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

We give the global mathematical formulation of the coupling of four-dimensional scalar sigma models to Abelian gauge fields on a Lorentzian four-manifold, for the generalized situation when the duality structure of the Abelian gauge theory is described by a flat symplectic vector bundle (S, D, ω) defined over the scalar manifold M. The construction uses a taming of (S, ω), which we find to be the correct mathematical object globally encoding 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 M to the bundle 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 M and gives rise to a smooth bundle of polarized Abelian varieties, endowed with a flat symplectic connection. This shows, in particular, that a generalization of part of the mathematical structure familiar from N = 2 supergravity is already present in such purely bosonic models, without any coupling to fermions and hence without any supersymmetr