The $SL_2(\mathbb{R})$ duality and the non-invertible $U(1)$ symmetry of Maxwell theory

TL;DR

This paper explores how classical $U(1)$ symmetry in Maxwell theory can be realized as a non-invertible symmetry through discrete gauging, illustrating the reemergence of classical symmetries in quantum regimes.

Contribution

It demonstrates the realization of $SL_2( ext{R})$ duality automorphisms in the SymTFT and shows how to restore the $U(1)$ symmetry as a non-invertible symmetry via infinite discrete gauging.

Findings

Classical $U(1)$ symmetry can be non-invertibly restored in quantum Maxwell theory.

The process involves an infinite series of discrete gaugings.

Continuous condensates trivialize all line operators.

Abstract

Recent proposals for the Symmetry Topological Field Theory (SymTFT) of Maxwell theory admit a 0-form symmetry compatible with the classical $SL_2(\mathbb{R})$ duality of electromagnetism. We describe how to realize these automorphisms of the SymTFT in terms of its operators and we detail 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 $SL_2(\mathbb{R})$, can be restored as a non-invertible one, by means of an infinite series of discrete gauging. This provides 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.