Nonlocal Operators and Duality in Abelian Gauge Theory on a Four-Manifold

We generalize our picture in [arXiv:0904.1744], and consider a pure abelian gauge theory on a four-manifold with nonlocal operators of every codimension arbitrarily and simultaneously inserted. We explicitly show that (i) the theory enjoys exact S-duality for certain choices of operator parameters; (ii) if there are only trivially-embedded surface operators and Wilson loop operators, or if there are only Wilson-'t Hooft loop operators, the theory enjoys a more general and exact SL(2,Z) or \Gamma_0(2) duality; (iii) the parameters of the loop and surface operators transform like electric-magnetic charges under the SL(2,Z) or \Gamma_0(2) duality of the theory. Through the formalism of duality walls, we derive the transformation of loop and surface operators embedded in a Chern-Simons operator. Via a Hamiltonian perspective, we furnish an alternative understanding of the SL(2,Z) duality. Last but not least, we also compute the partition function and correlation function of gauge-invariant local operators, and find that they transform as generalized modular forms under the respective duality groups.