The Sum Over Topological Sectors and θ in the 2+1-Dimensional CP¹σ -Model

We discuss the three spacetime dimensional CP^N model and specialize to the CP¹ model. Because of the Hopf map π₃(CP¹) = Z one might try to couple the model to a periodic θ parameter. However, we argue that only the values θ = 0 and θ = π 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 θ = π theory on arbitrary 3-manifolds, and use recent results about invertible field theories to prove that no other values of θ satisfy the necessary locality.