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

TL;DR

This paper links loop braid group representations to Aharonov-Bohm effects in 3+1D higher gauge theory, introducing W-bikoids from finite 2-groups to construct new unitary representations.

Contribution

It introduces W-bikoids as a categorification of biracks and constructs loop braid group representations from finite 2-groups in higher gauge theory.

Findings

Representations arise from Aharonov-Bohm effects in higher gauge theory

W-bikoids provide a new algebraic framework for loop braid groups

Constructs a candidate for higher quantum groups

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'.