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

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