  We develop the Dirac-Schwinger-Zwanziger (DSZ) quantization of
four-dimensional bosonic ungauged supergravity on an oriented four-manifold $M$
of arbitrary topology and use it to obtain its manifestly duality-covariant
gauge-theoretic geometric formulation. Classical bosonic supergravity is
completely determined by a submersion $\pi$ over $M$ equipped with a complete
Ehresmann connection, a vertical euclidean metric, and a vertically-polarized
flat symplectic vector bundle $\Xi$. We implement the Dirac-Schwinger-Zwanziger
quantization condition in the aforementioned classical supergravity through the
choice of an element in the degree-two sheaf cohomology group with coefficients
in a locally constant sheaf $\mathcal{L}\subset \Xi$ valued in the groupoid of
integral symplectic spaces. We show that this data determines a Siegel
principal bundle $P_{\mathfrak{t}}$ of fixed type $\mathfrak{t}\in \mathbb{Z}$
whose connections provide the global geometric description of the
electromagnetic guage potentials of the theory. The Maxwell gauge equations of
the theory reduce to the polarized self-duality condition determined by $\Xi$
on the connections of $P_{\mathfrak{t}}$. We investigate the continuous and
discrete U-duality groups of the theory, characterizing them through short
exact sequences and realizing the latter through the gauge group of
$P_{\mathfrak{t}}$ acting on its adjoint bundle. This elucidates the geometric
origin of U-duality, which we explore in several examples, illustrating its
dependence on the topology of the fiber bundles $\pi$ and $P_{\mathfrak{t}}$ as
well as on the isomorphism type of $\mathcal{L}$.
