The Geometry of SO(3), SO(5), and SO(6) models

SO(3), SO(5), and SO(6)-models are singular elliptic fibrations with Mordell--Weil torsion Z/2Z and singular fibers whose dual fibers correspond to affine Dynkin diagrams of type A1, C2, and A3 respectively, where we emphasize the distinction between SO(n) and its universal cover Spin(n). While the SO(3)-model has been studied before, the SO(5) and SO(6)-models are studied here for the first time. By computing crepant resolutions of their Weierstrass models, we study their fiber structures and topological invariants. In the special case that the SO(n)-model is an elliptically fibered Calabi-Yau threefold, we compute the Chern-Simons couplings and matter content of a 5D N=1 supergravity theory with gauge group SO(n), which is related to M-theory compactified on this Calabi-Yau threefold. We also verify the 6D lift of the 5D matter content is necessary and sufficient for anomaly cancellation in 6D (1,0) supergravity theories geometrically engineered by F-theory compactified on the same threefold. We find that the associated 5D and 6D supergravity theories with SO(n) gauge symmetry indeed differ from their Spin(n) cousins, with one striking consequence of this distinction being that all such theories must include adjoint matter.