Skip to main content
Springer Nature Link
Account
Menu
Find a journal Publish with us Track your research
Search
Cart
  1. Home
  2. Journal of High Energy Physics
  3. Article

Non-invertible global symmetries and completeness of the spectrum

  • Regular Article - Theoretical Physics
  • Open access
  • Published: 29 September 2021
  • Volume 2021, article number 203, (2021)
  • Cite this article
Download PDF

You have full access to this open access article

Journal of High Energy Physics Aims and scope Submit manuscript
Non-invertible global symmetries and completeness of the spectrum
Download PDF
  • Ben Heidenreich1,
  • Jacob McNamara2,
  • Miguel Montero2,
  • Matthew Reece2,
  • Tom Rudelius3,4 &
  • …
  • Irene Valenzuela2 

  • 697 Accesses

  • 117 Citations

  • 8 Altmetric

  • Explore all metrics

A preprint version of the article is available at arXiv.

Abstract

It is widely believed that consistent theories of quantum gravity satisfy two basic kinematic constraints: they are free from any global symmetry, and they contain a complete spectrum of gauge charges. For compact, abelian gauge groups, completeness follows from the absence of a 1-form global symmetry. However, this correspondence breaks down for more general gauge groups, where the breaking of the 1-form symmetry is insufficient to guarantee a complete spectrum. We show that the correspondence may be restored by broadening our notion of symmetry to include non-invertible topological operators, and prove that their absence is sufficient to guarantee a complete spectrum for any compact, possibly disconnected gauge group. In addition, we prove an analogous statement regarding the completeness of twist vortices: codimension-2 objects defined by a discrete holonomy around their worldvolume, such as cosmic strings in four dimensions. We discuss how this correspondence is modified in various, more general contexts, including non-compact gauge groups, Higgsing of gauge theories, and the addition of Chern-Simons terms. Finally, we discuss the implications of our results for the Swampland program, as well as the phenomenological implications of the existence of twist strings.

Article PDF

Download to read the full article text

Similar content being viewed by others

Topological operators and completeness of spectrum in discrete gauge theories

Article Open access 28 December 2020

Generalized symmetries and Noether’s theorem in QFT

Article Open access 31 August 2022

Non-invertible symmetries from discrete gauging and completeness of the spectrum

Article Open access 20 April 2023
Use our pre-submission checklist

Avoid common mistakes on your manuscript.

References

  1. D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized Global Symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. J. Polchinski, Monopoles, duality, and string theory, Int. J. Mod. Phys. A 19S1 (2004) 145 [hep-th/0304042] [INSPIRE].

  3. T. Banks and L.J. Dixon, Constraints on String Vacua with Space-Time Supersymmetry, Nucl. Phys. B 307 (1988) 93 [INSPIRE].

    Article  ADS  Google Scholar 

  4. R. Kallosh, A.D. Linde, D.A. Linde and L. Susskind, Gravity and global symmetries, Phys. Rev. D 52 (1995) 912 [hep-th/9502069] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  5. T. Banks and N. Seiberg, Symmetries and Strings in Field Theory and Gravity, Phys. Rev. D 83 (2011) 084019 [arXiv:1011.5120] [INSPIRE].

    Article  ADS  Google Scholar 

  6. D. Harlow and H. Ooguri, Symmetries in quantum field theory and quantum gravity, Commun. Math. Phys. 383 (2021) 1669 [arXiv:1810.05338] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. D. Harlow and E. Shaghoulian, Global symmetry, Euclidean gravity, and the black hole information problem, JHEP 04 (2021) 175 [arXiv:2010.10539] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Y. Chen and H.W. Lin, Signatures of global symmetry violation in relative entropies and replica wormholes, JHEP 03 (2021) 040 [arXiv:2011.06005] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. A. Belin, J. De Boer, P. Nayak and J. Sonner, Charged Eigenstate Thermalization, Euclidean Wormholes and Global Symmetries in Quantum Gravity, arXiv:2012.07875 [INSPIRE].

  10. K. Yonekura, Topological violation of global symmetries in quantum gravity, arXiv:2011.11868 [INSPIRE].

  11. H. Casini, M. Huerta, J.M. Magan and D. Pontello, Entropic order parameters for the phases of QFT, JHEP 04 (2021) 277 [arXiv:2008.11748] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. T. Rudelius and S.-H. Shao, Topological Operators and Completeness of Spectrum in Discrete Gauge Theories, JHEP 12 (2020) 172 [arXiv:2006.10052] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. S. Gukov and E. Witten, Gauge theory, ramification, and the geometric langlands program, Curr. Dev. Math. 2006 (2006) 35.

    Article  MATH  Google Scholar 

  14. S. Gukov and E. Witten, Rigid Surface Operators, Adv. Theor. Math. Phys. 14 (2010) 87 [arXiv:0804.1561] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  15. C. Córdova, T.T. Dumitrescu and K. Intriligator, Exploring 2-Group Global Symmetries, JHEP 02 (2019) 184 [arXiv:1802.04790] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Y. Hidaka, M. Nitta and R. Yokokura, Higher-form symmetries and 3-group in axion electrodynamics, Phys. Lett. B 808 (2020) 135672 [arXiv:2006.12532] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  17. Y. Hidaka, M. Nitta and R. Yokokura, Global 3-group symmetry and ’t Hooft anomalies in axion electrodynamics, JHEP 01 (2021) 173 [arXiv:2009.14368] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. E.P. Verlinde, Fusion Rules and Modular Transformations in 2D Conformal Field Theory, Nucl. Phys. B 300 (1988) 360 [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  19. G.W. Moore and N. Seiberg, Classical and Quantum Conformal Field Theory, Commun. Math. Phys. 123 (1989) 177 [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. J. Fröhlich, J. Fuchs, I. Runkel and C. Schweigert, Duality and defects in rational conformal field theory, Nucl. Phys. B 763 (2007) 354 [hep-th/0607247] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. A. Davydov, L. Kong and I. Runkel, Invertible Defects and Isomorphisms of Rational CFTs, Adv. Theor. Math. Phys. 15 (2011) 43 [arXiv:1004.4725] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  22. L. Bhardwaj and Y. Tachikawa, On finite symmetries and their gauging in two dimensions, JHEP 03 (2018) 189 [arXiv:1704.02330] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. C.-M. Chang, Y.-H. Lin, S.-H. Shao, Y. Wang and X. Yin, Topological Defect Lines and Renormalization Group Flows in Two Dimensions, JHEP 01 (2019) 026 [arXiv:1802.04445] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. W. Ji and X.-G. Wen, Categorical symmetry and noninvertible anomaly in symmetry-breaking and topological phase transitions, Phys. Rev. Res. 2 (2020) 033417 [arXiv:1912.13492] [INSPIRE].

    Article  Google Scholar 

  25. L. Kong, T. Lan, X.-G. Wen, Z.-H. Zhang and H. Zheng, Classification of topological phases with finite internal symmetries in all dimensions, JHEP 09 (2020) 093 [arXiv:2003.08898] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. L. Kong, T. Lan, X.-G. Wen, Z.-H. Zhang and H. Zheng, Algebraic higher symmetry and categorical symmetry: A holographic and entanglement view of symmetry, Phys. Rev. Res. 2 (2020) 043086.

    Article  Google Scholar 

  27. Z. Komargodski, K. Ohmori, K. Roumpedakis and S. Seifnashri, Symmetries and strings of adjoint QCD2, JHEP 03 (2021) 103 [arXiv:2008.07567] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. C. Nayak, S.H. Simon, A. Stern, M. Freedman and S. Das Sarma, Non-Abelian anyons and topological quantum computation, Rev. Mod. Phys. 80 (2008) 1083 [arXiv:0707.1889] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. R. Dijkgraaf, C. Vafa, E.P. Verlinde and H.L. Verlinde, The Operator Algebra of Orbifold Models, Commun. Math. Phys. 123 (1989) 485 [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. M.G. Alford, K.-M. Lee, J. March-Russell and J. Preskill, Quantum field theory of nonAbelian strings and vortices, Nucl. Phys. B 384 (1992) 251 [hep-th/9112038] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  31. W. Ambrose and I. M. Singer, A theorem on holonomy, Trans. Am. Math. Soc. 75 (1953) 428.

    Article  MathSciNet  MATH  Google Scholar 

  32. G. ’t Hooft, On the Phase Transition Towards Permanent Quark Confinement, Nucl. Phys. B 138 (1978) 1 [INSPIRE].

  33. A. Kapustin, Wilson-’t Hooft operators in four-dimensional gauge theories and S-duality, Phys. Rev. D 74 (2006) 025005 [hep-th/0501015] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  34. A. Kapustin and E. Witten, Electric-Magnetic Duality And The Geometric Langlands Program, Commun. Num. Theor. Phys. 1 (2007) 1 [hep-th/0604151] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  35. P. Goddard, J. Nuyts and D.I. Olive, Gauge Theories and Magnetic Charge, Nucl. Phys. B 125 (1977) 1.

    Article  ADS  MathSciNet  Google Scholar 

  36. O. Aharony, N. Seiberg and Y. Tachikawa, Reading between the lines of four-dimensional gauge theories, JHEP 08 (2013) 115 [arXiv:1305.0318] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. W. Fulton and J. Harris, Readings in Mathematics. Vol. 129: Representation Theory: A First Course, Springer Science & Business Media, New York U.S.A. (2013).

  38. N. Yamatsu, Finite-Dimensional Lie Algebras and Their Representations for Unified Model Building, arXiv:1511.08771 [INSPIRE].

  39. L.M. Krauss and F. Wilczek, Discrete Gauge Symmetry in Continuum Theories, Phys. Rev. Lett. 62 (1989) 1221 [INSPIRE].

    Article  ADS  Google Scholar 

  40. M.G. Alford, J. March-Russell and F. Wilczek, Discrete Quantum Hair on Black Holes and the Nonabelian Aharonov-Bohm Effect, Nucl. Phys. B 337 (1990) 695 [INSPIRE].

    Article  ADS  Google Scholar 

  41. J. Preskill and L.M. Krauss, Local Discrete Symmetry and Quantum Mechanical Hair, Nucl. Phys. B 341 (1990) 50 [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. M. Müger, On the structure of modular categories, Proc. Lond. Math. Soc. 87 (2003) 291.

    Article  MathSciNet  MATH  Google Scholar 

  43. M. Nguyen, Y. Tanizaki and M. Ünsal, Semi-Abelian gauge theories, non-invertible symmetries, and string tensions beyond N-ality, JHEP 03 (2021) 238 [arXiv:2101.02227] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. J.E. Kiskis, Disconnected Gauge Groups and the Global Violation of Charge Conservation, Phys. Rev. D 17 (1978) 3196 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  45. A.S. Schwarz, Field theories with no local conservation of the electric charge, Nucl. Phys. B 208 (1982) 141 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  46. F. Benini, C. Córdova and P.-S. Hsin, On 2-Group Global Symmetries and their Anomalies, JHEP 03 (2019) 118 [arXiv:1803.09336] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  47. H. Moradi and X.-G. Wen, Universal Topological Data for Gapped Quantum Liquids in Three Dimensions and Fusion Algebra for Non-Abelian String Excitations, Phys. Rev. B 91 (2015) 075114 [arXiv:1404.4618] [INSPIRE].

    Article  ADS  Google Scholar 

  48. M.G. Alford, K. Benson, S.R. Coleman, J. March-Russell and F. Wilczek, The Interactions and Excitations of Nonabelian Vortices, Phys. Rev. Lett. 64 (1990) 1632 [Erratum ibid. 65 (1990) 668] [INSPIRE].

  49. M. Dine, R.G. Leigh and D.A. MacIntire, Of CP and other gauge symmetries in string theory, Phys. Rev. Lett. 69 (1992) 2030 [hep-th/9205011] [INSPIRE].

    Article  ADS  Google Scholar 

  50. B. Heidenreich, Improved classification of compact Lie groups, MathOverflow, https://mathoverflow.net/q/378257 (version: 2020-12-08).

  51. L. Spice, Classification of (not necessarily connected) compact Lie groups, MathOverflow, https://mathoverflow.net/q/378141 (version: 2020-12-05).

  52. L. Spice, Does Aut(G) → Out(G) always split for a compact, connected Lie group G?, MathOverflow, https://mathoverflow.net/q/378220 (version: 2020-12-06).

  53. K. Bou-Rabee, In any Lie group with finitely many connected components, does there exist a finite subgroup which meets every component?, MathOverflow, https://mathoverflow.net/q/150949 (version: 2013-12-05).

  54. A.M. Polyakov, Particle Spectrum in Quantum Field Theory, JETP Lett. 20 (1974) 194 [INSPIRE].

    ADS  Google Scholar 

  55. G. ’t Hooft, Magnetic Monopoles in Unified Gauge Theories, Nucl. Phys. B 79 (1974) 276 [INSPIRE].

  56. A.A. Abrikosov, On the Magnetic properties of superconductors of the second group, Sov. Phys. JETP 5 (1957) 1174 [INSPIRE].

    Google Scholar 

  57. H.B. Nielsen and P. Olesen, Vortex Line Models for Dual Strings, Nucl. Phys. B 61 (1973) 45 [INSPIRE].

    Article  ADS  Google Scholar 

  58. T. Bröcker and T. Dieck, Graduate Texts in Mathematics. Vol. 98: Representations of Compact Lie Groups, Springer, Heidelberg Germany (2003), https://books.google.com/books?id=AfBzWL5bIIQC.

  59. N. Bourbaki, Lie Groups and Lie Algebras: Chapters 7-9, Springer-Verlag, Heidelberg Germany (2008).

    MATH  Google Scholar 

  60. T.D. Brennan and C. Cordova, Axions, Higher-Groups, and Emergent Symmetry, arXiv:2011.09600 [INSPIRE].

  61. B. Heidenreich, J. McNamara, M. Montero, M. Reece, T. Rudelius and I. Valenzuela, Chern-Weil Global Symmetries and How Quantum Gravity Avoids Them, arXiv:2012.00009 [INSPIRE].

  62. E. Witten, Dyons of Charge e theta/2 pi, Phys. Lett. B 86 (1979) 283 [INSPIRE].

    Article  ADS  Google Scholar 

  63. J. McNamara, Gravitational Solitons and Completeness, arXiv:2108.02228 [INSPIRE].

  64. C.G. Callan, Jr. and J.A. Harvey, Anomalies and Fermion Zero Modes on Strings and Domain Walls, Nucl. Phys. B 250 (1985) 427 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  65. C. Córdova, D.S. Freed, H.T. Lam and N. Seiberg, Anomalies in the Space of Coupling Constants and Their Dynamical Applications I, SciPost Phys. 8 (2020) 001 [arXiv:1905.09315] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  66. R. Jackiw, Charge and Mass Spectrum of Quantum Solitons, in Gauge Theories and Modern Field Theory. Proceedings of Northeastern University, Boston, R.L. Arnowitt and P. Nath eds., MIT Press, Cambridge U.S.A. (1976), pg. 377.

  67. E. Witten, Three lectures on topological phases of matter, Riv. Nuovo Cim. 39 (2016) 313 [arXiv:1510.07698] [INSPIRE].

    ADS  Google Scholar 

  68. J. McNamara and C. Vafa, Baby Universes, Holography, and the Swampland, arXiv:2004.06738 [INSPIRE].

  69. E. Witten, Symmetry and Emergence, Nature Phys. 14 (2018) 116 [arXiv:1710.01791] [INSPIRE].

    Article  ADS  Google Scholar 

  70. L.E. Ibáñez and G.G. Ross, Discrete gauge symmetry anomalies, Phys. Lett. B 260 (1991) 291 [INSPIRE].

    Article  ADS  Google Scholar 

  71. L.E. Ibáñez and G.G. Ross, Discrete gauge symmetries and the origin of baryon and lepton number conservation in supersymmetric versions of the standard model, Nucl. Phys. B 368 (1992) 3 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  72. H.K. Dreiner, C. Luhn and M. Thormeier, What is the discrete gauge symmetry of the MSSM?, Phys. Rev. D 73 (2006) 075007 [hep-ph/0512163] [INSPIRE].

  73. Y.B. Zeldovich, I.Y. Kobzarev and L.B. Okun, Cosmological Consequences of the Spontaneous Breakdown of Discrete Symmetry, Zh. Eksp. Teor. Fiz. 67 (1974) 3 [INSPIRE].

    ADS  Google Scholar 

  74. T.W.B. Kibble, Topology of Cosmic Domains and Strings, J. Phys. A 9 (1976) 1387 [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  75. T.W.B. Kibble, G. Lazarides and Q. Shafi, Walls Bounded by Strings, Phys. Rev. D 26 (1982) 435 [INSPIRE].

    Article  ADS  Google Scholar 

  76. Z. Chacko, H.-S. Goh and R. Harnik, The Twin Higgs: Natural electroweak breaking from mirror symmetry, Phys. Rev. Lett. 96 (2006) 231802 [hep-ph/0506256] [INSPIRE].

  77. M. Geller and O. Telem, Holographic Twin Higgs Model, Phys. Rev. Lett. 114 (2015) 191801 [arXiv:1411.2974] [INSPIRE].

    Article  ADS  Google Scholar 

  78. H. Beauchesne, K. Earl and T. Grégoire, The spontaneous ℤ2 breaking Twin Higgs, JHEP 01 (2016) 130 [arXiv:1510.06069] [INSPIRE].

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, University of Massachusetts, Amherst, MA, 01003, USA

    Ben Heidenreich

  2. Department of Physics, Harvard University, Cambridge, MA, 02138, USA

    Jacob McNamara, Miguel Montero, Matthew Reece & Irene Valenzuela

  3. Department of Physics, University of California, Berkeley, CA, 94720, USA

    Tom Rudelius

  4. School of Natural Sciences, Institute for Advanced Study, Princeton, NJ, 08540, USA

    Tom Rudelius

Authors
  1. Ben Heidenreich
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jacob McNamara
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Miguel Montero
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Matthew Reece
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Tom Rudelius
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Irene Valenzuela
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ben Heidenreich.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

ArXiv ePrint: 2104.07036

Rights and permissions

Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Heidenreich, B., McNamara, J., Montero, M. et al. Non-invertible global symmetries and completeness of the spectrum. J. High Energ. Phys. 2021, 203 (2021). https://doi.org/10.1007/JHEP09(2021)203

Download citation

  • Received: 25 August 2021

  • Accepted: 06 September 2021

  • Published: 29 September 2021

  • DOI: https://doi.org/10.1007/JHEP09(2021)203

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Gauge Symmetry
  • Global Symmetries
  • Topological Field Theories
  • Effective Field Theories
Use our pre-submission checklist

Avoid common mistakes on your manuscript.

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us
  • Track your research

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Journal finder
  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support
  • Cancel contracts here

Not affiliated

Springer Nature

© 2025 Springer Nature