Vanishing of Categorical Obstructions for Permutation Orbifolds

Abstract

The orbifold construction A↦ AG 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 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, o3∈ H3(G) and o4∈ H4(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≤ Sn acts by permutations on 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.