The Sum Over Topological Sectors and θ in the 2+1-Dimensional $\mathbb{CP}^1σ$-Model

Freed, Daniel S.; Komargodski, Zohar; Seiberg, Nathan

doi:10.1007/s00220-018-3093-0

Title: 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:

Freed, Daniel S. ^[1];

^[2]; Seiberg, Nathan ^[3]

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:: Tue Jan 30 00:00:00 EST 2018

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:: 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.

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                    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

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                    Freed, Daniel S., Komargodski, Zohar, and Seiberg, Nathan. Tue .  
"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.

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@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},

  journal      = {Communications in Mathematical Physics},

  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}

}

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Works referencing / citing this record:

Metastable vacua in large-N QCD3
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Journal of High Energy Physics, Vol. 2020, Issue 1
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Title: The Sum Over Topological Sectors and θ in the 2+1-Dimensional $$\mathbb{CP}^1σ$$-Model

Abstract

Citation Formats

Modifying the sum over topological sectors and constraints on supergravity journal, July 2010

Quadratic functions in geometry, topology, and M-theory journal, July 2005

Linking Numbers, Spin, and Statistics of Solitons journal, December 1983

Neutral fermions in paramagnetic insulators journal, February 1988

Spectral asymmetry and Riemannian Geometry. I journal, January 1975

Topological quantum field theories journal, January 1988

D-brane charges in five-brane backgrounds journal, October 2001

Pions and generalized cohomology journal, September 2008

On the Classification of Topological Field Theories journal, January 2008

A duality web in 2 + 1 dimensions and condensed matter physics journal, November 2016

Topological Disorder Operators in Three-Dimensional Conformal Field Theory journal, November 2002

Higher algebraic structures and quantization journal, January 1994

Setting the Quantum Integrand of M-Theory journal, January 2006

The Uncertainty of Fluxes journal, January 2007

Coupling a QFT to a TQFT and duality journal, April 2014

Global aspects of current algebra journal, August 1983

Theta-terms in nonlinear sigma-models journal, March 2000

Fermi-Bose Transmutations Induced by Gauge Fields journal, February 1988

Topological gauge theories and group cohomology journal, April 1990

Current algebra, baryons, and quark confinement journal, August 1983

Reading between the lines of four-dimensional gauge theories journal, August 2013

Fermi-Bose Transmutations Induced by Gauge Fields journal, February 1988

On the Classification of Topological Field Theories preprint, January 2009

Reading between the lines of four-dimensional gauge theories text, January 2013

Topological Disorder Operators in Three-Dimensional Conformal Field Theory text, January 2002

The Uncertainty of Fluxes text, January 2006

Higher Algebraic Structures and Quantization text, January 1992

Topological Disorder Operators in Three-Dimensional Conformal Field Theory journal, November 2002

Modifying the Sum Over Topological Sectors and Constraints on Supergravity text, January 2010

Reading between the lines of four-dimensional gauge theories text, January 2013

Coupling a QFT to a TQFT and Duality text, January 2014

Theta-terms in nonlinear sigma-models text, January 1999

Metastable vacua in large-N QCD3 journal, January 2020

Charting the phase diagram of QCD3 journal, August 2019

Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups journal, January 2018

Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups text, January 2017

Modifying the sum over topological sectors and constraints on supergravity
journal, July 2010

Quadratic functions in geometry, topology, and M-theory
journal, July 2005

Linking Numbers, Spin, and Statistics of Solitons
journal, December 1983

Neutral fermions in paramagnetic insulators
journal, February 1988

Spectral asymmetry and Riemannian Geometry. I
journal, January 1975

Topological quantum field theories
journal, January 1988

D-brane charges in five-brane backgrounds
journal, October 2001

Pions and generalized cohomology
journal, September 2008

On the Classification of Topological Field Theories
journal, January 2008

A duality web in $2 + 1$ dimensions and condensed matter physics
journal, November 2016

Topological Disorder Operators in Three-Dimensional Conformal Field Theory
journal, November 2002

Higher algebraic structures and quantization
journal, January 1994

Setting the Quantum Integrand of M-Theory
journal, January 2006

The Uncertainty of Fluxes
journal, January 2007

Coupling a QFT to a TQFT and duality
journal, April 2014

Global aspects of current algebra
journal, August 1983

Theta-terms in nonlinear sigma-models
journal, March 2000

Fermi-Bose Transmutations Induced by Gauge Fields
journal, February 1988

Topological gauge theories and group cohomology
journal, April 1990

Current algebra, baryons, and quark confinement
journal, August 1983

Reading between the lines of four-dimensional gauge theories
journal, August 2013

Fermi-Bose Transmutations Induced by Gauge Fields
journal, February 1988

On the Classification of Topological Field Theories
preprint, January 2009

Reading between the lines of four-dimensional gauge theories
text, January 2013

Topological Disorder Operators in Three-Dimensional Conformal Field Theory
text, January 2002

The Uncertainty of Fluxes
text, January 2006

Higher Algebraic Structures and Quantization
text, January 1992

Topological Disorder Operators in Three-Dimensional Conformal Field Theory
journal, November 2002

Modifying the Sum Over Topological Sectors and Constraints on Supergravity
text, January 2010

Reading between the lines of four-dimensional gauge theories
text, January 2013

Coupling a QFT to a TQFT and Duality
text, January 2014

Theta-terms in nonlinear sigma-models
text, January 1999

Metastable vacua in large-N QCD3
journal, January 2020

Charting the phase diagram of QCD3
journal, August 2019

Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups
journal, January 2018

Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups
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