A SymTFT for Continuous Symmetries

Symmetry is a powerful tool for studying dynamics in QFT as they provide selection rules, constrain RG flows, and allow for simplified dynamics. Currently, our understanding is that the most general form of symmetry is described by categorical symmetries which can be realized via Symmetry TQFTs or ``SymTFTs." In this paper, we show how the framework of the SymTFT, which is understood for discrete symmetries (i.e. finite categorical symmetries), can be generalized to continuous symmetries. In addition to demonstrating how $U(1)$ global symmetries can be incorporated into the paradigm of the SymTFT, we apply our formalism to construct the SymTFT for the $\mathbb{Q}/\mathbb{Z}$ non-invertible chiral symmetry in $4d$ theories, demonstrate how symmetry fractionalization is realized SymTFTs, and conjecture the SymTFT for general continuous $G^{(0)}$ global symmetries.