The geometry and DSZ quantization four-dimensional supergravity

We implement the Dirac–Schwinger–Zwanziger integrality condition on four-dimensional classical ungauged supergravity and use it to obtain its duality-covariant, gauge-theoretic, differential-geometric model on an oriented four-manifold M of arbitrary topology. Classical bosonic supergravity is completely determined by a submersion π over M equipped with a complete Ehresmann connection, a vertical Euclidean metric and a vertically polarized flat symplectic vector bundle Ξ . Building on these structures, we implement the Dirac–Schwinger–Zwanziger integrality condition through the choice of an element in the degree-two sheaf cohomology group with coefficients in a locally constant sheaf ℒ⊂Ξ valued in the groupoid of integral symplectic spaces. We show that these data determine a Siegel principal bundle P_𝔱 of fixed type 𝔱∈ℤ^n_v whose connections provide the global geometric description of the local electromagnetic gauge potentials of the theory. Furthermore, we prove that the Maxwell gauge equations of the theory reduce to the polarized self-duality condition determined by Ξ on the connections of P_𝔱 . In addition, 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_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 π and P_𝔱 as well as on the isomorphism type of ℒ .