SymTh for non-finite symmetries

A bstract Symmetry topological field theory (SymTFT) is a convenient tool for studying finite generalized symmetries of a given quantum field theory (QFT). In particular, SymTFTs encode all the symmetry structures and properties, including anomalies. Recently, this tool has been applied for non-finite symmetries as well. In this paper, we take a different route, which consists of considering a free theory rather than a topological field theory in the bulk. We call it Symmetry Theory (SymTh). We study its topological operators together with the free boundary conditions. We also propose a procedure that is analogous to the sandwich construction of SymTFTs and allows us to obtain the physical QFT. We apply this to many examples, ranging from abelian p -form symmetries to 2-groups, and the (solvable) case of group-like symmetries in quantum mechanics. Finally, we provide a derivation of the SymTh of ℚ / ℤ non-invertible symmetries from the dimensional reduction of IIB supergravity on the conifold. In addition, we give an ultraviolet interpretation of the quantum Hall states dressing the non-invertible ℚ / ℤ topological defects, in terms of branes in the IIB supergravity background.