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

## TL;DR

This paper calculates finite N corrections to the superconformal index of S-fold theories using AdS/CFT, revealing how wrapped D3-branes contribute to deviations from the large N limit.

## Contribution

It provides a novel method to compute finite N corrections by analyzing wrapped D3-brane fluctuations, extending the understanding of superconformal indices beyond the large N approximation.

## Key findings

- Finite N corrections are of order q^N.
- The derived formula matches known results up to order q^{2N}.
- Wrapped D3-branes account for Pfaffian-like operator contributions.

## 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 $q^{2N}$.