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

17 pages. v2 minor corrections and additions. v3 more minor corrections

References in corpus (2)

• A Duality Web in 2+1 Dimensions and Condensed Matter Physics
• The Uncertainty of Fluxes

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