Aspects of Categorical Symmetries from Branes: SymTFTs and Generalized Charges

TL;DR

The paper proposes a comprehensive brane-based construction of the Symmetry Topological Field Theory (SymTFT) for QFTs realized in geometric engineering or holography. It shows that branes encode both symmetry generators and the topological data of the SymTFT, including BF terms and anomaly couplings, with the Drinfeld center organizing the topological defects. Hanany-Witten brane configurations provide a concrete mechanism to realize generalized charges and their transformation properties under symmetry actions, linking bulk linking numbers to charges of end points and defect junctions. The work develops a unified framework connecting brane physics to categorical symmetry data, and illustrates it through numerous holographic and geometric-engineering examples, including various forms of 4d SYM and duality/triality defects. The results offer a powerful and unified lens for understanding symmetries and anomalies in high-dimensional QFTs, with broad implications for non-invertible symmetries and their realizations in string theory.

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.

Figures (14)

• Figure 1: The SymTFT sandwich and interval compactification to a $d$-dimensional physical theory $\mathcal{T}$ with symmetry $\mathcal{S}$. The SymTFT is a $(d+1)$-dimensional theory, $\text{SymTFT} (\mathcal{S})$, which is shown on the left: it has two boundaries, the gapped, symmetry boundary $\mathcal{B}^{\text{sym}}$, and the physical boundary $\mathcal{B}^{\text{phys}}$, which is not gapped (unless the theory $\mathcal{T}$ was topological as well).
• Figure 2: Symmetries from the SymTFT: the parallel projection of topological defects $\bm{Q}_{p+1}$ gives rise to topological defects on the symmetry boundary $\mathcal{B}^{\text{sym}}$. Put differently, the associated background fields, have Neumann boundary conditions. In general this is not the complete set of symmetry operators, but for abelian higher-form fields, for which we consider the SymTFT, this is the case.
• Figure 3: Generalized charges from bulk topological defects that end on the symmetry and physical boundaries: genuine $q$-charge $\mathcal{O}_q$. The left hand side shows the SymTFT sandwich, with the bulk topological operator $\mathcal{O}_{q+1}$ ending on both physical and symmetry boundaries. After interval compactification it gives rise to a $q$-charge in the theory $\mathcal{T}$.
• Figure 4: Twisted sector operators: L-shape projection of a bulk topological defect $\bm{Q}_{q+1}$ onto the symmetry boundary, creates a junction $\mathcal{E}_p$ which is attached to a topological defect $D_{q+1}$.
• Figure 5: Generalized symmetry acting on generalized charge via linking. The topological operator $\bm{Q}_{p+1}$ in the bulk SymTFT ends and gives rise to the (genuine) $q$-charge $\mathcal{O}_q$ in $\mathcal{T}$. In turn, the topoglogical operator $\bm{Q}_{d-q-1}$ projects onto the symmetry boundary and gives rise to a symmetry generator after the interval compactification. The non-trivial linking of these topological defects in the SymTFT results in the generalized charge.