Journal of High Energy Physics

, 2015:172 | Cite as

Generalized global symmetries

  • Davide Gaiotto
  • Anton KapustinEmail author
  • Nathan Seiberg
  • Brian Willett
Open Access
Regular Article - Theoretical Physics


A q-form global symmetry is a global symmetry for which the charged operators are of space-time dimension q; e.g. Wilson lines, surface defects, etc., and the charged excitations have q spatial dimensions; e.g. strings, membranes, etc. Many of the properties of ordinary global symmetries (q = 0) apply here. They lead to Ward identities and hence to selection rules on amplitudes. Such global symmetries can be coupled to classical background fields and they can be gauged by summing over these classical fields. These generalized global symmetries can be spontaneously broken (either completely or to a sub-group). They can also have ’t Hooft anomalies, which prevent us from gauging them, but lead to ’t Hooft anomaly matching conditions. Such anomalies can also lead to anomaly inflow on various defects and exotic Symmetry Protected Topological phases. Our analysis of these symmetries gives a new unified perspective of many known phenomena and uncovers new results.


Global Symmetries Wilson ’t Hooft and Polyakov loops Topological States of Matter Anomalies in Field and String Theories 


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.


  1. [1]
    M. Kalb and P. Ramond, Classical direct interstring action, Phys. Rev. D 9 (1974) 2273 [INSPIRE].ADSGoogle Scholar
  2. [2]
    J. Villain, Theory of one-dimensional and two-dimensional magnets with an easy magnetization plane. 2. The planar, classical, two-dimensional magnet, J. Phys. (France) 36 (1975) 581 [INSPIRE].CrossRefGoogle Scholar
  3. [3]
    R. Savit, Topological excitations in U(1) invariant theories, Phys. Rev. Lett. 39 (1977) 55 [INSPIRE].CrossRefADSGoogle Scholar
  4. [4]
    P. Orland, Instantons and disorder in antisymmetric tensor gauge fields, Nucl. Phys. B 205 (1982) 107 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  5. [5]
    C. Teitelboim, Gauge invariance for extended objects, Phys. Lett. B 167 (1986) 63 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  6. [6]
    C. Teitelboim, Monopoles of higher rank, Phys. Lett. B 167 (1986) 69 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  7. [7]
    J. Cheeger and J. Simons, Differential characters and geometric invariants, Lect. Notes Math. 1167 (1985) 50.CrossRefMathSciNetGoogle Scholar
  8. [8]
    P. Deligne, Théorie de Hodge II, Publ. Math. Inst. Hautes Études Sci. 40 (1971) 5.CrossRefzbMATHMathSciNetGoogle Scholar
  9. [9]
    A. Beilinson, Higher regulators and values of L-functions, J. Sov. Math. 30 (1985) 2036.CrossRefzbMATHGoogle Scholar
  10. [10]
    A. Kapustin and R. Thorngren, Higher symmetry and gapped phases of gauge theories, arXiv:1309.4721 [INSPIRE].
  11. [11]
    A. Kapustin and N. Seiberg, Coupling a QFT to a TQFT and duality, JHEP 04 (2014) 001 [arXiv:1401.0740] [INSPIRE].CrossRefADSGoogle Scholar
  12. [12]
    Z. Nussinov and G. Ortiz, A symmetry principle for topological quantum order, Annals Phys. 324 (2009) 977 [cond-mat/0702377].CrossRefADSzbMATHMathSciNetGoogle Scholar
  13. [13]
    T. Banks and N. Seiberg, Symmetries and strings in field theory and gravity, Phys. Rev. D 83 (2011) 084019 [arXiv:1011.5120] [INSPIRE].ADSGoogle Scholar
  14. [14]
    T.T. Dumitrescu and N. Seiberg, Supercurrents and brane currents in diverse dimensions, JHEP 07 (2011) 095 [arXiv:1106.0031] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  15. [15]
    O. Aharony, N. Seiberg and Y. Tachikawa, Reading between the lines of four-dimensional gauge theories, JHEP 08 (2013) 115 [arXiv:1305.0318] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  16. [16]
    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].CrossRefMathSciNetGoogle Scholar
  17. [17]
    J.D. Bjorken, A dynamical origin for the electromagnetic field, Annals Phys. 24 (1963) 174 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  18. [18]
    E. Witten, SL(2, Z) action on three-dimensional conformal field theories with Abelian symmetry, in From fields to strings, vol. 2, M. Shifman et al. eds., World Scientific, Singapore (2005), pg. 1173 [hep-th/0307041] [INSPIRE].
  19. [19]
    R. Dijkgraaf and E. Witten, Topological gauge theories and group cohomology, Commun. Math. Phys. 129 (1990) 393 [INSPIRE].CrossRefADSzbMATHMathSciNetGoogle Scholar
  20. [20]
    F. Cachazo, N. Seiberg and E. Witten, Phases of N = 1 supersymmetric gauge theories and matrices, JHEP 02 (2003) 042 [hep-th/0301006] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  21. [21]
    D.S. Freed, G.W. Moore and G. Segal, The uncertainty of fluxes, Commun. Math. Phys. 271 (2007) 247 [hep-th/0605198] [INSPIRE].CrossRefADSzbMATHMathSciNetGoogle Scholar
  22. [22]
    D.S. Freed, G.W. Moore and G. Segal, Heisenberg groups and noncommutative fluxes, Annals Phys. 322 (2007) 236 [hep-th/0605200] [INSPIRE].CrossRefADSzbMATHMathSciNetGoogle Scholar
  23. [23]
    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].CrossRefADSGoogle Scholar
  24. [24]
    A. Davydov, L. Kong and I. Runkel, Invertible defects and isomorphisms of rational CFTs, Adv. Theor. Math. Phys. 15 (2011) [arXiv:1004.4725] [INSPIRE].
  25. [25]
    I. Brunner, N. Carqueville and D. Plencner, Discrete torsion defects, arXiv:1404.7497 [INSPIRE].
  26. [26]
    C. Vafa, Modular invariance and discrete torsion on orbifolds, Nucl. Phys. B 273 (1986) 592 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  27. [27]
    X. Chen, Z.-C. Gu, Z.-X. Liu and X.-G. Wen, Symmetry protected topological orders and the group cohomology of their symmetry group, Phys. Rev. B 87 (2013) 155114 [arXiv:1106.4772] [INSPIRE].CrossRefADSGoogle Scholar
  28. [28]
    A. Kapustin, Symmetry protected topological phases, anomalies and cobordisms: beyond group cohomology, arXiv:1403.1467 [INSPIRE].
  29. [29]
    A. Kapustin, R. Thorngren, A. Turzillo and Z. Wang, Fermionic symmetry protected topological phases and cobordisms, arXiv:1406.7329 [INSPIRE].
  30. [30]
    Z.-C. Gu and X.-G. Wen, Symmetry-protected topological orders for interacting fermions: fermionic topological nonlinear σ models and a special group supercohomology theory, Phys. Rev. B 90 (2014) 115141 [arXiv:1201.2648] [INSPIRE].CrossRefADSGoogle Scholar
  31. [31]
    D.S. Freed, Short-range entanglement and invertible field theories, arXiv:1406.7278 [INSPIRE].
  32. [32]
    A. Kapustin and R. Thorngren, Anomalies of discrete symmetries in various dimensions and group cohomology, arXiv:1404.3230 [INSPIRE].
  33. [33]
    S. Gukov and E. Witten, Gauge theory, ramification, and the geometric Langlands program, hep-th/0612073 [INSPIRE].
  34. [34]
    S. Gukov and E. Witten, Rigid surface operators, Adv. Theor. Math. Phys. 14 (2010) [arXiv:0804.1561] [INSPIRE].
  35. [35]
    J.M. Maldacena, G.W. Moore and N. Seiberg, D-brane charges in five-brane backgrounds, JHEP 10 (2001) 005 [hep-th/0108152] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  36. [36]
    N. Seiberg, Modifying the sum over topological sectors and constraints on supergravity, JHEP 07 (2010) 070 [arXiv:1005.0002] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  37. [37]
    Z. Komargodski and N. Seiberg, Comments on the Fayet-Iliopoulos term in field theory and supergravity, JHEP 06 (2009) 007 [arXiv:0904.1159] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  38. [38]
    N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [Erratum ibid. B 430 (1994) 485] [hep-th/9407087] [INSPIRE].
  39. [39]
    N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  40. [40]
    A. Kapustin, Wilson-t Hooft operators in four-dimensional gauge theories and S-duality, Phys. Rev. D 74 (2006) 025005 [hep-th/0501015] [INSPIRE].ADSMathSciNetGoogle Scholar
  41. [41]
    E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].CrossRefADSzbMATHMathSciNetGoogle Scholar
  42. [42]
    D. Gepner, Foundations of rational quantum field theory. 1, hep-th/9211100 [INSPIRE].
  43. [43]
    M. Mariño and C. Vafa, Framed knots at large-N, Contemp. Math. 310 (2002) 185 [hep-th/0108064] [INSPIRE].CrossRefGoogle Scholar
  44. [44]
    D. Gaiotto, G.W. Moore and A. Neitzke, Framed BPS states, Adv. Theor. Math. Phys. 17 (2013) 241 [arXiv:1006.0146] [INSPIRE].CrossRefzbMATHMathSciNetGoogle Scholar
  45. [45]
    G. Moore, Lecture notes for Felix Klein lectures,
  46. [46]
    E. Witten, Geometric Langlands from six dimensions, arXiv:0905.2720 [INSPIRE].
  47. [47]
    N. Seiberg and W. Taylor, Charge lattices and consistency of 6D supergravity, JHEP 06 (2011) 001 [arXiv:1103.0019] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  48. [48]
    E.H. Fradkin and S.H. Shenker, Phase diagrams of lattice gauge theories with Higgs fields, Phys. Rev. D 19 (1979) 3682 [INSPIRE].ADSGoogle Scholar
  49. [49]
    T. Banks and E. Rabinovici, Finite temperature behavior of the lattice Abelian Higgs model, Nucl. Phys. B 160 (1979) 349 [INSPIRE].CrossRefADSGoogle Scholar
  50. [50]
    J.H.C. Whitehead, On simply connected, 4-dimensional polyhedra, Comm. Math. Helv. 22 (1949) 48.CrossRefzbMATHMathSciNetGoogle Scholar
  51. [51]
    S.M. Kravec and J. McGreevy, A gauge theory generalization of the fermion-doubling theorem, Phys. Rev. Lett. 111 (2013) 161603 [arXiv:1306.3992] [INSPIRE].CrossRefADSGoogle Scholar
  52. [52]
    P. Novotný and J. Hrivnák, On orbits of the ring \( {\mathrm{\mathbb{Z}}}_m^n \) under the action of the group SL(m, \( {\mathrm{\mathbb{Z}}}_m \)), arXiv:0710.0326.
  53. [53]
    N. Seiberg, Electric-magnetic duality in supersymmetric non-Abelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  54. [54]
    K.A. Intriligator and N. Seiberg, Duality, monopoles, dyons, confinement and oblique confinement in supersymmetric SO(N c) gauge theories, Nucl. Phys. B 444 (1995) 125 [hep-th/9503179] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  55. [55]
    E. Witten, Lecture II-9, in Quantum fields and strings: a course for mathematicians. Vol. 1 and 2, P. Deligne et al. eds., AMS, Providence U.S.A. (1999).Google Scholar
  56. [56]
    M. Dierigl and A. Pritzel, Topological model for domain walls in (super-)Yang-Mills theories, Phys. Rev. D 90 (2014) 105008 [arXiv:1405.4291] [INSPIRE].ADSGoogle Scholar
  57. [57]
    B.S. Acharya and C. Vafa, On domain walls of N = 1 supersymmetric Yang-Mills in four-dimensions, hep-th/0103011 [INSPIRE].
  58. [58]
    S. Gukov and A. Kapustin, Topological quantum field theory, nonlocal operators and gapped phases of gauge theories, arXiv:1307.4793 [INSPIRE].
  59. [59]
    R.J. Milgram, Surgery with coefficients, Ann. Math. 100 (1974) 194.CrossRefzbMATHMathSciNetGoogle Scholar
  60. [60]
    S.S. Razamat and B. Willett, Global properties of supersymmetric theories and the lens space, arXiv:1307.4381 [INSPIRE].

Copyright information

© The Author(s) 2015

Authors and Affiliations

  • Davide Gaiotto
    • 1
  • Anton Kapustin
    • 2
    Email author
  • Nathan Seiberg
    • 3
  • Brian Willett
    • 3
  1. 1.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  2. 2.Simons Center for Geometry and PhysicsState University of New YorkStony BrookU.S.A.
  3. 3.School of Natural SciencesInstitute for Advanced StudyPrincetonU.S.A.

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