We summarize basic features of quantum field theories with discrete symmetry $\mathbb{ Q}/\mathbb{ Z}$ (possibly higher form, global or gauged). The classification of representations and anomalies is quite rich and involves the ring of profinite integers. As a main example we consider in detail 3d topological Dijkgraaf-Witten $\mathbb{ Q}/\mathbb{ Z}$ gauge theories. We also briefly discuss relevance for some previously considered physical systems. In particular we comment on a relation to the recently discovered non-invertible symmetry in 4d QED and the problem of categorification of Chern-Simons TQFT.
Categories: hep-th cond-mat.str-el math.AT math.GT math.NT