Title: The Sum Over Topological Sectors and $θ$ in the 2+1-Dimensional $\mathbb{C}\mathbb{P}^1$ $σ$-Model,
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.