Skip to main content

On 2-group global symmetries and their anomalies

A preprint version of the article is available at arXiv.

Abstract

In general quantum field theories (QFTs), ordinary (0-form) global symmetries and 1-form symmetries can combine into 2-group global symmetries. We describe this phenomenon in detail using the language of symmetry defects. We exhibit a simple procedure to determine the (possible) 2-group global symmetry of a given QFT, and provide a classification of the related ’t Hooft anomalies (for symmetries not acting on spacetime). We also describe how QFTs can be coupled to extrinsic backgrounds for symmetry groups that differ from the intrinsic symmetry acting faithfully on the theory. Finally, we provide a variety of examples, ranging from TQFTs (gapped systems) to gapless QFTs. Along the way, we stress that the “obstruction to symmetry fractionalization” discussed in some condensed matter literature is really an instance of 2-group global symmetry.

References

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

    ADS  MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  3. C.-M. Chang et al., Topological defect lines and renormalization group flows in two dimensions, JHEP 01 (2019) 026 [arXiv:1802.04445] [INSPIRE].

    ADS  Article  Google Scholar 

  4. J.C. Baez and A.D. Lauda, Higher-dimensional algebra V: 2-groups, Theory Appl. Categ. 12 (2004) 423 [math/0307200].

  5. J. Baez and U. Schreiber, Higher gauge theory: 2-connections on 2-bundles, hep-th/0412325 [INSPIRE].

  6. J.C. Baez and U. Schreiber, Higher gauge theory, in Categories in algebra, geometry and mathematical physics, A. Davydov et al. eds., Contemporary Mathematics volume 431, AMS, Providence U.S.A. (2007) [math//0511710].

  7. U. Schreiber and K. Waldorf, Connections on non-Abelian Gerbes and their holonomy, Theory Appl. Categ. 28 (2013) 476 [arXiv:0808.1923].

    MathSciNet  MATH  Google Scholar 

  8. A. Kapustin and R. Thorngren, Higher symmetry and gapped phases of gauge theories, arXiv:1309.4721 [INSPIRE].

  9. S. Gukov and A. Kapustin, Topological quantum field theory, nonlocal operators and gapped phases of gauge theories, arXiv:1307.4793 [INSPIRE].

  10. A. Kapustin and R. Thorngren, Topological field theory on a lattice, discrete theta-angles and confinement, Adv. Theor. Math. Phys. 18 (2014) 1233 [arXiv:1308.2926] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  11. R. Thorngren and C. von Keyserlingk, Higher SPTs and a generalization of anomaly in-flow, arXiv:1511.02929 [INSPIRE].

  12. D. Gaiotto and T. Johnson-Freyd, Symmetry protected topological phases and generalized cohomology, arXiv:1712.07950 [INSPIRE].

  13. L. Bhardwaj, D. Gaiotto and A. Kapustin, State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter, JHEP 04 (2017) 096 [arXiv:1605.01640] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  14. Y. Tachikawa, On gauging finite subgroups, arXiv:1712.09542 [INSPIRE].

  15. C. Delcamp and A. Tiwari, From gauge to higher gauge models of topological phases, JHEP 10 (2018) 049 [arXiv:1802.10104] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  16. C. Córdova, T.T. Dumitrescu and K. Intriligator, Exploring 2-group global symmetries, JHEP 02 (2019) 184 [arXiv:1802.04790] [INSPIRE].

    Article  Google Scholar 

  17. E. Sharpe, Notes on generalized global symmetries in QFT, Fortsch. Phys. 63 (2015) 659 [arXiv:1508.04770] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  18. M. Bauer, G. Girardi, R. Stora and F. Thuillier, A class of topological actions, JHEP 08 (2005) 027 [hep-th/0406221] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  19. A. Kapustin and R. Thorngren, Anomalies of discrete symmetries in various dimensions and group cohomology, arXiv:1404.3230 [INSPIRE].

  20. M.B. Green and J.H. Schwarz, Anomaly cancellation in supersymmetric D = 10 gauge theory and superstring theory, Phys. Lett. B 149 (1984) 117.

  21. S. Eilenberg and S. Mac Lane, On the groups H, n), I, Ann. Math. 58 (1953) 55.

  22. M. Barkeshli, P. Bonderson, M. Cheng and Z. Wang, Symmetry, defects and gauging of topological phases, arXiv:1410.4540 [INSPIRE].

  23. 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].

  24. G. ’t Hooft, Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking, NATO Sci. Ser. B 59 (1980) 135.

  25. A. Kapustin and R. Thorngren, Anomalies of discrete symmetries in three dimensions and group cohomology, Phys. Rev. Lett. 112 (2014) 231602 [arXiv:1403.0617] [INSPIRE].

  26. A. Kitaev, On the classification of short-range entangled states, eminar at Simons Center for Geometry and Physics , June 6, Stony Brook, U.S.A. (2013).

  27. A. Kapustin, Symmetry protected topological phases, anomalies and cobordisms: beyond group cohomology, arXiv:1403.1467 [INSPIRE].

  28. D.S. Freed and M.J. Hopkins, Reflection positivity and invertible topological phases, arXiv:1604.06527 [INSPIRE].

  29. P. Etingof, D. Nikshych, V. Ostrik and E. Meir, Fusion categories and homotopy theory, Quant. Topol. 1 (2010) 209 [arXiv:0909.3140].

    MathSciNet  Article  MATH  Google Scholar 

  30. J.C.Y. Teo, T.L. Hughes and E. Fradkin, Theory of twist liquids: gauging an anyonic symmetry, Annals Phys. 360 (2015) 349 [arXiv:1503.06812] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  31. M. Barkeshli and M. Cheng, Time-reversal and spatial-reflection symmetry localization anomalies in (2 + 1)-dimensional topological phases of matter, Phys. Rev. B 98 (2018) 115129 [arXiv:1706.09464] [INSPIRE].

  32. G.W. Moore and N. Seiberg, Classical and quantum conformal field theory, Commun. Math. Phys. 123 (1989) 177 [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  33. G.W. Moore and G. Segal, D-branes and k-theory in 2D topological field theory, hep-th/0609042 [INSPIRE].

  34. Z. Komargodski, A. Sharon, R. Thorngren and X. Zhou, Comments on abelian higgs models and persistent order, SciPost Phys. 6 (2019) 003 [arXiv:1705.04786] [INSPIRE].

    ADS  Article  Google Scholar 

  35. A. Kapustin and N. Seiberg, Coupling a QFT to a TQFT and duality, JHEP 04 (2014) 001 [arXiv:1401.0740] [INSPIRE].

    ADS  Article  Google Scholar 

  36. G.W. Moore and N. Seiberg, Polynomial equations for rational conformal field theories, Phys. Lett. B 212 (1988) 451 [INSPIRE].

  37. G. Segal, Cohomology of topological groups, in Symposia Mathematica, Volume IV (INDAM, Rome, 1968/69), Academic Press, London U.K. (1970).

  38. J.-L. Brylinski, Differentiable cohomology of gauge groups, math/0011069.

  39. X. Chen, F.J. Burnell, A. Vishwanath and L. Fidkowski, Anomalous Symmetry Fractionalization and Surface Topological Order, Phys. Rev. X 5 (2015) 041013 [arXiv:1403.6491] [INSPIRE].

  40. J. Gomis, Z. Komargodski, and N. Seiberg, unpublished (2017).

  41. A. Kitaev, Anyons in an exactly solved model and beyond, Annals Phys. 321 (2006) 2 [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  42. J.H.C. Whitehead, On simply connected, 4-dimensional polyhedra, Comm. Math. Helv. 22 (1949) 48.

    MathSciNet  Article  MATH  Google Scholar 

  43. A.M. Essin and M. Hermele, Classifying fractionalization: symmetry classification of gapped2 spin liquids in two dimensions, Phys. Rev. B 87 (2013) 104406 [arXiv:1212.0593].

  44. N. Tarantino, N.H. Lindner and L. Fidkowski, Symmetry fractionalization and twist defects, New J. Phys. 18 (2016) 035006 [arXiv:1506.06754].

    ADS  MathSciNet  Article  Google Scholar 

  45. B. Bakalov and A.A. Kirillov, Lectures on tensor categories and modular functors, University Lecture Series volume 21, American Mathematical Society, U.S.A. (2001).

  46. C. Vafa, Toward classification of conformal theories, Phys. Lett. B 206 (1988) 421 [INSPIRE].

  47. A. Bernevig and T. Neupert, Topological superconductors and category theory, 2015, arXiv:1506.05805 [INSPIRE].

  48. J.M. Maldacena, G.W. Moore and N. Seiberg, D-brane charges in five-brane backgrounds, JHEP 10 (2001) 005 [hep-th/0108152] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  49. T. Banks and N. Seiberg, Symmetries and strings in field theory and gravity, Phys. Rev. D 83 (2011) 084019 [arXiv:1011.5120] [INSPIRE].

  50. F. Benini, P.-S. Hsin and N. Seiberg, Comments on global symmetries, anomalies and duality in (2 + 1)d, JHEP 04 (2017) 135 [arXiv:1702.07035] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  51. O. Aharony, F. Benini, P.-S. Hsin and N. Seiberg, Chern-Simons-matter dualities with SO and USp gauge groups, JHEP 02 (2017) 072 [arXiv:1611.07874] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  52. Z. Komargodski and N. Seiberg, A symmetry breaking scenario for QCD 3, JHEP 01 (2018) 109 [arXiv:1706.08755] [INSPIRE].

  53. C. Cordova, P.-S. Hsin and N. Seiberg, Global symmetries, counterterms and duality in Chern-Simons matter theories with orthogonal gauge groups, SciPost Phys. 4 (2018) 021 [arXiv:1711.10008] [INSPIRE].

    ADS  Article  Google Scholar 

  54. E. Witten, unpublished (2016).

  55. F. Benini, Three-dimensional dualities with bosons and fermions, JHEP 02 (2018) 068 [arXiv:1712.00020] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  56. C. Córdova, P.-S. Hsin and N. Seiberg, Time-reversal symmetry, anomalies and dualities in (2 + 1)d, SciPost Phys. 5 (2018) 006 [arXiv:1712.08639] [INSPIRE].

  57. N. Seiberg and E. Witten, Gapped boundary phases of topological insulators via weak coupling, PTEP 2016 (2016) 12C101 [arXiv:1602.04251] [INSPIRE].

  58. P.S. Hsin, unpublished (2017).

  59. C. Wang and M. Levin, Anomaly indicators for time-reversal symmetric topological orders, Phys. Rev. Lett. 119 (2017) 136801 [arXiv:1610.04624] [INSPIRE].

    ADS  Article  Google Scholar 

  60. M. Barkeshli et al., Reflection and time reversal symmetry enriched topological phases of matter: path integrals, non-orientable manifolds and anomalies, arXiv:1612.07792 [INSPIRE].

  61. A. Vishwanath and T. Senthil, Physics of three dimensional bosonic topological insulators: surface deconfined criticality and quantized magnetoelectric effect, Phys. Rev. X 3 (2013) 011016 [arXiv:1209.3058] [INSPIRE].

  62. R. Thorngren, Framed Wilson operators, fermionic strings and gravitational anomaly in 4d, JHEP 02 (2015) 152 [arXiv:1404.4385] [INSPIRE].

  63. J. Milnor, Construction of universal bundles, II, Ann. Math 63 (1965) 430.

    MathSciNet  Article  MATH  Google Scholar 

  64. A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge U.K. (2002).

  65. N.E. Steenrod, Products of cocycles and Extensions of Mappings, Ann. Math 48 (1947) 290.

    MathSciNet  Article  MATH  Google Scholar 

  66. S. Eilenberg and S. Mac Lane, On the groups H, n), II: methods of computation, Ann. Math. 60 (1954) 49.

  67. J.H.C. Whitehead, A certain exact sequence, Ann. Math. 52 (1950) 51.

    MathSciNet  Article  MATH  Google Scholar 

  68. C. Closset et al., Contact terms, unitarity and f-maximization in three-dimensional superconformal theories, JHEP 10 (2012) 053 [arXiv:1205.4142] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Francesco Benini.

Additional information

ArXiv ePrint: 1803.09336

Rights and permissions

Open Access  This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.

The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Benini, F., Córdova, C. & Hsin, PS. On 2-group global symmetries and their anomalies. J. High Energ. Phys. 2019, 118 (2019). https://doi.org/10.1007/JHEP03(2019)118

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP03(2019)118

Keywords

  • Global Symmetries
  • Anomalies in Field and String Theories
  • Chern-Simons Theories
  • Topological Field Theories