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We develop the Dirac-Schwinger-Zwanziger (DSZ) quantization of classicalabelian gauge theories with general duality structure on oriented Lorentzianfour-manifolds $(M,g)$ of arbitrary topology, obtaining, as a result, theduality-covariant geometric formulation of such theories through connections onprincipal bundles. We implement the DSZ condition by restricting the fieldstrengths of the theory to those which define classes originating in thedegree-two cohomology of a local system valued in the groupoid of integralsymplectic spaces. We prove that such field strengths are curvatures ofconnections $\mathcal{A}$ defined on principal bundles $P$ whose structuregroup $G$ is the disconnected group of automorphisms of an integral affinesymplectic torus. The connected component of the identity of $G$ is a torusgroup, while its group of connected components is a modified Siegel modulargroup. This formulation includes electromagnetic and magnetoelectric gaugepotentials on an equal footing and describes the equations of motion through afirst-order polarized self-duality condition for the curvature of$\mathcal{A}$. The condition involves a combination of the Hodge operator of$(M,g)$ with a taming of the duality structure determined by $P$, whose choiceencodes the self-couplings of the theory. This description is reminiscent ofthe theory of four-dimensional euclidean instantons, even though we consider atwo-derivative theory in Lorentzian signature. We use this formulation tocharacterize the hierarchy of duality groups of abelian gauge theory, providinga gauge-theoretic description of the electromagnetic duality group as thediscrete remnant of the gauge group of $P$. We also perform the time-likereduction of the polarized self-duality condition to a Riemannianthree-manifold, obtaining a new type of Bogomolny equation which we solveexplicitly in a particular case.

The duality covariant geometry and DSZ quantization of abelian gauge theory