# Compactifications of 6d $N = (1, 0)$ SCFTs with non-trivial Stiefel-Whitney classes

## TL;DR

This paper explores how compactifying certain 6d superconformal field theories with non-trivial Stiefel-Whitney classes on a torus yields specific 4d N=2 SCFTs, revealing new connections and classification rules.

## Contribution

It demonstrates that all rank-1 4d N=2 SCFTs with particular flavor symmetries can be derived from a single 6d theory with Stiefel-Whitney twists, providing a unified construction.

## Key findings

- All rank-1 4d SCFTs with specified flavor symmetries originate from a single 6d theory.
- A rule is proposed to determine Coulomb branch dimensions of 4d theories from 6d compactifications.
- The study links twisted compactifications to the classification of 4d N=2 SCFTs.

## Abstract

We consider compactifications of very Higgsable 6d $N =(1,0)$ SCFTs on $T^2$ with non-trivial Stiefel-Whitney classes (or equivalently 't Hooft magnetic fluxes) introduced for their flavor symmetry groups. These systems can also be studied as twisted $S^1$ compactifications of the corresponding 5d theories. We deduce various properties of the resulting 4d $N=2$ SCFTs by combining these two viewpoints. In particular, we find that all 4d rank-1 $N =2$ SCFTs with a dimension-6 Coulomb branch operator with flavor symmetry $\mathfrak{e}_8$, $\mathfrak{usp}(10)$, $\mathfrak{su}(4) $ and $\mathfrak{su}(3)$ can be uniformly obtained by starting from a single-tensor theory in 6d. We also have a mostly independent appendix where we propose a rule to determine the Coulomb branch dimensions of 4d $N =2$ theories obtained by $T^2$ compactifications of 6d very Higgsable theories with and without Stiefel-Whitney twist.