Generalized Charges, Part II: Non-Invertible Symmetries and the Symmetry TFT

TL;DR

This paper characterizes $q$-charges in quantum field theories with generalized symmetries, including non-invertible ones, using topological defects of the Symmetry TFT, extending previous results to broader symmetry types.

Contribution

It generalizes the classification of charges to non-invertible and categorical symmetries via the SymTFT framework, unifying previous invertible symmetry results.

Findings

$q$-charges correspond to topological defects of the SymTFT.

The framework applies to finite, non-invertible, and categorical symmetries.

Explicit development for 2d and 3d quantum field theories.

Abstract

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