We define new deformable families of vertex operator algebras $\mathfrak{A}[\mathfrak{g}, \Psi, \sigma]$ associated to a large set of S-duality operations in four-dimensional supersymmetric gauge theory. They are defined as algebras of protected operators for two-dimensional supersymmetric junctions which interpolate between a Dirichlet boundary condition and its S-duality image. The $\mathfrak{A}[\mathfrak{g}, \Psi, \sigma]$ VOAs are equipped with two $\mathfrak{g}$ affine vertex subalgebras whose levels are related by the S-duality operation. They compose accordingly under a natural convolution operation and can be used to define an action of the S-duality operations on a certain space of VOAs equipped with a $\mathfrak{g}$ affine vertex subalgebra. We give a self-contained definition of the S-duality action on that space of VOAs. The space of conformal blocks (in the derived sense, i.e. chiral homology) for $\mathfrak{A}[\mathfrak{g}, \Psi, \sigma]$ is expected to play an important role in a broad generalization of the quantum Geometric Langlands program. Namely, we expect the S-duality action on VOAs to extend to an action on the corresponding spaces of conformal blocks. This action should coincide with and generalize the usual quantum Geometric Langlands correspondence. The strategy we use to define the $\mathfrak{A}[\mathfrak{g}, \Psi, \sigma]$ VOAs is of broader applicability and leads to many new results and conjectures about deformable families of VOAs.

中文翻译：

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S-对偶的顶点代数

我们定义了新的顶点算子代数的可变形族 $\mathfrak{A}[\mathfrak{g}, \Psi, \sigma]$ 与四维超对称规范理论中的大量 S 对偶运算相关。它们被定义为二维超对称结的受保护算子的代数，这些结在狄利克雷边界条件及其 S 对偶图像之间进行插值。$\mathfrak{A}[\mathfrak{g}, \Psi, \sigma]$ VOA 配备了两个 $\mathfrak{g}$ 仿射顶点子代数，其层级通过 S 对偶运算相关联。它们在自然卷积运算下相应地构成，可用于定义 S-对偶运算在配备 $\mathfrak{g}$ 仿射顶点子代数的 VOA 的某个空间上的动作。我们给出了 VOA 空间上的 S-对偶动作的独立定义。$\mathfrak{A}[\mathfrak{g}, \Psi, \sigma]$ 的共形块空间（在派生意义上，即手性同源）预计将在量子的广泛推广中发挥重要作用几何朗兰兹计划。也就是说，我们期望对 VOA 的 S-对偶动作扩展到对保形块的相应空间的动作。这个动作应该与通常的量子几何朗兰兹对应一致并概括。我们用来定义 $\mathfrak{A}[\mathfrak{g}, \Psi, \sigma]$ VOA 的策略具有更广泛的适用性，并导致了许​​多关于可变形 VOA 族的新结果和猜想。也就是说，我们期望对 VOA 的 S-对偶动作扩展到对保形块的相应空间的动作。这个动作应该与通常的量子几何朗兰兹对应一致并概括。我们用来定义 $\mathfrak{A}[\mathfrak{g}, \Psi, \sigma]$ VOA 的策略具有更广泛的适用性，并导致了许​​多关于可变形 VOA 族的新结果和猜想。也就是说，我们期望对 VOA 的 S-对偶动作扩展到对保形块的相应空间的动作。这个动作应该与通常的量子几何朗兰兹对应一致并概括。我们用来定义 $\mathfrak{A}[\mathfrak{g}, \Psi, \sigma]$ VOA 的策略具有更广泛的适用性，并导致了许​​多关于可变形 VOA 族的新结果和猜想。