Generalized Einstein-Scalar-Maxwell theories and locally geometric U-folds

We give a global formulation of the coupling of four-dimensional scalar sigma models to Abelian gauge fields for the generalized situation when the "duality structure" of the Abelian gauge theory is described by a flat symplectic vector bundle $(\mathcal{S},D,\omega)$ defined over the scalar manifold $\mathcal{M}$. The construction uses a taming of $(\mathcal{S}, \omega)$, which encodes globally the inverse gauge couplings and theta angles of the "twisted" Abelian gauge theory in a manner that makes no use of duality frames. We show that global solutions of the equations of motion of such models give classical locally geometric U-folds. We also describe the groups of duality transformations and scalar-electromagnetic symmetries arising in such models, which involve lifting isometries of $\mathcal{M}$ to a particular class of flat automorphisms of the bundle $\mathcal{S}$ and hence differ from expectations based on local analysis. The appropriate version of the Dirac quantization condition involves a discrete local system defined over $\mathcal{M}$ and gives rise to a smooth bundle of polarized Abelian varieties, endowed with a flat symplectic connection. This shows that a generalization of part of the mathematical structure familiar from $\mathcal{N}=2$ supergravity is already present in such purely bosonic models, without any coupling to fermions and hence without any supersymmetry.