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On the global symmetries of 6D superconformal field theories

Marco Bertolini, Peter R. Merkx, David R. Morrison

TL;DR

This paper investigates global symmetries of $6$-dimensional SCFTs by comparing field-theoretic predictions on the Coulomb branch with constraints from F-theory realizations, focusing on theories with a single tensor multiplet. It develops a framework using orbifold bases, local F-theory models, and Tate-type analyses to determine which global-symmetry algebras can be manifest in F-theory, and contrasts these with anomaly-driven predictions from the field theory. In most cases, F-theory can realize the field-theory global symmetry, but several notable mismatches arise, reflecting either deeper field-theoretic structure or geometric limitations of F-theory realizations. The work culminates in tables of relatively maximal global-symmetry algebras for various Kodaira types, highlighting both concordance and discrepancy, and outlines a program to extend these results to the broader atomic classification of $6$D SCFTs and to configurations with multiple curves. Overall, the study provides a concrete bridge between Coulomb-branch field theory and geometric realizations, offering a roadmap for extracting symmetry data from the 6D SCFT dictionary.

Abstract

We study global symmetry groups of six-dimensional superconformal field theories (SCFTs). In the Coulomb branch we use field theoretical arguments to predict an upper bound for the global symmetry of the SCFT. We then analyze global symmetry groups of F-theory constructions of SCFTs with a one-dimensional Coulomb branch. While in the vast majority of cases, all of the global symmetries allowed by our Coulomb branch analysis can be realized in F-theory, in a handful of cases we find that F-theory models fail to realize the full symmetry of the theory on the Coulomb branch. In one particularly mysterious case, F-theory models realize several distinct maximal subgroups of the predicted group, but not the predicted group itself.

On the global symmetries of 6D superconformal field theories

TL;DR

This paper investigates global symmetries of -dimensional SCFTs by comparing field-theoretic predictions on the Coulomb branch with constraints from F-theory realizations, focusing on theories with a single tensor multiplet. It develops a framework using orbifold bases, local F-theory models, and Tate-type analyses to determine which global-symmetry algebras can be manifest in F-theory, and contrasts these with anomaly-driven predictions from the field theory. In most cases, F-theory can realize the field-theory global symmetry, but several notable mismatches arise, reflecting either deeper field-theoretic structure or geometric limitations of F-theory realizations. The work culminates in tables of relatively maximal global-symmetry algebras for various Kodaira types, highlighting both concordance and discrepancy, and outlines a program to extend these results to the broader atomic classification of D SCFTs and to configurations with multiple curves. Overall, the study provides a concrete bridge between Coulomb-branch field theory and geometric realizations, offering a roadmap for extracting symmetry data from the 6D SCFT dictionary.

Abstract

We study global symmetry groups of six-dimensional superconformal field theories (SCFTs). In the Coulomb branch we use field theoretical arguments to predict an upper bound for the global symmetry of the SCFT. We then analyze global symmetry groups of F-theory constructions of SCFTs with a one-dimensional Coulomb branch. While in the vast majority of cases, all of the global symmetries allowed by our Coulomb branch analysis can be realized in F-theory, in a handful of cases we find that F-theory models fail to realize the full symmetry of the theory on the Coulomb branch. In one particularly mysterious case, F-theory models realize several distinct maximal subgroups of the predicted group, but not the predicted group itself.

Paper Structure

This paper contains 16 sections, 1 theorem, 152 equations, 7 tables.

Table of Contents

  1. Introduction
  2. F-theory on orbifolds
  3. 6D SCFTs from local F-theory models
  4. Global symmetries in the Coulomb branch
  5. Prediction from field theory
  6. Constraints from F-theory
  7. Type I${}_n$
  8. Type III
  9. Type IV
  10. Type I${}_0^\ast$
  11. Type I${}_n^\ast$
  12. Type IV${}^\ast$, III${}^\ast$ and II${}^\ast$
  13. Summary
  14. Conclusions and outlook
  15. The Tate algorithm for the I${}_n$ case
  16. ...and 1 more sections

Key Result

Lemma 1

Given $\Sigma=\{z=0\}$ carrying type I${}_0^\ast$ with gauge algebra ${\mathfrak{g}}_2$, any configuration ${\cal C}_{2n_1,\dots,2n_N}$ is such that $n_i\leq10+3\Sigma^2$.

Theorems & Definitions (1)

  • Lemma 1