We review and extend the vertex algebra framework linking gauge theory constructions and a quantum deformation of the Geometric Langlands Program. The relevant vertex algebras are associated to junctions of two boundary conditions in a 4d gauge theory and can be constructed from the basic ones by following certain standard procedures. Conformal blocks of modules over these vertex algebras give rise to twisted D-modules on the moduli stacks of G-bundles on Riemann surfaces which have applications to the Langlands Program. In particular, we construct a series of vertex algebras for every simple Lie group G which we expect to yield D-module kernels of various quantum Geometric Langlands dualities. We pay particular attention to the full duality group of gauge theory, which enables us to extend the standard qGL duality to a larger duality groupoid. We also discuss various subtleties related to the spin and gerbe structures and present a detailed analysis for the U(1) and SU(2) gauge theories.

中文翻译：

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边界条件、$D$-模和共形块的量子朗兰兹对偶性

我们回顾并扩展了连接规范理论构造和几何朗兰兹计划的量子变形的顶点代数框架。相关的顶点代数与 4d 规范理论中的两个边界条件的连接点相关联，并且可以通过遵循某些标准程序从基本条件构建。这些顶点代数上的共形模块会在黎曼曲面上的 G 丛的模堆上产生扭曲的 D 模，这适用于朗兰兹计划。特别是，我们为每个简单的李群 G 构造了一系列顶点代数，我们期望它们产生各种量子几何朗兰兹对偶的 D 模核。我们特别关注规范理论的全对偶群，这使我们能够将标准 qGL 对偶扩展到更大的对偶群。