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 greater-than-or-equal-to 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 greater-than-or-equal-to 0 only anyons with rational statistical parameter theta/pi = p/q are allowed, and their number is restricted to be sq-g+1 (s-epsilon-Z). For nonorientable surfaces only-theta = 0, pi 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 non-linear sigma model with space manifold M.