On the SymTFTs of Finite Non-Abelian Symmetries

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.

Figures (7)

• Figure 1: In the SymTFT framework an absolute $D$-dimensional QFT$_D$ is "decompressed" to a relative QFT$_{D}$, (i.e., a physical boundary condition) and a topological boundary condition, with a bulk SymTFT$_{D+1}$ capturing the symmetry category $\mathcal{C}$ of the absolute QFT$_{D}$. Symmetry operators of the absolute theory specify objects in the Drinfeld center $\mathcal{Z}(\mathcal{C})$, and topologically link with defects which stretch from the physical to topological boundaries. A change of polarization in the topological boundary condition amounts to modifying the roles of the symmetry operators and defects. For non-Abelian symmetries, some of the symmetry operators of the absolute QFT$_D$ fail to fully detach from the topological boundary; these are instead captured by a flux tube which extends back to the boundary. The appearance of such non-genuine operators in the bulk SymTFT$_{D+1}$ which are not captured by just $\mathcal{Z}(\mathcal{C})$ is a common feature of non-Abelian symmetries.
• Figure 2: Subfigure (i) depicts two genuine codimension-2 operators whose fusion ring is commutative. Subfigure (ii) depicts two non-genuine codimension-2 operators each fixed to the boundary of a codimension-1 surface which further terminates on the topological boundary condition (blue) of the SymTFT slab. Non-commutative structure is manifest in the fusion ring of such operators.
• Figure 3: Non-commutativity is manifest in the bulk via the spectrum of non-genuine operators and their interactions. Subfigure (i) shows two non-genuine codimension-2 operators supported on the boundary of a single-boundary codimension-1 surface. Subfigure (ii) deforms these such that the surface attaching to $b$ engulfs $a$. Subfigure (iii) shows a contraction of $b$'s surface onto $a$ resulting in $c$ assuming the self-fusion of the codimension-1 surface is invertible.
• Figure 4: Triangulation of the 3-manifold $X$ with vertices $i,j,k,\dots$ and oriented edges labeled by group elements $g_{ij},g_{jk},g_{ki},\dots$.
• Figure 5: In subfigure (i) we sketch the reference configuration with Dirichlet boundary condition DBC. In subfigure (ii) we insert an invertible symmetry operator $O_g$ labeled by $g\in G$ across which the Dirichlet boundary condition jumps from DBC to DBC$^{\:\!(g)}$. In subfigure (iii) we deform this insertion into the SymTFT. In subfigure (iv) we fuse / fold the boundary condition onto itself resulting in a surface $B_g$ terminating on the boundary on the junction $J_g$.