SymTFT Entanglement and Holographic (Non)-Factorization

TL;DR

The paper introduces SymTFT entanglement (S-entanglement) as a principled way to couple multiple QFTs sharing a symmetry subcategory by entangling their SymTFT boundaries, effectively gauging a diagonal symmetry. In holographic settings, S-entanglement ties bulk gauge data across disconnected AdS spacetimes, leading to non-factorization of CFT data and enabling a bulk interpretation of wormhole-like connections and emergent bulk global symmetries. By studying various explicit examples across dimensions (D=4,6,3,2) including non-invertible symmetries (Rep(S_N)) and ABJM theories, the work shows how partial traces of S-entangled states realize ensemble averaging via α-states and how this connects to baby universe Hilbert spaces and bulk charges. The framework further clarifies refinements to thermofield double constructions for eternal AdS black holes and provides a path toward understanding factorization puzzles, diagonal gauging, and the emergence (or breaking) of bulk global symmetries in a UV-complete, string-theoretic setting. Overall, S-entanglement offers a unified, top-down mechanism linking symmetry gauging, ensemble averaging, and bulk/global structure across holographic dualities, with broad implications for quantum gravity, string theory, and the study of generalized global symmetries.

Abstract

Given two otherwise decoupled $D$-dimensional CFTs which possess a common (finite) symmetry subcategory, one can consider entangled boundary states of their $(D+1)$-dimensional SymTFTs. This roughly corresponds to performing a gauging of the tensor product of two CFTs, and we call this phenomena ``SymTFT entanglement" (or ``S-entanglement" for short). In the case when these CFTs have semiclassical holographic duals, the S-entanglement relates the bulk gauge charges between two otherwise disconnected AdS spacetimes as we highlight in several top-down examples. We show that taking partial traces of such S-entangled states leads to a streamlined approach to preparing ensemble-averaged CFTs in string theory. This ensemble averaging coincides with that generated by $α$-states in the baby universe Hilbert space, and we propose a symmetry-enriched generalization of this Hilbert space via generalized global symmetries. We quantify how this symmetry-governed averaging violates holographic factorization and leads to the emergence of bulk global symmetries. We also consider the eternal (two-sided) AdS black hole geometries, where our SymTFT entanglement considerations imply that there exist refinements of the usual theromofield double state preparation of the system. We show that one may prepare the system in such a way that the total CFT data does not factorize into left and right copies. As anticipated by Marolf and Wall \cite{Marolf:2012xe}, we highlight that such considerations are necessary to define the gauge charges of eternal black holes, and in certain cases, can imply that there exist extended bulk objects stretching across the wormhole which cannot be expressed in terms of a product of left and right CFT operators.

Figures (26)

• Figure 1: Illustration of a basic SymTFT "sandwich" construction whereby a $(D+1)$-dimensional TFT has a physical boundary, where typically gapless degrees of freedom are localized, and a topological boundary. Compactifying the interval leads to an absolute $D$-dimensional QFT, which one can denote $\mathcal{T}^{(\mathcal{B}_{\mathrm{top.}})}_D$, $\mathcal{U}$ is then a topological symmetry operator and $\mathcal{O}$ denotes a charged operator.
• Figure 2: Two equivalent ways of presenting a disconnected SymTFT construction. In $(a)$, the worldvolume of the SymTFT $\mathcal{S}_{D+1}$ is disconnected and the topological boundary condition is specified by a possibly entangled state $\ket{\mathcal{B}}$. This is the convention we will most commonly take in this work. In (b), the data topological coupling caused by the entanglement in $\ket{\mathcal{B}}$ is given by a topological interface $\mathcal{I}_{\mathcal{B}}$. When $\mathcal{I}_{\mathcal{B}}$ being trivial implies $\ket{\mathcal{B}}$ is maximally entangled.
• Figure 3: At least naively, a gravitational path integral for the case of two asymptotic boundaries (blue circles) includes both disconnected and connected contributions.
• Figure 4:
• Figure 5: Left: SymTFT setup for 4D $\mathcal{N}=4$$\mathfrak{su}(2)$ theory with topological boundary condition $\ket{0}$. Right: Closing the sandwich fixes the gauge group to $SU(2)$ where minimal charge Wilson/'t Hooft line $W_{\mathbf{2}}$/$H_{\mathbf{2}}$ is genuine/non-genuine.