A remark about the anomalies of cyclic holomorphic permutation orbifolds

Abstract

Using a result of Longo and Xu, we show that the anomaly arising from a cyclic permutation orbifold of order 3 of a holomorphic conformal net $\mathcal{A}$ with central charge $c=8k$ depends on the "gravitational anomaly" $k(\mathrm{mod}3)$. In particular, the conjecture that holomorphic permutation orbifolds are non-anomalous and therefore a stronger conjecture of Müger about braided crossed $S_n$-categories arising from permutation orbifolds of completely rational conformal nets are wrong. More general, we show that cyclic permutations of order $n$ are non-anomalous if and only if $3\nmid n$ or $24|c$. We also show that all cyclic permutation gaugings of $\mathrm{Rep}(\mathcal{A})$ arise from conformal nets.

11 pages, 2 figures. Extension of an unpublished note. Comments are welcome!