We study surface defects in 4d $\mathcal{N}=1$ $SU(N)$ superconformal gauge theories of class ${\mathcal{S}}_k$ obtained from the 6d (1,0) theories of type $A_{N-1}$, which are worldvolume theories on N M5-branes at ${\mathbb{C}}^2/{\mathbb{Z}}_k$ singularities, compactified on Riemann surfaces with punctures. We evaluate the superconformal indices with surface defects for the orbifold of linear quiver gauge theories by two methods. First we apply a method based on Riemann surface description and obtain the superconformal index of the theories in the presence of surface defects labelled by arbitrary symmetric representations of $su(N)$. Then we propose another description for the same surface defects, which involves 4d-2d coupled systems, by identifying which 2d $\mathcal{N}=(0,2)$ theories should be coupled. We compute the index of the 4d-2d systems and reproduce the results obtained from the first method. Finally we study the 2d TQFT structure of the index for class ${\mathcal{S}}_k$ theories by obtaining several eigenfunctions and eigenvalues of the difference operators that capture the surface defects and checking their relation.

我们研究4D表面缺陷 $\mathcal{ñ}=\mathrm{1个}$ $\mathrm{小号}ü（ñ）$ 类的超共形规范理论 ${\mathcal{小号}}_{ķ}$ 从类型的6d（1,0）理论获得 ${\mathrm{一种}}_{ñ-\mathrm{1个}}$，这是有关N M5-大脑的世界量理论，${\mathbb{C}}^2/{\mathbb{ž}}_{ķ}$奇异性，在带刺的黎曼表面上被压实。我们通过两种方法对线性颤动量规理论的圆度评估具有表面缺陷的超保形指数。首先，我们应用基于Riemann表面描述的方法，并在存在以任意对称表示形式标记的表面缺陷的情况下获得理论的超保形指数。$sü（ñ）$。然后，我们通过识别哪一个2d来提出对相同表面缺陷的另一种描述，涉及4d-2d耦合系统。$\mathcal{ñ}=（0，2）$理论应该耦合。我们计算4d-2d系统的索引，并重现从第一种方法获得的结果。最后，我们研究了类的索引的2d TQFT结构${\mathcal{小号}}_{ķ}$ 通过获取捕获表面缺陷的差算子的几个特征函数和特征值并检查它们之间的关系来建立理论。