Department of Applied Mathematics & Physics, Kyoto Univiversity

### Technical Report #97004 (May, 1997) Quantum Mechanically Induced Hopf Term in the O(3) Nonlinear Sigma Model by Hiroyuki Kobayashi, Izumi Tsutsui and Shogo Tanimura

The Hopf term in the $2+1$ dimensional $O(3)$ nonlinear sigma model, which is known to be responsible for fractional spin and statistics, is re-examined from the viewpoint of quantization ambiguity. It is confirmed that the Hopf term can be understood as a quantum mechanically induced term, in precisely the same manner as the $\theta$-term in QCD. We present a detailed analysis of the topological aspect of the model based on the adjoint orbit parametrization of the spin vectors, which not only is very useful in handling topological (soliton and/or Hopf) numbers, but also plays a crucial role in defining the Hopf term for configurations of nonvanishing soliton numbers. The Hopf term is seen to arise explicitly as a quantum effect acquired by quantizing an $S^1$ degree of freedom hidden in the configuration space.