[Paper Review] Generalized Charges, Part II: Non-Invertible Symmetries and the Symmetry TFT

Lakshya Bhardwaj, Sakura Schäfer‐Nameki|arXiv (Cornell University)|May 26, 2023

Black Holes and Theoretical PhysicsPhysics and Astronomy30 citations

TL;DR

Proposes that q-charges of a generalized symmetry S are precisely the topological defects of the Symmetry TFT Z(S), i.e., the Drinfeld center of S, valid for both invertible and non-invertible (categorical) symmetries, with detailed analysis in 2d and 3d.

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).

Motivation & Objective

• Characterize generalized charges (q-charges) for non-invertible symmetries in any dimension.
• Show that q-charges are given by topological defects of the Symmetry TFT Z(S) (the Drinfeld center of the symmetry category S).
• Extend Part I results to non-invertible, higher-categorical symmetries and provide concrete 2d and 3d realizations.

Proposed method

• Introduce the Symmetry TFT Z(S) and the sandwich construction with boundary conditions Bsym_S and Bphys_T.
• Identify q-charges with topological defects in Z(S), i.e., the Drinfeld center Z(S) of the symmetry category S.
• Compute generalized charges by interval compactification of bulk defects Qq+1, yielding q-charges Oq on the physical theory boundary.
• Use examples from BF-type (abelian p-form) theories and non-abelian 2d symmetry S3 to illustrate defect labels and charges.
• Explain gauging and its effect on the SymTFT and Drinfeld center, showing Z(S) = Z(S′) under gauging.

Experimental results

Research questions

• RQ1 How are generalized charges defined for non-invertible symmetries S in d dimensions?
• RQ2 What is the precise relationship between q-charges and the topological defects of the SymTFT Z(S)?
• RQ3 How does the Drinfeld center encode charges for both invertible and non-invertible symmetries?
• RQ4 How can one compute q-charges in low-dimensional examples (2d and 3d) and relate them to known symmetry structures?
• RQ5 What is the impact of gauging on the symmetry and its charges within the SymTFT framework?

Key findings

• q-charges of a symmetry S are (q+1)-dimensional topological operators in the SymTFT Z(S).
• The collection of q-charges forms the (d−1)-category Z(S), i.e., the Drinfeld center of the symmetry category S.
• For abelian BF-type symmetries, the Drinfeld center reproduces standard condensation/theta defects and their projections relevant to charges.
• In 2d with non-abelian finite G(0), topological defects of Z(S) are labeled by conjugacy classes and stabilizers, yielding genuine and twisted sector charges.
• Gauging a non-anomalous part of S leaves Z(S) invariant, changing only the symmetry boundary condition, hence leaving generalized charges unchanged.
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