# Exceptional Chern-Simons-Matter dualities

## Abstract

We use conformal embeddings involving exceptional affine Kac-Moody algebras to derive new dualities of three-dimensional topological field theories. These generalize the familiar level-rank duality of Chern-Simons theories based on classical gauge groups to the setting of exceptional gauge groups. For instance, one duality sequence we discuss is (E_{N})_{1}\leftrightarrow SU(9-N)_{-1} $({E}_{N}{)}_{1}\leftrightarrow SU(9-N{)}_{-1}$ . Others such as SO(3)_{8}\leftrightarrow PSU(3)_{-6}, $SO(3{)}_{8}\leftrightarrow PSU(3{)}_{-6},$ are dualities among theories with classical gauge groups that arise due to their embedding into an exceptional chiral algebra. We apply these equivalences between topological field theories to conjecture new boson-boson Chern-Simons-matter dualities. We also use them to determine candidate phase diagrams of time-reversal invariant G_{2} ${G}_{2}$ gauge theory coupled to either an adjoint fermion, or two fundamental fermions.

- Authors:

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

- Publication Date:

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1572130

- Grant/Contract Number:
- SC0009988; SC0011632

- Resource Type:
- Published Article

- Journal Name:
- SciPost Physics Proceedings

- Additional Journal Information:
- Journal Name: SciPost Physics Proceedings Journal Volume: 7 Journal Issue: 4; Journal ID: ISSN 2542-4653

- Publisher:
- Stichting SciPost

- Country of Publication:
- Netherlands

- Language:
- English

### Citation Formats

```
Cordova, Clay, Hsin, Po-Shen, and Ohmori, Kantaro. Exceptional Chern-Simons-Matter dualities. Netherlands: N. p., 2019.
Web. doi:10.21468/SciPostPhys.7.4.056.
```

```
Cordova, Clay, Hsin, Po-Shen, & Ohmori, Kantaro. Exceptional Chern-Simons-Matter dualities. Netherlands. doi:10.21468/SciPostPhys.7.4.056.
```

```
Cordova, Clay, Hsin, Po-Shen, and Ohmori, Kantaro. Mon .
"Exceptional Chern-Simons-Matter dualities". Netherlands. doi:10.21468/SciPostPhys.7.4.056.
```

```
@article{osti_1572130,
```

title = {Exceptional Chern-Simons-Matter dualities},

author = {Cordova, Clay and Hsin, Po-Shen and Ohmori, Kantaro},

abstractNote = {We use conformal embeddings involving exceptional affine Kac-Moody algebras to derive new dualities of three-dimensional topological field theories. These generalize the familiar level-rank duality of Chern-Simons theories based on classical gauge groups to the setting of exceptional gauge groups. For instance, one duality sequence we discuss is (E_{N})_{1}\leftrightarrow SU(9-N)_{-1} ( E N ) 1 ↔ S U ( 9 − N ) − 1 . Others such as SO(3)_{8}\leftrightarrow PSU(3)_{-6}, S O ( 3 ) 8 ↔ P S U ( 3 ) − 6 , are dualities among theories with classical gauge groups that arise due to their embedding into an exceptional chiral algebra. We apply these equivalences between topological field theories to conjecture new boson-boson Chern-Simons-matter dualities. We also use them to determine candidate phase diagrams of time-reversal invariant G_{2} G 2 gauge theory coupled to either an adjoint fermion, or two fundamental fermions.},

doi = {10.21468/SciPostPhys.7.4.056},

journal = {SciPost Physics Proceedings},

number = 4,

volume = 7,

place = {Netherlands},

year = {2019},

month = {10}

}

DOI: 10.21468/SciPostPhys.7.4.056

Works referenced in this record:

##
Eigenvalue inequalities for fermions in gauge theories

journal, September 1984

- Vafa, Cumrun; Witten, Edward
- Communications in Mathematical Physics, Vol. 95, Issue 3

##
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

##
Chern Simons duality with a fundamental boson and fermion

journal, November 2013

- Jain, Sachin; Minwalla, Shiraz; Yokoyama, Shuichi
- Journal of High Energy Physics, Vol. 2013, Issue 11

##
Bosonization and mirror symmetry

journal, October 2016

- Kachru, Shamit; Mulligan, Michael; Torroba, Gonzalo
- Physical Review D, Vol. 94, Issue 8

##
Conformal subalgebras of Kac-Moody algebras

journal, November 1986

- Schellekens, A. N.; Warner, N. P.
- Physical Review D, Vol. 34, Issue 10

##
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

##
Anomaly Indicators for Time-Reversal Symmetric Topological Orders

journal, September 2017

- Wang, Chenjie; Levin, Michael
- Physical Review Letters, Vol. 119, Issue 13

##
Baryons, monopoles and dualities in Chern-Simons-matter theories

journal, February 2016

- Aharony, Ofer
- Journal of High Energy Physics, Vol. 2016, Issue 2

##
A study of time reversal symmetry of abelian anyons

journal, July 2018

- Lee, Yasunori; Tachikawa, Yuji
- Journal of High Energy Physics, Vol. 2018, Issue 7

##
Parity Conservation in Quantum Chromodynamics

journal, August 1984

- Vafa, Cumrun; Witten, Edward
- Physical Review Letters, Vol. 53, Issue 6

##
Extremal transitions and five-dimensional supersymmetric field theories

journal, January 1997

- Morrison, David R.; Seiberg, Nathan
- Nuclear Physics B, Vol. 483, Issue 1-2

##
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

##
Generalized global symmetries

journal, February 2015

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

##
Fermion path integrals and topological phases

journal, July 2016

- Witten, Edward
- Reviews of Modern Physics, Vol. 88, Issue 3

##
Level-rank duality of the U( *N* ) WZW model, Chern-Simons theory, and 2d qYM theory

journal, June 2007

- Naculich, Stephen G.; Schnitzer, Howard J.
- Journal of High Energy Physics, Vol. 2007, Issue 06

##
Holographic aspects of three dimensional QCD from string theory

journal, May 2010

- Ki Hong, Deog; Yee, Ho-Ung
- Journal of High Energy Physics, Vol. 2010, Issue 5

##
Curious aspects of three-dimensional N = 1 $$ \mathcal{N}=1 $$ SCFTs

journal, August 2018

- Gaiotto, Davide; Komargodski, Zohar; Wu, Jingxiang
- Journal of High Energy Physics, Vol. 2018, Issue 8

##
Decompositions of representations of exceptional affine algebras with respect to conformal subalgebras

journal, April 1988

- Ka, Victor G.; Sanielevici, Mihaela Niculescu
- Physical Review D, Vol. 37, Issue 8

##
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

##
Phases Of Adjoint QCD$_3$ And Dualities

journal, January 2018

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

##
Group-level duality in WZW models and Chern-Simons theory

journal, August 1990

- Naculich, S. G.; Riggs, H. A.; Schnitzer, H. J.
- Physics Letters B, Vol. 246, Issue 3-4

##
Chern–Simons theory with vector fermion matter

journal, August 2012

- Giombi, Simone; Minwalla, Shiraz; Prakash, Shiroman
- The European Physical Journal C, Vol. 72, Issue 8

##
Phases of N = 1 $$ \mathcal{N}=1 $$ theories in 2 + 1 dimensions

journal, July 2018

- Bashmakov, Vladimir; Gomis, Jaume; Komargodski, Zohar
- Journal of High Energy Physics, Vol. 2018, Issue 7

##
Naturality in conformal field theory

journal, January 1989

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

##
Conformal embeddings, rank-level duality and exceptional modular invariants

journal, April 1991

- Verstegen, D.
- Communications in Mathematical Physics, Vol. 137, Issue 3

##
Correlation functions of large N Chern-Simons-Matter theories and bosonization in three dimensions

journal, December 2012

- Aharony, Ofer; Gur-Ari, Guy; Yacoby, Ran
- Journal of High Energy Physics, Vol. 2012, Issue 12

##
Anyons in an exactly solved model and beyond

journal, January 2006

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

##
Conformal Embeddings and Simple Current Extensions

journal, June 2014

- Kac, Victor G.; Frajria, Pierluigi Möseneder; Papi, Paolo
- International Mathematics Research Notices, Vol. 2015, Issue 14

##
Dualities and phases of 3d N = 1 $$ \mathcal{N}=1 $$ SQCD

journal, October 2018

- Choi, Changha; Roček, Martin; Sharon, Adar
- Journal of High Energy Physics, Vol. 2018, Issue 10

##
A symmetry breaking scenario for QCD3

journal, January 2018

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

##
Strange Duality for Verlinde Spaces of Exceptional Groups at Level One

journal, September 2009

- Boysal, A.; Pauly, C.
- International Mathematics Research Notices

##
N $$ \mathcal{N} $$ = 1 dualities in 2+1 dimensions

journal, November 2018

- Benini, Francesco; Benvenuti, Sergio
- Journal of High Energy Physics, Vol. 2018, Issue 11

##
Assembling Fibonacci anyons from a ${\mathbb{Z}}_{3}$ parafermion lattice model

journal, June 2015

- Stoudenmire, E. M.; Clarke, David J.; Mong, Roger S. K.
- Physical Review B, Vol. 91, Issue 23

##
Five dimensional SUSY field theories, non-trivial fixed points and string dynamics

journal, November 1996

- Seiberg, Nathan
- Physics Letters B, Vol. 388, Issue 4

##
Restrictions on symmetry breaking in vector-like gauge theories

journal, March 1984

- Vafa, C.; Witten, E.
- Nuclear Physics B, Vol. 234, Issue 1

##
Spin module versions of Weyl's reciprocity theorem for classical Kac-Moody Lie algebras---an application to branching rule duality

journal, January 1989

- Hasegawa, Koji
- Publications of the Research Institute for Mathematical Sciences, Vol. 25, Issue 5

##
Group-level duality of WZW fusion coefficients and Chern-Simons link observables

journal, April 1991

- Mlawer, Eli J.; Naculich, Stephen G.; Riggs, Harold A.
- Nuclear Physics B, Vol. 352, Issue 3

##
Particle-vortex duality of two-dimensional Dirac fermion from electric-magnetic duality of three-dimensional topological insulators

journal, June 2016

- Metlitski, Max A.; Vishwanath, Ashvin
- Physical Review B, Vol. 93, Issue 24

##
d = 3 bosonic vector models coupled to Chern-Simons gauge theories

journal, March 2012

- Aharony, Ofer; Gur-Ari, Guy; Yacoby, Ran
- Journal of High Energy Physics, Vol. 2012, Issue 3

##
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

##
Three dimensional bosonization from supersymmetry

journal, November 2015

- Gur-Ari, Guy; Yacoby, Ran
- Journal of High Energy Physics, Vol. 2015, Issue 11

##
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

##
Interacting Anyons in Topological Quantum Liquids: The Golden Chain

journal, April 2007

- Feiguin, Adrian; Trebst, Simon; Ludwig, Andreas W. W.
- Physical Review Letters, Vol. 98, Issue 16

##
Non-Abelian anyons and topological quantum computation

journal, September 2008

- Nayak, Chetan; Simon, Steven H.; Stern, Ady
- Reviews of Modern Physics, Vol. 80, Issue 3

##
Non-Abelian 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

##
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

##
Phase Transition in a Lattice Model of Superconductivity

journal, November 1981

- Dasgupta, C.; Halperin, B. I.
- Physical Review Letters, Vol. 47, Issue 21

##
Three-dimensional dualities with bosons and fermions

journal, February 2018

- Benini, Francesco
- Journal of High Energy Physics, Vol. 2018, Issue 2

##
Particle-Vortex Duality from 3D Bosonization

journal, September 2016

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

##
Mandelstam-'t Hooft duality in abelian lattice models

journal, July 1978

- Peskin, Michael E.
- Annals of Physics, Vol. 113, Issue 1

##
Level-rank duality of WZW models in conformal field theory

journal, February 1992

- Nakanishi, Tomoki; Tsuchiya, Akihiro
- Communications in Mathematical Physics, Vol. 144, 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

##
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

##
Exploring 2-group global symmetries

journal, February 2019

- Córdova, Clay; Dumitrescu, Thomas T.; Intriligator, Kenneth
- Journal of High Energy Physics, Vol. 2019, Issue 2