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# Advances in Theoretical and Mathematical Physics

## Volume 26 (2022)

### Number 7

### The duality covariant geometry and DSZ quantization of abelian gauge theory

Pages: 2213 – 2312

DOI: https://dx.doi.org/10.4310/ATMP.2022.v26.n7.a5

#### Authors

#### Abstract

We develop the Dirac–Schwinger–Zwanziger (DSZ) quantization of classical abelian gauge theories with general duality structure on oriented and connected Lorentzian four-manifolds $(M,g)$ of arbitrary topology, obtaining as a result the duality-covariant geometric formulation of such theories through connections on principal bundles. We implement the DSZ condition by restricting the field strengths of the theory to those which define classes originating in the degree-two cohomology of a local system valued in the groupoid of integral symplectic spaces. We prove that such field strengths are curvatures of connections $\mathcal{A}$ defined on principal bundles P whose structure group $G$ is the disconnected *non-abelian* group of automorphisms of an integral affine symplectic torus. The connected component of the identity of $G$ is a torus group, while its group of connected components is a modified Siegel modular group which coincides with the group of local duality transformations of the theory. This formulation includes electromagnetic and magnetoelectric gauge potentials on an equal footing and describes the equations of motion through a first-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 choice encodes the self-couplings of the theory. This description is reminiscent of the theory of fourdimensional euclidean instantons, even though we consider a twoderivative theory in Lorentzian signature. We use this formulation to characterize the hierarchy of duality groups of abelian gauge theory, providing a gauge-theoretic description of the electromagnetic duality group as the discrete remnant of the gauge group of $P$. We also perform the time-like reduction of the polarized self-duality condition to a Riemannian three-manifold, obtaining a new type of Bogomolny equation which is modified by the given taming and duality structure induced by $P$. We give explicit examples of such solutions, which we call *polarized dyons*.

Published 30 August 2023