A SymTFT for Continuous Symmetries

A SymTFT for Continuous Symmetries

5 Nov 2024 | T. Daniel Brennan and Zhengdi Sun
This paper explores the extension of Symmetry TQFTs (SymTFTs) to describe continuous symmetries, building upon the existing framework for discrete symmetries. The authors demonstrate how to incorporate $U(1)^{(0)}$ global symmetries into the SymTFT framework and apply this 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 a SymTFT for general continuous $G^{(0)}$ global symmetries. The paper begins by reviewing the basics of SymTFTs, including the $\mathbb{Z}_N$ SymTFT, and then delves into the reduction of SymTFTs from $\mathbb{Z}_N$ to $\mathbb{Z}_M$ when $M$ divides $N$. It also discusses how anomalies of $\mathbb{Z}_N^{(0)}$ symmetries can be encoded in the SymTFT, preventing the existence of certain boundary conditions. The main section focuses on the SymTFT for $U(1)^{(0)}$ symmetry, detailing its topological operators, canonical quantization, and the behavior of bulk operators on boundaries. The authors conclude by discussing how the SymTFT can couple to non-flat connections and the dynamical gauging of $U(1)$ symmetries.This paper explores the extension of Symmetry TQFTs (SymTFTs) to describe continuous symmetries, building upon the existing framework for discrete symmetries. The authors demonstrate how to incorporate $U(1)^{(0)}$ global symmetries into the SymTFT framework and apply this 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 a SymTFT for general continuous $G^{(0)}$ global symmetries. The paper begins by reviewing the basics of SymTFTs, including the $\mathbb{Z}_N$ SymTFT, and then delves into the reduction of SymTFTs from $\mathbb{Z}_N$ to $\mathbb{Z}_M$ when $M$ divides $N$. It also discusses how anomalies of $\mathbb{Z}_N^{(0)}$ symmetries can be encoded in the SymTFT, preventing the existence of certain boundary conditions. The main section focuses on the SymTFT for $U(1)^{(0)}$ symmetry, detailing its topological operators, canonical quantization, and the behavior of bulk operators on boundaries. The authors conclude by discussing how the SymTFT can couple to non-flat connections and the dynamical gauging of $U(1)$ symmetries.
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