Charge Lattices and Consistency of 6D Supergravity

TL;DR

This work shows that any consistent six-dimensional $\mathcal{N}=1$ supergravity theory with chiral two-form fields must have a self-dual, unimodular dyonic string charge lattice $\Gamma$, and that the Green–Schwarz anomaly lattice $\Lambda$ must embed into $\Gamma$; reductions to $2$D and $4$D, along with compactifications on $\mathbb{CP}^2$ and tori, force $\Gamma=\Gamma^*$ and thereby impose strong constraints on admissible theories. The authors connect anomaly cancellation data to a global lattice consistency condition, ruling out many apparently consistent models that fail unimodular embedding, and discuss implications for F-theory constructions and the possible role of discrete gauge data. The results sharply constrain the landscape of viable 6D supergravity theories and guide future string-theory realizations, while suggesting avenues for a more rigorous field-theoretic or worldsheet formulation of dyonic strings. Overall, the paper establishes self-duality of the 6D charge lattice as a fundamental consistency requirement with broad implications for model-building and UV completions.

Abstract

We extend the known consistency conditions on the low-energy theory of six-dimensional N = 1 supergravity. We review some facts about the theory of two-form gauge fields and conclude that the charge lattice Gamma for such a theory has to be self-dual. The Green-Schwarz anomaly cancellation conditions in the supergravity theory determine a sublattice of Gamma. The condition that this sublattice can be extended to a self-dual lattice Gamma leads to a strong constraint on theories that otherwise appear to be self-consistent.