In this article we show that the conformal nets corresponding to WZW models are rational, resolving a long-standing open problem. Specifically, we show that the Jones-Wassermann subfactors associated with these models have finite index. This result was first conjectured in the early 90s but had previously only been proven in special cases, beginning with Wassermann’s landmark results in type A. The proof relies on a new framework for the systematic comparison of tensor products (a.k.a. ‘fusion’) of conformal net representations with the corresponding tensor product of vertex operator algebra modules. This framework is based on the geometric technique of ‘bounded localized vertex operators,’ which realizes algebras of observables via insertion operators localized in partially thin Riemann surfaces. We obtain a general method for showing that Jones-Wassermann subfactors have finite index, and apply it to additional families of important examples beyond WZW models. We also consider applications to a class of positivity phenomena for VOAs, and use this to outline a program for identifying unitary tensor product theories of VOAs and conformal nets even for badly-behaved models.

在本文中，我们证明了与 WZW 模型相对应的共形网络是合理的，解决了一个长期存在的开放问题。具体来说，我们表明与这些模型相关的 Jones-Wassermann 子因子具有有限指数。这个结果最初是在 90 年代初被推测出来的，但之前只在特殊情况下得到证明，从 Wassermann 在 A 类中具有里程碑意义的结果开始。证明依赖于一个用于系统比较保形张量积（也称为“融合”）的新框架网络表示与顶点算子代数模块的相应张量积。该框架基于“有界局部顶点算子”的几何技术，该技术通过局部薄黎曼曲面中的插入算子实现可观测量的代数。我们获得了证明 Jones-Wassermann 子因子具有有限指数的通用方法，并将其应用于 WZW 模型之外的其他重要示例系列。我们还考虑了 VOA 的一类正现象的应用，并用它来概述一个程序，用于识别 VOA 的酉张量积理论和共形网络，甚至对于行为不良的模型也是如此。