# A remark about the anomalies of cyclic holomorphic permutation orbifolds

## TL;DR

This paper investigates anomalies in cyclic permutation orbifolds of holomorphic conformal nets, revealing their dependence on the gravitational anomaly and disproving previous conjectures about their non-anomalous nature.

## Contribution

It demonstrates that cyclic permutation orbifolds are anomalous depending on the central charge and provides conditions for non-anomalous cases, challenging existing conjectures.

## Key findings

- Anomalies depend on the gravitational anomaly mod 3.
- Cyclic permutations are non-anomalous iff 3 does not divide n or 24 divides c.
- All cyclic permutation gaugings originate from conformal nets.

## 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" $k\pmod 3$. In particular, the conjecture that holomorphic permutation orbifolds are non-anomalous and therefore a stronger conjecture of M\"uger 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.