Balanced tensor categories of representations of fixed-points conformal nets

Abstract

Let $\mathcal{A}$ be a (not necessarily rational) conformal net with a faithful action of a finite group $G$. Let ${\text{Rep}}^G(\mathcal{A})$ be the $G$-crossed balanced ${\mathrm{W}}^{\ast }$-tensor category of $G$-twisted representations of $\mathcal{A}$ as introduced in arXiv:2606.03623. We show that there is an equivalence of balanced ${\mathrm{W}}^{\ast }$-tensor categories $({\text{Rep}}^G(\mathcal{A}){)}^G\cong \text{Rep}({\mathcal{A}}^G)$ between the $G$-equivariantization of ${\text{Rep}}^G(\mathcal{A})$ and the category of representations of the fixed-points conformal net ${\mathcal{A}}^G$. This generalizes to the non-rational case the equivalence of braided tensor categories $({\text{Rep}}^G(\mathcal{A}){)}^G\cong \text{Rep}({\mathcal{A}}^G)$ for $\mathcal{A}$ rational appearing (in the language of localized endomorphisms) in arXiv:math/0403322, and it also includes the balances.

50 pages, references updated