Superconformal index with surface defects for class ${\mathcal{S}}_k$

Abstract

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.