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Vanishing of Categorical Obstructions for Permutation Orbifolds

  • Terry Gannon
  • Corey JonesEmail author
Article
  • 7 Downloads

Abstract

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.

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Notes

Acknowledgement

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.

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

Authors and Affiliations

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

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