# 5d Superconformal Field Theories and Graphs

## TL;DR

This paper introduces a graph-based method called Combined Fiber Diagrams to classify and analyze 5d superconformal field theories derived from 6d theories, revealing their flavor symmetries and BPS states.

## Contribution

It develops a novel graph-theoretic framework to characterize 5d SCFTs and their flows from 6d theories, including predictions of unknown flavor symmetry enhancements.

## Key findings

- Complete classification of 5d SCFTs descending from specific 6d theories.
- Graph encoding of flavor symmetries and BPS states.
- Predictions of new flavor symmetry enhancements.

## Abstract

We propose a graph-theoretic description to determine and characterize 5d superconformal field theories (SCFTs) that arise as circle reductions of 6d $\mathcal{N} = (1,0)$ SCFTs. Each 5d SCFT is captured by a graph, called a Combined Fiber Diagram (CFD). Transitions between CFDs encode mass deformations that trigger flows between SCFTs. In this way, the complete set of descendants of a given 6d theory are obtained from a single marginal CFD. The graphs encode key physical information like the superconformal flavor symmetry and BPS states. As an illustration, we ascertain the aforementioned data associated to all the 5d SCFTs descending from 6d minimal $(E_6, E_6)$ and $(D_k, D_k)$ conformal matter for any $k$. This includes predictions for thus far unknown flavor symmetry enhancements.