Abstract
We discuss the three spacetime dimensional \({\mathbb{CP}^N}\) model and specialize to the \({\mathbb{CP}^1}\) model. Because of the Hopf map \({\pi_3(\mathbb{CP}^1)=\mathbb{Z}}\) 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 \({\theta}\) satisfy the necessary locality.
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Freed, D.S., Komargodski, Z. & Seiberg, N. The Sum Over Topological Sectors and θ in the 2+1-Dimensional \({\mathbb{CP}^1 \sigma}\)-Model. Commun. Math. Phys. 362, 167–183 (2018). https://doi.org/10.1007/s00220-018-3093-0
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DOI: https://doi.org/10.1007/s00220-018-3093-0