Topological Aspects of Spin and Statistics in Nonlinear Sigma Models

TL;DR

This paper investigates the topological constraints on the spin and statistics of solitons in nonlinear sigma models, revealing how the topology of the space and target manifold restrict possible quantum statistics.

Contribution

It introduces a framework linking topological solitons to quotient groups of framed braids, and explicitly computes these groups for specific models like the $O(3)$ sigma model.

Findings

For $ ext{Stat}_n( ext{R}^2, ext{α})$, the group is $ ext{Z}$.

For $ ext{Stat}_n( ext{S}^2, ext{α})$, the group is $ ext{Z}_{2n}$.

Allowed phase factors for soliton exchange are quantized as $ heta = krac{ ext{π}}{n}$.

Abstract

We study the purely topological restrictions on allowed spin and statistics of topological solitons in nonlinear sigma models. Taking as space the connected $d$-manifold $X$, and considering nonlinear sigma models with the connected manifold $M$ as target space, topological solitons are given by elements of $pi_d(M)$. Any topological soliton $\alpha \in \pi_d(M)$ determines a quotient $\Stat_n(X,\alpha)$ of the group of framed braids on $X$, such that choices of allowed statistics for solitons of type $\alpha$ are given by unitary representations of $\Stat_n(X,\alpha)$ when $n$ solitons are present. In particular, when $M = S^2$, as in the $O(3)$ nonlinear sigma model with Hopf term, and $\alpha \in \pi_2(S^2)$ is a generator, we compute that $\Stat_n(\R^2,\alpha) = \Z$, while $\Stat_n(S^2,\alpha) = \Z_{2n}$. It follows that phase $\exp(i\theta)$ for interchanging two solitons of type $\alpha$ on $S^2$ must satisfy the constraint $\theta = k\pi/n$, $k \in \Z$, when $n$ such solitons are present.