SymTrees and Multi-Sector QFTs

TL;DR

The paper develops a framework in which the global symmetry data of a D-dimensional QFT, including multi-sector decoupled blocks, is captured by a D+1 dimensional symmetry TFT (SymTFT). It introduces SymTrees, treelike networks of SymTFTs fused along possibly non-topological junctions, to encode topological couplings between sectors and the dressing of defects and symmetry operators by junction data. A top-down string-theoretic picture shows how these SymTrees arise from extra-dimensional geometries, with heavy defects and topological operators understood as branes at infinity or in the bulk, and their dressing manifest at junctions. The authors illustrate the construction through Adjoint Higgsing in 7D SYM and generalize to many QFTs across dimensions (6D, 5D, 4D), including non-supersymmetric examples, brane setups, and moduli-space flows, culminating in an application to large M holographic ensemble averaging. The work suggests a richer categorical structure underlying symmetry data beyond a single bulk SymTFT and points to future directions in higher-categorical formulations, gravity couplings, and looped SymTrees. All statements include precise topological data, differential cohomology, and Mayer-Vietoris machinery to describe junction conditions, and consistently track how bulk and boundary fields glue across the SymTree.

Abstract

The global symmetries of a $D$-dimensional QFT can, in many cases, be captured in terms of a $(D+1)$-dimensional symmetry topological field theory (SymTFT). In this work we construct a $(D+1)$-dimensional theory which governs the symmetries of QFTs with multiple sectors which have connected correlators that admit a decoupling limit. The associated symmetry field theory decomposes into a SymTree, namely a treelike structure of SymTFTs fused along possibly non-topological junctions. In string-realized multi-sector QFTs, these junctions are smoothed out in the extra-dimensional geometry, as we demonstrate in examples. We further use this perspective to study the fate of higher-form symmetries in the context of holographic large $M$ averaging where the topological sectors of different large $M$ replicas become dressed by additional extended operators associated with the SymTree.

Figures (29)

• Figure 1: Standard SymTFT setup. Topological symmetry operators (green, $\mathcal{U}$) link heavy defect operators (grey, $D$) in the $(D+1)$-dimensional slab. The defects stretch from the topological boundary (blue, $\mathcal{B}_{\mathrm{top}}$) to the physical boundary (red, $\mathcal{B}_{\mathrm{phys}}$).
• Figure 2: We depict a trivalent junction $\mathcal{J}$ of symmetry TFTs. The junctions supports the $D$-dimensional theory $\mathcal{G}_{J} \otimes \mathsf{TFT}_{J}$. Color conventions: Junctions are purple.
• Figure 3: Junctions can be assembled into trees (i). The tree $\Upsilon$ can be visualized as a horizontal cross-section. Junctions can have arbitrary valency (ii).
• Figure 4: Depiction of retracting a SymTree to produce the corresponding SymTFT $\mathcal{S}_{\mathrm{full}}$ for the multi-sector QFT with topological couplings between the different sectors. In terms of the SymTree, this amounts to pulling in the different branches into the physical boundaries.
• Figure 5: Subfigures (ii),(iii),(iv),(v) show various degenerations of (i) achieved by contracting one or more of the three symmetry theory slabs. Here $\mathcal{B}_{\textnormal{hyb}}$ denotes a hybrid boundary condition which occurs whenever the branch of the SymTree connecting to $\mathcal{B}_{\textnormal{top}}$ is contracted. Hybrid boundary are generically not purely topological. When a physical boundary condition is fused with a junction a new junction $\mathcal{J}'$ emerges.