We explore the non-perturbative Dyson–Schwinger equations obeyed by the partition functions of the $\Omega$-deformed $\mathcal{N}=2, d=4$ supersymmetric linear quiver gauge theories in the presence of surface defects. We demonstrate that the partition functions of different types of defects (orbifold or vortex strings) are related by analytic continuation. We introduce Darboux coordinates on a patch of the moduli space of flat $SL(N)$-connections on a sphere with special punctures, which generalize the NRS coordinates defined in the $SL(2)$ case. Finally, we compare the generating function of the Lagrangian variety of opers in these Darboux coordinates with the effective twisted superpotential of the linear quiver theory in the two-dimensional $\Omega$-background, thereby proving the NRS conjecture and its generalization to the $SL(3)$ case.

中文翻译：

---

操作、表面缺陷和阳阳功能

我们探索了在存在表面缺陷的情况下，$\Omega$ 变形 $\mathcal{N}=2, d=4$ 超对称线性颤振规范理论的配分函数所遵循的非微扰 Dyson-Schwinger 方程。我们证明了不同类型缺陷（orbifold 或 vortex 弦）的配分函数是通过解析延拓相关的。我们在具有特殊穿孔的球体上的平面 $SL(N)$-连接的模空间的补丁上引入 Darboux 坐标，这概括了 $SL(2)$ 情况下定义的 NRS 坐标。最后，我们将这些 Darboux 坐标中的拉格朗日算子的生成函数与二维 $\Omega$ 背景下线性颤动理论的有效扭曲超势进行比较，从而证明 NRS 猜想及其对 $ SL(3)$ 案例。