The SymTFT for $N$-ality defects: Part I

TL;DR

This work constructs a comprehensive Symmetry TFT framework for N-ality defects, showing how bulk $G$-gauge theories plus a bulk $\ Z_N$ gauging encode all boundary $F$-symbols for a boundary non-invertible $N$-ality symmetry. By analyzing condensation defects, twist defects, and bimodules in the bulk, the authors derive boundary data and then propagate it to the boundary via anyon condensation, with explicit treatment for prime $N$ (triality). The paper provides explicit bulk and boundary $F$-symbols, classifies triality defects, and outlines how to obtain the full set of fusion rules and line spectra, setting the stage for a complete $(2+1)$d SymTFT for $N$-ality in forthcoming work. The results extend Tambara–Yamagami-like structures to higher-dimensional, non-invertible symmetries and give concrete computational tools for boundary phenomena arising from bulk gaugings and higher-categorical data.

Abstract

In order to obtain the SymTFT for a theory with an $N$-ality extension of a discrete, Abelian group $G$, one begins by considering a bulk $G$-gauge theory, and then gauges an appropriate $\mathbb{Z}_N$ symmetry. This procedure involves three choices: the choice of a suitable bulk $\mathbb{Z}_N$ symmetry, of a fractionalization class, and of a discrete torsion. The first choice is, somewhat surprisingly, the most involved, and in this paper we discuss it in detail. In particular, we show that the choice of bulk $\mathbb{Z}_N$ symmetry determines all boundary $F$-symbols with a single incoming $N$-ality defect, and that any theory with an $N$-ality symmetry is invariant under a certain twisted gauging given in terms of these $F$-symbols. These $F$-symbols can furthermore be input into the pentagon identities to obtain the other $F$-symbols, up to freedoms related to the choices appearing in the second and third steps of bulk gauging. Although many of our results hold for general $N$, we restrict ourselves in some places to the case of $N=p$ prime. In particular, for generic triality defects, we acquire explicit $F$-symbols which are reminiscent of those in Tambara-Yamagami fusion categories.

Figures (13)

• Figure 1: A $(1+1)$d QFT $\mathcal{X}$ with $\mathbb{Z}_M^n$ symmetry and another $(1+1)$d QFT $\mathcal{X}/\mathbb{Z}_M^n$ are separated by a topological interface $Q$. This setup can be expanded into a $(2+1)$d slab, where the $(2+1)$d $\mathbb{Z}_N$ SymTFT has an insertion of a twist defect parallel to the Dirichlet boundary. The particular choice of gauging (discrete torsion etc.) is determined by the choice of bulk condensation defect ending on the twist defect.
• Figure 2: Condensing the algebra object $(\mathcal{A}_L, \mu_L)$ on the left and $(\mathcal{A}_R, \mu_R)$ on the right gives an interface between the corresponding condensation defects. Mathematically, these interfaces are given by $(\mathcal{A}_L, \mu_L)$-$(\mathcal{A}_R, \mu_R)$ bimodules.
• Figure 3: F- and R-moves of the line operators $L_{\mathbf{{\mathbf{v}}}}$.
• Figure 4: The defect $D_{{A}}$ viewed from the front is equivalent to the defect $\overline{D_{{A}}}$ viewed from the back. In the rest of the figures in this paper, we will always assume that we are viewing the defects from the front, and hence will drop the $\odot$ and $\otimes$.
• Figure 5: Our choice for the orientation of the twist defect. As mentioned before, each surface is viewed from the front, i.e. there is an implicit $\odot$. The orientation reversal twist defects $\overline \Sigma_{A}^{[\mathbf{0}]}$ can be obtained by flipping these pictures upside down.