Bisognano–Wichmann Property for Rigid Categorical Extensions and Non-local Extensions of Conformal Nets

Given an (irreducible) Mobius covariant net $\mathcal A$, we prove a Bisognano-Wichmann theorem for its categorical extension $\mathscr E^{\textrm{d}}$ associated to the braided $C^*$-tensor category $\textrm{Rep}^{\textrm{d}}(\mathcal A)$ of dualizable (more precisely "dualized") A$-modules. As closely related result, (modified) any (possibly) non-local A$ obtained by $C^*$-Frobenius algebra $Q$ in A)$. application, discuss relation between domains modular operators and preclosedness certain unbounded E^{\textrm{d}}$.