On finite symmetries and their gauging in two dimensions

  • Lakshya Bhardwaj
  • Yuji Tachikawa
Open Access
Regular Article - Theoretical Physics


It is well-known that if we gauge a ℤn symmetry in two dimensions, a dual ℤn symmetry appears, such that re-gauging this dual ℤn symmetry leads back to the original theory. We describe how this can be generalized to non-Abelian groups, by enlarging the concept of symmetries from those defined by groups to those defined by unitary fusion categories. We will see that this generalization is also useful when studying what happens when a non-anomalous subgroup of an anomalous finite group is gauged: for example, the gauged theory can have non-Abelian group symmetry even when the original symmetry is an Abelian group. We then discuss the axiomatization of two-dimensional topological quantum field theories whose symmetry is given by a category. We see explicitly that the gauged version is a topological quantum field theory with a new symmetry given by a dual category.


Anyons Discrete Symmetries Global Symmetries Topological Field Theories 


Open Access

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  1. [1]
    C. Vafa, Quantum Symmetries of String Vacua, Mod. Phys. Lett. A 4 (1989) 1615 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized Global Symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    I. Brunner, N. Carqueville and D. Plencner, Discrete torsion defects, Commun. Math. Phys. 337 (2015) 429 [arXiv:1404.7497] [INSPIRE].
  4. [4]
    E. Sharpe, Notes on generalized global symmetries in QFT, Fortsch. Phys. 63 (2015) 659 [arXiv:1508.04770] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, Theta, Time Reversal and Temperature, JHEP 05 (2017) 091 [arXiv:1703.00501] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    M. Bischoff, R. Longo, Y. Kawahigashi and K.-H. Rehren, Tensor categories of endomorphisms and inclusions of von Neumann algebras, arXiv:1407.4793 [INSPIRE].
  7. [7]
    P. Etingof, S. Gelaki, D. Nikshych and V. Ostrik, Tensor categories, Mathematical Surveys and Monographs, vol. 205, AMS, Providence, RI (2015).Google Scholar
  8. [8]
    N. Carqueville and I. Runkel, Orbifold completion of defect bicategories, Quantum Topol. 7 (2016) 203 [arXiv:1210.6363] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    I. Brunner, N. Carqueville and D. Plencner, A quick guide to defect orbifolds, Proc. Symp. Pure Math. 88 (2014) 231 [arXiv:1310.0062] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    G.W. Moore and G. Segal, D-branes and k-theory in 2D topological field theory, hep-th/0609042 [INSPIRE].
  11. [11]
    G.W. Moore and N. Seiberg, Lectures on RCFT, in Strings ’89, Proceedings of the Trieste Spring School on Superstrings, World Scientific (1990) [] [INSPIRE].
  12. [12]
    R. Dijkgraaf and E. Witten, Topological Gauge Theories and Group Cohomology, Commun. Math. Phys. 129 (1990) 393 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    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].ADSCrossRefGoogle Scholar
  14. [14]
    M.R. Douglas, D-branes and discrete torsion, hep-th/9807235 [INSPIRE].
  15. [15]
    D. Tambara and S. Yamagami, Tensor categories with fusion rules of self-duality for finite abelian groups, J. Algebra 209 (1998) 692.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    P. Etingof and S. Gelaki, Isocategorical groups, Int. Math. Res. Not. (2001) 59, [math/0007196].
  17. [17]
    M. Izumi and H. Kosaki, On a subfactor analogue of the second cohomology, Rev. Math. Phys. 14 (2002) 733.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    J. Fuchs, I. Runkel and C. Schweigert, TFT construction of RCFT correlators 1. Partition functions, Nucl. Phys. B 646 (2002) 353 [hep-th/0204148] [INSPIRE].
  19. [19]
    G. Schaumann, Traces on module categories over fusion categories, J. Algebra 379 (2013) 382 [arXiv:1206.5716].MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    P. Etingof, D. Nikshych and V. Ostrik, Fusion categories and homotopy theory, Quantum Topol. 1 (2010) 209 [arXiv:0909.3140].MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    L. Kong, Anyon condensation and tensor categories, Nucl. Phys. B 886 (2014) 436 [arXiv:1307.8244] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    I. Brunner, N. Carqueville and D. Plencner, Orbifolds and topological defects, Commun. Math. Phys. 332 (2014) 669 [arXiv:1307.3141] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    G.W. Moore and N. Seiberg, Classical and Quantum Conformal Field Theory, Commun. Math. Phys. 123 (1989) 177 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    V. Ostrik, Fusion categories of rank 2, Math. Res. Lett. 10 (2003) 177 [math/0203255].
  26. [26]
    V. Ostrik, Pivotal fusion categories of rank 3, Mosc. Math. J. 15 (2015) 373 [arXiv:1309.4822].MathSciNetzbMATHGoogle Scholar
  27. [27]
    J. Fröhlich, J. Fuchs, I. Runkel and C. Schweigert, Defect lines, dualities and generalised orbifolds, in Proceedings, 16th International Congress on Mathematical Physics (ICMP09), Prague, Czech Republic, August 3-8, 2009 [DOI:] [arXiv:0909.5013] [INSPIRE].
  28. [28]
    V. Ostrik, Module categories, weak Hopf algebras and modular invariants, Transform. Groups 8 (2003) 177 [math/0111139].
  29. [29]
    D. Naidu, Categorical Morita equivalence for group-theoretical categories, Comm. Algebra 35 (2007) 3544 [math/0605530].
  30. [30]
    D. Naidu and D. Nikshych, Lagrangian subcategories and braided tensor equivalences of twisted quantum doubles of finite groups, Commun. Math. Phys. 279 (2008) 845 [arXiv:0705.0665].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    B. Uribe, On the classification of pointed fusion categories up to weak morita equivalence, arXiv:1511.05522.
  32. [32]
    P. Etingof, S. Gelaki and V. Ostrik, Classification of fusion categories of dimension pq, Int. Math. Res. Not. 57 (2004) 3041 [math/0304194].
  33. [33]
    H.I. Blau, Fusion rings with few degrees, J. Algebra 396 (2013) 220.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    G.I. Kac and V.G. Paljutkin, Finite ring groups, Trans. Moscow Math. Soc. (1966) 251 [Tr. Mosk. Mat. Obs. 15 (1966) 224].Google Scholar
  35. [35]
    E. Meir and E. Musicantov, Module categories over graded fusion categories, J. Pure Appl. Algebra 216 (2012) 2449 [arXiv:1010.4333].MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    M. Buican and A. Gromov, Anyonic Chains, Topological Defects and Conformal Field Theory, Commun. Math. Phys. 356 (2017) 1017 [arXiv:1701.02800] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    D. Handel, On products in the cohomology of the dihedral groups, Tohoku Math. J. (2) 45 (1993) 13.Google Scholar
  38. [38]
    M.D.F. de Wild Propitius, Topological interactions in broken gauge theories, Ph.D. Thesis, University of Amsterdam (1995) [hep-th/9511195] [INSPIRE].
  39. [39]
    T. Hayami and K. Sanada, Cohomology ring of the generalized quaternion group with coefficients in an order, Commun. Algebra 30 (2002) 3611.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  2. 2.Kavli Institute for the Physics and Mathematics of the UniverseUniversity of TokyoKashiwaJapan

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