Comparison of Extensions of Unitary Vertex Operator Algebras and Conformal Nets

Let $V$ be one of the following unitary strongly-rational VOAs: unitary WZW models, discrete series W-algebras of type ADE, even lattice VOAs, parafermion VOAs, their tensor products, and their strongly-rational cosets. Let $U$ be a (unitary) VOA extension of $V$, described by a Q-system $Q$. We prove that $U$ is strongly local. Let $\mathcal A_V,\mathcal A_U$ be the conformal nets associated to $V,U$ in the sense of Carpi-Kawahigashi-Longo-Weiner (CKLW). We prove that $\mathcal A_U$ is canonically isomorphic to the conformal net extension of $\mathcal A_V$ defined by the Q-system $Q$. We prove that all unitary $U$-modules are strongly integrable in the sense of Carpi-Weiner-Xu (CWX). We show that the CWX $*$-functor from the $C^*$-category of unitary $U$-modules to the $C^*$-category of finite-index $\mathcal A_U$-modules is naturally isomorphic to $*$-functor defined by $Q$.