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

TL;DR

This paper develops methods to compute the superconformal index with surface defects in class s_k theories, revealing a 2d TQFT structure and providing new insights into surface defect representations.

Contribution

It introduces a novel approach to calculate superconformal indices with surface defects in class s_k theories using Riemann surface and 4d-2d coupled systems methods.

Findings

Superconformal index with surface defects expressed via symmetric representations.

Equivalence of Riemann surface and 4d-2d coupled system descriptions.

Identification of eigenfunctions and eigenvalues of difference operators related to surface defects.

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. 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.