Recently, I am trying to read the paper Generalized Global Symmetries. In the Preliminaries part, authors formulated ordinary symmetries by network of defects (PP6-7). It seems to be related to constructing a modular invariant CFT partition function. In this formalism, they attach phase factors (which is labelled by group elements) to the junction point of defect networks and these phases turn out to be cocycles. This clearly has something to do with SPT phases of matter. Actually, modular (non)invariance is used to detect nontrivial SPT phase in this paper.
This diagrammatic representation is also used in the TASI 2019 lectures. The actual operation mentioned by Yuji is similar to pentagon identity and hexagon identity in CFT, but I am not clear about this connection. Similar representation is also used in some researches of topological order (and I have no idea about this...).
However, I can't understand how this defect network is constructed and why this can resolve the question about "discrete torsion". Then I checked the ref of Generalized Global Symmetries, and I guess this method comes from CFT community, see this article and ref therein. It seems CFT people call this TFT formalism. But the series of paper by Fuchs et al. is too comprehensive and I have no idea about modular tensor category so I found extremely hard to digest their paper.
So can someone tell me what is the use of defects in CFT and TFT? And systematically, how the network of defects is constructed? And how this related to discrete torsion? Any help is welcome, thanks!
