Finite $N$ Corrections to the Superconformal Index of S-fold Theories

TL;DR

This work analyzes finite-N corrections to the superconformal index of S-fold theories using AdS/CFT, showing that in the large-N limit the index is captured by bulk KK modes while finite-N effects arise from D3-branes wrapped on nontrivial cycles and giant gravitons. The authors introduce a patch-wise method to quantize wrapped D3-branes, derive universal single-particle indices for ground-state and fluctuation modes, and assemble the full index via plethystic exponentials and S-fold projections. They demonstrate that leading finite-N corrections begin at order $q^N$ and that the combined KK plus D3 contributions reproduce localization results up to $\mathcal O(q^{2N})$ for several cases, including the orientifold and select $k\ge3$ S-folds, with supporting dualities for small N. The findings provide a robust framework for understanding finite-N structure in S-fold theories and offer a pathway to systematic extensions to higher-order corrections and more general backgrounds.

Abstract

We study the superconformal index of S-fold theories by using AdS/CFT correspondence. It has been known that the index in the large $N$ limit is reproduced as the contribution of bulk Kaluza-Klein modes. For finite $N$ D3-branes wrapped around the non-trivial cycle in $\boldsymbol{S}^5/\mathbb{Z}_k$, which corresponds to Pfaffian-like operators, give the corrections of order $q^N$ to the index. We calculate the finite $N$ corrections by analyzing the fluctuations of wrapped D3-branes. Comparisons to known results show that our formula correctly reproduces the corrections up to errors of order $q^{2N}$.

Figures (2)

• Figure 1: The structure of the ground state configuration spaces for $k=2$ (a) and $k\geq3$ (b) are shown.
• Figure 2: The spaces of ground state configurations contributing to the Schur index are shown. (a) is the space for $k=2$ and (b) is that for $k\geq3$.