Candidate Gaugings of Categorical Continuous Symmetry

TL;DR

This paper investigates potential gaugeable symmetries in quantum field theories with continuous global symmetry, using a kernel-theoretic approach involving $BF$ and Chern-Simons theories to identify candidate gaugings.

Contribution

It introduces a novel kernel-theoretic framework for analyzing candidate gaugings of continuous symmetries via modular kernels derived from topological field theories.

Findings

Derived candidate modular $S$- and $T$-kernels from semi-classical correlators.

Obtained candidate modular invariants and gaugings consistent with known cases.

Suggested a possible extension of the framework to compact Lie groups.

Abstract

Different gaugings of the global symmetry of a quantum field theory are closely related to its various phases. In this work, we study candidate gaugeable symmetries by analyzing candidate Lagrangian algebra data in the Drinfeld center of a symmetry category $\mathscr{C}^k(G)$ associated to a QFT with continuous global $G$-symmetry and possible 't Hooft anomaly labeled by an integer $k$. We use the combination of the $BF$ theory and the level-$k$ Chern-Simons theory with gauge group $G$ as a semiclassical kernel-theoretic model for the corresponding SymTFT. Under two explicit assumptions, namely that this $BF{+}k$CS theory provides the relevant SymTFT model and that the common $+1$ eigenspaces of the resulting modular kernels detect candidate Lagrangian algebra data in the continuous setting, we derive candidate modular $S$- and $T$-kernels from Hopf-link and framing correlators in $S^3$ semi-classically. We then use these kernels to obtain candidate modular invariants and candidate gaugings. The resulting formulas recover the established cases and suggest a possible extension of this kernel-theoretic picture to compact Lie groups.