Superconformal Index with Duality Domain Wall

TL;DR

This work develops a framework to compute the superconformal index of 4d ${\cal N}=4$ (and related ${\cal N}=2$) theories in the presence of a half-BPS duality domain wall by gluing a 3d generalized index on the wall to 4d half-indices. It introduces the concept of the duality wall as a kernel that implements S-duality on the index and extends it to configurations with line operators, providing integral equations that relate dual configurations. The authors explicitly compute and check these relations in ${\cal N}=4$ $SU(2)$ via the theories $T[SU(2),\varphi]$, explore mass-deformations to ${\cal N}=2^*$, and compare to dual 3d descriptions, thereby supporting a generalized AGT-like correspondence for domain walls. They also outline a path to generalize to $A_1$ Gaiotto theories and demonstrate a concrete ${\cal N}=2$ ${\cal SU}(2)$ with $N_F=4$ example, showing S-duality acts on line operators as expected. Overall, the paper provides a concrete, testable framework for coupling 3d wall theories to 4d bulk indices and for understanding duality kernels in supersymmetric gauge theories with boundaries or defects, with potential implications for the 3d-3d/AGT program.

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.

Figures (4)

• Figure 1: A Janus domain wall of 4d, ${\cal N}=4$ SYM with gauge group $G$ is a configuration that the holomorphic coupling $\tau$ varies as a step function at $x^4=0$ (left). For $\tau^{ \prime}$ related with $\tau$ by S-duality, one can take S-duality on the right-half plane (red region), which results in the S-duality domain wall supported at $x^4=0$ (right).
• Figure 2: The quiver diagram of $T[SU(N)]$ theory (left). A unitary gauge group is denoted by a circle, $SU(N)$ flavor symmetry by a square, and a bi-fundamental hypermultiplet by a line. $T[SU(N)]$ theory has a global symmetry $SU(N)\times {}^L SU(N)$ and we draw the quiver diagram for the theory in a simpler form (right).
• Figure 3: The quiver diagram for $T[G, \varphi]$ with $\varphi =S T^{k_1} S T^{k_2}$. An integer $k$ under a hexagon denotes the Chern-Simons level for the gauge field (or background gauge field coupled to global symmetry) in the hexagon. An internal node denotes a gauge symmetry and external two nodes denote global symmetries.
• Figure 4: Consider a Janus domain wall at the equator of 3-sphere in $S^3 \times S^1$ with a line operator $L^{ \prime}$ inserted at the south pole of the 3-sphere (left). $\tau^{ \prime}$ and $L^{ \prime}$ are related with $\tau$ and $L$ by a duality element $\varphi$. Taking $\varphi^{-1}$-daulity on the southern hemisphere results a duality domain wall at the equator and the line operator $L$ at the south pole (right).