On topological terms in the O(3) nonlinear sigma model. (English) Zbl 0987.81054
Summary: Topological terms in the \(O(3)\) nonlinear sigma model in \((1+1)\) and \((2+1)\) dimensions are re-examined based on the description of the SU(2)-valued field \(g\). We first show that the topological soliton term in \((1+1)\) dimensions arises from the unitary representations of the group characterizing the global structure of the symmetry inherent in the description, in a manner analogous to the appearance of the \(\theta\)-term in Yang-Mills theory in \((3+1)\) dimensions. We then present a detailed argument as to why the conventional Hopf term, which is the topological counterpart in \((2+1)\) dimensions and has been widely used to realize fractional spin and statistics, is ill-defined unless the soliton charge vanishes. We show how this restriction can be lifted by means of a procedure proposed recently, and provide its physical interpretation as well.
MSC:
| 81T10 | Model quantum field theories |
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