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5d Superconformal Field Theories and Graphs

Fabio Apruzzi, Craig Lawrie, Ling Lin, Sakura Schafer-Nameki, Yi-Nan Wang

TL;DR

This work develops Combined Fiber Diagrams (CFDs), a graph-theoretic framework that encodes 5d $\mathcal{N}=1$ SCFTs arising from circle reductions of 6d $\mathcal{N}=(1,0)$ theories, including their UV flavor symmetries and BPS spectra. CFDs organize RG flows as mass-deformation transitions between CFDs, enabling a systematic construction of all descendant 5d SCFTs from marginal CFDs associated to 6d conformal matter. The authors work out the descendant structure for 6d $(D_k,D_k)$ and $(E_6,E_6)$ conformal matter, predicting flavor enhancements and spin-0 BPS spectra, even in cases without known gauge-theory descriptions, and clarifying UV dualities within a purely combinatorial, geometric approach.

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.

5d Superconformal Field Theories and Graphs

TL;DR

This work develops Combined Fiber Diagrams (CFDs), a graph-theoretic framework that encodes 5d SCFTs arising from circle reductions of 6d theories, including their UV flavor symmetries and BPS spectra. CFDs organize RG flows as mass-deformation transitions between CFDs, enabling a systematic construction of all descendant 5d SCFTs from marginal CFDs associated to 6d conformal matter. The authors work out the descendant structure for 6d and conformal matter, predicting flavor enhancements and spin-0 BPS spectra, even in cases without known gauge-theory descriptions, and clarifying UV dualities within a purely combinatorial, geometric approach.

Abstract

We propose a graph-theoretic description to determine and characterize 5d superconformal field theories (SCFTs) that arise as circle reductions of 6d 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 and conformal matter for any . This includes predictions for thus far unknown flavor symmetry enhancements.

Paper Structure

This paper contains 6 sections, 16 equations, 2 figures.

Table of Contents

  1. Introduction
  2. SCFTs from Graphs
  3. Rank One Theories
  4. 5d SCFTs from (Dk,Dk) CM
  5. 5d SCFTs from (E6,E6) CM
  6. BPS States

Figures (2)

  • Figure 1: CFD-transition tree for rank one 5d SCFTs including the superconformal flavor symmetries $G_F$.
  • Figure 2: CFDs for all 5d SCFTs descending from 6d $(D_k, D_k)$ CM. Each box contains the 5d strongly coupled flavor symmetry, $G_F$, and the $G_F$ representations of the spin $0$ BPS states (right upper corner). In cases when there is a weakly coupled gauge theory description, this is noted at the bottom of each box. Connecting lines between boxes indicate transitions.