# Fibers add Flavor, Part I: Classification of 5d SCFTs, Flavor Symmetries and BPS States

## TL;DR

This paper introduces a graph-based method to classify 5d superconformal field theories derived from 6d theories, revealing new flavor symmetries and SCFTs through combinatorial analysis of Calabi--Yau geometries.

## Contribution

It develops Combined Fiber Diagrams as a novel combinatorial tool to characterize 5d SCFTs from 6d origins, capturing their flavor symmetries and BPS states.

## Key findings

- Derived all rank one and two 5d SCFTs.
- Discovered new flavor symmetry enhancements.
- Identified previously unknown 5d SCFTs.

## Abstract

We propose a graph-based approach to 5d superconformal field theories (SCFTs) based on their realization as M-theory compactifications on singular elliptic Calabi--Yau threefolds. Field-theoretically, these 5d SCFTs descend from 6d $\mathcal{N}=(1,0)$ SCFTs by circle compactification and mass deformations. We derive a description of these theories in terms of graphs, so-called Combined Fiber Diagrams, which encode salient features of the partially resolved Calabi--Yau geometry, and provides a combinatorial way of characterizing all 5d SCFTs that descend from a given 6d theory. Remarkably, these graphs manifestly capture strongly coupled data of the 5d SCFTs, such as the superconformal flavor symmetry, BPS states, and mass deformations. The capabilities of this approach are demonstrated by deriving all rank one and rank two 5d SCFTs. The full potential, however, becomes apparent when applied to theories with higher rank. Starting with the higher rank conformal matter theories in 6d, we are led to the discovery of previously unknown flavor symmetry enhancements and new 5d SCFTs.