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Remarks on the Green-Schwarz terms of six-dimensional supergravity theories

Samuel Monnier, Gregory W. Moore

TL;DR

This work constructs a comprehensive Green-Schwarz mechanism for six-dimensional $N=(1,0)$ supergravity on manifolds with nontrivial topology, using a Wu-Chern-Simons framework to cancel both local and global anomalies. It introduces a differential-cohomology model for the self-dual tensor fields and defines a shifted Wu Chern-Simons theory whose partition function matches the anomaly field theory of the bare 6d theory, enabling anomaly cancellation via a well-defined Green-Schwarz term. For gauge groups built from $U(n)$, $SU(n)$, $Sp(n)$ factors and $E_8$, all anomalies cancel; for other groups the cancellation reduces to triviality in a 7D spin TFT, with residual global anomalies in the finite Abelian case that align with known F-theory constraints. A crucial result is that the gravitational anomaly coefficient must be a characteristic element of the string-charge lattice, a condition naturally met in F-theory but shown here as a low-energy consistency requirement. The work also reveals a new torsion-related anomaly coefficient in theories with disconnected gauge groups, and establishes unimodularity of the string lattice as a consistency condition, linking high-energy consistency to topological invariants.

Abstract

We construct the Green-Schwarz terms of six-dimensional supergravity theories on spacetimes with non-trivial topology and gauge bundle. We prove the cancellation of all global gauge and gravitational anomalies for theories with gauge groups given by products of $U(n)$, $SU(n)$ and $Sp(n)$ factors, as well as for $E_8$. For other gauge groups, anomaly cancellation is equivalent to the triviality of a certain 7-dimensional spin topological field theory. We show in the case of a finite Abelian gauge group that there are residual global anomalies imposing constraints on the 6d supergravity. These constraints are compatible with the known F-theory models. Interestingly, our construction requires that the gravitational anomaly coefficient of the 6d supergravity theory is a characteristic element of the lattice of string charges, a fact true in six-dimensional F-theory compactifications but that until now was lacking a low-energy explanation. We also discover a new anomaly coefficient associated with a torsion characteristic class in theories with a disconnected gauge group.

Remarks on the Green-Schwarz terms of six-dimensional supergravity theories

TL;DR

This work constructs a comprehensive Green-Schwarz mechanism for six-dimensional supergravity on manifolds with nontrivial topology, using a Wu-Chern-Simons framework to cancel both local and global anomalies. It introduces a differential-cohomology model for the self-dual tensor fields and defines a shifted Wu Chern-Simons theory whose partition function matches the anomaly field theory of the bare 6d theory, enabling anomaly cancellation via a well-defined Green-Schwarz term. For gauge groups built from , , factors and , all anomalies cancel; for other groups the cancellation reduces to triviality in a 7D spin TFT, with residual global anomalies in the finite Abelian case that align with known F-theory constraints. A crucial result is that the gravitational anomaly coefficient must be a characteristic element of the string-charge lattice, a condition naturally met in F-theory but shown here as a low-energy consistency requirement. The work also reveals a new torsion-related anomaly coefficient in theories with disconnected gauge groups, and establishes unimodularity of the string lattice as a consistency condition, linking high-energy consistency to topological invariants.

Abstract

We construct the Green-Schwarz terms of six-dimensional supergravity theories on spacetimes with non-trivial topology and gauge bundle. We prove the cancellation of all global gauge and gravitational anomalies for theories with gauge groups given by products of , and factors, as well as for . For other gauge groups, anomaly cancellation is equivalent to the triviality of a certain 7-dimensional spin topological field theory. We show in the case of a finite Abelian gauge group that there are residual global anomalies imposing constraints on the 6d supergravity. These constraints are compatible with the known F-theory models. Interestingly, our construction requires that the gravitational anomaly coefficient of the 6d supergravity theory is a characteristic element of the lattice of string charges, a fact true in six-dimensional F-theory compactifications but that until now was lacking a low-energy explanation. We also discover a new anomaly coefficient associated with a torsion characteristic class in theories with a disconnected gauge group.

Paper Structure

This paper contains 87 sections, 2 theorems, 144 equations.

Table of Contents

  1. Introduction and summary
  2. Preliminaries
  3. Six-dimensional supergravity theories
  4. Field content
  5. Gauge group
  6. Self-dual fields
  7. Green-Schwarz mechanism
  8. Anomaly coefficients
  9. Some facts about anomalies
  10. Generalities
  11. Anomaly field theories
  12. Complex Weyl fermions
  13. Majorana-Weyl fermions
  14. Symplectic Majorana-Weyl fermions
  15. Self-dual fields
  16. ...and 72 more sections

Key Result

Proposition 7.1

Let $\mathcal{S}$ be a 6-dimensional supergravity theory with gauge group $G$, string charge lattice $\Lambda$ and anomaly coefficients $(a,b, b_T = 0)$, where $b_T$ is the torsion anomaly coefficient in EqCharYGen. Let $A_8$ be the degree 8 anomaly polynomial of the theory and $Y$ the degree 4 form Then all anomalies of $\mathcal{S}$, local and global, cancel.

Theorems & Definitions (2)

  • Proposition 7.1
  • Proposition 7.2