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

TL;DR

This paper re-examines topological terms in the O(3) nonlinear sigma model across different dimensions, clarifying their origins and limitations, especially concerning the Hopf term's well-definedness and fractional spin realization.

Contribution

It clarifies the origin of topological soliton terms in (1+1) dimensions and addresses the ill-defined nature of the Hopf term in (2+1) dimensions, proposing a method to overcome this issue.

Findings

Topological soliton term in (1+1) dimensions arises from group representations.

The Hopf term in (2+1) dimensions is ill-defined unless soliton charge vanishes.

A recent procedure can lift the restriction on the Hopf term, enabling fractional spin realization.

Abstract

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.