Twisted Fibrations in M/F-theory

In this work we investigate 5-dimensional theories obtained from M-theory on genus one fibered threefolds which exhibit twisted algebras in their fibers. We provide a base-independent algebraic description of the threefolds and compute light 5D BPS states charged under finite sub-algebras of the twisted algebras. We further construct the Jacobian fibrations that are associated to 6-dimensional F-theory lifts, where the twisted algebra is absent. These 6/5-dimensional theories are compared via twisted circle reductions of F-theory to M-theory. In the 5-dimensional theories we discuss several geometric transitions that connect twisted with untwisted fibrations. We present detailed discussions of ${\mathfrak{e}}_6^{\left(2\right)}$ \( {\mathfrak{e}}_6^{(2)} \) , ${\mathfrak{so}}_8^{\left(3\right)}$ \( {\mathfrak{so}}_8^{(3)} \) and ${\mathfrak{su}}_3^{\left(2\right)}$ \( {\mathfrak{su}}_3^{(2)} \) twisted fibers and provide several explicit example threefolds via toric constructions. Finally, limits are considered in which gravity is decoupled, including Little String Theories for which we match 2-group symmetries across twisted T-dual theories.