\rm Homeo}(D^3,\emptyset))$ of $D^3$.
  • ...and 3 more figures

Theorems & Definitions (79)

  • Remark 1: Higher gauge flux metamorphosis
  • Definition 2: Action groupoid
  • Example 3: ${\rm AUT}(G)$
  • Definition 4: Groupoid algebra
  • Remark 5
  • Definition 6
  • Example 7
  • Example 8: Object regular representation
  • Definition 9: Virtual braid diagram
  • Definition 10: The monoid ${\rm M V}{{[n]}}$ of virtual braid diagrams
  • ...and 69 more