# Anomalies and bounds on charged operators

## Abstract

We study the implications of the ’t Hooft anomaly (i.e., obstruction to gauging) on conformal field theory, focusing on the case when the global symmetry is ${\mathbb{Z}}_{2}$. Using the modular bootstrap, universal bounds on ( $1+1$)-dimensional bosonic conformal field theories with an internal ${\mathbb{Z}}_{2}$ global symmetry are derived. The bootstrap bounds depend dramatically on the ’t Hooft anomaly. In particular, there is a universal upper bound on the lightest ${\mathbb{Z}}_{2}$ odd operator if the symmetry is anomalous, but there is no bound if the symmetry is nonanomalous. In the nonanomalous case, we find that the lightest ${\mathbb{Z}}_{2}$ odd state and the defect ground state cannot both be arbitrarily heavy. We also consider theories with a $U\left(1\right)$ global symmetry, and comment that there is no bound on the lightest $U\left(1\right)$ charged operator if the symmetry is nonanomalous.

- Authors:

- Publication Date:

- Research Org.:
- California Institute of Technology (CalTech), Pasadena, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP)

- OSTI Identifier:
- 1545466

- Alternate Identifier(s):
- OSTI ID: 1595838

- Grant/Contract Number:
- SC0011632

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Name: Physical Review D Journal Volume: 100 Journal Issue: 2; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS

### Citation Formats

```
Lin, Ying-Hsuan, and Shao, Shu-Heng. Anomalies and bounds on charged operators. United States: N. p., 2019.
Web. doi:10.1103/PhysRevD.100.025013.
```

```
Lin, Ying-Hsuan, & Shao, Shu-Heng. Anomalies and bounds on charged operators. United States. https://doi.org/10.1103/PhysRevD.100.025013
```

```
Lin, Ying-Hsuan, and Shao, Shu-Heng. Thu .
"Anomalies and bounds on charged operators". United States. https://doi.org/10.1103/PhysRevD.100.025013.
```

```
@article{osti_1545466,
```

title = {Anomalies and bounds on charged operators},

author = {Lin, Ying-Hsuan and Shao, Shu-Heng},

abstractNote = {We study the implications of the ’t Hooft anomaly (i.e., obstruction to gauging) on conformal field theory, focusing on the case when the global symmetry is Z2. Using the modular bootstrap, universal bounds on (1+1)-dimensional bosonic conformal field theories with an internal Z2 global symmetry are derived. The bootstrap bounds depend dramatically on the ’t Hooft anomaly. In particular, there is a universal upper bound on the lightest Z2 odd operator if the symmetry is anomalous, but there is no bound if the symmetry is nonanomalous. In the nonanomalous case, we find that the lightest Z2 odd state and the defect ground state cannot both be arbitrarily heavy. We also consider theories with a U(1) global symmetry, and comment that there is no bound on the lightest U(1) charged operator if the symmetry is nonanomalous.},

doi = {10.1103/PhysRevD.100.025013},

journal = {Physical Review D},

number = 2,

volume = 100,

place = {United States},

year = {2019},

month = {7}

}

https://doi.org/10.1103/PhysRevD.100.025013

*Citation information provided by*

Web of Science

Web of Science

Works referenced in this record:

##
Bootstrapping the spectral function: on the uniqueness of Liouville and the universality of BTZ

journal, September 2018

- Collier, Scott; Kravchuk, Petr; Lin, Ying-Hsuan
- Journal of High Energy Physics, Vol. 2018, Issue 9

##
Parity and the modular bootstrap

journal, January 2018

- Anous, Tarek; Mahajan, Raghu; Shaghoulian, Edgar
- SciPost Physics, Vol. 5, Issue 3

##
Constraints on 2d CFT partition functions

journal, October 2013

- Friedan, Daniel; Keller, Christoph A.
- Journal of High Energy Physics, Vol. 2013, Issue 10

##
The Weak Gravity Conjecture in three dimensions

journal, October 2016

- Montero, Miguel; Shiu, Gary; Soler, Pablo
- Journal of High Energy Physics, Vol. 2016, Issue 10

##
Intrinsic and emergent anomalies at deconfined critical points

journal, August 2018

- Metlitski, Max A.; Thorngren, Ryan
- Physical Review B, Vol. 98, Issue 8

##
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

##
Monster symmetry and extremal CFTs

journal, November 2012

- Gaiotto, Davide
- Journal of High Energy Physics, Vol. 2012, Issue 11

##
TFT construction of RCFT correlators I: partition functions

journal, December 2002

- Fuchs, Jürgen; Runkel, Ingo; Schweigert, Christoph
- Nuclear Physics B, Vol. 646, Issue 3

##
Fermions, strings, and gauge fields in lattice spin models

journal, June 2003

- Levin, Michael; Wen, Xiao-Gang
- Physical Review B, Vol. 67, Issue 24

##
Symmetry protected topological orders and the group cohomology of their symmetry group

journal, April 2013

- Chen, Xie; Gu, Zheng-Cheng; Liu, Zheng-Xin
- Physical Review B, Vol. 87, Issue 15

##
Symmetries and strings in field theory and gravity

journal, April 2011

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

##
On finite symmetries and their gauging in two dimensions

journal, March 2018

- Bhardwaj, Lakshya; Tachikawa, Yuji
- Journal of High Energy Physics, Vol. 2018, Issue 3

##
Modular constraints on superconformal field theories

journal, January 2019

- Bae, Jin-Beom; Lee, Sungjay; Song, Jaewon
- Journal of High Energy Physics, Vol. 2019, Issue 1

##
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

##
The string landscape, black holes and gravity as the weakest force

journal, June 2007

- Arkani-Hamed, Nima; Motl, Luboš; Nicolis, Alberto
- Journal of High Energy Physics, Vol. 2007, Issue 06

##
Modular invariance and discrete torsion on orbifolds

journal, September 1986

- Vafa, Cumrun
- Nuclear Physics B, Vol. 273, Issue 3-4

##
Anti de Sitter space and holography

journal, January 1998

- Witten, Edward
- Advances in Theoretical and Mathematical Physics, Vol. 2, Issue 2

##
Modular bootstrap revisited

journal, September 2018

- Collier, Scott; Lin, Ying-Hsuan; Yin, Xi
- Journal of High Energy Physics, Vol. 2018, Issue 9

##
Evidence for a sublattice weak gravity conjecture

journal, August 2017

- Heidenreich, Ben; Reece, Matthew; Rudelius, Tom
- Journal of High Energy Physics, Vol. 2017, Issue 8

##
Bounds on operator dimensions in 2D conformal field theories

journal, May 2014

- Qualls, Joshua D.; Shapere, Alfred D.
- Journal of High Energy Physics, Vol. 2014, Issue 5

##
A PCAC puzzle: π0→γγ in the σ-model

journal, March 1969

- Bell, J. S.; Jackiw, R.
- Il Nuovo Cimento A, Vol. 60, Issue 1

##
A semidefinite program solver for the conformal bootstrap

journal, June 2015

- Simmons-Duffin, David
- Journal of High Energy Physics, Vol. 2015, Issue 6

##
Coupling a QFT to a TQFT and duality

journal, April 2014

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

##
Statistics of the Two-Dimensional Ferromagnet. Part I

journal, August 1941

- Kramers, H. A.; Wannier, G. H.
- Physical Review, Vol. 60, Issue 3

##
Symmetry-protected topological phases and orbifolds: Generalized Laughlin's argument

journal, August 2013

- Sule, Olabode Mayodele; Chen, Xiao; Ryu, Shinsei
- Physical Review B, Vol. 88, Issue 7

##
Fusion rules and modular transformations in 2D conformal field theory

journal, January 1988

- Verlinde, Erik
- Nuclear Physics B, Vol. 300

##
Topological gauge theories and group cohomology

journal, April 1990

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

##
Operator content of two-dimensional conformally invariant theories

journal, January 1986

- Cardy, John L.
- Nuclear Physics B, Vol. 270

##
The Fusion Algebra of Bimodule Categories

journal, July 2007

- Fuchs, Jürgen; Runkel, Ingo; Schweigert, Christoph
- Applied Categorical Structures, Vol. 16, Issue 1-2

##
The conformal bootstrap: Theory, numerical techniques, and applications

journal, January 2019

- Poland, David; Rychkov, Slava; Vichi, Alessandro
- Reviews of Modern Physics, Vol. 91, Issue 1

##
Fast conformal bootstrap and constraints on 3d gravity

journal, May 2019

- Afkhami-Jeddi, Nima; Hartman, Thomas; Tajdini, Amirhossein
- Journal of High Energy Physics, Vol. 2019, Issue 5

##
Invertible defects and isomorphisms of rational CFTs

journal, January 2011

- Davydov, Alexei; Kong, Liang; Runkel, Ingo
- Advances in Theoretical and Mathematical Physics, Vol. 15, Issue 1

##
Global anomalies on orbifolds

journal, September 1987

- Freed, Daniel S.; Vafa, Cumrun
- Communications in Mathematical Physics, Vol. 110, Issue 3

##
Critical behaviour of SU(n) quantum chains and topological non-linear σ-models

journal, December 1988

- Affleck, Ian
- Nuclear Physics B, Vol. 305, Issue 4

##
Global anomalies on orbifolds

journal, June 1988

- Freed, Daniel S.; Vafa, Cumrun
- Communications in Mathematical Physics, Vol. 117, Issue 2

##
A universal inequality for CFT and quantum gravity

journal, August 2011

- Hellerman, Simeon
- Journal of High Energy Physics, Vol. 2011, Issue 8

##
Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations

journal, December 2011

- Chen, Xie; Liu, Zheng-Xin; Wen, Xiao-Gang
- Physical Review B, Vol. 84, Issue 23

##
TFT construction of RCFT correlators II: unoriented world sheets

journal, February 2004

- Fuchs, Jürgen; Runkel, Ingo; Schweigert, Christoph
- Nuclear Physics B, Vol. 678, Issue 3

##
Axial-Vector Vertex in Spinor Electrodynamics

journal, January 1969

- Adler, Stephen L.
- Physical Review, Vol. 177, Issue 5

##
Topological defect lines and renormalization group flows in two dimensions

journal, January 2019

- Chang, Chi-Ming; Lin, Ying-Hsuan; Shao, Shu-Heng
- Journal of High Energy Physics, Vol. 2019, Issue 1

##
Mixed global anomalies and boundary conformal field theories

journal, November 2018

- Numasawa, Tokiro; Yamaguch, Satoshi
- Journal of High Energy Physics, Vol. 2018, Issue 11

##
Comments on the N = 2,3,4 superconformal algebras in two dimensions

journal, January 1987

- Schwimmer, A.; Seiberg, N.
- Physics Letters B, Vol. 184, Issue 2-3

##
Symmetry Protection of Critical Phases and a Global Anomaly in $1+1$ Dimensions

journal, January 2017

- Furuya, Shunsuke C.; Oshikawa, Masaki
- Physical Review Letters, Vol. 118, Issue 2

##
TFT construction of RCFT correlators IV:

journal, May 2005

- Fuchs, Jürgen; Runkel, Ingo; Schweigert, Christoph
- Nuclear Physics B, Vol. 715, Issue 3

##
Duality and defects in rational conformal field theory

journal, February 2007

- Fröhlich, Jürg; Fuchs, Jürgen; Runkel, Ingo
- Nuclear Physics B, Vol. 763, Issue 3

##
Spectra of Wess-Zumino-Witten models with arbitrary simple groups

journal, March 1988

- Felder, G.; Gawedzki, K.; Kupiainen, A.
- Communications in Mathematical Physics, Vol. 117, Issue 1

##
Sigma models on flags

journal, January 2019

- Ohmori, Kantaro; Seiberg, Nathan; Shao, Shu-Heng
- SciPost Physics, Vol. 6, Issue 2

##
Generalised twisted partition functions

journal, April 2001

- Petkova, V. B.; Zuber, J. -B.
- Physics Letters B, Vol. 504, Issue 1-2

##
Universal bounds on charged states in 2d CFT and 3d gravity

journal, August 2016

- Benjamin, Nathan; Dyer, Ethan; Fitzpatrick, A. Liam
- Journal of High Energy Physics, Vol. 2016, Issue 8

##
Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions

journal, August 2018

- Wang, Juven; Wen, Xiao-Gang; Witten, Edward
- Physical Review X, Vol. 8, Issue 3

##
The conformal bootstrap

journal, June 2016

- Poland, David; Simmons-Duffin, David
- Nature Physics, Vol. 12, Issue 6

##
Kramers-Wannier Duality from Conformal Defects

journal, August 2004

- Fröhlich, Jürg; Fuchs, Jürgen; Runkel, Ingo
- Physical Review Letters, Vol. 93, Issue 7

##
Universal symmetry-protected topological invariants for symmetry-protected topological states

journal, February 2014

- Hung, Ling-Yan; Wen, Xiao-Gang
- Physical Review B, Vol. 89, Issue 7