[Paper Review] Aspects of Categorical Symmetries from Branes: SymTFTs and Generalized Charges

Fabio Apruzzi, F Bonetti+2 more|arXiv (Cornell University)|Jun 28, 2023

Black Holes and Theoretical PhysicsPhysics and Astronomy3 citations

TL;DR

This paper establishes a direct connection between branes in geometric engineering and holography and the mathematical structure of Symmetry Topological Field Theories (SymTFTs), showing that branes in a topological limit realize both symmetry generators and generalized charges as topological defects. The key result is that these defects form the Drinfeld center of the symmetry category, providing a unified framework for categorical symmetries in QFTs via brane configurations and Hanany-Witten effects.

ABSTRACT

Recently it has been observed that branes in geometric engineering and holography have a striking connection with generalized global symmetries. In this paper we argue that branes, in a certain topological limit, not only furnish the symmetry generators, but also encode the so-called Symmetry Topological Field Theory (or SymTFT). For a $d$-dimensional QFT, this is a $(d+1)$-dimensional topological field theory, whose topological defects encode both the symmetry generators (invertible or non-invertible) and the generalized charges. Mathematically, the topological defects form the Drinfeld center of the symmetry category of the QFT. In this paper we derive the SymTFT and the Drinfeld center topological defects directly from branes. Central to the identification of these are Hanany-Witten brane configurations, which encode both topological couplings in the SymTFT and the generalized charges under the symmetries. We exemplify the general analysis with examples of QFTs realized in geometric engineering or holography.

Motivation & Objective

• To unify the description of global symmetries and generalized charges in QFTs by deriving them from a single framework based on branes.
• To show that the Symmetry Topological Field Theory (SymTFT) and its topological defects arise naturally from brane configurations in geometric engineering and holography.
• To establish that the topological defects of the SymTFT correspond to the Drinfeld center of the symmetry category, providing a mathematically consistent structure.
• To demonstrate how Hanany-Witten brane transitions encode generalized charges and anomaly couplings, linking brane physics to topological invariants.
• To provide explicit realizations of SymTFTs in 4d $ N=4$ $rak{so}(4n)$ SYM and $ N=2$ $[A_2,D_4]$ theories, verifying the framework through concrete examples.

Proposed method

• Derive the SymTFT as a (d+1)-dimensional topological field theory from branes in a topological limit, using dimensional reduction and boundary conditions.
• Use Hanany-Witten brane configurations to encode topological couplings and generalized charges, with fluxes and brane sources formulated in a democratic, gauge-invariant way.
• Construct BF-theory terms from brane sources to derive topological couplings in the SymTFT, linking them to symmetry generators and defects.
• Identify topological defects in the SymTFT as junctions of brane configurations, where anomaly couplings and condensation defects emerge from brane transitions.
• Apply the Drinfeld center construction to the symmetry category of the QFT, showing that its topological defects classify both symmetries and generalized charges.
• Use half-space gauging and redefinitions of gauge fields to compute line operator transformations across non-invertible defects, verifying the action of non-invertible symmetries.

Experimental results

Research questions

• RQ1 How do branes in geometric engineering and holography realize the Symmetry Topological Field Theory (SymTFT) and its topological defects?
• RQ2 What is the precise mathematical structure of the topological defects in the SymTFT, and how does it relate to the Drinfeld center of the symmetry category?
• RQ3 How do Hanany-Witten brane transitions encode generalized charges and anomaly couplings in the SymTFT?
• RQ4 How do non-invertible symmetries act on line operators in the presence of brane defects, and what is the role of junctions in this action?
• RQ5 Can explicit examples such as 4d $ N=4$ $rak{so}(4n)$ SYM and $ N=2$ $[A_2,D_4]$ theories be used to verify the proposed SymTFT framework?

Key findings

• The SymTFT for a d-dimensional QFT is realized as a (d+1)-dimensional topological field theory directly from brane configurations in geometric engineering and holography.
• Topological defects in the SymTFT, which encode both symmetry generators and generalized charges, are shown to form the Drinfeld center of the symmetry category of the QFT.
• Hanany-Witten brane transitions provide a physical realization of anomaly couplings and generalized charges, with junctions of defects encoding the action of non-invertible symmetries.
• In the 4d $ N=4$ $rak{so}(4n)$ SYM theory, the SymTFT is derived via dimensional reduction and flux compactification, with the Drinfeld center structure confirmed through explicit path integral computation.
• For the $ N=2$ $[A_2,D_4]$ theory, duality and triality defects are realized as non-invertible symmetries via brane junctions, with their action on line operators verified through gauge field redefinitions and half-space gauging.
• The action of non-invertible symmetries on line operators is shown to produce surface operators attached to the line, with explicit expressions derived from gauge field constraints at brane junctions.