# BPS Partition Functions for S-folds

## TL;DR

This paper derives a unified formula for BPS partition functions in S-fold theories, extending known results from ${
m ext{U}}(N)$ to ${
m SO}$, ${
m Sp}$, and ${
m N}=3$ theories, confirming supersymmetry enhancements.

## Contribution

It generalizes BPS partition function formulas to include ${
m SO}$, ${
m Sp}$, and ${
m N}=3$ theories, and derives the same from D3-brane quantization.

## Key findings

- Results are consistent with supersymmetry enhancement from ${
m N}=3$ to ${
m N}=4$.
- Derived formulas match for rank 1 and 2 cases.
- Unified approach from field theory and string theory perspectives.

## Abstract

We derive a formula for the BPS partition functions of arbitrary S-fold theories. We first generalize the known result for the ${\cal N}=4$ $U(N)$ supersymmetric Yang-Mills theory to $SO$ and $Sp$ theories, and then we extend the formula to ${\cal N}=3$ theories. We confirm that the results for rank $1$ and $2$ are consistent to the supersymmetry enhancement from ${\cal N}=3$ to ${\cal N}=4$. We also derive the same formula from the quantization of D3-branes in $\boldsymbol{S}^5/\mathbb{Z}_k$.