标题： 计算 $G$-交叉扩张和顶点算子代数的 orbifolds 显示英文标题

标题： Computing $G$-Crossed Extensions and Orbifolds of Vertex Operator Algebras

摘要： 本文中，我们发展了计算辫子张量范畴的$G$-交叉扩张的工具。 它们的等变化出现在顶点算子代数的不动点子代数（或orbifold）的模范畴中，并且通常难以确定。 作为第一个工具，我们展示了Etingof、Nikshych和Ostrik关于$G$-交叉扩张唯一性的开创性工作如何能够用于确定orbifold顶点算子代数的模范畴。 作为一个应用，我们确定了具有奇阶判别形式的格顶点算子代数在$-\mathrm{id}$的提升下的orbifold的模张量范畴。 在这种情况下，去等变化属于Tambara-Yamagami类型。 作为第二个工具，我们描述了$G$-交叉扩张与交换代数的凝聚在某种意义上是如何相容的。 这导致了一种有效的方法来计算新的$G$-交叉扩张。 作为一个应用，我们生成了在arXiv:2411.12251中用于定义具有多个简单对象的扭曲部分的Tambara-Yamagami范畴推广的协调数据。 这也给出了任意格顶点算子代数在$-\mathrm{id}$提升下的orbifold的模张量范畴。 最后，我们概述了如何用范畴论方法处理任意格自同构提升下的格orbifold的一般问题。 显示英文摘要

摘要： In this article, we develop tools for computing $G$-crossed extensions of braided tensor categories. Their equivariantisations appear as categories of modules of fixed-point subalgebras (or orbifolds) of vertex operator algebras and are often difficult to determine. As the first tool, we show how the seminal work of Etingof, Nikshych and Ostrik on the uniqueness of $G$-crossed extensions can be used to determine the category of modules of orbifold vertex operator algebras. As an application, we determine the modular tensor category of the orbifold of a lattice vertex operator algebra under a lift of $-\mathrm{id}$ for a lattice with odd-order discriminant form. In that case, the de-equivariantisation is of Tambara-Yamagami type. As the second tool, we describe how $G$-crossed extensions and condensations by commutative algebras commute in a suitable sense. This leads to an effective approach to compute new $G$-crossed extensions. As one application, we produce the coherence data that are then used in arXiv:2411.12251 to define a generalisation of the Tambara-Yamagami categories with more than one simple object in the twisted sector. This also yields the modular tensor category of the orbifold of an arbitrary lattice vertex operator algebra under a lift of $-\mathrm{id}$. Finally, we sketch how to categorically approach the general problem of lattice orbifolds under lifts of arbitrary lattice involutions.