论文标题
3D镜像对称和$βγ$ VOA
3d Mirror Symmetry and the $βγ$ VOA
论文作者
论文摘要
我们研究了3D $ \ MATHCAL n = 4 $ SUSY GAUGE理论的最简单的镜子对称性示例：免费的Hypermultiplet的A TWIST和SQED的B-Twist。我们特别关注每个理论中的线路运营商类别。使用Costello-Gaiotto的工作，我们将这些类别定义为适当的模块类别，用于每个理论中存在的边界顶点操作员代数。对于自由hyper的a圈，这将是$βγ$ voa的一定类别的模块，适当地包含艾伦 - 伍德先前研究的类别。应用Creutzig-Kanade-McRae和Creutzig-McRae-Yang的工作，我们表明，A侧的线算子的类别具有编织张量的类别的结构，从而扩展了艾伦林的结果。此外，我们证明，A侧和B侧的线路运算符类别之间存在编织的张量等效性，完成了3D镜像对称性猜想的非平凡检查。由于这种等效性，我们得出了明确的融合规则，并与相关的量子组表示有有趣的关系。
We study the simplest example of mirror symmetry for 3d $\mathcal N=4$ SUSY gauge theories: the A-twist of a free hypermultiplet and the B-twist of SQED. We particularly focus on the category of line operators in each theory. Using the work of Costello-Gaiotto, we define these categories as appropriate categories of modules for the boundary vertex operator algebras present in each theory. For the A-twist of a free hyper, this will be a certain category of modules for the $βγ$ VOA, properly containing the category previously studied by Allen-Wood. Applying the work of Creutzig-Kanade-McRae and Creutzig-McRae-Yang, we show that the category of line operators on the A side possesses the structure of a braided tensor category, extending the result of Allen-Wood. In addition, we prove that there is a braided tensor equivalence between the categories of line operators on the A side and B side, completing a non-trivial check of the 3d mirror symmetry conjecture. We derive explicit fusion rules as a consequence of this equivalence and obtain interesting relations with associated quantum group representations.
