Exploring new constraints on Kähler moduli space of 6d \( \mathcal{N} \) = 1 supergravity

We propose new constraints for 6d (1, 0) supergravity theories based on consistency conditions on the Kähler moduli spaces of their 5d reductions. The requirement that both the metric and the BPS string tensions in the Kähler moduli space are positive imposes specific restrictions on the Chern-Simons coefficients in the 5d effective Lagrangians that are derived from the Kaluza-Klein reductions of 6d theories. Moreover, the emergence of local interacting 5d CFTs when the moduli space metric degenerates introduces additional constraints coming from the analysis of 5d SCFTs. Focusing on the moduli spaces of 6d supergravity theories without a tensor multiplet and their Higgsings, we show that these constraints require the presence of certain primary states in the 2d worldvolume CFTs on 1/2 BPS strings. We specifically analyze a class of SU(2) models and infinite families of U(1) models using these constraints, and demonstrate that the theories featuring a 1-form symmetry in their massless spectra, unless the 1-form symmetry is gauged, fail to satisfy the constraints and therefore belong to the Swampland.