Representations of the loop braid group and Aharonov–Bohm like effects in discrete $(3+1)$-dimensional higher gauge theory

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’.

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Published 15 May 2020