Generalized charges, part II: Non-invertible symmetries and the symmetry TFT

Abstract

Consider a d-dimensional quantum field theory (QFT) (formula presenetd), with a generalized symmetry S, which may or may not be invertible. We study the action of S on generalized or qcharges, i.e. q-dimensional operators. The main result of this paper is that q-charges are characterized in terms of the topological defects of the Symmetry Topological Field Theory (SymTFT) of S, also known as the “Sandwich Construction”. The SymTFT is a (d + 1)-dimensional topological field theory, which encodes the symmetry S and the physical theory in terms of its boundary conditions. Our proposal applies quite generally to any finite symmetry S, including non-invertible, categorical symmetries. Mathematically, the topological defects of the SymTFT form the Drinfeld Center of the symmetry category S. Applied to invertible symmetries, we recover the result of Part I [1] of this series of papers. After providing general arguments for the identification of q-charges with the topological defects of the SymTFT, we develop this program in detail for QFTs in 2d (for general fusion category symmetries) and 3d (for fusion 2-category symmetries).