Vanishing of categorical obstructions for permutation orbifolds

TL;DR

This paper proves that for permutation group actions on braided fusion categories, certain categorical obstructions vanish, confirming a conjecture related to the construction of orbifold theories in conformal field theory.

Contribution

It demonstrates the vanishing of key obstructions in the permutation case, supporting the conjecture that all modular tensor categories originate from vertex operator algebras or conformal nets.

Findings

Both obstructions o_3 and o_4 vanish for permutation actions.

Verifies M"uger's conjecture in the permutation case.

Supports the idea that all modular tensor categories may come from VOAs or conformal nets.

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 $\mathcal{C}^{\boxtimes n}$, both of these obstructions vanish. This verifies a conjecture of M\"uger, and constitutes a nontrivial test of the conjecture that all modular tensor categories come from vertex operator algebras or conformal nets.