On Topological Terms in the O(3) Nonlinear Sigma Model

Details On Topological Terms in the O(3) Nonlinear Sigma Model On Topological Terms in the O(3) Nonlinear Sigma Model

1999-05-24

arxiv.org/abs/hep-th/9905166v1

Phys.Lett. B460 (1999) 94-102

Physics

High Energy Physics

High Energy Physics - Theory

15 pages, 1 figure

Scientific paper

10.1016/S0370-2693(99)00740-6

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.

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