Decomposition and (non-invertible) (−1)-form symmetries from the symmetry topological field theory

A bstract We investigate the interplay between ( − 1)-form symmetries and their quantum-dual ( d − 1)-form counterparts within the framework of Symmetry Topological Field Theories (SymTFTs). In this framework the phenomenon of decomposition — a d -dimensional quantum field theory with ( d − 1)-form symmetry being the disjoint union of other theories (or “universes”) — arises naturally from manipulations of topological boundary conditions of the SymTFT. We corroborate our findings with various examples, including a generalization of “instanton-restricted” 4d Yang-Mills theories with no sum over instanton sectors. Furthermore, we construct a 3d SymTFT with a non-invertible ( − 1)-form symmetry. The absolute 2d quantum field theory includes a 0-form global symmetry that depends on a parameter whose value gets shifted by the action of the ( − 1)-form symmetry, and we show that the non-invertibility of the latter is needed to encode this modification of the 0-form symmetry.