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We give a gentle introduction to the global geometric formulation of thebosonic sector of four-dimensional supergravity on an oriented four-manifold$M$ of arbitrary topology, providing a geometric characterization of itsU-duality group. The geometric formulation of four-dimensional supergravity isbased on a choice of a vertically Riemannian submersion $\pi$ over $M$ equippedwith a flat Ehresmann connection, which determines the non-linear section sigmamodel of the theory, and a choice of flat symplectic vector bundle$\mathcal{S}$ equipped with a positive complex polarization over the totalspace of $\pi$, which encodes the inverse gauge couplings and theta angles ofthe theory and determines its gauge sector. The classical fields of the theoryconsist of Lorentzian metrics on $M$, global sections of $\pi$ and two-formsvalued in $\mathcal{S}$ that satisfy an algebraic relation which defines thenotion of \emph{twisted} self-duality in four Lorentzian dimensions. We usethis geometric formulation to investigate the group of electromagnetic dualitytransformations of supergravity, also known as the continuous classicalU-duality group, which we characterize using a certain short exact sequence ofautomorphism groups of vector bundles. Moreover, we discuss the generalstructure of the Killing spinor equations of four-dimensional supergravity,providing several explicit examples and remarking on a few open mathematicalproblems. This presentation is aimed at mathematicians working in differentialgeometry.

Four-dimensional geometric supergravity and electromagnetic duality: a brief guide for mathematicians