# Lorentz symmetry fractionalization and dualities in (2+1)d

## Abstract

We discuss symmetry fractionalization of the Lorentz group in (2+1) d $d$ non-spin quantum field theory (QFT), and its implications for dualities. We prove that two inequivalent non-spin QFTs are dual as spin QFTs if and only if they are related by a Lorentz symmetry fractionalization with respect to an anomalous \mathbb{Z}_2 ${\mathbb{Z}}_{2}$ one-form symmetry. Moreover, if the framing anomalies of two non-spin QFTs differ by a multiple of 8, then they are dual as spin QFTs if and only if they are also dual as non-spin QFTs. Applications to summing over the spin structures, time-reversal symmetry, and level/rank dualities are explored. The Lorentz symmetry fractionalization naturally arises in Chern-Simons matter dualities that obey certain spin/charge relations, and is instrumental for the dualities to hold when viewed as non-spin theories.

- Authors:

- Walter Burke Institute for Theoretical Physics
- Institute for Advanced Study, Princeton University

- Publication Date:

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1597129

- Grant/Contract Number:
- SC0011632

- Resource Type:
- Published Article

- Journal Name:
- SciPost Physics Proceedings

- Additional Journal Information:
- Journal Name: SciPost Physics Proceedings Journal Volume: 8 Journal Issue: 2; Journal ID: ISSN 2542-4653

- Publisher:
- Stichting SciPost

- Country of Publication:
- Netherlands

- Language:
- English

### Citation Formats

```
Hsin, Po-Shen, and Shao, Shu-Heng. Lorentz symmetry fractionalization and dualities in (2+1)d. Netherlands: N. p., 2020.
Web. doi:10.21468/SciPostPhys.8.2.018.
```

```
Hsin, Po-Shen, & Shao, Shu-Heng. Lorentz symmetry fractionalization and dualities in (2+1)d. Netherlands. doi:10.21468/SciPostPhys.8.2.018.
```

```
Hsin, Po-Shen, and Shao, Shu-Heng. Tue .
"Lorentz symmetry fractionalization and dualities in (2+1)d". Netherlands. doi:10.21468/SciPostPhys.8.2.018.
```

```
@article{osti_1597129,
```

title = {Lorentz symmetry fractionalization and dualities in (2+1)d},

author = {Hsin, Po-Shen and Shao, Shu-Heng},

abstractNote = {We discuss symmetry fractionalization of the Lorentz group in (2+1) d d non-spin quantum field theory (QFT), and its implications for dualities. We prove that two inequivalent non-spin QFTs are dual as spin QFTs if and only if they are related by a Lorentz symmetry fractionalization with respect to an anomalous \mathbb{Z}_2 ℤ 2 one-form symmetry. Moreover, if the framing anomalies of two non-spin QFTs differ by a multiple of 8, then they are dual as spin QFTs if and only if they are also dual as non-spin QFTs. Applications to summing over the spin structures, time-reversal symmetry, and level/rank dualities are explored. The Lorentz symmetry fractionalization naturally arises in Chern-Simons matter dualities that obey certain spin/charge relations, and is instrumental for the dualities to hold when viewed as non-spin theories.},

doi = {10.21468/SciPostPhys.8.2.018},

journal = {SciPost Physics Proceedings},

number = 2,

volume = 8,

place = {Netherlands},

year = {2020},

month = {2}

}

DOI: 10.21468/SciPostPhys.8.2.018

Works referenced in this record:

##
On 2-group global symmetries and their anomalies

journal, March 2019

- Benini, Francesco; Córdova, Clay; Hsin, Po-Shen
- Journal of High Energy Physics, Vol. 2019, Issue 3

##
Interaction effect on topological classification of superconductors in two dimensions

journal, August 2013

- Yao, Hong; Ryu, Shinsei
- Physical Review B, Vol. 88, Issue 6

##
Fermionic SPT phases in higher dimensions and bosonization

journal, October 2017

- Kapustin, Anton; Thorngren, Ryan
- Journal of High Energy Physics, Vol. 2017, Issue 10

##
New higher anomalies, SU(N) Yang–Mills gauge theory and ${\mathbb{CP}}^{\mathrm{N}-1}$ sigma model

journal, January 2020

- Wan, Zheyan; Wang, Juven; Zheng, Yunqin
- Annals of Physics

##
Comments on one-form global symmetries and their gauging in 3d and 4d

journal, January 2019

- Hsin, Po-Shen; Lam, Ho Tat; Seiberg, Nathan
- SciPost Physics, Vol. 6, Issue 3

##
Classifying fractionalization: Symmetry classification of gapped ${\mathbb{Z}}_{2}$ spin liquids in two dimensions

journal, March 2013

- Essin, Andrew M.; Hermele, Michael
- Physical Review B, Vol. 87, Issue 10

##
A new SU(2) anomaly

journal, May 2019

- Wang, Juven; Wen, Xiao-Gang; Witten, Edward
- Journal of Mathematical Physics, Vol. 60, Issue 5

##
D-brane charges in five-brane backgrounds

journal, October 2001

- Maldacena, Juan; Seiberg, Nathan; Moore, Gregory
- Journal of High Energy Physics, Vol. 2001, Issue 10

##
Symmetries and strings in field theory and gravity

journal, April 2011

- Banks, Tom; Seiberg, Nathan
- Physical Review D, Vol. 83, Issue 8

##
TFT construction of RCFT correlators

journal, August 2004

- Fuchs, Jürgen; Runkel, Ingo; Schweigert, Christoph
- Nuclear Physics B, Vol. 694, Issue 3

##
Generalized global symmetries

journal, February 2015

- Gaiotto, Davide; Kapustin, Anton; Seiberg, Nathan
- Journal of High Energy Physics, Vol. 2015, Issue 2

##
Simple Currents, Modular Invariants and Fixed Points

journal, August 1990

- Schellekens, A. N.; Yankielowicz, S.
- International Journal of Modern Physics A, Vol. 05, Issue 15

##
Effect of interactions on two-dimensional fermionic symmetry-protected topological phases with ${Z}_{2}$ symmetry

journal, May 2014

- Gu, Zheng-Cheng; Levin, Michael
- Physical Review B, Vol. 89, Issue 20

##
Symmetry enriched U(1) quantum spin liquids

journal, May 2018

- Zou, Liujun; Wang, Chong; Senthil, T.
- Physical Review B, Vol. 97, Issue 19

##
Time-reversal symmetry, anomalies, and dualities in (2+1)$d$

journal, January 2018

- Cordova, Clay; Hsin, Po-Shen; Seiberg, Nathan
- SciPost Physics, Vol. 5, Issue 1

##
Topological field theory on a lattice, discrete theta-angles and confinement

journal, January 2014

- Kapustin, Anton; Thorngren, Ryan
- Advances in Theoretical and Mathematical Physics, Vol. 18, Issue 5

##
Phases Of Adjoint QCD$_3$ And Dualities

journal, January 2018

- Gomis, Jaume; Komargodski, Zohar; Seiberg, Nathan
- SciPost Physics, Vol. 5, Issue 1

##
On simply connected, 4-dimensional polyhedra

journal, December 1949

- Whitehead, J. H. C.
- Commentarii Mathematici Helvetici, Vol. 22, Issue 1

##
Time-Reversal Symmetric $U(1)$ Quantum Spin Liquids

journal, March 2016

- Wang, Chong; Senthil, T.
- Physical Review X, Vol. 6, Issue 1

##
Contact terms, unitarity, and F -maximization in three-dimensional superconformal theories

journal, October 2012

- Closset, Cyril; Dumitrescu, Thomas T.; Festuccia, Guido
- Journal of High Energy Physics, Vol. 2012, Issue 10

##
Comments on Chern-Simons contact terms in three dimensions

journal, September 2012

- Closset, Cyril; Dumitrescu, Thomas T.; Festuccia, Guido
- Journal of High Energy Physics, Vol. 2012, Issue 9

##
A theory of 2+1D bosonic topological orders

journal, November 2015

- Wen, Xiao-Gang
- National Science Review, Vol. 3, Issue 1

##
Exceptional Chern-Simons-Matter dualities

journal, January 2019

- Cordova, Clay; Hsin, Po-Shen; Ohmori, Kantaro
- SciPost Physics, Vol. 7, Issue 4

##
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

##
Coupling a QFT to a TQFT and duality

journal, April 2014

- Kapustin, Anton; Seiberg, Nathan
- Journal of High Energy Physics, Vol. 2014, Issue 4

##
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

##
Topological gauge theories and group cohomology

journal, April 1990

- Dijkgraaf, Robbert; Witten, Edward
- Communications in Mathematical Physics, Vol. 129, Issue 2

##
Reading between the lines of four-dimensional gauge theories

journal, August 2013

- Aharony, Ofer; Seiberg, Nathan; Tachikawa, Yuji
- Journal of High Energy Physics, Vol. 2013, Issue 8

##
All-fermion electrodynamics and fermion number anomaly inflow

journal, October 2015

- Kravec, Shauna; McGreevy, John; Swingle, Brian
- Physical Review D, Vol. 92, Issue 8

##
Naturality in conformal field theory

journal, January 1989

- Moore, Gregory; Seiberg, Nathan
- Nuclear Physics B, Vol. 313, Issue 1

##
Symmetry fractionalization and twist defects

journal, March 2016

- Tarantino, Nicolas; Lindner, Netanel H.; Fidkowski, Lukasz
- New Journal of Physics, Vol. 18, Issue 3

##
Quantum 4d Yang-Mills theory and time-reversal symmetric 5d higher-gauge topological field theory

journal, October 2019

- Wan, Zheyan; Wang, Juven; Zheng, Yunqin
- Physical Review D, Vol. 100, Issue 8

##
Anyons in an exactly solved model and beyond

journal, January 2006

- Kitaev, Alexei
- Annals of Physics, Vol. 321, Issue 1

##
Fusion categories and homotopy theory

journal, January 2010

- Etingof, Pavel; Nikshych, Dmitri; Ostrik, Viktor
- Quantum Topology

##
On the Cohomology of the Real Grassmann Complexes and the Characteristic Classes of n-Plane Bundles

journal, July 1960

- Thomas, Emery
- Transactions of the American Mathematical Society, Vol. 96, Issue 1

##
A symmetry breaking scenario for QCD3

journal, January 2018

- Komargodski, Zohar; Seiberg, Nathan
- Journal of High Energy Physics, Vol. 2018, Issue 1

##
Symmetry fractionalization, defects, and gauging of topological phases

journal, September 2019

- Barkeshli, Maissam; Bonderson, Parsa; Cheng, Meng
- Physical Review B, Vol. 100, Issue 11

##
Metastable vacua in large-N QCD3

journal, January 2020

- Armoni, Adi; Dumitrescu, Thomas T.; Festuccia, Guido
- Journal of High Energy Physics, Vol. 2020, Issue 1

##
Interacting topological phases and modular invariance

journal, June 2012

- Ryu, Shinsei; Zhang, Shou-Cheng
- Physical Review B, Vol. 85, Issue 24

##
Framed Wilson operators, fermionic strings, and gravitational anomaly in 4d

journal, February 2015

- Thorngren, Ryan
- Journal of High Energy Physics, Vol. 2015, Issue 2

##
Spin TQFTs and fermionic phases of matter

journal, October 2016

- Gaiotto, Davide; Kapustin, Anton
- International Journal of Modern Physics A, Vol. 31, Issue 28n29

##
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

##
Effective-Field-Theory Model for the Fractional Quantum Hall Effect

journal, January 1989

- Zhang, S. C.; Hansson, T. H.; Kivelson, S.
- Physical Review Letters, Vol. 62, Issue 1

##
Level/rank duality and Chern-Simons-matter theories

journal, September 2016

- Hsin, Po-Shen; Seiberg, Nathan
- Journal of High Energy Physics, Vol. 2016, Issue 9

##
Classical and quantum conformal field theory

journal, June 1989

- Moore, Gregory; Seiberg, Nathan
- Communications in Mathematical Physics, Vol. 123, Issue 2

##
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

##
Extended chiral algebras and modular invariant partition functions

journal, December 1989

- Schellekens, A. N.; Yankielowicz, S.
- Nuclear Physics B, Vol. 327, Issue 3

##
Theory of twist liquids: Gauging an anyonic symmetry

journal, September 2015

- Teo, Jeffrey C. Y.; Hughes, Taylor L.; Fradkin, Eduardo
- Annals of Physics, Vol. 360

##
State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter

journal, April 2017

- Bhardwaj, Lakshya; Gaiotto, Davide; Kapustin, Anton
- Journal of High Energy Physics, Vol. 2017, Issue 4

##
Interacting fermionic topological insulators/superconductors in three dimensions

journal, May 2014

- Wang, Chong; Senthil, T.
- Physical Review B, Vol. 89, Issue 19

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

journal, January 2018

- Cordova, Clay; Hsin, Po-Shen; Seiberg, Nathan
- SciPost Physics, Vol. 4, Issue 4

##
Taming the conformal zoo

journal, April 1989

- Moore, Gregory; Seiberg, Nathan
- Physics Letters B, Vol. 220, Issue 3

##
Particle-Vortex Duality from 3D Bosonization

journal, September 2016

- Karch, Andreas; Tong, David
- Physical Review X, Vol. 6, Issue 3

##
On Classification of Modular Tensor Categories

journal, August 2009

- Rowell, Eric; Stong, Richard; Wang, Zhenghan
- Communications in Mathematical Physics, Vol. 292, Issue 2

##
Time-reversal and spatial-reflection symmetry localization anomalies in (2+1)-dimensional topological phases of matter

journal, September 2018

- Barkeshli, Maissam; Cheng, Meng
- Physical Review B, Vol. 98, Issue 11

##
Quantum field theory and the Jones polynomial

journal, September 1989

- Witten, Edward
- Communications in Mathematical Physics, Vol. 121, Issue 3

##
Chern-Simons-matter dualities with SO and USp gauge groups

journal, February 2017

- Aharony, Ofer; Benini, Francesco; Hsin, Po-Shen
- Journal of High Energy Physics, Vol. 2017, Issue 2