Timereversal symmetry, anomalies, and dualities in (2+1)$$d$$
Abstract
We study continuum quantum field theories in 2+1 dimensions with timereversal symmetry \cal T . The standard relation {\cal T}^2=(1)^F is satisfied on all the “perturbative operators” i.e. polynomials in the fundamental fields and their derivatives. However, we find that it is often the case that acting on more complicated operators {\cal T}^2=(1)^F {\cal M} with \cal M a nontrivial global symmetry. For example, acting on monopole operators, \cal M could be \pm1 $\pm 1$ depending on the magnetic charge. We study in detail U(1) $U\left(1\right)$ gauge theories with fermions of various charges. Such a modification of the timereversal algebra happens when the number of odd charge fermions is 2 ~{\rm mod }~4 , e.g. in QED with two fermions. Our work also clarifies the dynamics of QED with fermions of higher charges. In particular, we argue that the longdistance behavior of QED with a single fermion of charge 2 $2$ is a free theory consisting of a Dirac fermion and a decoupled topological quantum field theory. The extension to an arbitrary even charge is straightforward. The generalization of these abelian theories to SO(N) $SO\left(N\right)$ gauge theories with fermions in the vector or in twoindex tensor representations leads to new results and new consistency conditions on previously suggested scenarios for the dynamics of these theories. Among these new results is a surprising nonabelian symmetry involving timereversal.
 Authors:

 Institute for Advanced Study, Princeton University
 Princeton University
 Publication Date:
 Sponsoring Org.:
 USDOE
 OSTI Identifier:
 1461012
 Grant/Contract Number:
 SC0009988
 Resource Type:
 Published Article
 Journal Name:
 SciPost Physics Proceedings
 Additional Journal Information:
 Journal Name: SciPost Physics Proceedings Journal Volume: 5 Journal Issue: 1; Journal ID: ISSN 25424653
 Publisher:
 Stichting SciPost
 Country of Publication:
 Netherlands
 Language:
 English
Citation Formats
Cordova, Clay, Hsin, PoShen, and Seiberg, Nathan. Timereversal symmetry, anomalies, and dualities in (2+1)$d$. Netherlands: N. p., 2018.
Web. doi:10.21468/SciPostPhys.5.1.006.
Cordova, Clay, Hsin, PoShen, & Seiberg, Nathan. Timereversal symmetry, anomalies, and dualities in (2+1)$d$. Netherlands. https://doi.org/10.21468/SciPostPhys.5.1.006
Cordova, Clay, Hsin, PoShen, and Seiberg, Nathan. Fri .
"Timereversal symmetry, anomalies, and dualities in (2+1)$d$". Netherlands. https://doi.org/10.21468/SciPostPhys.5.1.006.
@article{osti_1461012,
title = {Timereversal symmetry, anomalies, and dualities in (2+1)$d$},
author = {Cordova, Clay and Hsin, PoShen and Seiberg, Nathan},
abstractNote = {We study continuum quantum field theories in 2+1 dimensions with timereversal symmetry \cal T . The standard relation {\cal T}^2=(1)^F is satisfied on all the “perturbative operators” i.e. polynomials in the fundamental fields and their derivatives. However, we find that it is often the case that acting on more complicated operators {\cal T}^2=(1)^F {\cal M} with \cal M a nontrivial global symmetry. For example, acting on monopole operators, \cal M could be \pm1 ± 1 depending on the magnetic charge. We study in detail U(1) U ( 1 ) gauge theories with fermions of various charges. Such a modification of the timereversal algebra happens when the number of odd charge fermions is 2 ~{\rm mod }~4 , e.g. in QED with two fermions. Our work also clarifies the dynamics of QED with fermions of higher charges. In particular, we argue that the longdistance behavior of QED with a single fermion of charge 2 2 is a free theory consisting of a Dirac fermion and a decoupled topological quantum field theory. The extension to an arbitrary even charge is straightforward. The generalization of these abelian theories to SO(N) S O ( N ) gauge theories with fermions in the vector or in twoindex tensor representations leads to new results and new consistency conditions on previously suggested scenarios for the dynamics of these theories. Among these new results is a surprising nonabelian symmetry involving timereversal.},
doi = {10.21468/SciPostPhys.5.1.006},
journal = {SciPost Physics Proceedings},
number = 1,
volume = 5,
place = {Netherlands},
year = {2018},
month = {7}
}
https://doi.org/10.21468/SciPostPhys.5.1.006
Web of Science
Works referenced in this record:
Is the Composite Fermion a Dirac Particle?
journal, September 2015
 Son, Dam Thanh
 Physical Review X, Vol. 5, Issue 3
Selfdual quantum electrodynamics as boundary state of the threedimensional bosonic topological insulator
journal, December 2015
 Xu, Cenke; You, YiZhuang
 Physical Review B, Vol. 92, Issue 22
Theta, time reversal and temperature
journal, May 2017
 Gaiotto, Davide; Kapustin, Anton; Komargodski, Zohar
 Journal of High Energy Physics, Vol. 2017, Issue 5
Parity violation and gauge noninvariance of the effective gauge field action in three dimensions
journal, May 1984
 Redlich, A. N.
 Physical Review D, Vol. 29, Issue 10
A symmetry breaking scenario for QCD3
journal, January 2018
 Komargodski, Zohar; Seiberg, Nathan
 Journal of High Energy Physics, Vol. 2018, Issue 1
Anomaly Indicators for TimeReversal Symmetric Topological Orders
journal, September 2017
 Wang, Chenjie; Levin, Michael
 Physical Review Letters, Vol. 119, Issue 13
The “parity” anomaly on an unorientable manifold
journal, November 2016
 Witten, Edward
 Physical Review B, Vol. 94, Issue 19
Deconfined Quantum Critical Points: Symmetries and Dualities
journal, September 2017
 Wang, Chong; Nahum, Adam; Metlitski, Max A.
 Physical Review X, Vol. 7, Issue 3
Boson topological insulators: A window into highly entangled quantum phases
journal, June 2013
 Wang, Chong; Senthil, T.
 Physical Review B, Vol. 87, Issue 23
Derivation of the TimeReversal Anomaly for ( $2+1$ )Dimensional Topological Phases
journal, September 2017
 Tachikawa, Yuji; Yonekura, Kazuya
 Physical Review Letters, Vol. 119, Issue 11
Comments on global symmetries, anomalies, and duality in (2 + 1)d
journal, April 2017
 Benini, Francesco; Hsin, PoShen; Seiberg, Nathan
 Journal of High Energy Physics, Vol. 2017, Issue 4
Particlevortex duality of twodimensional Dirac fermion from electricmagnetic duality of threedimensional topological insulators
journal, June 2016
 Metlitski, Max A.; Vishwanath, Ashvin
 Physical Review B, Vol. 93, Issue 24
Anomalies and odd dimensions
journal, September 1985
 AlvarezGaumé, L.; Della Pietra, S.; Moore, G.
 Annals of Physics, Vol. 163, Issue 2
Dual Dirac Liquid on the Surface of the Electron Topological Insulator
journal, November 2015
 Wang, Chong; Senthil, T.
 Physical Review X, Vol. 5, Issue 4
A duality web in $2+1$ dimensions and condensed matter physics
journal, November 2016
 Seiberg, Nathan; Senthil, T.; Wang, Chong
 Annals of Physics, Vol. 374
A timereversal invariant topological phase at the surface of a 3D topological insulator
journal, September 2013
 Bonderson, Parsa; Nayak, Chetan; Qi, XiaoLiang
 Journal of Statistical Mechanics: Theory and Experiment, Vol. 2013, Issue 09
Symmetry enforced nonAbelian topological order at the surface of a topological insulator
journal, April 2014
 Chen, Xie; Fidkowski, Lukasz; Vishwanath, Ashvin
 Physical Review B, Vol. 89, Issue 16
Fermion path integrals and topological phases
journal, July 2016
 Witten, Edward
 Reviews of Modern Physics, Vol. 88, Issue 3
Level/rank duality and ChernSimonsmatter theories
journal, September 2016
 Hsin, PoShen; Seiberg, Nathan
 Journal of High Energy Physics, Vol. 2016, Issue 9
Erratum: Interacting fermionic topological insulators/superconductors in three dimensions [Phys. Rev. B 89 , 195124 (2014)]
journal, June 2015
 Wang, Chong; Senthil, T.
 Physical Review B, Vol. 91, Issue 23
NonAbelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model
journal, November 2013
 Fidkowski, Lukasz; Chen, Xie; Vishwanath, Ashvin
 Physical Review X, Vol. 3, Issue 4
Global Symmetries, Counterterms, and Duality in ChernSimons Matter Theories with Orthogonal Gauge Groups
journal, January 2018
 Cordova, Clay; Hsin, PoShen; Seiberg, Nathan
 SciPost Physics, Vol. 4, Issue 4
Phases Of Adjoint QCD$_3$ And Dualities
journal, January 2018
 Gomis, Jaume; Komargodski, Zohar; Seiberg, Nathan
 SciPost Physics, Vol. 5, Issue 1
Microscopic Theory of Surface Topological Order for Topological Crystalline Superconductors
journal, January 2018
 Cheng, Meng
 Physical Review Letters, Vol. 120, Issue 3
Fermionic symmetry protected topological phases and cobordisms
journal, December 2015
 Kapustin, Anton; Thorngren, Ryan; Turzillo, Alex
 Journal of High Energy Physics, Vol. 2015, Issue 12
AxialAnomalyInduced Fermion Fractionization and Effective GaugeTheory Actions in OddDimensional SpaceTimes
journal, December 1983
 Niemi, A. J.; Semenoff, G. W.
 Physical Review Letters, Vol. 51, Issue 23
On timereversal anomaly of 2+1d topological phases
journal, March 2017
 Tachikawa, Yuji; Yonekura, Kazuya
 Progress of Theoretical and Experimental Physics, Vol. 2017, Issue 3
Gapped boundary phases of topological insulators via weak coupling
journal, November 2016
 Seiberg, Nathan; Witten, Edward
 Progress of Theoretical and Experimental Physics, Vol. 2016, Issue 12
ChernSimonsmatter dualities with SO and USp gauge groups
journal, February 2017
 Aharony, Ofer; Benini, Francesco; Hsin, PoShen
 Journal of High Energy Physics, Vol. 2017, Issue 2