The Sum Over Topological Sectors and $\theta$ in the 2+1-Dimensional $\mathbb{C}\mathbb{P}^1$ $\sigma$-Model

TL;DR

This paper analyzes the 3D $ ext{CP}^1$ sigma model, showing only $ heta=0$ and $ heta= ext{pi}$ are consistent, and explores the topological and quantum properties of Skyrmions, including their statistics.

Contribution

It demonstrates the constraints on the $ heta$ parameter in the 3D $ ext{CP}^1$ model and extends the theory to include Skyrmions as anyons using topological quantum field theory techniques.

Findings

Only $ heta=0$ and $ heta= ext{pi}$ are consistent in the model.

Skyrmions are bosons at $ heta=0$ and fermions at $ heta= ext{pi}$.

Coupling to TQFT allows Skyrmions to be anyons.

Abstract

We discuss the three spacetime dimensional $\mathbb{C}\mathbb{P}^N$ model and specialize to the $\mathbb{C}\mathbb{P}^1$ model. Because of the Hopf map $\pi_3(\mathbb{C}\mathbb{P}^1)=\mathbb{Z}$ one might try to couple the model to a periodic $\theta$ parameter. However, we argue that only the values $\theta=0$ and $\theta=\pi$ are consistent. For these values the Skyrmions in the model are bosons and fermions respectively, rather than being anyons. We also extend the model by coupling it to a topological quantum field theory, such that the Skyrmions are anyons. We use techniques from geometry and topology to construct the $\theta =\pi $ theory on arbitrary 3-manifolds, and use recent results about invertible field theories to prove that no other values of $\theta $ satisfy the necessary locality.