The Sum Over Topological Sectors and θ in the 2+1-Dimensional $$\mathbb{CP}^1σ$$-Model
Abstract
Here 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 θ satisfy the necessary locality.
- Authors:
-
- Univ. of Texas, Austin, TX (United States)
- Weizmann Inst. of Science, Rehovot (Israel); Stony Brook Univ., NY (United States)
- Institute for Advanced Study, Princeton, NJ (United States)
- Publication Date:
- Research Org.:
- Institute for Advanced Study, Princeton, NJ (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC); National Science Foundation (NSF); Israel Science Foundation; European Research Council (ERC); Simons Foundation
- OSTI Identifier:
- 1537661
- Grant/Contract Number:
- SC0009988; DMS-1611957; PHY-1607611; 1937/12; 335182; 488657
- Resource Type:
- Journal Article: Accepted Manuscript
- Journal Name:
- Communications in Mathematical Physics
- Additional Journal Information:
- Journal Volume: 362; Journal Issue: 1; Journal ID: ISSN 0010-3616
- Publisher:
- Springer
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY
Citation Formats
Freed, Daniel S., Komargodski, Zohar, and Seiberg, Nathan. The Sum Over Topological Sectors and θ in the 2+1-Dimensional $\mathbb{CP}^1σ$-Model. United States: N. p., 2018.
Web. doi:10.1007/s00220-018-3093-0.
Freed, Daniel S., Komargodski, Zohar, & Seiberg, Nathan. The Sum Over Topological Sectors and θ in the 2+1-Dimensional $\mathbb{CP}^1σ$-Model. United States. https://doi.org/10.1007/s00220-018-3093-0
Freed, Daniel S., Komargodski, Zohar, and Seiberg, Nathan. 2018.
"The Sum Over Topological Sectors and θ in the 2+1-Dimensional $\mathbb{CP}^1σ$-Model". United States. https://doi.org/10.1007/s00220-018-3093-0. https://www.osti.gov/servlets/purl/1537661.
@article{osti_1537661,
title = {The Sum Over Topological Sectors and θ in the 2+1-Dimensional $\mathbb{CP}^1σ$-Model},
author = {Freed, Daniel S. and Komargodski, Zohar and Seiberg, Nathan},
abstractNote = {Here 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 θ satisfy the necessary locality.},
doi = {10.1007/s00220-018-3093-0},
url = {https://www.osti.gov/biblio/1537661},
journal = {Communications in Mathematical Physics},
issn = {0010-3616},
number = 1,
volume = 362,
place = {United States},
year = {Tue Jan 30 00:00:00 EST 2018},
month = {Tue Jan 30 00:00:00 EST 2018}
}
Web of Science
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