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, Rimanyi, 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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.