Comparison of Extensions of Unitary Vertex Operator Algebras and Conformal Nets
Bin Gui
TL;DR
The paper builds a unified framework linking unitary VOA extensions and conformal nets through categorical tools, notably haploid commutative $C^*$-Frobenius algebras (Q-systems). It proves that any unitary extension $U$ of a strongly local, completely unitary VOA $V$ is itself strongly local and that the CKLW-net extensions $ cal A_V$ and $ cal A_U$ correspond canonically via the Q-system, with unitary $U$-modules being strongly integrable. By translating VOA data into categorical extensions and employing the CWX functor along with the Wassermann tensorator, the authors establish equivalences between the representation categories of VOAs and those of conformal nets, and they formulate a main comparison theorem that gives a commutative diagram of braided $*$-functors. These results lead to a precise algebraic-analytic correspondence between VOA and net extensions, enabling surjectivity results for CWX functors and clarifying how extensions preserve and reflect braiding structures in both settings. The findings significantly advance the understanding of 2D rational CFTs by unifying analytic locality with categorical extension theory and demonstrating that algebraic extensions in VOAs align with finite-index net extensions in conformal field theory.
Abstract
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 $Q$.