SL₂(ℝ) symmetries of SymTFT and non-invertible U(1) symmetries of Maxwell theory

Recent proposals for the Symmetry Topological Filed Theory (SymTFT) of Maxwell theory admit a 0-form symmetry compatible with the classical SL₂(ℝ) duality of electromagnetism. We describe how to realize these automorphisms of the SymTFT in terms of its operators and we describe their effects on the dynamical theory and its global variants. In the process, we show that the classical U(1) symmetry, corresponding to the stabilizer of SL₂(ℝ), can be restored as a non-invertible one, by means of an infinite series of discrete gauging. This provide an example of the reemergence of a classical symmetry in the quantum regime, which was not broken by anomalies, but rather by the quantization of electromagnetic fluxes. However, this procedure comes at the price of introducing “continuous” condensates that trivialize all line operators.