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 stringlike 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 XiaoSong 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 4d BF theory this group naturally acts on the moduli space of flat Gbundles on the complement of a collection of unlinked unknotted circles in 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2006
 arXiv:
 arXiv:grqc/0603085
 Bibcode:
 2006gr.qc.....3085B
 Keywords:

 General Relativity and Quantum Cosmology;
 High Energy Physics  Theory;
 Mathematics  Geometric Topology
 EPrint:
 41 pages, many figures. New version has minor corrections and clarifications, and some added references