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

TL;DR

This paper explores the self-duality and non-invertible $SO(2)$ symmetry in 4d Maxwell theory using topological methods, revealing new symmetry structures and their realization through higher gauging and condensation defects.

Contribution

It introduces a novel topological framework for understanding Maxwell's duality symmetries, including non-invertible symmetries, via higher gauging and symTFT techniques.

Findings

Realization of Maxwell duality through trivial gauging operations.

Construction of condensation defects implementing non-invertible $SO(2)$ symmetry.

Encoding of continuous symmetries using non-compact symTFT.

Abstract

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.