A SymTFT for Continuous Symmetries

TL;DR

This paper extends the Symmetry TQFT framework from discrete to continuous symmetries, enabling new insights into $U(1)$ symmetries, anomalies, and non-invertible symmetries in quantum field theories.

Contribution

It generalizes the SymTFT framework to continuous symmetries, including $U(1)$, and explores their implications in 4d QFTs.

Findings

Incorporation of $U(1)$ symmetries into SymTFT.

Analysis of cubic $U(1)$ anomalies in 4d QFTs.

Description of $ ext{Q}/ ext{Z}$ non-invertible chiral symmetry.

Abstract

Symmetry is a powerful tool for studying dynamics in QFT: it provides selection rules, constrains RG flows, and often simplifies analysis. 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 study cubic $U(1)$ anomalies in $4d$ QFTs, describe the $\mathbb{Q}/\mathbb{Z}$ non-invertible chiral symmetry in $4d$ theories, and conjecture the SymTFT for general continuous $G^{(0)}$ global symmetries.