Comparison of Extensions of Unitary Vertex Operator Algebras and Conformal Nets

TL;DR

This paper establishes a correspondence between unitary VOA extensions and conformal net extensions, proving strong locality, integrability of modules, and functor isomorphisms in the context of various rational VOAs.

Contribution

It demonstrates that VOA extensions described by Q-systems correspond to conformal net extensions, with all modules being strongly integrable and functor equivalences established.

Findings

U is strongly local for various rational VOAs

Conformal net extension is canonically isomorphic to VOA extension defined by Q-system

All unitary U-modules are strongly integrable

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 $Q$.