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

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