Quantization of anomaly coefficients in 6D $\mathcal{N}=(1,0)$ supergravity

TL;DR

The paper derives new quantization constraints on the anomaly coefficients of 6D $\mathcal{N}=(1,0)$ supergravity by combining local and global anomaly cancellation with a strong completeness hypothesis, culminating in the statement that the gauge-anomaly data must lie in $2 H^4(BG;\mathbb{Z}) \otimes \Lambda_S$. It then shows these strongest conditions hold in F-theory vacua and identifies the cocharacter lattice of the gauge group with a sublattice of the Calabi–Yau resolution’s homology, thereby fixing the global form of the gauge group. The work provides a unified framework linking field-theory anomaly constraints to geometric data, and demonstrates consistency in F-theory while clarifying how global structure constraints arise from cocharacter lattices and string-charge quantization. These results have potential implications for understanding the string landscape and swampland in 6D, and guide how anomaly data should be realized in concrete string constructions.

Abstract

We obtain new constraints on the anomaly coefficients of 6D $\mathcal{N}=(1,0)$ supergravity theories using local and global anomaly cancellation conditions. We show how these constraints can be strengthened if we assume that the theory is well-defined on any spin space-time with an arbitrary gauge bundle. We distinguish the constraints depending on the gauge algebra only from those depending on the global structure of the gauge group. Our main constraint states that the coefficients of the anomaly polynomial for the gauge group $G$ should be an element of $2 H^4(BG;\mathbb{Z}) \otimes Λ_S$ where $Λ_S$ is the unimodular string charge lattice. We show that the constraints in their strongest form are realized in F-theory compactifications. In the process, we identify the cocharacter lattice, which determines the global structure of the gauge group, within the homology lattice of the compactification manifold.