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

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.