Non-Invertible $SO(2)$ Symmetry of 4d Maxwell from Continuous Gaugings

We describe the self-duality symmetries for 4d Maxwell theory at any value of the coupling $\tau$ via topological manipulations that include gauging continuous symmetries with flat connections. Moreover, we demonstrate that the $SL(2,\mathbb{Z})$ duality of Maxwell can be realized by trivial gauging operations. Using a non-compact symmetry topological field theory (symTFT) to encode continuous global symmetries of the boundary theory, we reproduce the symTFT for Maxwell and find within this framework condensation defects that implement the non-invertible $SO(2)$ self-duality symmetry. These defects are systematically constructed by higher gauging subsets of the bulk $\mathbb{R}\times \mathbb{R}$ symmetry with appropriate discrete torsion.