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

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.