Communications in Mathematical Physics

, Volume 369, Issue 1, pp 245–259 | Cite as

Vanishing of Categorical Obstructions for Permutation Orbifolds

  • Terry Gannon
  • Corey JonesEmail author


The orbifold construction \({A\mapsto A^G}\) for a finite group G is fundamental in rational conformal field theory. The construction of Rep(AG) from Rep(A) on the categorical level, often called gauging, is also prominent in the study of topological phases of matter. Given a non-degenerate braided fusion category \({\mathcal{C}}\) with a G-action, the key step in this construction is to find a braided G-crossed extension compatible with the action. The extension theory of Etingof–Nikshych–Ostrik gives two obstructions for this problem, \({o_3\in H^3(G)}\) and \({o_4\in H^4(G)}\) for certain coefficients, the latter of which depends on a categorical lifting of the action and is notoriously difficult to compute. We show that in the case where \({G\le S_n}\) acts by permutations on \({\mathcal{C}^{\boxtimes n}}\), both of these obstructions vanish. This verifies a conjecture of Müger, and constitutes a nontrivial test of the conjecture that all modular tensor categories come from vertex operator algebras or conformal nets.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors would like to thank Michael Müger, Andrew Schopieray and Zhenghan Wang for helpful comments. This research was conducted while the first author was visiting the Australian National University. The first author was supported by an NSERC Discovery grant. The second author was supported by Discovery Projects subfactors and symmetries DP140100732 and Low dimensional categories DP160103479 from the Australian Research Council.


  1. 1.
    Adem A., Maginnis J., Milgram R.J.: Symmetric invariants and cohomology of groups. Math. Ann. 287(3), 391–411 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Adem, A., Milgram, R.J.: Cohomology of Finite Groups, Volume 309 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1994)Google Scholar
  3. 3.
    Bantay P.: Characters and modular properties of permutation orbifolds.. Phys. Lett. B 419(1–4), 175–178 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bantay P.: Frobenius-Schur indicators, the Klein-bottle amplitude, and the principle of orbifold covariance. Phys. Lett. B. 488(2), 207–210 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Barkeshli, M., Bonderson, P., Cheng, M., Wang, Z.: Symmetry, defects, and gauging of topological phases (2014) arXiv:1410.4540
  6. 6.
    Barmeier T., Schweigert C.: A geometric construction for permutation equivariant categories from modular functors. Transform. Groups 16(2), 287–337 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Borisov L., Halpern M.B., Schweigert C.: Systematic approach to cyclic orbifolds.. Int. J. Modern Phys. A 13(1), 125–168 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Brown, K.S.: Cohomology of Groups, Volume 87 of Graduate Texts in Mathematics. Springer, New York (1982)Google Scholar
  9. 9.
    Cui S.X., Galindo C., Plavnik J.Y., Wang Z.: On gauging symmetry of modular categories. Commun. Math. Phys. 348(3), 1043–1064 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Davydov Alexei, Nikshych Dmitri: The Picard crossed module of a braided tensor category. Algebra Number Theory. 7(6), 1365–1403 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Drinfeld, V., Gelaki, S., Nikshych, D., Ostrik, V.: On braided fusion categories. I. Selecta Math. (N.S.), 16(1) 1–119, (2010) arXiv:0906.0620
  12. 12.
    Edie-Michell, C.: Equivalences of graded categories (2017) arXiv:1711.00645
  13. 13.
    Edie-Michell, C., Jones, C., Plavnik, J. Y.: Fusion rules for \({\mathbb{Z}/2\mathbb{Z}}\) permutation gauging (2017) arXiv:1804.01657
  14. 14.
    Etingof, P., Gelaki, S., Nikshych, D., Ostrik, V.: Tensor categories.
  15. 15.
    Etingof, P., Gelaki, S., Nikshych, D., Ostrik, V.: Tensor Categories, Volume 205 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2015)
  16. 16.
    Etingof, P., Nikshych, D., Ostrik, V.: Fusion categories and homotopy theory. Quantum Topol. 1(3):209–273 (2010) (with an appendix by Ehud Meir). arXiv:0909.3140
  17. 17.
    Evans, D., Gannon, T.: Reconstruction and local extensions for twisted group doubles, and permutation orbifolds. arXiv:1804.11145
  18. 18.
    Galindo, C., Venegas-Ramírez, C. F.: Categorical fermionic actions and minimal modular extensions (2017) arXiv:1712.07097
  19. 19.
    Johnson-Freyd, T.: The moonshine anomaly (2017). arXiv:1707.08388
  20. 20.
    Kac V. G., Longo R., Xu F.: Solitons in affine and permutation orbifolds. Commun. Math. Phys. 253(3), 723–764 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Müger M.: Conformal orbifold theories and braided crossed G-categories. Commun. Math. Phys. 260(3), 727–762 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Nakaoka M.: Decomposition theorem for homology groups of symmetric groups. Ann. of Math. (2) 71, 16–42 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Turaev, V.: Homotopy Quantum Field Theory, Volume 10 of EMS Tracts in Mathematics. European Mathematical Society (EMS), Zürich (2010). Appendix 5 by Michael Müger and Appendices 6 and 7 by Alexis Virelizier Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of AlbertaEdmonton, AlbertaCanada
  2. 2.Ohio State UniversityColumbusUSA

Personalised recommendations