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

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arxiv 1707.05448 v3 pith:PW5AJWEI submitted 2017-07-18 cond-mat.str-el hep-thmath.AT

Daniel S. Freed , Zohar Komargodski , Nathan Seiberg This is my paper
classification cond-mat.str-elhep-thmath.AT
keywords mathbbmodelthetavaluesanyonsdimensionalfieldskyrmions
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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.

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