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

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).

144 pages; v2: References added

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• Fusion 3-Categories for Duality Defects
• Non-invertible Global Symmetries in the Standard Model
• Drinfeld Centers and Morita Equivalence Classes of Fusion 2-Categories
• Coupling a Cosmic String to a TQFT

Cited by in corpus (12)

• Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries
• Quantum Cluster State Model with Haagerup Fusion Category Symmetry
• Weak Hopf non-invertible symmetry-protected topological spin liquid and lattice realization of (1+1)D symmetry topological field theory
• The 3d $A$-model and generalised symmetries, Part I: bosonic Chern-Simons theories
• (2+1)d Lattice Models and Tensor Networks for Gapped Phases with Categorical Symmetry
• (-1)-Form Symmetries and Anomaly Shifting from SymTFT
• The surface code beyond Pauli channels: Logical noise coherence, information-theoretic measures, and errorfield-double phenomenology
• Spacetime symmetry-enriched SymTFT: from LSM anomalies to modulated symmetries and beyond
• Entropic order parameters and topological holography
• Topological Manipulations On $\mathbb{R}$ Symmetries Of Abelian Gauge Theory
• Spontaneously Broken Non-Invertible Symmetries in Transverse-Field Ising Qudit Chains
• Symmetric entanglers for non-invertible SPT phases