Connection between topology and statistics

Abstract

We introduce topological magnetic field in two-dimensional flat space, which admits a solution of scalar monopole that describes the nontrivial topology. In the Chern-Simons gauge field theory of anyons, we interpret the anyons as the quasi-particles composed of fermions and scalar monopoles in such a form that each fermion is surrounded by infinite number of scalar monopoles. It is the monopole charge that determines the statistics of anyons. We re-analyze the conventional arguments of the connection between topology and statistics in three-dimensional space, and find that those arguments are based on the global topology, which is relatively trivial compared with the monopole structure. Through a simple model, we formulate the three-dimensional anyon field using infinite number of Dirac's magnetic monopoles that change the ordinary spacetime topology. The quasi-particle picture of the three-dimensional anyons is quite similar to that of the usual two-dimensional anyons. However, the exotic statistics there is not restricted to the usual fractional statistics, but a functional statistics.