Massive 3d Abelian gauge theories and electric-magnetic duality

We study a model of massive photons with a parity invariant and non-local mass term. We identify a discrete symmetry of the classical equations of motion and show that this symmetry can be thought of as an electric-magnetic duality valid only in the presence of a non-zero photon mass. We explore this duality and discuss some features of the theory. We define notions of electric and magnetic charge that can be computed in any spatial domain as a volume integral of a density. These integrals can be transformed into integrals on the boundary. For instance, the electric charge is given by the sum of the electric flux through the boundary of the domain and a line integral of the dual gauge field along that same boundary. We show that electric and magnetic charge is quantized $q_mq_e = 2\pi n m_\gamma/e^2$, where $e$ is the coupling constant and $m_\gamma$ is the mass of the photon. Classical field configurations associated with external electric and magnetic charges as well as the classical forces between electric and magnetic charges are computed.