Vanishing of categorical obstructions for permutation orbifolds

Abstract

The orbifold construction $A\mapsto A^G$ for a finite group $G$ is fundamental in rational conformal field theory. The construction of $Rep(A^G)$ 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 depending 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 ${\mathcal{C}}^{⊠n}$, 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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References in corpus (6)

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Cited by in corpus (6)

• A mathematical theory of gapless edges of 2d topological orders. Part I
• Bosonic Rational Conformal Field Theories in Small Genera, Chiral Fermionization, and Symmetry/Subalgebra Duality
• Galois Orbits of TQFTs: Symmetries and Unitarity
• Fusion Rules for $\mathbb{Z}/2\mathbb{Z}$ Permutation Gauging
• Anomalies in mirror symmetry enriched topological orders
• Modular ${\mathbb{Z}}_2$-Crossed Tambara-Yamagami-like Categories for Even Groups