Superconformal Index with Duality Domain Wall

TL;DR

This paper develops a superconformal index framework for ${ m extbf{N}=4}$ super Yang-Mills with duality domain walls, incorporating line operators and proposing integral equations that encode duality transformations, with explicit checks in specific theories.

Contribution

It introduces a novel superconformal index involving duality domain walls and line operators, providing integral equations that serve as duality kernels, with explicit validation in ${ m extbf{N}=4}$ SU(2) theory.

Findings

Derived superconformal index with duality domain walls.

Proposed integral equations as duality kernels for line operators.

Validated equations in ${ m extbf{N}=4}$ SU(2) theory.

Abstract

We study a superconformal index for ${\cal N}=4$ super Yang-Mills on $S^1 \times S^3$ with a half BPS duality domain wall inserted at the great two-sphere in $S^3$. The index is obtained by coupling the 3d generalized superconformal index on the duality domain wall with 4d half-indices. We further consider insertions of line operators to the configuration and propose integral equations which express that the 3d index on duality domain wall is a duality kernel relating half indices of two line operators related by the duality map. We explicitly check the proposed integral equations for various duality domain walls and line operators in the ${\cal N}=4$ SU(2) theory. We also briefly comment on a generalization to $\mathcal{N}=2$ $A_1$ Gaiotto theories with a simple example, ${\cal N}=2$ SU(2) SYM with four flavors.