Lattice Translation Modulated Symmetries and TFTs

Abstract: Modulated symmetries are internal symmetries that are not invariant under spacetime symmetry actions. We propose a general way to describe the lattice translation modulated symmetries in 1+1D, including the non-invertible ones, via the tensor network language. We demonstrate that the modulations can be described by some autoequivalences of the categories. Although the topological behaviors are broken because of the presence of modulations, we can still construct the modulated version of the symmetry TFT bulks by inserting a series of domain walls described by invertible bimodule categories. This structure not only recovers some known results on invertible modulated symmetries but also provides a general framework to tackle modulated symmetries in a more general setting.