Cyclic statistics in three dimensions

Abstract

The existence of anyons in two-dimensional systems is a well-known example of nonpermutation group statistics. In higher dimensions, however, it is expected that statistics is dictated solely by representations of the permutation group. Using basic elements from representation theory we show that this expectation is false in three-dimensions for a certain nongravitational system. Namely, we demonstrate the existence of "cyclic," or [openface Z]n, nonpermutation group statistics for a system of n>2 identical, unknotted rings embedded in [openface R]3. We make crucial use of a theorem due to Goldsmith in conjunction with the Fuchs–Rabinovitch relations for the automorphisms of the free product group on n elements.