Exotic statistics for strings in 4d BF theory

Abstract

After a review of exotic statistics for point particles in 3d BF theory, and especially 3d quantum gravity, we show that string-like defects in 4d BF theory obey exotic statistics governed by the “loop braid group”. This group has a set of generators that switch two strings just as one would normally switch point particles, but also a set of generators that switch two strings by passing one through the other. The first set generates a copy of the symmetric group, while the second generates a copy of the braid group. Thanks to recent work of Xiao-Song Lin, we can give a presentation of the whole loop braid group, which turns out to be isomorphic to the “braid permutation group” of Fenn, Rimányi, and Rourke. In the context of 4d BF theory, this group naturally acts on the moduli space of flat $G$-bundles on the complement of a collection of unlinked unknotted circles in $\mathbb{R}^3$. When $G$ is unimodular, this gives a unitary representation of the loop braid group. We also discuss “quandle field theory”, in which the gauge group $G$ is replaced by a quandle.