Cornering relative symmetry theories

The symmetry data of a d-dimensional quantum field theory (QFT) can often be captured in terms of a higher-dimensional symmetry topological field theory. In top-down (i.e., stringy) realizations of this structure, the QFT in question is localized in a higher-dimensional bulk. In many cases of interest, however, the associated (d+1)-dimensional bulk is not fully gapped and one must instead consider a filtration of theories to reach a gapped bulk in D=d+m dimensions. Overall, this leads us to a nested structure of relative symmetry theories which descend to coupled edge modes, with the original QFT degrees of freedom localized at a corner of this D-dimensional bulk system. We present a bottom-up characterization of this structure and also show how it naturally arises in a number of string-based constructions of QFTs with both finite and continuous symmetries.