On the SymTFTs of Finite Non-Abelian Symmetries

TL;DR

This paper develops a framework for describing 3D symmetry topological field theories of finite non-Abelian groups, enabling explicit construction of operators and fusion rules that reveal non-Abelian symmetry structures.

Contribution

It introduces discrete BF-like theories for finite groups, facilitating a clearer understanding of non-Abelian symmetries and their associated operators in SymTFTs.

Findings

Constructed BF-like Lagrangians for finite groups with extensions by Abelian groups.

Reconstructed fusion rules of the Drinfeld center for these symmetries.

Built surface-attaching operators linked directly to non-Abelian group elements.

Abstract

The $(D+1)$-dimensional symmetry topological field theory (SymTFT$_{D+1}$) of a $D$-dimensional absolute quantum field theory (QFT$_D$) provides a topological characterization of symmetry data. In this framework, the SymTFT comes equipped with a physical boundary specifying a relative QFT, and a topological boundary which specifies the global form of symmetries. In general, there need not be a unique bulk theory which encodes this information but it is often helpful to have a more manifest presentation of symmetries in terms of bulk degrees of freedom. For the case of a finite non-Abelian symmetry group $G$, the bulk SymTFT may be described by a Dijkgraaf-Witten TFT with gauge group $G$. This makes manifest the ``electric'' presentation of the symmetry data but can obscure some of the magnetic data as well as non-Abelian structure present in the absolute QFT$_D$ such as symmetry operators which cannot fully detach from the topological boundary. We address these issues for 3D SymTFTs by constructing discrete BF-like theory Lagrangians for finite groups which admit a presentation as an extension by a finite Abelian group and a finite (possibly non-Abelian) group. This enables us to give a streamlined approach to reconstructing the fusion rules of the accompanying Drinfeld center, but also allows us to construct surface-attaching non-genuine line operators associated directly with non-Abelian group elements rather than just their conjugacy classes. We also sketch how our treatment generalizes to higher-dimensional SymTFTs.