Exotic statistics on surfaces

We investigate the allowed spectrum of statistics for n identical spinless particles on an arbitrary closed two-manifold M, by using a powerful topological approach to the study of quantum kinematics. On a surface of genus g ≥ 1 statistics other than Bose or Fermi can only be obtained by utilizing multi-component state vectors transforming as an irreducible unitary representation of the fundamental group of the n-particle configuration space. These multi-component (or nonscalar) quantizations allow the possibility of fractional statistics, as well as other exotic, nonfractional statistics some of whose properties we discuss. On an orientable surface of genus g ≥ 0 only anyons with rational statistical parameter θ/π=p/q are allowed, and their number is restricted to be sq-g+1 (s∈ℤ). For nonorientable surfaces only θ=0, π are allowed. Finally, we briefly comment on systems of spinning particles and make a comparison with the results for solitons in the O(3)-invariant nonlinear sigma model with space manifold M