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Higher-form symmetries of 6d and 5d theories

Lakshya Bhardwaj, Sakura Schafer-Nameki

TL;DR

The paper develops a unified, geometry-informed method to determine higher-form symmetries in 5d and 6d theories, including SCFTs and LSTs, by combining IR tensor-multiplet data with M-theory/F-theory geometry and brane-web constructions. The 2-form symmetry is read from the Green–Schwarz charge matrix $\Omega^{ij}$ via Smith normal form, yielding a product of cyclic groups that encodes BPS string charges; the 1-form symmetry is set by the centers of simple gauge factors, further reduced by hypermultiplet and instanton/string data, with explicit formulas and examples. The framework extends to 5d KK theories arising from circle compactifications, including twisted cases, where discrete twists modify the symmetry groups in a controlled way; geometric and brane-web tools (including SNF calculations and GTP analyses) provide cross-checks and practical computation rules. Collectively, the results connect IR effective descriptions, stringy excitations, and UV geometric data to deliver precise catalogs of higher-form symmetries across broad classes of 5d/6d theories, enabling refined consistency checks and applications to dualities and defect structures.

Abstract

We describe general methods for determining higher-form symmetry groups of known 5d and 6d superconformal field theories (SCFTs), and 6d little string theories (LSTs). The 6d theories can be described as supersymmetric gauge theories in 6d which include both ordinary non-abelian 1-form gauge fields and also abelian 2-form gauge fields. Similarly, the 5d theories can also be often described as supersymmetric non-abelian gauge theories in 5d. Naively, the 1-form symmetry of these 6d and 5d theories is captured by those elements of the center of ordinary gauge group which leave the matter content of the gauge theory invariant. However, an interesting subtlety is presented by the fact that some massive BPS excitations, which includes the BPS instantons, are charged under the center of the gauge group, thus resulting in a further reduction of the 1-form symmetry. We use the geometric construction of these theories in M/F-theory to determine the charges of these BPS excitations under the center. We also provide an independent algorithm for the determination of 1-form symmetry for 5d theories that admit a generalized toric construction. The 2-form symmetry group of 6d theories, on the other hand, is captured by those elements of the center of the abelian 2-form gauge group that leave all the massive BPS string excitations invariant, which is much more straightforward to compute as it is encoded in the Green-Schwarz coupling associated to the 6d theory.

Higher-form symmetries of 6d and 5d theories

TL;DR

The paper develops a unified, geometry-informed method to determine higher-form symmetries in 5d and 6d theories, including SCFTs and LSTs, by combining IR tensor-multiplet data with M-theory/F-theory geometry and brane-web constructions. The 2-form symmetry is read from the Green–Schwarz charge matrix via Smith normal form, yielding a product of cyclic groups that encodes BPS string charges; the 1-form symmetry is set by the centers of simple gauge factors, further reduced by hypermultiplet and instanton/string data, with explicit formulas and examples. The framework extends to 5d KK theories arising from circle compactifications, including twisted cases, where discrete twists modify the symmetry groups in a controlled way; geometric and brane-web tools (including SNF calculations and GTP analyses) provide cross-checks and practical computation rules. Collectively, the results connect IR effective descriptions, stringy excitations, and UV geometric data to deliver precise catalogs of higher-form symmetries across broad classes of 5d/6d theories, enabling refined consistency checks and applications to dualities and defect structures.

Abstract

We describe general methods for determining higher-form symmetry groups of known 5d and 6d superconformal field theories (SCFTs), and 6d little string theories (LSTs). The 6d theories can be described as supersymmetric gauge theories in 6d which include both ordinary non-abelian 1-form gauge fields and also abelian 2-form gauge fields. Similarly, the 5d theories can also be often described as supersymmetric non-abelian gauge theories in 5d. Naively, the 1-form symmetry of these 6d and 5d theories is captured by those elements of the center of ordinary gauge group which leave the matter content of the gauge theory invariant. However, an interesting subtlety is presented by the fact that some massive BPS excitations, which includes the BPS instantons, are charged under the center of the gauge group, thus resulting in a further reduction of the 1-form symmetry. We use the geometric construction of these theories in M/F-theory to determine the charges of these BPS excitations under the center. We also provide an independent algorithm for the determination of 1-form symmetry for 5d theories that admit a generalized toric construction. The 2-form symmetry group of 6d theories, on the other hand, is captured by those elements of the center of the abelian 2-form gauge group that leave all the massive BPS string excitations invariant, which is much more straightforward to compute as it is encoded in the Green-Schwarz coupling associated to the 6d theory.

Paper Structure

This paper contains 20 sections, 108 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Higher-form symmetries of $6d$ SCFTs and LSTs
  3. 2-form symmetry
  4. 2-form symmetry of $6d$ SCFTs
  5. 2-form symmetry of $6d$ LSTs
  6. Examples
  7. Relative nature of $6d$ SCFTs and LSTs
  8. 1-form symmetry of $6d$ SCFTs and LSTs
  9. Examples
  10. 1-form symmetry of $5d$ $\mathcal{N}=1$ theories
  11. 1-form symmetry from the Coulomb branch
  12. 1-form symmetry from M-theory geometry
  13. 1-form symmetry of $5d$ $\mathcal{N}=1$ non-abelian gauge theories
  14. 1-form symmetry of non-abelian gauge theories from geometry
  15. 1-form symmetry of $5d$ KK theories
  16. ...and 5 more sections

Figures (1)

  • Figure 1: On the left hand side is shown a general 5-brane web (blue) indicating the external $(p,q)$ 5-branes, ending on 7-branes (cyan). From these emanate $(p,q)$-strings (green), that end on D3-branes (yellow). Given a pair of external 5-branes, the strings can only form a junction, if they satisfy (\ref{['junctioncond']}). This is shown on the right hand side. The resulting string can be moved into the brane-web, by moving the D3-brane inside the web, and becomes a local operator. This is the screening of the Wilson lines by local operators, realized in the brane-web.